Introduction
The Ryu–Takayanagi (RT) formula translates quantum entanglement into bulk geometry [1 , 2 ] . On a static bulk slice, its bit-thread formulation replaces the RT minimal surface by the maximum flux of a divergenceless vector field [3 , 4 ] ,
S ( A ) = max v ∫ A v i n i d A E , ∇ i E v i = 0 , ∣ v ∣ E ≤ 1 4 G N , (1) S(A) = \max_{v} \int_{A} v^{i}n_{i}\,\mathrm{d}A_{E}, \qquad\nabla^{E}_{i}v^{i} = 0, \qquad|v|_{E} \leq\frac{1}{4G_{N}}, \tag*{(1)} S ( A ) = v max ∫ A v i n i d A E , ∇ i E v i = 0 , ∣ v ∣ E ≤ 4 G N 1 , (1) where the metric, norm, and area element are written in the Einstein frame. The two constraints in this configuration have distinct roles. The divergence condition makes the flux invariant under deformations among homologous cuts, while the norm bound sets the largest entropy flux that can pass through a unit area. Their combination turns the RT surface into the bottleneck of a maximal flow and has led to direct geometric proofs of holographic entropy inequalities. Related developments include geometric constructions, higher-curvature extensions, thermal bit threads, and recent string-diagrammatic and Lorentzian thread formulations [5 , 6 , 7 , 8 , 9 ] . The vector v i v^{i} v i
String theory supplies a natural candidate. A fundamental string is electrically charged under the Kalb–Ramond two-form. In the three-dimensional setting, varying its worldsheet coupling to B a b B_{ab} B ab [10 , 11 , 12 , 13 ]
j a b = 1 − g ∫ d 2 σ ( ∂ X a ∂ τ ∂ X b ∂ σ − ∂ X b ∂ τ ∂ X a ∂ σ ) δ ( 3 ) ( x − X ( τ , σ ) ) . (2) j^{ab} = \frac{1}{\sqrt{-g}}\int d^{2}\sigma\left(\frac{\partial X^{a}}{\partial\tau}\frac{\partial X^{b}}{\partial\sigma}-\frac{\partial X^{b}}{\partial\tau}\frac{\partial X^{a}}{\partial\sigma}\right)\delta^{(3)}\left(x-X(\tau,\sigma)\right). \tag*{(2)} j ab = − g 1 ∫ d 2 σ ( ∂ τ ∂ X a ∂ σ ∂ X b − ∂ τ ∂ X b ∂ σ ∂ X a ) δ ( 3 ) ( x − X ( τ , σ ) ) . (2) For a closed worldsheet, or locally away from open-string endpoints, this current is conserved. On a static slice with induced string-frame metric h i j h_{ij} h ij N A D M N_{\mathrm{ADM}} N ADM
q i = N A D M j 0 i , ∂ i ( h q i ) = 0. (3) q^{i}=N_{\mathrm{ADM}}j^{0i},\qquad\partial_{i}\left(\sqrt{h}q^{i}\right)=0. \tag*{(3)} q i = N ADM j 0 i , ∂ i ( h q i ) = 0. (3) The vector q i q^{i} q i
Motivated by these properties, in ref. [14 ] we proposed the string charge density/bit threads correspondence
v i = C g e o m q i . (4) v^{i}=C_{\mathrm{geom}}q^{i}. \tag*{(4)} v i = C geom q i . (4) The spacetime metric determines the transverse geometry and the minimal surface that forms the bottleneck, while the Kalb–Ramond current supplies conserved lines passing through it. Schematically,
g a b ⟷ R T g e o m e t r y , B a b ⟷ s t r i n g c h a r g e f l o w ⟷ b i t t h r e a d s . (5) g_{ab}\ \longleftrightarrow\ \mathrm{RT\ geometry},\qquad B_{ab}\ \longleftrightarrow\ \mathrm{string\ charge\ flow}\ \longleftrightarrow\ \mathrm{bit\ threads}. \tag*{(5)} g ab ⟷ RT geometry , B ab ⟷ string charge flow ⟷ bit threads . (5) In the original BTZ construction, C g e o m C_{\mathrm{geom}} C geom q i q^{i} q i v i v^{i} v i
This correspondence is also naturally related to the Susskind–Uglum worldsheet picture of gravitational entropy [15 ] . In that picture, black hole entropy is described by closed string sphere diagrams whose target space embeddings intersect the horizon. A closed string that crosses the horizon twice appears to an exterior Rindler observer as an open segment with endpoints on the horizon. Related entangling brane constructions make this open-string endpoint picture precise in two-dimensional string theory [16 ] . The closed string description associates entropy with the response of the sphere amplitude to a conical deformation, while the open string picture suggests physical lines attached to the entangling surface. The string charge density construction gives these lines a spacetime current and turns their spatial projection into a conserved entropy-flow candidate.
Figure 1. The string charge density/bit threads correspondence. The projected Kalb–Ramond charge density is tangent to the strings and supplies conserved flow lines. The metric determines the transverse geometry and the entropy bottleneck. In the original construction, both the string arrangement and its entropy normalization were chosen to reproduce a known bit-thread flow. The red entangling brane/D-brane shown here imposes an auxiliary endpoint condition associated with the RT surface γ A \gamma_A γ A
Ahmadain and Wall placed the closed string part of this picture on a precise off-shell footing [17 , 18 ] , and it was subsequently extended with Frenkel [19 ] . Using Tseytlin’s sphere prescription, they derived the leading closed string effective action from a regulated genus-zero worldsheet functional and evaluated it on conical target space geometries. The cone is off shell because its tip carries a localized worldsheet beta function, or equivalently a target space graviton tadpole. Differentiating the sphere-derived action with respect to the angular period gives the Bekenstein–Hawking entropy at leading order in α ′ \alpha' α ′ [20 , 21 , 22 , 23 , 24 ] .
Our previous work supplied the complementary conserved-current description. The correspondence was first formulated in an effective three-dimensional setting and was subsequently extended to ten-dimensional three-charge black holes. In the Type IIB S-duality analysis of the F1–NS5–P and D1–D5–P systems [25 ] , the F1 current is mapped to a D1 current. The metric, dilaton, local current, and pointwise flow norm change, while the integrated maximal entropy flux is preserved. Under T-duality [26 ] , a radial F1 remains an F1 along the NS–NS duality chain, whereas a D1 transverse to the duality circle becomes a D2 wrapped on the dual circle. Contracting the projected D2 current with the normalized one-form along that circle gives an effective conserved vector flow. The local representatives again change, but the dilaton-weighted transverse capacity and the maximal entropy flux remain invariant. These duality checks show that distinct string and brane currents can represent the same physical entropy flux.
The original BTZ construction provided a conserved-current representation of entropy flow, while taking the source embeddings and the local entropy capacity as inputs. Specifically, as illustrated in fig. (1), it looks as if we artificially place parallel strings along the known bit-thread lines, which would naturally give the desired flow. Current conservation ensures that such prescribed strings define a divergenceless vector field, but it does not show that their embeddings solve the F1 equations in the specified background. Moreover, the conversion coefficient in Eq. (4) was fixed by imposing
∣ C g e o m q ∣ ≤ e − 2 ϕ ( x ) 4 G N ( D ) (6) \lvert C_{\mathrm{geom}}q\rvert\le\frac{e^{-2\phi(x)}}{4G_N^{(D)}} \tag*{(6)} ∣ C geom q ∣ ≤ 4 G N ( D ) e − 2 ϕ ( x ) (6) obtained from bit-thread formulation and demanding saturation at the horizon. The entropy normalization was therefore provided by the geometric capacity constraint, rather than independently determined within the string construction. The present work derives this capacity from the off-shell sphere functional, together with the gravitational replica construction and flow–cut duality.
Therefore, the aim of this paper is to resolve these two issues at genus zero and two-derivative order through the following steps:
Figure 1 (PDF p. 5).
We first derive the local entropy capacity from the off-shell sphere functional. The conical variation of the sphere-derived target space action assigns a smooth codimension-two surface m m m
S l o c ( 0 ) [ m ] = 1 4 G N ∫ m e − 2 ϕ d A S = ∫ m ρ S ( x ) d A S , ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (7) S_{\mathrm{loc}}^{(0)}[m]=\frac{1}{4G_N}\int_m e^{-2\phi}dA_S=\int_m \rho_S(x)dA_S,\qquad\rho_S(x)=\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(7)} S loc ( 0 ) [ m ] = 4 G N 1 ∫ m e − 2 ϕ d A S = ∫ m ρ S ( x ) d A S , ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (7) The factor e − 2 ϕ e^{-2\phi} e − 2 ϕ ρ S ( x ) \rho_S(x) ρ S ( x )
For a static holographic region A A A
S A ( 0 ) = min ∂ m = ∂ A m ∼ A ∫ m ρ S ( x ) d A S = max ∇ i v i = 0 ∣ v ∣ ≤ ρ S ∫ A v i n i d A S . (8) S_A^{(0)}=\min_{\substack{\partial m=\partial A\\m\sim A}}\int_m \rho_S(x)dA_S=\max_{\substack{\nabla_i v^i=0\\\lvert v\rvert\le\rho_S}}\int_A v^i n_i dA_S. \tag*{(8)} S A ( 0 ) = ∂ m = ∂ A m ∼ A min ∫ m ρ S ( x ) d A S = ∇ i v i = 0 ∣ v ∣ ≤ ρ S max ∫ A v i n i d A S . (8) This chain gives the string-frame norm bound
∣ v ∣ ≤ ρ S ( x ) = e − 2 ϕ ( x ) 4 G N (9) |v| \le\rho_S(x) = \frac{e^{-2\phi(x)}}{4G_N} \tag*{(9)} ∣ v ∣ ≤ ρ S ( x ) = 4 G N e − 2 ϕ ( x ) (9) from the sphere-derived entropy density, the gravitational replica saddle, and convex duality. The off-shell sphere calculation and the charge flow construction consequently play different roles. The former fixes the local amount of entropy that a transverse area element can support. On the other hand, the latter supplies a conserved spatial realization of that capacity. Their combination applies the same local density to both a black hole horizon and the weighted RT surface of a general static boundary region.
We then determine the flow lines from physical F1 dynamics. For nonrotating BTZ with constant dilaton and NS–NS flux, the radial embedding used in the original construction is a conserved external source but is not a free stationary string. The first variation of its Kalb–Ramond coupling produces an angular force. The regular stationary solution is instead a logarithmic spiral. Classical string propagation in BTZ backgrounds was analyzed in [27 , 28 ] . In this work, we find the horizon regular branches by solving the probe F1 equations:
θ ( r ) = θ h + ℓ r h log r r h . (10) \theta(r) = \theta_h + \frac{\ell}{r_h}\log\frac{r}{r_h}. \tag*{(10)} θ ( r ) = θ h + r h ℓ log r h r . (10) This spatial profile is determined directly by the classical F1 equations in the given background, independently of any prescribed bit-thread configuration. The resulting on-shell probe strings therefore provide a dynamical realization of the entropy-flow lines. For a uniform ensemble of N \mathcal{N} N
∣ q ∣ = N 2 π r h , q ^ i = q i ∣ q ∣ , ∇ i q ^ i = 0. (11) |q| = \frac{\mathcal{N}}{2\pi r_h}, \qquad\hat{q}^{i} = \frac{q^{i}}{|q|}, \qquad\nabla_i\hat{q}^{i} = 0. \tag*{(11)} ∣ q ∣ = 2 π r h N , q ^ i = ∣ q ∣ q i , ∇ i q ^ i = 0. (11) Combining the dynamically selected direction with the independently derived capacity gives
v p h y s i = ρ S q ^ i = C g e o m q i , C g e o m = ρ S ∣ q ∣ . (12) v_{\mathrm{phys}}^{i} = \rho_S\hat{q}^{i} = C_{\mathrm{geom}}q^{i}, \qquad C_{\mathrm{geom}} = \frac{\rho_S}{|q|}. \tag*{(12)} v phys i = ρ S q ^ i = C geom q i , C geom = ∣ q ∣ ρ S . (12) This flow is conserved, saturates the string-frame norm bound throughout the BTZ exterior, and becomes normal to the bifurcation circle. Its flux is
Φ [ v p h y s ] = 2 π r h ρ S = e − 2 ϕ 0 ( 2 π r h ) 4 G N ( 3 ) = S B H ( 0 ) . (13) \Phi[v_{\mathrm{phys}}] = 2\pi r_h\rho_S = \frac{e^{-2\phi_0}(2\pi r_h)}{4G_N^{(3)}} = S_{\mathrm{BH}}^{(0)}. \tag*{(13)} Φ [ v phys ] = 2 π r h ρ S = 4 G N ( 3 ) e − 2 ϕ 0 ( 2 π r h ) = S BH ( 0 ) . (13) The horizon capacity bounds every admissible flow, so the stationary F1 current realizes a global maximal flow. Both factors in C g e o m C_{\mathrm{geom}} C geom ρ S \rho_S ρ S ∣ q ∣ |q| ∣ q ∣ Figure 2 illustrates how stationary F1 dynamics provides a physical maximal-flow representative with the same entropy flux as a radial bit-thread flow.
Figure 2. This figure illustrates the comparison of a radial bit-thread maximal flow (blue dashed curves) with the flow constructed from stationary F1 sources (red solid curves) in NS–NS-supported BTZ. The upper panels show one exterior in ( r , θ ) (r,\theta) ( r , θ ) S B H ( 0 ) S_{\mathrm{BH}}^{(0)} S BH ( 0 )
This result gives the thread picture a concrete physical interpretation. The string worldsheet is a solution of the source dynamics, its Kalb–Ramond charge supplies a conserved spacetime current, and the sphere-derived capacity assigns an entropy weight to that current. The stationary strings are therefore physical channels carrying the entropic flux represented by bit threads. For a uniform charge ensemble,
C g e o m = ρ S 2 π r h N = S B H ( 0 ) N , (14) C_{\mathrm{geom}}=\rho_S\frac{2\pi r_h}{\mathcal{N}}=\frac{S_{\mathrm{BH}}^{(0)}}{\mathcal{N}}, \tag*{(14)} C geom = ρ S N 2 π r h = N S BH ( 0 ) , (14) so the entropy weight per unit string charge equals the local capacity multiplied by the portion of the horizon circumference assigned to that charge. This is the precise sense in which the physical strings carry entropic information in the present construction.
We finally extend the calculation to BTZ backgrounds with mixed NS–NS and RR flux, providing a further test of the construction while holding the metric and dilaton fixed. We use the BTZ quotient of the type IIB A d S 3 × S 3 × T 4 \mathrm{AdS}_3 \times S^3 \times T^4 AdS 3 × S 3 × T 4 η \eta η 1 − η 2 \sqrt{1-\eta^2} 1 − η 2 [30 , 31 ] , where the parameter 0 ≤ η ≤ 1 0 \le\eta\le1 0 ≤ η ≤ 1 B B B
D η 2 = ( 1 − η 2 ) r 2 + η 2 r h 2 , θ η ′ = η ℓ r D η , ∣ q η ∣ = N 2 π D η . (15) D_\eta^2=(1-\eta^2)r^2+\eta^2 r_h^2,\qquad\theta_\eta'=\frac{\eta\ell}{rD_\eta},\qquad|q_\eta|=\frac{\mathcal{N}}{2\pi D_\eta}. \tag*{(15)} D η 2 = ( 1 − η 2 ) r 2 + η 2 r h 2 , θ η ′ = r D η η ℓ , ∣ q η ∣ = 2 π D η N . (15) It interpolates between a radial F1 in the pure RR background and the logarithmic spiral in the pure NS–NS background. For generic mixed flux, ∣ q η ∣ |q_\eta| ∣ q η ∣ r r r
v p h y s , η i = C g e o m q η i , C g e o m = ρ S ∣ q η ∣ h = 2 π r h ρ S N , ∣ v p h y s , η ∣ ρ S = r h D η ≤ 1. (16) v_{\mathrm{phys},\eta}^{i}=C_{\mathrm{geom}}q_{\eta}^{i},\qquad C_{\mathrm{geom}}=\frac{\rho_S}{|q_\eta|_h}=\frac{2\pi r_h\rho_S}{\mathcal{N}},\qquad\frac{|v_{\mathrm{phys},\eta}|}{\rho_S}=\frac{r_h}{D_\eta}\leq1. \tag*{(16)} v phys , η i = C geom q η i , C geom = ∣ q η ∣ h ρ S = N 2 π r h ρ S , ρ S ∣ v phys , η ∣ = D η r h ≤ 1. (16) For every 0 ≤ η ≤ 1 0\leq\eta\leq1 0 ≤ η ≤ 1 v p h y s , η i v_{\mathrm{phys},\eta}^{i} v phys , η i 2 π r h ρ S = S B H ( 0 ) 2\pi r_h\rho_S=S_{\mathrm{BH}}^{(0)} 2 π r h ρ S = S BH ( 0 )
In this paper, we show that the off-shell sphere functional determines the local entropy capacity. The gravitational replica construction selects the entropy bottleneck. The F1 equations determine physical charge-current lines. The horizon flux bound proves that the resulting capacity-normalized current carries the maximal allowed flux. These results extend the string charge density/bit threads correspondence by deriving its local entropy capacity and constructing stationary on-shell F1 representatives of maximal entropy flow. It is worth noting that all statements in this paper are made at genus zero and two-derivative order, and the source realizations use classical probe strings and continuum coarse graining.
The paper is organized as follows. Section 2 reviews the string charge density/bit-thread correspondence and the normalization issue in the original BTZ construction. Section 3 derives the two-derivative target-space action and the dilaton-weighted entropy from the regulated off-shell sphere. Section 4 combines this local entropy functional with the gravitational replica construction and weighted flow–cut duality to obtain the string-frame norm bound. Section 5 constructs the stationary BTZ strings and proves that their projected current realizes a maximal entropy flow. Section 6 specifies the mixed-flux bulk and F1 source actions, derives the regular stationary strings and their projected currents, and proves that the corresponding flows reproduce the BTZ entropy throughout the NS–NS/RR interpolation. Section 7 summarizes the results.
String charge density/bit threads correspondence
In this section, we briefly review the string charge density/bit threads correspondence introduced in our previous work [14 ] . The central proposal is that the spatial projection of a conserved string charge density can provide a physical interpretation of a bit-thread flow. The construction has two logically distinct ingredients. The worldsheet coupling to the Kalb–Ramond field determines a conserved charge current and hence the direction and flux lines of the candidate flow. Its overall entropy normalization, however, is introduced through a conversion coefficient C g e o m C_{\mathrm{geom}} C geom
From the worldsheet current to a conserved spatial flow
Consider a macroscopic fundamental string propagating in a background metric, dilaton, and Kalb–Ramond field [10 , 11 , 12 , 13 ] . It can be described by the sigma model
S σ = − 1 4 π α ′ ∫ d 2 σ ( − γ γ m n ∂ m X a ∂ n X b g a b + ϵ m n ∂ m X a ∂ n X b B a b ) . (17) S_{\sigma}=-\frac{1}{4\pi\alpha'}\int d^2\sigma\left(\sqrt{-\gamma}\gamma^{mn}\partial_mX^a\partial_nX^b g_{ab}+\epsilon^{mn}\partial_mX^a\partial_nX^b B_{ab}\right). \tag*{(17)} S σ = − 4 π α ′ 1 ∫ d 2 σ ( − γ γ mn ∂ m X a ∂ n X b g ab + ϵ mn ∂ m X a ∂ n X b B ab ) . (17) Varying the coupling to B a b B_{ab} B ab
j a b = 1 − g ( x ) ∫ d 2 σ ( ∂ X a ∂ τ ∂ X b ∂ σ − ∂ X b ∂ τ ∂ X a ∂ σ ) δ ( 3 ) ( x − X ( τ , σ ) ) . (18) j^{ab}=\frac{1}{\sqrt{-g(x)}}\int d^2\sigma\left(\frac{\partial X^a}{\partial\tau}\frac{\partial X^b}{\partial\sigma}-\frac{\partial X^b}{\partial\tau}\frac{\partial X^a}{\partial\sigma}\right)\delta^{(3)}(x-X(\tau,\sigma)). \tag*{(18)} j ab = − g ( x ) 1 ∫ d 2 σ ( ∂ τ ∂ X a ∂ σ ∂ X b − ∂ τ ∂ X b ∂ σ ∂ X a ) δ ( 3 ) ( x − X ( τ , σ )) . (18) For a closed worldsheet, or locally away from the endpoints of an open worldsheet, this current obeys
∂ a ( − g j a b ) = 0. (19) \partial_a\left(\sqrt{-g}j^{ab}\right)=0. \tag*{(19)} ∂ a ( − g j ab ) = 0. (19) The current in Eq. (18) is normalized so that one consistently oriented fundamental string carries one unit of Kalb–Ramond charge.
We now restrict to a static spacetime with vanishing shift and write
d S 2 = − N A D M 2 d t 2 + h i j d x i d x j , − g = N A D M h . (20) dS^2=-N_{\mathrm{ADM}}^2dt^2+h_{ij}dx^idx^j,\qquad\sqrt{-g}=N_{\mathrm{ADM}}\sqrt{h}. \tag*{(20)} d S 2 = − N ADM 2 d t 2 + h ij d x i d x j , − g = N ADM h . (20) The projected string charge density on a constant time slice Σ \Sigma Σ
q i ≡ N A D M j 0 i = − g h j 0 i . (21) q^i\equiv N_{\mathrm{ADM}}j^{0i}=\frac{\sqrt{-g}}{\sqrt{h}}j^{0i}. \tag*{(21)} q i ≡ N ADM j 0 i = h − g j 0 i . (21) It also obeys
∇ i q i = 1 h ∂ i ( h q i ) = 0. (22) \nabla_iq^i=\frac{1}{\sqrt{h}}\partial_i\left(\sqrt{h}q^i\right)=0. \tag*{(22)} ∇ i q i = h 1 ∂ i ( h q i ) = 0. (22) In other words, the projected string charge density automatically has the conservation property required of a bit-thread flow.
Let us consider the nonrotating BTZ black hole supported by NS–NS flux [32 , 33 ] . In string-frame coordinates adapted to one exterior region,
d S B T Z 2 = − r h 2 ℓ 2 sinh 2 ρ d t 2 + ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 , θ ∼ θ + 2 π , (23) dS_{\mathrm{BTZ}}^2=-\frac{r_h^2}{\ell^2}\sinh^2\rho\,dt^2+\ell^2d\rho^2+r_h^2\cosh^2\rho\,d\theta^2,\qquad\theta\sim\theta+2\pi, \tag*{(23)} d S BTZ 2 = − ℓ 2 r h 2 sinh 2 ρ d t 2 + ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 , θ ∼ θ + 2 π , (23) B t θ = − r h 2 ℓ sinh 2 ρ , ϕ = ϕ 0 . (24) B_{t\theta}=-\frac{r_h^2}{\ell}\sinh^2\rho,\qquad\phi=\phi_0. \tag*{(24)} B tθ = − ℓ r h 2 sinh 2 ρ , ϕ = ϕ 0 . (24) The bifurcation circle is at ρ = 0 \rho=0 ρ = 0 2 π r h 2\pi r_h 2 π r h
In the original construction, a radial source embedding is prescribed in static gauge:
X t = τ , X ρ = σ , X θ = θ 0 . (25) X^t=\tau,\qquad X^\rho=\sigma,\qquad X^\theta=\theta_0. \tag*{(25)} X t = τ , X ρ = σ , X θ = θ 0 . (25) For one string, Eq. (18) gives
j B T Z 0 ρ = 1 − g δ 2 π ( θ − θ 0 ) . (26) j_{\mathrm{BTZ}}^{0\rho}=\frac{1}{\sqrt{-g}}\delta_{2\pi}(\theta-\theta_0). \tag*{(26)} j BTZ 0 ρ = − g 1 δ 2 π ( θ − θ 0 ) . (26) For N \mathcal{N} N N / ( 2 π ) \mathcal{N}/(2\pi) N / ( 2 π ) N \mathcal{N} N
j B T Z 0 ρ = 1 − g ∑ k = 0 N − 1 δ 2 π ( θ − 2 π k N ) . (27) j_{\mathrm{BTZ}}^{0\rho}=\frac{1}{\sqrt{-g}}\sum_{k=0}^{\mathcal{N}-1}\delta_{2\pi}\left(\theta-\frac{2\pi k}{\mathcal{N}}\right). \tag*{(27)} j BTZ 0 ρ = − g 1 k = 0 ∑ N − 1 δ 2 π ( θ − N 2 π k ) . (27) The illustration of this configuration can be found in Figure 3 . Denoting the angular separation between
Figure 3. This figure illustrates the radial string-source ensembles on static BTZ slices. Blue dashed lines represent prescribed radial string sources with X θ = θ 0 X^\theta=\theta_0 X θ = θ 0 ( r , θ ) (r,\theta) ( r , θ ) r = r h r=r_h r = r h r = r c r=r_c r = r c
any two adjacent strings as θ s = 2 π / N \theta_s=2\pi/\mathcal{N} θ s = 2 π / N
j B T Z 0 ρ = 1 − g 1 θ s [ θ s ∑ k = 0 N − 1 δ 2 π ( θ − k θ s ) ] = 1 − g N 2 π [ θ s ∑ k = 0 N − 1 δ 2 π ( θ − k θ s ) ] . (28) j_{\mathrm{BTZ}}^{0\rho}=\frac{1}{\sqrt{-g}}\frac{1}{\theta_s}\left[\theta_s\sum_{k=0}^{\mathcal{N}-1}\delta_{2\pi}\left(\theta-k\theta_s\right)\right]=\frac{1}{\sqrt{-g}}\frac{\mathcal{N}}{2\pi}\left[\theta_s\sum_{k=0}^{\mathcal{N}-1}\delta_{2\pi}\left(\theta-k\theta_s\right)\right]. \tag*{(28)} j BTZ 0 ρ = − g 1 θ s 1 [ θ s k = 0 ∑ N − 1 δ 2 π ( θ − k θ s ) ] = − g 1 2 π N [ θ s k = 0 ∑ N − 1 δ 2 π ( θ − k θ s ) ] . (28) Hence, coarse-graining over angular scales larger than the separation between neighboring strings gives
j B T Z 0 ρ = 1 − g N 2 π = 2 r h 2 sinh ( 2 ρ ) N 2 π , j 0 θ = 0. (29) j_{\mathrm{BTZ}}^{0\rho}=\frac{1}{\sqrt{-g}}\frac{\mathcal{N}}{2\pi}=\frac{2}{r_h^2\sinh(2\rho)}\frac{\mathcal{N}}{2\pi},\qquad j^{0\theta}=0. \tag*{(29)} j BTZ 0 ρ = − g 1 2 π N = r h 2 sinh ( 2 ρ ) 2 2 π N , j 0 θ = 0. (29) The lapse and the induced spatial metric are
N A D M = r h ℓ sinh ρ , h i j d x i d x j = ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 . (30) N_{\mathrm{ADM}}=\frac{r_h}{\ell}\sinh\rho,\qquad h_{ij}dx^idx^j=\ell^2d\rho^2+r_h^2\cosh^2\rho\,d\theta^2. \tag*{(30)} N ADM = ℓ r h sinh ρ , h ij d x i d x j = ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 . (30) The projected string charge density is therefore
q B T Z i = N A D M j 0 i = N 2 π ℓ r h cosh ρ ( 1 , 0 ) , i = ρ , θ . (31) q_{\mathrm{BTZ}}^i=N_{\mathrm{ADM}}j^{0i}=\frac{\mathcal{N}}{2\pi\ell r_h\cosh\rho}(1,0),\qquad i=\rho,\theta. \tag*{(31)} q BTZ i = N ADM j 0 i = 2 π ℓ r h cosh ρ N ( 1 , 0 ) , i = ρ , θ . (31) and its norm is
∣ q B T Z ∣ = h i j q B T Z i q B T Z j = N 2 π r h cosh ρ . (32) \lvert q_{\mathrm{BTZ}}\rvert= \sqrt{h_{ij}q_{\mathrm{BTZ}}^{i}q_{\mathrm{BTZ}}^{j}} = \frac{\mathcal{N}}{2\pi r_h\cosh\rho}. \tag*{(32)} ∣ q BTZ ∣ = h ij q BTZ i q BTZ j = 2 π r h cosh ρ N . (32) The candidate bit-thread/entropy flow can be given by using the dimensionless conversion coefficient C g e o m B T Z C_{\mathrm{geom}}^{\mathrm{BTZ}} C geom BTZ
v B T Z i = C g e o m B T Z q B T Z i , (33) v_{\mathrm{BTZ}}^{i} = C_{\mathrm{geom}}^{\mathrm{BTZ}}q_{\mathrm{BTZ}}^{i}, \tag*{(33)} v BTZ i = C geom BTZ q BTZ i , (33) which is divergenceless, and its norm is
∣ v B T Z ∣ = C g e o m B T Z N 2 π r h cosh ρ . (34) \lvert v_{\mathrm{BTZ}}\rvert= C_{\mathrm{geom}}^{\mathrm{BTZ}}\frac{\mathcal{N}}{2\pi r_h\cosh\rho}. \tag*{(34)} ∣ v BTZ ∣ = C geom BTZ 2 π r h cosh ρ N . (34) Because ( 1 / cosh ρ ) ≤ 1 (1/\cosh\rho) \leq1 ( 1/ cosh ρ ) ≤ 1 ρ = 0 \rho= 0 ρ = 0 [3 , 4 , 14 ]
C g e o m B T Z N 2 π r h = e − 2 ϕ 0 4 G N ( 3 ) , (35) C_{\mathrm{geom}}^{\mathrm{BTZ}}\frac{\mathcal{N}}{2\pi r_h} = \frac{e^{-2\phi_0}}{4G_N^{(3)}}, \tag*{(35)} C geom BTZ 2 π r h N = 4 G N ( 3 ) e − 2 ϕ 0 , (35) and therefore
C g e o m B T Z = 2 π r h N e − 2 ϕ 0 4 G N ( 3 ) = e − 2 ϕ 0 π r h 2 G N ( 3 ) N . (36) C_{\mathrm{geom}}^{\mathrm{BTZ}} = \frac{2\pi r_h}{\mathcal{N}}\frac{e^{-2\phi_0}}{4G_N^{(3)}} = \frac{e^{-2\phi_0}\pi r_h}{2G_N^{(3)}\mathcal{N}}. \tag*{(36)} C geom BTZ = N 2 π r h 4 G N ( 3 ) e − 2 ϕ 0 = 2 G N ( 3 ) N e − 2 ϕ 0 π r h . (36) In the convention ϕ 0 = 0 \phi_0 = 0 ϕ 0 = 0
v B T Z i = e − 2 ϕ 0 4 G N ( 3 ) ℓ cosh ρ ( 1 , 0 ) . (37) v_{\mathrm{BTZ}}^{i} = \frac{e^{-2\phi_0}}{4G_N^{(3)}\ell\cosh\rho}(1,0). \tag*{(37)} v BTZ i = 4 G N ( 3 ) ℓ cosh ρ e − 2 ϕ 0 ( 1 , 0 ) . (37) For a circle of constant ρ \rho ρ
n i d x i = ℓ d ρ , d A S = r h cosh ρ d θ . (38) n_i dx^i = \ell d\rho,\qquad dA_S = r_h\cosh\rho\,d\theta. \tag*{(38)} n i d x i = ℓ d ρ , d A S = r h cosh ρ d θ . (38) Its flux is consequently independent of ρ \rho ρ
Φ [ v B T Z ] = ∫ 0 2 π v B T Z i n i d A S = e − 2 ϕ 0 ( 2 π r h ) 4 G N ( 3 ) = S B H ( 0 ) . (39) \Phi[v_{\mathrm{BTZ}}] = \int_{0}^{2\pi}v_{\mathrm{BTZ}}^{i}n_i\,dA_S = \frac{e^{-2\phi_0}(2\pi r_h)}{4G_N^{(3)}} = S_{\mathrm{BH}}^{(0)}. \tag*{(39)} Φ [ v BTZ ] = ∫ 0 2 π v BTZ i n i d A S = 4 G N ( 3 ) e − 2 ϕ 0 ( 2 π r h ) = S BH ( 0 ) . (39) Thus the projected string charge density determines a conserved set of radial flow lines, while the norm-bound saturation determines how much entropy flux is assigned to those lines. The two requirements combine to reproduce the BTZ entropy.
The correspondence was subsequently extended to ten-dimensional three charge black holes and tested under string dualities. In the Type IIB S-duality analysis of the F1–NS5–P and D1–D5–P systems [25 ] , an F1 current is mapped to a D1 current. The local metric, dilaton, charge current, and flow norm change between the two descriptions, but the maximal integrated entropy flux is preserved. In the T-duality analysis [26 ] , a radial F1 source remains an F1 source along the NS–NS duality chain, whereas a radial D1 source transverse to the duality circle becomes a D2 source wrapped on the dual circle. Contracting the projected D2 bivector with the normalized one-form along that circle produces an effective conserved vector flow. Again, the local representatives change, while the dilaton-weighted transverse capacity and the integrated maximal entropy flux remain invariant. These results provide nontrivial duality checks that string and brane charge currents can furnish different microscopic representatives of the same entropy flow.
However, two ingredients of the original construction require a firmer foundation from string theory.
Independent determination of the entropy capacity. In the original identification,
v B T Z i = C g e o m B T Z q B T Z i , (40) v_{\mathrm{BTZ}}^{i}=C_{\mathrm{geom}}^{\mathrm{BTZ}}q_{\mathrm{BTZ}}^{i}, \tag*{(40)} v BTZ i = C geom BTZ q BTZ i , (40) the coefficient C g e o m B T Z C_{\mathrm{geom}}^{\mathrm{BTZ}} C geom BTZ
Dynamical selection of the string embeddings. In the original BTZ construction, the parallel radial strings are introduced through embeddings described above. Although their projected charge density is conserved and reproduces the desired flow, current conservation alone does not establish that these embeddings satisfy the stationary fundamental-string equations of motion. This motivates us to determine the string arrangement directly from the F1 dynamics and then establish whether its projected current realizes a maximal entropy flow. In this way, the string configuration is selected by the worldsheet equations rather than prescribed using a known bit-thread solution.
Black hole entropy from off-shell strings
The purpose of this section is to determine the local entropy capacity independently of the F1 source construction. We first review Tseytlin’s regulated sphere prescription [34 , 35 , 17 , 18 ] and show how the logarithmic cutoff dependence of the genus-zero worldsheet functional produces the two-derivative string-frame effective action. We then evaluate the same off-shell functional on a regulated conical target-space geometry. The contribution linear in the conical deficit localizes on the bifurcation surface Γ H \Gamma_H Γ H
S B H ( 0 ) = 1 4 G N ∫ Γ H e − 2 ϕ d A S . (41) S_{\mathrm{BH}}^{(0)}=\frac{1}{4G_N}\int_{\Gamma_H}e^{-2\phi}\,\mathrm{d}A_S. \tag*{(41)} S BH ( 0 ) = 4 G N 1 ∫ Γ H e − 2 ϕ d A S . (41) This result identifies
ρ S ( x ) = e − 2 ϕ ( x ) 4 G N (42) \rho_S(x)=\frac{e^{-2\phi(x)}}{4G_N} \tag*{(42)} ρ S ( x ) = 4 G N e − 2 ϕ ( x ) (42) as the entropy capacity per unit string-frame area. Note that the fixed-cone calculation determines this density for a given surface. It does not yet identify the surface that computes the entropy of a general boundary region. Section 4 provides that selection through the gravitational replica construction and then converts the weighted minimal cut into a maximal flow.
Brief review of low energy effective action from off-shell strings
Let
Ψ = { g μ ν , B μ ν , ϕ , … } (43) \Psi= \{g_{\mu\nu}, B_{\mu\nu}, \phi, \ldots\} \tag*{(43)} Ψ = { g μν , B μν , ϕ , … } (43) denote the target-space fields. We use g μ ν g_{\mu\nu} g μν γ a b \gamma_{ab} γ ab W W W [36 , 37 ]
I σ [ X ; Ψ ] = 1 4 π α ′ ∫ W d 2 σ [ γ γ a b g μ ν ( X ) ∂ a X μ ∂ b X ν + i ϵ a b B μ ν ( X ) ∂ a X μ ∂ b X ν + α ′ γ ϕ ( X ) R ( 2 ) [ γ ] ] , (44) I_{\sigma}[X;\Psi] = \frac{1}{4\pi\alpha'} \int_W d^2\sigma\left[\sqrt{\gamma}\gamma^{ab}g_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+ i\epsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+ \alpha'\sqrt{\gamma}\phi(X)R^{(2)}[\gamma]\right], \tag*{(44)} I σ [ X ; Ψ ] = 4 π α ′ 1 ∫ W d 2 σ [ γ γ ab g μν ( X ) ∂ a X μ ∂ b X ν + i ϵ ab B μν ( X ) ∂ a X μ ∂ b X ν + α ′ γ ϕ ( X ) R ( 2 ) [ γ ] ] , (44) where ϵ a b \epsilon^{ab} ϵ ab ϵ 12 = + 1 \epsilon^{12}=+1 ϵ 12 = + 1 ϕ = ϕ ∞ + φ \phi=\phi_\infty+\varphi ϕ = ϕ ∞ + φ g s = e ϕ ∞ g_s=e^{\phi_\infty} g s = e ϕ ∞
exp [ − ϕ ∞ 4 π ∫ W γ R ( 2 ) ] = e − ϕ ∞ χ ( W ) . (45) \exp\left[-\frac{\phi_\infty}{4\pi}\int_W\sqrt{\gamma}R^{(2)}\right] = e^{-\phi_\infty\chi(W)}. \tag*{(45)} exp [ − 4 π ϕ ∞ ∫ W γ R ( 2 ) ] = e − ϕ ∞ χ ( W ) . (45) For the sphere, χ ( S 2 ) = 2 \chi(S^2)=2 χ ( S 2 ) = 2 g s − 2 g_s^{-2} g s − 2 Y Y Y
g s − 2 e − 2 φ ( Y ) = e − 2 ϕ ( Y ) . (46) g_s^{-2}e^{-2\varphi(Y)}=e^{-2\phi(Y)}. \tag*{(46)} g s − 2 e − 2 φ ( Y ) = e − 2 ϕ ( Y ) . (46) On the sphere, χ = 2 \chi=2 χ = 2 e − 2 ϕ e^{-2\phi} e − 2 ϕ I 0 = O ( g s − 2 ) I_0=O(g_s^{-2}) I 0 = O ( g s − 2 ) (42) . We retain B μ ν B_{\mu\nu} B μν B B B
To make the prescription explicit, we expand locally in theory space around a reference worldsheet CFT [17 , 18 , 19 ] . Let O a O_a O a ( h a , h a ) (h_a,h_a) ( h a , h a )
I Q F T = I C F T + ∑ a ϵ 2 ( h a − 1 ) ω a V a , V a = ∫ S 2 d 2 z g O a ( z , z ˉ ) , (47) I_{\mathrm{QFT}}=I_{\mathrm{CFT}}+\sum_a\epsilon^{2(h_a-1)}\omega^a V_a,\qquad V_a=\int_{S^2}d^2z\sqrt{g}O_a(z,\bar z), \tag*{(47)} I QFT = I CFT + a ∑ ϵ 2 ( h a − 1 ) ω a V a , V a = ∫ S 2 d 2 z g O a ( z , z ˉ ) , (47) where Δ a = 2 h a \Delta_a=2h_a Δ a = 2 h a ω a \omega^a ω a ( 1 , 1 ) (1,1) ( 1 , 1 ) h a ≠ 1 h_a\ne1 h a = 1 ω a \omega^a ω a g μ ν g_{\mu\nu} g μν B μ ν B_{\mu\nu} B μν ϕ \phi ϕ n n n
K g ( τ ) = ∑ n = 0 ∞ K g , n ( τ ) , (48) K_g(\tau)=\sum_{n=0}^{\infty}K_{g,n}(\tau), \tag*{(48)} K g ( τ ) = n = 0 ∑ ∞ K g , n ( τ ) , (48) K g , n = ∑ a 1 … a n ( ∏ i = 1 n e 2 ( h a i − 1 ) ω a i ) 1 n ! ⟨ ⟨ V a 1 ⋯ V a n ⟩ ⟩ C F T . (49) K_{g,n}=\sum_{a_1\ldots a_n}\left(\prod_{i=1}^{n}e^{2(h_{a_i}-1)\omega^{a_i}}\right)\frac{1}{n!}\langle\!\langle V_{a_1}\cdots V_{a_n}\rangle\!\rangle_{\mathrm{CFT}}. \tag*{(49)} K g , n = a 1 … a n ∑ ( i = 1 ∏ n e 2 ( h a i − 1 ) ω a i ) n ! 1 ⟨ ⟨ V a 1 ⋯ V a n ⟩ ⟩ CFT . (49) where τ \tau τ g = 0 g=0 g = 0 K 0 = ∑ n K 0 , n K_0=\sum_n K_{0,n} K 0 = ∑ n K 0 , n n n n V a V_a V a g = 0 g=0 g = 0 ϵ \epsilon ϵ
d ( z i , z j ) ≥ 2 ϵ ( i ≠ j ) . (50) d(z_i,z_j)\ge2\epsilon\qquad(i\ne j). \tag*{(50)} d ( z i , z j ) ≥ 2 ϵ ( i = j ) . (50) Now, let us recall Tseytlin’s two sphere prescriptions [34 , 35 , 18 ] , which act separately on every K 0 , n K_{0,n} K 0 , n
T 1 : A 0 , n = ∂ K 0 , n ∂ log ϵ , (51) \mathrm{T1}:\qquad A_{0,n}=\frac{\partial K_{0,n}}{\partial\log\epsilon}, \tag*{(51)} T1 : A 0 , n = ∂ log ϵ ∂ K 0 , n , (51) T 2 : A 0 , n = ( ∂ ∂ log ϵ + 1 2 ∂ 2 ( ∂ log ϵ ) 2 ) K 0 , n . (52) \mathrm{T2}:\qquad A_{0,n}=\left(\frac{\partial}{\partial\log\epsilon}+\frac{1}{2}\frac{\partial^2}{(\partial\log\epsilon)^2}\right)K_{0,n}. \tag*{(52)} T2 : A 0 , n = ( ∂ log ϵ ∂ + 2 1 ( ∂ log ϵ ) 2 ∂ 2 ) K 0 , n . (52) Resumming the connected sphere amplitudes in n n n
− I 0 = Z 0 = ∑ n = 0 ∞ A 0 , n . (53) -I_0=Z_0=\sum_{n=0}^{\infty}A_{0,n}. \tag*{(53)} − I 0 = Z 0 = n = 0 ∑ ∞ A 0 , n . (53) Thus, the n n n [18 ]
I 0 ( T 1 ) = − ( ∂ ∂ log ϵ ) K 0 , I 0 ( T 2 ) = − ( ∂ ∂ log ϵ + 1 2 ∂ 2 ( ∂ log ϵ ) 2 ) K 0 . (54) \begin{aligned}
I_0^{(\mathrm{T1})}&=-\left(\frac{\partial}{\partial\log\epsilon}\right)K_0,\\
I_0^{(\mathrm{T2})}&=-\left(\frac{\partial}{\partial\log\epsilon}+\frac{1}{2}\frac{\partial^2}{(\partial\log\epsilon)^2}\right)K_0. \tag*{(54)}
\end{aligned} I 0 ( T1 ) I 0 ( T2 ) = − ( ∂ log ϵ ∂ ) K 0 , = − ( ∂ log ϵ ∂ + 2 1 ( ∂ log ϵ ) 2 ∂ 2 ) K 0 . (54) Moreover, this action also can be written in terms of β \beta β ω a \omega^a ω a log ϵ \log\epsilon log ϵ β \beta β
β a ≔ − d ω a d log ϵ . (55) \beta^a\coloneqq-\frac{d\omega^a}{d\log\epsilon}. \tag*{(55)} β a : = − d log ϵ d ω a . (55) In this case, the T1 action can be rewritten as
I 0 ( T 1 ) = − ∂ K 0 ∂ log ϵ = ∑ a β a ∂ K 0 ∂ ω a . (56) I_0^{(\mathrm{T1})}=-\frac{\partial K_0}{\partial\log\epsilon}=\sum_a\beta^a\frac{\partial K_0}{\partial\omega^a}. \tag*{(56)} I 0 ( T1 ) = − ∂ log ϵ ∂ K 0 = a ∑ β a ∂ ω a ∂ K 0 . (56) And the T2 action can be given by
I 0 ( T 2 ) = I 0 ( T 1 ) − 1 2 ( ∑ a , b β a β b ∂ 2 K 0 ∂ ω a ∂ ω b + ∑ a , b β b ∂ β a ∂ ω b ∂ K 0 ∂ ω a ) . (57) I_0^{(\mathrm{T2})}=I_0^{(\mathrm{T1})}-\frac{1}{2}\left(\sum_{a,b}\beta^a\beta^b\frac{\partial^2 K_0}{\partial\omega^a\partial\omega^b}+\sum_{a,b}\beta^b\frac{\partial\beta^a}{\partial\omega^b}\frac{\partial K_0}{\partial\omega^a}\right). \tag*{(57)} I 0 ( T2 ) = I 0 ( T1 ) − 2 1 a , b ∑ β a β b ∂ ω a ∂ ω b ∂ 2 K 0 + a , b ∑ β b ∂ ω b ∂ β a ∂ ω a ∂ K 0 . (57) It is worth noting that every term of action is proportional to at least one beta function. To obtain the equations of motion and low energy effective action, it is sufficient to use the T1 prescription. Indeed, after the cancellation of the power-law divergences between the matter measure and the Gaussian determinant, and the cancellation of the remaining critical-dimension contribution against the ghosts, the cutoff dependence of K 0 K_0 K 0 log ϵ \log\epsilon log ϵ O ( α ′ ) O(\alpha') O ( α ′ )
I 0 ( T 2 ) = I 0 ( T 1 ) + O ( α ′ 2 ) . (58) I_0^{(\mathrm{T2})}=I_0^{(\mathrm{T1})}+O(\alpha'^2). \tag*{(58)} I 0 ( T2 ) = I 0 ( T1 ) + O ( α ′2 ) . (58) The T2 prescription becomes essential when the bosonic closed-string tachyon is included, since its additional second-derivative term automatically removes the spurious contribution proportional to the worldsheet cosmological-constant divergence ϵ − 2 \epsilon^{-2} ϵ − 2 T1 prescription is sufficient at this order.
Now, let us see how the equations of motion and the corresponding low-energy effective action are obtained using Ahmadain and Wall’s method [17 , 18 , 19 ] . We first consider the bosonic nonlinear sigma model on a compact Euclidean worldsheet W \mathcal{W} W
Z ( B ) ( g , ϕ ; ϵ ) = ∫ [ d X ] e − I Q F T [ X ] , (59) Z^{(B)}(g,\phi;\epsilon)=\int[dX]e^{-I_{\mathrm{QFT}}[X]}, \tag*{(59)} Z ( B ) ( g , ϕ ; ϵ ) = ∫ [ d X ] e − I QFT [ X ] , (59) where [ d X ] = ∏ z d x ( z ) g ( X ( z ˉ ) ) [dX]=\prod_z dx(z)\sqrt{g(X(\bar z))} [ d X ] = ∏ z d x ( z ) g ( X ( z ˉ ))
I Q F T [ X ] = 1 4 π α ′ ∫ W d 2 z γ [ g μ ν ( X ) ∂ a X μ ∂ a X ν + α ′ R ( 2 ) [ γ ] ϕ ( X ) ] . (60) I_{\mathrm{QFT}}[X]=\frac{1}{4\pi\alpha'}\int_{\mathcal{W}}d^2z\sqrt{\gamma}\left[g_{\mu\nu}(X)\partial_aX^\mu\partial^aX^\nu+\alpha'R^{(2)}[\gamma]\phi(X)\right]. \tag*{(60)} I QFT [ X ] = 4 π α ′ 1 ∫ W d 2 z γ [ g μν ( X ) ∂ a X μ ∂ a X ν + α ′ R ( 2 ) [ γ ] ϕ ( X ) ] . (60) Here g μ ν ( X ) g_{\mu\nu}(X) g μν ( X ) ϕ ( X ) \phi(X) ϕ ( X ) T1 prescription is
K 0 ( ϵ ) = Z ( B ) Z g h o s t . (61) K_0(\epsilon)=Z^{(B)}Z_{\mathrm{ghost}}. \tag*{(61)} K 0 ( ϵ ) = Z ( B ) Z ghost . (61) If we can show the two-derivative part of the regulated functional has the schematic form
K 0 ( ϵ ) = K 0 ( 0 ) + ( log ϵ ) ∫ d D Y L log ( Y ) + finite + O ( α ′ 2 ) , (62) K_0(\epsilon)=K_0^{(0)}+(\log\epsilon)\int d^D Y\,\mathcal{L}_{\log}(Y)+\text{finite}+O(\alpha'^2), \tag*{(62)} K 0 ( ϵ ) = K 0 ( 0 ) + ( log ϵ ) ∫ d D Y L l o g ( Y ) + finite + O ( α ′2 ) , (62) then the T1 action gives
I 0 ( T 1 ) = − ( ∂ ∂ log ϵ ) K 0 = − ∫ d D Y L log ( Y ) . (63) I_0^{(\mathrm{T1})}=-\left(\frac{\partial}{\partial\log\epsilon}\right)K_0=-\int d^D Y\,\mathcal{L}_{\log}(Y). \tag*{(63)} I 0 ( T1 ) = − ( ∂ log ϵ ∂ ) K 0 = − ∫ d D Y L l o g ( Y ) . (63) It gives us low energy effective action. Therefore, the main aim of following calculations is to obtain K 0 ( ϵ ) K_0(\epsilon) K 0 ( ϵ ) L log ( Y ) \mathcal{L}_{\log}(Y) L l o g ( Y )
To approach this aim, we split the embedding field X μ ( z ) X^\mu(z) X μ ( z )
X μ ( z ) = Y μ + η μ ( z ) , ∫ W d 2 z γ η μ ( z ) = 0 , (64) X^\mu(z)=Y^\mu+\eta^\mu(z), \qquad\int_W \mathrm{d}^2z\sqrt{\gamma}\eta^\mu(z)=0, \tag*{(64)} X μ ( z ) = Y μ + η μ ( z ) , ∫ W d 2 z γ η μ ( z ) = 0 , (64) where Y μ Y^\mu Y μ η μ \eta^\mu η μ W W W
X μ ( z ) = ∑ m = 0 ∞ x m μ ϕ m ( z ) , (65) X^\mu(z)=\sum_{m=0}^{\infty}x^\mu_m\phi_m(z), \tag*{(65)} X μ ( z ) = m = 0 ∑ ∞ x m μ ϕ m ( z ) , (65) where
Δ ϕ m = λ m ϕ m , ∫ W d 2 z γ ϕ m ϕ n = δ m n . (66) \Delta\phi_m=\lambda_m\phi_m, \qquad\int_W \mathrm{d}^2z\sqrt{\gamma}\phi_m\phi_n=\delta_{mn}. \tag*{(66)} Δ ϕ m = λ m ϕ m , ∫ W d 2 z γ ϕ m ϕ n = δ mn . (66) For the non-bounded compact worldsheet, we have
λ m = ∫ W d 2 z γ ∇ A ϕ m ∇ A ϕ n ≥ 0. (67) \lambda_m=\int_W \mathrm{d}^2z\sqrt{\gamma}\nabla_A\phi_m\nabla^A\phi_n\ge0. \tag*{(67)} λ m = ∫ W d 2 z γ ∇ A ϕ m ∇ A ϕ n ≥ 0. (67) Moreover, if λ 0 = 0 \lambda_0=0 λ 0 = 0 ∇ A ϕ 0 = 0 \nabla_A\phi_0=0 ∇ A ϕ 0 = 0 ϕ 0 \phi_0 ϕ 0
1 = ∫ W d 2 z γ ϕ 0 2 = V ϕ 0 2 , V ≡ ∫ W d 2 z γ , (68) 1=\int_W \mathrm{d}^2z\sqrt{\gamma}\phi_0^2=V\phi_0^2, \qquad V\equiv\int_W \mathrm{d}^2z\sqrt{\gamma}, \tag*{(68)} 1 = ∫ W d 2 z γ ϕ 0 2 = V ϕ 0 2 , V ≡ ∫ W d 2 z γ , (68) which gives ϕ 0 = 1 / V \phi_0=1/\sqrt{V} ϕ 0 = 1/ V
X μ ( z ) = x 0 μ ϕ 0 ( z ) + ∑ m ≠ 0 x m μ ϕ m ( z ) = x 0 μ V + ∑ m ≠ 0 x m μ ϕ m ( z ) ≡ Y μ + η μ ( z ) . (69) \begin{aligned}
X^\mu(z) &= x^\mu_0\phi_0(z)+\sum_{m\ne0}x^\mu_m\phi_m(z) \\
&=\frac{x^\mu_0}{\sqrt{V}}+\sum_{m\ne0}x^\mu_m\phi_m(z) \\
&\equiv Y^\mu+\eta^\mu(z). \tag*{(69)}
\end{aligned} X μ ( z ) = x 0 μ ϕ 0 ( z ) + m = 0 ∑ x m μ ϕ m ( z ) = V x 0 μ + m = 0 ∑ x m μ ϕ m ( z ) ≡ Y μ + η μ ( z ) . (69) Now, the regulated scalar propagator for non-zero modes can be given by
D ( z , z ′ ; ϵ ) = ∑ m ≠ 0 e − ϵ 2 λ m λ m ϕ m ( z ) ϕ m ( z ′ ) , (70) D(z,z';\epsilon)=\sum_{m\ne0}\frac{e^{-\epsilon^2\lambda_m}}{\lambda_m}\phi_m(z)\phi_m(z'), \tag*{(70)} D ( z , z ′ ; ϵ ) = m = 0 ∑ λ m e − ϵ 2 λ m ϕ m ( z ) ϕ m ( z ′ ) , (70) Acting with ∂ / ∂ ϵ 2 \partial/\partial\epsilon^2 ∂ / ∂ ϵ 2
∂ D ( z , z ′ ; ϵ ) ∂ ϵ 2 = − ∑ m ≠ 0 e − ϵ 2 λ m ϕ m ( z ) ϕ m ( z ′ ) , (71) \frac{\partial D(z,z';\epsilon)}{\partial\epsilon^2}=-\sum_{m\ne0}e^{-\epsilon^2\lambda_m}\phi_m(z)\phi_m(z'), \tag*{(71)} ∂ ϵ 2 ∂ D ( z , z ′ ; ϵ ) = − m = 0 ∑ e − ϵ 2 λ m ϕ m ( z ) ϕ m ( z ′ ) , (71) where the heat kernel is given by [38 ]
K ( z , z ′ ; ϵ ) = ∑ m ≥ 0 e − ϵ 2 λ m ϕ m ( z ) ϕ m ( z ′ ) . (72) K(z,z';\epsilon)=\sum_{m\ge0}e^{-\epsilon^2\lambda_m}\phi_m(z)\phi_m(z'). \tag*{(72)} K ( z , z ′ ; ϵ ) = m ≥ 0 ∑ e − ϵ 2 λ m ϕ m ( z ) ϕ m ( z ′ ) . (72) Since ϕ 0 ( z ) ϕ 0 ( z ′ ) = 1 / V \phi_0(z)\phi_0(z')=1/V ϕ 0 ( z ) ϕ 0 ( z ′ ) = 1/ V
∂ D ( z , z ; ϵ ) ∂ ϵ 2 = − K ( z , z ; ϵ ) + 1 V . (73) \frac{\partial D(z,z;\epsilon)}{\partial\epsilon^2}=-K(z,z;\epsilon)+\frac{1}{V}. \tag*{(73)} ∂ ϵ 2 ∂ D ( z , z ; ϵ ) = − K ( z , z ; ϵ ) + V 1 . (73) where the coincident heat kernel has the small-ϵ \epsilon ϵ
K ( z , z ; ϵ ) = 1 4 π ϵ 2 + 1 24 π R ( 2 ) + O ( ϵ 2 ) . (74) K(z,z;\epsilon)=\frac{1}{4\pi\epsilon^{2}}+\frac{1}{24\pi}R^{(2)}+O(\epsilon^{2}). \tag*{(74)} K ( z , z ; ϵ ) = 4 π ϵ 2 1 + 24 π 1 R ( 2 ) + O ( ϵ 2 ) . (74) Integrating with respect to ϵ 2 \epsilon^{2} ϵ 2
D ( z , z ; ϵ ) = − 1 4 π log ϵ 2 + h = − 1 2 π log ϵ + h . (75) D(z,z;\epsilon)=-\frac{1}{4\pi}\log\epsilon^{2}+h=-\frac{1}{2\pi}\log\epsilon+h. \tag*{(75)} D ( z , z ; ϵ ) = − 4 π 1 log ϵ 2 + h = − 2 π 1 log ϵ + h . (75) Therefore, the term log ϵ \log\epsilon log ϵ N N N
N ≡ Tr e − ϵ 2 Δ = ∫ d 2 z g γ K ( z , z ; ϵ ) = V 4 π ϵ 2 + 1 6 χ + O ( ϵ 2 ) , (76) N\equiv\operatorname{Tr}e^{-\epsilon^{2}\Delta}=\int d^{2}z\sqrt{g}\gamma K(z,z;\epsilon)=\frac{V}{4\pi\epsilon^{2}}+\frac{1}{6}\chi+O(\epsilon^{2}), \tag*{(76)} N ≡ Tr e − ϵ 2 Δ = ∫ d 2 z g γ K ( z , z ; ϵ ) = 4 π ϵ 2 V + 6 1 χ + O ( ϵ 2 ) , (76) Equivalently, the regulated number of propagating nonzero modes is
N ′ = Tr ′ e − ϵ 2 Δ = N − 1. (77) N'=\operatorname{Tr}'e^{-\epsilon^{2}\Delta}=N-1. \tag*{(77)} N ′ = Tr ′ e − ϵ 2 Δ = N − 1. (77) Using the full dilaton ϕ = ϕ ∞ + φ \phi=\phi_{\infty}+\varphi ϕ = ϕ ∞ + φ
Z ( B ) = Z f ( 0 ) ∫ d D Y g e − χ ϕ Z D ( 1 ) Z g ( 1 ) Z M ( 1 ) ( Y ) = Z f ( 0 ) ∫ d D Y g e − χ ϕ ( 1 + δ Z ϕ + δ Z g + δ Z M + O ( α ′ 2 ) ) , (78) \begin{aligned}
Z^{(B)}=Z_f^{(0)}\int d^{D}Y\sqrt{g}e^{-\chi\phi}Z_D^{(1)}Z_g^{(1)}Z_M^{(1)}(Y) \\
&=Z_f^{(0)}\int d^{D}Y\sqrt{g}e^{-\chi\phi}\left(1+\delta Z_{\phi}+\delta Z_g+\delta Z_M+O(\alpha'^{2})\right), \tag*{(78)}
\end{aligned} Z ( B ) = Z f ( 0 ) ∫ d D Y g e − χ ϕ Z D ( 1 ) Z g ( 1 ) Z M ( 1 ) ( Y ) = Z f ( 0 ) ∫ d D Y g e − χ ϕ ( 1 + δ Z ϕ + δ Z g + δ Z M + O ( α ′ 2 ) ) , (78) where Z D ( 1 ) Z_D^{(1)} Z D ( 1 ) Z g ( 1 ) Z_g^{(1)} Z g ( 1 ) Z M ( 1 ) Z_M^{(1)} Z M ( 1 ) X μ ( z ) = Y μ + η μ ( z ) X^\mu(z)=Y^\mu+\eta^\mu(z) X μ ( z ) = Y μ + η μ ( z ) Y μ Y^\mu Y μ η μ \eta^\mu η μ O ( α ′ ) O(\alpha') O ( α ′ )
δ Z ϕ = − π χ α ′ D ( z , z ; ϵ ) g μ ν ∂ μ ∂ ν ϕ . (79) \delta Z_{\phi}=-\pi\chi\alpha'D(z,z;\epsilon)g^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi. \tag*{(79)} δ Z ϕ = − π χ α ′ D ( z , z ; ϵ ) g μν ∂ μ ∂ ν ϕ . (79) On the other hand, the metric and measure sectors Z g ( 1 ) Z_g^{(1)} Z g ( 1 ) Z M ( 1 ) Z_M^{(1)} Z M ( 1 ) ∂ λ ∂ ρ g μ ν \partial_{\lambda}\partial_{\rho}g_{\mu\nu} ∂ λ ∂ ρ g μν ∂ ρ g μ ν ∂ λ g α β \partial_{\rho}g_{\mu\nu}\partial_{\lambda}g_{\alpha\beta} ∂ ρ g μν ∂ λ g α β [18 ] . Using
D ( z , z ; ϵ ) = − 1 2 π log ϵ + h , N − N ′ = 1 , (80) D(z,z;\epsilon)=-\frac{1}{2\pi}\log\epsilon+h,\qquad N-N'=1, \tag*{(80)} D ( z , z ; ϵ ) = − 2 π 1 log ϵ + h , N − N ′ = 1 , (80) and combining the dilaton, metric, and measure contributions, one obtains
δ Z ϕ + δ Z g + δ Z M ≐ α ′ 2 ( log ϵ + h ) ( R + χ 2 ∂ μ ϕ ∂ μ ϕ ) . (81) \delta Z_{\phi}+\delta Z_g+\delta Z_M\doteq\frac{\alpha'}{2}(\log\epsilon+h)\left(R+\chi^{2}\partial_{\mu}\phi\partial^{\mu}\phi\right). \tag*{(81)} δ Z ϕ + δ Z g + δ Z M ≐ 2 α ′ ( log ϵ + h ) ( R + χ 2 ∂ μ ϕ ∂ μ ϕ ) . (81) Here ≐ \doteq ≐ d D Y g e − χ ϕ d^{D}Y\sqrt{g}e^{-\chi\phi} d D Y g e − χ ϕ
Z ( B ) = g s − χ Z f ( 0 ) ∫ d D Y g e − χ ϕ [ 1 + α ′ 2 ( log ϵ + h ) ( R + χ 2 ∂ μ ϕ ∂ μ ϕ ) + O ( α ′ 2 ) ] . (82) Z^{(B)}=g_s^{-\chi}Z_f^{(0)}\int d^{D}Y\sqrt{g}e^{-\chi\phi}\left[1+\frac{\alpha'}{2}(\log\epsilon+h)\left(R+\chi^{2}\partial_{\mu}\phi\partial^{\mu}\phi\right)+O(\alpha'^{2})\right]. \tag*{(82)} Z ( B ) = g s − χ Z f ( 0 ) ∫ d D Y g e − χ ϕ [ 1 + 2 α ′ ( log ϵ + h ) ( R + χ 2 ∂ μ ϕ ∂ μ ϕ ) + O ( α ′ 2 ) ] . (82) Multiplying by the ghost contribution K 0 ( ϵ ) = Z ( B ) Z g h o s t K_0(\epsilon)=Z^{(B)}Z_{\mathrm{ghost}} K 0 ( ϵ ) = Z ( B ) Z ghost D = 26 D=26 D = 26 χ = 2 \chi=2 χ = 2 log ϵ \log\epsilon log ϵ Z f ( 0 ) Z_f^{(0)} Z f ( 0 )
I 0 = − ∂ K 0 ∂ log ϵ ∝ − α ′ ∫ d 26 Y g e − 2 ϕ ( R + 2 ∇ 2 ϕ ) . (83) I_0=-\frac{\partial K_0}{\partial\log\epsilon}\propto-\alpha'\int d^{26}Y\sqrt{g}e^{-2\phi}\left(R+2\nabla^2\phi\right). \tag*{(83)} I 0 = − ∂ log ϵ ∂ K 0 ∝ − α ′ ∫ d 26 Y g e − 2 ϕ ( R + 2 ∇ 2 ϕ ) . (83) Using covariant total derivatives,
∫ g e − 2 ϕ ∇ 2 ϕ = 2 ∫ g e − 2 ϕ ( ∇ ϕ ) 2 + ∫ ∂ M h e − 2 ϕ n μ ∇ μ ϕ . (84) \int\sqrt{g}e^{-2\phi}\nabla^2\phi=2\int\sqrt{g}e^{-2\phi}(\nabla\phi)^2+\int_{\partial\mathcal{M}}\sqrt{h}e^{-2\phi}n^\mu\nabla_\mu\phi. \tag*{(84)} ∫ g e − 2 ϕ ∇ 2 ϕ = 2 ∫ g e − 2 ϕ ( ∇ ϕ ) 2 + ∫ ∂ M h e − 2 ϕ n μ ∇ μ ϕ . (84) the bulk integrand becomes R + 4 ( ∇ ϕ ) 2 R+4(\nabla\phi)^2 R + 4 ( ∇ ϕ ) 2
Finally, we need to fix the normalization constant. The proportionality sign in (83) implies that the heat kernel calculation only determines the target space structure and its dependence on g s g_s g s α ′ \alpha' α ′ g μ ν g_{\mu\nu} g μν e − 2 ϕ ∞ / ( 16 π G N ) e^{-2\phi_\infty}/(16\pi G_N) e − 2 ϕ ∞ / ( 16 π G N )
I 0 = − 1 16 π G N ∫ d 26 Y g e − 2 ϕ ( R + 2 ∇ 2 ϕ + O ( α ′ ) ) . (85) I_0=-\frac{1}{16\pi G_N}\int d^{26}Y\sqrt{g}e^{-2\phi}\left(R+2\nabla^2\phi+O(\alpha')\right). \tag*{(85)} I 0 = − 16 π G N 1 ∫ d 26 Y g e − 2 ϕ ( R + 2 ∇ 2 ϕ + O ( α ′ ) ) . (85) Thus the overall gravitational coefficient is fixed by the normalized sphere graviton amplitude, independently of entropy matching. In our full-dilaton convention, a constant background ϕ 0 \phi_0 ϕ 0 G N , e f f = e 2 ϕ 0 G N G_{N,\mathrm{eff}}=e^{2\phi_0}G_N G N , eff = e 2 ϕ 0 G N
For a target space with boundary, I 0 I_0 I 0
The displayed heat-kernel derivation is bosonic. For the later type IIB application we use the universal metric–dilaton term of the type II tree-level NS–NS action, whose coefficient is likewise fixed by superstring sphere amplitudes. Smooth NS–NS and RR matter terms contain no explicit curvature coupling at two-derivative order, so they do not modify the local area density.
Dilaton-weighted black hole entropy from off-shell effects
We now apply this framework to the calculation of black hole entropy by varying the conical deficit at the tip of the Euclidean cigar. This requires an off-shell extension of the conventional worldsheet formulation [37 , 17 ] . In the usual on-shell formulation, the background fields are required to satisfy the conditions of Weyl invariance,
β μ ν g = 0 , β ϕ = 0 , (86) \beta^g_{\mu\nu}=0,\qquad\beta^\phi=0, \tag*{(86)} β μν g = 0 , β ϕ = 0 , (86) together with β μ ν B = 0 \beta^B_{\mu\nu}=0 β μν B = 0
( β μ ν g , β μ ν B , β ϕ ) ≠ 0 , (87) \left(\beta^g_{\mu\nu},\beta^B_{\mu\nu},\beta^\phi\right)\ne0, \tag*{(87)} ( β μν g , β μν B , β ϕ ) = 0 , (87) the worldsheet theory is no longer a CFT, but it remains a well-defined two-dimensional quantum field theory once a worldsheet ultraviolet cutoff ϵ \epsilon ϵ K 0 ( g , ϕ ; ϵ ) K_0(g,\phi;\epsilon) K 0 ( g , ϕ ; ϵ )
Z 0 = ∂ K 0 ∂ log ϵ , I 0 = − Z 0 (88) Z_0=\frac{\partial K_0}{\partial\log\epsilon},\qquad I_0=-Z_0 \tag*{(88)} Z 0 = ∂ log ϵ ∂ K 0 , I 0 = − Z 0 (88) Schematically, if ω i \omega^i ω i g μ ν g_{\mu\nu} g μν B μ ν B_{\mu\nu} B μν ϕ \phi ϕ
E i ≔ ∂ I 0 ∂ ω i = − κ i j β j , (89) E_i\coloneqq\frac{\partial I_0}{\partial\omega^i}=-\kappa_{ij}\beta^j, \tag*{(89)} E i : = ∂ ω i ∂ I 0 = − κ ij β j , (89) where κ i j \kappa_{ij} κ ij
β i = 0 ⟺ δ I 0 δ ω i = 0 , (90) \beta^i=0\qquad\Longleftrightarrow\qquad\frac{\delta I_0}{\delta\omega^i}=0, \tag*{(90)} β i = 0 ⟺ δ ω i δ I 0 = 0 , (90) The term “off shell” therefore does not mean that this prescription produces different physical equations of motion. Rather, it means that it defines the complete functional I 0 [ g , ϕ , … ] I_0[g,\phi,\ldots] I 0 [ g , ϕ , … ]
In the notation of Ahmadain and Wall, the sphere contribution to the black hole entropy is
S B H = ( 1 − β ∂ β ) Z 0 [ M β ] ∣ β = 2 π = ( 1 − β ∂ β ) ∂ K 0 [ M β ] ∂ log ϵ ∣ β = 2 π = ( β ∂ β − 1 ) I 0 [ M β ] ∣ β = 2 π . (91) \begin{aligned}
S_{\mathrm{BH}}&=(1-\beta\partial_\beta)Z_0[\mathcal{M}_\beta]\big|_{\beta=2\pi}\\
&=(1-\beta\partial_\beta)\left.\frac{\partial K_0[\mathcal{M}_\beta]}{\partial\log\epsilon}\right|_{\beta=2\pi}\\
&=(\beta\partial_\beta-1)I_0[\mathcal{M}_\beta]\big|_{\beta=2\pi}. \tag*{(91)}
\end{aligned} S BH = ( 1 − β ∂ β ) Z 0 [ M β ] β = 2 π = ( 1 − β ∂ β ) ∂ log ϵ ∂ K 0 [ M β ] β = 2 π = ( β ∂ β − 1 ) I 0 [ M β ] β = 2 π . (91) The parameter β \beta β I 0 [ M β ] I_0[\mathcal{M}_\beta] I 0 [ M β ] I 0 [ g , ϕ , … ] I_0[g,\phi,\ldots] I 0 [ g , ϕ , … ] M β \mathcal{M}_\beta M β I 0 [ M 2 π ] I_0[\mathcal{M}_{2\pi}] I 0 [ M 2 π ] I 0 [ M β ] I_0[\mathcal{M}_\beta] I 0 [ M β ]
Near the bifurcation surface Γ H \Gamma_H Γ H
d s 2 = d ρ 2 + ρ 2 d τ 2 + γ i j ( Y ) d Y i d Y j + O ( ρ 2 ) , τ ∼ τ + 2 π , (92) \mathrm{d}s^2=\mathrm{d}\rho^2+\rho^2\mathrm{d}\tau^2+\gamma_{ij}(Y)\mathrm{d}Y^i\mathrm{d}Y^j+O\left(\rho^2\right),\qquad\tau\sim\tau+2\pi, \tag*{(92)} d s 2 = d ρ 2 + ρ 2 d τ 2 + γ ij ( Y ) d Y i d Y j + O ( ρ 2 ) , τ ∼ τ + 2 π , (92) where Y i Y^i Y i Γ H \Gamma_H Γ H ρ = 0 \rho=0 ρ = 0 τ \tau τ M β \mathcal{M}_\beta M β
τ ∼ τ + β . (93) \tau\sim\tau+\beta. \tag*{(93)} τ ∼ τ + β . (93) At proper radius ρ \rho ρ β ρ \beta\rho β ρ 2 π ρ 2\pi\rho 2 π ρ β ≠ 2 π \beta\ne2\pi β = 2 π 2 π − β 2\pi-\beta 2 π − β Γ H \Gamma_H Γ H [39 ] . Since the action Eq. (85) is initially defined for smooth target space fields, we regulate the conical tip by replacing it with a smooth cap of radius r ∗ r_* r ∗
d s 2 = d ρ 2 + f β ( ρ ) 2 d τ 2 + γ i j ( Y ) d Y i d Y j + O ( ρ 2 ) , 0 ≤ τ < β , (94) \mathrm{d}s^2=\mathrm{d}\rho^2+f_\beta(\rho)^2\mathrm{d}\tau^2+\gamma_{ij}(Y)\mathrm{d}Y^i\mathrm{d}Y^j+O\left(\rho^2\right),\qquad0\le\tau<\beta, \tag*{(94)} d s 2 = d ρ 2 + f β ( ρ ) 2 d τ 2 + γ ij ( Y ) d Y i d Y j + O ( ρ 2 ) , 0 ≤ τ < β , (94) where
f β ( 0 ) = 0 , f β ′ ( 0 ) = 2 π β , f β ( ρ ) = ρ ( ρ ≥ r ∗ ) . (95) f_\beta(0)=0,\qquad f'_\beta(0)=\frac{2\pi}{\beta},\qquad f_\beta(\rho)=\rho\quad(\rho\ge r_*). \tag*{(95)} f β ( 0 ) = 0 , f β ′ ( 0 ) = β 2 π , f β ( ρ ) = ρ ( ρ ≥ r ∗ ) . (95) For the two-dimensional metric g a b g_{ab} g ab
d s ( 2 ) 2 = d ρ 2 + f β ( ρ ) 2 d τ 2 . (96) \mathrm{d}s^2_{(2)}=\mathrm{d}\rho^2+f_\beta(\rho)^2\mathrm{d}\tau^2. \tag*{(96)} d s ( 2 ) 2 = d ρ 2 + f β ( ρ ) 2 d τ 2 . (96) where a , b a,b a , b
∫ M β ( 2 ) d 2 X g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) , (97) \int_{\mathcal{M}^{(2)}_\beta}\mathrm{d}^2X\sqrt{g^{(2)}}R^{(2)}=2(2\pi-\beta), \tag*{(97)} ∫ M β ( 2 ) d 2 X g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) , (97) which implies
g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) δ ( 2 ) ( X ) (98) \sqrt{g^{(2)}}R^{(2)}=2(2\pi-\beta)\delta^{(2)}(X) \tag*{(98)} g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) δ ( 2 ) ( X ) (98) The distributional formula denotes the singular part in the sharp-cap limit. The conical response is computed to first order in β − 2 π \beta-2\pi β − 2 π r ∗ r_* r ∗ r ∗ → 0 r_*\to0 r ∗ → 0 M β \mathcal{M}_\beta M β r ∗ r_* r ∗ β \beta β H H H
β a b g = α ′ R a b + O ( α ′ 2 ) . (99) \beta^g_{ab}=\alpha'R_{ab}+O\left(\alpha'^2\right). \tag*{(99)} β ab g = α ′ R ab + O ( α ′2 ) . (99) In two dimensions R a b ( 2 ) = 1 2 g a b ( 2 ) R ( 2 ) R^{(2)}_{ab}=\frac{1}{2}g^{(2)}_{ab}R^{(2)} R ab ( 2 ) = 2 1 g ab ( 2 ) R ( 2 ) (98) and then taking r ∗ → 0 r_*\to0 r ∗ → 0
β a b g ∣ t i p = α ′ ( 2 π − β ) g a b ( 2 ) δ ( 2 ) ( X ) g ( 2 ) + O ( ( β − 2 π ) 2 , α ′ 2 ) . (100) \left.\beta^g_{ab}\right|_{\mathrm{tip}}=\alpha'(2\pi-\beta)g^{(2)}_{ab}\frac{\delta^{(2)}(X)}{\sqrt{g^{(2)}}}+O\left((\beta-2\pi)^2,\alpha'^2\right). \tag*{(100)} β ab g ∣ tip = α ′ ( 2 π − β ) g ab ( 2 ) g ( 2 ) δ ( 2 ) ( X ) + O ( ( β − 2 π ) 2 , α ′2 ) . (100) This delta function is a distribution in the target space, and the worldsheet itself remains a single smooth sphere. Smooth background fluxes and a smooth dilaton contribute regular terms to the beta functions and do not cancel this singular curvature contribution. Based on the gradient relation introduced above,
E i = − κ i j β j (101) E_i=-\kappa_{ij}\beta^j \tag*{(101)} E i = − κ ij β j (101) this localized beta function is equivalent to a delta-function contribution to the target-space equation-of-motion:
E i ∣ t i p ≠ 0 , β ≠ 2 π . (102) \left.E_i\right|_{\mathrm{tip}}\ne0,\qquad\beta\ne2\pi. \tag*{(102)} E i ∣ tip = 0 , β = 2 π . (102) In the sphere-amplitude description, the same nonzero effect appears as a localized graviton one-point function, or graviton tadpole
A 0 , 1 g = ∂ ∂ log ϵ K 0 , 1 g ∼ − ∫ M β d D X g δ g a b ( X ) E ( g ) a b ( X ) , (103) A^g_{0,1}=\frac{\partial}{\partial\log\epsilon}K^g_{0,1}\sim-\int_{\mathcal{M}_\beta}d^D X\sqrt{g}\,\delta g^{ab}(X)E_{(g)ab}(X), \tag*{(103)} A 0 , 1 g = ∂ log ϵ ∂ K 0 , 1 g ∼ − ∫ M β d D X g δ g ab ( X ) E ( g ) ab ( X ) , (103) We now return to the action (85) and evaluate the corresponding black hole entropy. Near the horizon of this metric, the measure factorizes as g = g ( 2 ) γ + O ( ρ ) 2 \sqrt{g}=\sqrt{g^{(2)}}\sqrt{\gamma}+O(\rho)^2 g = g ( 2 ) γ + O ( ρ ) 2
ϕ ( ρ , τ , Y ) = ϕ H ( Y ) + O ( ρ ) 2 , (104) \phi(\rho,\tau,Y)=\phi_H(Y)+O(\rho)^2, \tag*{(104)} ϕ ( ρ , τ , Y ) = ϕ H ( Y ) + O ( ρ ) 2 , (104) where ϕ H ( Y ) ≡ ϕ ∣ Γ H \phi_H(Y)\equiv\left.\phi\right|_{\Gamma_H} ϕ H ( Y ) ≡ ϕ ∣ Γ H (85) localized in the cap gives
∫ M β d D X g e − 2 ϕ R ∣ c a p = ( ∫ M β ( 2 ) d 2 X g ( 2 ) R ( 2 ) ) ( ∫ Γ H d D − 2 Y γ e − 2 ϕ H ) = 2 ( 2 π − β ) A ϕ ( Γ H ) + O ( r ∗ 2 ) + O ( ( β − 2 π ) 2 ) , (105) \begin{aligned}
\left.\int_{\mathcal{M}_\beta}d^D X\sqrt{g}e^{-2\phi}R\right|_{\mathrm{cap}}=\left(\int_{\mathcal{M}^{(2)}_\beta}d^2X\sqrt{g^{(2)}}R^{(2)}\right)\left(\int_{\Gamma_H}d^{D-2}Y\sqrt{\gamma}e^{-2\phi_H}\right) \\
&=2(2\pi-\beta)A_\phi(\Gamma_H)+O(r_*^2)+O\left((\beta-2\pi)^2\right), \tag*{(105)}
\end{aligned} ∫ M β d D X g e − 2 ϕ R cap = ( ∫ M β ( 2 ) d 2 X g ( 2 ) R ( 2 ) ) ( ∫ Γ H d D − 2 Y γ e − 2 ϕ H ) = 2 ( 2 π − β ) A ϕ ( Γ H ) + O ( r ∗ 2 ) + O ( ( β − 2 π ) 2 ) , (105) where
A ϕ ( Γ H ) : = ∫ Γ H d D − 2 Y γ e − 2 ϕ H , g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) δ ( 2 ) ( X ) . (106) A_\phi(\Gamma_H):=\int_{\Gamma_H}d^{D-2}Y\sqrt{\gamma}e^{-2\phi_H},\qquad\sqrt{g^{(2)}}R^{(2)}=2(2\pi-\beta)\delta^{(2)}(X). \tag*{(106)} A ϕ ( Γ H ) := ∫ Γ H d D − 2 Y γ e − 2 ϕ H , g ( 2 ) R ( 2 ) = 2 ( 2 π − β ) δ ( 2 ) ( X ) . (106) Moreover, since ∇ 2 ϕ \nabla^2\phi ∇ 2 ϕ ϕ H ( Y ) \phi_H(Y) ϕ H ( Y ) (85) becomes
∫ M β d D X g e − 2 ϕ ∇ 2 ϕ ∣ c a p = O ( r ∗ 2 ) , as r ∗ → 0. (107) \left.\int_{\mathcal{M}_\beta}d^D X\sqrt{g}e^{-2\phi}\nabla^2\phi\right|_{\mathrm{cap}}=O(r_*^2),\qquad\text{as }r_*\to0. \tag*{(107)} ∫ M β d D X g e − 2 ϕ ∇ 2 ϕ cap = O ( r ∗ 2 ) , as r ∗ → 0. (107) Thus the dilaton enters the leading tip term only through the weight e − 2 ϕ H e^{-2\phi_H} e − 2 ϕ H A ϕ A_\phi A ϕ
Away from the cap, changing the range of the τ \tau τ 2 π 2\pi 2 π β \beta β β / ( 2 π ) \beta/(2\pi) β / ( 2 π ) (105) into the action (85) gives
I 0 [ M β ] = β 2 π I 0 [ M 2 π ] + I t i p ( β ) + O ( ( β − 2 π ) 2 , r ∗ 2 , α ′ ) , (108) I_0[\mathcal{M}_\beta]=\frac{\beta}{2\pi}I_0[\mathcal{M}_{2\pi}]+I_{\mathrm{tip}}(\beta)+O\left((\beta-2\pi)^2,r_*^2,\alpha'\right), \tag*{(108)} I 0 [ M β ] = 2 π β I 0 [ M 2 π ] + I tip ( β ) + O ( ( β − 2 π ) 2 , r ∗ 2 , α ′ ) , (108) where
I t i p ( β ) = − 1 16 π G N [ 2 ( 2 π − β ) A ϕ ( Γ H ) ] = β − 2 π 8 π G N A ϕ ( Γ H ) . (109) I_{\mathrm{tip}}(\beta)=-\frac{1}{16\pi G_N}\left[2(2\pi-\beta)A_\phi(\Gamma_H)\right]=\frac{\beta-2\pi}{8\pi G_N}A_\phi(\Gamma_H). \tag*{(109)} I tip ( β ) = − 16 π G N 1 [ 2 ( 2 π − β ) A ϕ ( Γ H ) ] = 8 π G N β − 2 π A ϕ ( Γ H ) . (109) Equivalently,
I 0 [ M β ] − β 2 π I 0 [ M 2 π ] = β − 2 π 8 π G N A ϕ ( Γ H ) + O ( ( β − 2 π ) 2 , r ∗ 2 , α ′ ) . (110) I_0[\mathcal{M}_{\beta}] - \frac{\beta}{2\pi} I_0[\mathcal{M}_{2\pi}] = \frac{\beta-2\pi}{8\pi G_N} A_{\phi}(\Gamma_H) + O\!\left((\beta-2\pi)^2,r_*^2,\alpha'\right). \tag*{(110)} I 0 [ M β ] − 2 π β I 0 [ M 2 π ] = 8 π G N β − 2 π A ϕ ( Γ H ) + O ( ( β − 2 π ) 2 , r ∗ 2 , α ′ ) . (110) Since − I 0 = Z 0 -I_0=Z_0 − I 0 = Z 0
S B H = ( 1 − β ∂ β ) Z 0 [ M β ] ∣ β = 2 π = ( β ∂ β − 1 ) I 0 [ M β ] ∣ β = 2 π = A ϕ ( Γ H ) 4 G N . (111) \begin{aligned}
S_{\mathrm{BH}} &= (1-\beta\partial_{\beta})Z_0[\mathcal{M}_{\beta}]\big|_{\beta=2\pi}\\
&= (\beta\partial_{\beta}-1)I_0[\mathcal{M}_{\beta}]\big|_{\beta=2\pi}\\
&= \frac{A_{\phi}(\Gamma_H)}{4G_N}. \tag*{(111)}
\end{aligned} S BH = ( 1 − β ∂ β ) Z 0 [ M β ] β = 2 π = ( β ∂ β − 1 ) I 0 [ M β ] β = 2 π = 4 G N A ϕ ( Γ H ) . (111) For the constant-dilaton Rindler background, A ϕ = A ⊥ A_{\phi}=A_{\perp} A ϕ = A ⊥ S B H = A ⊥ / 4 G N S_{\mathrm{BH}}=A_{\perp}/4G_N S BH = A ⊥ /4 G N [40 , 41 ] .
The stringy origin of the norm bound
The calculation in the previous section already completes the genus-zero entropy derivation for a prescribed stationary Killing horizon. Its result can be written as
S B H = A ϕ ( Γ H ) 4 G N , A ϕ ( Γ H ) : = ∫ Γ H e − 2 ϕ H d A S , (112) S_{\mathrm{BH}}=\frac{A_{\phi}(\Gamma_H)}{4G_N},\qquad A_{\phi}(\Gamma_H):=\int_{\Gamma_H}e^{-2\phi_H}dA_S, \tag*{(112)} S BH = 4 G N A ϕ ( Γ H ) , A ϕ ( Γ H ) := ∫ Γ H e − 2 ϕ H d A S , (112) where d A S = γ d D − 2 Y dA_S=\sqrt{\gamma}\,d^{D-2}Y d A S = γ d D − 2 Y Γ H \Gamma_H Γ H
Because the genus-zero two-derivative target-space action is a local Einstein–dilaton functional, the change in the action caused by a small conical deficit is determined entirely by the fields near the codimension-two defect. The same regulated-cone calculation can therefore be applied to any smooth codimension-two candidate surface Γ \Gamma Γ Γ \Gamma Γ
S l o c ( 0 ) [ Γ ] = 1 4 G N ∫ Γ e − 2 ϕ d A S = ∫ Γ ρ S ( x ) d A S , ρ S ( x ) : = e − 2 ϕ ( x ) 4 G N . (113) S_{\mathrm{loc}}^{(0)}[\Gamma]=\frac{1}{4G_N}\int_{\Gamma}e^{-2\phi}dA_S=\int_{\Gamma}\rho_S(x)dA_S,\qquad\rho_S(x):=\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(113)} S loc ( 0 ) [ Γ ] = 4 G N 1 ∫ Γ e − 2 ϕ d A S = ∫ Γ ρ S ( x ) d A S , ρ S ( x ) := 4 G N e − 2 ϕ ( x ) . (113) Thus ρ S ( x ) \rho_S(x) ρ S ( x )
ρ S ( x ) = δ S l o c ( 0 ) δ A S ( x ) . (114) \rho_S(x)=\frac{\delta S_{\mathrm{loc}}^{(0)}}{\delta A_S(x)}. \tag*{(114)} ρ S ( x ) = δ A S ( x ) δ S loc ( 0 ) . (114) Evaluating this functional on the prescribed bifurcation surface, Γ = Γ H \Gamma=\Gamma_H Γ = Γ H
S B H = S l o c ( 0 ) [ Γ H ] = ∫ Γ H ρ S ( x ) d A S . (115) S_{\mathrm{BH}}=S_{\mathrm{loc}}^{(0)}[\Gamma_H]=\int_{\Gamma_H}\rho_S(x)dA_S. \tag*{(115)} S BH = S loc ( 0 ) [ Γ H ] = ∫ Γ H ρ S ( x ) d A S . (115) This equality gives an evaluation of S l o c ( 0 ) S_{\mathrm{loc}}^{(0)} S loc ( 0 )
But for the generic boundary region A A A S l o c ( 0 ) [ Γ ] S_{\mathrm{loc}}^{(0)}[\Gamma] S loc ( 0 ) [ Γ ] A A A Γ \Gamma Γ ∂ Γ = ∂ A \partial\Gamma= \partial A ∂ Γ = ∂ A Γ ∼ A \Gamma\sim A Γ ∼ A S l o c ( 0 ) [ Γ ] S_{\mathrm{loc}}^{(0)}[\Gamma] S loc ( 0 ) [ Γ ]
S l o c ( 0 ) [ Γ ] ⇏ S A ( 0 ) = min Γ ∼ A S l o c ( 0 ) [ Γ ] , (116) S_{\mathrm{loc}}^{(0)}[\Gamma]\ \not\Rightarrow\ S_A^{(0)} = \min_{\Gamma\sim A} S_{\mathrm{loc}}^{(0)}[\Gamma], \tag*{(116)} S loc ( 0 ) [ Γ ] ⇒ S A ( 0 ) = Γ ∼ A min S loc ( 0 ) [ Γ ] , (116) without an additional surface-selection argument. In other words, it cannot, by itself, provide a minimization principle over arbitrary codimension-two surfaces. Therefore, the purpose of the following subsection is to derive this selection rule from the gravitational replica construction. In other words, the Ahmadain–Wall off-shell calculation determines the entropy density assigned to each candidate surface, while the replica construction selects the corresponding weighted minimal cut. Then, in the following subsections, we will show that the convex duality converts this weighted minimal cut into a divergenceless flow obeying a norm bound which provides the string derived norm bound. Completing this chain relates the sphere derived entropy functional to the bit-thread formulation and provides the norm bound required for the subsequent identification of the bit threads with the appropriately string charge density.
From the replica saddle to the weighted minimal cut
We now combine the entropy density
ρ S ( x ) = e − 2 ϕ ( x ) 4 G N , (117) \rho_S(x) = \frac{e^{-2\phi(x)}}{4G_N}, \tag*{(117)} ρ S ( x ) = 4 G N e − 2 ϕ ( x ) , (117) obtained from the Ahmadain–Wall sphere calculation with the standard Lewkowycz–Maldacena gravitational replica argument [42 ] . This combination requires an additional holographic step. The fixed-cone calculation above does not, by itself, introduce a boundary density matrix or its replica partition functions. To pass from the entropy of a prescribed Killing horizon to that of a generic boundary region, we assume that the target-space string background admits a semiclassical holographic description. We also assume that replica symmetry is unbroken in the bulk and that the dominant replica saddles admit an analytic continuation to a neighborhood of n = 1 n = 1 n = 1 n n n K 0 K_0 K 0
The connection between the worldsheet calculation and the gravitational replica construction comes from the same tree-level target space action I 0 I_0 I 0
Z 0 [ Ψ ] = ∂ K 0 [ Ψ ; ϵ ] ∂ log ϵ = − I 0 [ Ψ ] , (118) Z_0[\Psi] = \frac{\partial K_0[\Psi;\epsilon]}{\partial\log\epsilon} = -I_0[\Psi], \tag*{(118)} Z 0 [ Ψ ] = ∂ log ϵ ∂ K 0 [ Ψ ; ϵ ] = − I 0 [ Ψ ] , (118) where Ψ = { g μ ν , B μ ν , ϕ , … } \Psi= \{g_{\mu\nu}, B_{\mu\nu}, \phi, \ldots\} Ψ = { g μν , B μν , ϕ , … } Z 0 [ Ψ ] Z_0[\Psi] Z 0 [ Ψ ] Ψ \Psi Ψ B \mathcal{B} B Ψ c l ( B ) \Psi_{\mathrm{cl}}(\mathcal{B}) Ψ cl ( B ) l l l
Z b u l k ( 0 ) [ B ] : = ∑ l = 0 ∞ 1 l ! ( Z 0 [ Ψ c l ( B ) ] ) l = exp ( Z 0 [ Ψ c l ( B ) ] ) = exp ( − I 0 [ Ψ c l ( B ) ] ) . (119) \mathcal{Z}_{\mathrm{bulk}}^{(0)}[\mathcal{B}] := \sum_{l=0}^{\infty} \frac{1}{l!}\left(Z_0[\Psi_{\mathrm{cl}}(\mathcal{B})]\right)^l = \exp\left(Z_0[\Psi_{\mathrm{cl}}(\mathcal{B})]\right) = \exp\left(-I_0[\Psi_{\mathrm{cl}}(\mathcal{B})]\right). \tag*{(119)} Z bulk ( 0 ) [ B ] := l = 0 ∑ ∞ l ! 1 ( Z 0 [ Ψ cl ( B )] ) l = exp ( Z 0 [ Ψ cl ( B )] ) = exp ( − I 0 [ Ψ cl ( B )] ) . (119) where Z b u l k ( 0 ) [ B ] \mathcal{Z}_{\mathrm{bulk}}^{(0)}[\mathcal{B}] Z bulk ( 0 ) [ B ] B \mathcal{B} B B n \mathcal{B}_n B n n n n A A A
Z ( n ) : = Tr ρ A n , (120) Z(n) := \operatorname{Tr}\rho_A^n, \tag*{(120)} Z ( n ) := Tr ρ A n , (120) where ρ A \rho_A ρ A B n \mathcal{B}_n B n M n \mathcal{M}_n M n
Z ( n ) = Z b o u n d a r y [ B n ] ≃ Z b u l k ( 0 ) [ B n ] = exp ( Z 0 [ M n ] ) = exp ( − I 0 [ M n ] ) . (121) Z(n) = Z_{\mathrm{boundary}}[\mathcal{B}_n] \simeq\mathcal{Z}_{\mathrm{bulk}}^{(0)}[\mathcal{B}_n] = \exp\left(Z_0[\mathcal{M}_n]\right) = \exp\left(-I_0[\mathcal{M}_n]\right). \tag*{(121)} Z ( n ) = Z boundary [ B n ] ≃ Z bulk ( 0 ) [ B n ] = exp ( Z 0 [ M n ] ) = exp ( − I 0 [ M n ] ) . (121) Equivalently,
log Z ( n ) = Z 0 [ M n ] = − I 0 [ M n ] . (122) \log Z(n) = Z_0[\mathcal{M}_n] = -I_0[\mathcal{M}_n]. \tag*{(122)} log Z ( n ) = Z 0 [ M n ] = − I 0 [ M n ] . (122) In particular,
Z ( 1 ) ≃ exp ( Z 0 [ M 1 ] ) = exp ( − I 0 [ M 1 ] ) . (123) Z(1) \simeq\exp\left(Z_0[\mathcal{M}_1]\right) = \exp\left(-I_0[\mathcal{M}_1]\right). \tag*{(123)} Z ( 1 ) ≃ exp ( Z 0 [ M 1 ] ) = exp ( − I 0 [ M 1 ] ) . (123) Note that Z 0 ≠ Z ( 1 ) Z_0 \ne Z(1) Z 0 = Z ( 1 ) Z 0 Z_0 Z 0 Z ( 1 ) Z(1) Z ( 1 ) n = 1 n = 1 n = 1
ρ ^ A : = ρ A Tr ρ A = ρ A Z ( 1 ) , (124) \widehat{\rho}_A := \frac{\rho_A}{\operatorname{Tr}\rho_A} = \frac{\rho_A}{Z(1)}, \tag*{(124)} ρ A := Tr ρ A ρ A = Z ( 1 ) ρ A , (124) and hence
Tr ρ ^ A n = Z ( n ) Z ( 1 ) n . (125) \operatorname{Tr}\widehat{\rho}_A^n = \frac{Z(n)}{Z(1)^n}. \tag*{(125)} Tr ρ A n = Z ( 1 ) n Z ( n ) . (125) The von Neumann entropy is therefore
S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] ∣ n = 1 = n ∂ n ( I 0 [ M n ] − n I 0 [ M 1 ] ) ∣ n = 1 = ( n ∂ n − 1 ) I 0 [ M n ] ∣ n = 1 . (126) \begin{aligned}
S_A^{(0)} = -n\partial_n\left[\log Z(n) - n\log Z(1)\right]\big|_{n=1} \\
&= n\partial_n\left(I_0[\mathcal{M}_n] - nI_0[\mathcal{M}_1]\right)\big|_{n=1} \\
&= (n\partial_n - 1)I_0[\mathcal{M}_n]\big|_{n=1}. \tag*{(126)}
\end{aligned} S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] n = 1 = n ∂ n ( I 0 [ M n ] − n I 0 [ M 1 ] ) n = 1 = ( n ∂ n − 1 ) I 0 [ M n ] n = 1 . (126) Moreover, it is worth noting that M n \mathcal{M}_n M n M β \mathcal{M}_\beta M β
For integer n > 1 n>1 n > 1 M n \mathcal{M}_n M n Z n \mathbb{Z}_n Z n [42 ] , we assume that the family of replica saddles admits an analytic continuation toward n = 1 n=1 n = 1 Γ ⊂ M 1 \Gamma\subset\mathcal{M}_1 Γ ⊂ M 1 n = 1 n=1 n = 1 Γ \Gamma Γ n n n M n \mathcal{M}_n M n n = 1 n=1 n = 1 M ~ n ( r ∗ ) \widetilde{\mathcal{M}}_n(r_*) M n ( r ∗ ) r ∗ r_* r ∗ Γ \Gamma Γ n n n n = 1 n=1 n = 1 [43 , 44 ] . Therefore, M ~ n \widetilde{\mathcal{M}}_n M n β = 2 π n \beta=2\pi n β = 2 π n M n \mathcal{M}_n M n M ~ n \widetilde{\mathcal{M}}_n M n n = 1 n=1 n = 1 M 1 \mathcal{M}_1 M 1 n = 1 n=1 n = 1 I 0 I_0 I 0 ρ S ( x ) \rho_S(x) ρ S ( x )
To calculate the entropy S A ( 0 ) S_A^{(0)} S A ( 0 )
log Z o f f ( n ) : = − I 0 [ M ~ n ( r ∗ ) ] . (127) \log Z^{\mathrm{off}}(n):=-I_0\left[\widetilde{\mathcal{M}}_n(r_*)\right]. \tag*{(127)} log Z off ( n ) := − I 0 [ M n ( r ∗ ) ] . (127) The auxiliary configuration has the same asymptotic sources as M n \mathcal{M}_n M n
δ Ψ n : = Ψ ~ n − Ψ n = O ( n − 1 ) , (128) \delta\Psi_n:=\widetilde{\Psi}_n-\Psi_n=O(n-1), \tag*{(128)} δ Ψ n := Ψ n − Ψ n = O ( n − 1 ) , (128) where Ψ \Psi Ψ n → 1 n\to1 n → 1 r ∗ → 0 r_*\to0 r ∗ → 0
Based on S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] ∣ n = 1 S_A^{(0)}=-n\partial_n[\log Z(n)-n\log Z(1)]\big|_{n=1} S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 )] n = 1
log Z ( n ) − n log Z ( 1 ) = ( log Z ( n ) − log Z o f f ( n ) ) + ( log Z o f f ( n ) − n log Z ( 1 ) ) . (129) \log Z(n)-n\log Z(1)=\left(\log Z(n)-\log Z^{\mathrm{off}}(n)\right)+\left(\log Z^{\mathrm{off}}(n)-n\log Z(1)\right). \tag*{(129)} log Z ( n ) − n log Z ( 1 ) = ( log Z ( n ) − log Z off ( n ) ) + ( log Z off ( n ) − n log Z ( 1 ) ) . (129) Expanding the auxiliary configuration around the true on-shell saddle gives
I 0 [ Ψ ~ n ] − I 0 [ Ψ n ] = ∫ M n E Ψ ( Ψ ~ n ) δ Ψ n + ∫ ∂ M n Θ ( Ψ n , δ Ψ n ) + O ( δ Ψ n 2 ) . (130) I_0[\widetilde{\Psi}_n]-I_0[\Psi_n]=\int_{\mathcal{M}_n}E_\Psi(\widetilde{\Psi}_n)\,\delta\Psi_n+\int_{\partial\mathcal{M}_n}\Theta(\Psi_n,\delta\Psi_n)+O(\delta\Psi_n^2). \tag*{(130)} I 0 [ Ψ n ] − I 0 [ Ψ n ] = ∫ M n E Ψ ( Ψ n ) δ Ψ n + ∫ ∂ M n Θ ( Ψ n , δ Ψ n ) + O ( δ Ψ n 2 ) . (130) The bulk term vanishes because M n \mathcal{M}_n M n E Ψ ( Ψ n ) = 0 E_\Psi(\Psi_n)=0 E Ψ ( Ψ n ) = 0 (130) . It follows that
log Z ( n ) − log Z o f f ( n ) = I 0 [ M ~ n ] − I 0 [ M n ] = O ( ( n − 1 ) 2 ) . (131) \log Z(n) - \log Z^{\mathrm{off}}(n) = I_0\left[\widetilde{\mathcal{M}}_n\right] - I_0\left[\mathcal{M}_n\right] = O\left((n-1)^2\right). \tag*{(131)} log Z ( n ) − log Z off ( n ) = I 0 [ M n ] − I 0 [ M n ] = O ( ( n − 1 ) 2 ) . (131) For the second bracket in Eq. (129) , setting β = 2 π n \beta= 2\pi n β = 2 π n
log Z o f f ( n ) − n log Z ( 1 ) = − 2 π ( n − 1 ) 8 π G N ∫ Γ e − 2 ϕ d A S + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) = − ( n − 1 ) S l o c ( 0 ) [ Γ ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) , (132) \begin{aligned}
\log Z^{\mathrm{off}}(n) - n\log Z(1) &= -\frac{2\pi(n-1)}{8\pi G_N}\int_{\Gamma}e^{-2\phi}dA_S + O\left((n-1)^2\right) + (n-1)O(\alpha') \\
&= -(n-1)S_{\mathrm{loc}}^{(0)}[\Gamma] + O\left((n-1)^2\right) + (n-1)O(\alpha'), \tag*{(132)}
\end{aligned} log Z off ( n ) − n log Z ( 1 ) = − 8 π G N 2 π ( n − 1 ) ∫ Γ e − 2 ϕ d A S + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) = − ( n − 1 ) S loc ( 0 ) [ Γ ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) , (132) where we use the important result
I 0 [ M ~ n ] − n I 0 [ M 1 ] = I t i p ( n ) + O ( ( n − 1 ) 2 ) . (133) I_0[\widetilde{\mathcal{M}}_n] - nI_0[\mathcal{M}_1] = I_{\mathrm{tip}}(n) + O\left((n-1)^2\right). \tag*{(133)} I 0 [ M n ] − n I 0 [ M 1 ] = I tip ( n ) + O ( ( n − 1 ) 2 ) . (133) Combining Eqs. (129) , (131) , and (132) gives
log Z ( n ) − n log Z ( 1 ) = − ( n − 1 ) S l o c ( 0 ) [ Γ ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (134) \log Z(n) - n\log Z(1) = -(n-1)S_{\mathrm{loc}}^{(0)}[\Gamma] + O\left((n-1)^2\right) + (n-1)O(\alpha'). \tag*{(134)} log Z ( n ) − n log Z ( 1 ) = − ( n − 1 ) S loc ( 0 ) [ Γ ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (134) Using S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] ∣ n = 1 S_A^{(0)}=-n\partial_n[\log Z(n)-n\log Z(1)]|_{n=1} S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 )] ∣ n = 1
S A ( 0 ) = S l o c ( 0 ) [ Γ ] = 1 4 G N ∫ Γ e − 2 ϕ d A S , (135) S_A^{(0)} = S_{\mathrm{loc}}^{(0)}[\Gamma] = \frac{1}{4G_N}\int_{\Gamma}e^{-2\phi}dA_S, \tag*{(135)} S A ( 0 ) = S loc ( 0 ) [ Γ ] = 4 G N 1 ∫ Γ e − 2 ϕ d A S , (135) up to higher-derivative corrections. Finally, it remains to determine the location of Γ \Gamma Γ z , z ˉ z,\bar z z , z ˉ n = 1 + ϵ n=1+\epsilon n = 1 + ϵ [42 ] :
δ R z z E = − ϵ K z E z − 1 2 ∂ z 2 δ γ + ⋯ , δ R z ˉ z ˉ E = − ϵ K z ˉ E z ˉ − 1 2 ∂ z ˉ 2 δ γ + ⋯ , (136) \begin{aligned}
\delta R^{E}_{zz} &= -\epsilon\frac{K^{E}_{z}}{z} - \frac{1}{2}\partial_z^2\delta\gamma+\cdots,\\
\delta R^{E}_{\bar z\bar z} &= -\epsilon\frac{K^{E}_{\bar z}}{\bar z} - \frac{1}{2}\partial_{\bar z}^2\delta\gamma+\cdots, \tag*{(136)}
\end{aligned} δ R z z E δ R z ˉ z ˉ E = − ϵ z K z E − 2 1 ∂ z 2 δ γ + ⋯ , = − ϵ z ˉ K z ˉ E − 2 1 ∂ z ˉ 2 δ γ + ⋯ , (136) where δ γ = γ E i j δ g i j E \delta\gamma=\gamma^{ij}_{E}\delta g^{E}_{ij} δ γ = γ E ij δ g ij E K z E = 1 2 ( K 1 E − i K 2 E ) K^{E}_{z}=\frac{1}{2}(K^{E}_{1}-iK^{E}_{2}) K z E = 2 1 ( K 1 E − i K 2 E )
R z z E ⊃ − ϵ K z E z , R z ˉ z ˉ E ⊃ − ϵ K z ˉ E z ˉ . (137) R^{E}_{zz}\supset-\epsilon\frac{K^{E}_{z}}{z},\qquad R^{E}_{\bar z\bar z}\supset-\epsilon\frac{K^{E}_{\bar z}}{\bar z}. \tag*{(137)} R z z E ⊃ − ϵ z K z E , R z ˉ z ˉ E ⊃ − ϵ z ˉ K z ˉ E . (137) display the geometrically important poles, but they are not by themselves the complete derivation. The perturbation δ γ \delta\gamma δ γ 2 π 2\pi 2 π z z zz z z z ˉ z ˉ \bar z\bar z z ˉ z ˉ
K z E = K z ˉ E = 0 , (138) K^{E}_{z}=K^{E}_{\bar z}=0, \tag*{(138)} K z E = K z ˉ E = 0 , (138) Thus replica regularity selects an extremal surface in the Einstein-frame geometry.
The same condition can be expressed directly in string-frame variables by varying Eq. (113) . For a normal displacement δ X μ = δ X a n a μ \delta X^\mu= \delta X^a n_a^\mu δ X μ = δ X a n a μ δ ( d A S ) = K a δ X a d A S \delta(dA_S) = K_a\delta X^a dA_S δ ( d A S ) = K a δ X a d A S
δ Γ ∫ Γ e − 2 ϕ d A S = ∫ Γ e − 2 ϕ ( K a − 2 n a μ ∇ μ ϕ ) δ X a d A S . (139) \delta_\Gamma\int_\Gamma e^{-2\phi}dA_S = \int_\Gamma e^{-2\phi}\left(K_a - 2n_a^\mu\nabla_\mu\phi\right)\delta X^a dA_S. \tag*{(139)} δ Γ ∫ Γ e − 2 ϕ d A S = ∫ Γ e − 2 ϕ ( K a − 2 n a μ ∇ μ ϕ ) δ X a d A S . (139) Hence the string-frame replica-extremality equation is
K a − 2 n a μ ∇ μ ϕ = 0. (140) K_a - 2n_a^\mu\nabla_\mu\phi= 0. \tag*{(140)} K a − 2 n a μ ∇ μ ϕ = 0. (140) The boundary replica construction also supplies the anchoring and homology conditions
∂ Γ = ∂ A , Γ ∼ A . (141) \partial\Gamma= \partial A,\qquad\Gamma\sim A. \tag*{(141)} ∂ Γ = ∂ A , Γ ∼ A . (141) Now suppose that the bulk geometry is static and invariant under reflection about a time-symmetric Riemannian slice. The surface-selection problem can then be formulated entirely on this slice. The replica boundary-value problem may admit several smooth replica-symmetric saddle branches
M n ( 1 ) , M n ( 2 ) , … , (142) \mathcal{M}_n^{(1)},\qquad\mathcal{M}_n^{(2)},\qquad\ldots, \tag*{(142)} M n ( 1 ) , M n ( 2 ) , … , (142) with the same asymptotic replica boundary data B n \mathcal{B}_n B n i i i M n ( i ) \mathcal{M}_n^{(i)} M n ( i ) M 1 \mathcal{M}_1 M 1 n → 1 n \to1 n → 1
Γ n ( i ) ⟶ m i ( n → 1 ) . (143) \Gamma_n^{(i)} \longrightarrow m_i \qquad(n \to1). \tag*{(143)} Γ n ( i ) ⟶ m i ( n → 1 ) . (143) Each m i m_i m i
∂ m i = ∂ A , m i ∼ A , K a − 2 n a μ ∇ μ ϕ = 0 , (144) \partial m_i = \partial A,\qquad m_i \sim A,\qquad K_a - 2n_a^\mu\nabla_\mu\phi= 0, \tag*{(144)} ∂ m i = ∂ A , m i ∼ A , K a − 2 n a μ ∇ μ ϕ = 0 , (144) where A A A
I 0 [ M n ( i ) ] = n I 0 [ M 1 ] + ( n − 1 ) S l o c ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (145) I_0\left[\mathcal{M}_n^{(i)}\right] = nI_0[\mathcal{M}_1] + (n-1)S_{\mathrm{loc}}^{(0)}[m_i] + O\left((n-1)^2\right) + (n-1)O(\alpha'). \tag*{(145)} I 0 [ M n ( i ) ] = n I 0 [ M 1 ] + ( n − 1 ) S loc ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (145) where
S l o c ( 0 ) [ m i ] = 1 4 G N ∫ m i e − 2 ϕ d A S (146) S_{\mathrm{loc}}^{(0)}[m_i] = \frac{1}{4G_N}\int_{m_i}e^{-2\phi}dA_S \tag*{(146)} S loc ( 0 ) [ m i ] = 4 G N 1 ∫ m i e − 2 ϕ d A S (146) In the semiclassical approximation, the replica partition function is a sum over these saddle contributions:
Z ( n ) ≃ ∑ i exp ( − I 0 [ M n ( i ) ] ) = e − n I 0 [ M 1 ] ∑ i exp [ − ( n − 1 ) S l o c ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) ] . (147) \begin{aligned}
Z(n) &\simeq\sum_i \exp\left(-I_0\left[\mathcal{M}_n^{(i)}\right]\right) \\
&= e^{-nI_0[\mathcal{M}_1]}\sum_i \exp\left[-(n-1)S_{\mathrm{loc}}^{(0)}[m_i] + O\left((n-1)^2\right) + (n-1)O(\alpha')\right]. \tag*{(147)}
\end{aligned} Z ( n ) ≃ i ∑ exp ( − I 0 [ M n ( i ) ] ) = e − n I 0 [ M 1 ] i ∑ exp [ − ( n − 1 ) S loc ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) ] . (147) For n > 1 n > 1 n > 1 n − 1 n - 1 n − 1 S l o c ( 0 ) [ m i ] = O ( G N − 1 ) S_{\mathrm{loc}}^{(0)}[m_i] = O\left(G_N^{-1}\right) S loc ( 0 ) [ m i ] = O ( G N − 1 ) S l o c ( 0 ) [ m i ] S_{\mathrm{loc}}^{(0)}[m_i] S loc ( 0 ) [ m i ] i i i j j j
exp [ − I 0 [ M n ( j ) ] ] exp [ − I 0 [ M n ( i ) ] ] = exp [ − ( n − 1 ) ( S l o c ( 0 ) [ m j ] − S l o c ( 0 ) [ m i ] + ⋯ ) ] . (148) \frac{\exp\left[-I_0\left[\mathcal{M}_n^{(j)}\right]\right]}{\exp\left[-I_0\left[\mathcal{M}_n^{(i)}\right]\right]} = \exp\left[-(n-1)\left(S_{\mathrm{loc}}^{(0)}[m_j]-S_{\mathrm{loc}}^{(0)}[m_i]+\cdots\right)\right]. \tag*{(148)} exp [ − I 0 [ M n ( i ) ] ] exp [ − I 0 [ M n ( j ) ] ] = exp [ − ( n − 1 ) ( S loc ( 0 ) [ m j ] − S loc ( 0 ) [ m i ] + ⋯ ) ] . (148) Hence, if
S l o c ( 0 ) [ m i ] < S l o c ( 0 ) [ m j ] , (149) S_{\mathrm{loc}}^{(0)}[m_i] < S_{\mathrm{loc}}^{(0)}[m_j], \tag*{(149)} S loc ( 0 ) [ m i ] < S loc ( 0 ) [ m j ] , (149) the j j j i i i n → 1 + n \to1^{+} n → 1 +
log Z ( n ) = − n I 0 [ M 1 ] − ( n − 1 ) min i S l o c ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (150) \log Z(n) = -nI_0[\mathcal{M}_1] - (n-1)\min_i S_{\mathrm{loc}}^{(0)}[m_i] + O\left((n-1)^2\right) + (n-1)O(\alpha'). \tag*{(150)} log Z ( n ) = − n I 0 [ M 1 ] − ( n − 1 ) i min S loc ( 0 ) [ m i ] + O ( ( n − 1 ) 2 ) + ( n − 1 ) O ( α ′ ) . (150) Since log Z ( 1 ) = − I 0 [ M 1 ] \log Z(1) = -I_0[\mathcal{M}_1] log Z ( 1 ) = − I 0 [ M 1 ]
S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] ∣ n = 1 = min i 1 4 G N ∫ m i e − 2 ϕ d A S . (151) S_A^{(0)} = -n\partial_n\left[\log Z(n)-n\log Z(1)\right]\big|_{n=1}
= \min_i \frac{1}{4G_N}\int_{m_i} e^{-2\phi}dA_S. \tag*{(151)} S A ( 0 ) = − n ∂ n [ log Z ( n ) − n log Z ( 1 ) ] n = 1 = i min 4 G N 1 ∫ m i e − 2 ϕ d A S . (151) Therefore, assuming that all relevant dominant extremal branches admit the required replica continuation, the minimization over saddle branches may equivalently be written as a minimization over all admissible cuts:
S A ( 0 ) = min ∂ m = ∂ A m ∼ A ∫ m ρ S ( x ) d A S , ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (152) S_A^{(0)} = \min_{\substack{\partial m=\partial A\\m\sim A}}\int_m \rho_S(x)\,dA_S,\qquad\rho_S(x)=\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(152)} S A ( 0 ) = ∂ m = ∂ A m ∼ A min ∫ m ρ S ( x ) d A S , ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (152) at leading order in both the genus and derivative expansions. Here m ∼ A m\sim A m ∼ A
Convex optimization and the origin of the string-frame norm bound
We now translate the weighted minimal cut obtained above into a flow formulation. We work first at a finite radial cutoff with regulated entangling anchors, so that the geometric optimization problem is well defined. The cutoff may be removed after the usual entropy regularization. To avoid confusion between the worldsheet and the bulk spatial slice, we denote the genus-zero worldsheet by Σ w s ≃ S 2 \Sigma_{\mathrm{ws}}\simeq S^2 Σ ws ≃ S 2 Σ \Sigma Σ Σ \Sigma Σ h i j h_{ij} h ij m ⊂ Σ m\subset\Sigma m ⊂ Σ
The origin of the norm bound can already be seen on an infinitesimal surface element. Let n i n^i n i v i v^i v i
d Φ [ v i ] = v i n i d A S , (153) d\Phi[v^i] = v^i n_i dA_S, \tag*{(153)} d Φ [ v i ] = v i n i d A S , (153) whereas the entropy capacity obtained from the sphere-derived functional previously is
d S c a p = ρ S ( x ) d A S , ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (154) dS_{\mathrm{cap}}=\rho_S(x)\,dA_S,\qquad\rho_S(x)=\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(154)} d S cap = ρ S ( x ) d A S , ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (154) Demanding d Φ [ v i ] ≤ d S c a p d\Phi[v^i]\le dS_{\mathrm{cap}} d Φ [ v i ] ≤ d S cap n i n^i n i
∣ v i n i ∣ ≤ ρ S ( x ) for every unit n i . (155) \left|v^i n_i\right|\le\rho_S(x)\qquad\text{for every unit }n^i. \tag*{(155)} v i n i ≤ ρ S ( x ) for every unit n i . (155) which is equivalent to
∣ v ∣ ≤ ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (156) |v|\le\rho_S(x)=\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(156)} ∣ v ∣ ≤ ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (156) For completeness, we now derive the weighted Riemannian max-flow/min-cut relation by following the convex-duality argument of ref. [4 ] . The only modification is that the unit capacity used there is replaced by the positive position-dependent capacity ρ S ( x ) \rho_S(x) ρ S ( x )
Let Σ \Sigma Σ h i j h_{ij} h ij A A A A ˉ \bar{A} A ˉ R R R
sup v Φ [ A ] , Φ [ A ] : = ∫ A v i n i d A S , (157) \underset{v}{\sup}\ \Phi[A],\qquad\Phi[A]:=\int_A v^i n_i\,dA_S, \tag*{(157)} v sup Φ [ A ] , Φ [ A ] := ∫ A v i n i d A S , (157) subject to
∇ i v i = 0 , ∣ v ∣ ≤ ρ S ( x ) , v i n i ∣ R = 0. (158) \nabla_i v^i=0,\qquad|v|\le\rho_S(x),\qquad\left.v^i n_i\right|_R=0. \tag*{(158)} ∇ i v i = 0 , ∣ v ∣ ≤ ρ S ( x ) , v i n i R = 0. (158) all norms and volume elements are defined using the string-frame metric. We assume that ρ S ( x ) \rho_S(x) ρ S ( x ) v i = 0 v^i=0 v i = 0 [4 , 45 ] .
To dualize the program, we introduce a scalar Lagrange multiplier ψ \psi ψ λ \lambda λ [4 ] . Keeping the no-flux condition on R R R
L [ v , ψ , λ ] = ∫ A v i n i d A S + ∫ Σ [ − ψ ∇ i v i + λ ( ρ S − ∣ v ∣ ) ] d V S . (159) L[v,\psi,\lambda]=\int_A v^i n_i\,dA_S+\int_\Sigma\left[-\psi\nabla_i v^i+\lambda\left(\rho_S-|v|\right)\right]dV_S. \tag*{(159)} L [ v , ψ , λ ] = ∫ A v i n i d A S + ∫ Σ [ − ψ ∇ i v i + λ ( ρ S − ∣ v ∣ ) ] d V S . (159) Integrating the first bulk term by parts gives
L [ v , ψ , λ ] = ∫ ∂ Σ ∖ R v i n i ( χ A − ψ ) d A S + ∫ Σ [ v i ∇ i ψ − λ ∣ v ∣ + λ ρ S ] d V S . (160) L[v,\psi,\lambda]=\int_{\partial\Sigma\setminus R}v^i n_i\left(\chi_A-\psi\right)dA_S+\int_\Sigma\left[v^i\nabla_i\psi-\lambda|v|+\lambda\rho_S\right]dV_S. \tag*{(160)} L [ v , ψ , λ ] = ∫ ∂ Σ ∖ R v i n i ( χ A − ψ ) d A S + ∫ Σ [ v i ∇ i ψ − λ ∣ v ∣ + λ ρ S ] d V S . (160) where χ A \chi_A χ A A A A ∂ Σ ∖ R \partial\Sigma\setminus R ∂ Σ ∖ R
χ A = { 1 , on A , 0 , on A ˉ . (161) \chi_A=
\begin{cases}
1,&\text{on }A,\\
0,&\text{on }\bar{A}.
\end{cases} \tag*{(161)} χ A = { 1 , 0 , on A , on A ˉ . (161) Note that ∂ Σ = A ∪ A ˉ ∪ R \partial\Sigma= A \cup\bar{A} \cup R ∂ Σ = A ∪ A ˉ ∪ R A ˉ : = ( ∂ Σ ∖ R ) ∖ A \bar{A} := (\partial\Sigma\setminus R) \setminus A A ˉ := ( ∂ Σ ∖ R ) ∖ A v i v^i v i
ψ ∣ A = 1 , ψ ∣ A ˉ = 0. (162) \left.\psi\right|_A = 1,\qquad\left.\psi\right|_{\bar{A}} = 0. \tag*{(162)} ψ ∣ A = 1 , ψ ∣ A ˉ = 0. (162) In the interior, using Cauchy–Schwarz inequality v i ∇ i ψ ≤ ∣ v ∣ ∣ ∇ ψ ∣ v^i\nabla_i\psi\le|v||\nabla\psi| v i ∇ i ψ ≤ ∣ v ∣∣∇ ψ ∣ v i v^i v i
v i ∇ i ψ − λ ∣ v ∣ ≤ ( ∣ ∇ ψ ∣ − λ ) ∣ v ∣ . (163) v^i\nabla_i\psi-\lambda|v| \le(|\nabla\psi|-\lambda)|v|. \tag*{(163)} v i ∇ i ψ − λ ∣ v ∣ ≤ ( ∣∇ ψ ∣ − λ ) ∣ v ∣. (163) The pointwise supremum over v i v^i v i
λ ≥ ∣ ∇ ψ ∣ , (164) \lambda\ge|\nabla\psi|, \tag*{(164)} λ ≥ ∣∇ ψ ∣ , (164) in which case the supremum of the v i v^i v i
inf ψ ∣ A = 1 , ψ ∣ A ˉ = 0 λ ≥ ∣ ∇ ψ ∣ ∫ Σ ρ S ( x ) λ d V S . (165) \underset{\substack{\left.\psi\right|_A=1,\ \left.\psi\right|_{\bar{A}}=0\\ \lambda\ge|\nabla\psi|}}{\inf}\int_\Sigma\rho_S(x)\lambda\,dV_S. \tag*{(165)} ψ ∣ A = 1 , ψ ∣ A ˉ = 0 λ ≥ ∣∇ ψ ∣ inf ∫ Σ ρ S ( x ) λ d V S . (165) Because ρ S ( x ) > 0 \rho_S(x)>0 ρ S ( x ) > 0 λ \lambda λ
λ = ∣ ∇ ψ ∣ . (166) \lambda=|\nabla\psi|. \tag*{(166)} λ = ∣∇ ψ ∣. (166) The dual problem therefore reduces to
inf ψ ∣ A = 1 , ψ ∣ A ˉ = 0 ∫ Σ ρ S ( x ) ∣ ∇ ψ ∣ d V S . (167) \underset{\substack{\left.\psi\right|_A=1,\ \left.\psi\right|_{\bar{A}}=0}}{\inf}\int_\Sigma\rho_S(x)|\nabla\psi|\,dV_S. \tag*{(167)} ψ ∣ A = 1 , ψ ∣ A ˉ = 0 inf ∫ Σ ρ S ( x ) ∣∇ ψ ∣ d V S . (167) This is the weighted counterpart of the convex relaxation in ref. [4 ] . To identify this convex program with a weighted minimal cut, define the one-parameter family of bulk regions
r ( p ) : = { x ∈ Σ : ψ ( x ) ≥ p } , p ∈ R , (168) r(p):=\{x\in\Sigma:\psi(x)\ge p\},\qquad p\in\mathbb{R}, \tag*{(168)} r ( p ) := { x ∈ Σ : ψ ( x ) ≥ p } , p ∈ R , (168) and let bulk surface
m ( p ) : = ∂ r ( p ) ∖ ∂ Σ , (169) m(p):=\partial r(p)\setminus\partial\Sigma, \tag*{(169)} m ( p ) := ∂ r ( p ) ∖ ∂ Σ , (169) be the corresponding bulk level set. The weighted coarea formula gives
∫ Σ ρ S ( x ) ∣ ∇ ψ ∣ d V S = ∫ − ∞ ∞ d p ∫ m ( p ) ρ S ( x ) d A S . (170) \int_\Sigma\rho_S(x)|\nabla\psi|\,dV_S=\int_{-\infty}^{\infty}dp\int_{m(p)}\rho_S(x)\,dA_S. \tag*{(170)} ∫ Σ ρ S ( x ) ∣∇ ψ ∣ d V S = ∫ − ∞ ∞ d p ∫ m ( p ) ρ S ( x ) d A S . (170) For 0 < p < 1 0<p<1 0 < p < 1 ψ \psi ψ r ( p ) ∩ ( ∂ Σ ∖ R ) = A r(p)\cap(\partial\Sigma\setminus R)=A r ( p ) ∩ ( ∂ Σ ∖ R ) = A m ( p ) ∼ A m(p)\sim A m ( p ) ∼ A R R R ρ S ( x ) \rho_S(x) ρ S ( x )
∫ Σ ρ S ∣ ∇ ψ ∣ d V S ≥ ∫ 0 1 d p ∫ m ( p ) ρ S d A S ≥ inf m ∼ A r e l R ∫ m ρ S d A . (171) \int_\Sigma\rho_S|\nabla\psi|\,dV_S\ge\int_0^1dp\int_{m(p)}\rho_SdA_S\ge\inf_{\substack{m\sim A\\ \mathrm{rel}\ R}}\int_m\rho_SdA. \tag*{(171)} ∫ Σ ρ S ∣∇ ψ ∣ d V S ≥ ∫ 0 1 d p ∫ m ( p ) ρ S d A S ≥ m ∼ A rel R inf ∫ m ρ S d A . (171) Taking the infimum over ψ \psi ψ
inf ψ ∣ A = 1 , ψ ∣ A ˉ = 0 ∫ Σ ρ S ∣ ∇ ψ ∣ d V S ≥ inf m ∼ A r e l R ∫ m ρ S d A S . (172) \inf_{\substack{\psi|_A=1,\ \psi|_{\bar A}=0}} \int_{\Sigma}\rho_S|\nabla\psi|dV_S \ge\inf_{\substack{m\sim A\\ \mathrm{rel}\ R}} \int_m \rho_S dA_S. \tag*{(172)} ψ ∣ A = 1 , ψ ∣ A ˉ = 0 inf ∫ Σ ρ S ∣∇ ψ ∣ d V S ≥ m ∼ A rel R inf ∫ m ρ S d A S . (172) Conversely, consider any cut m ∼ A m\sim A m ∼ A r r r A − m A-m A − m R R R χ r \chi_r χ r m m m
∫ Σ ρ S ∣ ∇ ψ ∣ d V S ⟶ ∫ m ρ S d A S . (173) \int_{\Sigma}\rho_S|\nabla\psi|dV_S \longrightarrow\int_m \rho_S dA_S. \tag*{(173)} ∫ Σ ρ S ∣∇ ψ ∣ d V S ⟶ ∫ m ρ S d A S . (173) It follows that
inf ψ ∣ A = 1 , ψ ∣ A ˉ = 0 ∫ Σ ρ S ∣ ∇ ψ ∣ d V S = inf m ∼ A r e l R ∫ m ρ S d A S . (174) \inf_{\substack{\psi|_A=1,\ \psi|_{\bar A}=0}} \int_{\Sigma}\rho_S|\nabla\psi|dV_S = \inf_{\substack{m\sim A\\ \mathrm{rel}\ R}} \int_m \rho_S dA_S. \tag*{(174)} ψ ∣ A = 1 , ψ ∣ A ˉ = 0 inf ∫ Σ ρ S ∣∇ ψ ∣ d V S = m ∼ A rel R inf ∫ m ρ S d A S . (174) Combining this result with strong convex duality yields the weighted Riemannian max-flow/min-cut relation,
sup ∇ i v i = 0 ∣ v ∣ ≤ ρ S v i n i ∣ R = 0 ∫ A v i n i d A S = inf m ∼ A r e l R ∫ m ρ S d A S . (175) \sup_{\substack{\nabla_i v^i=0\\ |v|\le\rho_S\\ v^i n_i|_{R=0}}} \int_A v^i n_i dA_S = \inf_{\substack{m\sim A\\ \mathrm{rel}\ R}} \int_m \rho_S dA_S. \tag*{(175)} ∇ i v i = 0 ∣ v ∣ ≤ ρ S v i n i ∣ R = 0 sup ∫ A v i n i d A S = m ∼ A rel R inf ∫ m ρ S d A S . (175) When the optimal flow and cut exist, the supremum and infimum may be replaced by a maximum and a minimum. Substituting the sphere-derived string-frame capacity, ρ S ( x ) \rho_S(x) ρ S ( x )
min m ∼ A r e l R 1 4 G N ∫ m e − 2 ϕ ( x ) d A S = max ∇ i v i = 0 ∣ v ∣ ≤ ρ S v i n i ∣ R = 0 ∫ A v i n i d A S . (176) \min_{\substack{m\sim A\\ \mathrm{rel}\ R}} \frac{1}{4G_N}\int_m e^{-2\phi(x)}dA_S = \max_{\substack{\nabla_i v^i=0\\ |v|\le\rho_S\\ v^i n_i|_{R=0}}}\int_A v^i n_i dA_S. \tag*{(176)} m ∼ A rel R min 4 G N 1 ∫ m e − 2 ϕ ( x ) d A S = ∇ i v i = 0 ∣ v ∣ ≤ ρ S v i n i ∣ R = 0 max ∫ A v i n i d A S . (176) Together with the weighted minimal-cut formula derived from the genus-zero off-shell string functional, this gives
S A ( 0 ) = max v Φ [ A ] = max v ∫ A v i n i d A S , (177) S_A^{(0)}=\max_v\Phi[A]=\max_v\int_A v^i n_i dA_S, \tag*{(177)} S A ( 0 ) = v max Φ [ A ] = v max ∫ A v i n i d A S , (177) subject to
∇ i v i = 0 , ∣ v ∣ ≤ e − 2 ϕ ( x ) 4 G N , v i n i ∣ R = 0 if R ≠ ∅ . (178) \nabla_i v^i=0,\qquad|v|\le\frac{e^{-2\phi(x)}}{4G_N},\qquad v^i n_i|_R=0\ \text{ if }R\ne\varnothing. \tag*{(178)} ∇ i v i = 0 , ∣ v ∣ ≤ 4 G N e − 2 ϕ ( x ) , v i n i ∣ R = 0 if R = ∅ . (178) Thus the weighted flow–cut duality yields the string-frame norm bound from the replica-selected entropy functional. Moreover, for a prescribed stationary horizon Γ H \Gamma_H Γ H
Φ ( Γ H ) = ∫ Γ H v i n i d A S = ∫ Γ H ρ S d A S = A ϕ ( Γ H ) 4 G N = S B H ( 0 ) . (179) \begin{aligned}
\Phi(\Gamma_H)=\int_{\Gamma_H}v^i n_i dA_S \\
&= \int_{\Gamma_H} \rho_S dA_S = \frac{A_\phi(\Gamma_H)}{4G_N} = S_{\mathrm{BH}}^{(0)}. \tag*{(179)}
\end{aligned} Φ ( Γ H ) = ∫ Γ H v i n i d A S = ∫ Γ H ρ S d A S = 4 G N A ϕ ( Γ H ) = S BH ( 0 ) . (179) This is the precise relation between the Ahmadain–Wall generalized area and the flow bound: A ϕ / ( 4 G N ) A_\phi/(4G_N) A ϕ / ( 4 G N ) ρ S ( x ) = e − 2 ϕ / ( 4 G N ) \rho_S(x) = e^{-2\phi}/(4G_N) ρ S ( x ) = e − 2 ϕ / ( 4 G N ) R R R
Physical interpretations
We now summarize the physical interpretations of the results presented above. At genus zero and to two-derivative order, the regulated sphere functional determines the tree-level target-space action through
Z 0 = ∂ K 0 ∂ log ϵ , I 0 = − Z 0 . (180) Z_0 = \frac{\partial K_0}{\partial\log\epsilon}, \qquad I_0 = -Z_0. \tag*{(180)} Z 0 = ∂ log ϵ ∂ K 0 , I 0 = − Z 0 . (180) The conical variation of this action gives the local string-frame entropy functional
S l o c ( 0 ) [ m ] = ∫ m ρ S ( x ) d A S , ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (181) S_{\mathrm{loc}}^{(0)}[m] = \int_m \rho_S(x)\,dA_S, \qquad\rho_S(x) = \frac{e^{-2\phi(x)}}{4G_N}. \tag*{(181)} S loc ( 0 ) [ m ] = ∫ m ρ S ( x ) d A S , ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (181) The gravitational replica construction then selects the weighted minimal cut,
S A ( 0 ) = min m ∼ A ∫ m ρ S ( x ) d A S , (182) S_A^{(0)} = \min_{m\sim A} \int_m \rho_S(x)\,dA_S, \tag*{(182)} S A ( 0 ) = m ∼ A min ∫ m ρ S ( x ) d A S , (182) and the weighted Riemannian max-flow/min-cut theorem gives
S A ( 0 ) = max v ∫ A v i n i d A S , ∇ i v i = 0 , ∣ v ∣ ≤ e − 2 ϕ ( x ) 4 G N . (183) S_A^{(0)} = \max_v \int_A v^i n_i\,dA_S, \qquad\nabla_i v^i = 0, \qquad|v| \le\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(183)} S A ( 0 ) = v max ∫ A v i n i d A S , ∇ i v i = 0 , ∣ v ∣ ≤ 4 G N e − 2 ϕ ( x ) . (183) In this qualified but precise sense, the string-frame norm bound is derived from the genus zero string sphere through the following sequence:
S l o c ( 0 ) [ m ] → off-shell string S A ( 0 ) → gravi-replica ∣ v ∣ ≤ e − 2 ϕ ( x ) 4 G N max-flow/min-cut (184) S_{\mathrm{loc}}^{(0)}[m]
\xrightarrow{\text{off-shell string}}
S_A^{(0)}
\xrightarrow{\text{gravi-replica}}
|v| \le\frac{e^{-2\phi(x)}}{4G_N}
\quad\text{max-flow/min-cut} \tag*{(184)} S loc ( 0 ) [ m ] off-shell string S A ( 0 ) gravi-replica ∣ v ∣ ≤ 4 G N e − 2 ϕ ( x ) max-flow/min-cut (184) In other words, the off-shell worldsheet calculation first fixes the local entropy density ρ S ( x ) \rho_S(x) ρ S ( x )
∣ v ∣ ≤ e − 2 ϕ ( x ) 4 G N . (185) |v| \le\frac{e^{-2\phi(x)}}{4G_N}. \tag*{(185)} ∣ v ∣ ≤ 4 G N e − 2 ϕ ( x ) . (185) This is the principal physical content of the construction. Flow conservation,
∇ i v i = 0 , (186) \nabla_i v^i = 0, \tag*{(186)} ∇ i v i = 0 , (186) is the dual constraint associated with the unrestricted interior values of the scalar variable ψ \psi ψ
ρ S ( x ) = e − 2 ϕ ( x ) 4 G N (187) \rho_{S}(x)=\frac{e^{-2\phi(x)}}{4G_{N}} \tag*{(187)} ρ S ( x ) = 4 G N e − 2 ϕ ( x ) (187) has the direct interpretation of an entropy capacity.
This interpretation also gives a precise flow meaning to the Ahmadain–Wall generalized area,
A ϕ ( m ) : = ∫ m e − 2 ϕ ( x ) d A S . (188) A_{\phi}(m):=\int_{m}e^{-2\phi(x)}dA_{S}. \tag*{(188)} A ϕ ( m ) := ∫ m e − 2 ϕ ( x ) d A S . (188) Its relation to the flow capacity is
A ϕ ( m ) 4 G N = ∫ m ρ S ( x ) d A S . (189) \frac{A_{\phi}(m)}{4G_{N}}=\int_{m}\rho_{S}(x)dA_{S}. \tag*{(189)} 4 G N A ϕ ( m ) = ∫ m ρ S ( x ) d A S . (189) Consequently, A ϕ / ( 4 G N ) A_{\phi}/(4G_{N}) A ϕ / ( 4 G N ) ρ S ( x ) \rho_{S}(x) ρ S ( x ) Γ H \Gamma_{H} Γ H
max v Φ [ Γ H ] = 1 4 G N ∫ Γ H e − 2 ϕ ( x ) d A S = A ϕ ( Γ H ) 4 G N = S B H ( 0 ) . (190) \max_{v}\Phi[\Gamma_{H}]=\frac{1}{4G_{N}}\int_{\Gamma_{H}}e^{-2\phi(x)}dA_{S}=\frac{A_{\phi}(\Gamma_{H})}{4G_{N}}=S_{\mathrm{BH}}^{(0)}. \tag*{(190)} v max Φ [ Γ H ] = 4 G N 1 ∫ Γ H e − 2 ϕ ( x ) d A S = 4 G N A ϕ ( Γ H ) = S BH ( 0 ) . (190) The same construction applies to a general static boundary region A A A The black hole entropy and the entanglement entropy are therefore governed by the same capacity density, while the relevant homology condition determines the corresponding bottleneck.
The dilaton factor is also precisely the factor required by the relation between the string and Einstein frames. In D D D
g μ ν E = e − 4 ϕ D − 2 g μ ν S , (191) g^{E}_{\mu\nu}=e^{-\frac{4\phi}{D-2}}g^{S}_{\mu\nu}, \tag*{(191)} g μν E = e − D − 2 4 ϕ g μν S , (191) and the induced codimension-two area elements obey
d A E = e − 2 ϕ d A S . (192) dA_{E}=e^{-2\phi}dA_{S}. \tag*{(192)} d A E = e − 2 ϕ d A S . (192) It follows immediately that
d A E 4 G N = e − 2 ϕ d A S 4 G N . (193) \frac{dA_{E}}{4G_{N}}=\frac{e^{-2\phi}dA_{S}}{4G_{N}}. \tag*{(193)} 4 G N d A E = 4 G N e − 2 ϕ d A S . (193) More explicitly, introducing
Ω = e − 2 ϕ D − 2 , h i j E = Ω 2 h i j S , (194) \Omega=e^{-\frac{2\phi}{D-2}},\qquad h^{E}_{ij}=\Omega^{2}h^{S}_{ij}, \tag*{(194)} Ω = e − D − 2 2 ϕ , h ij E = Ω 2 h ij S , (194) the flux-preserving relation between the two flow fields is
v S i = Ω D − 1 v E i . (195) v_S^i = \Omega^{D-1}v_E^i. \tag*{(195)} v S i = Ω D − 1 v E i . (195) Indeed,
h S v S i = h E v E i , (196) \sqrt{h_S}v_S^i = \sqrt{h_E}v_E^i, \tag*{(196)} h S v S i = h E v E i , (196) so the integrated flux is frame invariant. Moreover,
∇ i S v S i = 0 ⟺ ∇ i E v E i = 0 , (197) \nabla_i^S v_S^i = 0 \quad\Longleftrightarrow\quad\nabla_i^E v_E^i = 0, \tag*{(197)} ∇ i S v S i = 0 ⟺ ∇ i E v E i = 0 , (197) and their norms are related by
∣ v S ∣ = Ω D − 2 ∣ v E ∣ = e − 2 ϕ ∣ v E ∣ . (198) |v_S| = \Omega^{D-2}|v_E| = e^{-2\phi}|v_E|. \tag*{(198)} ∣ v S ∣ = Ω D − 2 ∣ v E ∣ = e − 2 ϕ ∣ v E ∣. (198) Hence the standard Einstein-frame norm bound
∣ v E ∣ ≤ 1 4 G N , (199) |v_E| \leq\frac{1}{4G_N}, \tag*{(199)} ∣ v E ∣ ≤ 4 G N 1 , (199) is equivalent to the string-frame bound
∣ v S ∣ ≤ e − 2 ϕ 4 G N . (200) |v_S| \leq\frac{e^{-2\phi}}{4G_N}. \tag*{(200)} ∣ v S ∣ ≤ 4 G N e − 2 ϕ . (200) The sphere derivation explains why this frame transformation appears locally in the flow description: the factor e − 2 ϕ e^{-2\phi} e − 2 ϕ
We now relate this geometrically determined flow capacity to the projected string charge density. Varying the worldsheet coupling to the Kalb–Ramond field produces the antisymmetric spacetime current j a b j^{ab} j ab
q i = N A D M j 0 i = − g h j 0 i , (201) q^i = N_{\mathrm{ADM}}j^{0i} = \frac{\sqrt{-g}}{\sqrt{h}}j^{0i}, \tag*{(201)} q i = N ADM j 0 i = h − g j 0 i , (201) and current conservation implies
∂ i ( h q i ) = 0. (202) \partial_i\left(\sqrt{h}q^i\right) = 0. \tag*{(202)} ∂ i ( h q i ) = 0. (202) The projected string charge density therefore has precisely the conservation property required to define candidate flow lines. The proposed string charge density/bit threads correspondence introduces the additional identification
v i = C g e o m q i , (203) v^i = C_{\mathrm{geom}}q^i, \tag*{(203)} v i = C geom q i , (203) Under this identification, the sphere-derived geometric capacity imposes the constraint
C g e o m ∣ q ( x ) ∣ ≤ ρ S ( x ) = e − 2 ϕ ( x ) 4 G N . (204) C_{\mathrm{geom}}|q(x)| \leq\rho_S(x) = \frac{e^{-2\phi(x)}}{4G_N}. \tag*{(204)} C geom ∣ q ( x ) ∣ ≤ ρ S ( x ) = 4 G N e − 2 ϕ ( x ) . (204) For the homology class under consideration, conservation makes the string-charge flux cut independent:
N = ∫ m q i n i d A S , m ∼ A . (205) \mathcal{N} = \int_m q^i n_i dA_S,\qquad m \sim A. \tag*{(205)} N = ∫ m q i n i d A S , m ∼ A . (205) The capacity bound then implies
C g e o m N = C g e o m ∫ m q i n i d A S ≤ ∫ m ρ S ( x ) d A S . (206) C_{\mathrm{geom}}\mathcal{N} = C_{\mathrm{geom}}\int_m q^i n_i dA_S \le\int_m \rho_S(x)\,dA_S. \tag*{(206)} C geom N = C geom ∫ m q i n i d A S ≤ ∫ m ρ S ( x ) d A S . (206) Since this inequality holds for every cut m ∼ A m \sim A m ∼ A
C g e o m N ≤ min m ∼ A ∫ m ρ S ( x ) d A S = S A ( 0 ) . (207) C_{\mathrm{geom}}\mathcal{N} \le\min_{m\sim A}\int_m \rho_S(x)\,dA_S = S_A^{(0)}. \tag*{(207)} C geom N ≤ m ∼ A min ∫ m ρ S ( x ) d A S = S A ( 0 ) . (207) When the projected string current realizes the corresponding maximal flow,
C g e o m N = S A ( 0 ) , (208) C_{\mathrm{geom}}\mathcal{N} = S_A^{(0)}, \tag*{(208)} C geom N = S A ( 0 ) , (208) At the minimizing cut m A m_A m A
v i ∣ m A = ρ S n i ∣ m A , (209) v^i\big|_{m_A} = \rho_S n^i\big|_{m_A}, \tag*{(209)} v i m A = ρ S n i m A , (209) Using v i = C g e o m q i v^i = C_{\mathrm{geom}}q^i v i = C geom q i
C g e o m ∣ q ( x ) ∣ ∣ m A = ρ S ( x ) ∣ m A = e − 2 ϕ ( x ) 4 G N ∣ m A . (210) C_{\mathrm{geom}}|q(x)|\big|_{m_A} = \rho_S(x)\big|_{m_A} = \left.\frac{e^{-2\phi(x)}}{4G_N}\right|_{m_A}. \tag*{(210)} C geom ∣ q ( x ) ∣ m A = ρ S ( x ) m A = 4 G N e − 2 ϕ ( x ) m A . (210) For a black hole bifurcation surface, this becomes
C g e o m N = S B H ( 0 ) , (211) C_{\mathrm{geom}}\mathcal{N} = S_{\mathrm{BH}}^{(0)}, \tag*{(211)} C geom N = S BH ( 0 ) , (211) when the projected physical string charge density provides the maximal-flow representative.
Physical string sources and dynamical selection of a maximal flow
The results in the previous sections determine the string-frame entropy capacity, or norm bound
ρ S ( x ) = e − 2 ϕ ( x ) 4 G N , ∇ i v i = 0 , ∣ v ∣ ≤ ρ S ( x ) . (212) \rho_S(x) = \frac{e^{-2\phi(x)}}{4G_N},\qquad\nabla_i v^i = 0,\qquad|v| \le\rho_S(x). \tag*{(212)} ρ S ( x ) = 4 G N e − 2 ϕ ( x ) , ∇ i v i = 0 , ∣ v ∣ ≤ ρ S ( x ) . (212) We now construct a dynamical F1 realization of the corresponding maximal flow. In the classical probe regime adopted here, the BTZ fields are held fixed while the string embedding is determined by the full classical F1 equations,
δ S F 1 δ X a = 0. (213) \frac{\delta S_{\mathrm{F1}}}{\delta X^a} = 0. \tag*{(213)} δ X a δ S F1 = 0. (213) In this precise sense, a physical string denotes an on-shell F1 probe. Varying its Wess-Zumino coupling with respect to B a b B_{ab} B ab N \mathcal{N} N
Stationary strings in the NS–NS BTZ background
We consider the three-dimensional NS–NS action
S b u l k ( 0 ) = 1 16 π G N ( 3 ) ∫ d 3 x − g e − 2 ϕ [ R + 4 ( ∇ ϕ ) 2 − 1 12 H 2 + 4 ℓ 2 ] , H = d B , (214) S_{\mathrm{bulk}}^{(0)}=\frac{1}{16\pi G_N^{(3)}}\int d^3x\sqrt{-g}e^{-2\phi}\left[R+4(\nabla\phi)^2-\frac{1}{12}H^2+\frac{4}{\ell^2}\right],\qquad H=dB, \tag*{(214)} S bulk ( 0 ) = 16 π G N ( 3 ) 1 ∫ d 3 x − g e − 2 ϕ [ R + 4 ( ∇ ϕ ) 2 − 12 1 H 2 + ℓ 2 4 ] , H = d B , (214) with H = ( 2 / ℓ ) ϵ B T Z H=(2/\ell)\epsilon_{\mathrm{BTZ}} H = ( 2/ ℓ ) ϵ BTZ [32 , 33 ]
d s B T Z 2 = − f ( r ) d t 2 + d r 2 f ( r ) + r 2 d θ 2 , f ( r ) = r 2 − r h 2 ℓ 2 , θ ∼ θ + 2 π , (215) ds_{\mathrm{BTZ}}^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\theta^2,\qquad f(r)=\frac{r^2-r_h^2}{\ell^2},\qquad\theta\sim\theta+2\pi, \tag*{(215)} d s BTZ 2 = − f ( r ) d t 2 + f ( r ) d r 2 + r 2 d θ 2 , f ( r ) = ℓ 2 r 2 − r h 2 , θ ∼ θ + 2 π , (215) ϕ = ϕ 0 , B = − b ( r ) d t ∧ d θ , b ( r ) = r 2 − r h 2 ℓ . (216) \phi=\phi_0,\qquad B=-b(r)\,dt\wedge d\theta,\qquad b(r)=\frac{r^2-r_h^2}{\ell}. \tag*{(216)} ϕ = ϕ 0 , B = − b ( r ) d t ∧ d θ , b ( r ) = ℓ r 2 − r h 2 . (216) The additive constant in b b b
The classical F1 action is given by [10 , 11 , 12 ]
S F 1 = − 1 4 π α ′ ∫ d 2 σ ( − γ γ m n ∂ m X a ∂ n X b g a b + ϵ m n ∂ m X a ∂ n X b B a b ) , (217) S_{\mathrm{F1}}=-\frac{1}{4\pi\alpha'}\int d^2\sigma\left(\sqrt{-\gamma}\gamma^{mn}\partial_mX^a\partial_nX^b g_{ab}+\epsilon^{mn}\partial_mX^a\partial_nX^b B_{ab}\right), \tag*{(217)} S F1 = − 4 π α ′ 1 ∫ d 2 σ ( − γ γ mn ∂ m X a ∂ n X b g ab + ϵ mn ∂ m X a ∂ n X b B ab ) , (217) where γ m n \gamma_{mn} γ mn γ m n i n d = g a b ∂ m X a ∂ n X b \gamma^{\mathrm{ind}}_{mn}=g_{ab}\partial_mX^a\partial_nX^b γ mn ind = g ab ∂ m X a ∂ n X b
X t = τ , X r = σ , X θ = θ ( r ) . (218) X^t=\tau,\qquad X^r=\sigma,\qquad X^\theta=\theta(r). \tag*{(218)} X t = τ , X r = σ , X θ = θ ( r ) . (218) Using the notations ϵ τ σ = + 1 \epsilon^{\tau\sigma}=+1 ϵ τ σ = + 1 θ ′ ( r ) ≡ d θ / d r \theta'(r)\equiv d\theta/dr θ ′ ( r ) ≡ d θ / d r
− det γ m n i n d = 1 + f r 2 θ ′ 2 , S F 1 = − 1 2 π α ′ ∫ d t d r ( 1 + f r 2 θ ′ 2 − b θ ′ ) . (219) -\det\gamma^{\mathrm{ind}}_{mn}=1+fr^2\theta'^2,\qquad S_{\mathrm{F1}}=-\frac{1}{2\pi\alpha'}\int dt\,dr\left(\sqrt{1+fr^2\theta'^2}-b\theta'\right). \tag*{(219)} − det γ mn ind = 1 + f r 2 θ ′2 , S F1 = − 2 π α ′ 1 ∫ d t d r ( 1 + f r 2 θ ′2 − b θ ′ ) . (219) Since θ \theta θ
f ( r ) r 2 θ ′ ( r ) 1 + f ( r ) r 2 θ ′ 2 ( r ) − b ( r ) = k θ . (220) \frac{f(r)r^2\theta'(r)}{\sqrt{1+f(r)r^2\theta'^2(r)}}-b(r)=k_\theta. \tag*{(220)} 1 + f ( r ) r 2 θ ′2 ( r ) f ( r ) r 2 θ ′ ( r ) − b ( r ) = k θ . (220) The remaining embedding equations follow from worldsheet reparametrization invariance once the transverse equation and the auxiliary-metric equation hold. A local constant gauge shift b → b + b 0 b \to b+b_0 b → b + b 0 k θ → k θ − b 0 k_\theta\to k_\theta-b_0 k θ → k θ − b 0 b + k θ b+k_\theta b + k θ k θ = − b ( r h ) k_\theta=-b(r_h) k θ = − b ( r h ) θ ( r ) \theta(r) θ ( r ) θ ′ \theta' θ ′ r = r h r=r_h r = r h f ( r h ) = b ( r h ) = 0 f(r_h)=b(r_h)=0 f ( r h ) = b ( r h ) = 0 k θ = 0 k_\theta=0 k θ = 0 (220) gives
θ ′ 2 ( r ) = b 2 f r 2 ( f r 2 − b 2 ) , f r 2 − b 2 = r h 2 ( r 2 − r h 2 ) ℓ 2 , (221) \theta'^{2}(r)=\frac{b^{2}}{fr^{2}\left(fr^{2}-b^{2}\right)},\qquad fr^{2}-b^{2}=\frac{r_h^{2}\left(r^{2}-r_h^{2}\right)}{\ell^{2}}, \tag*{(221)} θ ′ 2 ( r ) = f r 2 ( f r 2 − b 2 ) b 2 , f r 2 − b 2 = ℓ 2 r h 2 ( r 2 − r h 2 ) , (221) and hence, for our orientation,
θ ′ ( r ) = ℓ r r h , θ ( r ) = θ h + ℓ r h log r r h , − det γ i n d = r r h . (222) \theta'(r)=\frac{\ell}{r r_h},\qquad\theta(r)=\theta_h+\frac{\ell}{r_h}\log\frac{r}{r_h},\qquad\sqrt{-\det\gamma^{\mathrm{ind}}}=\frac{r}{r_h}. \tag*{(222)} θ ′ ( r ) = r r h ℓ , θ ( r ) = θ h + r h ℓ log r h r , − det γ ind = r h r . (222) The solution is regular at the horizon. Its angular winding grows without bound as r → ∞ r\to\infty r → ∞ r c r_c r c r c r_c r c θ ′ ( r ) \theta'(r) θ ′ ( r ) Figure 4 .
Figure 4. Comparison between the prescribed radial string sources and the stationary physical F1 sources on a constant-time slice of one BTZ exterior and two-sided BTZ geometry. The blue dashed curves represent the radially embedded sources X μ = ( τ , r , θ 0 ) X^\mu=(\tau,r,\theta_0) X μ = ( τ , r , θ 0 ) X μ = ( τ , r , θ ( r ) ) X^\mu=(\tau,r,\theta(r)) X μ = ( τ , r , θ ( r )) v p h y s i = ρ S n i v_{\mathrm{phys}}^i=\rho S n^i v phys i = ρS n i
Classical string dynamics in BTZ backgrounds, including fundamental string probes in the presence of NS–NS flux, has been studied previously [27 , 28 ] . Our purpose here is to relate the regular stationary branch to its projected charge density and to the independently determined entropy capacity.
The distinction from the purely radial source is essential. Setting θ ′ = 0 \theta'=0 θ ′ = 0 (220) would require b ′ = 0 b'=0 b ′ = 0
b ′ ( r ) = 2 r ℓ 2 ≠ 0. (223) b'(r)=\frac{2r}{\ell^{2}}\ne0. \tag*{(223)} b ′ ( r ) = ℓ 2 2 r = 0. (223) Although the term ϵ m n ∂ m X a ∂ n X b B a b \epsilon^{mn}\partial_mX^a\partial_nX^bB_{ab} ϵ mn ∂ m X a ∂ n X b B ab
δ S B ∣ θ ′ = 0 = − 1 2 π α ′ ∫ d t d r b ′ ( r ) δ θ + boundary terms . (224) \left.\delta S_B\right|_{\theta'=0}=-\frac{1}{2\pi\alpha'}\int dt\,dr\,b'(r)\,\delta\theta+\text{boundary terms}. \tag*{(224)} δ S B ∣ θ ′ = 0 = − 2 π α ′ 1 ∫ d t d r b ′ ( r ) δ θ + boundary terms . (224) Consequently, the radially embedded current used in our previous construction,
X μ = ( τ , r , θ 0 ) , (225) X^\mu=(\tau,r,\theta_0), \tag*{(225)} X μ = ( τ , r , θ 0 ) , (225) defines a conserved radial charge density, but it is not a free stationary F1 solution of the embedding equations in the present NS-NS background. It may instead be interpreted as a fixed external source, or instantaneous initial data for a time-dependent string. This distinction concerns the source dynamics:
The physical string current and black hole entropy
We normalize the antisymmetric charge current so that a single oriented string carries one unit of charge. With ϵ τ σ = + 1 \epsilon^{\tau\sigma}=+1 ϵ τ σ = + 1
j a b ( x ) = 1 − g ( x ) ∑ k = 1 N ∫ d τ d σ ϵ m n ∂ m X k a ∂ n X k b δ ( 3 ) ( x − X k ) . (226) j^{ab}(x)=\frac{1}{\sqrt{-g(x)}}\sum_{k=1}^{\mathcal{N}}\int d\tau d\sigma\,\epsilon^{mn}\partial_m X_k^a\partial_n X_k^b\delta^{(3)}(x-X_k). \tag*{(226)} j ab ( x ) = − g ( x ) 1 k = 1 ∑ N ∫ d τ d σ ϵ mn ∂ m X k a ∂ n X k b δ ( 3 ) ( x − X k ) . (226) The physical source appearing in the variation of the B B B 1 / ( 2 π α ′ ) 1/(2\pi\alpha') 1/ ( 2 π α ′ ) (226) . Thus N \mathcal{N} N
Consider the ensemble of Eq. (222) , with θ h \theta_h θ h
∑ k δ 2 π ( θ − θ k ( r ) ) ⟶ N 2 π . (227) \sum_k\delta_{2\pi}(\theta-\theta_k(r))\longrightarrow\frac{\mathcal{N}}{2\pi}. \tag*{(227)} k ∑ δ 2 π ( θ − θ k ( r )) ⟶ 2 π N . (227) Since − g = r \sqrt{-g}=r − g = r
j 0 r = N 2 π r , j 0 θ = N 2 π r θ ′ ( r ) , j r θ = 0. (228) j^{0r}=\frac{\mathcal{N}}{2\pi r},\qquad j^{0\theta}=\frac{\mathcal{N}}{2\pi r}\theta'(r),\qquad j^{r\theta}=0. \tag*{(228)} j 0 r = 2 π r N , j 0 θ = 2 π r N θ ′ ( r ) , j r θ = 0. (228) The induced metric and lapse on a static exterior slice Σ \Sigma Σ
h i j d x i d x j = d r 2 f + r 2 d θ 2 , h = r f , N A D M = f . (229) h_{ij}dx^i dx^j = \frac{dr^2}{f} + r^2 d\theta^2,\qquad\sqrt{h} = \frac{r}{\sqrt{f}},\qquad N_{\mathrm{ADM}} = \sqrt{f}. \tag*{(229)} h ij d x i d x j = f d r 2 + r 2 d θ 2 , h = f r , N ADM = f . (229) Projection with the future-directed unit normal n μ n_\mu n μ q μ = − n ν j ν μ q^\mu=-n_\nu j^{\nu\mu} q μ = − n ν j ν μ
q r = N A D M j 0 r = N f 2 π r , q θ = N A D M j 0 θ = N ℓ f 2 π r h r 2 = θ ′ q r . (230) \begin{aligned}
q^r &= N_{\mathrm{ADM}}j^{0r} = \frac{\mathcal{N}\sqrt{f}}{2\pi r},\\
q^\theta&= N_{\mathrm{ADM}}j^{0\theta} = \frac{\mathcal{N}\ell\sqrt{f}}{2\pi r_h r^2} = \theta' q^r. \tag*{(230)}
\end{aligned} q r q θ = N ADM j 0 r = 2 π r N f , = N ADM j 0 θ = 2 π r h r 2 N ℓ f = θ ′ q r . (230) The slice projection retains the angular component. Current conservation and the flux through a constant-r r r
∇ i q i = 1 h [ ∂ r ( h q r ) + ∂ θ ( h q θ ) ] = f r ∂ r [ r f N f 2 π r ] = f r ∂ r ( N 2 π ) = 0 , (231) \begin{aligned}
\nabla_i q^i &= \frac{1}{\sqrt{h}}\left[\partial_r\left(\sqrt{h}q^r\right)+\partial_\theta\left(\sqrt{h}q^\theta\right)\right]\\
&= \frac{\sqrt{f}}{r}\partial_r\left[\frac{r}{\sqrt{f}}\frac{\mathcal{N}\sqrt{f}}{2\pi r}\right]\\
&= \frac{\sqrt{f}}{r}\partial_r\left(\frac{\mathcal{N}}{2\pi}\right)=0, \tag*{(231)}
\end{aligned} ∇ i q i = h 1 [ ∂ r ( h q r ) + ∂ θ ( h q θ ) ] = r f ∂ r [ f r 2 π r N f ] = r f ∂ r ( 2 π N ) = 0 , (231) and
∫ r = c o n s t q i n i d A S = N . (232) \int_{r=\mathrm{const}} q^i n_i\,dA_S=\mathcal{N}. \tag*{(232)} ∫ r = const q i n i d A S = N . (232) Here the circle’s outward unit normal is n i = f ( ∂ r ) i n^i=\sqrt{f}(\partial_r)^i n i = f ( ∂ r ) i d A S = r d θ dA_S=r\,d\theta d A S = r d θ
∣ q ∣ 2 = h i j q i q j = ( q r ) 2 f + r 2 ( q θ ) 2 = ( N 2 π r ) 2 ( 1 + f r 2 θ ′ 2 ) = ( N 2 π r h ) 2 . (233) |q|^2=h_{ij}q^i q^j=\frac{(q^r)^2}{f}+r^2(q^\theta)^2=\left(\frac{\mathcal{N}}{2\pi r}\right)^2\left(1+fr^2\theta'^2\right)=\left(\frac{\mathcal{N}}{2\pi r_h}\right)^2. \tag*{(233)} ∣ q ∣ 2 = h ij q i q j = f ( q r ) 2 + r 2 ( q θ ) 2 = ( 2 π r N ) 2 ( 1 + f r 2 θ ′2 ) = ( 2 π r h N ) 2 . (233) Therefore, we successfully obtain the constant norm that comes from the stationary embedding equation and the uniform distribution, rather than an imposed choice of radial flow profile.
We now define the unit-normalized current
q ^ i ≡ q i ∣ q ∣ . (234) \hat{q}^i\equiv\frac{q^i}{|q|}. \tag*{(234)} q ^ i ≡ ∣ q ∣ q i . (234) Its components are
q ^ r = r h f r , q ^ θ = ℓ f r 2 , (235) \hat{q}^r=\frac{r_h\sqrt{f}}{r},\qquad\hat{q}^\theta=\frac{\ell\sqrt{f}}{r^2}, \tag*{(235)} q ^ r = r r h f , q ^ θ = r 2 ℓ f , (235) and it satisfies
∣ q ^ ∣ = 1. (236) |\hat{q}|=1. \tag*{(236)} ∣ q ^ ∣ = 1. (236) The constancy of ∣ q ∣ |q| ∣ q ∣
∇ i q i = 0 , (237) \nabla_i q^i = 0, \tag*{(237)} ∇ i q i = 0 , (237) normalization does not necessarily preserve conservation. In fact,
∇ i ( q i ∣ q ∣ ) = ∇ i q i ∣ q ∣ − q i ∂ i ∣ q ∣ ∣ q ∣ 2 . (238) \nabla_i\left(\frac{q^i}{|q|}\right)=\frac{\nabla_iq^i}{|q|}-\frac{q^i\partial_i|q|}{|q|^2}. \tag*{(238)} ∇ i ( ∣ q ∣ q i ) = ∣ q ∣ ∇ i q i − ∣ q ∣ 2 q i ∂ i ∣ q ∣ . (238) Consequently, when ∇ i q i = 0 \nabla_iq^i=0 ∇ i q i = 0
∇ i ( q i ∣ q ∣ ) = − q i ∂ i ∣ q ∣ ∣ q ∣ 2 , (239) \nabla_i\left(\frac{q^i}{|q|}\right)=-\frac{q^i\partial_i|q|}{|q|^2}, \tag*{(239)} ∇ i ( ∣ q ∣ q i ) = − ∣ q ∣ 2 q i ∂ i ∣ q ∣ , (239) which is generally nonzero. For the stationary spiral solution, however,
∂ i ∣ q ∣ = 0. (240) \partial_i|q|=0. \tag*{(240)} ∂ i ∣ q ∣ = 0. (240) Therefore,
∇ i q ^ i = ∇ i ( q i ∣ q ∣ ) = 0. (241) \nabla_i\hat q^i=\nabla_i\left(\frac{q^i}{|q|}\right)=0. \tag*{(241)} ∇ i q ^ i = ∇ i ( ∣ q ∣ q i ) = 0. (241) It implies that the physical F1 equation gives more than a conserved charge current: it produces a current whose unit normalization remains divergenceless.
We next compute the flux of the unit current. For a constant-r r r C r C_r C r
n i = f ( ∂ r ) i , n i d x i = d r f , d A S = r d θ . (242) n^i=\sqrt{f}\,(\partial_r)^i,\qquad n_i dx^i=\frac{dr}{\sqrt{f}},\qquad dA_S=r\,d\theta. \tag*{(242)} n i = f ( ∂ r ) i , n i d x i = f d r , d A S = r d θ . (242) It follows that
q ^ i n i = q ^ r f = r h r . (243) \hat q^i n_i=\frac{\hat q^r}{\sqrt{f}}=\frac{r_h}{r}. \tag*{(243)} q ^ i n i = f q ^ r = r r h . (243) Therefore,
Φ [ q ^ ] = ∫ 0 2 π q ^ i n i d A S = 2 π r h = A H , (244) \Phi[\hat q]=\int_0^{2\pi}\hat q^i n_i\,dA_S=2\pi r_h=A_H, \tag*{(244)} Φ [ q ^ ] = ∫ 0 2 π q ^ i n i d A S = 2 π r h = A H , (244) which implies the flux of the unit-normalized physical string current equals the string-frame circumference of the BTZ horizon. We now recall the string-frame norm bound derived from the off-shell sphere and replica construction:
ρ S ( x ) = e − 2 ϕ 0 4 G N ( 3 ) . (245) \rho_S(x)=\frac{e^{-2\phi_0}}{4G_N^{(3)}}. \tag*{(245)} ρ S ( x ) = 4 G N ( 3 ) e − 2 ϕ 0 . (245) Because the dilaton is constant in the present BTZ background, ρ S \rho_S ρ S
v p h y s i ≡ ρ S q ^ i = ρ S q i ∣ q ∣ . (246) v_{\mathrm{phys}}^i\equiv\rho_S\hat q^i=\rho_S\frac{q^i}{|q|}. \tag*{(246)} v phys i ≡ ρ S q ^ i = ρ S ∣ q ∣ q i . (246) This is the important relation that shows the correspondence between the physical string charge density and bit threads v p h y s i v_{\mathrm{phys}}^i v phys i ρ S \rho_S ρ S
v p h y s = ρ S r h f r ∂ r + ρ S ℓ f r 2 ∂ θ . (247) v_{\mathrm{phys}}=\rho_S\frac{r_h\sqrt{f}}{r}\partial_r+\rho_S\frac{\ell\sqrt{f}}{r^2}\partial_\theta. \tag*{(247)} v phys = ρ S r r h f ∂ r + ρ S r 2 ℓ f ∂ θ . (247) Since ρ S \rho_S ρ S q ^ i \hat{q}^i q ^ i v p h y s v_{\mathrm{phys}} v phys
∇ i v p h y s i = ρ S ∇ i q ^ i = 0. (248) \nabla_i v_{\mathrm{phys}}^i=\rho_S\nabla_i\hat{q}^i=0. \tag*{(248)} ∇ i v phys i = ρ S ∇ i q ^ i = 0. (248) Furthermore,
∣ v p h y s ∣ = ρ S ∣ q ^ ∣ = ρ S . (249) \left|v_{\mathrm{phys}}\right|=\rho_S\left|\hat{q}\right|=\rho_S. \tag*{(249)} ∣ v phys ∣ = ρ S ∣ q ^ ∣ = ρ S . (249) Thus, we have
∣ v p h y s ∣ = e − 2 ϕ 0 4 G N ( 3 ) . (250) \left|v_{\mathrm{phys}}\right|=\frac{e^{-2\phi_0}}{4G_N^{(3)}}. \tag*{(250)} ∣ v phys ∣ = 4 G N ( 3 ) e − 2 ϕ 0 . (250) throughout the exterior region. The physical string flow saturates the string-frame norm bound everywhere. Its flux is
Φ [ v p h y s ] = ρ S Φ [ q ^ ] = e − 2 ϕ 0 ( 2 π r h ) 4 G N ( 3 ) = S B H ( 0 ) . (251) \Phi[v_{\mathrm{phys}}]=\rho_S\Phi[\hat{q}]=\frac{e^{-2\phi_0}(2\pi r_h)}{4G_N^{(3)}}=S_{\mathrm{BH}}^{(0)}. \tag*{(251)} Φ [ v phys ] = ρ S Φ [ q ^ ] = 4 G N ( 3 ) e − 2 ϕ 0 ( 2 π r h ) = S BH ( 0 ) . (251) In other words
S B H ( 0 ) = ∫ v p h y s i n i d A S = e − 2 ϕ 0 4 G N ( 3 ) ∫ q i ∣ q ∣ n i d A S . (252) S_{\mathrm{BH}}^{(0)}=\int v_{\mathrm{phys}}^i n_i\,dA_S=\frac{e^{-2\phi_0}}{4G_N^{(3)}}\int\frac{q^i}{|q|}n_i\,dA_S. \tag*{(252)} S BH ( 0 ) = ∫ v phys i n i d A S = 4 G N ( 3 ) e − 2 ϕ 0 ∫ ∣ q ∣ q i n i d A S . (252) On the other hand, it remains to verify that v p h y s i v_{\mathrm{phys}}^i v phys i
v p h y s r ^ ( r ) = v p h y s r f = ρ S r h r , v p h y s θ ^ ( r ) = r v p h y s θ = ρ S r 2 − r h 2 r . (253) v_{\mathrm{phys}}^{\hat r}(r)=\frac{v_{\mathrm{phys}}^r}{\sqrt{f}}=\rho_S\frac{r_h}{r},\qquad v_{\mathrm{phys}}^{\hat\theta}(r)=rv_{\mathrm{phys}}^\theta=\rho_S\frac{\sqrt{r^2-r_h^2}}{r}. \tag*{(253)} v phys r ^ ( r ) = f v phys r = ρ S r r h , v phys θ ^ ( r ) = r v phys θ = ρ S r r 2 − r h 2 . (253) Therefore, we have
v p h y s i ( r h ) = ρ S n i , (254) v_{\mathrm{phys}}^i(r_h)=\rho_S n^i, \tag*{(254)} v phys i ( r h ) = ρ S n i , (254) on the horizon. It implies
Φ [ v p h y s ] = ρ S Φ [ q ^ ] = 2 π r h ρ S , ∣ v p h y s i ( r h ) ∣ = ρ S (255) \Phi[v_{\mathrm{phys}}]=\rho_S\Phi[\hat{q}]=2\pi r_h\rho_S,\qquad\left|v_{\mathrm{phys}}^i(r_h)\right|=\rho_S \tag*{(255)} Φ [ v phys ] = ρ S Φ [ q ^ ] = 2 π r h ρ S , v phys i ( r h ) = ρ S (255) The flow is normal to the bottleneck and saturates the capacity there. For any admissible flow w i w^i w i
∇ i w i = 0 , ∣ w ∣ ≤ ρ S . (256) \nabla_i w^i=0,\qquad|w|\leq\rho_S. \tag*{(256)} ∇ i w i = 0 , ∣ w ∣ ≤ ρ S . (256) Its flux is bounded by the horizon cut:
Φ [ w ] = ∫ Γ H w ⋅ n d A S ≤ ∫ Γ H ∣ w ∣ d A S ≤ ∫ Γ H ρ S d A S = 2 π r h ρ S . (257) \Phi[w] = \int_{\Gamma_H} w \cdot n\,dA_S \le\int_{\Gamma_H} |w|dA_S \le\int_{\Gamma_H} \rho_S dA_S = 2\pi r_h\rho_S. \tag*{(257)} Φ [ w ] = ∫ Γ H w ⋅ n d A S ≤ ∫ Γ H ∣ w ∣ d A S ≤ ∫ Γ H ρ S d A S = 2 π r h ρ S . (257) Therefore, v p h y s i v_{\mathrm{phys}}^i v phys i
Φ [ w ] ≤ 2 π r h ρ S = Φ [ v p h y s ] , ∣ w ∣ ≤ ∣ v p h y s i ( r h ) ∣ . (258) \Phi[w] \le2\pi r_h\rho_S = \Phi[v_{\mathrm{phys}}], \qquad|w| \le|v_{\mathrm{phys}}^i(r_h)|. \tag*{(258)} Φ [ w ] ≤ 2 π r h ρ S = Φ [ v phys ] , ∣ w ∣ ≤ ∣ v phys i ( r h ) ∣. (258) The horizon is also a weighted minimal cut. For any homologous cut m m m
∫ m ρ S d A S ≥ ∫ m v p h y s i n i d A S = 2 π r h ρ S = ∫ Γ H ρ S d A S . (259) \int_m \rho_S dA_S \ge\int_m v_{\mathrm{phys}}^i n_i dA_S = 2\pi r_h\rho_S = \int_{\Gamma_H} \rho_S dA_S. \tag*{(259)} ∫ m ρ S d A S ≥ ∫ m v phys i n i d A S = 2 π r h ρ S = ∫ Γ H ρ S d A S . (259) This argument does not restrict competing cuts to circles. The entropy problem here concerns the complete boundary of a BTZ exterior, with the horizon as the other terminal. Its flux is the thermal entropy of the BTZ saddle, or equivalently the entropy of either boundary CFT in the pure thermofield-double description of two-sided BTZ [29 ] . Note that a radial cutoff regulates the string endpoints, and the smooth horizon is a crossing surface, not a physical endpoint of the string.
Comparison with the original string charge density/bit threads correspondence
We can now compare the present construction with the original string charge density/bit threads correspondence,
v i = C g e o m q i , C g e o m > 0. (260) v^i = C_{\mathrm{geom}}q^i, \qquad C_{\mathrm{geom}} > 0. \tag*{(260)} v i = C geom q i , C geom > 0. (260) where q i q^i q i
∣ q ∣ = N 2 π r h . (261) |q| = \frac{\mathcal{N}}{2\pi r_h}. \tag*{(261)} ∣ q ∣ = 2 π r h N . (261) The corresponding unit-normalized current is
q ^ i = q i ∣ q ∣ . (262) \hat{q}^i = \frac{q^i}{|q|}. \tag*{(262)} q ^ i = ∣ q ∣ q i . (262) Identifying the flow v i v^i v i
v i ≡ v p h y s i , (263) v^i \equiv v_{\mathrm{phys}}^i, \tag*{(263)} v i ≡ v phys i , (263) we have
v p h y s i = ρ S q ^ i = ρ S q i ∣ q ∣ . (264) v_{\mathrm{phys}}^i = \rho_S\hat{q}^i = \rho_S\frac{q^i}{|q|}. \tag*{(264)} v phys i = ρ S q ^ i = ρ S ∣ q ∣ q i . (264) The same flow also can be written as
v p h y s i = ( ρ S ∣ q ∣ ) q i . (265) v_{\mathrm{phys}}^{i}=\left(\frac{\rho_{S}}{\lvert q\rvert}\right)q^{i}. \tag*{(265)} v phys i = ( ∣ q ∣ ρ S ) q i . (265) Comparing this expression with v i = C g e o m q i v^{i}=C_{\mathrm{geom}}q^{i} v i = C geom q i
C g e o m = ρ S ∣ q ∣ = 2 π r h ρ S N = S B H ( 0 ) N . (266) C_{\mathrm{geom}}=\frac{\rho_{S}}{\lvert q\rvert}=\frac{2\pi r_{h}\rho_{S}}{\mathcal{N}}=\frac{S_{\mathrm{BH}}^{(0)}}{\mathcal{N}}. \tag*{(266)} C geom = ∣ q ∣ ρ S = N 2 π r h ρ S = N S BH ( 0 ) . (266) Equivalently,
ρ S q ^ i = C g e o m q i . (267) \rho_{S}\hat{q}^{i}=C_{\mathrm{geom}}q^{i}. \tag*{(267)} ρ S q ^ i = C geom q i . (267) The two expressions therefore do not represent different correspondences. Rather, they are two normalizations of the same entropy-flow field. The form
v p h y s i = ρ S q ^ i , (268) v_{\mathrm{phys}}^{i}=\rho_{S}\hat{q}^{i}, \tag*{(268)} v phys i = ρ S q ^ i , (268) separates the direction selected by the stationary F1 dynamics from the entropy capacity ρ S \rho_{S} ρ S
d S max = ρ S d L S , ρ S = e − 2 ϕ 0 4 G N ( 3 ) , (269) dS_{\max}=\rho_{S}dL_{S},\qquad\rho_{S}=\frac{e^{-2\phi_{0}}}{4G_{N}^{(3)}}, \tag*{(269)} d S m a x = ρ S d L S , ρ S = 4 G N ( 3 ) e − 2 ϕ 0 , (269) so that ρ S \rho_{S} ρ S S max S_{\max} S m a x L S L_{S} L S q ^ i \hat{q}^{i} q ^ i
On the other hand, the form
v p h y s i = C g e o m q i , (270) v_{\mathrm{phys}}^{i}=C_{\mathrm{geom}}q^{i}, \tag*{(270)} v phys i = C geom q i , (270) makes explicit how the unnormalized string-charge current is converted into an entropy flow. If every string carries one unit of Kalb–Ramond charge, then
C g e o m = S B H ( 0 ) N , (271) C_{\mathrm{geom}}=\frac{S_{\mathrm{BH}}^{(0)}}{\mathcal{N}}, \tag*{(271)} C geom = N S BH ( 0 ) , (271) can be interpreted as the entropy capacity allocated per unit of string charge. Under the assumptions of unit charge, common orientation, and uniform coarse graining, this is also the entropy capacity assigned to each string. More generally, C g e o m C_{\mathrm{geom}} C geom
This relation admits a direct coarse-grained geometric interpretation. For N \mathcal{N} N
A H ( S ) = 2 π r h , (272) A_{H}^{(S)}=2\pi r_{h}, \tag*{(272)} A H ( S ) = 2 π r h , (272) and the portion assigned to each string is
Δ L H = A H ( S ) N = 2 π r h N , (273) \Delta L_H = \frac{A_H^{(S)}}{\mathcal{N}} = \frac{2\pi r_h}{\mathcal{N}}, \tag*{(273)} Δ L H = N A H ( S ) = N 2 π r h , (273) Since the entropy capacity per unit string-frame length is ρ S \rho_S ρ S
C g e o m = ρ S Δ L H = ρ S 2 π r h N = S B H ( 0 ) N . (274) C_{\mathrm{geom}} = \rho_S \Delta L_H = \rho_S \frac{2\pi r_h}{\mathcal{N}} = \frac{S_{\mathrm{BH}}^{(0)}}{\mathcal{N}}. \tag*{(274)} C geom = ρ S Δ L H = ρ S N 2 π r h = N S BH ( 0 ) . (274) The relation between the two normalizations may therefore be summarized as
ρ S ⏟ entropy capacity per unit string-frame length × 2 π r h N ⏟ horizon length assigned to each string = C g e o m ⏟ entropy capacity per unit string charge . (275) \underbrace{\rho_S}_{\text{entropy capacity per unit string-frame length}
}
\times
\underbrace{\frac{2\pi r_h}{\mathcal{N}}}_{\text{horizon length assigned to each string}}
=
\underbrace{C_{\mathrm{geom}}}_{\text{entropy capacity per unit string charge}}. \tag*{(275)} entropy capacity per unit string-frame length ρ S × horizon length assigned to each string N 2 π r h = entropy capacity per unit string charge C geom . (275) In other words, C g e o m C_{\mathrm{geom}} C geom
As a consistency check, the total flux of the resulting entropy flow is
Φ [ v p h y s ] = C g e o m Φ [ q ] = ρ S ∣ q ∣ N = 2 π r h ρ S = S B H ( 0 ) . (276) \Phi[v_{\mathrm{phys}}] = C_{\mathrm{geom}}\Phi[q] = \frac{\rho_S}{\lvert q\rvert}\mathcal{N} = 2\pi r_h\rho_S = S_{\mathrm{BH}}^{(0)}. \tag*{(276)} Φ [ v phys ] = C geom Φ [ q ] = ∣ q ∣ ρ S N = 2 π r h ρ S = S BH ( 0 ) . (276) The central result is therefore that, for the stationary BTZ string ensemble, C g e o m C_{\mathrm{geom}} C geom ρ S \rho_S ρ S 1 / ∣ q ∣ 1/\lvert q\rvert 1/ ∣ q ∣
C g e o m = ρ S ∣ q ∣ . (277) C_{\mathrm{geom}} = \frac{\rho_S}{\lvert q\rvert}. \tag*{(277)} C geom = ∣ q ∣ ρ S . (277) The global relation C g e o m N = S B H ( 0 ) C_{\mathrm{geom}}\mathcal{N} = S_{\mathrm{BH}}^{(0)} C geom N = S BH ( 0 )
One more example: BTZ with mixed flux
We now extend the construction of § 5 to BTZ supported by both NS–NS and Ramond–Ramond (RR) three-form fluxes. We keep the string-frame geometry and the dilaton fixed and vary the fraction of NS–NS flux. This separates the effect of the background on the F1 embedding from the geometric entropy capacity.
The mixed-flux action and the BTZ background
The mixed-flux solution is a type IIB background. The fields include the string-frame metric, the dilaton, the Kalb–Ramond field B B B C ( 2 ) C^{(2)} C ( 2 )
S I I B ( 0 ) = 1 16 π G N ( 10 ) ∫ d 10 x − g 10 [ e − 2 ϕ ( R 10 + 4 ( ∇ ϕ ) 2 − 1 12 H M N P H M N P ) − 1 12 F M N P ( 3 ) F ( 3 ) M N P ] , (278) S_{\mathrm{IIB}}^{(0)} = \frac{1}{16\pi G_N^{(10)}} \int d^{10}x \sqrt{-g_{10}} \left[e^{-2\phi}\left(R_{10} + 4(\nabla\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP}\right) - \frac{1}{12}F_{MNP}^{(3)}F^{(3)MNP}\right], \tag*{(278)} S IIB ( 0 ) = 16 π G N ( 10 ) 1 ∫ d 10 x − g 10 [ e − 2 ϕ ( R 10 + 4 ( ∇ ϕ ) 2 − 12 1 H M N P H M N P ) − 12 1 F M N P ( 3 ) F ( 3 ) M N P ] , (278) where H = d B H = dB H = d B F ( 3 ) = d C ( 2 ) F^{(3)} = dC^{(2)} F ( 3 ) = d C ( 2 ) A d S 3 × S 3 × T 4 \mathrm{AdS}_3 \times S^3 \times T^4 AdS 3 × S 3 × T 4 [30 , 31 ]
d S 10 2 = d s B T Z 2 + ℓ 2 d Ω 3 2 + d s T 4 2 , ϕ = ϕ 0 , (279) dS_{10}^{2} = ds_{\mathrm{BTZ}}^{2} + \ell^{2}d\Omega_{3}^{2} + ds_{T^4}^{2}, \qquad\phi= \phi_{0}, \tag*{(279)} d S 10 2 = d s BTZ 2 + ℓ 2 d Ω 3 2 + d s T 4 2 , ϕ = ϕ 0 , (279) where
d s B T Z 2 = − f ( r ) d t 2 + d r 2 f ( r ) + r 2 d θ 2 , f ( r ) = r 2 − r h 2 ℓ 2 , θ ∼ θ + 2 π . (280) ds_{\mathrm{BTZ}}^{2} = -f(r)\,dt^{2} + \frac{dr^{2}}{f(r)} + r^{2}d\theta^{2}, \qquad f(r) = \frac{r^{2}-r_{h}^{2}}{\ell^{2}}, \qquad\theta\sim\theta+ 2\pi. \tag*{(280)} d s BTZ 2 = − f ( r ) d t 2 + f ( r ) d r 2 + r 2 d θ 2 , f ( r ) = ℓ 2 r 2 − r h 2 , θ ∼ θ + 2 π . (280) Let ε B T Z \varepsilon_{\mathrm{BTZ}} ε BTZ ε S 3 \varepsilon_{S^3} ε S 3 ℓ \ell ℓ ε B T Z = r d t ∧ d r ∧ d θ \varepsilon_{\mathrm{BTZ}} = rdt \wedge dr \wedge d\theta ε BTZ = r d t ∧ d r ∧ d θ
H = 2 η ℓ ( ε B T Z + ε S 3 ) , H = \frac{2\eta}{\ell}\left(\varepsilon_{\mathrm{BTZ}}+\varepsilon_{S^3}\right), H = ℓ 2 η ( ε BTZ + ε S 3 ) , F ( 3 ) = 2 e − ϕ 0 1 − η 2 ℓ ( ε B T Z + ε S 3 ) , 0 ≤ η ≤ 1. (281) F^{(3)} = \frac{2e^{-\phi_0}\sqrt{1-\eta^{2}}}{\ell}\left(\varepsilon_{\mathrm{BTZ}}+\varepsilon_{S^3}\right), \qquad0 \le\eta\le1. \tag*{(281)} F ( 3 ) = ℓ 2 e − ϕ 0 1 − η 2 ( ε BTZ + ε S 3 ) , 0 ≤ η ≤ 1. (281) The parameter η \eta η q i q^{i} q i − 3 ! -3! − 3 ! + 3 ! +3! + 3 !
H 2 = ( F ( 3 ) ) 2 = H ⋅ F ( 3 ) = 0 , H ∧ F ( 3 ) = 0. (282) H^{2} = \left(F^{(3)}\right)^{2} = H \cdot F^{(3)} = 0, \qquad H \wedge F^{(3)} = 0. \tag*{(282)} H 2 = ( F ( 3 ) ) 2 = H ⋅ F ( 3 ) = 0 , H ∧ F ( 3 ) = 0. (282) The last two identities ensure that the axion and five-form equations are satisfied when those fields are set to zero. The Bianchi identities and the three-form equations hold because the physical volume forms are closed and covariantly constant.
On this solution, the dilaton is constant and both three-form scalar contractions vanish. The metric equation therefore reduces to
R M N = 1 4 H M P Q H N P Q + e 2 ϕ 0 4 F M P Q ( 3 ) F N ( 3 ) P Q . (283) R_{MN} = \frac{1}{4}H_{MPQ}H_{N}^{\;\;PQ} + \frac{e^{2\phi_0}}{4}F^{(3)}_{MPQ}F_{N}^{(3)\,PQ}. \tag*{(283)} R M N = 4 1 H M P Q H N P Q + 4 e 2 ϕ 0 F M P Q ( 3 ) F N ( 3 ) P Q . (283) For BTZ indices a, b this gives
R a b = − 2 ℓ 2 [ η 2 + ( 1 − η 2 ) ] g a b = − 2 ℓ 2 g a b . (284) R_{ab} = -\frac{2}{\ell^{2}}\left[\eta^{2}+\left(1-\eta^{2}\right)\right]g_{ab} = -\frac{2}{\ell^{2}}g_{ab}. \tag*{(284)} R ab = − ℓ 2 2 [ η 2 + ( 1 − η 2 ) ] g ab = − ℓ 2 2 g ab . (284) The sphere equation has the opposite curvature sign, and the torus is flat. The dilaton equation is satisfied since R 10 = H 2 = 0 R_{10} = H^{2} = 0 R 10 = H 2 = 0 η \eta η
Integrating over the fixed internal geometry gives the following restricted three-dimensional action:
S 3 D ( 0 ) = 1 16 π G N ( 3 ) ∫ d 3 x − g e − 2 ϕ { R + 4 ( ∇ ϕ ) 2 − 1 12 H a b c H a b c + 6 − 2 η 2 ℓ 2 − 1 12 F a b c ( 3 ) F ( 3 ) a b c − 2 e − 2 ϕ 0 ( 1 − η 2 ) ℓ 2 } , 1 G N ( 3 ) = V i n t ( S ) G N ( 10 ) , (285) S_{3D}^{(0)} = \frac{1}{16\pi G_{N}^{(3)}}\int d^{3}x\sqrt{-g}\,e^{-2\phi}\left\{R+4(\nabla\phi)^{2}-\frac{1}{12}H_{abc}H^{abc}+\frac{6-2\eta^{2}}{\ell^{2}}-\frac{1}{12}F_{abc}^{(3)}F^{(3)\,abc}-\frac{2e^{-2\phi_0}\left(1-\eta^{2}\right)}{\ell^{2}}\right\}, \qquad\frac{1}{G_{N}^{(3)}}=\frac{V_{\mathrm{int}}^{(S)}}{G_{N}^{(10)}}, \tag*{(285)} S 3 D ( 0 ) = 16 π G N ( 3 ) 1 ∫ d 3 x − g e − 2 ϕ { R + 4 ( ∇ ϕ ) 2 − 12 1 H ab c H ab c + ℓ 2 6 − 2 η 2 − 12 1 F ab c ( 3 ) F ( 3 ) ab c − ℓ 2 2 e − 2 ϕ 0 ( 1 − η 2 ) } , G N ( 3 ) 1 = G N ( 10 ) V int ( S ) , (285) where V i n t ( S ) ≡ V S 3 V T 4 V_{\mathrm{int}}^{(S)} \equiv V_{S^3}V_{T^4} V int ( S ) ≡ V S 3 V T 4 6 / ℓ 2 6/\ell^{2} 6/ ℓ 2 ϕ ( x ) \phi(x) ϕ ( x ) ϕ 0 \phi_0 ϕ 0 η = 1 \eta= 1 η = 1 4 / ℓ 2 4/\ell^2 4/ ℓ 2
The BTZ metric together with ϕ = ϕ 0 \phi= \phi_0 ϕ = ϕ 0
B ∣ B T Z = − b η ( r ) d t ∧ d θ , b η ( r ) = η ( r 2 − r h 2 ) ℓ , C ( 2 ) ∣ B T Z = − c η ( r ) d t ∧ d θ , c η ( r ) = e − ϕ 0 1 − η 2 ( r 2 − r h 2 ) ℓ . (286) \begin{aligned}
B\big|_{\mathrm{BTZ}} &= -b_\eta(r)\,dt \wedge d\theta, \qquad b_\eta(r) = \frac{\eta\left(r^2-r_h^2\right)}{\ell},\\
C^{(2)}\big|_{\mathrm{BTZ}} &= -c_\eta(r)\,dt \wedge d\theta, \qquad c_\eta(r) = \frac{e^{-\phi_0}\sqrt{1-\eta^2}\left(r^2-r_h^2\right)}{\ell}. \tag*{(286)}
\end{aligned} B BTZ C ( 2 ) BTZ = − b η ( r ) d t ∧ d θ , b η ( r ) = ℓ η ( r 2 − r h 2 ) , = − c η ( r ) d t ∧ d θ , c η ( r ) = ℓ e − ϕ 0 1 − η 2 ( r 2 − r h 2 ) . (286) Both potentials vanish at the horizon and reproduce the external parts of Eq. (281) . Their internal components produce the sphere fluxes and have zero pullback to the worldsheets with fixed internal coordinates.
For clarity, the scalar contractions in three dimensions are nonzero:
H a b c H a b c = − 24 η 2 ℓ 2 , ( F e x t ( 3 ) ) 2 = − 24 e − 2 ϕ 0 ( 1 − η 2 ) ℓ 2 . (287) H_{abc}H^{abc}=-\frac{24\eta^2}{\ell^2}, \qquad\left(F_{\mathrm{ext}}^{(3)}\right)^2=-\frac{24e^{-2\phi_0}\left(1-\eta^2\right)}{\ell^2}. \tag*{(287)} H ab c H ab c = − ℓ 2 24 η 2 , ( F ext ( 3 ) ) 2 = − ℓ 2 24 e − 2 ϕ 0 ( 1 − η 2 ) . (287) Writing U η = ( 6 − 2 η 2 ) / ℓ 2 U_\eta=(6-2\eta^2)/\ell^2 U η = ( 6 − 2 η 2 ) / ℓ 2 ϕ = ϕ 0 \phi=\phi_0 ϕ = ϕ 0
R − 1 12 H e x t 2 + U η = − 6 + 2 η 2 + 6 − 2 η 2 ℓ 2 = 0. (288) R-\frac{1}{12}H_{\mathrm{ext}}^2+U_\eta=\frac{-6+2\eta^2+6-2\eta^2}{\ell^2}=0. \tag*{(288)} R − 12 1 H ext 2 + U η = ℓ 2 − 6 + 2 η 2 + 6 − 2 η 2 = 0. (288) If V R R = 2 e − 2 ϕ 0 ( 1 − η 2 ) / ℓ 2 V_{\mathrm{RR}}=2e^{-2\phi_0}(1-\eta^2)/\ell^2 V RR = 2 e − 2 ϕ 0 ( 1 − η 2 ) / ℓ 2 (285) , the constant-dilaton metric equation, after using the dilaton equation, is
R a b = 1 4 H a c d H b c d + e 2 ϕ 0 4 F a c d ( 3 ) F b ( 3 ) c d − e 2 ϕ 0 24 g a b ( F e x t ( 3 ) ) 2 − e 2 ϕ 0 2 V R R g a b = − 2 ℓ 2 g a b . (289) R_{ab}=\frac{1}{4}H_{acd}H_b{}^{cd}+\frac{e^{2\phi_0}}{4}F_{acd}^{(3)}F_b{}^{(3)cd}-\frac{e^{2\phi_0}}{24}g_{ab}\left(F_{\mathrm{ext}}^{(3)}\right)^2-\frac{e^{2\phi_0}}{2}V_{\mathrm{RR}}g_{ab}=-\frac{2}{\ell^2}g_{ab}. \tag*{(289)} R ab = 4 1 H a c d H b c d + 4 e 2 ϕ 0 F a c d ( 3 ) F b ( 3 ) c d − 24 e 2 ϕ 0 g ab ( F ext ( 3 ) ) 2 − 2 e 2 ϕ 0 V RR g ab = − ℓ 2 2 g ab . (289) The two RR trace terms cancel on the background. This verifies the BTZ solution directly in the restricted three-dimensional action. The continuous family is understood classically. Its flux quantization restricts the allowed values in a quantum compactification.
The F1 source action and its regular stationary solution
We add fundamental string sources to the ten-dimensional type IIB action (278) :
S t o t ( 0 ) = S I I B ( 0 ) + S σ , S σ = ∑ k = 1 N S F 1 [ X k ] . (290) S_{\mathrm{tot}}^{(0)}=S_{\mathrm{IIB}}^{(0)}+S_\sigma,\qquad S_\sigma=\sum_{k=1}^{\mathcal{N}}S_{\mathrm{F1}}[X_k]. \tag*{(290)} S tot ( 0 ) = S IIB ( 0 ) + S σ , S σ = k = 1 ∑ N S F1 [ X k ] . (290) with
S F 1 = − 1 4 π α ′ ∫ d 2 σ ( − γ γ m n ∂ m X a ∂ n X b g a b + ϵ m n ∂ m X a ∂ n X b B a b ) , ϵ τ σ = + 1 , (291) S_{\mathrm{F1}}=-\frac{1}{4\pi\alpha'}\int d^2\sigma\left(\sqrt{-\gamma}\,\gamma^{mn}\partial_mX^a\partial_nX^b g_{ab}+\epsilon^{mn}\partial_mX^a\partial_nX^bB_{ab}\right),\qquad\epsilon^{\tau\sigma}=+1, \tag*{(291)} S F1 = − 4 π α ′ 1 ∫ d 2 σ ( − γ γ mn ∂ m X a ∂ n X b g ab + ϵ mn ∂ m X a ∂ n X b B ab ) , ϵ τ σ = + 1 , (291) where the worldsheet fermions are set to zero. RR fields enter the full superstring action through fermionic couplings, so the F1 action has no bosonic electric coupling to C ( 2 ) C^{(2)} C ( 2 ) [31 ] . In other words, the RR field supports the background, while B B B (291) . Moreover, we also use the probe limit in our calculations. The embeddings solve the F1 equations in this background, and the response of the background to S σ S_{\sigma} S σ
Choose static gauge separately on each exterior worldsheet,
X t = τ , X r = σ = r , X θ = θ η ( r ) , X i n t e r n a l = constant . (292) X^{t}=\tau,\qquad X^{r}=\sigma=r,\qquad X^{\theta}=\theta_{\eta}(r),\qquad X^{\mathrm{internal}}=\text{constant}. \tag*{(292)} X t = τ , X r = σ = r , X θ = θ η ( r ) , X internal = constant . (292) Writing the induced worldsheet metric as γ m n i n d = g a b ∂ m X a ∂ n X b \gamma^{\mathrm{ind}}_{mn}=g_{ab}\partial_m X^a\partial_n X^b γ mn ind = g ab ∂ m X a ∂ n X b
γ τ τ i n d = − f , γ σ σ i n d = f − 1 + r 2 θ η ′ 2 , γ τ σ i n d = 0 , − det γ i n d = 1 + f r 2 θ η ′ 2 . (293) \gamma^{\mathrm{ind}}_{\tau\tau}=-f,\qquad\gamma^{\mathrm{ind}}_{\sigma\sigma}=f^{-1}+r^2\theta_{\eta}^{\prime2},\qquad\gamma^{\mathrm{ind}}_{\tau\sigma}=0,\qquad-\det\gamma^{\mathrm{ind}}=1+fr^2\theta_{\eta}^{\prime2}. \tag*{(293)} γ τ τ ind = − f , γ σ σ ind = f − 1 + r 2 θ η ′2 , γ τ σ ind = 0 , − det γ ind = 1 + f r 2 θ η ′2 . (293) Eliminating the auxiliary metric in Eq. (291) gives
S F 1 = − 1 2 π α ′ ∫ d t d r ( 1 + f r 2 θ η ′ 2 − b η θ η ′ ) . (294) S_{\mathrm{F1}}=-\frac{1}{2\pi\alpha'}\int dt dr\left(\sqrt{1+fr^2\theta_{\eta}^{\prime2}}-b_{\eta}\theta_{\eta}'\right). \tag*{(294)} S F1 = − 2 π α ′ 1 ∫ d t d r ( 1 + f r 2 θ η ′2 − b η θ η ′ ) . (294) The sign of the second term follows from B = − b η d t ∧ d θ B=-b_{\eta}dt\wedge d\theta B = − b η d t ∧ d θ θ η \theta_{\eta} θ η
f r 2 θ η ′ 1 + f r 2 θ η ′ 2 − b η = k θ . (295) \frac{fr^2\theta_{\eta}'}{\sqrt{1+fr^2\theta_{\eta}^{\prime2}}}-b_{\eta}=k_{\theta}. \tag*{(295)} 1 + f r 2 θ η ′2 f r 2 θ η ′ − b η = k θ . (295) Squaring this equation gives
θ η ′ 2 = ( b η + k θ ) 2 f r 2 [ f r 2 − ( b η + k θ ) 2 ] . (296) \theta_{\eta}^{\prime2}=\frac{(b_{\eta}+k_{\theta})^2}{fr^2\left[fr^2-(b_{\eta}+k_{\theta})^2\right]}. \tag*{(296)} θ η ′2 = f r 2 [ f r 2 − ( b η + k θ ) 2 ] ( b η + k θ ) 2 . (296) A branch reaching the horizon with finite θ η ′ \theta_{\eta}' θ η ′ k θ = 0 k_{\theta}=0 k θ = 0 f ( r h ) = b η ( r h ) = 0 f(r_h)=b_{\eta}(r_h)=0 f ( r h ) = b η ( r h ) = 0
D η 2 ( r ) ≡ r 2 − b η 2 f = ( 1 − η 2 ) r 2 + η 2 r h 2 . (297) D_{\eta}^{2}(r)\equiv r^2-\frac{b_{\eta}^{2}}{f}=(1-\eta^2)r^2+\eta^2r_h^2. \tag*{(297)} D η 2 ( r ) ≡ r 2 − f b η 2 = ( 1 − η 2 ) r 2 + η 2 r h 2 . (297) and solving Eq. (295) on the regular branch gives
θ η ′ = b η f r D η = η ℓ r D η , 1 + f r 2 θ η ′ 2 = r D η . (298) \theta_{\eta}'=\frac{b_{\eta}}{frD_{\eta}}=\frac{\eta\ell}{rD_{\eta}},\qquad\sqrt{1+fr^2\theta_{\eta}^{\prime2}}=\frac{r}{D_{\eta}}. \tag*{(298)} θ η ′ = f r D η b η = r D η η ℓ , 1 + f r 2 θ η ′2 = D η r . (298) In particular, D η ( r h ) = r h D_{\eta}(r_h)=r_h D η ( r h ) = r h θ η ′ ( r h ) = η ℓ / r h 2 \theta_{\eta}'(r_h)=\eta\ell/r_h^2 θ η ′ ( r h ) = η ℓ / r h 2 − det γ i n d ∣ r h = 1 \left.\sqrt{-\det\gamma^{\mathrm{ind}}}\right|_{r_h}=1 − det γ ind r h = 1 v = t + ∫ d r / f v=t+\int dr/f v = t + ∫ d r / f X v = τ X^v=\tau X v = τ X r = σ X^r=\sigma X r = σ
d s w s 2 = − f d τ 2 + 2 d τ d r + r 2 θ η ′ 2 d r 2 , det γ i n d = − ( 1 + f r 2 θ η ′ 2 ) . (299) ds_{\mathrm{ws}}^2=-f d\tau^2+2d\tau dr+r^2\theta_{\eta}^{\prime2}dr^2,\qquad\det\gamma^{\mathrm{ind}}=-\left(1+fr^2\theta_{\eta}^{\prime2}\right). \tag*{(299)} d s ws 2 = − f d τ 2 + 2 d τ d r + r 2 θ η ′2 d r 2 , det γ ind = − ( 1 + f r 2 θ η ′2 ) . (299) which is nondegenerate at the future horizon. The pullback of the potential is regular as well. These are on-shell probe F1 sources, with the remaining embedding equations following from the worldsheet reparametrization identities once the transverse equation and auxiliary-metric equation hold.
Integrating with θ η ( r h ) = θ h \theta_\eta(r_h)=\theta_h θ η ( r h ) = θ h
θ η ( r ) = θ h + ℓ r h log ( 1 + η ) r D η ( r ) + η r h . (300) \theta_\eta(r)=\theta_h+\frac{\ell}{r_h}\log\frac{(1+\eta)r}{D_\eta(r)+\eta r_h}. \tag*{(300)} θ η ( r ) = θ h + r h ℓ log D η ( r ) + η r h ( 1 + η ) r . (300) The limiting profiles η = 0 , 1 \eta=0,1 η = 0 , 1
θ 0 ( r ) = θ h , θ 1 ( r ) = θ h + ℓ r h log r r h . (301) \theta_0(r)=\theta_h,\qquad\theta_1(r)=\theta_h+\frac{\ell}{r_h}\log\frac{r}{r_h}. \tag*{(301)} θ 0 ( r ) = θ h , θ 1 ( r ) = θ h + r h ℓ log r h r . (301) Thus a radial F1 is an on-shell probe in the pure RR background. It still carries Kalb–Ramond charge, defined by varying its source coupling to B B B H H H η > 0 \eta>0 η > 0 b η ′ = 2 η r / ℓ = 0 b'_\eta=2\eta r/\ell=0 b η ′ = 2 η r / ℓ = 0 η \eta η Figure 5 .
Figure 5. Red solid curves show stationary F1 strings for η = 0 , 0.5 , 0.9 \eta=0,0.5,0.9 η = 0 , 0.5 , 0.9 1 1 1
For 0 ≤ η < 1 0\leq\eta<1 0 ≤ η < 1
θ η ( ∞ ) − θ h = ℓ r h arctanh η . (302) \theta_\eta(\infty)-\theta_h=\frac{\ell}{r_h}\operatorname{arctanh}\eta. \tag*{(302)} θ η ( ∞ ) − θ h = r h ℓ arctanh η . (302) It diverges as η → 1 \eta\to1 η → 1
1 + f r 2 θ η ′ 2 − b η θ η ′ = D η r . (303) \sqrt{1+fr^2\theta_\eta^{\prime2}-b_\eta\theta_\eta'}=\frac{D_\eta}{r}. \tag*{(303)} 1 + f r 2 θ η ′2 − b η θ η ′ = r D η . (303) The physical string current, spatial projection and black hole entropy
Now, we calculate the physical string current and its spatial projection based on this solution. The source variation and the corresponding current are
j η a b ( x ) = 1 − g ( x ) ∑ k = 1 N ∫ d r d σ ϵ m n ∂ m X k a ∂ n X k b δ ( 3 ) ( x − X k ) . (304) j_{\eta}^{ab}(x)=\frac{1}{\sqrt{-g(x)}}\sum_{k=1}^{\mathcal{N}}\int drd\sigma\epsilon^{mn}\partial_mX_k^a\partial_nX_k^b\delta^{(3)}(x-X_k). \tag*{(304)} j η ab ( x ) = − g ( x ) 1 k = 1 ∑ N ∫ d r d σ ϵ mn ∂ m X k a ∂ n X k b δ ( 3 ) ( x − X k ) . (304) These are the three-dimensional currents obtained after integrating the ten-dimensional distribution over the internal coordinates. Take θ h , k \theta_{h,k} θ h , k (300) , so the continuum replacement
∑ k = 1 N δ 2 π ( θ − θ η , k ( r ) ) ⟶ N 2 π . (305) \sum_{k=1}^{\mathcal{N}}\delta_{2\pi}(\theta-\theta_{\eta,k}(r))\longrightarrow\frac{\mathcal{N}}{2\pi}. \tag*{(305)} k = 1 ∑ N δ 2 π ( θ − θ η , k ( r )) ⟶ 2 π N . (305) Using − g = r \sqrt{-g}=r − g = r
j η 0 r = N 2 π r , j η 0 θ = N 2 π r θ η ′ , j η r θ = 0. (306) j_{\eta}^{0r}=\frac{\mathcal{N}}{2\pi r},\qquad j_{\eta}^{0\theta}=\frac{\mathcal{N}}{2\pi r}\theta'_{\eta},\qquad j_{\eta}^{r\theta}=0. \tag*{(306)} j η 0 r = 2 π r N , j η 0 θ = 2 π r N θ η ′ , j η r θ = 0. (306) The angular component is therefore part of the physical current, just as in the pure NS–NS calculation. On the static slice,
h i j d x i d x j = d r 2 f + r 2 d θ 2 , h = r f , N A D M = f . (307) h_{ij}dx^idx^j=\frac{dr^2}{f}+r^2d\theta^2,\qquad\sqrt{h}=\frac{r}{\sqrt{f}},\qquad N_{\mathrm{ADM}}=\sqrt{f}. \tag*{(307)} h ij d x i d x j = f d r 2 + r 2 d θ 2 , h = f r , N ADM = f . (307) If n μ n_\mu n μ q η μ = − n ν j η ν μ q^\mu_\eta=-n_\nu j^{\nu\mu}_\eta q η μ = − n ν j η ν μ
q η r = N A D M j η 0 r = N f 2 π r , q η θ = N A D M j η 0 θ = N η ℓ f 2 π r 2 D η = θ η ′ q η r . (308) q^r_\eta=N_{\mathrm{ADM}}j_{\eta}^{0r}=\frac{\mathcal{N}\sqrt{f}}{2\pi r},\qquad q^\theta_\eta=N_{\mathrm{ADM}}j_{\eta}^{0\theta}=\frac{\mathcal{N}\eta\ell\sqrt{f}}{2\pi r^2D_\eta}=\theta'_\eta q^r_\eta. \tag*{(308)} q η r = N ADM j η 0 r = 2 π r N f , q η θ = N ADM j η 0 θ = 2 π r 2 D η N η ℓ f = θ η ′ q η r . (308) This current is conserved in the exterior, away from any string endpoints:
∇ i q η i = 1 h [ ∂ r ( h q η r ) + ∂ θ ( h q η θ ) ] = f r ∂ r ( N 2 π ) = 0. (309) \nabla_iq^i_\eta=\frac{1}{\sqrt{h}}\left[\partial_r\left(\sqrt{h}q^r_\eta\right)+\partial_\theta\left(\sqrt{h}q^\theta_\eta\right)\right]=\frac{\sqrt{f}}{r}\partial_r\left(\frac{\mathcal{N}}{2\pi}\right)=0. \tag*{(309)} ∇ i q η i = h 1 [ ∂ r ( h q η r ) + ∂ θ ( h q η θ ) ] = r f ∂ r ( 2 π N ) = 0. (309) For a circle C r C_r C r r r r n i = f ( ∂ r ) i n^i=\sqrt{f}(\partial_r)^i n i = f ( ∂ r ) i d A S = r d θ dA_S=rd\theta d A S = r d θ
Φ [ q η ] = ∫ 0 2 π q η r f r d θ = N . (310) \Phi[q_\eta]=\int_0^{2\pi}\frac{q^r_\eta}{\sqrt{f}}rd\theta=\mathcal{N}. \tag*{(310)} Φ [ q η ] = ∫ 0 2 π f q η r r d θ = N . (310) The full current norm follows directly from the embedding equation:
∣ q η ∣ 2 = ( q η r ) 2 f + r 2 ( q η θ ) 2 = ( N 2 π r ) 2 ( 1 + f r 2 θ η ′ 2 ) = ( N 2 π D η ) 2 . (311) \begin{aligned}
|q_\eta|^2=\frac{(q^r_\eta)^2}{f}+r^2(q^\theta_\eta)^2 \\
&=\left(\frac{\mathcal{N}}{2\pi r}\right)^2\left(1+fr^2\theta_\eta'^2\right)=\left(\frac{\mathcal{N}}{2\pi D_\eta}\right)^2. \tag*{(311)}
\end{aligned} ∣ q η ∣ 2 = f ( q η r ) 2 + r 2 ( q η θ ) 2 = ( 2 π r N ) 2 ( 1 + f r 2 θ η ′2 ) = ( 2 π D η N ) 2 . (311) Moreover, it is worth noting that its horizon value is independent of η \eta η
∣ q η ∣ h = N 2 π r h , ∣ q 0 ∣ = N 2 π r , ∣ q 1 ∣ = N 2 π r h . (312) |q_{\eta}|_{h} = \frac{\mathcal{N}}{2\pi r_{h}}, \qquad|q_{0}| = \frac{\mathcal{N}}{2\pi r}, \qquad|q_{1}| = \frac{\mathcal{N}}{2\pi r_{h}}. \tag*{(312)} ∣ q η ∣ h = 2 π r h N , ∣ q 0 ∣ = 2 π r N , ∣ q 1 ∣ = 2 π r h N . (312) Unlike the pure NS–NS result of § 5 , the pointwise unit normalization of the mixed-flux current fails to preserve conservation. For q ^ η i = q η i / ∣ q η ∣ \hat{q}_{\eta}^{i} = q_{\eta}^{i}/|q_{\eta}| q ^ η i = q η i /∣ q η ∣
∇ i ( ρ S q η i ) = ρ S f r D η ′ = ρ S ( 1 − η 2 ) f D η , (313) \nabla_{i}\left(\rho_{S}q_{\eta}^{i}\right) = \frac{\rho_{S}\sqrt{f}}{r}\mathcal{D}_{\eta}' = \frac{\rho_{S}\left(1-\eta^{2}\right)\sqrt{f}}{\mathcal{D}_{\eta}}, \tag*{(313)} ∇ i ( ρ S q η i ) = r ρ S f D η ′ = D η ρ S ( 1 − η 2 ) f , (313) which is nonzero in the exterior when η < 1 \eta< 1 η < 1
v p h y s , η i = C g e o m q η i . (314) v_{\mathrm{phys},\eta}^{i} = C_{\mathrm{geom}}q_{\eta}^{i}. \tag*{(314)} v phys , η i = C geom q η i . (314) In general, conservation requires q η i ∂ i C g e o m = 0 q_{\eta}^{i}\partial_{i}C_{\mathrm{geom}} = 0 q η i ∂ i C geom = 0 C g e o m = C g e o m ( r ) C_{\mathrm{geom}} = C_{\mathrm{geom}}(r) C geom = C geom ( r ) q η r ∂ r C g e o m = 0 q_{\eta}^{r}\partial_{r}C_{\mathrm{geom}} = 0 q η r ∂ r C geom = 0
C g e o m max = inf r ≥ r h ρ S ∣ q η ( r ) ∣ = 2 π ρ S N inf r ≥ r h D η ( r ) = 2 π r h ρ S N = ρ S ∣ q η ∣ h , (315) \begin{aligned}
C_{\mathrm{geom}}^{\max} &= \inf_{r\ge r_{h}}\frac{\rho_{S}}{|q_{\eta}(r)|} = \frac{2\pi\rho_{S}}{\mathcal{N}}\inf_{r\ge r_{h}}\mathcal{D}_{\eta}(r) \\
&= \frac{2\pi r_{h}\rho_{S}}{\mathcal{N}} = \frac{\rho_{S}}{|q_{\eta}|_{h}}, \tag*{(315)}
\end{aligned} C geom m a x = r ≥ r h inf ∣ q η ( r ) ∣ ρ S = N 2 π ρ S r ≥ r h inf D η ( r ) = N 2 π r h ρ S = ∣ q η ∣ h ρ S , (315) where we use
D η 2 − r h 2 = ( 1 − η 2 ) ( r 2 − r h 2 ) ≥ 0. (316) \mathcal{D}_{\eta}^{2} - r_{h}^{2} = \left(1-\eta^{2}\right)\left(r^{2}-r_{h}^{2}\right) \ge0. \tag*{(316)} D η 2 − r h 2 = ( 1 − η 2 ) ( r 2 − r h 2 ) ≥ 0. (316) Thus the coefficient follows from the local norm constraint and the computed physical current before its total entropy flux is evaluated. We henceforth write this maximal constant simply as C g e o m C_{\mathrm{geom}} C geom
The resulting physical flow is
v p h y s , η = ρ S r h f r ( ∂ r + η ℓ r D η ∂ θ ) = ρ S r h D η q ^ η , (317) v_{\mathrm{phys},\eta} = \rho_{S}\frac{r_{h}\sqrt{f}}{r}\left(\partial_{r}+\frac{\eta^{\ell}}{r\mathcal{D}_{\eta}}\partial_{\theta}\right) = \rho_{S}\frac{r_{h}}{\mathcal{D}_{\eta}}\hat{q}_{\eta}, \tag*{(317)} v phys , η = ρ S r r h f ( ∂ r + r D η η ℓ ∂ θ ) = ρ S D η r h q ^ η , (317) and obeys
∇ i v p h y s , η i = 0 , ∣ v p h y s , η ∣ ρ S = r h D η ≤ 1. (318) \nabla_{i}v_{\mathrm{phys},\eta}^{i}=0,\qquad\frac{|v_{\mathrm{phys},\eta}|}{\rho_{S}}=\frac{r_{h}}{\mathcal{D}_{\eta}}\le1. \tag*{(318)} ∇ i v phys , η i = 0 , ρ S ∣ v phys , η ∣ = D η r h ≤ 1. (318) For η < 1 \eta< 1 η < 1 r > r h r > r_{h} r > r h η = 1 \eta= 1 η = 1 N \mathcal{N} N
We now compute the entropy flux directly, using the same circle normals as in Eq. (310) :
Φ [ v p h y s , η ] = ∫ 0 2 π v p h y s , η r f r d θ = 2 π r h ρ S = e − 2 ϕ 0 ( 2 π r h ) 4 G N ( 3 ) = S B H ( 0 ) . (319) \begin{aligned}
\Phi\left[v_{\mathrm{phys},\eta}\right] = \int_{0}^{2\pi}\frac{v_{\mathrm{phys},\eta}^{r}}{\sqrt{f}}r\,\mathrm{d}\theta= 2\pi r_{h}\rho_{S} \\
&= \frac{e^{-2\phi_0}(2\pi r_h)}{4G_N^{(3)}} = S_{\mathrm{BH}}^{(0)}. \tag*{(319)}
\end{aligned} Φ [ v phys , η ] = ∫ 0 2 π f v phys , η r r d θ = 2 π r h ρ S = 4 G N ( 3 ) e − 2 ϕ 0 ( 2 π r h ) = S BH ( 0 ) . (319) This equals the ten-dimensional entropy e − 2 ϕ 0 ( 2 π r h ) V i n t ( S ) / ( 4 G N ( 10 ) ) e^{-2\phi_0}(2\pi r_h)V_{\mathrm{int}}^{(S)}/(4G_N^{(10)}) e − 2 ϕ 0 ( 2 π r h ) V int ( S ) / ( 4 G N ( 10 ) )
The maximality of this flow follows from the same argument used for pure NS–NS BTZ. The flow is conserved and satisfies the norm bound throughout the exterior,
∇ i v p h y s , η i = 0 , ∣ v p h y s , η ∣ ≤ ρ S . (320) \nabla_i v_{\mathrm{phys},\eta}^{i}=0,\qquad\lvert v_{\mathrm{phys},\eta}\rvert\leq\rho_S. \tag*{(320)} ∇ i v phys , η i = 0 , ∣ v phys , η ∣ ≤ ρ S . (320) Its orthonormal components are
v p h y s , η r ^ = ρ S r h r , v p h y s , η θ ^ = ρ S η r h r 2 − r h 2 r D η . (321) v_{\mathrm{phys},\eta}^{\hat r}=\rho_S\frac{r_h}{r},\qquad v_{\mathrm{phys},\eta}^{\hat\theta}=\frac{\rho_S\eta r_h\sqrt{r^2-r_h^2}}{rD_\eta}. \tag*{(321)} v phys , η r ^ = ρ S r r h , v phys , η θ ^ = r D η ρ S η r h r 2 − r h 2 . (321) Taking the horizon limit shows that the flow is normal to the bifurcation circle and saturates its capacity:
v p h y s , η i ∣ Γ H = ρ S n i . (322) \left.v_{\mathrm{phys},\eta}^{i}\right|_{\Gamma_H}=\rho_S n^i. \tag*{(322)} v phys , η i Γ H = ρ S n i . (322) For any admissible flow w i w^i w i
Φ [ w ] ≤ ∫ Γ H ρ S d A S = 2 π r h ρ S = Φ [ v p h y s , η ] . (323) \Phi[w]\leq\int_{\Gamma_H}\rho_S dA_S=2\pi r_h\rho_S=\Phi[v_{\mathrm{phys},\eta}]. \tag*{(323)} Φ [ w ] ≤ ∫ Γ H ρ S d A S = 2 π r h ρ S = Φ [ v phys , η ] . (323) Thus, the physical F1 current realizes a global maximal flow for every 0 ≤ η ≤ 1 0\leq\eta\leq1 0 ≤ η ≤ 1 Φ [ v p h y s , η ] = S B H ( 0 ) \Phi[v_{\mathrm{phys},\eta}]=S_{\mathrm{BH}}^{(0)} Φ [ v phys , η ] = S BH ( 0 )
Finally, we wish to note that the entropy flow has a smooth extension across the two-sided spatial slice. Set r = r h cosh ρ r=r_h\cosh\rho r = r h cosh ρ ρ \rho ρ
h i j d x i d x j = ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 , W η ( ρ ) = D η r h = 1 + ( 1 − η 2 ) sinh 2 ρ . (324) h_{ij}dx^idx^j=\ell^2d\rho^2+r_h^2\cosh^2\rho d\theta^2,\qquad W_\eta(\rho)=\frac{D_\eta}{r_h}=\sqrt{1+(1-\eta^2)\sinh^2\rho}. \tag*{(324)} h ij d x i d x j = ℓ 2 d ρ 2 + r h 2 cosh 2 ρ d θ 2 , W η ( ρ ) = r h D η = 1 + ( 1 − η 2 ) sinh 2 ρ . (324) For a flow oriented from the left exterior to the right,
v p h y s , η ρ = ρ S ℓ cosh ρ , v p h y s , η θ = η ρ S sinh ρ r h cosh 2 ρ W η , ∣ v p h y s , η ∣ = ρ S W η . (325) v_{\mathrm{phys},\eta}^{\rho}=\frac{\rho_S}{\ell\cosh\rho},\qquad v_{\mathrm{phys},\eta}^{\theta}=\frac{\eta\rho_S\sinh\rho}{r_h\cosh^2\rho W_\eta},\qquad\lvert v_{\mathrm{phys},\eta}\rvert=\frac{\rho_S}{W_\eta}. \tag*{(325)} v phys , η ρ = ℓ cosh ρ ρ S , v phys , η θ = r h cosh 2 ρ W η η ρ S sinh ρ , ∣ v phys , η ∣ = W η ρ S . (325) These are smooth functions at ρ = 0 \rho=0 ρ = 0 − S B H ( 0 ) -S_{\mathrm{BH}}^{(0)} − S BH ( 0 ) + S B H ( 0 ) +S_{\mathrm{BH}}^{(0)} + S BH ( 0 ) S L = Φ L [ − v p h y s , η ] = S B H ( 0 ) S_L=\Phi_L[-v_{\mathrm{phys},\eta}]=S_{\mathrm{BH}}^{(0)} S L = Φ L [ − v phys , η ] = S BH ( 0 ) S R = Φ R [ v p h y s , η ] = S B H ( 0 ) S_R=\Phi_R[v_{\mathrm{phys},\eta}]=S_{\mathrm{BH}}^{(0)} S R = Φ R [ v phys , η ] = S BH ( 0 )
The mixed-flux result highlights the role of physical string dynamics in determining entropy-flow lines. Varying the NS–NS fraction 0 ≤ η ≤ 1 0\leq\eta\leq1 0 ≤ η ≤ 1 2 π r h ρ S = S B H ( 0 ) 2\pi r_h\rho_S = S_{\mathrm{BH}}^{(0)} 2 π r h ρ S = S BH ( 0 )
Discussion and conclusion
In this paper, we have connected the local capacity of an entropy flow to the off-shell closed string sphere functional and constructed maximal-flow representatives from stationary fundamental-string sources. At genus zero and two-derivative order, the conical variation of the sphere-derived target space action gives
S l o c ( 0 ) [ m ] = ∫ m ρ S d A S , ρ S = e − 2 ϕ 4 G N . (326) S_{\mathrm{loc}}^{(0)}[m] = \int_m \rho_S dA_S,\qquad\rho_S = \frac{e^{-2\phi}}{4G_N}. \tag*{(326)} S loc ( 0 ) [ m ] = ∫ m ρ S d A S , ρ S = 4 G N e − 2 ϕ . (326) In a static holographic background, the semiclassical replica construction selects the weighted minimal cut, and flow–cut duality turns its entropy functional into the string-frame bound ∣ v ∣ ≤ ρ S |v| \le\rho_S ∣ v ∣ ≤ ρ S
The NS–NS BTZ calculation gives the dynamical realization. A regular stationary F1 forms a logarithmic spiral, and a uniform oriented ensemble has the spatially constant current norm ∣ q ∣ = N / ( 2 π r h ) |q| = \mathcal{N}/(2\pi r_h) ∣ q ∣ = N / ( 2 π r h )
v p h y s i = ρ S q ^ i , C g e o m = ρ S ∣ q ∣ , Φ [ v p h y s ] = 2 π r h ρ S = S B H ( 0 ) . (327) v_{\mathrm{phys}}^i = \rho_S \hat{q}^i,\qquad C_{\mathrm{geom}} = \frac{\rho_S}{|q|},\qquad\Phi[v_{\mathrm{phys}}] = 2\pi r_h\rho_S = S_{\mathrm{BH}}^{(0)}. \tag*{(327)} v phys i = ρ S q ^ i , C geom = ∣ q ∣ ρ S , Φ [ v phys ] = 2 π r h ρ S = S BH ( 0 ) . (327) The flow is normal to the horizon, saturates its capacity, and attains the horizon upper bound on every admissible flow. Its flux is therefore the dilaton-weighted BTZ entropy. Moreover, the conversion coefficient C g e o m C_{\mathrm{geom}} C geom
We then extended the calculation to BTZ with mixed NS–NS/RR flux in type IIB theory. We specify the bulk action, the complementary fluxes supporting the fixed geometry, and the classical F1 source action. The RR field supports the metric alongside the NS–NS field, while the bosonic F1 embedding couples directly to the metric and B B B
θ η ′ = η ℓ r D η , D η 2 = ( 1 − η 2 ) r 2 + η 2 r h 2 , ∣ q η ∣ = N 2 π D η . (328) \theta'_\eta= \frac{\eta^\ell}{rD_\eta},\qquad D_\eta^2 = (1-\eta^2)r^2+\eta^2r_h^2,\qquad|q_\eta| = \frac{\mathcal{N}}{2\pi D_\eta}. \tag*{(328)} θ η ′ = r D η η ℓ , D η 2 = ( 1 − η 2 ) r 2 + η 2 r h 2 , ∣ q η ∣ = 2 π D η N . (328) The regular profile interpolates from a physical radial F1 at pure RR flux to a logarithmic spiral at pure NS–NS flux. For η < 1 \eta< 1 η < 1 ρ S \rho_S ρ S
v p h y s , η i = ρ S ∣ q η ∣ h q η i , ∣ v p h y s , η ∣ ρ S = r h D η ≤ 1 , v_{\mathrm{phys},\eta}^i = \frac{\rho_S}{|q_\eta|_h}q_\eta^i,\qquad\frac{|v_{\mathrm{phys},\eta}|}{\rho_S} = \frac{r_h}{D_\eta} \le1, v phys , η i = ∣ q η ∣ h ρ S q η i , ρ S ∣ v phys , η ∣ = D η r h ≤ 1 , Φ [ v p h y s , η ] = e − 2 ϕ 0 ( 2 π r h ) 4 G N ( 3 ) = S B H ( 0 ) . (329) \Phi[v_{\mathrm{phys},\eta}] = \frac{e^{-2\phi_0}(2\pi r_h)}{4G_N^{(3)}} = S_{\mathrm{BH}}^{(0)}. \tag*{(329)} Φ [ v phys , η ] = 4 G N ( 3 ) e − 2 ϕ 0 ( 2 π r h ) = S BH ( 0 ) . (329) The coefficient is the largest constant weight compatible with the local capacity, determined from the physical current before its integrated entropy flux is evaluated. The horizon upper bound, together with the conserved admissible flow that saturates it, establishes global maximality and horizon minimality. All flux fractions therefore give the same entropy at fixed geometry, even though their physical strings and the exterior norms of their flows differ. This distinguishes saturation of the entropy bottleneck from saturation at every exterior point.
In other words, the mixed-flux extension exhibits a family of physical realizations of the same BTZ entropy. Changing the NS–NS/RR mixture changes the stationary F1 configurations, while the fixed metric and dilaton keep the entropy capacity unchanged. For every flux fraction, a constant normalization preserves current conservation, respects the local norm bound, and saturates the horizon capacity. The resulting maximal entropy flux is therefore independent of the flux mixture. This result demonstrates how physical string dynamics determines the flow lines while the geometric capacity fixes the maximal entropy they carry.
The resulting physical picture is that the off-shell sphere functional fixes the transverse entropy capacity, while the source worldsheets provide dynamically determined lines carrying conserved string charge. Assigning each line the capacity of its horizon share turns that charge current into a maximal entropy flow. Physical strings thus furnish explicit carriers of the flux representing entropy in the correspondence. At the stated order, the stationary F1 ensembles constructed here provide conserved currents that attain the entropy capacity fixed by the sphere functional. Their geometric conversion coefficient is fixed for the specified ensemble, while the microscopic entropy of individual strings and the effects of backreaction, interactions, higher genus, and higher derivatives remain outside this calculation. Extending the explicit on-shell source construction to general boundary subregions is a further problem beyond the complete-boundary BTZ examples treated here.
Acknowledgements
Acknowledgements HW is supported by NSFC Grant No.12105191. SY is supported by NSFC Grant No.12105031 and No.12547101.
