Introduction
By the state-operator correspondence, the thermal partition function of a C F T d \mathrm{CFT}_d CFT d S d − 1 × S 1 S^{d-1} \times S^1 S d − 1 × S 1
Z ( β ; ω i ) : = Tr S d − 1 e − β ( D − ω 1 J 1 − ⋯ − ω κ J κ ) , κ : = ⌊ d 2 ⌋ (1) Z(\beta;\omega_i) := \operatorname{Tr}_{S^{d-1}} e^{-\beta(D-\omega_1J_1-\cdots-\omega_\kappa J_\kappa)}, \qquad\kappa:= \left\lfloor\frac{d}{2}\right\rfloor \tag*{(1)} Z ( β ; ω i ) := Tr S d − 1 e − β ( D − ω 1 J 1 − ⋯ − ω κ J κ ) , κ := ⌊ 2 d ⌋ (1) fully encodes the scaling dimensions Δ \Delta Δ J i J_i J i Z ( β ; ω i ) Z(\beta;\omega_i) Z ( β ; ω i ) [1 ] that its high-temperature behavior always takes the hydrodynamic form
ln Z = c ′ β d − 1 ∏ i = 1 κ ( 1 − ω i 2 ) + O ( β 3 − d ) , β → 0 + , ω i fixed . (2) \ln Z = \frac{c'}{\beta^{d-1}\prod_{i=1}^{\kappa}(1-\omega_i^2)}+O\left(\beta^{3-d}\right),\qquad\beta\to0^+,\ \omega_i\ \text{fixed}. \tag*{(2)} ln Z = β d − 1 ∏ i = 1 κ ( 1 − ω i 2 ) c ′ + O ( β 3 − d ) , β → 0 + , ω i fixed . (2) This remarkably universal result was subsequently rederived using the formalism of ‘equilibrium partition functions’ [2 , 3 , 4 ] , an approach that was further developed and named thermal effective theory in [5 ] .
A systematic analysis of subleading corrections to Eq. (2) was presented in the recent paper [6 ] . The analysis of [6 ] was largely performed in d = 4 d=4 d = 4 1
d = 2 κ d=2\kappa d = 2 κ
ln Z = ln Z r o t + ln Z s h , \ln Z=\ln Z^{\mathrm{rot}}+\ln Z^{\mathrm{sh}}, ln Z = ln Z rot + ln Z sh , ln Z r o t = − 1 β d − 1 x 1 x 2 … x κ ( ∑ q = 0 ∞ β 2 q S q ( x i ) ) , \ln Z^{\mathrm{rot}}=-\frac{1}{\beta^{d-1}x_1x_2\ldots x_\kappa}\left(\sum_{q=0}^{\infty}\beta^{2q}S_q(x_i)\right), ln Z rot = − β d − 1 x 1 x 2 … x κ 1 ( q = 0 ∑ ∞ β 2 q S q ( x i ) ) , ln Z s h β = κ ! a 2 κ [ ( − 1 ) κ + 1 2 2 κ − 1 ∑ i = 1 κ ∫ 0 1 d s ∏ j = 1 j ≠ i κ ( s x i − 2 x j ) 2 x j ( x i − x j ) ] . (3) \frac{\ln Z^{\mathrm{sh}}}{\beta}=\kappa!a_{2\kappa}\left[(-1)^\kappa+\frac{1}{2^{2\kappa-1}}\sum_{i=1}^{\kappa}\int_0^1ds\prod_{\substack{j=1\\j\ne i}}^\kappa\frac{(sx_i-2x_j)^2}{x_j(x_i-x_j)}\right]. \tag*{(3)} β ln Z sh = κ ! a 2 κ ( − 1 ) κ + 2 2 κ − 1 1 i = 1 ∑ κ ∫ 0 1 d s j = 1 j = i ∏ κ x j ( x i − x j ) ( s x i − 2 x j ) 2 . (3) ln Z s h \ln Z^{\mathrm{sh}} ln Z sh a a a a 2 κ a_{2\kappa} a 2 κ 12 ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot S q S_q S q q q q 13 ω i 2 \omega_i^2 ω i 2
x i : = 1 − ω i 2 , ( i = 1 … κ ) . (4) x_i:=1-\omega_i^2,\qquad(i=1\ldots\kappa). \tag*{(4)} x i := 1 − ω i 2 , ( i = 1 … κ ) . (4) While the polynomials S q S_q S q [6 ] . These constraints force S q S_q S q d q d_q d q [6 ] (of the space of arbitrary symmetric degree q q q d q d_q d q 14
t κ ∏ i = 1 κ + 1 ( 1 − t i ) = ∑ q = 0 ∞ d q t q . (5) \frac{t^\kappa}{\prod_{i=1}^{\kappa+1}(1-t^i)}=\sum_{q=0}^{\infty}d_qt^q \, . \tag*{(5)} ∏ i = 1 κ + 1 ( 1 − t i ) t κ = q = 0 ∑ ∞ d q t q . (5) at large q q q d q ∼ q κ κ ! ( κ + 1 ) ! d_q\sim\frac{q^\kappa}{\kappa!(\kappa+1)!} d q ∼ κ ! ( κ + 1 )! q κ S q S_q S q [6 ] at the first nontrivial value, q = κ q=\kappa q = κ q ≥ κ + 1 q\geq\kappa+1 q ≥ κ + 1 κ ≥ 2 \kappa\geq2 κ ≥ 2 Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 Ω \Omega Ω S 2 κ − 1 × S 1 S^{2\kappa-1}\times S^1 S 2 κ − 1 × S 1
Ω : ( b 0 , b 1 , … , b κ ) ⟶ ( b 1 , … , b κ , b 0 ) , \Omega:(b_0,b_1,\ldots,b_\kappa)\longrightarrow(b_1,\ldots,b_\kappa,b_0), Ω : ( b 0 , b 1 , … , b κ ) ⟶ ( b 1 , … , b κ , b 0 ) , where $(b_0,b_1,\ldots,b_\kappa):=(\beta,\beta\omega_1,\ldots,\beta\omega_\kappa).
{#eq:6}
Our central equation asserts the vanishing of the Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 ln Z r o t β \frac{\ln Z^{\mathrm{rot}}}{\beta} β l n Z rot
( 1 + Ω + Ω 2 + … + Ω κ ) ∘ [ ln Z r o t β ] = 0. (6) (1+\Omega+\Omega^2+\ldots+\Omega^\kappa)\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0. \tag*{(6)} ( 1 + Ω + Ω 2 + … + Ω κ ) ∘ [ β ln Z rot ] = 0. (6) Eq. (6) , together with our (conjectured) expression for ln Z s h \ln Z^{\mathrm{sh}} ln Z sh
( 1 + Ω + Ω 2 + … + Ω κ ) ∘ [ ln Z β ] = ( − 1 ) κ κ ! a 2 κ . (7) (1+\Omega+\Omega^2+\ldots+\Omega^\kappa)\circ\left[\frac{\ln Z}{\beta}\right]=(-1)^\kappa\kappa!a_{2\kappa}. \tag*{(7)} ( 1 + Ω + Ω 2 + … + Ω κ ) ∘ [ β ln Z ] = ( − 1 ) κ κ ! a 2 κ . (7) Our demonstration of Eq. (7) applies only in an order-by-order sense in β 2 \beta^2 β 2 (7) is thus best viewed as a generating function for an infinite tower of constraints on coefficient polynomials (see Eq. (3) )
T q : = β 2 q S q , (8) T_q:=\beta^{2q}S_q, \tag*{(8)} T q := β 2 q S q , (8) that are homogeneous of degree q q q ( b 0 2 , b 1 2 , … , b κ 2 ) (b_0^2,b_1^2,\ldots,b_\kappa^2) ( b 0 2 , b 1 2 , … , b κ 2 ) T q T_q T q b i 2 ↔ b j 2 b_i^2\leftrightarrow b_j^2 b i 2 ↔ b j 2 i , j ≠ 0 i,j\neq0 i , j = 0 b 0 2 b_0^2 b 0 2
T q = I q ( 0 ) + b 0 2 I q ( 1 ) + b 0 4 I q ( 2 ) + … + b 0 2 κ I q ( κ ) (9) T_q=I_q^{(0)}+b_0^2I_q^{(1)}+b_0^4I_q^{(2)}+\ldots+b_0^{2\kappa}I_q^{(\kappa)} \tag*{(9)} T q = I q ( 0 ) + b 0 2 I q ( 1 ) + b 0 4 I q ( 2 ) + … + b 0 2 κ I q ( κ ) (9) where I q ( r ) I_q^{(r)} I q ( r ) ( q − r ) (q-r) ( q − r ) ( b 0 2 , b 1 2 , … , b κ 2 ) (b_0^2,b_1^2,\ldots,b_\kappa^2) ( b 0 2 , b 1 2 , … , b κ 2 ) r > q r>q r > q (6) is equivalent to
I q ( κ ) = 0 , ∀ q ≥ κ , (10) I_q^{(\kappa)}=0,\quad\forall q\geq\kappa, \tag*{(10)} I q ( κ ) = 0 , ∀ q ≥ κ , (10) effectively imposing d q d_q d q S q S_q S q (5) . We demonstrate the constraining power below explicitly for κ = 2 \kappa=2 κ = 2 (10) for κ > 2 \kappa>2 κ > 2
As a highly nontrivial check on the results of this paper, we verify that the partition function of Kerr-AdS2 κ + 1 _{2\kappa+1} 2 κ + 1 15 (7) as an exact non-perturbative identity between functions of ( b 0 2 , b 1 2 , … , b κ 2 ) (b_0^2,b_1^2,\ldots,b_\kappa^2) ( b 0 2 , b 1 2 , … , b κ 2 ) d = 4 d=4 d = 4
Derivation of the central equation
Simplifying the measure
As explained in [6 ] , the most general local thermal effective action for a d = 2 κ d=2\kappa d = 2 κ (3) , with
ln Z r o t = ∫ d d x g ( υ 2 ) κ I . (11) \ln Z^{\mathrm{rot}}=\int d^dx\frac{\sqrt{g}}{(\upsilon^2)^\kappa}\mathcal{I}. \tag*{(11)} ln Z rot = ∫ d d x ( υ 2 ) κ g I . (11) Specializing to d = 4 d=4 d = 4 I \mathcal{I} I S 3 × R S^3\times R S 3 × R
d s 2 = d τ 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 , (12) ds^2=d\tau^2+d\theta^2+\sin^2\theta\,d\phi_1^2+\cos^2\theta\,d\phi_2^2, \tag*{(12)} d s 2 = d τ 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 , (12) as well as the ‘identification vector’
v = β ( ∂ τ − i ω 1 ∂ ϕ 1 − i ω 2 ∂ ϕ 2 ) = b 0 ∂ τ + b 1 ( − i ∂ ϕ 1 ) + b 2 ( − i ∂ ϕ 2 ) (13) v=\beta\left(\partial_\tau-i\omega_1\partial_{\phi_1}-i\omega_2\partial_{\phi_2}\right)
=b_0\partial_\tau+b_1(-i\partial_{\phi_1})+b_2(-i\partial_{\phi_2}) \tag*{(13)} v = β ( ∂ τ − i ω 1 ∂ ϕ 1 − i ω 2 ∂ ϕ 2 ) = b 0 ∂ τ + b 1 ( − i ∂ ϕ 1 ) + b 2 ( − i ∂ ϕ 2 ) (13) which determines the boundary conditions of all dynamical fields χ \chi χ
χ ( τ , ϕ 1 , ϕ 2 , θ ) = e L v χ ( τ , ϕ 1 , ϕ 2 , θ ) . (14) \chi(\tau,\phi_1,\phi_2,\theta)=e^{\mathcal{L}_v}\chi(\tau,\phi_1,\phi_2,\theta). \tag*{(14)} χ ( τ , ϕ 1 , ϕ 2 , θ ) = e L v χ ( τ , ϕ 1 , ϕ 2 , θ ) . (14) Translational invariance allows the integrals over τ \tau τ ϕ 1 \phi_1 ϕ 1 ϕ 2 \phi_2 ϕ 2 θ \theta θ θ 6 \theta^6 θ 6
z : = β 2 ( 1 − x 1 x 2 γ 2 ) , (15) z:=\beta^2(1-x_1x_2\gamma^2), \tag*{(15)} z := β 2 ( 1 − x 1 x 2 γ 2 ) , (15) where
γ 2 : = 1 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) , \gamma^2:=\frac{1}{1-\omega_2^2+\sin^2\theta(\omega_2^2-\omega_1^2)}, γ 2 := 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) 1 , we find (see around (S1) in SM)
g 4 ( v 2 ) 2 ∫ 0 β d θ ∫ 0 β d τ ∫ 0 2 π d ϕ 1 ∫ 0 2 π d ϕ 2 = β P d z , (16) \frac{\sqrt{g_4}}{(v^2)^2}\int_0^\beta d\theta\int_0^\beta d\tau\int_0^{2\pi}d\phi_1\int_0^{2\pi}d\phi_2=\beta P\,dz, \tag*{(16)} ( v 2 ) 2 g 4 ∫ 0 β d θ ∫ 0 β d τ ∫ 0 2 π d ϕ 1 ∫ 0 2 π d ϕ 2 = β P d z , (16) where P : = 2 π 2 ( b 0 2 − b 1 2 ) ( b 1 2 − b 2 2 ) ( b 2 2 − b 0 2 ) P:=\frac{2\pi^2}{(b_0^2-b_1^2)(b_1^2-b_2^2)(b_2^2-b_0^2)} P := ( b 0 2 − b 1 2 ) ( b 1 2 − b 2 2 ) ( b 2 2 − b 0 2 ) 2 π 2
Eq. (12) can thus be rewritten as
ln Z r o t β = P ∫ b 1 2 b 2 2 d z I b 0 , b 1 , b 2 ( z ) . (17) \frac{\ln Z^{\mathrm{rot}}}{\beta}=P\int_{b_1^2}^{b_2^2}dz\,\mathcal{I}_{b_0,b_1,b_2}(z). \tag*{(17)} β ln Z rot = P ∫ b 1 2 b 2 2 d z I b 0 , b 1 , b 2 ( z ) . (17) Notice that while P P P b i 2 b_i^2 b i 2 b j 2 b_j^2 b j 2 Z 3 \mathbb{Z}_3 Z 3 (17) we have renamed I = I b 0 , b 1 , b 2 \mathcal{I}=\mathcal{I}_{b_0,b_1,b_2} I = I b 0 , b 1 , b 2 ( b 0 , b 1 , b 2 ) (b_0,b_1,b_2) ( b 0 , b 1 , b 2 )
The Z 3 \mathbb{Z}_3 Z 3 I b 0 , b 1 , b 2 \mathcal{I}_{b_0,b_1,b_2} I b 0 , b 1 , b 2
As explained in [6 ] , Weyl invariance and locality together ensure that I \mathcal{I} I tr ( ω 2 ) \operatorname{tr}(\omega^2) tr ( ω 2 ) v 2 S v^2S v 2 S v 2 B μ ( ω 2 ) μ ν B ν v^2B_\mu(\omega^2)^{\mu\nu}B_\nu v 2 B μ ( ω 2 ) μν B ν [6 ] for definitions). The coefficients of this expansion are theory dependent but independent of the ‘compactification data’. As a consequence, I \mathcal{I} I ( b 0 , b 1 , b 2 ) (b_0,b_1,b_2) ( b 0 , b 1 , b 2 )
tr ( ω 2 ) 8 = z , v 2 S = b 0 2 + b 1 2 + b 2 2 − 3 z , \frac{\operatorname{tr}(\omega^2)}{8}=z,\qquad v^2S=b_0^2+b_1^2+b_2^2-3z, 8 tr ( ω 2 ) = z , v 2 S = b 0 2 + b 1 2 + b 2 2 − 3 z , v 2 B μ ( ω 2 ) μ ν B ν = − 4 ( z − b 0 2 ) ( z − b 1 2 ) ( z − b 2 2 ) . (18) v^2B_\mu(\omega^2)^{\mu\nu}B_\nu=-4(z-b_0^2)(z-b_1^2)(z-b_2^2). \tag*{(18)} v 2 B μ ( ω 2 ) μν B ν = − 4 ( z − b 0 2 ) ( z − b 1 2 ) ( z − b 2 2 ) . (18) Note that each of the quantities in (18) is a polynomial both in z z z b i 2 b_i^2 b i 2 (29) for a structural explanation of this feature). Each of the scalars listed in (18) , and so I b 0 , b 1 , b 2 \mathcal{I}_{b_0,b_1,b_2} I b 0 , b 1 , b 2 S 3 S_3 S 3 ( b 0 , b 1 , b 2 ) (b_0,b_1,b_2) ( b 0 , b 1 , b 2 )
I b 0 , b 1 , b 2 ( z ) = I b 1 , b 2 , b 0 ( z ) = I b 2 , b 0 , b 1 ( z ) = I ( z ) . (19) \mathcal{I}_{b_0,b_1,b_2}(z)=\mathcal{I}_{b_1,b_2,b_0}(z)=\mathcal{I}_{b_2,b_0,b_1}(z)=\mathcal{I}(z). \tag*{(19)} I b 0 , b 1 , b 2 ( z ) = I b 1 , b 2 , b 0 ( z ) = I b 2 , b 0 , b 1 ( z ) = I ( z ) . (19) Derivation using Cauchy’s theorem
When d = 4 d=4 d = 4 (7) abbreviates
3 S ∘ [ ln Z β ] = 2 a 4 , (20) 3\mathcal{S}\circ\left[\frac{\ln Z}{\beta}\right]=2a_4, \tag*{(20)} 3 S ∘ [ β ln Z ] = 2 a 4 , (20) where S \mathcal{S} S Z 3 \mathbb{Z}_3 Z 3
S : = ( 1 + Ω + Ω 2 ) 3 . (21) \mathcal{S}:=\frac{(1+\Omega+\Omega^2)}{3}. \tag*{(21)} S := 3 ( 1 + Ω + Ω 2 ) . (21) Since P P P Z 3 \mathbb{Z}_3 Z 3 S ∘ P = P \mathcal{S}\circ P=P S ∘ P = P (17)
3 P × S ∘ [ ln Z r o t β ] = 3 S ∘ [ ln Z r o t β × 1 P ] = ∫ b 1 2 b 2 2 d z I b 0 , b 1 , b 2 ( z ) + ∫ b 2 2 b 0 2 d z I b 1 , b 2 , b 0 ( z ) + ∫ b 0 2 b 1 2 d z I b 2 , b 0 , b 1 ( z ) = ∮ d z I ( z ) = 0. (22) \begin{aligned}
\frac{3}{P}\times\mathcal{S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=3\mathcal{S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\times\frac{1}{P}\right] \\
&=\int_{b_1^2}^{b_2^2}dz\,\mathcal{I}_{b_0,b_1,b_2}(z)+\int_{b_2^2}^{b_0^2}dz\,\mathcal{I}_{b_1,b_2,b_0}(z) \\
&+\int_{b_0^2}^{b_1^2}dz\,\mathcal{I}_{b_2,b_0,b_1}(z) \\
&=\oint dz\,\mathcal{I}(z)=0. \tag*{(22)}
\end{aligned} P 3 × S ∘ [ β ln Z rot ] = 3 S ∘ [ β ln Z rot × P 1 ] = ∫ b 1 2 b 2 2 d z I b 0 , b 1 , b 2 ( z ) + ∫ b 2 2 b 0 2 d z I b 1 , b 2 , b 0 ( z ) + ∫ b 0 2 b 1 2 d z I b 2 , b 0 , b 1 ( z ) = ∮ d z I ( z ) = 0. (22) ∴ S ∘ [ ln Z r o t β ] = 0. \therefore\mathcal{S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0. ∴ S ∘ [ β ln Z rot ] = 0. The equality in the fourth line above follows order by order in β 2 \beta^2 β 2 I \mathcal{I} I z z z b i 2 b_i^2 b i 2 [ b 1 2 , b 2 2 ] ∪ [ b 2 2 , b 0 2 ] ∪ [ b 0 2 , b 1 2 ] [b_1^2,b_2^2]\cup[b_2^2,b_0^2]\cup[b_0^2,b_1^2] [ b 1 2 , b 2 2 ] ∪ [ b 2 2 , b 0 2 ] ∪ [ b 0 2 , b 1 2 ]
3 S ∘ ln Z s h β = 3 S ∘ [ a 4 12 ( x 1 x 2 + x 2 x 1 + 7 ) ] = 2 a 4 . (23) 3\mathcal{S}\circ\frac{\ln Z^{\mathrm{sh}}}{\beta}=3\mathcal{S}\circ\left[\frac{a_4}{12}\left(\frac{x_1}{x_2}+\frac{x_2}{x_1}+7\right)\right]=2a_4. \tag*{(23)} 3 S ∘ β ln Z sh = 3 S ∘ [ 12 a 4 ( x 2 x 1 + x 1 x 2 + 7 ) ] = 2 a 4 . (23) Plugging Eq. (23) and Eq. (24) into the first of Eq. (3), we recover Eq. (21). In Appendix B we demonstrate that Eq. (21) simplifies at the special angular velocities ( ω 1 2 , ω 2 2 ) = ( e 2 π i 3 , e 4 π i 3 ) (\omega_1^2,\omega_2^2)=\left(e^{\frac{2\pi i}{3}},e^{\frac{4\pi i}{3}}\right) ( ω 1 2 , ω 2 2 ) = ( e 3 2 π i , e 3 4 π i ) [6 ] .
In Appendix C we present a physically intriguing rederivation of (19) that does not invoke the computational result (18) . Briefly, we construct a Z 3 \mathbb{Z}_3 Z 3 ( τ , i ϕ 1 , i ϕ 2 ) (\tau,i\phi_1,i\phi_2) ( τ , i ϕ 1 , i ϕ 2 ) z z z Figure 5 in SM)
Figure 5. The map from spheres to hyperbolic spaces
S 3 × R → Ω A d S 3 × S 1 → Ω A d S 3 ′ × S ′ 1 → Ω S 3 × R . (24) S^3\times R\xrightarrow{\Omega}AdS_3\times S^1\xrightarrow{\Omega}AdS'_3\times S'^1\xrightarrow{\Omega}S^3\times R. \tag*{(24)} S 3 × R Ω A d S 3 × S 1 Ω A d S 3 ′ × S ′1 Ω S 3 × R . (24) where A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1 A d S 3 ′ × S ′ 1 AdS'_3\times S'^1 A d S 3 ′ × S ′1 z z z A d S × S 1 AdS\times S^1 A d S × S 1 S 3 × R S^3\times R S 3 × R v v v (19) then follows from the Weyl invariance of I \mathcal{I} I
The vanishing of I q ( 2 ) \mathcal{I}^{(2)}_q I q ( 2 ) d = 4 d=4 d = 4
We will now show that the small β 2 \beta^2 β 2 (6) on ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot (10) in d = 4 d=4 d = 4 (3) can be rewritten as 17
ln Z r o t β = − P 2 π 2 ∑ q = 0 ∞ ( b 2 2 − b 1 2 ) T q = − 1 2 π 2 ∑ q = 0 ∞ [ P I q ( 0 ) ( b 2 2 − b 1 2 ) + P I q ( 1 ) ( b 2 2 b 0 2 − b 1 2 b 0 2 ) + P I q ( 2 ) ( b 2 2 b 0 4 − b 1 2 b 0 4 ) ] , (25) \begin{aligned}
\frac{\ln Z^{\mathrm{rot}}}{\beta}=-\frac{P}{2\pi^2}\sum_{q=0}^{\infty}(b_2^2-b_1^2)T_q \\
&=-\frac{1}{2\pi^2}\sum_{q=0}^{\infty}\left[P I_q^{(0)}\left(b_2^2-b_1^2\right)+P I_q^{(1)}\left(b_2^2b_0^2-b_1^2b_0^2\right)+P I_q^{(2)}\left(b_2^2b_0^4-b_1^2b_0^4\right)\right], \tag*{(25)}
\end{aligned} β ln Z rot = − 2 π 2 P q = 0 ∑ ∞ ( b 2 2 − b 1 2 ) T q = − 2 π 2 1 q = 0 ∑ ∞ [ P I q ( 0 ) ( b 2 2 − b 1 2 ) + P I q ( 1 ) ( b 2 2 b 0 2 − b 1 2 b 0 2 ) + P I q ( 2 ) ( b 2 2 b 0 4 − b 1 2 b 0 4 ) ] , (25) where we have used Eq. (9) . As P P P I q ( 0 ) I_q^{(0)} I q ( 0 ) I q ( 1 ) I_q^{(1)} I q ( 1 ) I q ( 2 ) I_q^{(2)} I q ( 2 ) Z 3 \mathbb{Z}_3 Z 3
3 S ∘ [ ln Z r o t β ] = 3 2 π 2 ∑ q = 0 ∞ [ { S ∘ ( b 1 2 − b 2 2 ) } P I q ( 0 ) + { S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) } P I q ( 1 ) + { S ∘ ( b 1 2 b 0 4 − b 2 2 b 0 4 ) } P I q ( 2 ) ] = − ∑ q = 0 ∞ I q ( 2 ) . (26) \begin{aligned}
3\mathcal{S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=\frac{3}{2\pi^2}\sum_{q=0}^{\infty}\left[\{\mathcal{S}\circ\left(b_1^2-b_2^2\right)\}PI_q^{(0)}+\{\mathcal{S}\circ\left(b_1^2b_0^2-b_2^2b_0^2\right)\}PI_q^{(1)}+\{\mathcal{S}\circ\left(b_1^2b_0^4-b_2^2b_0^4\right)\}PI_q^{(2)}\right] \\
&=-\sum_{q=0}^{\infty}I_q^{(2)}. \tag*{(26)}
\end{aligned} 3 S ∘ [ β ln Z rot ] = 2 π 2 3 q = 0 ∑ ∞ [ { S ∘ ( b 1 2 − b 2 2 ) } P I q ( 0 ) + { S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) } P I q ( 1 ) + { S ∘ ( b 1 2 b 0 4 − b 2 2 b 0 4 ) } P I q ( 2 ) ] = − q = 0 ∑ ∞ I q ( 2 ) . (26) (The vanishing of the first two terms and the simplification of the last term follow from the identities Eq. (95) in the SM). Since I q ( 2 ) I_q^{(2)} I q ( 2 ) ( q − 2 ) (q-2) ( q − 2 ) q q q
See around (S51) in SM for a generalization of this derivation to d = 2 κ d=2\kappa d = 2 κ
Generalization to d = 2 κ d=2\kappa d = 2 κ κ \kappa κ
The generalization of our discussion to arbitrary even dimensions (see SM, between (S63) and (S157) for all details) proceeds in two steps.
We first express all local EFT terms using a common set of scalar variables. In the embedding space R 2 κ , 2 \mathbb{R}^{2\kappa,2} R 2 κ , 2 N N N v v v SO ( 2 κ , 2 ) \operatorname{SO}(2\kappa,2) SO ( 2 κ , 2 ) A A A
m ( a ) ≡ N ⋅ B a N N ⋅ B N , B = − A 2 , (27) m^{(a)}\equiv\frac{N\cdot B^aN}{N\cdot BN},\qquad B=-A^2, \tag*{(27)} m ( a ) ≡ N ⋅ B N N ⋅ B a N , B = − A 2 , (27) we find that Eq. (11) becomes
( ln Z r o t β ) = ( − 1 ) κ 2 π κ V ∫ D 0 I \left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=\frac{(-1)^{\kappa}2\pi^{\kappa}}{\mathcal{V}}\int_{D_0}\mathcal{I} ( β ln Z rot ) = V ( − 1 ) κ 2 π κ ∫ D 0 I I = d m 2 ∧ d m 3 … ∧ d m κ I ( E r , m ( a ) ) \mathcal{I}=dm_2\wedge dm_3\ldots\wedge dm_\kappa\,\mathcal{I}\left(E_r,m^{(a)}\right) I = d m 2 ∧ d m 3 … ∧ d m κ I ( E r , m ( a ) ) V = ∏ 0 ≤ i < j ≤ κ ( b j 2 − b i 2 ) (28) \mathcal{V}=\prod_{0\leq i<j\leq\kappa}\left(b_j^2-b_i^2\right) \tag*{(28)} V = 0 ≤ i < j ≤ κ ∏ ( b j 2 − b i 2 ) (28) where E r E_r E r D 0 D_0 D 0 κ \kappa κ
( p 0 b ) j = b 0 2 j − b b 2 j b 0 2 − b b 2 , ( b = 1 … κ ) . (29) (p_{0b})_j=\frac{b_0^{2j}-b_b^{2j}}{b_0^2-b_b^2},\qquad(b=1\ldots\kappa). \tag*{(29)} ( p 0 b ) j = b 0 2 − b b 2 b 0 2 j − b b 2 j , ( b = 1 … κ ) . (29) In the Weyl frame defined by the condition N ⋅ B N = − 1 N\cdot BN=-1 N ⋅ B N = − 1 m ( a ) m^{(a)} m ( a ) N N N B B B Π \Pi Π R 2 κ , 2 \mathbb{R}^{2\kappa,2} R 2 κ , 2 I ( E r , m ( a ) ) \mathcal{I}(E_r,m^{(a)}) I ( E r , m ( a ) ) (28) that ln Z \ln Z ln Z (3) .
Next, we act on Eq. (28) with the Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 E r E_r E r m i m_i m i Ω a ∘ V = ( − 1 ) κ a V \Omega^a\circ\mathcal{V}=(-1)^{\kappa a}\mathcal{V} Ω a ∘ V = ( − 1 ) κa V Ω a \Omega^a Ω a D 0 D_0 D 0 D a D_a D a
( p a b ) j = b a 2 j − b b 2 j b a 2 − b b 2 , ( b = 0 … κ , b ≠ a ) , (30) (p_{ab})_j=\frac{b_a^{2j}-b_b^{2j}}{b_a^2-b_b^2},\qquad(b=0\ldots\kappa,\ b\ne a), \tag*{(30)} ( p ab ) j = b a 2 − b b 2 b a 2 j − b b 2 j , ( b = 0 … κ , b = a ) , (30) with orientation and signs described around Eq. (194) in SM.
The ( κ + 1 ) (\kappa+1) ( κ + 1 ) R κ + 1 \mathbb{R}^{\kappa+1} R κ + 1
Δ ( 2 , κ + 1 ) = { ( t 0 , … , t κ ) : 0 ≤ t a ≤ 1 , ∑ a t a = 2 } . (31) \Delta(2,\kappa+1)=\left\{(t_0,\ldots,t_\kappa):0\le t_a\le1,\ \sum_a t_a=2\right\}. \tag*{(31)} Δ ( 2 , κ + 1 ) = { ( t 0 , … , t κ ) : 0 ≤ t a ≤ 1 , a ∑ t a = 2 } . (31) (see Figure 3 in SM for a visualization when κ = 2 \kappa=2 κ = 2 κ ≥ 3 \kappa\ge3 κ ≥ 3 κ \kappa κ 2 ( κ + 1 ) 2(\kappa+1) 2 ( κ + 1 ) ( κ + 1 ) (\kappa+1) ( κ + 1 ) t a = 1 t_a=1 t a = 1 a = 0 … κ a=0\ldots\kappa a = 0 … κ D a \mathcal{D}_a D a ( κ + 1 ) (\kappa+1) ( κ + 1 ) t a = 0 t_a=0 t a = 0 a = 0 … κ a=0\ldots\kappa a = 0 … κ H a \mathcal{H}_a H a d = 6 d=6 d = 6 D a \mathcal{D}_a D a H a \mathcal{H}_a H a Figure 1 .
Figure 1. The d = 6 d=6 d = 6 Δ ( 2 , 4 ) \Delta(2,4) Δ ( 2 , 4 ) ( a , b ) (a,b) ( a , b ) H a \mathcal{H}_a H a
Figure 3. The polytope Δ ( 2 , κ + 1 ) \Delta(2,\kappa+1) Δ ( 2 , κ + 1 ) κ = 2 \kappa=2 κ = 2 ∑ a t a = 2 \sum_a t_a=2 ∑ a t a = 2 t a = 1 t_a=1 t a = 1 D a \mathcal{D}_a D a t a = 0 t_a=0 t a = 0 H a \mathcal{H}_a H a κ = 2 \kappa=2 κ = 2 D a \mathcal{D}_a D a
As I ( E r , m ( a ) ) \mathcal{I}(E_r,m^{(a)}) I ( E r , m ( a ) ) m ( a ) m^{(a)} m ( a ) Υ \Upsilon Υ Υ \Upsilon Υ D a \mathcal{D}_a D a H a \mathcal{H}_a H a H a \mathcal{H}_a H a ( κ − 2 ) (\kappa-2) ( κ − 2 ) D a \mathcal{D}_a D a H a \mathcal{H}_a H a D a \mathcal{D}_a D a (6) .
Checks
Exact verification from the Kerr-AdS2 κ + 1 _{2\kappa+1} 2 κ + 1
Consider Kerr–AdS in ( 2 κ + 1 ) (2\kappa+1) ( 2 κ + 1 ) r + r_+ r + a i a_i a i [8 , 9 ] . In d = 2 κ d=2\kappa d = 2 κ ( q 0 , q 1 , ⋯ , q κ ) (q_0,q_1,\cdots,q_\kappa) ( q 0 , q 1 , ⋯ , q κ )
q 0 : = 1 r + , q i : = a i r + , for i = 1 , … , κ . (32) q_0:=\frac{1}{r_+},\qquad q_i:=\frac{a_i}{r_+},\qquad\text{for }i=1,\ldots,\kappa. \tag*{(32)} q 0 := r + 1 , q i := r + a i , for i = 1 , … , κ . (32) The black hole thermodynamic potentials ( b 0 , b 1 , ⋯ , b κ ) (b_0,b_1,\cdots,b_\kappa) ( b 0 , b 1 , ⋯ , b κ ) q i q_i q i
b i = 2 π q i ∏ j = 0 j ≠ i κ ( 1 + q j 2 ) Ξ , for i = 0 , 1 , … , κ (33) b_i=\frac{2\pi q_i\prod_{\substack{j=0\\j\ne i}}^\kappa(1+q_j^2)}{\Xi},\qquad\text{for }i=0,1,\ldots,\kappa \tag*{(33)} b i = Ξ 2 π q i ∏ j = 0 j = i κ ( 1 + q j 2 ) , for i = 0 , 1 , … , κ (33) where Ξ \Xi Ξ
Ξ : = ∑ i = 0 κ + 1 ( κ − i ) E i ( q 0 2 , … , q κ 2 ) . (34) \Xi:=\sum_{i=0}^{\kappa+1}(\kappa-i)E_i(q_0^2,\ldots,q_\kappa^2). \tag*{(34)} Ξ := i = 0 ∑ κ + 1 ( κ − i ) E i ( q 0 2 , … , q κ 2 ) . (34) Here E i E_i E i (38) . The Kerr-AdS2 κ + 1 _{2\kappa+1} 2 κ + 1
ln Z B H = − I 2 κ + 1 t h = β π κ − 1 8 ( κ − 1 ) ! G 2 κ + 1 ( 1 − q 0 2 ) ∏ i = 1 κ 1 + q i 2 q 0 2 − q i 2 . (35) \ln Z_{\mathrm{BH}}=-I^{\mathrm{th}}_{2\kappa+1}=\frac{\beta\pi^{\kappa-1}}{8(\kappa-1)!G_{2\kappa+1}}(1-q_0^2)\prod_{i=1}^{\kappa}\frac{1+q_i^2}{q_0^2-q_i^2}. \tag*{(35)} ln Z BH = − I 2 κ + 1 th = 8 ( κ − 1 )! G 2 κ + 1 β π κ − 1 ( 1 − q 0 2 ) i = 1 ∏ κ q 0 2 − q i 2 1 + q i 2 . (35) Now, consider a Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 ( q 0 , q 1 , ⋯ , q κ ) (q_0,q_1,\cdots,q_\kappa) ( q 0 , q 1 , ⋯ , q κ )
Ω b h : ( q 0 , q 1 , … , q κ ) → ( q 1 , … , q κ , q 0 ) . (36) \Omega_{\mathrm{bh}}:(q_0,q_1,\ldots,q_\kappa)\to(q_1,\ldots,q_\kappa,q_0). \tag*{(36)} Ω bh : ( q 0 , q 1 , … , q κ ) → ( q 1 , … , q κ , q 0 ) . (36) It follows immediately from Eq. (33) and Eq. (34) (which is Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 (36) induces the action Eq. (6) on ( b 0 , b 1 , ⋯ , b κ ) (b_0,b_1,\cdots,b_\kappa) ( b 0 , b 1 , ⋯ , b κ ) Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 Ω b h \Omega_{\mathrm{bh}} Ω bh Ω \Omega Ω
Using Eq. (36) we have checked (see around (S165) in SM) that
( 1 + Ω + ⋯ + Ω κ ) ∘ [ ln Z B H β ] = ( − 1 ) κ π κ − 1 4 ( κ − 1 ) ! G 2 κ + 1 = ( − 1 ) κ κ ! a 2 κ (37) \begin{aligned}
\left(1+\Omega+\cdots+\Omega^\kappa\right)\circ\left[\frac{\ln Z_{\mathrm{BH}}}{\beta}\right]=(-1)^\kappa\frac{\pi^{\kappa-1}}{4(\kappa-1)!G_{2\kappa+1}} \\
&=(-1)^\kappa\kappa!a_{2\kappa} \tag*{(37)}
\end{aligned} ( 1 + Ω + ⋯ + Ω κ ) ∘ [ β ln Z BH ] = ( − 1 ) κ 4 ( κ − 1 )! G 2 κ + 1 π κ − 1 = ( − 1 ) κ κ ! a 2 κ (37) in perfect agreement18 (7) . Eq. (37) holds as an exact (non-perturbative) identity, and is true at all values of the parameters.
Direct checks at low orders in the small x i x_i x i
While this paper has focused on the study of ln Z \ln Z ln Z [7 ] (see also [10 , 11 , 12 , 13 ] ) that ln Z \ln Z ln Z x i x_i x i ln Z \ln Z ln Z [6 ] . In SM (see around (S18)) we have used the methods of [6 ] to work out the explicit form of the constraints on ln Z r o t β \frac{\ln Z^{\mathrm{rot}}}{\beta} β l n Z rot x i x_i x i d = 4 d=4 d = 4 x i x_i x i (10) of [6 ] ) into Eq. (6) , Taylor expanding up to the same order in x i x_i x i x i x_i x i (76) in SM, in perfect agreement with Eq. (15) of [6 ] at a = 0 a=0 a = 0
Discussion
The authors of [6 ] had predicted the existence of an infinite tower of constraints on coefficients of the high temperature expansion, but had no expectation for the precise form of these equations. In this paper we have demonstrated that this entire tower of relations is equivalent to the single, beautiful constraint equation, Eq. (6) .
The order-by-order expansion of this equation in β 2 \beta^2 β 2 (10) .
It would be good to understand the precise conditions on CFTs that guarantee that ln Z \ln Z ln Z [6 , 7 ] for relevant discussion). It would also be useful to generalize our analysis to CP violating theories in dimensions d = 6 , 10 , … d = 6, 10, \ldots d = 6 , 10 , … [14 , 15 , 16 , 17 , 18 , 19 , 20 ] where thermal EFT has been useful.
Our striking central equation Eq. (6) , which asserts a relationship between thermodynamics in the time direction and in the rotation planes, calls for a direct physical interpretation, perhaps along the lines of Appendix C. However, we emphasize that Eq. (6) has only been derived order-by-order in the high temperature expansion (i.e. in an expansion around ( b 0 , b 1 , … , b κ ) = ( 0 , 0 , … , 0 ) (b_0,b_1,\ldots,b_\kappa)=(0,0,\ldots,0) ( b 0 , b 1 , … , b κ ) = ( 0 , 0 , … , 0 ) ln Z \ln Z ln Z (1) converges only when ∣ Re ( b i ) ∣ < b 0 ∀ i ≠ 0 |\operatorname{Re}(b_i)|<b_0\ \forall i\ne0 ∣ Re ( b i ) ∣ < b 0 ∀ i = 0 [7 , 21 , 22 , 23 , 24 ] ), and as Ω \Omega Ω b i b_i b i b i b_i b i (6) would seem to require an analytic continuation of ln Z \ln Z ln Z [25 , 26 , 27 , 28 ] for relevant discussion).
Relatedly, it was explained in [29 ] that all ‘bulk’ terms (see [29 ] for terminology) in the all-orders expansion of ln Z \ln Z ln Z ( b 0 , b 1 … b κ ) = ( 0 , 2 π i r 1 s 1 , 2 π i r 2 s 2 , … , 2 π i r κ s κ ) (b_0,b_1\ldots b_\kappa)=\left(0,\frac{2\pi i r_1}{s_1},\frac{2\pi i r_2}{s_2},\ldots,\frac{2\pi i r_\kappa}{s_\kappa}\right) ( b 0 , b 1 … b κ ) = ( 0 , s 1 2 π i r 1 , s 2 2 π i r 2 , … , s κ 2 π i r κ ) 1 S \frac{1}{S} S 1 S δ b i S\delta b_i S δ b i S S S s 1 , s 2 , … , s κ s_1,s_2,\ldots,s_\kappa s 1 , s 2 , … , s κ δ b i \delta b_i δ b i b i b_i b i ln Z \ln Z ln Z [30 ] for a potentially related study of free theories).
Finally, it is possible that an analysis similar to that of [6 ] and the current paper can also be carried out for the superconformal indices [31 , 32 ] in the so-called Cardy limit (see e.g. [33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 ] ). Such a study would require the generalization to partition functions with additional conserved R R R [23 , 48 ] ).
Acknowledgments
We thank E. D’Hoker, T. Dumitrescu, H. T. Lam, O. Parrikar, P. Rath, V. Schomerus, S. Trivedi, and Y. Wang for very useful discussions. We would also like to thank K. Bajaj, J. Choi, K. Jensen, Z. Komargodski, A. Miscioscia, C. Patel, F. Popov, S. Van Leeuwen, and S. Wadia for comments on a preliminary version of this manuscript. N.B. is supported by the U.S. Department of Energy, Office of Science, under grant Contract Number DE-SC0026324. The work of A.A., H.A., A.G., V.K., S.M., J.M., S.P., A.R. and P.R. was supported by the Department of Atomic Energy, Government of India, under Project Identification Number RTI-4012 and the Infosys Endowment for the study of the Quantum Structure of Spacetime. A.A., H.A., A.G., V.K., S.M., J.M., S.P., A.R. and P.R. would also like to acknowledge their debt to the people of India for their steady support of the study of basic science.
AI Disclosure Statement
The formula in the last line of Eq. (3) was developed in discussions with Claude Fable 5.1; the study of κ ≥ 3 \kappa\ge3 κ ≥ 3 (121) and Eq. (216) in SM) was developed in discussions with ChatGPT Astra 6.1 and independently with Claude Fable 5.1; and the material presented between Eq. (80) and Eq. (93) in SM were developed in discussions with ChatGPT Sol 6.1. In addition, the authors used Claude Opus 5.5 to assist with checks of all calculations and preparation of the figures. Every part of this manuscript (including subsections whose material was developed in discussion with AI tools) is written entirely by the authors. The authors take full responsibility for the scientific content of the paper.
References
Appendix A: Constraining polynomials
Every symmetric polynomial of ( y 0 , y 1 , ⋯ , y κ ) (y_0,y_1,\cdots,y_\kappa) ( y 0 , y 1 , ⋯ , y κ )
E 0 ( y 0 , y 1 , ⋯ , y κ ) : = 1 , E_0(y_0,y_1,\cdots,y_\kappa):=1, E 0 ( y 0 , y 1 , ⋯ , y κ ) := 1 , E r ( y 0 , y 1 , ⋯ , y κ ) : = ∑ 0 ≤ i 1 < ⋯ < i r ≤ κ y i 1 ⋯ y i r , for r = 1 , ⋯ , κ + 1. (38) E_r(y_0,y_1,\cdots,y_\kappa):=\sum_{0\leq i_1<\cdots<i_r\leq\kappa}y_{i_1}\cdots y_{i_r},\quad\text{for }r=1,\cdots,\kappa+1. \tag*{(38)} E r ( y 0 , y 1 , ⋯ , y κ ) := 0 ≤ i 1 < ⋯ < i r ≤ κ ∑ y i 1 ⋯ y i r , for r = 1 , ⋯ , κ + 1. (38) Note that each of ( y 0 , y 1 , ⋯ , y κ ) (y_0,y_1,\cdots,y_\kappa) ( y 0 , y 1 , ⋯ , y κ )
w κ + 1 − E 1 w κ + ⋯ + ( − 1 ) r E r w κ + 1 − r + ⋯ + ( − 1 ) κ + 1 E κ + 1 = 0. (39) w^{\kappa+1}-E_1w^\kappa+\cdots+(-1)^rE_rw^{\kappa+1-r}+\cdots+(-1)^{\kappa+1}E_{\kappa+1}=0. \tag*{(39)} w κ + 1 − E 1 w κ + ⋯ + ( − 1 ) r E r w κ + 1 − r + ⋯ + ( − 1 ) κ + 1 E κ + 1 = 0. (39) Now consider a polynomial of the above variables that is completely symmetric in y 1 ⋯ y κ y_1\cdots y_\kappa y 1 ⋯ y κ y 0 y_0 y 0 E 1 , E 2 , ⋯ , E κ + 1 E_1,E_2,\cdots,E_{\kappa+1} E 1 , E 2 , ⋯ , E κ + 1 y 0 y_0 y 0 y 0 y_0 y 0 (39) , however, powers of y 0 y_0 y 0 ≥ κ + 1 \geq\kappa+1 ≥ κ + 1 y 0 y_0 y 0 E r E_r E r y 1 ⋯ y κ y_1\cdots y_\kappa y 1 ⋯ y κ (9) , with y i = b i 2 y_i=b_i^2 y i = b i 2
In SM (see around (S51)) we insert (10) into Eq. (3) , and then insert this expansion into Eq. (6) , and equate powers of β 2 \beta^2 β 2 I q ( 0 ) , I q ( 1 ) , … , I q ( κ − 1 ) I_q^{(0)},I_q^{(1)},\ldots,I_q^{(\kappa-1)} I q ( 0 ) , I q ( 1 ) , … , I q ( κ − 1 )
I q ( κ ) = 0 , ∀ q ≥ κ . (40) I_q^{(\kappa)}=0,\quad\forall q\geq\kappa. \tag*{(40)} I q ( κ ) = 0 , ∀ q ≥ κ . (40) Eq. (40) expresses a simple universal constraint on the coefficient functions of the high temperature expansion.
Appendix B: Simplification at cyclotomic points
Throughout this Appendix we work at the special values
( ω 1 2 , ω 2 2 … ω κ 2 ) = ( e 2 π i κ + 1 , e 4 π i κ + 1 , … , e 2 π κ i κ + 1 ) (41) (\omega_1^2,\omega_2^2\ldots\omega_\kappa^2)=\left(e^{\frac{2\pi i}{\kappa+1}},e^{\frac{4\pi i}{\kappa+1}},\ldots,e^{\frac{2\pi\kappa i}{\kappa+1}}\right) \tag*{(41)} ( ω 1 2 , ω 2 2 … ω κ 2 ) = ( e κ + 1 2 π i , e κ + 1 4 π i , … , e κ + 1 2 π κi ) (41) which implies
( b 0 2 , b 1 2 , … b κ 2 ) = β 2 ( 1 , e 2 π i κ + 1 , … , e 2 π κ i κ + 1 ) (b_0^2,b_1^2,\ldots b_\kappa^2)=\beta^2\left(1,e^{\frac{2\pi i}{\kappa+1}},\ldots,e^{\frac{2\pi\kappa i}{\kappa+1}}\right) ( b 0 2 , b 1 2 , … b κ 2 ) = β 2 ( 1 , e κ + 1 2 π i , … , e κ + 1 2 π κi ) ⟹ Ω ∘ ( b 0 2 , b 1 2 , … , b κ 2 ) = β 2 ( e 2 π i κ + 1 , e 4 π i κ + 1 , … , e 2 π κ i κ + 1 , 1 ) = e 2 π i κ + 1 ( b 0 2 , b 1 2 , b 2 2 , … , b κ 2 ) . (42) \Longrightarrow\Omega\circ(b_0^2,b_1^2,\ldots,b_\kappa^2)=\beta^2\left(e^{\frac{2\pi i}{\kappa+1}},e^{\frac{4\pi i}{\kappa+1}},\ldots,e^{\frac{2\pi\kappa i}{\kappa+1}},1\right)
=e^{\frac{2\pi i}{\kappa+1}}(b_0^2,b_1^2,b_2^2,\ldots,b_\kappa^2). \tag*{(42)} ⟹ Ω ∘ ( b 0 2 , b 1 2 , … , b κ 2 ) = β 2 ( e κ + 1 2 π i , e κ + 1 4 π i , … , e κ + 1 2 π κi , 1 ) = e κ + 1 2 π i ( b 0 2 , b 1 2 , b 2 2 , … , b κ 2 ) . (42) Consider any degree h h h f f f b μ 2 b_\mu^2 b μ 2 μ = 0 ⋯ κ \mu=0\cdots\kappa μ = 0 ⋯ κ h h h (42) that the action of the cyclic symmetrization operator
S = 1 + Ω + Ω 2 + ⋯ + Ω κ κ + 1 (43) \mathcal{S}=\frac{1+\Omega+\Omega^2+\cdots+\Omega^\kappa}{\kappa+1} \tag*{(43)} S = κ + 1 1 + Ω + Ω 2 + ⋯ + Ω κ (43) on f f f f f f h h h κ + 1 \kappa+1 κ + 1 ln Z β \frac{\ln Z}{\beta} β l n Z b μ 2 b_\mu^2 b μ 2 β 2 \beta^2 β 2 S \mathcal{S} S ω i 2 \omega_i^2 ω i 2 ( β 2 ) q ( κ + 1 ) (\beta^2)^{q(\kappa+1)} ( β 2 ) q ( κ + 1 ) q q q ω i 2 \omega_i^2 ω i 2
( κ + 1 ) S ∘ ( ln Z β ) = ( κ + 1 ) ∑ q = 0 ∞ ( β 2 ) ( κ + 1 ) q [ ln Z β ] ( β 2 ) ( κ + 1 ) q , (44) (\kappa+1)\mathcal{S}\circ\left(\frac{\ln Z}{\beta}\right)
=(\kappa+1)\sum_{q=0}^{\infty}(\beta^2)^{(\kappa+1)q}\left[\frac{\ln Z}{\beta}\right]_{(\beta^2)^{(\kappa+1)q}}, \tag*{(44)} ( κ + 1 ) S ∘ ( β ln Z ) = ( κ + 1 ) q = 0 ∑ ∞ ( β 2 ) ( κ + 1 ) q [ β ln Z ] ( β 2 ) ( κ + 1 ) q , (44) where ∣ ∣ β n \left.\vphantom{\big|}\right|_{\beta^n} β n β n \beta^n β n (44) into (8) and comparing powers of β 2 \beta^2 β 2
( κ + 1 ) × [ ln Z β ] ( β 2 ) ( κ + 1 ) q = ( − 1 ) κ κ ! a 2 κ δ q , 0 , ∀ q ∈ Z ≥ 0 (45) (\kappa+1)\times\left[\frac{\ln Z}{\beta}\right]_{(\beta^2)^{(\kappa+1)q}}=(-1)^\kappa\kappa!a_{2\kappa}\delta_{q,0},\quad\forall q\in\mathbb{Z}_{\geq0} \tag*{(45)} ( κ + 1 ) × [ β ln Z ] ( β 2 ) ( κ + 1 ) q = ( − 1 ) κ κ ! a 2 κ δ q , 0 , ∀ q ∈ Z ≥ 0 (45) in perfect agreement with Eq. (6) and Eq. (21) of [6 ] .
In this Appendix, we construct three large conformal maps that cycle through S 3 × R S^3\times R S 3 × R A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1 A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1 (24) ) closing in to the Z 3 \mathbb{Z}_3 Z 3 b i b_i b i
Consider the metric g μ ν g_{\mu\nu} g μν
d s 2 = d τ 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 = d τ 2 + d l 2 4 ( 1 − l ) l + l d ϕ 1 2 + ( 1 − l ) d ϕ 2 2 , (46) \begin{aligned}
ds^2=d\tau^2+d\theta^2+\sin^2\theta\,d\phi_1^2+\cos^2\theta\,d\phi_2^2 \\
&=d\tau^2+\frac{dl^2}{4(1-l)l}+l\,d\phi_1^2+(1-l)d\phi_2^2, \tag*{(46)}
\end{aligned} d s 2 = d τ 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 = d τ 2 + 4 ( 1 − l ) l d l 2 + l d ϕ 1 2 + ( 1 − l ) d ϕ 2 2 , (46) where l = sin 2 θ l=\sin^2\theta l = sin 2 θ l ∈ ( 0 , 1 ) l\in(0,1) l ∈ ( 0 , 1 )
Map to A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1
Pulling out an overall factor of l l l ∼ \sim ∼
d s 2 = d τ 2 l + d ϕ 1 2 + 1 − l l d ϕ 2 2 + d l 2 4 l 2 ( 1 − l ) (47) ds^2=\frac{d\tau^2}{l}+d\phi_1^2+\frac{1-l}{l}d\phi_2^2+\frac{dl^2}{4l^2(1-l)} \tag*{(47)} d s 2 = l d τ 2 + d ϕ 1 2 + l 1 − l d ϕ 2 2 + 4 l 2 ( 1 − l ) d l 2 (47) where ∼ \sim ∼ ϕ 1 \phi_1 ϕ 1 (47) . Changing coordinates,
l = 1 1 − l ~ , l ~ ∈ ( − ∞ , 0 ) , (48) l=\frac{1}{1-\tilde l},\quad\tilde l\in(-\infty,0), \tag*{(48)} l = 1 − l ~ 1 , l ~ ∈ ( − ∞ , 0 ) , (48) d s 2 = ( 1 − l ~ ) d τ 2 + d ϕ 1 2 − l ~ d ϕ 2 2 − d l ~ 2 4 l ~ ( 1 − l ~ ) . ds^2=(1-\tilde l)d\tau^2+d\phi_1^2-\tilde l\,d\phi_2^2-\frac{d\tilde l^2}{4\tilde l(1-\tilde l)}. d s 2 = ( 1 − l ~ ) d τ 2 + d ϕ 1 2 − l ~ d ϕ 2 2 − 4 l ~ ( 1 − l ~ ) d l ~ 2 . The further redefinition l ~ = − sinh 2 ρ \tilde l=-\sinh^2\rho l ~ = − sinh 2 ρ ρ ∈ ( 0 , ∞ ) \rho\in(0,\infty) ρ ∈ ( 0 , ∞ )
d s 2 = d ϕ 1 2 + d ρ 2 + cosh 2 ρ d τ 2 + sinh 2 ρ d ϕ 2 2 , (49) ds^2=d\phi_1^2+d\rho^2+\cosh^2\rho\,d\tau^2+\sinh^2\rho\,d\phi_2^2, \tag*{(49)} d s 2 = d ϕ 1 2 + d ρ 2 + cosh 2 ρ d τ 2 + sinh 2 ρ d ϕ 2 2 , (49) the metric of (Euclidean) A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1 ρ → ∞ \rho\to\infty ρ → ∞ A d S 3 AdS_3 A d S 3 l = 0 l=0 l = 0 ϕ 1 \phi_1 ϕ 1 (46) .
The CFT path integral on S 3 × R S^3 \times R S 3 × R H 3 × S 1 H_3 \times S^1 H 3 × S 1 (14) carries over to the H 3 × S 1 H_3 \times S^1 H 3 × S 1
Tr H 3 × S 1 e − β H + β ω 2 J + β ω 1 Q (50) \operatorname{Tr}_{H_3 \times S^1} e^{-\beta H+\beta\omega_2 J+\beta\omega_1 Q} \tag*{(50)} Tr H 3 × S 1 e − β H + β ω 2 J + β ω 1 Q (50) where H H H A d S 3 AdS_3 A d S 3 J J J S 1 S^1 S 1 A d S 3 AdS_3 A d S 3 Q Q Q S 1 S^1 S 1 ϕ 1 \phi_1 ϕ 1 (50) is identical to the S 3 × S 1 S^3 \times S^1 S 3 × S 1 (50) is a nonperturbative statement; the partition function on S 3 × R S^3 \times R S 3 × R A d S 3 × S 1 AdS_3 \times S^1 A d S 3 × S 1 (14) , are literally the same function. In SM, we check this equivalence for a free conformally coupled scalar.
Analytic continuation to S 3 × R S^3 \times R S 3 × R
The complex variable change
( τ , ϕ 1 , ϕ 2 ) = ( i ϕ ~ 2 , − i τ ~ , ϕ ~ 1 ) (51) (\tau,\phi_1,\phi_2) = (i\widetilde{\phi}_2,-i\widetilde{\tau},\widetilde{\phi}_1) \tag*{(51)} ( τ , ϕ 1 , ϕ 2 ) = ( i ϕ 2 , − i τ , ϕ 1 ) (51) turns the A d S 3 × S 1 AdS_3 \times S^1 A d S 3 × S 1
d s 2 = − ( d τ ~ 2 + d l ~ 2 4 l ~ ( 1 − l ~ ) + l ~ d ϕ ~ 1 2 + ( 1 − l ~ ) d ϕ ~ 2 2 ) . (52) ds^2=-\left(d\widetilde{\tau}^2+\frac{d\widetilde{l}^2}{4\widetilde{l}(1-\widetilde{l})}+\widetilde{l}\,d\widetilde{\phi}_1^2+(1-\widetilde{l})d\widetilde{\phi}_2^2\right). \tag*{(52)} d s 2 = − ( d τ 2 + 4 l ( 1 − l ) d l 2 + l d ϕ 1 2 + ( 1 − l ) d ϕ 2 2 ) . (52) Up to a Weyl transformation that removes the overall minus sign, Eq. (52) has the same form in tilde coordinates, as the metric Eq. (46) had in the old coordinates, and so yields S 3 × R S^3 \times R S 3 × R l ~ \widetilde{l} l (48) to the ‘unphysical’ range l ~ ∈ ( 0 , 1 ) \widetilde{l} \in(0,1) l ∈ ( 0 , 1 ) A d S 3 × S 1 AdS_3 \times S^1 A d S 3 × S 1 S 3 × R S^3 \times R S 3 × R
Since
( ∂ τ , − i ∂ ϕ 1 , − i ∂ ϕ 2 ) = ( − i ∂ ϕ ~ 2 , ∂ τ ~ , − i ∂ ϕ ~ 1 ) , (53) (\partial_\tau,-i\partial_{\phi_1},-i\partial_{\phi_2})=(-i\partial_{\widetilde{\phi}_2},\partial_{\widetilde{\tau}},-i\partial_{\widetilde{\phi}_1}), \tag*{(53)} ( ∂ τ , − i ∂ ϕ 1 , − i ∂ ϕ 2 ) = ( − i ∂ ϕ 2 , ∂ τ , − i ∂ ϕ 1 ) , (53) it follows that
v = b 0 ∂ τ + b 1 ( − i ∂ ϕ 1 ) + b 2 ( − i ∂ ϕ 2 ) = b ~ 0 ∂ τ ~ + b ~ 1 ( − i ∂ ϕ ~ 1 ) + b ~ 2 ( − i ∂ ϕ ~ 2 ) (54) v=b_0\partial_\tau+b_1(-i\partial_{\phi_1})+b_2(-i\partial_{\phi_2})
=\widetilde{b}_0\partial_{\widetilde{\tau}}+\widetilde{b}_1(-i\partial_{\widetilde{\phi}_1})+\widetilde{b}_2(-i\partial_{\widetilde{\phi}_2}) \tag*{(54)} v = b 0 ∂ τ + b 1 ( − i ∂ ϕ 1 ) + b 2 ( − i ∂ ϕ 2 ) = b 0 ∂ τ + b 1 ( − i ∂ ϕ 1 ) + b 2 ( − i ∂ ϕ 2 ) (54) where ( b ~ 0 , b ~ 1 , b ~ 2 ) = ( b 1 , b 2 , b 0 ) (\widetilde{b}_0,\widetilde{b}_1,\widetilde{b}_2)=(b_1,b_2,b_0) ( b 0 , b 1 , b 2 ) = ( b 1 , b 2 , b 0 )
The variable z z z (15) is related to l l l
z = ( 1 − l ) b 0 2 b 1 2 − b 1 2 b 2 2 + l b 2 2 b 0 2 b 0 2 − l b 1 2 + ( l − 1 ) b 2 2 , z ~ = ( 1 − l ~ ) b ~ 0 2 b ~ 1 2 − b ~ 1 2 b ~ 2 2 + l ~ b ~ 2 2 b ~ 0 2 b ~ 0 2 − l ~ b ~ 1 2 + ( l ~ − 1 ) b ~ 2 2 . (55) z=\frac{(1-l)b_0^2b_1^2-b_1^2b_2^2+lb_2^2b_0^2}{b_0^2-lb_1^2+(l-1)b_2^2},
\qquad
\widetilde{z}=\frac{(1-\widetilde{l})\widetilde{b}_0^2\widetilde{b}_1^2-\widetilde{b}_1^2\widetilde{b}_2^2+\widetilde{l}\widetilde{b}_2^2\widetilde{b}_0^2}{\widetilde{b}_0^2-\widetilde{l}\widetilde{b}_1^2+(\widetilde{l}-1)\widetilde{b}_2^2}. \tag*{(55)} z = b 0 2 − l b 1 2 + ( l − 1 ) b 2 2 ( 1 − l ) b 0 2 b 1 2 − b 1 2 b 2 2 + l b 2 2 b 0 2 , z = b 0 2 − l b 1 2 + ( l − 1 ) b 2 2 ( 1 − l ) b 0 2 b 1 2 − b 1 2 b 2 2 + l b 2 2 b 0 2 . (55) Inserting Eq. (48) into Eq. (55) , it is easily verified that z ~ = z \widetilde{z}=z z = z
Now consider the quantity I \mathcal{I} I I \mathcal{I} I
I b 0 , b 1 , b 2 ( z ) = I b 1 , b 2 , b 0 ( z ) (56) \mathcal{I}_{b_0,b_1,b_2}(z)=\mathcal{I}_{b_1,b_2,b_0}(z) \tag*{(56)} I b 0 , b 1 , b 2 ( z ) = I b 1 , b 2 , b 0 ( z ) (56) yielding a structural explanation for Eq. (19) .
We can now rerun this analysis with the role of ϕ 1 \phi_1 ϕ 1 ϕ 2 \phi_2 ϕ 2 ( 1 − l ) (1-l) ( 1 − l ) l l l (46) , to obtain a second A d S 3 × S 1 AdS_3 \times S^1 A d S 3 × S 1 S 3 × R S^3 \times R S 3 × R Z 3 Z_3 Z 3
The workout of this Appendix generalizes in a straightforward manner to d = 2 κ d=2\kappa d = 2 κ κ ≥ 3 \kappa\ge3 κ ≥ 3
Appendix D: Counting constraints Eq. (40)
Completely Bose symmetric degree-q q q ( b 0 2 , b 1 2 … b κ 2 ) (b_0^2,b_1^2\ldots b_\kappa^2) ( b 0 2 , b 1 2 … b κ 2 ) 1 , 2 , … , κ + 1 1,2,\ldots,\kappa+1 1 , 2 , … , κ + 1 E 1 E_1 E 1 E 2 … E κ + 1 E_2\ldots E_{\kappa+1} E 2 … E κ + 1 (38) ). Such polynomials thus constitute an n q s y m n_q^{\mathrm{sym}} n q sym
1 ∏ i = 1 κ + 1 ( 1 − t i ) = ∑ q = 0 ∞ n q s y m t q . (57) \frac{1}{\prod_{i=1}^{\kappa+1}(1-t^i)}=\sum_{q=0}^{\infty}n_q^{\mathrm{sym}}t^q. \tag*{(57)} ∏ i = 1 κ + 1 ( 1 − t i ) 1 = q = 0 ∑ ∞ n q sym t q . (57) It follows from Eq. (9) that polynomials symmetrical only under permutations of ( b 1 2 , b 2 2 , … , b κ 2 ) (b_1^2,b_2^2,\ldots,b_\kappa^2) ( b 1 2 , b 2 2 , … , b κ 2 ) n q b n_q^b n q b
∑ i = 0 κ t i ∏ i = 1 κ + 1 ( 1 − t i ) = 1 ( 1 − t ) 2 ∏ i = 2 κ ( 1 − t i ) = ∑ q = 0 ∞ n q b t q (58) \frac{\sum_{i=0}^{\kappa}t^i}{\prod_{i=1}^{\kappa+1}(1-t^i)}=\frac{1}{(1-t)^2\prod_{i=2}^{\kappa}(1-t^i)}=\sum_{q=0}^{\infty}n_q^b t^q \tag*{(58)} ∏ i = 1 κ + 1 ( 1 − t i ) ∑ i = 0 κ t i = ( 1 − t ) 2 ∏ i = 2 κ ( 1 − t i ) 1 = q = 0 ∑ ∞ n q b t q (58) in agreement with the ‘naive count’, Eq. (C.3) of [6 ] .
Similarly, degree q q q (9) , but with I q ( κ ) = 0 I_q^{(\kappa)}=0 I q ( κ ) = 0 n q m n_q^m n q m
∑ i = 0 κ − 1 t i ∏ i = 1 κ − 1 ( 1 − t i ) = 1 ( 1 − t ) 2 ( 1 − t κ + 1 ) ∏ i = 2 κ − 1 ( 1 − t i ) = ∑ q = 0 ∞ n q m t q (59) \begin{aligned}
\frac{\sum_{i=0}^{\kappa-1}t^i}{\prod_{i=1}^{\kappa-1}(1-t^i)}=\frac{1}{(1-t)^2(1-t^{\kappa+1})\prod_{i=2}^{\kappa-1}(1-t^i)} \\
&=\sum_{q=0}^{\infty}n_q^m t^q \tag*{(59)}
\end{aligned} ∏ i = 1 κ − 1 ( 1 − t i ) ∑ i = 0 κ − 1 t i = ( 1 − t ) 2 ( 1 − t κ + 1 ) ∏ i = 2 κ − 1 ( 1 − t i ) 1 = q = 0 ∑ ∞ n q m t q (59) in agreement with the local count Eq. C.4 of [6 ] .
By subtracting Eq. (59) from Eq. (58) , we see by comparison with Eq. (5) that the condition that I q ( κ ) = 0 I_q^{(\kappa)}=0 I q ( κ ) = 0 n q b − n q m = d q n_q^b-n_q^m=d_q n q b − n q m = d q T q T_q T q [6 ] . See around Eq. (120) in SM for an S 3 S_3 S 3
Supplemental Material
Computation of the measure factor
In the coordinates Eq. (12) , the measure factor in Eq. (11) equals
d 4 x g 4 ( V 2 ) 2 = cos θ sin θ d ϕ 1 d ϕ 2 d τ d θ β 4 ( 1 − ω 1 2 sin 2 θ − ω 2 2 cos 2 θ ) 2 = 1 2 β 4 d ϕ 1 d ϕ 2 d τ d ( sin 2 θ ) [ 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) ] 2 = d ϕ 1 d ϕ 2 d τ 2 β 4 ( ω 1 2 − ω 2 2 ) d ( 1 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) ) = d γ 2 d ϕ 1 d ϕ 2 d τ 2 β 4 ( ω 1 2 − ω 2 2 ) = − d z d ϕ 1 d ϕ 2 d τ 2 ( b 1 2 − b 2 2 ) ( b 0 2 − b 1 2 ) ( b 0 2 − b 2 2 ) (60) \frac{d^4x\sqrt{g_4}}{(\mathcal{V}^2)^2}
= \frac{\cos\theta\sin\theta\,d\phi_1d\phi_2d\tau d\theta}{\beta^4(1-\omega_1^2\sin^2\theta-\omega_2^2\cos^2\theta)^2}
= \frac{1}{2\beta^4}\frac{d\phi_1d\phi_2d\tau d(\sin^2\theta)}{[1-\omega_2^2+\sin^2\theta(\omega_2^2-\omega_1^2)]^2}
= \frac{d\phi_1d\phi_2d\tau}{2\beta^4(\omega_1^2-\omega_2^2)}d\left(\frac{1}{1-\omega_2^2+\sin^2\theta(\omega_2^2-\omega_1^2)}\right)
= \frac{d\gamma^2d\phi_1d\phi_2d\tau}{2\beta^4(\omega_1^2-\omega_2^2)}
= -\frac{dzd\phi_1d\phi_2d\tau}{2(b_1^2-b_2^2)(b_0^2-b_1^2)(b_0^2-b_2^2)} \tag*{(60)} ( V 2 ) 2 d 4 x g 4 = β 4 ( 1 − ω 1 2 sin 2 θ − ω 2 2 cos 2 θ ) 2 cos θ sin θ d ϕ 1 d ϕ 2 d τ d θ = 2 β 4 1 [ 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) ] 2 d ϕ 1 d ϕ 2 d τ d ( sin 2 θ ) = 2 β 4 ( ω 1 2 − ω 2 2 ) d ϕ 1 d ϕ 2 d τ d ( 1 − ω 2 2 + sin 2 θ ( ω 2 2 − ω 1 2 ) 1 ) = 2 β 4 ( ω 1 2 − ω 2 2 ) d γ 2 d ϕ 1 d ϕ 2 d τ = − 2 ( b 1 2 − b 2 2 ) ( b 0 2 − b 1 2 ) ( b 0 2 − b 2 2 ) d z d ϕ 1 d ϕ 2 d τ (60) where γ 2 \gamma^2 γ 2 (15) and z z z (15) . 19 (11) reduces to Eq. (17) .
χ = x 1 ℓ x 2 + ( x 1 − x 2 ) ℓ , ( ℓ = sin 2 θ ) \chi=\frac{x_1\ell}{x_2+(x_1-x_2)\ell},\qquad(\ell=\sin^2\theta) χ = x 2 + ( x 1 − x 2 ) ℓ x 1 ℓ , ( ℓ = sin 2 θ ) d χ = x 1 x 2 ( x 2 + ( x 1 − x 2 ) ℓ ) 2 d ℓ = x 1 x 2 ( x 2 − x 1 ) d γ 2 (61) \begin{aligned}
d\chi=\frac{x_1x_2}{(x_2+(x_1-x_2)\ell)^2}d\ell\\
&=\frac{x_1x_2}{(x_2-x_1)}d\gamma^2 \tag*{(61)}
\end{aligned} d χ = ( x 2 + ( x 1 − x 2 ) ℓ ) 2 x 1 x 2 d ℓ = ( x 2 − x 1 ) x 1 x 2 d γ 2 (61) so that
γ 2 = ( 1 − χ ) x 1 + x 2 χ x 1 x 2 (62) \gamma^2=\frac{(1-\chi)x_1+x_2\chi}{x_1x_2} \tag*{(62)} γ 2 = x 1 x 2 ( 1 − χ ) x 1 + x 2 χ (62) Explicit constraints at low orders
In the main text of this paper, we have verified that the expansion of Eq. (6) , at order β 2 q \beta^{2q} β 2 q d q d_q d q (5) for a definition of d q d_q d q [6 ] using a different method. Recall that the constraints derived in [6 ] follow from the fact that I \mathcal{I} I (18) , up to total derivatives that do not affect the final partition function. Since (as we have demonstrated around Eq. (22) ), Eq. (6) also follows from exactly the same assumptions, it follows immediately that the partition functions built out of such I \mathcal{I} I (6) . As the space of partition functions that obey Eq. (6) has exactly the same dimension as the space of partition functions with a local I \mathcal{I} I (18) , it follows that Eq. (6) imposes no additional constraints on I \mathcal{I} I [6 ] .
As a sanity check, in this section we provide some direct calculational evidence for the conclusion of the previous paragraph. We work in the simplest dimension d = 4 d=4 d = 4
x 1 , x 2 → 0 , ( x 1 x 2 and β ) fixed (63) x_1,x_2\to0,\qquad\left(\frac{x_1}{x_2}\ \text{and}\ \beta\right)\ \text{fixed} \tag*{(63)} x 1 , x 2 → 0 , ( x 2 x 1 and β ) fixed (63) (see [6 , 7 ] ), rather than the ‘high temperature’ expansion in small β 2 \beta^2 β 2 (3) , the coefficient of β 2 q \beta^{2q} β 2 q q q q x i x_i x i q q q d q d_q d q q q q x i x_i x i x 1 m x 2 n x_1^mx_2^n x 1 m x 2 n h m , n ( β 2 ) h_{m,n}(\beta^2) h m , n ( β 2 ) 20 x i q x_i^q x i q ⌊ q 2 ⌋ − ⌊ q 3 ⌋ \left\lfloor\frac{q}{2}\right\rfloor-\left\lfloor\frac{q}{3}\right\rfloor ⌊ 2 q ⌋ − ⌊ 3 q ⌋ x i q x_i^q x i q [6 ] ). 21 β 2 \beta^{2} β 2 infinite number of constraints on coefficients. At a practical level, therefore, the small x i x_i x i β 2 \beta^{2} β 2 q q q x i x_i x i β 2 q \beta^{2q} β 2 q
In order to extract the constraints that follow from Eq. (6) in a small x i x_i x i (64)
( 1 + Ω + Ω 2 ) ∘ [ ln Z r o t ( b 0 2 , b 1 2 , b 2 2 ) β ] = 0 (64) \left(1+\Omega+\Omega^{2}\right)\circ\left[\frac{\ln Z^{\mathrm{rot}}\left(b_{0}^{2},b_{1}^{2},b_{2}^{2}\right)}{\beta}\right]=0 \tag*{(64)} ( 1 + Ω + Ω 2 ) ∘ [ β ln Z rot ( b 0 2 , b 1 2 , b 2 2 ) ] = 0 (64) where the Z 3 \mathbb{Z}_{3} Z 3 Ω \Omega Ω ( b 0 , b 1 , b 2 ) (b_{0},b_{1},b_{2}) ( b 0 , b 1 , b 2 ) H ( b 0 2 , b 1 2 , b 2 2 ) H(b_{0}^{2},b_{1}^{2},b_{2}^{2}) H ( b 0 2 , b 1 2 , b 2 2 )
ln Z r o t β = − H ( b 0 2 − b 1 2 ) ( b 0 2 − b 2 2 ) . (65) \frac{\ln Z^{\mathrm{rot}}}{\beta}=-\frac{H}{\left(b_{0}^{2}-b_{1}^{2}\right)\left(b_{0}^{2}-b_{2}^{2}\right)}. \tag*{(65)} β ln Z rot = − ( b 0 2 − b 1 2 ) ( b 0 2 − b 2 2 ) H . (65) ( b 1 2 − b 2 2 ) H ( b 0 2 , b 1 2 , b 2 2 ) + ( b 2 2 − b 0 2 ) H ( b 1 2 , b 2 2 , b 0 2 ) + ( b 0 2 − b 1 2 ) H ( b 2 2 , b 0 2 , b 1 2 ) = 0. (66) \left(b_{1}^{2}-b_{2}^{2}\right)H\left(b_{0}^{2},b_{1}^{2},b_{2}^{2}\right)+\left(b_{2}^{2}-b_{0}^{2}\right)H\left(b_{1}^{2},b_{2}^{2},b_{0}^{2}\right)+\left(b_{0}^{2}-b_{1}^{2}\right)H\left(b_{2}^{2},b_{0}^{2},b_{1}^{2}\right)=0. \tag*{(66)} ( b 1 2 − b 2 2 ) H ( b 0 2 , b 1 2 , b 2 2 ) + ( b 2 2 − b 0 2 ) H ( b 1 2 , b 2 2 , b 0 2 ) + ( b 0 2 − b 1 2 ) H ( b 2 2 , b 0 2 , b 1 2 ) = 0. (66) We now choose to view H H H ( β 2 , x 1 , x 2 ) (\beta^{2},x_{1},x_{2}) ( β 2 , x 1 , x 2 ) ( b 0 2 , b 1 2 , b 2 2 ) (b_{0}^{2},b_{1}^{2},b_{2}^{2}) ( b 0 2 , b 1 2 , b 2 2 ) (66) becomes
( x 2 − x 1 ) H ( β 2 , x 1 , x 2 ) − x 2 H ( β 2 ( 1 − x 1 ) , Ω x 1 , Ω x 2 ) + x 1 H ( β 2 ( 1 − x 2 ) , Ω 2 x 1 , Ω 2 x 2 ) = 0 , (67) (x_{2}-x_{1})H(\beta^{2},x_{1},x_{2})-x_{2}H\left(\beta^{2}(1-x_{1}),\Omega x_{1},\Omega x_{2}\right)+x_{1}H\left(\beta^{2}(1-x_{2}),\Omega^{2}x_{1},\Omega^{2}x_{2}\right)=0, \tag*{(67)} ( x 2 − x 1 ) H ( β 2 , x 1 , x 2 ) − x 2 H ( β 2 ( 1 − x 1 ) , Ω x 1 , Ω x 2 ) + x 1 H ( β 2 ( 1 − x 2 ) , Ω 2 x 1 , Ω 2 x 2 ) = 0 , (67) where Ω x i \Omega x_i Ω x i Ω \Omega Ω x i x_i x i
Ω ( x 1 , x 2 ) = ( x 2 − x 1 1 − x 1 , x 1 x 1 − 1 ) , \Omega(x_{1},x_{2})=\left(\frac{x_{2}-x_{1}}{1-x_{1}},\frac{x_{1}}{x_{1}-1}\right), Ω ( x 1 , x 2 ) = ( 1 − x 1 x 2 − x 1 , x 1 − 1 x 1 ) , Ω 2 ( x 1 , x 2 ) = ( x 2 x 2 − 1 , x 1 − x 2 1 − x 2 ) . (68) \Omega^{2}(x_{1},x_{2})=\left(\frac{x_{2}}{x_{2}-1},\frac{x_{1}-x_{2}}{1-x_{2}}\right). \tag*{(68)} Ω 2 ( x 1 , x 2 ) = ( x 2 − 1 x 2 , 1 − x 2 x 1 − x 2 ) . (68) In the second and third terms in Eq. (67) , the first argument β 2 \beta^{2} β 2 β 2 ( 1 − x 1 ) \beta^{2}(1-x_{1}) β 2 ( 1 − x 1 ) β 2 ( 1 − x 2 ) \beta^{2}(1-x_{2}) β 2 ( 1 − x 2 ) (67) is effectively ‘non-local’ in β 2 \beta^{2} β 2 β 2 \beta^{2} β 2 x i x_i x i F ( β 2 ) F(\beta^{2}) F ( β 2 )
F ( β 2 ( 1 − x i ) ) = ( 1 − x i ) θ F ( β 2 ) , θ = β 2 d d β 2 (69) F\left(\beta^{2}(1-x_i)\right)=(1-x_i)^{\theta}F(\beta^{2}),\qquad\theta=\beta^{2}\frac{d}{d\beta^{2}} \tag*{(69)} F ( β 2 ( 1 − x i ) ) = ( 1 − x i ) θ F ( β 2 ) , θ = β 2 d β 2 d (69) (one way to verify this equation is to expand F F F (67) can be rewritten in the form
( x 2 − x 1 ) H ( β 2 , x 1 , x 2 ) − x 2 ( 1 − x 1 ) θ H ( β 2 , Ω x 1 , Ω x 2 ) + x 1 ( 1 − x 2 ) θ H ( β 2 , Ω 2 x 1 , Ω 2 x 2 ) = 0 (70) (x_{2}-x_{1})H(\beta^{2},x_{1},x_{2})-x_{2}(1-x_{1})^{\theta}H(\beta^{2},\Omega x_{1},\Omega x_{2})+x_{1}(1-x_{2})^{\theta}H(\beta^{2},\Omega^{2}x_{1},\Omega^{2}x_{2})=0 \tag*{(70)} ( x 2 − x 1 ) H ( β 2 , x 1 , x 2 ) − x 2 ( 1 − x 1 ) θ H ( β 2 , Ω x 1 , Ω x 2 ) + x 1 ( 1 − x 2 ) θ H ( β 2 , Ω 2 x 1 , Ω 2 x 2 ) = 0 (70) where
( 1 − x i ) θ = 1 − θ x i + θ ( θ − 1 ) 2 ! x i 2 − θ ( θ − 1 ) ( θ − 2 ) 3 ! x i 3 + … (71) (1-x_i)^{\theta}=1-\theta x_i+\frac{\theta(\theta-1)}{2!}x_i^{2}-\frac{\theta(\theta-1)(\theta-2)}{3!}x_i^{3}+\ldots \tag*{(71)} ( 1 − x i ) θ = 1 − θ x i + 2 ! θ ( θ − 1 ) x i 2 − 3 ! θ ( θ − 1 ) ( θ − 2 ) x i 3 + … (71) and θ \theta θ (69) .
We now follow [6 ] (See Eq. (11) of [6 ] ) to expand
H = ∑ m , n = 0 ∞ h m , n ( β 2 ) x 1 m x 2 n (72) H=\sum_{m,n=0}^{\infty}h_{m,n}(\beta^{2})x_{1}^{m}x_{2}^{n} \tag*{(72)} H = m , n = 0 ∑ ∞ h m , n ( β 2 ) x 1 m x 2 n (72) with h m , n ( β 2 ) = h n , m ( β 2 ) h_{m,n}(\beta^2)=h_{n,m}(\beta^2) h m , n ( β 2 ) = h n , m ( β 2 ) (72) into (70) , Taylor expanding Ω x i \Omega x_i Ω x i Ω 2 x i \Omega^2 x_i Ω 2 x i x i x_i x i x 1 m x 2 n x_1^m x_2^n x 1 m x 2 n m + n = p m+n=p m + n = p
∑ a , b > 0 , a ≥ b a + b ≤ p D a , b [ h a , b ( β 2 ) ] = 0. (73) \sum_{\substack{a,b>0,\ a\ge b\\a+b\le p}}D_{a,b}\left[h_{a,b}(\beta^2)\right]=0 . \tag*{(73)} a , b > 0 , a ≥ b a + b ≤ p ∑ D a , b [ h a , b ( β 2 ) ] = 0. (73) Here D a , b D_{a,b} D a , b θ \theta θ p − ( a + b ) p-(a+b) p − ( a + b ) β \beta β
We have implemented this procedure for all monomials with p = m + n ≤ 7 p=m+n\le7 p = m + n ≤ 7 D a , b D_{a,b} D a , b p p p D a , b D_{a,b} D a , b a + b ≤ p a+b\le p a + b ≤ p
( b , a ) = ( 0 , 0 ) , ( 0 , 1 ) , … ( 0 , p ) , ( 1 , 1 ) , ( 1 , 2 ) … ( 1 , p − 1 ) , ( 2 , 2 ) … (74) (b,a)=(0,0),\ (0,1),\ldots(0,p),\ (1,1),\ (1,2)\ldots(1,p-1),\ (2,2)\ldots \tag*{(74)} ( b , a ) = ( 0 , 0 ) , ( 0 , 1 ) , … ( 0 , p ) , ( 1 , 1 ) , ( 1 , 2 ) … ( 1 , p − 1 ) , ( 2 , 2 ) … (74) In other words we first list D a , b D_{a,b} D a , b b = 0 b=0 b = 0 a a a p p p b = 1 b=1 b = 1 a a a D a , b D_{a,b} D a , b
( p + 1 ) + ( p − 1 ) + ( p − 3 ) + … ( p + 1 − 2 ⌊ p + 1 2 ⌋ ) = ⌊ ( p + 2 ) 2 4 ⌋ . (75) (p+1)+(p-1)+(p-3)+\ldots\left(p+1-2\left\lfloor\frac{p+1}{2}\right\rfloor\right)=\left\lfloor\frac{(p+2)^2}{4}\right\rfloor. \tag*{(75)} ( p + 1 ) + ( p − 1 ) + ( p − 3 ) + … ( p + 1 − 2 ⌊ 2 p + 1 ⌋ ) = ⌊ 4 ( p + 2 ) 2 ⌋ . (75) So we have 4 terms at p = 2 p=2 p = 2 p = 4 p=4 p = 4 p = 5 p=5 p = 5 p = 6 p=6 p = 6 p = 7 p=7 p = 7
As an example of the use of the table Eq. (S18), we use the p = 2 p=2 p = 2 p = 2 p=2 p = 2
( θ 2 ) h 0 , 0 + 3 ( θ − 1 1 ) h 1 , 0 + 3 ( h 1 , 1 + h 2 , 0 ) = 0 ⇒ ( β 2 d d β 2 − 1 ) ( β 2 d d β 2 ) h 0 , 0 + 6 ( β 2 d d β 2 − 1 ) h 1 , 0 + 6 ( h 2 , 0 + h 1 , 1 ) = 0 (76) \begin{aligned}
\binom{\theta}{2}h_{0,0}+3\binom{\theta-1}{1}h_{1,0}+3(h_{1,1}+h_{2,0})=0 \\
&\Rightarrow\left(\beta^2\frac{d}{d\beta^2}-1\right)\left(\beta^2\frac{d}{d\beta^2}\right)h_{0,0}+6\left(\beta^2\frac{d}{d\beta^2}-1\right)h_{1,0}+6(h_{2,0}+h_{1,1})=0 \tag*{(76)}
\end{aligned} ( 2 θ ) h 0 , 0 + 3 ( 1 θ − 1 ) h 1 , 0 + 3 ( h 1 , 1 + h 2 , 0 ) = 0 ⇒ ( β 2 d β 2 d − 1 ) ( β 2 d β 2 d ) h 0 , 0 + 6 ( β 2 d β 2 d − 1 ) h 1 , 0 + 6 ( h 2 , 0 + h 1 , 1 ) = 0 (76) (S17) is in perfect agreement with Eq. (15) of [6 ] after setting a = 0 a=0 a = 0 ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot ln Z \ln Z ln Z
Table S18.
As we have explained above, the table Eq. (S18) lists the form of the semi-universal constraints (for p ≤ 7 p \le7 p ≤ 7 (70) . We now compare these constraints with those obtained using the methods of [6 ] , i.e. by proceeding as follows. We simply plug the expansion Eq. (S.13) of [6 ] into Eq. (17) , and then plug (19) into the resultant equation, and perform the integral over z z z z z z H H H (72) . While the number of distinct functions h m , n ( β 2 ) h_{m,n}(\beta^2) h m , n ( β 2 ) m + n = p m+n=p m + n = p ⌊ p 2 + 1 ⌋ \left\lfloor\frac{p}{2}+1\right\rfloor ⌊ 2 p + 1 ⌋ [6 ] ) that contribute to the partition function at this order equals the smaller number ⌊ p 3 + 1 ⌋ \left\lfloor\frac{p}{3}+1\right\rfloor ⌊ 3 p + 1 ⌋ h m , n ( β 2 ) h_{m,n}(\beta^2) h m , n ( β 2 ) m + n ≤ p m+n\le p m + n ≤ p ⌊ ( p + 2 ) 2 4 ⌋ \left\lfloor\frac{(p+2)^2}{4}\right\rfloor ⌊ 4 ( p + 2 ) 2 ⌋ [6 ] with m + 3 n ≤ p m+3n\le p m + 3 n ≤ p ⌊ ( p + 2 ) ( p + 3 ) 6 ⌋ \left\lfloor\frac{(p+2)(p+3)}{6}\right\rfloor ⌊ 6 ( p + 2 ) ( p + 3 ) ⌋
Consequently, the integral described above can be thought of as generating a rectangular matrix with ⌊ ( p + 2 ) 2 4 ⌋ \left\lfloor\frac{(p+2)^2}{4}\right\rfloor ⌊ 4 ( p + 2 ) 2 ⌋ ⌊ ( p + 2 ) ( p + 3 ) 6 ⌋ \left\lfloor\frac{(p+2)(p+3)}{6}\right\rfloor ⌊ 6 ( p + 2 ) ( p + 3 ) ⌋ ⌊ ( p + 2 ) ( p + 3 ) 6 ⌋ \left\lfloor\frac{(p+2)(p+3)}{6}\right\rfloor ⌊ 6 ( p + 2 ) ( p + 3 ) ⌋ [6 ] , and creates a ⌊ ( p + 2 ) 2 4 ⌋ \left\lfloor\frac{(p+2)^2}{4}\right\rfloor ⌊ 4 ( p + 2 ) 2 ⌋ h m , n ( β 2 ) h_{m,n}(\beta^2) h m , n ( β 2 ) m + n ≤ p m+n\le p m + n ≤ p 24
Now any matrix with r r r c c c r − c r-c r − c r − c r-c r − c r − c = ⌊ ( p + 2 ) 2 4 ⌋ − ⌊ ( p + 2 ) ( p + 3 ) 6 ⌋ r-c=\left\lfloor\frac{(p+2)^2}{4}\right\rfloor-\left\lfloor\frac{(p+2)(p+3)}{6}\right\rfloor r − c = ⌊ 4 ( p + 2 ) 2 ⌋ − ⌊ 6 ( p + 2 ) ( p + 3 ) ⌋ > 0 >0 > 0 p ≥ 2 p\ge2 p ≥ 2 (73) , with D a , b D_{a,b} D a , b w w w p = p ′ p=p' p = p ′ p ≥ p ′ p\ge p' p ≥ p ′
We have used Mathematica to perform the integrals necessary to evaluate the matrices above up to p = 7 p=7 p = 7 p p p p = 7 p=7 p = 7 p p p
We have thus explicitly verified that the semi-universal constraints that follow from Eq. (64) are identical to those obtained using the methods of [6 ] , at least for p ≤ 7 p\le7 p ≤ 7
More about ln Z s h \ln Z^{\mathrm{sh}} ln Z sh
In Eq. (3) we have presented a conjecture for the form of the quantity ln Z s h \ln Z^{\mathrm{sh}} ln Z sh
The explicit form of the trace of the stress tensor in terms of the various Weyl anomalies is, of course, very well known. By integrating this expression, it is, in principle, not difficult to evaluate ln Z s h \ln Z^{\mathrm{sh}} ln Z sh ln Z s h β \frac{\ln Z^{\mathrm{sh}}}{\beta} β l n Z sh d d d 16 16 16 (3) , which reproduced all 65 65 65 d ≤ 16 d\le16 d ≤ 16 d d d d = 18 , 20 d=18,20 d = 18 , 20
In low dimensions, we have also compared the expression in Eq. (3) against results in the published literature. In d = 4 , 6 , 8 d=4,6,8 d = 4 , 6 , 8 s s s ln Z s h \ln Z^{\mathrm{sh}} ln Z sh (3) is easily evaluated, yielding the following more explicit versions of our conjectured expression
ln Z s h β ∣ d = 4 = a 4 12 [ x 1 x 2 + x 2 x 1 + 7 ] \left.\frac{\ln Z^{\mathrm{sh}}}{\beta}\right|_{d=4}=\frac{a_4}{12}\left[\frac{x_1}{x_2}+\frac{x_2}{x_1}+7\right] β ln Z sh d = 4 = 12 a 4 [ x 2 x 1 + x 1 x 2 + 7 ] ln Z s h β ∣ d = 6 = 3 a 6 80 [ x 1 2 x 2 x 3 + x 2 2 x 1 x 3 + x 3 2 x 1 x 2 − 4 ( x 1 x 2 + x 1 x 3 + x 2 x 1 + x 2 x 3 + x 3 x 1 + x 3 x 2 ) − 29 ] \left.\frac{\ln Z^{\mathrm{sh}}}{\beta}\right|_{d=6}=\frac{3a_6}{80}\left[\frac{x_1^2}{x_2x_3}+\frac{x_2^2}{x_1x_3}+\frac{x_3^2}{x_1x_2}-4\left(\frac{x_1}{x_2}+\frac{x_1}{x_3}+\frac{x_2}{x_1}+\frac{x_2}{x_3}+\frac{x_3}{x_1}+\frac{x_3}{x_2}\right)-29\right] β ln Z sh d = 6 = 80 3 a 6 [ x 2 x 3 x 1 2 + x 1 x 3 x 2 2 + x 1 x 2 x 3 2 − 4 ( x 2 x 1 + x 3 x 1 + x 1 x 2 + x 3 x 2 + x 1 x 3 + x 2 x 3 ) − 29 ] ln Z s h β ∣ d = 8 = a 8 560 [ 15 ( x 1 3 x 2 x 3 x 4 + x 2 3 x 1 x 3 x 4 + x 3 3 x 1 x 2 x 4 + x 4 3 x 1 x 2 x 3 ) − 55 ( x 1 2 x 2 x 3 + x 1 2 x 2 x 4 + x 1 2 x 3 x 4 + x 2 2 x 1 x 3 + x 2 2 x 1 x 4 + x 2 2 x 3 x 4 + x 3 2 x 1 x 2 + x 3 2 x 1 x 4 + x 3 2 x 2 x 4 + x 4 2 x 1 x 2 + x 4 2 x 1 x 3 + x 4 2 x 2 x 3 ) + 29 ( x 1 x 2 x 3 x 4 + x 1 x 3 x 2 x 4 + x 1 x 4 x 2 x 3 + x 2 x 3 x 1 x 4 + x 2 x 4 x 1 x 3 + x 3 x 4 x 1 x 2 ) + 211 ( x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 1 + x 2 x 3 + x 2 x 4 + x 3 x 1 + x 3 x 2 + x 3 x 4 + x 4 x 1 + x 4 x 2 + x 4 x 3 ) + 1569 ] . (77) \begin{aligned}
\left.\frac{\ln Z^{\mathrm{sh}}}{\beta}\right|_{d=8}
={}&\frac{a_8}{560}\Bigg[15\left(\frac{x_1^3}{x_2x_3x_4}+\frac{x_2^3}{x_1x_3x_4}+\frac{x_3^3}{x_1x_2x_4}+\frac{x_4^3}{x_1x_2x_3}\right)\\
&-55\left(\frac{x_1^2}{x_2x_3}+\frac{x_1^2}{x_2x_4}+\frac{x_1^2}{x_3x_4}+\frac{x_2^2}{x_1x_3}+\frac{x_2^2}{x_1x_4}+\frac{x_2^2}{x_3x_4}+\frac{x_3^2}{x_1x_2}+\frac{x_3^2}{x_1x_4}+\frac{x_3^2}{x_2x_4}+\frac{x_4^2}{x_1x_2}+\frac{x_4^2}{x_1x_3}+\frac{x_4^2}{x_2x_3}\right)\\
&+29\left(\frac{x_1x_2}{x_3x_4}+\frac{x_1x_3}{x_2x_4}+\frac{x_1x_4}{x_2x_3}+\frac{x_2x_3}{x_1x_4}+\frac{x_2x_4}{x_1x_3}+\frac{x_3x_4}{x_1x_2}\right)\\
&+211\left(\frac{x_1}{x_2}+\frac{x_1}{x_3}+\frac{x_1}{x_4}+\frac{x_2}{x_1}+\frac{x_2}{x_3}+\frac{x_2}{x_4}+\frac{x_3}{x_1}+\frac{x_3}{x_2}+\frac{x_3}{x_4}+\frac{x_4}{x_1}+\frac{x_4}{x_2}+\frac{x_4}{x_3}\right)+1569\Bigg]. \tag*{(77)}
\end{aligned} β ln Z sh d = 8 = 560 a 8 [ 15 ( x 2 x 3 x 4 x 1 3 + x 1 x 3 x 4 x 2 3 + x 1 x 2 x 4 x 3 3 + x 1 x 2 x 3 x 4 3 ) − 55 ( x 2 x 3 x 1 2 + x 2 x 4 x 1 2 + x 3 x 4 x 1 2 + x 1 x 3 x 2 2 + x 1 x 4 x 2 2 + x 3 x 4 x 2 2 + x 1 x 2 x 3 2 + x 1 x 4 x 3 2 + x 2 x 4 x 3 2 + x 1 x 2 x 4 2 + x 1 x 3 x 4 2 + x 2 x 3 x 4 2 ) + 29 ( x 3 x 4 x 1 x 2 + x 2 x 4 x 1 x 3 + x 2 x 3 x 1 x 4 + x 1 x 4 x 2 x 3 + x 1 x 3 x 2 x 4 + x 1 x 2 x 3 x 4 ) + 211 ( x 2 x 1 + x 3 x 1 + x 4 x 1 + x 1 x 2 + x 3 x 2 + x 4 x 2 + x 1 x 3 + x 2 x 3 + x 4 x 3 + x 1 x 4 + x 2 x 4 + x 3 x 4 ) + 1569 ] . (77) The above three expressions for d = 4 , 6 , 8 d=4,6,8 d = 4 , 6 , 8 15 of the paper [49 ] . It should be noted that ln Z s h β ∣ d = 4 , 6 , 8 = E v a c ( d = 4 , 6 , 8 ) \left.\frac{\ln Z^{\mathrm{sh}}}{\beta}\right|_{d=4,6,8}=E_{\mathrm{vac}}^{(d=4,6,8)} β l n Z sh d = 4 , 6 , 8 = E vac ( d = 4 , 6 , 8 ) [49 ] and a 4 = π 2 48 G 5 a_4=\frac{\pi^2}{48G_5} a 4 = 48 G 5 π 2 a 6 = π 3 48 G 7 a_6=\frac{\pi^3}{48G_7} a 6 = 48 G 7 π 3 a 8 = π 4 576 G 9 a_8=\frac{\pi^4}{576G_9} a 8 = 576 G 9 π 4 (224) .
The Z κ + 1 Z_{\kappa+1} Z κ + 1 ( ln Z s h ) / β (\ln Z^{\mathrm{sh}})/\beta ( ln Z sh ) / β
In this section we evaluate the orbit sum of ln Z s h β \frac{\ln Z^{\mathrm{sh}}}{\beta} β l n Z sh κ \kappa κ b i b_i b i i = 0 , 1 , … , κ i=0,1,\ldots,\kappa i = 0 , 1 , … , κ
ln Z s h β = κ ! a 2 κ [ ( − 1 ) κ + 1 2 2 κ − 1 ∑ i = 0 κ ∫ 0 1 d s ∏ j = 1 j ≠ i κ [ ( s − 2 ) b 0 2 + 2 b j 2 − s b i 2 ] 2 ( b 0 2 − b j 2 ) ( b j 2 − b i 2 ) ] . (78) \frac{\ln Z^{\mathrm{sh}}}{\beta}=\kappa!a_{2\kappa}\left[(-1)^\kappa+\frac{1}{2^{2\kappa-1}}\sum_{i=0}^{\kappa}\int_0^1 ds\prod_{\substack{j=1\\j\ne i}}^\kappa\frac{\left[(s-2)b_0^2+2b_j^2-sb_i^2\right]^2}{(b_0^2-b_j^2)(b_j^2-b_i^2)}\right]. \tag*{(78)} β ln Z sh = κ ! a 2 κ ( − 1 ) κ + 2 2 κ − 1 1 i = 0 ∑ κ ∫ 0 1 d s j = 1 j = i ∏ κ ( b 0 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b 0 2 + 2 b j 2 − s b i 2 ] 2 . (78) Applying Ω p \Omega^p Ω p
Ω p ∘ [ ln Z s h β ] = κ ! a 2 κ [ ( − 1 ) κ + 1 2 2 κ − 1 ∑ i = 0 κ ∫ 0 1 d s ∏ j = 0 j ≠ i , p κ [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) ] , for p = 0 , 1 , … , κ . (79) \Omega^p\circ\left[\frac{\ln Z^{\mathrm{sh}}}{\beta}\right]=\kappa!a_{2\kappa}\left[(-1)^\kappa+\frac{1}{2^{2\kappa-1}}\sum_{i=0}^{\kappa}\int_0^1 ds\prod_{\substack{j=0\\j\ne i,p}}^\kappa\frac{\left[(s-2)b_p^2+2b_j^2-sb_i^2\right]^2}{(b_p^2-b_j^2)(b_j^2-b_i^2)}\right],\qquad\text{for }p=0,1,\ldots,\kappa. \tag*{(79)} Ω p ∘ [ β ln Z sh ] = κ ! a 2 κ ( − 1 ) κ + 2 2 κ − 1 1 i = 0 ∑ κ ∫ 0 1 d s j = 0 j = i , p ∏ κ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 , for p = 0 , 1 , … , κ . (79) Before going into the actual proof, we will define notation. Set
Q ( z ) : = ∏ i = 0 κ ( z − b i 2 ) , L i ( z ) : = ∏ j = 0 j ≠ i κ z − b j 2 b i 2 − b j 2 = Q ( z ) ( z − b i 2 ) Q ′ ( b i 2 ) . (80) Q(z):=\prod_{i=0}^{\kappa}(z-b_i^2),\qquad L_i(z):=\prod_{\substack{j=0\\j\ne i}}^\kappa\frac{z-b_j^2}{b_i^2-b_j^2}=\frac{Q(z)}{(z-b_i^2)Q'(b_i^2)}. \tag*{(80)} Q ( z ) := i = 0 ∏ κ ( z − b i 2 ) , L i ( z ) := j = 0 j = i ∏ κ b i 2 − b j 2 z − b j 2 = ( z − b i 2 ) Q ′ ( b i 2 ) Q ( z ) . (80) so that L i ( z ) L_i(z) L i ( z ) L i ( b j 2 ) = δ i j L_i(b_j^2)=\delta_{ij} L i ( b j 2 ) = δ ij
f ( ζ ) = 1 ( z − ζ ) Q ( ζ ) . (81) f(\zeta)=\frac{1}{(z-\zeta)Q(\zeta)}. \tag*{(81)} f ( ζ ) = ( z − ζ ) Q ( ζ ) 1 . (81) Its residues are 1 ( z − b i 2 ) Q ′ ( b i 2 ) \frac{1}{(z-b_i^2)Q'(b_i^2)} ( z − b i 2 ) Q ′ ( b i 2 ) 1 ζ = b i 2 \zeta=b_i^2 ζ = b i 2 − 1 Q ( z ) -\frac{1}{Q(z)} − Q ( z ) 1 ζ = z ≠ b i 2 \zeta=z\ne b_i^2 ζ = z = b i 2 i = 0 , 1 , ⋯ , κ i=0,1,\cdots,\kappa i = 0 , 1 , ⋯ , κ f ( ζ ) ∼ − ζ − κ − 2 f(\zeta)\sim-\zeta^{-\kappa-2} f ( ζ ) ∼ − ζ − κ − 2
∑ i = 0 κ L i ( z ) = 1. (82) \sum_{i=0}^{\kappa}L_i(z)=1 . \tag*{(82)} i = 0 ∑ κ L i ( z ) = 1. (82) Now consider the distinguished variable in slot p p p ln Z s h β \frac{\ln Z^{\mathrm{sh}}}{\beta} β l n Z sh p = 0 p=0 p = 0 i ≠ p i\ne p i = p
z = b p 2 + s 2 ( b i 2 − b p 2 ) . (83) z=b_p^2+\frac{s}{2}(b_i^2-b_p^2). \tag*{(83)} z = b p 2 + 2 s ( b i 2 − b p 2 ) . (83) The integration range of s s s z z z b p 2 b_p^2 b p 2 1 2 ( b p 2 + b i 2 ) \frac{1}{2}(b_p^2+b_i^2) 2 1 ( b p 2 + b i 2 )
( s − 2 ) b p 2 + 2 b i 2 − s b i 2 = 2 ( b i 2 − z ) , (84) (s-2)b_p^2+2b_i^2-sb_i^2=2(b_i^2-z), \tag*{(84)} ( s − 2 ) b p 2 + 2 b i 2 − s b i 2 = 2 ( b i 2 − z ) , (84) ∏ j ≠ p , i ( b j 2 − z ) 2 = Q ( z ) 2 ( z − b p 2 ) 2 ( z − b i 2 ) 2 , (85) \prod_{j\ne p,i}(b_j^2-z)^2=\frac{Q(z)^2}{(z-b_p^2)^2(z-b_i^2)^2}, \tag*{(85)} j = p , i ∏ ( b j 2 − z ) 2 = ( z − b p 2 ) 2 ( z − b i 2 ) 2 Q ( z ) 2 , (85) ∏ j ≠ p , i ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ Q ′ ( b p 2 ) Q ′ ( b i 2 ) ( b i 2 − b p 2 ) 2 . (86) \prod_{j\ne p,i}(b_p^2-b_j^2)(b_j^2-b_i^2)=(-1)^\kappa\frac{Q'(b_p^2)Q'(b_i^2)}{(b_i^2-b_p^2)^2}. \tag*{(86)} j = p , i ∏ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ ( b i 2 − b p 2 ) 2 Q ′ ( b p 2 ) Q ′ ( b i 2 ) . (86) The integrand of (S21) therefore becomes
1 2 2 κ − 1 ∏ j ≠ p , i [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ 2 ( b i 2 − b p 2 ) 2 L p ( z ) L i ( z ) ( z − b p 2 ) ( z − b i 2 ) . (87) \frac{1}{2^{2\kappa-1}}\prod_{j\ne p,i}\frac{\left[(s-2)b_p^2+2b_j^2-sb_i^2\right]^2}{(b_p^2-b_j^2)(b_j^2-b_i^2)}
=\frac{(-1)^\kappa}{2}(b_i^2-b_p^2)^2\frac{L_p(z)L_i(z)}{(z-b_p^2)(z-b_i^2)}. \tag*{(87)} 2 2 κ − 1 1 j = p , i ∏ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 = 2 ( − 1 ) κ ( b i 2 − b p 2 ) 2 ( z − b p 2 ) ( z − b i 2 ) L p ( z ) L i ( z ) . (87) Changing variables with d s = 2 d z / ( b i 2 − b p 2 ) ds=2\,dz/(b_i^2-b_p^2) d s = 2 d z / ( b i 2 − b p 2 ) 1 2 \frac{1}{2} 2 1 b i 2 − b p 2 ( z − b p 2 ) ( z − b i 2 ) \frac{b_i^2-b_p^2}{(z-b_p^2)(z-b_i^2)} ( z − b p 2 ) ( z − b i 2 ) b i 2 − b p 2
1 2 2 κ − 1 ∫ 0 1 d s ∏ j ≠ i , p [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ ∫ b p 2 1 2 ( b p 2 + b i 2 ) d z L p ( z ) L i ( z ) [ 1 z − b i 2 − 1 z − b p 2 ] . (88) \frac{1}{2^{2\kappa-1}}\int_0^1ds\prod_{j\ne i,p}\frac{\left[(s-2)b_p^2+2b_j^2-sb_i^2\right]^2}{(b_p^2-b_j^2)(b_j^2-b_i^2)}
=(-1)^\kappa\int_{b_p^2}^{\frac{1}{2}(b_p^2+b_i^2)}dz\,L_p(z)L_i(z)\left[\frac{1}{z-b_i^2}-\frac{1}{z-b_p^2}\right]. \tag*{(88)} 2 2 κ − 1 1 ∫ 0 1 d s j = i , p ∏ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 = ( − 1 ) κ ∫ b p 2 2 1 ( b p 2 + b i 2 ) d z L p ( z ) L i ( z ) [ z − b i 2 1 − z − b p 2 1 ] . (88) We will use the following identity
L i ′ ( z ) L i ( z ) = Q ′ ( z ) Q ( z ) − 1 z − b i 2 , for i = 0 , 1 , ⋯ , κ . (89) \frac{L_i'(z)}{L_i(z)}=\frac{Q'(z)}{Q(z)}-\frac{1}{z-b_i^2},\qquad\text{for }i=0,1,\cdots,\kappa. \tag*{(89)} L i ( z ) L i ′ ( z ) = Q ( z ) Q ′ ( z ) − z − b i 2 1 , for i = 0 , 1 , ⋯ , κ . (89) So we can write
1 2 2 κ − 1 ∫ 0 1 d s ∏ j ≠ i , p [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ ∫ b p 2 1 2 ( b p 2 + b i 2 ) d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] . (90) \frac{1}{2^{2\kappa-1}}\int_0^1ds\prod_{j\ne i,p}\frac{\left[(s-2)b_p^2+2b_j^2-sb_i^2\right]^2}{(b_p^2-b_j^2)(b_j^2-b_i^2)}
=(-1)^\kappa\int_{b_p^2}^{\frac{1}{2}(b_p^2+b_i^2)}dz\,\left[L_p'(z)L_i(z)-L_p(z)L_i'(z)\right]. \tag*{(90)} 2 2 κ − 1 1 ∫ 0 1 d s j = i , p ∏ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 = ( − 1 ) κ ∫ b p 2 2 1 ( b p 2 + b i 2 ) d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] . (90) We will sum the indices i , p i,p i , p
∑ i , p i ≠ p 1 2 2 κ − 1 ∫ 0 1 d s ∏ j ≠ i , p [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) = ( − 1 ) κ 2 ∑ i , p i ≠ p [ ∫ b p 2 1 2 ( b p 2 + b i 2 ) d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] + ∫ b i 2 1 2 ( b p 2 + b i 2 ) d z [ L i ′ ( z ) L p ( z ) − L i ( z ) L p ′ ( z ) ] ] = ( − 1 ) κ 2 ∑ i , p i ≠ p ∫ b p 2 b i 2 d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] = ( − 1 ) κ ∑ i , p = 0 κ ∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) . (91) \begin{aligned}
\sum_{\substack{i,p\\i\ne p}}\frac{1}{2^{2\kappa-1}}\int_0^1ds\prod_{j\ne i,p}\frac{\left[(s-2)b_p^2+2b_j^2-sb_i^2\right]^2}{(b_p^2-b_j^2)(b_j^2-b_i^2)}
={}&\frac{(-1)^\kappa}{2}\sum_{\substack{i,p\\i\ne p}}\left[\int_{b_p^2}^{\frac{1}{2}(b_p^2+b_i^2)}dz\,\left[L_p'(z)L_i(z)-L_p(z)L_i'(z)\right]\right.\\
&\left.+\int_{b_i^2}^{\frac{1}{2}(b_p^2+b_i^2)}dz\,\left[L_i'(z)L_p(z)-L_i(z)L_p'(z)\right]\right]\\
={}&\frac{(-1)^\kappa}{2}\sum_{\substack{i,p\\i\ne p}}\int_{b_p^2}^{b_i^2}dz\,\left[L_p'(z)L_i(z)-L_p(z)L_i'(z)\right]\\
={}&(-1)^\kappa\sum_{i,p=0}^{\kappa}\int_{b_p^2}^{b_i^2}dz\,L_p'(z)L_i(z). \tag*{(91)}
\end{aligned} i , p i = p ∑ 2 2 κ − 1 1 ∫ 0 1 d s j = i , p ∏ ( b p 2 − b j 2 ) ( b j 2 − b i 2 ) [ ( s − 2 ) b p 2 + 2 b j 2 − s b i 2 ] 2 = = = 2 ( − 1 ) κ i , p i = p ∑ [ ∫ b p 2 2 1 ( b p 2 + b i 2 ) d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] + ∫ b i 2 2 1 ( b p 2 + b i 2 ) d z [ L i ′ ( z ) L p ( z ) − L i ( z ) L p ′ ( z ) ] ] 2 ( − 1 ) κ i , p i = p ∑ ∫ b p 2 b i 2 d z [ L p ′ ( z ) L i ( z ) − L p ( z ) L i ′ ( z ) ] ( − 1 ) κ i , p = 0 ∑ κ ∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) . (91) where in the second step of the manipulation, we have symmetrized the index ( i , p ) (i,p) ( i , p ) 1 2 ( b p 2 + b i 2 ) \frac{1}{2}(b_p^2+b_i^2) 2 1 ( b p 2 + b i 2 ) i ≠ p i\ne p i = p i = p i=p i = p z = c ≠ b i 2 z=c\ne b_i^2 z = c = b i 2
∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) = ∫ c b i 2 d z L p ′ ( z ) L i ( z ) − ∫ c b p 2 d z L p ′ ( z ) L i ( z ) . (92) \int_{b_p^2}^{b_i^2} \mathrm{d}z\,L_p'(z)L_i(z)=\int_c^{b_i^2}\mathrm{d}z\,L_p'(z)L_i(z)-\int_c^{b_p^2}\mathrm{d}z\,L_p'(z)L_i(z). \tag*{(92)} ∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) = ∫ c b i 2 d z L p ′ ( z ) L i ( z ) − ∫ c b p 2 d z L p ′ ( z ) L i ( z ) . (92) Hence the summation can be written as
∑ i , p = 0 κ ∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) = ∑ i = 0 κ ∫ c b i 2 d z ( ∑ p = 0 κ L p ( z ) ) ′ L i ( z ) − ∑ p = 0 κ ∫ c b p 2 d z L p ′ ( z ) ( ∑ i = 0 κ L i ( z ) ) = 0 − ∑ p = 0 κ [ L p ( b p 2 ) − L p ( c ) ] = − κ . (93) \begin{aligned}
\sum_{i,p=0}^{\kappa}\int_{b_p^2}^{b_i^2}\mathrm{d}z\,L_p'(z)L_i(z)=\sum_{i=0}^{\kappa}\int_c^{b_i^2}\mathrm{d}z\left(\sum_{p=0}^{\kappa}L_p(z)\right)'L_i(z)-\sum_{p=0}^{\kappa}\int_c^{b_p^2}\mathrm{d}z\,L_p'(z)\left(\sum_{i=0}^{\kappa}L_i(z)\right) \\
&=0-\sum_{p=0}^{\kappa}\left[L_p(b_p^2)-L_p(c)\right]=-\kappa. \tag*{(93)}
\end{aligned} i , p = 0 ∑ κ ∫ b p 2 b i 2 d z L p ′ ( z ) L i ( z ) = i = 0 ∑ κ ∫ c b i 2 d z ( p = 0 ∑ κ L p ( z ) ) ′ L i ( z ) − p = 0 ∑ κ ∫ c b p 2 d z L p ′ ( z ) ( i = 0 ∑ κ L i ( z ) ) = 0 − p = 0 ∑ κ [ L p ( b p 2 ) − L p ( c ) ] = − κ . (93) Summing over p p p (79) , we have
( 1 + Ω + ⋯ + Ω κ ) ∘ [ ln Z s h β ] = ( − 1 ) κ κ ! a 2 κ ( κ + 1 − κ ) = ( − 1 ) κ κ ! a 2 κ , (94) \left(1+\Omega+\cdots+\Omega^\kappa\right)\circ\left[\frac{\ln Z^{\mathrm{sh}}}{\beta}\right]=(-1)^\kappa\kappa!a_{2\kappa}(\kappa+1-\kappa)=(-1)^\kappa\kappa!a_{2\kappa}, \tag*{(94)} ( 1 + Ω + ⋯ + Ω κ ) ∘ [ β ln Z sh ] = ( − 1 ) κ κ ! a 2 κ ( κ + 1 − κ ) = ( − 1 ) κ κ ! a 2 κ , (94) which is our cyclic identity for the anomalous part.
Simple symmetrization identities
In the last step in Eq. (26) we have used
3 S ∘ ( b 1 2 − b 2 2 ) = ( b 1 2 − b 2 2 ) + ( b 2 2 − b 0 2 ) + ( b 0 2 − b 1 2 ) = 0 , 3\mathfrak{S}\circ(b_1^2-b_2^2)=(b_1^2-b_2^2)+(b_2^2-b_0^2)+(b_0^2-b_1^2)=0, 3 S ∘ ( b 1 2 − b 2 2 ) = ( b 1 2 − b 2 2 ) + ( b 2 2 − b 0 2 ) + ( b 0 2 − b 1 2 ) = 0 , 3 S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = ( b 1 2 b 0 2 − b 2 2 b 0 2 ) + ( b 2 2 b 1 2 − b 0 2 b 1 2 ) + ( b 0 2 b 2 2 − b 1 2 b 2 2 ) = 0 , 3\mathfrak{S}\circ(b_1^2b_0^2-b_2^2b_0^2)=(b_1^2b_0^2-b_2^2b_0^2)+(b_2^2b_1^2-b_0^2b_1^2)+(b_0^2b_2^2-b_1^2b_2^2)=0, 3 S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = ( b 1 2 b 0 2 − b 2 2 b 0 2 ) + ( b 2 2 b 1 2 − b 0 2 b 1 2 ) + ( b 0 2 b 2 2 − b 1 2 b 2 2 ) = 0 , 3 S ∘ ( b 1 2 b 0 4 − b 2 2 b 0 4 ) = ( b 1 2 b 0 4 − b 2 2 b 0 4 ) + ( b 2 2 b 1 4 − b 0 2 b 1 4 ) + ( b 0 2 b 2 4 − b 1 2 b 2 4 ) = − ( b 0 2 − b 1 2 ) ( b 1 2 − b 2 2 ) ( b 2 2 − b 0 2 ) = − 2 π 2 P . (95) 3\mathfrak{S}\circ(b_1^2b_0^4-b_2^2b_0^4)=(b_1^2b_0^4-b_2^2b_0^4)+(b_2^2b_1^4-b_0^2b_1^4)+(b_0^2b_2^4-b_1^2b_2^4)=-(b_0^2-b_1^2)(b_1^2-b_2^2)(b_2^2-b_0^2)=-\frac{2\pi^2}{P}. \tag*{(95)} 3 S ∘ ( b 1 2 b 0 4 − b 2 2 b 0 4 ) = ( b 1 2 b 0 4 − b 2 2 b 0 4 ) + ( b 2 2 b 1 4 − b 0 2 b 1 4 ) + ( b 0 2 b 2 4 − b 1 2 b 2 4 ) = − ( b 0 2 − b 1 2 ) ( b 1 2 − b 2 2 ) ( b 2 2 − b 0 2 ) = − P 2 π 2 . (95) Equating the coefficients of distinct powers of β 2 q \beta^{2q} β 2 q (26) yields Eq. (40) .
The vanishing of I q ( κ ) I_q^{(\kappa)} I q ( κ ) 2 κ 2\kappa 2 κ κ ≥ 2 \kappa\ge2 κ ≥ 2
In this Appendix we will demonstrate that the first of Eq. (6) implies Eq. (40) is true in arbitrary even dimensions, i.e. at arbitrary integer values of κ \kappa κ
ln Z r o t β = − 1 ∏ i = 1 κ ( b 0 2 − b i 2 ) ( ∑ q = 0 ∞ T q ) . (96) \frac{\ln Z^{\mathrm{rot}}}{\beta}=-\frac{1}{\prod_{i=1}^{\kappa}(b_0^2-b_i^2)}\left(\sum_{q=0}^{\infty}T_q\right). \tag*{(96)} β ln Z rot = − ∏ i = 1 κ ( b 0 2 − b i 2 ) 1 ( q = 0 ∑ ∞ T q ) . (96) The denominator in (S38) does not transform in a particularly simple manner under either Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 S κ + 1 \mathbb{S}_{\kappa+1} S κ + 1
V 0 = det ( 1 1 ⋯ 1 b 1 2 b 2 2 ⋯ b κ 2 ⋮ ⋮ ⋱ ⋮ b 1 2 ( κ − 1 ) b 2 2 ( κ − 1 ) ⋯ b κ 2 ( κ − 1 ) ) = ∏ 1 ≤ i < j ≤ κ ( b j 2 − b i 2 ) . (97) \mathcal{V}_0=\det\begin{pmatrix}
1 & 1 & \cdots& 1\\
b_1^2 & b_2^2 & \cdots& b_\kappa^2\\
\vdots& \vdots& \ddots& \vdots\\
b_1^{2(\kappa-1)} & b_2^{2(\kappa-1)} & \cdots& b_\kappa^{2(\kappa-1)}
\end{pmatrix}=\prod_{1\le i<j\le\kappa}(b_j^2-b_i^2). \tag*{(97)} V 0 = det 1 b 1 2 ⋮ b 1 2 ( κ − 1 ) 1 b 2 2 ⋮ b 2 2 ( κ − 1 ) ⋯ ⋯ ⋱ ⋯ 1 b κ 2 ⋮ b κ 2 ( κ − 1 ) = 1 ≤ i < j ≤ κ ∏ ( b j 2 − b i 2 ) . (97) On doing this, (S38) can be rewritten as
ln Z r o t β = ( − 1 ) κ + 1 V ( ∑ q = 0 ∞ V 0 T q ) , (98) \frac{\ln Z^{\mathrm{rot}}}{\beta}=\frac{(-1)^{\kappa+1}}{\mathcal{V}}\left(\sum_{q=0}^{\infty}\mathcal{V}_0T_q\right), \tag*{(98)} β ln Z rot = V ( − 1 ) κ + 1 ( q = 0 ∑ ∞ V 0 T q ) , (98) where the denominator is now the Vandermonde
V = det ( 1 1 ⋯ 1 b 0 2 b 1 2 ⋯ b κ 2 ⋮ ⋮ ⋱ ⋮ b 0 2 κ b 1 2 κ ⋯ b κ 2 κ ) = ∏ 0 ≤ i < j ≤ κ ( b j 2 − b i 2 ) . (99) \mathcal{V}=\det\begin{pmatrix}
1 & 1 & \cdots& 1 \\
b_0^2 & b_1^2 & \cdots& b_\kappa^2 \\
\vdots& \vdots& \ddots& \vdots\\
b_0^{2\kappa} & b_1^{2\kappa} & \cdots& b_\kappa^{2\kappa}
\end{pmatrix}
=\prod_{0\le i<j\le\kappa}(b_j^2-b_i^2). \tag*{(99)} V = det 1 b 0 2 ⋮ b 0 2 κ 1 b 1 2 ⋮ b 1 2 κ ⋯ ⋯ ⋱ ⋯ 1 b κ 2 ⋮ b κ 2 κ = 0 ≤ i < j ≤ κ ∏ ( b j 2 − b i 2 ) . (99) which lies in the completely antisymmetric representation of S κ + 1 S_{\kappa+1} S κ + 1 Ω \Omega Ω Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 κ \kappa κ V \mathcal{V} V
Ω ∘ V = ( − 1 ) κ V . (100) \Omega\circ\mathcal{V}=(-1)^\kappa\mathcal{V}. \tag*{(100)} Ω ∘ V = ( − 1 ) κ V . (100) For later use we also define
V a = ∏ 0 ≤ i < j ≤ κ i , j ≠ a ( b j 2 − b i 2 ) . (101) \mathcal{V}_a=\prod_{\substack{0\le i<j\le\kappa\\i,j\ne a}}(b_j^2-b_i^2). \tag*{(101)} V a = 0 ≤ i < j ≤ κ i , j = a ∏ ( b j 2 − b i 2 ) . (101) Inserting (10) into (S40) we see that ln Z r o t β \frac{\ln Z^{\mathrm{rot}}}{\beta} β l n Z rot
ln Z r o t β = ( − 1 ) κ + 1 V ( ∑ q = 0 ∞ V 0 ( ∑ m = 0 κ b 0 2 m I q ( m ) ) ) . (102) \frac{\ln Z^{\mathrm{rot}}}{\beta}=\frac{(-1)^{\kappa+1}}{\mathcal{V}}\left(\sum_{q=0}^{\infty}\mathcal{V}_0\left(\sum_{m=0}^{\kappa}b_0^{2m}I_q^{(m)}\right)\right). \tag*{(102)} β ln Z rot = V ( − 1 ) κ + 1 ( q = 0 ∑ ∞ V 0 ( m = 0 ∑ κ b 0 2 m I q ( m ) ) ) . (102) We now insert (S40) into (7). Using (S42), the permutation invariance of I q ( m ) I_q^{(m)} I q ( m )
Ω a ∘ [ V 0 V ] = ( − 1 ) a V a V , (103) \Omega^a\circ\left[\frac{\mathcal{V}_0}{\mathcal{V}}\right]=(-1)^a\frac{\mathcal{V}_a}{\mathcal{V}}, \tag*{(103)} Ω a ∘ [ V V 0 ] = ( − 1 ) a V V a , (103) G ∘ [ ln Z r o t β ] = ( − 1 ) κ + 1 V ( κ + 1 ) ∑ q = 0 ∞ ∑ m = 0 κ ∑ a = 0 κ [ V a ( − 1 ) a I q ( m ) b a 2 m ] . (104) \mathcal{G}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=\frac{(-1)^{\kappa+1}}{\mathcal{V}(\kappa+1)}\sum_{q=0}^{\infty}\sum_{m=0}^{\kappa}\sum_{a=0}^{\kappa}\left[\mathcal{V}_a(-1)^a I_q^{(m)}b_a^{2m}\right]. \tag*{(104)} G ∘ [ β ln Z rot ] = V ( κ + 1 ) ( − 1 ) κ + 1 q = 0 ∑ ∞ m = 0 ∑ κ a = 0 ∑ κ [ V a ( − 1 ) a I q ( m ) b a 2 m ] . (104) Where
G = 1 κ + 1 ∑ a = 0 κ Ω a . (105) \mathcal{G}=\frac{1}{\kappa+1}\sum_{a=0}^{\kappa}\Omega^a. \tag*{(105)} G = κ + 1 1 a = 0 ∑ κ Ω a . (105) Now recall that any n × n n\times n n × n A i j A_{ij} A ij
∑ j = 1 n ( − 1 ) j + n A m j A ^ n j = δ m n det ( A ) (106) \sum_{j=1}^{n}(-1)^{j+n}A_{mj}\widehat{A}_{nj}=\delta_{mn}\det(A) \tag*{(106)} j = 1 ∑ n ( − 1 ) j + n A mj A nj = δ mn det ( A ) (106) where A ^ i j \widehat{A}_{ij} A ij A i j A_{ij} A ij m = 0 , 1 , 2 , … , κ − 1 m=0,1,2,\ldots,\kappa-1 m = 0 , 1 , 2 , … , κ − 1 n = κ n=\kappa n = κ
∑ a = 0 κ b a 2 n V a ( − 1 ) a = 0 ( n = 0 , 1 , 2 , … , κ − 1 ) . (107) \sum_{a=0}^{\kappa}b_a^{2n}\mathcal{V}_a(-1)^a=0\qquad(n=0,1,2,\ldots,\kappa-1). \tag*{(107)} a = 0 ∑ κ b a 2 n V a ( − 1 ) a = 0 ( n = 0 , 1 , 2 , … , κ − 1 ) . (107) Applying the same identity with m = κ m=\kappa m = κ
∑ a = 0 κ b a 2 κ ( − 1 ) a V a = ( − 1 ) κ V . (108) \sum_{a=0}^{\kappa}b_a^{2\kappa}(-1)^a\mathcal{V}_a=(-1)^\kappa\mathcal{V}. \tag*{(108)} a = 0 ∑ κ b a 2 κ ( − 1 ) a V a = ( − 1 ) κ V . (108) Inserting (S49) and (S50) into (S46), we see that the coefficients of I q ( 0 ) I_q^{(0)} I q ( 0 ) I q ( 1 ) I_q^{(1)} I q ( 1 ) I q ( κ − 1 ) I_q^{(\kappa-1)} I q ( κ − 1 )
G ∘ [ ln Z r o t β ] = − 1 κ + 1 ∑ q = 0 ∞ I q ( κ ) . (109) \mathcal{G}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=-\frac{1}{\kappa+1}\sum_{q=0}^{\infty}I_q^{(\kappa)}. \tag*{(109)} G ∘ [ β ln Z rot ] = − κ + 1 1 q = 0 ∑ ∞ I q ( κ ) . (109) Expanding (S51) term by term, (and using that I q ( κ ) I_q^{(\kappa)} I q ( κ ) ( q − κ ) (q-\kappa) ( q − κ ) b μ 2 b_\mu^2 b μ 2 μ = 0 … κ \mu=0\ldots\kappa μ = 0 … κ
Relation to S 3 S_3 S 3
We have demonstrated in the main text that if the ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot 4 _4 4 (22) , which asserts that
S ∘ [ ln Z r o t β ] = 0 , (110) \mathcal{S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0, \tag*{(110)} S ∘ [ β ln Z rot ] = 0 , (110) where S \mathcal{S} S (21) . In this section we will view this condition (and resultant equations) from the lens of S 3 S_3 S 3
The permutation group S 3 S_3 S 3 ( b 0 2 , b 1 2 , b 2 2 ) (b_0^2,b_1^2,b_2^2) ( b 0 2 , b 1 2 , b 2 2 ) b i 2 b_i^2 b i 2 S 3 S_3 S 3 1 S \mathbf{1}_S 1 S 1 A \mathbf{1}_A 1 A 2 M \mathbf{2}_M 2 M 26
P 1 S = 1 6 [ 1 + ( 01 ) + ( 02 ) + ( 12 ) + ( 012 ) + ( 021 ) ] , P_{1_S}=\frac{1}{6}[1+(01)+(02)+(12)+(012)+(021)], P 1 S = 6 1 [ 1 + ( 01 ) + ( 02 ) + ( 12 ) + ( 012 ) + ( 021 )] , P 1 A = 1 6 [ 1 − ( 01 ) − ( 02 ) − ( 12 ) + ( 012 ) + ( 021 ) ] , P_{1_A}=\frac{1}{6}[1-(01)-(02)-(12)+(012)+(021)], P 1 A = 6 1 [ 1 − ( 01 ) − ( 02 ) − ( 12 ) + ( 012 ) + ( 021 )] , P 2 M = 1 − P 1 S − P 1 A = 1 3 [ 2 − ( 012 ) − ( 021 ) ] . (111) P_{2_M}=1-P_{1_S}-P_{1_A}=\frac{1}{3}[2-(012)-(021)]. \tag*{(111)} P 2 M = 1 − P 1 S − P 1 A = 3 1 [ 2 − ( 012 ) − ( 021 )] . (111) act as projectors onto the relevant representation spaces. In Eq. (111) we have used the following notation: 1 1 1 ( i j ) (ij) ( ij ) i i i j j j ( i j k ) (ijk) ( ij k ) ( i → j → k → i ) (i\to j\to k\to i) ( i → j → k → i ) (111) is one sixth the sum over all 6 permutations in S 3 S_3 S 3 (111) is one sixth the signed sum over all 6 elements of the permutation group, and so projects onto the totally antisymmetric representation. The third line of Eq. (111) projects onto whatever is left after these two representations have been removed, and so projects onto the 2 M \mathbf{2}_M 2 M Ω = ( 012 ) \Omega=(012) Ω = ( 012 ) Ω 2 = ( 021 ) \Omega^2=(021) Ω 2 = ( 021 ) Ω \Omega Ω
S = 1 3 ( 1 + Ω + Ω 2 ) = P 1 S + P 1 A = 1 − P 2 M . (112) \mathcal{S}=\frac{1}{3}(1+\Omega+\Omega^2)=P_{1_S}+P_{1_A}=1-P_{2_M}. \tag*{(112)} S = 3 1 ( 1 + Ω + Ω 2 ) = P 1 S + P 1 A = 1 − P 2 M . (112) In other words, S \mathcal{S} S 2 M \mathbf{2}_M 2 M (110) can be rewritten as
( P 1 S + P 1 A ) ∘ [ ln Z r o t β ] = 0. (113) (P_{1_S}+P_{1_A})\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0. \tag*{(113)} ( P 1 S + P 1 A ) ∘ [ β ln Z rot ] = 0. (113) Now the condition P 1 A ∘ [ ln Z r o t β ] = 0 P_{1_A}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0 P 1 A ∘ [ β l n Z rot ] = 0 ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot b 1 2 b_1^2 b 1 2 b 2 2 b_2^2 b 2 2 (113) is the assertion
P 1 S ∘ [ ln Z r o t β ] = 0 , ln Z r o t β ∈ 2 M . (114) P_{1_S}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=0,\qquad\frac{\ln Z^{\mathrm{rot}}}{\beta}\in\mathbf{2}_M. \tag*{(114)} P 1 S ∘ [ β ln Z rot ] = 0 , β ln Z rot ∈ 2 M . (114) Note also that while the 2 M \mathbf{2}_M 2 M b 1 2 b_1^2 b 1 2 b 2 2 b_2^2 b 2 2 ln Z β \frac{\ln Z}{\beta} β l n Z
In order to make our discussion more quantitative, we now recall that ln Z r o t β \frac{\ln Z^{\mathrm{rot}}}{\beta} β l n Z rot (65) , where H ( y 0 , y 1 , y 2 ) H(y_0,y_1,y_2) H ( y 0 , y 1 , y 2 ) [50 ] for a review) that the space of polynomials of arbitrary degree, of the three variables ( b 0 2 , b 1 2 , b 2 2 ) (b_0^2,b_1^2,b_2^2) ( b 0 2 , b 1 2 , b 2 2 ) S 3 S_3 S 3 E 1 E_1 E 1 E 2 E_2 E 2 E 3 E_3 E 3 (38) ). The module of all polynomials is not irreducible; it has four distinct submodules (each of which is freely generated, i.e. has no null states) and so has four distinct primaries. The first of these (at ‘level zero’) is monomial unity, in the 1 S \mathbf{1}_{S} 1 S 2 M \mathbf{2}_{M} 2 M ( b 0 2 − b 1 2 ) (b_0^2-b_1^2) ( b 0 2 − b 1 2 ) ( b 1 2 − b 2 2 ) (b_1^2-b_2^2) ( b 1 2 − b 2 2 ) ( b 2 2 − b 0 2 ) (b_2^2-b_0^2) ( b 2 2 − b 0 2 ) ( b 0 2 b 2 2 − b 1 2 b 2 2 ) (b_0^2b_2^2-b_1^2b_2^2) ( b 0 2 b 2 2 − b 1 2 b 2 2 ) ( b 1 2 b 0 2 − b 2 2 b 0 2 ) (b_1^2b_0^2-b_2^2b_0^2) ( b 1 2 b 0 2 − b 2 2 b 0 2 ) ( b 2 2 b 1 2 − b 0 2 b 1 2 ) (b_2^2b_1^2-b_0^2b_1^2) ( b 2 2 b 1 2 − b 0 2 b 1 2 ) 1 A \mathbf{1}_{A} 1 A V V V (99) .
We can verify that the primaries listed above do indeed transform in the assigned representations by checking (for instance) that
P 1 S ∘ ( 1 ) = 1 , P 1 A ∘ ( 1 ) = P 2 M ∘ ( 1 ) = 0 , P 1 S ∘ ( b 1 2 − b 2 2 ) = P 1 A ∘ ( b 1 2 − b 2 2 ) = 0 , P 2 M ∘ ( b 1 2 − b 2 2 ) = b 1 2 − b 2 2 , P 1 S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = P 1 A ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = 0 , P 2 M ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = b 1 2 b 0 2 − b 2 2 b 0 2 , P 1 S ∘ ( P ) = 0 , P 1 A ∘ ( P ) = P , P 2 M ∘ ( P ) = 0. (115) \begin{aligned}
P_{\mathbf{1}_{S}}\circ(1)&=1,\quad P_{\mathbf{1}_{A}}\circ(1)=P_{\mathbf{2}_{M}}\circ(1)=0,\\
P_{\mathbf{1}_{S}}\circ(b_1^2-b_2^2)&=P_{\mathbf{1}_{A}}\circ(b_1^2-b_2^2)=0,\quad P_{\mathbf{2}_{M}}\circ(b_1^2-b_2^2)=b_1^2-b_2^2,\\
P_{\mathbf{1}_{S}}\circ(b_1^2b_0^2-b_2^2b_0^2)&=P_{\mathbf{1}_{A}}\circ(b_1^2b_0^2-b_2^2b_0^2)=0,\quad P_{\mathbf{2}_{M}}\circ(b_1^2b_0^2-b_2^2b_0^2)=b_1^2b_0^2-b_2^2b_0^2,\\
P_{\mathbf{1}_{S}}\circ(P)&=0,\quad P_{\mathbf{1}_{A}}\circ(P)=P,\quad P_{\mathbf{2}_{M}}\circ(P)=0. \tag*{(115)}
\end{aligned} P 1 S ∘ ( 1 ) P 1 S ∘ ( b 1 2 − b 2 2 ) P 1 S ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) P 1 S ∘ ( P ) = 1 , P 1 A ∘ ( 1 ) = P 2 M ∘ ( 1 ) = 0 , = P 1 A ∘ ( b 1 2 − b 2 2 ) = 0 , P 2 M ∘ ( b 1 2 − b 2 2 ) = b 1 2 − b 2 2 , = P 1 A ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = 0 , P 2 M ∘ ( b 1 2 b 0 2 − b 2 2 b 0 2 ) = b 1 2 b 0 2 − b 2 2 b 0 2 , = 0 , P 1 A ∘ ( P ) = P , P 2 M ∘ ( P ) = 0. (115) We will now count polynomials weighted by degree. Let z ( x ) z(x) z ( x ) Tr x L \operatorname{Tr}x^L Tr x L L L L z a l l ( x ) z_{\mathrm{all}}(x) z all ( x ) z 1 S ( x ) z_{\mathbf{1}_{S}}(x) z 1 S ( x ) z 1 A ( x ) z_{\mathbf{1}_{A}}(x) z 1 A ( x ) z 2 M ( x ) z_{\mathbf{2}_{M}}(x) z 2 M ( x )
z 1 S ( x ) = 1 ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) , z 1 A ( x ) = x 3 ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) , z 2 M ( x ) = 2 x + 2 x 2 ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) , z a l l ( x ) = 1 ( 1 − x ) 3 = z 1 S + z 1 A ( x ) + z 2 M ( x ) . (116) \begin{aligned}
z_{\mathbf{1}_{S}}(x)&=\frac{1}{(1-x)(1-x^2)(1-x^3)},\\
z_{\mathbf{1}_{A}}(x)&=\frac{x^3}{(1-x)(1-x^2)(1-x^3)},\\
z_{\mathbf{2}_{M}}(x)&=\frac{2x+2x^2}{(1-x)(1-x^2)(1-x^3)},\\
z_{\mathrm{all}}(x)&=\frac{1}{(1-x)^3}=z_{\mathbf{1}_{S}}+z_{\mathbf{1}_{A}}(x)+z_{\mathbf{2}_{M}}(x). \tag*{(116)}
\end{aligned} z 1 S ( x ) z 1 A ( x ) z 2 M ( x ) z all ( x ) = ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) 1 , = ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) x 3 , = ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) 2 x + 2 x 2 , = ( 1 − x ) 3 1 = z 1 S + z 1 A ( x ) + z 2 M ( x ) . (116) We will now use the formulae Eq. (116) to count both the number of polynomials H ( b 0 2 , b 1 2 , b 2 2 ) H(b_0^2,b_1^2,b_2^2) H ( b 0 2 , b 1 2 , b 2 2 ) 1 ↔ 2 1\leftrightarrow2 1 ↔ 2
Let z n a i v e ( x ) z_{\mathrm{naive}}(x) z naive ( x ) 1 ↔ 2 1\leftrightarrow2 1 ↔ 2 1 S \mathbf{1}_{S} 1 S 2 M \mathbf{2}_{M} 2 M 1 A \mathbf{1}_{A} 1 A
z n a i v e ( x ) = z 1 S ( x ) + 1 2 z 2 M ( x ) = 1 + x + x 2 ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) . (117) z_{\mathrm{naive}}(x)=z_{\mathbf{1}_{S}}(x)+\frac{1}{2}z_{\mathbf{2}_{M}}(x)=\frac{1+x+x^2}{(1-x)(1-x^2)(1-x^3)}. \tag*{(117)} z naive ( x ) = z 1 S ( x ) + 2 1 z 2 M ( x ) = ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) 1 + x + x 2 . (117) in agreement with (D2).
Of course the count Eq. (117) does not account for the fact that the polynomials in the function H H H (66) . In order to understand the S 3 S_3 S 3 ln Z r o t β = P H ( b 1 2 − b 2 2 ) 2 π 2 \frac{\ln Z^{\mathrm{rot}}}{\beta}=\frac{P H(b_1^2-b_2^2)}{2\pi^2} β l n Z rot = 2 π 2 P H ( b 1 2 − b 2 2 )
P 1 S ∘ [ ln Z r o t β ] = 1 2 π 2 P 1 S ∘ [ P ( b 1 2 − b 2 2 ) H ] = P 2 π 2 P 1 A ∘ [ ( b 1 2 − b 2 2 ) H ] . (118) P_{\mathbf{1}_{S}}\circ\left[\frac{\ln Z^{\mathrm{rot}}}{\beta}\right]=\frac{1}{2\pi^2}P_{\mathbf{1}_{S}}\circ\left[P(b_1^2-b_2^2)H\right]=\frac{P}{2\pi^2}P_{\mathbf{1}_{A}}\circ\left[(b_1^2-b_2^2)H\right]. \tag*{(118)} P 1 S ∘ [ β ln Z rot ] = 2 π 2 1 P 1 S ∘ [ P ( b 1 2 − b 2 2 ) H ] = 2 π 2 P P 1 A ∘ [ ( b 1 2 − b 2 2 ) H ] . (118) Note that pulling out the completely antisymmetric object P P P (118) into a projector onto the antisymmetric representation in the last of (S60). Inserting Eq. (118) into the first of Eq. (114) yields the equation
P 1 A ∘ [ ( b 1 2 − b 2 2 ) H ] = 0. (119) P_{\mathbf{1}_{A}}\circ\left[(b_1^2-b_2^2)H\right]=0. \tag*{(119)} P 1 A ∘ [ ( b 1 2 − b 2 2 ) H ] = 0. (119) But Eq. (119) is simply a rewriting of the equation Eq. (66) , as may be verified by expanding out the projector on the LHS of Eq. (119) and using the 1 ↔ 2 1\leftrightarrow2 1 ↔ 2 H H H
We see that the LHS of Eq. (66) lies in the 1 A \mathbf{1}_{A} 1 A 1 A \mathbf{1}_{A} 1 A (66) ) for a suitable choice of the function H H H (66) on the degree L L L H H H L + 1 L+1 L + 1 1 A \mathbf{1}_{A} 1 A z 1 A ( x ) x \frac{z_{1A}(x)}{x} x z 1 A ( x ) x x x L + 1 L+1 L + 1 H H H (66) is thus given by z a c c ( x ) z_{\mathrm{acc}}(x) z acc ( x )
z a c c ( x ) = z 1 S ( x ) + 1 2 z 2 M ( x ) − z 1 A ( x ) x = 1 + x ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) (120) z_{\mathrm{acc}}(x)=z_{1S}(x)+\frac{1}{2}z_{2M}(x)-\frac{z_{1A}(x)}{x}=\frac{1+x}{(1-x)(1-x^{2})(1-x^{3})} \tag*{(120)} z acc ( x ) = z 1 S ( x ) + 2 1 z 2 M ( x ) − x z 1 A ( x ) = ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) 1 + x (120) in perfect agreement with Eq. (59) .
Polynomial nature of I \mathcal{I} I 2 κ 2\kappa 2 κ κ ≥ 2 \kappa\ge2 κ ≥ 2
In this section we demonstrate that I \mathcal{I} I
The discussion of this subsection can be viewed as a generalization (also an independent rederivation) of (19) to arbitrary even dimensions. As in the rest of this paper, we work with CP even theories whose partition function is well described by a local thermal effective action.
Throughout this Appendix we work in the embedding space, R 2 κ , 2 \mathbb{R}^{2\kappa,2} R 2 κ , 2 [51 ] for a review). The manifold S 2 κ − 1 × R S^{2\kappa-1}\times\mathbb{R} S 2 κ − 1 × R N \mathcal{N} N N P N^{P} N P P = − 1 , 0 , 1 … 2 κ P=-1,0,1\ldots2\kappa P = − 1 , 0 , 1 … 2 κ N 2 = 0 N^{2}=0 N 2 = 0 N N N N N N ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot [6 ] ). We can reach this Weyl gauge starting with the metric g μ ν g_{\mu\nu} g μν
g μ ν v 2 (121) \frac{g_{\mu\nu}}{v^{2}} \tag*{(121)} v 2 g μν (121) where v 2 v^{2} v 2 g μ ν g_{\mu\nu} g μν 27
The vector v v v
Recall that the vector v v v S O ( 2 κ , 2 ) SO(2\kappa,2) S O ( 2 κ , 2 ) ( 2 κ + 2 ) × ( 2 κ + 2 ) (2\kappa+2)\times(2\kappa+2) ( 2 κ + 2 ) × ( 2 κ + 2 ) A Q P A^{P}_{Q} A Q P 28
η A T = − A η , ⟹ ( A η ) T = − A η (122) \eta A^{T}=-A\eta,\qquad\Longrightarrow\qquad(A\eta)^{T}=-A\eta \tag*{(122)} η A T = − A η , ⟹ ( A η ) T = − A η (122) where η = η − 1 \eta=\eta^{-1} η = η − 1 R 2 κ , 2 R^{2\kappa,2} R 2 κ , 2 A η A\eta A η A A A
Now let us suppose that λ \lambda λ A A A
det ( A − λ I ) = 0 ⟹ det ( A η − λ η ) = 0 ⟹ det ( ( A η ) T − λ η T ) = 0 ⟹ det ( − A η − λ η ) = 0 ⟹ det ( A + λ I ) = 0 (123) \begin{aligned}
\det(A-\lambda I)&=0\\
\Longrightarrow\det(A\eta-\lambda\eta)&=0\\
\Longrightarrow\det((A\eta)^{T}-\lambda\eta^{T})&=0\\
\Longrightarrow\det(-A\eta-\lambda\eta)&=0\\
\Longrightarrow\det(A+\lambda I)&=0 \tag*{(123)}
\end{aligned} det ( A − λ I ) ⟹ det ( A η − λ η ) ⟹ det (( A η ) T − λ η T ) ⟹ det ( − A η − λ η ) ⟹ det ( A + λ I ) = 0 = 0 = 0 = 0 = 0 (123) so that − λ -\lambda − λ A A A B B B
B : = − A 2 . (124) B:=-A^{2}. \tag*{(124)} B := − A 2 . (124) have degeneracy two (B B B A A A B Q P B^{P}_{Q} B Q P B B B κ + 1 \kappa+1 κ + 1
( c 0 , c 1 , … , c κ ) . (125) (c_0,c_1,\ldots,c_\kappa). \tag*{(125)} ( c 0 , c 1 , … , c κ ) . (125) In order to understand the physical meaning of c i c_i c i ( v L ) μ (v_L)_\mu ( v L ) μ
v L = i v E = β ( ∂ t + ω 1 ∂ ϕ 1 + ⋯ + ω κ ∂ ϕ κ ) (126) v_L=iv_E=\beta\left(\partial_t+\omega_1\partial_{\phi_1}+\cdots+\omega_\kappa\partial_{\phi_\kappa}\right) \tag*{(126)} v L = i v E = β ( ∂ t + ω 1 ∂ ϕ 1 + ⋯ + ω κ ∂ ϕ κ ) (126) where “L” denotes Lorentzian and “E” denotes Euclidean. This CKV clearly corresponds to a block diagonal matrix A A A 2 × 2 2\times2 2 × 2
\begin{pmatrix}0&\beta\\-\beta&0\end{pmatrix},
\qquad\text{and}\qquad
\begin{pmatrix}0&\beta\omega_i\\-\beta\omega_i&0\end{pmatrix}
\quad\text{for }i=1,2,\ldots,\kappa. \tag*{(127)}\end{pmatrix}
Computing the eigenvalues of B = − A 2 B=-A^2 B = − A 2
( c 0 , c 1 , … , c κ ) = ( b 0 2 , b 1 2 , … , b κ 2 ) = ( β 2 , β 2 ω 1 2 , … , β 2 ω κ 2 ) . (128) (c_0,c_1,\ldots,c_\kappa)=(b_0^2,b_1^2,\ldots,b_\kappa^2)=(\beta^2,\beta^2\omega_1^2,\ldots,\beta^2\omega_\kappa^2). \tag*{(128)} ( c 0 , c 1 , … , c κ ) = ( b 0 2 , b 1 2 , … , b κ 2 ) = ( β 2 , β 2 ω 1 2 , … , β 2 ω κ 2 ) . (128) Below we will also often use the completely symmetric polynomials E r E_r E r (38) ; or equivalently via
Q ( t ) : = ∏ i = 0 κ ( t − c i ) = t κ + 1 − E 1 t κ + E 2 t κ − 1 − ⋯ + ( − 1 ) κ + 1 E κ + 1 . (129) Q(t):=\prod_{i=0}^{\kappa}(t-c_i)=t^{\kappa+1}-E_1t^\kappa+E_2t^{\kappa-1}-\cdots+(-1)^{\kappa+1}E_{\kappa+1}. \tag*{(129)} Q ( t ) := i = 0 ∏ κ ( t − c i ) = t κ + 1 − E 1 t κ + E 2 t κ − 1 − ⋯ + ( − 1 ) κ + 1 E κ + 1 . (129) We emphasize that E r E_r E r S κ + 1 S_{\kappa+1} S κ + 1
Fixing the Weyl gauge
It is now a simple matter to fix Weyl ‘gauge’ as in Eq. (121) . Recall that N N N
N 2 = 0. (130) N^2=0. \tag*{(130)} N 2 = 0. (130) In the Weyl gauge (S63) it follows from definitions that v 2 = 1 v^2=1 v 2 = 1 v 2 v^2 v 2
v 2 = − ( A N ) 2 = N ⋅ A 2 N = − N ⋅ B N . (131) v^2=-(AN)^2=N\cdot A^2N=-N\cdot BN. \tag*{(131)} v 2 = − ( A N ) 2 = N ⋅ A 2 N = − N ⋅ B N . (131) (the minus sign converts the Lorentzian norm of A N AN A N v 2 v^2 v 2 (121) is implemented by rescaling all null rays to ensure that
N ⋅ B N = − 1 (132) N\cdot BN=-1 \tag*{(132)} N ⋅ B N = − 1 (132) (S74) implements our choice of Weyl gauge.
Coordinates in embedding space
Below, we will make frequent use of the position-dependent quantities
m j : = − N ⋅ B j N , m 0 = 0 , m 1 = 1. (133) m_j:=-N\cdot B^jN,\qquad m_0=0,\qquad m_1=1. \tag*{(133)} m j := − N ⋅ B j N , m 0 = 0 , m 1 = 1. (133) Recall that m j m_j m j E r E_r E r m j m_j m j j = 2 … ∞ j=2\ldots\infty j = 2 … ∞ tr ( B j ) \operatorname{tr}(B^j) tr ( B j ) j = 1 … ∞ j=1\ldots\infty j = 1 … ∞
E 1 , … , E κ + 1 , m 2 , … , m κ . (134) E_1,\ldots,E_{\kappa+1},\qquad m_2,\ldots,m_\kappa. \tag*{(134)} E 1 , … , E κ + 1 , m 2 , … , m κ . (134) The simple argument for this result goes as follows. As every matrix obeys its characteristic equation, it follows that
Q ( B ) = B κ + 1 − E 1 B κ + E 2 B κ − 1 − ⋯ + ( − 1 ) κ + 1 E κ + 1 = 0. (135) Q(B)=B^{\kappa+1}-E_1B^\kappa+E_2B^{\kappa-1}-\cdots+(-1)^{\kappa+1}E_{\kappa+1}=0 . \tag*{(135)} Q ( B ) = B κ + 1 − E 1 B κ + E 2 B κ − 1 − ⋯ + ( − 1 ) κ + 1 E κ + 1 = 0. (135) Multiplying Eq. (135) by B r B^r B r N N N
m r + κ + 1 = E 1 m r + κ − E 2 m r + κ − 1 + ⋯ + ( − 1 ) κ E κ + 1 m r . (136) m_{r+\kappa+1}=E_1m_{r+\kappa}-E_2m_{r+\kappa-1}+\cdots+(-1)^\kappa E_{\kappa+1}m_r . \tag*{(136)} m r + κ + 1 = E 1 m r + κ − E 2 m r + κ − 1 + ⋯ + ( − 1 ) κ E κ + 1 m r . (136) Using the same equation on every m s m_s m s s ≥ κ + 1 s\geq\kappa+1 s ≥ κ + 1 (136) , and repeatedly iterating this process, we conclude that every m j m_j m j j > κ j>\kappa j > κ Tr B j = 2 ∑ a c a j \operatorname{Tr} B^j=2\sum_a c_a^j Tr B j = 2 ∑ a c a j E r E_r E r
The m j m_j m j κ − 1 \kappa-1 κ − 1 ( S 2 κ − 1 × R ) / G v (S^{2\kappa-1}\times R)/G_v ( S 2 κ − 1 × R ) / G v G v G_v G v S O ( 2 κ , 2 ) SO(2\kappa,2) S O ( 2 κ , 2 ) v v v v v v G v G_v G v κ + 1 \kappa+1 κ + 1 ϕ i \phi_i ϕ i t t t ( κ − 1 ) (\kappa-1) ( κ − 1 ) m j m_j m j t t t ϕ i \phi_i ϕ i S 2 κ − 1 × R S^{2\kappa-1}\times R S 2 κ − 1 × R κ − 1 \kappa-1 κ − 1 m j m_j m j κ − 1 \kappa-1 κ − 1 S 2 κ − 1 S^{2\kappa-1} S 2 κ − 1 m j m_j m j S O ( d , 2 ) SO(d,2) S O ( d , 2 )
The projector
The metric of S 2 κ − 1 × R S^{2\kappa-1}\times R S 2 κ − 1 × R R 2 κ , 2 R^{2\kappa,2} R 2 κ , 2 N 0 N_0 N 0 N = N 0 + δ N N=N_0+\delta N N = N 0 + δ N δ N \delta N δ N δ N \delta N δ N
δ N ⋅ N 0 = 0 , δ N ⋅ B N 0 = 0. (137) \delta N\cdot N_0=0,\qquad\delta N\cdot BN_0=0 . \tag*{(137)} δ N ⋅ N 0 = 0 , δ N ⋅ B N 0 = 0. (137) (S79), viewed as two constraints on the vectors δ N \delta N δ N N 0 N_0 N 0 Π \Pi Π 2 κ 2\kappa 2 κ
Consider a general n n n m m m v ( a ) μ v^\mu_{(a)} v ( a ) μ a = 1 … m a=1\ldots m a = 1 … m G a b G_{ab} G ab m × m m\times m m × m G a b G^{ab} G ab G a b G_{ab} G ab v ( a ) v_{(a)} v ( a )
Π ν μ = δ ν μ − G a b v ( a ) μ v ( b ) ν . (138) \Pi^\mu_{\ \nu}=\delta^\mu_{\ \nu}-G^{ab}v^\mu_{(a)}v_{(b)\nu}. \tag*{(138)} Π ν μ = δ ν μ − G ab v ( a ) μ v ( b ) ν . (138) Note, in particular, that independent rescalings of v ( a ) v_{(a)} v ( a )
We can now apply the general formula (S80) to obtain the projector onto the tangent space of a general point N N N m = 2 m=2 m = 2 v ( a ) v_{(a)} v ( a ) N N N B N BN B N
G a b = ( N 2 N ⋅ B N B N ⋅ N ( B N ) 2 ) = ( 0 − 1 − 1 − m 2 ) . (139) G_{ab}=
\begin{pmatrix}
N^2 & N\cdot BN\\
BN\cdot N & (BN)^2
\end{pmatrix}
=
\begin{pmatrix}
0 & -1\\
-1 & -m_2
\end{pmatrix}. \tag*{(139)} G ab = ( N 2 B N ⋅ N N ⋅ B N ( B N ) 2 ) = ( 0 − 1 − 1 − m 2 ) . (139) A key simplification is that the determinant of this matrix is − 1 -1 − 1
G a b = ( m 2 − 1 − 1 0 ) . (140) G^{ab}=
\begin{pmatrix}
m_2 & -1\\
-1 & 0
\end{pmatrix}. \tag*{(140)} G ab = ( m 2 − 1 − 1 0 ) . (140) It follows from (S80) that projector onto our tangent space is given by
Π P Q = δ P Q − m 2 N Q N P + N Q ( B N ) P + ( B N ) Q N P \Pi^Q_{\ P}=\delta^Q_{\ P}-m_2N^QN_P+N^Q(BN)_P+(BN)^QN_P Π P Q = δ P Q − m 2 N Q N P + N Q ( B N ) P + ( B N ) Q N P ⟹ Π V = V − m 2 N ( N ⋅ V ) + N ( B N ⋅ V ) + B N ( N ⋅ V ) . (141) \Longrightarrow\quad\Pi V=V-m_2N(N\cdot V)+N(BN\cdot V)+BN(N\cdot V) . \tag*{(141)} ⟹ Π V = V − m 2 N ( N ⋅ V ) + N ( B N ⋅ V ) + B N ( N ⋅ V ) . (141) Covariant derivatives preserve the polynomial nature
Whenever a manifold can be viewed as a submanifold of a flat space, the covariant derivative (of any tensor on the submanifold) is obtained by taking the ordinary derivative of that tensor (along the relevant tangent derivatives of the manifold), and projecting the result of this computation back onto the tangent space of the manifold.
In the current context, our boundary spacetime consists of the codimension two submanifold of R 2 κ , 2 \mathbb{R}^{2\kappa,2} R 2 κ , 2 (141) . Applying the prescription spelled out in the previous paragraph, we conclude that
∇ ˉ U V = Π D U V , (142) \bar{\nabla}_{U}V = \Pi D_{U}V \,, \tag*{(142)} ∇ ˉ U V = Π D U V , (142) where D U D_U D U U U U
Now recall that A A A v v v B = − A 2 B=-A^2 B = − A 2
D U A = 0 , D U B = 0 . (143) D_U A = 0 \,, \qquad D_U B = 0 \,. \tag*{(143)} D U A = 0 , D U B = 0 . (143) Moreover, the covariant derivative of the vector field N A N^A N A
( D U N ) P = Π Q P U M ∂ M N Q = Π Q P U Q = U P , ⟹ D U N = U . (144) (D_U N)^P = \Pi^P_Q U^M \partial_M N^Q = \Pi^P_Q U^Q = U^P \,, \qquad\Longrightarrow\quad D_U N = U \,. \tag*{(144)} ( D U N ) P = Π Q P U M ∂ M N Q = Π Q P U Q = U P , ⟹ D U N = U . (144) It follows that differentiating a polynomial expression in A , B , N A,B,N A , B , N
Since the m j m_j m j N N N B B B m j m_j m j m j m_j m j E r E_r E r m j m_j m j
∇ ˉ P m j = Π P Q ∂ Q m j = − 2 Π P Q ( B j ) Q R N R = − 2 ( B j N ) P − 2 ( m 2 m j − m j + 1 ) N P + 2 m j ( B N ) P . (145) \bar{\nabla}_P m_j = \Pi_P^Q \partial_Q m_j = -2\Pi_P^Q (B^j)_{QR}N^R = -2(B^jN)_P - 2(m_2m_j-m_{j+1})N_P + 2m_j(BN)_P \,. \tag*{(145)} ∇ ˉ P m j = Π P Q ∂ Q m j = − 2 Π P Q ( B j ) QR N R = − 2 ( B j N ) P − 2 ( m 2 m j − m j + 1 ) N P + 2 m j ( B N ) P . (145) The RHS is polynomial, as expected. Repeated derivatives of (S87), followed by repeated use of (S83), therefore remain polynomial as well.
The Riemann Tensor
A similar conclusion also holds for all curvature invariants. As our submanifold is embedded in flat space, its Riemann tensor is given by the Gauss–Codazzi equations
R P Q R S = G a b ( K P R ( a ) K Q S ( b ) − K P S ( a ) K Q R ( b ) ) , (146) R_{PQRS}=G^{ab}\left(K^{(a)}_{PR}K^{(b)}_{QS}-K^{(a)}_{PS}K^{(b)}_{QR}\right) \,, \tag*{(146)} R P QR S = G ab ( K P R ( a ) K QS ( b ) − K P S ( a ) K QR ( b ) ) , (146) where G a b G^{ab} G ab K μ ν ( a ) K^{(a)}_{\mu\nu} K μν ( a )
K P Q ( a ) : = Π P P ′ Π Q Q ′ D P ′ n Q ′ ( a ) . (147) K^{(a)}_{PQ}:=\Pi_P^{P'}\Pi_Q^{Q'}D_{P'}n^{(a)}_{Q'} \,. \tag*{(147)} K P Q ( a ) := Π P P ′ Π Q Q ′ D P ′ n Q ′ ( a ) . (147) As around (S81) we choose our two normal vectors n ( a ) n^{(a)} n ( a ) N P N^P N P ( B N ) P (BN)^P ( B N ) P
K P Q ( 1 ) = Π P P ′ Π Q Q ′ D P ′ N Q ′ = Π P P ′ Π P ′ Q = Π P Q , K^{(1)}_{PQ}=\Pi_P^{P'}\Pi_Q^{Q'}D_{P'}N_{Q'}=\Pi_P^{P'}\Pi_{P'Q}=\Pi_{PQ} \,, K P Q ( 1 ) = Π P P ′ Π Q Q ′ D P ′ N Q ′ = Π P P ′ Π P ′ Q = Π P Q , K P Q ( 2 ) = Π P P ′ Π Q Q ′ D P ′ ( B N ) Q ′ = Π P P ′ Π Q Q ′ B P ′ Q ′ : = M P Q , (148) K^{(2)}_{PQ}=\Pi_P^{P'}\Pi_Q^{Q'}D_{P'}(BN)_{Q'}=\Pi_P^{P'}\Pi_Q^{Q'}B_{P'Q'}:=M_{PQ} \,, \tag*{(148)} K P Q ( 2 ) = Π P P ′ Π Q Q ′ D P ′ ( B N ) Q ′ = Π P P ′ Π Q Q ′ B P ′ Q ′ := M P Q , (148) where we have used D P N Q = η P Q D_PN_Q=\eta_{PQ} D P N Q = η P Q
R P Q R S = m 2 ( Π P R Π Q S − Π P S Π Q R ) − M P R Π Q S − Π P R M Q S + M P S Π Q R + Π P S M Q R . (149) R_{PQRS}=m_2\left(\Pi_{PR}\Pi_{QS}-\Pi_{PS}\Pi_{QR}\right)-M_{PR}\Pi_{QS}-\Pi_{PR}M_{QS}+M_{PS}\Pi_{QR}+\Pi_{PS}M_{QR} \,. \tag*{(149)} R P QR S = m 2 ( Π P R Π QS − Π P S Π QR ) − M P R Π QS − Π P R M QS + M P S Π QR + Π P S M QR . (149) The following identities can be obtained using (S90) and (S83)
M P P = M P Q η Q P = Tr Π B = Tr B − m 2 , M^P_P=M_{PQ}\eta^{QP}=\operatorname{Tr}\Pi B=\operatorname{Tr}B-m_2 \,, M P P = M P Q η QP = Tr Π B = Tr B − m 2 , M P Q M P Q = Tr [ Π B Π B ] = Tr B 2 + m 2 2 − 2 m 3 . (150) M_{PQ}M^{PQ}=\operatorname{Tr}\left[\Pi B\Pi B\right]=\operatorname{Tr}B^2+m_2^2-2m_3 \,. \tag*{(150)} M P Q M P Q = Tr [ Π B Π B ] = Tr B 2 + m 2 2 − 2 m 3 . (150) Hence, these scalars are also polynomials. Using (S91) and (S92), we get
R P Q = ( d m 2 − Tr B ) Π P Q − ( d − 2 ) M P Q , R_{PQ}=(d\,m_2-\operatorname{Tr}B)\Pi_{PQ}-(d-2)M_{PQ} \,, R P Q = ( d m 2 − Tr B ) Π P Q − ( d − 2 ) M P Q , R = ( d − 1 ) [ ( d + 2 ) m 2 − 2 Tr B ] . (151) R=(d-1)\left[(d+2)m_2-2\operatorname{Tr}B\right] \,. \tag*{(151)} R = ( d − 1 ) [ ( d + 2 ) m 2 − 2 Tr B ] . (151) Sample curvature invariants
The other curvature invariants are
R P Q R P Q = d ( d m 2 − Tr B ) 2 − 2 ( d − 2 ) ( d m 2 − Tr B ) ( Tr B − m 2 ) + ( d − 2 ) 2 [ Tr ( B 2 ) + m 2 2 − 2 m 3 ] (152) R_{PQ}R^{PQ}=d\left(d m_{2}-\operatorname{Tr} B\right)^{2}-2(d-2)\left(d m_{2}-\operatorname{Tr} B\right)\left(\operatorname{Tr} B-m_{2}\right)+(d-2)^{2}\left[\operatorname{Tr}\left(B^{2}\right)+m_{2}^{2}-2m_{3}\right] \tag*{(152)} R P Q R P Q = d ( d m 2 − Tr B ) 2 − 2 ( d − 2 ) ( d m 2 − Tr B ) ( Tr B − m 2 ) + ( d − 2 ) 2 [ Tr ( B 2 ) + m 2 2 − 2 m 3 ] (152) R P Q R S R P Q R S = 2 d ( d − 1 ) m 2 2 − 8 ( d − 1 ) m 2 ( Tr B − m 2 ) + 4 ( Tr B − m 2 ) 2 + 4 ( d − 2 ) [ Tr ( B 2 ) + m 2 2 − 2 m 3 ] . (153) R_{PQRS}R^{PQRS}=2d(d-1)m_{2}^{2}-8(d-1)m_{2}\left(\operatorname{Tr} B-m_{2}\right)+4\left(\operatorname{Tr} B-m_{2}\right)^{2}+4(d-2)\left[\operatorname{Tr}\left(B^{2}\right)+m_{2}^{2}-2m_{3}\right]. \tag*{(153)} R P QR S R P QR S = 2 d ( d − 1 ) m 2 2 − 8 ( d − 1 ) m 2 ( Tr B − m 2 ) + 4 ( Tr B − m 2 ) 2 + 4 ( d − 2 ) [ Tr ( B 2 ) + m 2 2 − 2 m 3 ] . (153) One can easily see that all these scalars are polynomials. Derivatives of curvature are also polynomial as the derivatives act only on m i m_i m i B B B
( ∇ ˉ m j ) ⋅ ( ∇ ˉ m k ) = 4 [ − m j + k + m j m k + 1 + m k m j + 1 − m 2 m j m k ] . (154) \left(\bar{\nabla}m_j\right)\cdot\left(\bar{\nabla}m_k\right)=4\left[-m_{j+k}+m_jm_{k+1}+m_km_{j+1}-m_2m_jm_k\right]. \tag*{(154)} ( ∇ ˉ m j ) ⋅ ( ∇ ˉ m k ) = 4 [ − m j + k + m j m k + 1 + m k m j + 1 − m 2 m j m k ] . (154) Since Tr B \operatorname{Tr} B Tr B
( ∇ ˉ R ) 2 = 4 ( d − 1 ) 2 ( d + 2 ) 2 [ − m 4 + 2 m 2 m 3 − m 2 3 ] . (155) \left(\bar{\nabla}R\right)^2=4(d-1)^2(d+2)^2\left[-m_4+2m_2m_3-m_2^3\right]. \tag*{(155)} ( ∇ ˉ R ) 2 = 4 ( d − 1 ) 2 ( d + 2 ) 2 [ − m 4 + 2 m 2 m 3 − m 2 3 ] . (155) When m 4 m_4 m 4
In the metric induced on the codimension two manifold (S79), the unit thermal vector is
u : = A N , u 2 = − N A 2 N = N B N = − 1 , (156) u:=AN,\qquad u^2=-NA^2N=NB N=-1, \tag*{(156)} u := A N , u 2 = − N A 2 N = N B N = − 1 , (156) (u 2 = − 1 u^2=-1 u 2 = − 1 u 2 = 1 u^2=1 u 2 = 1
The projection orthogonal to u u u N N N
Tr A 2 j = Tr ( − B ) j , N ⋅ A 2 j N = ( − 1 ) j + 1 m j . (157) \operatorname{Tr}A^{2j}=\operatorname{Tr}(-B)^j,\qquad N\cdot A^{2j}N=(-1)^{j+1}m_j. \tag*{(157)} Tr A 2 j = Tr ( − B ) j , N ⋅ A 2 j N = ( − 1 ) j + 1 m j . (157) Odd traces and N ⋅ A 2 j + 1 N N\cdot A^{2j+1}N N ⋅ A 2 j + 1 N A A A Tr B j = 2 ∑ i c i j \operatorname{Tr}B^j=2\sum_i c_i^j Tr B j = 2 ∑ i c i j E r E_r E r
I = I ( E 1 , … , E κ + 1 ; m 2 , … , m κ ) (158) \mathcal{I}=\mathcal{I}(E_1,\ldots,E_{\kappa+1};m_2,\ldots,m_\kappa) \tag*{(158)} I = I ( E 1 , … , E κ + 1 ; m 2 , … , m κ ) (158) is a polynomial. Its coefficients are theory-dependent Wilson coefficients and do not depend on the compactification data. In d = 4 d=4 d = 4 m 2 m_2 m 2 m 2 m_2 m 2 z z z (15) and used extensively in the main text) via
z = E 1 − m 2 . (159) z=E_1-m_2. \tag*{(159)} z = E 1 − m 2 . (159) Thus the fact that I \mathcal{I} I z z z b i 2 b_i^2 b i 2 (18) ) follows directly from the general discussion of this subsection.
The integral over S 2 κ − 1 S^{2\kappa-1} S 2 κ − 1 κ ≥ 2 \kappa\geq2 κ ≥ 2
In the previous subsection we have demonstrated that the integrand, I \mathcal{I} I m i m_i m i E r E_r E r I \mathcal{I} I (3) . The final formulae of this subsection will also form the starting point for our derivation of (6) in the next subsection.
Below, we will find it convenient to switch between several distinct coordinates for S 2 κ − 1 S^{2\kappa-1} S 2 κ − 1
Direction cosines
One familiar set of coordinates on S 2 κ − 1 S^{2\kappa-1} S 2 κ − 1 κ \kappa κ ϕ i \phi_i ϕ i κ \kappa κ J i J_i J i κ \kappa κ ℓ i \ell_i ℓ i C κ \mathbb{C}^{\kappa} C κ S 2 κ − 1 ⊂ C κ S^{2\kappa-1} \subset\mathbb{C}^{\kappa} S 2 κ − 1 ⊂ C κ
z i = ℓ i e i ϕ i , ℓ i ≥ 0 , ∑ i = 1 κ ℓ i = 1. (160) z_i = \sqrt{\ell_i}e^{i\phi_i}, \qquad\ell_i \ge0, \qquad\sum_{i=1}^{\kappa} \ell_i = 1 . \tag*{(160)} z i = ℓ i e i ϕ i , ℓ i ≥ 0 , i = 1 ∑ κ ℓ i = 1. (160) In these coordinates, the round sphere metric times R R R
d s 2 = d τ 2 + ∑ m = 1 κ ( d ℓ m 2 4 ℓ m + ℓ m d ϕ m 2 ) . (161) ds^2 = d\tau^2 + \sum_{m=1}^{\kappa} \left(\frac{d\ell_m^2}{4\ell_m} + \ell_m d\phi_m^2\right) . \tag*{(161)} d s 2 = d τ 2 + m = 1 ∑ κ ( 4 ℓ m d ℓ m 2 + ℓ m d ϕ m 2 ) . (161) In the coordinates (S102), the norm of the vector v v v (13) ) is given by
v 2 = β 2 − ∑ i = 1 κ ℓ i β 2 ω i 2 = β 2 ( ∑ i = 1 κ ℓ i ) − ∑ i = 1 κ ℓ i β 2 ω i 2 = ∑ i = 1 κ ( c 0 − c i ) ℓ i . (162) v^2 = \beta^2 - \sum_{i=1}^{\kappa} \ell_i\beta^2\omega_i^2 = \beta^2\left(\sum_{i=1}^{\kappa}\ell_i\right) - \sum_{i=1}^{\kappa}\ell_i\beta^2\omega_i^2 = \sum_{i=1}^{\kappa}(c_0-c_i)\ell_i . \tag*{(162)} v 2 = β 2 − i = 1 ∑ κ ℓ i β 2 ω i 2 = β 2 ( i = 1 ∑ κ ℓ i ) − i = 1 ∑ κ ℓ i β 2 ω i 2 = i = 1 ∑ κ ( c 0 − c i ) ℓ i . (162) In these coordinates, the thermal effective action is given in terms of the density I \mathcal{I} I
ln Z r o t β = ( 2 π ) κ 2 κ − 1 ∫ ℓ i ≥ 0 , ∑ i ℓ i = 1 d κ − 1 ℓ ( v 2 ) κ I ( E , m ) , (163) \frac{\ln Z^{\mathrm{rot}}}{\beta} = \frac{(2\pi)^{\kappa}}{2^{\kappa-1}}\int_{\substack{\ell_i\ge0,\\ \sum_i\ell_i=1}}\frac{d^{\kappa-1}\ell}{(v^2)^{\kappa}}\mathcal{I}(E,m) , \tag*{(163)} β ln Z rot = 2 κ − 1 ( 2 π ) κ ∫ ℓ i ≥ 0 , ∑ i ℓ i = 1 ( v 2 ) κ d κ − 1 ℓ I ( E , m ) , (163) (the factor of ( 2 π ) κ (2\pi)^{\kappa} ( 2 π ) κ κ \kappa κ ϕ i \phi_i ϕ i 2 κ − 1 2^{\kappa-1} 2 κ − 1 g \sqrt{g} g
We can visualize the integration region in ℓ i \ell_i ℓ i ∑ j ℓ j = 1 \sum_j\ell_j=1 ∑ j ℓ j = 1 κ \kappa κ ℓ i \ell_i ℓ i i = 1 … κ i=1\ldots\kappa i = 1 … κ κ \kappa κ ℓ i = 0 \ell_i=0 ℓ i = 0 i = 1 … κ i=1\ldots\kappa i = 1 … κ ϕ i \phi_i ϕ i
The coordinates λ i \lambda_i λ i
For reasons that will soon become clear, we will sometimes find it convenient to change coordinates from ℓ i \ell_i ℓ i λ i \lambda_i λ i
λ i : = ( c 0 − c i ) ℓ i v 2 , λ i ≥ 0 , ∑ i = 1 κ λ i = 1. (164) \lambda_i := \frac{(c_0-c_i)\ell_i}{v^2}, \qquad\lambda_i \ge0, \qquad\sum_{i=1}^{\kappa}\lambda_i = 1 . \tag*{(164)} λ i := v 2 ( c 0 − c i ) ℓ i , λ i ≥ 0 , i = 1 ∑ κ λ i = 1. (164) This coordinate change is easily inverted to find ℓ i \ell_i ℓ i λ i \lambda_i λ i
ℓ i = λ i / ( c 0 − c i ) ∑ j λ j / ( c 0 − c j ) . (165) \ell_i = \frac{\lambda_i/(c_0-c_i)}{\sum_j\lambda_j/(c_0-c_j)} . \tag*{(165)} ℓ i = ∑ j λ j / ( c 0 − c j ) λ i / ( c 0 − c i ) . (165) Inserting (S106) allows us to express v 2 v^2 v 2 λ i \lambda_i λ i
v 2 = ( ∑ j λ j c 0 − c j ) − 1 . (166) v^2 = \left(\sum_j\frac{\lambda_j}{c_0-c_j}\right)^{-1} . \tag*{(166)} v 2 = ( j ∑ c 0 − c j λ j ) − 1 . (166) The Jacobian of the variable change from ℓ i \ell_i ℓ i λ i \lambda_i λ i
d κ − 1 ℓ ( v 2 ) κ = d κ − 1 λ ∏ i = 1 κ ( c 0 − c i ) . (167) \frac{d^{\kappa-1}\ell}{(v^2)^{\kappa}} = \frac{d^{\kappa-1}\lambda}{\prod_{i=1}^{\kappa}(c_0-c_i)} . \tag*{(167)} ( v 2 ) κ d κ − 1 ℓ = ∏ i = 1 κ ( c 0 − c i ) d κ − 1 λ . (167) The key point here is that the λ i \lambda_i λ i ℓ i \ell_i ℓ i
The integration domain for the λ i \lambda_i λ i ℓ i \ell_i ℓ i ϕ i \phi_i ϕ i λ i = 0 \lambda_i=0 λ i = 0
The integration domain in the λ i \lambda_i λ i ( λ 1 , λ 2 … λ κ ) (\lambda_1,\lambda_2\ldots\lambda_\kappa) ( λ 1 , λ 2 … λ κ ) κ \kappa κ e 1 = ( 1 , 0 , … , 0 ) \mathbf{e}_1=(1,0,\ldots,0) e 1 = ( 1 , 0 , … , 0 ) e 2 = ( 0 , 1 , 0 , … , 0 ) , … e κ = ( 0 , 0 , 0 , … , 1 ) \mathbf{e}_2=(0,1,0,\ldots,0),\ldots\mathbf{e}_\kappa=(0,0,0,\ldots,1) e 2 = ( 0 , 1 , 0 , … , 0 ) , … e κ = ( 0 , 0 , 0 , … , 1 )
∑ j λ j e j , λ i ≥ 0 , ∑ j = 1 κ λ j = 1. (168) \sum_j\lambda_j\mathbf{e}_j,\quad\lambda_i\geq0,\quad\sum_{j=1}^{\kappa}\lambda_j=1. \tag*{(168)} j ∑ λ j e j , λ i ≥ 0 , j = 1 ∑ κ λ j = 1. (168) However (S109) is the definition of a polytope built out of the κ \kappa κ e 1 … e κ \mathbf{e}_1\ldots\mathbf{e}_\kappa e 1 … e κ λ i \lambda_i λ i
mi as coordinates
As we have already explained in the previous subsection, the quantities m i m_i m i S 2 κ − 1 S^{2\kappa-1} S 2 κ − 1 m i m_i m i
m j = N ⋅ B j N N ⋅ B N (169) m_j=\frac{N\cdot B^jN}{N\cdot BN} \tag*{(169)} m j = N ⋅ B N N ⋅ B j N (169) (S110) is the Weyl invariant generalization of (S75) because
We can now use the Weyl invariant definition (S110), to compute m j m_j m j N N N
N = ( cos t , sin t , l 1 cos ϕ 1 , l 1 sin ϕ 1 , l 2 cos ϕ 2 , l 2 sin ϕ 2 … l κ cos ϕ κ , l κ sin ϕ κ ) , (170) N=\left(\cos t,\sin t,\sqrt{l_1}\cos\phi_1,\sqrt{l_1}\sin\phi_1,\sqrt{l_2}\cos\phi_2,\sqrt{l_2}\sin\phi_2\ldots\sqrt{l_\kappa}\cos\phi_\kappa,\sqrt{l_\kappa}\sin\phi_\kappa\right), \tag*{(170)} N = ( cos t , sin t , l 1 cos ϕ 1 , l 1 sin ϕ 1 , l 2 cos ϕ 2 , l 2 sin ϕ 2 … l κ cos ϕ κ , l κ sin ϕ κ ) , (170) (where t t t (170) are timelike, while the remaining are spacelike). In the same basis, the matrix A A A B = − A 2 B=-A^2 B = − A 2
m j = c 0 j − ∑ i c i j l i v 2 , (171) m_j=\frac{c_0^j-\sum_i c_i^j l_i}{v^2}, \tag*{(171)} m j = v 2 c 0 j − ∑ i c i j l i , (171) 29
and hence
m j = ∑ i = 1 κ λ i c 0 j − c i j c 0 − c i = ∑ i = 1 κ λ i ( c 0 j − 1 + c 0 j − 2 c i + ⋯ + c i j − 1 ) . (172) \begin{aligned}
m_j&=\sum_{i=1}^{\kappa}\lambda_i\frac{c_0^j-c_i^j}{c_0-c_i}\\
&=\sum_{i=1}^{\kappa}\lambda_i\left(c_0^{j-1}+c_0^{j-2}c_i+\cdots+c_i^{j-1}\right). \tag*{(172)}
\end{aligned} m j = i = 1 ∑ κ λ i c 0 − c i c 0 j − c i j = i = 1 ∑ κ λ i ( c 0 j − 1 + c 0 j − 2 c i + ⋯ + c i j − 1 ) . (172) Note that the apparent denominator in the first line cancels. It follows that m j m_j m j λ i \lambda_i λ i c a c_a c a
Integration measure in m i m_i m i
The integration measure over the m a m_a m a
1 ∏ j = 1 κ ( c 0 − c j ) ∫ d λ 1 ∧ d λ 2 … d λ κ δ ( λ 1 + λ 2 + … λ κ − 1 ) . (173) \frac{1}{\prod_{j=1}^{\kappa}(c_0-c_j)}\int d\lambda_1\wedge d\lambda_2\ldots d\lambda_\kappa\delta(\lambda_1+\lambda_2+\ldots\lambda_\kappa-1). \tag*{(173)} ∏ j = 1 κ ( c 0 − c j ) 1 ∫ d λ 1 ∧ d λ 2 … d λ κ δ ( λ 1 + λ 2 + … λ κ − 1 ) . (173) Using the delta function to integrate out (say) λ 1 \lambda_1 λ 1
m j = c 0 j − c 1 j c 0 − c 1 + ∑ i = 2 κ λ i ( c 0 j − c i j c 0 − c i − c 0 j − c 1 j c 0 − c 1 ) , (174) m_j=\frac{c_0^j-c_1^j}{c_0-c_1}+\sum_{i=2}^{\kappa}\lambda_i\left(\frac{c_0^j-c_i^j}{c_0-c_i}-\frac{c_0^j-c_1^j}{c_0-c_1}\right), \tag*{(174)} m j = c 0 − c 1 c 0 j − c 1 j + i = 2 ∑ κ λ i ( c 0 − c i c 0 j − c i j − c 0 − c 1 c 0 j − c 1 j ) , (174) so that (S114) can be rewritten as
1 det A ~ ∏ j = 1 κ ( c 0 − c j ) d m 2 ∧ d m 3 … d m κ , (175) \frac{1}{\det\widetilde{A}\prod_{j=1}^{\kappa}(c_0-c_j)}dm_2\wedge dm_3\ldots dm_\kappa, \tag*{(175)} det A ∏ j = 1 κ ( c 0 − c j ) 1 d m 2 ∧ d m 3 … d m κ , (175) where the ( κ − 1 ) × ( κ − 1 ) (\kappa-1)\times(\kappa-1) ( κ − 1 ) × ( κ − 1 ) A ~ \widetilde{A} A
A ~ j i = B j i − B j 1 , ( i , j = 2 … κ ) \widetilde{A}_{ji}=B_{ji}-B_{j1},\qquad(i,j=2\ldots\kappa) A j i = B j i − B j 1 , ( i , j = 2 … κ ) B j i = c 0 j − c i j c 0 − c i , B j 1 = c 0 j − c 1 j c 0 − c 1 . (176) B_{ji}=\frac{c_0^j-c_i^j}{c_0-c_i},\qquad B_{j1}=\frac{c_0^j-c_1^j}{c_0-c_1}. \tag*{(176)} B j i = c 0 − c i c 0 j − c i j , B j 1 = c 0 − c 1 c 0 j − c 1 j . (176) As defined, B B B ( κ − 1 ) × κ (\kappa-1)\times\kappa ( κ − 1 ) × κ κ × κ \kappa\times\kappa κ × κ
B ^ = ( 1 1 ⋯ 1 B 2 , 1 B 2 , 2 ⋯ B 2 , κ ⋮ ⋮ ⋮ B κ , 1 B κ , 2 ⋯ B κ , κ ) . (177) \widehat{B}=
\begin{pmatrix}
1 & 1 & \cdots& 1\\
B_{2,1} & B_{2,2} & \cdots& B_{2,\kappa}\\
\vdots& \vdots& & \vdots\\
B_{\kappa,1} & B_{\kappa,2} & \cdots& B_{\kappa,\kappa}
\end{pmatrix}. \tag*{(177)} B = 1 B 2 , 1 ⋮ B κ , 1 1 B 2 , 2 ⋮ B κ , 2 ⋯ ⋯ ⋯ 1 B 2 , κ ⋮ B κ , κ . (177) By subtracting the first column of B ^ \widehat{B} B κ − 1 \kappa-1 κ − 1
det B ^ = det ( 1 0 0 ⋯ 0 B 2 , 1 A ~ 2 , 2 A ~ 2 , 3 ⋯ A ~ 2 , κ ⋮ ⋮ ⋮ B κ , 1 A ~ κ , 2 A ~ κ , 3 ⋯ A ~ κ , κ ) . (178) \det\widehat{B}=\det\begin{pmatrix}
1 & 0 & 0 & \cdots& 0\\
B_{2,1} & \widetilde{A}_{2,2} & \widetilde{A}_{2,3} & \cdots& \widetilde{A}_{2,\kappa}\\
\vdots& \vdots& & & \vdots\\
B_{\kappa,1} & \widetilde{A}_{\kappa,2} & \widetilde{A}_{\kappa,3} & \cdots& \widetilde{A}_{\kappa,\kappa}
\end{pmatrix}. \tag*{(178)} det B = det 1 B 2 , 1 ⋮ B κ , 1 0 A 2 , 2 ⋮ A κ , 2 0 A 2 , 3 A κ , 3 ⋯ ⋯ ⋯ 0 A 2 , κ ⋮ A κ , κ . (178) We conclude that
det B ^ = det A ~ . (179) \det\widehat{B}=\det\widetilde{A}. \tag*{(179)} det B = det A . (179) Now the determinant of B ^ \widehat{B} B 1 + 2 + … κ − 1 = κ ( κ − 1 ) 2 1+2+\ldots\kappa-1=\frac{\kappa(\kappa-1)}{2} 1 + 2 + … κ − 1 = 2 κ ( κ − 1 ) c 0 , c 1 … c κ c_0,c_1\ldots c_\kappa c 0 , c 1 … c κ c i = c j c_i=c_j c i = c j i , j ∈ 1 … κ i,j\in1\ldots\kappa i , j ∈ 1 … κ ∏ i < j ( c i − c j ) \prod_{i<j}(c_i-c_j) ∏ i < j ( c i − c j ) det B ^ \det\widehat{B} det B c 0 c_0 c 0
det B ^ = ∏ 1 ≤ i < j ≤ κ ( c j − c i ) . (180) \det\widehat{B}=\prod_{1\leq i<j\leq\kappa}(c_j-c_i). \tag*{(180)} det B = 1 ≤ i < j ≤ κ ∏ ( c j − c i ) . (180) Inserting (S121) and (S120) into (S116) we have
1 det A ~ ∏ j = 1 κ ( c 0 − c j ) d m 2 ∧ d m 3 … d m κ = ( − 1 ) κ ∏ 0 ≤ i < j ≤ κ ( c j − c i ) d m 2 ∧ d m 3 … d m κ . (181) \frac{1}{\det\widetilde{A}\prod_{j=1}^{\kappa}(c_0-c_j)}dm_2\wedge dm_3\ldots dm_\kappa
=
\frac{(-1)^\kappa}{\prod_{0\leq i<j\leq\kappa}(c_j-c_i)}dm_2\wedge dm_3\ldots dm_\kappa. \tag*{(181)} det A ∏ j = 1 κ ( c 0 − c j ) 1 d m 2 ∧ d m 3 … d m κ = ∏ 0 ≤ i < j ≤ κ ( c j − c i ) ( − 1 ) κ d m 2 ∧ d m 3 … d m κ . (181) Defining V \mathcal{V} V 2 π 2 P \frac{2\pi^2}{P} P 2 π 2
V : = ∏ 0 ≤ i < j ≤ κ ( c j − c i ) , (182) \mathcal{V} := \prod_{0\leq i<j\leq\kappa}(c_j-c_i), \tag*{(182)} V := 0 ≤ i < j ≤ κ ∏ ( c j − c i ) , (182) (note that V \mathcal{V} V V \mathcal{V} V ( 2 π ) κ (2\pi)^\kappa ( 2 π ) κ
( − 1 ) κ 2 π κ V d m 2 ∧ d m 3 … d m κ . (183) (-1)^\kappa\frac{2\pi^\kappa}{\mathcal{V}}\,dm_2\wedge dm_3\ldots dm_\kappa. \tag*{(183)} ( − 1 ) κ V 2 π κ d m 2 ∧ d m 3 … d m κ . (183) The integration range in m i m_i m i
As the m j m_j m j λ i \lambda_i λ i m j m_j m j κ \kappa κ V 0 ( a ) V_0^{(a)} V 0 ( a ) a = 1 … κ a=1\ldots\kappa a = 1 … κ
( m 2 , m 3 … m κ ) = ( ( V 0 ( a ) ) 2 , ( V 0 ( a ) ) 3 , … ( V 0 ( a ) ) κ ) , (184) (m_2,m_3\ldots m_\kappa)=\left(\left(V_0^{(a)}\right)_2,\left(V_0^{(a)}\right)_3,\ldots\left(V_0^{(a)}\right)_\kappa\right), \tag*{(184)} ( m 2 , m 3 … m κ ) = ( ( V 0 ( a ) ) 2 , ( V 0 ( a ) ) 3 , … ( V 0 ( a ) ) κ ) , (184) where (see (S113))
( V 0 ( b ) ) j = c 0 j − c b j c 0 − c b . (185) \left(V_0^{(b)}\right)_j=\frac{c_0^j-c_b^j}{c_0-c_b}. \tag*{(185)} ( V 0 ( b ) ) j = c 0 − c b c 0 j − c b j . (185) For use below, we define the quantities
s j ( a , b ) : = c a j − c b j c a − c b = c a j − 1 + c a j − 2 c b + ⋯ + c b j − 1 , b ≠ a , (186) s_j(a,b):=\frac{c_a^j-c_b^j}{c_a-c_b}=c_a^{j-1}+c_a^{j-2}c_b+\cdots+c_b^{j-1},\qquad b\ne a, \tag*{(186)} s j ( a , b ) := c a − c b c a j − c b j = c a j − 1 + c a j − 2 c b + ⋯ + c b j − 1 , b = a , (186) in terms of which the vertices of our polytope can be rewritten as
( V 0 ( b ) ) j = s j ( 0 , b ) . (187) \left(V_0^{(b)}\right)_j=s_j(0,b). \tag*{(187)} ( V 0 ( b ) ) j = s j ( 0 , b ) . (187) Consequently, the integration region in the variables m j m_j m j D 0 D_0 D 0
D 0 : m j = ∑ b = 1 κ λ b s j ( 0 , b ) , ∑ b = 1 κ λ b = 1 , λ b ≥ 0 , (188) D_0:\qquad m_j=\sum_{b=1}^{\kappa}\lambda_b s_j(0,b),\qquad\sum_{b=1}^{\kappa}\lambda_b=1,\qquad\lambda_b\geq0, \tag*{(188)} D 0 : m j = b = 1 ∑ κ λ b s j ( 0 , b ) , b = 1 ∑ κ λ b = 1 , λ b ≥ 0 , (188) with an orientation we will describe in more detail below.
The analytic structure of Z r o t Z^{\mathrm{rot}} Z rot
It follows from (S129) that
( ln Z r o t β ) = ( − 1 ) κ 2 π κ V ∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) (189) \left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=\frac{(-1)^\kappa2\pi^\kappa}{\mathcal{V}}\int_{D_0}dm_2\wedge dm_3\ldots\wedge dm_\kappa\ \mathcal{I}\left(E,m^{(a)}\right) \tag*{(189)} ( β ln Z rot ) = V ( − 1 ) κ 2 π κ ∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) (189) (S130) is the generalization (to arbitrary dimension) of the four dimensional formula (18).
In the previous subsection we have demonstrated that the integrand in (S130) is a polynomial in the E r E_r E r m i m_i m i c a c_a c a E r E_r E r c a c_a c a c b c_b c b b = 0 … κ b=0\ldots\kappa b = 0 … κ
We can actually say more. As we have explained above, at generic values of c a c_a c a D 0 D_0 D 0 κ − 1 \kappa-1 κ − 1 κ \kappa κ V ( a ) V^{(a)} V ( a ) a = 1 … κ a=1\ldots\kappa a = 1 … κ c a = c b c_a=c_b c a = c b a a a b b b V ( a ) V^{(a)} V ( a ) V ( b ) V^{(b)} V ( b ) κ − 2 \kappa-2 κ − 2 κ − 1 \kappa-1 κ − 1 κ − 2 \kappa-2 κ − 2
∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) \int_{\mathcal{D}_0} dm_2 \wedge dm_3 \ldots\wedge dm_\kappa\,\mathcal{I}\left(E,m^{(a)}\right) ∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) vanishes (and, in fact, generically develops a simple zero) in the limit c a → c b c^a \to c^b c a → c b ( a , b ) (a,b) ( a , b )
∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) = ∏ 1 ≤ a < b ≤ κ ( c a − c b ) × Q (190) \int_{\mathcal{D}_0} dm_2 \wedge dm_3 \ldots\wedge dm_\kappa\,\mathcal{I}\left(E,m^{(a)}\right)=\prod_{1\leq a<b\leq\kappa}(c_a-c_b)\times Q \tag*{(190)} ∫ D 0 d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) = 1 ≤ a < b ≤ κ ∏ ( c a − c b ) × Q (190) where Q Q Q c a c_a c a V \mathcal{V} V
( ln Z r o t β ) ∝ 1 ∏ b = 1 κ ( c 0 − c b ) × Q (191) \left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)\propto\frac{1}{\prod_{b=1}^{\kappa}(c_0-c_b)}\times Q \tag*{(191)} ( β ln Z rot ) ∝ ∏ b = 1 κ ( c 0 − c b ) 1 × Q (191) demonstrating that ln Z r o t \ln Z^{\mathrm{rot}} ln Z rot
Derivation of our central equation in every even dimension
In this subsection we demonstrate that (7) holds for any CP invariant CFT2 κ _{2\kappa} 2 κ κ ≥ 2 \kappa\geq2 κ ≥ 2 d = 4 d=4 d = 4 κ = 2 \kappa=2 κ = 2
Action of permutations
We now study the action of the Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 Ω \Omega Ω ln Z r o t β \frac{\ln Z^{\mathrm{rot}}}{\beta} β l n Z rot Ω a ∘ V = ( − 1 ) κ a V \Omega^a\circ\mathcal{V}=(-1)^{\kappa a}\mathcal{V} Ω a ∘ V = ( − 1 ) κa V I \mathcal{I} I E r E_r E r m i m_i m i S κ + 1 S_{\kappa+1} S κ + 1 Z κ + 1 \mathbb{Z}_{\kappa+1} Z κ + 1 Ω a ∘ I = I \Omega^a\circ\mathcal{I}=\mathcal{I} Ω a ∘ I = I Ω a \Omega^a Ω a D 0 \mathcal{D}_0 D 0 D a \mathcal{D}_a D a D a \mathcal{D}_a D a κ \kappa κ V a ( b ) V_a^{(b)} V a ( b ) b = 0 , … , κ b=0,\ldots,\kappa b = 0 , … , κ b ≠ a b\ne a b = a
( V a ( b ) ) j = c a j − c b j c a − c b = s j ( a , b ) . (192) \left(V_a^{(b)}\right)_j=\frac{c_a^j-c_b^j}{c_a-c_b}=s_j(a,b). \tag*{(192)} ( V a ( b ) ) j = c a − c b c a j − c b j = s j ( a , b ) . (192) Note that V a ( b ) = V b ( a ) V_a^{(b)}=V_b^{(a)} V a ( b ) = V b ( a )
p a b = V a ( b ) = V b ( a ) . (193) p_{ab}=V_a^{(b)}=V_b^{(a)}. \tag*{(193)} p ab = V a ( b ) = V b ( a ) . (193) In order to properly define the domain of integration described above, we need to specify both the polytope as well as its orientation. There is a natural way of associating an orientation of the polytope with an ordering of its defining vectors; given the ordered vector set ( v 1 , v 2 … v n + 1 ) (v^1,v^2\ldots v^{n+1}) ( v 1 , v 2 … v n + 1 ) n n n ( ( v 2 − v 1 ) , ( v 3 − v 1 ) , … , ( v n + 1 − v 1 ) ) \left((v^2-v^1),(v^3-v^1),\ldots,(v^{n+1}-v^1)\right) ( ( v 2 − v 1 ) , ( v 3 − v 1 ) , … , ( v n + 1 − v 1 ) ) n × n n\times n n × n D 0 \mathcal{D}_0 D 0 ( 1 , 2 , 3 , … κ ) (1,2,3,\ldots\kappa) ( 1 , 2 , 3 , … κ ) Ω a \Omega^a Ω a
( V a ( a + 1 ) , V a ( a + 2 ) , … , V a κ , V a ( 0 ) , … , V a ( a − 1 ) ) . (194) \left(V_a^{(a+1)},V_a^{(a+2)},\ldots,V_a^\kappa,V_a^{(0)},\ldots,V_a^{(a-1)}\right). \tag*{(194)} ( V a ( a + 1 ) , V a ( a + 2 ) , … , V a κ , V a ( 0 ) , … , V a ( a − 1 ) ) . (194) However, we choose to define the polytope D a \mathcal{D}_a D a V ( a ) V^{(a)} V ( a ) V ( 0 ) V^{(0)} V ( 0 ) V ( 1 ) V^{(1)} V ( 1 ) ( − 1 ) ( κ − a ) a (-1)^{(\kappa-a)a} ( − 1 ) ( κ − a ) a Ω a \Omega^a Ω a V \mathcal{V} V ( − 1 ) a 2 = ( − 1 ) a (-1)^{a^2}=(-1)^a ( − 1 ) a 2 = ( − 1 ) a D a \mathcal{D}_a D a
Ω a ∘ ( ln Z r o t β ) = 2 π κ ( − 1 ) κ − a V ∫ D a d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) (195) \Omega^a \circ\left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=\frac{2\pi^\kappa(-1)^{\kappa-a}}{\mathcal{V}}\int_{\mathcal{D}_a}dm_2\wedge dm_3\ldots\wedge dm_\kappa\,\mathcal{I}\left(E,m^{(a)}\right) \tag*{(195)} Ω a ∘ ( β ln Z rot ) = V 2 π κ ( − 1 ) κ − a ∫ D a d m 2 ∧ d m 3 … ∧ d m κ I ( E , m ( a ) ) (195) where the factor of ( − 1 ) κ (-1)^\kappa ( − 1 ) κ
( ln Z r o t β ) a = 2 π κ V ( − 1 ) κ − a ∫ D a Υ (196) \left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)_a=\frac{2\pi^\kappa}{\mathcal{V}}(-1)^{\kappa-a}\int_{\mathcal{D}_a}\Upsilon \tag*{(196)} ( β ln Z rot ) a = V 2 π κ ( − 1 ) κ − a ∫ D a Υ (196) where the κ − 1 \kappa-1 κ − 1 Υ \Upsilon Υ
Υ = d m 2 ∧ d m 3 … ∧ d m κ I . (197) \Upsilon=dm_2\wedge dm_3\ldots\wedge dm_\kappa\,\mathcal{I}. \tag*{(197)} Υ = d m 2 ∧ d m 3 … ∧ d m κ I . (197) d=4
In the simple special case of four dimensions, we have already seen in (17) that the contour for the original integral, the contour for the action of Ω \Omega Ω Ω 2 \Omega^2 Ω 2
[ c 1 , c 2 ] , [ c 2 , c 0 ] , [ c 0 , c 1 ] . (198) [c_1,c_2],\quad[c_2,c_0],\quad[c_0,c_1]. \tag*{(198)} [ c 1 , c 2 ] , [ c 2 , c 0 ] , [ c 0 , c 1 ] . (198) in the variable z = E 1 − m 2 z=E_1-m_2 z = E 1 − m 2 (17) , the integral then vanishes as a consequence of the fact that the intervals in (S139) double back on themselves. As a sanity check, we first rederive this result using (S136), specialized to d = 4 d=4 d = 4 D a \mathcal{D}_a D a κ = 2 \kappa=2 κ = 2 κ − 1 \kappa-1 κ − 1 V a ( b ) V_a^{(b)} V a ( b ) m 2 m_2 m 2
V a ( b ) = c a + c b (199) V_a^{(b)}=c_a+c_b \tag*{(199)} V a ( b ) = c a + c b (199) (S136) thus instructs us to integrate m 2 m_2 m 2
[ c 1 + c 0 , c 2 + c 0 ] , [ c 1 + c 0 , c 1 + c 2 ] , [ c 2 + c 0 , c 2 + c 1 ] , (200) [c_1+c_0,c_2+c_0],\quad[c_1+c_0,c_1+c_2],\quad[c_2+c_0,c_2+c_1], \tag*{(200)} [ c 1 + c 0 , c 2 + c 0 ] , [ c 1 + c 0 , c 1 + c 2 ] , [ c 2 + c 0 , c 2 + c 1 ] , (200) but to weight three integrals above with the relative signs + + + − - − + + + E 1 = c 0 + c 1 + c 2 E_1=c_0+c_1+c_2 E 1 = c 0 + c 1 + c 2 z z z
d=6
Let us now study the case d = 6 d=6 d = 6 c a c_a c a I \mathcal{I} I c a c_a c a c a c_a c a
In this case the vectors V a ( b ) = V b ( a ) = p a b V_a^{(b)}=V_b^{(a)}=p_{ab} V a ( b ) = V b ( a ) = p ab ( m 2 , m 3 ) (m_2,m_3) ( m 2 , m 3 ) D 0 \mathcal{D}_0 D 0 D 1 \mathcal{D}_1 D 1 D 2 \mathcal{D}_2 D 2 D 3 \mathcal{D}_3 D 3 m 2 , m 3 m_2,m_3 m 2 , m 3 D a \mathcal{D}_a D a p a b p_{ab} p ab b = 0 , … , 3 b=0,\ldots,3 b = 0 , … , 3 b ≠ a b\ne a b = a
p a b = ( c a + c b , c a 2 + c a c b + c b 2 ) . (201) p_{ab}=\left(c_a+c_b,c_a^2+c_ac_b+c_b^2\right). \tag*{(201)} p ab = ( c a + c b , c a 2 + c a c b + c b 2 ) . (201) Each of these four triangles shares exactly one vertex with every other triangle (the common vertex between D a \mathcal{D}_a D a D b \mathcal{D}_b D b p a b p_{ab} p ab S ∘ ln Z β \mathcal{S}\circ\frac{\ln Z}{\beta} S ∘ β l n Z
D 0 − D 1 + D 2 − D 3 . (202) \mathcal{D}_0-\mathcal{D}_1+\mathcal{D}_2-\mathcal{D}_3. \tag*{(202)} D 0 − D 1 + D 2 − D 3 . (202) As the 12 components of the 6 vectors p a b p_{ab} p ab c a c_a c a p a b p_{ab} p ab p a b p_{ab} p ab p a b p_{ab} p ab
When p a b p_{ab} p ab Figure 1 below. D 0 \mathcal{D}_0 D 0 D 1 \mathcal{D}_1 D 1 D 2 \mathcal{D}_2 D 2 D 3 \mathcal{D}_3 D 3 Figure 1 ; note that every triangle shares exactly one common vertex with every other triangle. While Figure 1 has been drawn in a 3 dimensional manner, to aid visualization, the reader should mentally visualize the points p 02 p_{02} p 02 p 13 p_{13} p 13 Figure 1 is flattened onto the single plane (the plane of p 12 , p 23 , p 03 , p 01 p_{12},p_{23},p_{03},p_{01} p 12 , p 23 , p 03 , p 01 ( m 2 , m 3 ) (m_2,m_3) ( m 2 , m 3 )
Continuing to work with generic p a b p_{ab} p ab Figure 1 (in addition to the four blue shaded triangles already studied above). Orientations are chosen so that the normal to every triangle is outward pointing. This augmented integration domain is a flat origami whose upper and lower flaps each consist of two blue and two white triangles, and can be thought of as a ‘closed kite shaped envelope’. When we paste this new flat origami onto the ( m 2 , m 3 ) (m_2,m_3) ( m 2 , m 3 ) p a b p_{ab} p ab
Now the augmented integration domain effectively coincides with the actual domain (S143) when each of the four holes (white triangles) in Figure 1 has zero area, i.e. when the three vectors ( p a b , p a c , p b c ) (p_{ab},p_{ac},p_{bc}) ( p ab , p a c , p b c ) ( a , b , c ) (a,b,c) ( a , b , c ) p a b p_{ab} p ab p a b p_{ab} p ab ( m 2 , m 3 ) (m_2,m_3) ( m 2 , m 3 ) p a b p_{ab} p ab p b c p_{bc} p b c p a c p_{ac} p a c
m 3 = ( c a + c b + c c ) m 2 − ( c a c b + c a c c + c b c c ) . (203) m_3=(c_a+c_b+c_c)m_2-(c_ac_b+c_ac_c+c_bc_c). \tag*{(203)} m 3 = ( c a + c b + c c ) m 2 − ( c a c b + c a c c + c b c c ) . (203) for each of the four choices of the triplet ( a , b , c ) (a,b,c) ( a , b , c )
S ∘ ( ln Z r o t β ) = 0 (204) \mathcal{S}\circ\left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=0 \tag*{(204)} S ∘ ( β ln Z rot ) = 0 (204) as claimed.
Before generalizing to higher dimensions, it is interesting to pause to visualize the integration region (S143) in the physically relevant collinear case. Upon mentally squashing the octahedron of Figure 1 down to plane, in this case, we obtain a flattened origami of the form depicted in Figure 2 . Figure 2 has been plotted for the illustrative choice ( c 0 , c 1 , c 2 , c 3 ) = ( 0 , 1 , 2 , 3 ) (c_0,c_1,c_2,c_3)=(0,1,2,3) ( c 0 , c 1 , c 2 , c 3 ) = ( 0 , 1 , 2 , 3 ) ζ = m 2 \zeta=m_2 ζ = m 2 η = m 3 − 4 m 2 + 4 \eta=m_3-4m_2+4 η = m 3 − 4 m 2 + 4
Figure 2. The four integration regions in d = 6 d=6 d = 6 ( c 0 , c 1 , c 2 , c 3 ) = ( 0 , 1 , 2 , 3 ) (c_0,c_1,c_2,c_3)=(0,1,2,3) ( c 0 , c 1 , c 2 , c 3 ) = ( 0 , 1 , 2 , 3 ) ξ = m 2 \xi=m_2 ξ = m 2 η = m 3 − 4 m 2 + 4 \eta=m_3-4m_2+4 η = m 3 − 4 m 2 + 4 p 01 , p 02 , p 12 p_{01},p_{02},p_{12} p 01 , p 02 , p 12
We have explained above that the collinearity condition on p a b p_{ab} p ab Figure 2 as follows. The integration region (S143) has four boundaries that lie along the four straight lines in Figure 2 . However these boundaries actually double back on each other, and so can (for the purposes of computing an integral) equally well be sewn together to form a closed surface (this closed surface is the augmented domain). For example, the part of the oriented boundary involving the labels 0 , 1 , 2 0,1,2 0 , 1 , 2 Figure 2 , and equals
[ p 01 , p 02 ] − [ p 01 , p 12 ] + [ p 02 , p 12 ] . (205) [p_{01},p_{02}]-[p_{01},p_{12}]+[p_{02},p_{12}]. \tag*{(205)} [ p 01 , p 02 ] − [ p 01 , p 12 ] + [ p 02 , p 12 ] . (205) A glance at Figure 2 will convince the reader that the three segments in (S146) lie on the same line, and the line segment (second term in (S146)) is the sum of the two shorter ones but with opposite orientation. Hence the integration domain does not really have a boundary. The same argument applies to each choice of three labels (each of the four straight lines in Figure 2 so the oriented boundary of
D 0 − D 1 + D 2 − D 3 (206) D_0-D_1+D_2-D_3 \tag*{(206)} D 0 − D 1 + D 2 − D 3 (206) effectively vanishes, and the integration domain is effectively closed.
The vanishing of the integral over the domain depicted in Figure 2 can also be verified very explicitly as follows. In Figure 2 , the triangles ( p 01 , p 02 , p 03 ) (p_{01},p_{02},p_{03}) ( p 01 , p 02 , p 03 ) ( p 20 , p 21 , p 23 ) (p_{20},p_{21},p_{23}) ( p 20 , p 21 , p 23 ) ( p 10 , p 12 , p 13 ) (p_{10},p_{12},p_{13}) ( p 10 , p 12 , p 13 ) ( p 30 , p 31 , p 32 ) (p_{30},p_{31},p_{32}) ( p 30 , p 31 , p 32 ) Figure 2 . We thus see that every point on the integration domain in Figure 2 occurs twice; once with positive and once with negative orientation. Consequently the integral vanishes.
Finally, the vanishing of the integral over the surface of Figure 2 can also be argued in the following equivalent fashion. As the real ( m 2 , m 3 ) (m_2,m_3) ( m 2 , m 3 ) Υ \Upsilon Υ Υ = d α \Upsilon=d\alpha Υ = d α
∫ D 0 Υ − ∫ D 1 Υ + ∫ D 2 Υ − ∫ D 3 Υ = 0 (207) \int_{\mathcal{D}_0}\Upsilon-\int_{\mathcal{D}_1}\Upsilon+\int_{\mathcal{D}_2}\Upsilon-\int_{\mathcal{D}_3}\Upsilon=0 \tag*{(207)} ∫ D 0 Υ − ∫ D 1 Υ + ∫ D 2 Υ − ∫ D 3 Υ = 0 (207) where we have used that the boundary of the integration region is effectively trivial.
General even dimensions
We now turn to the generalization of the discussion above to every even dimension. In each such dimension, we have
G ∘ ( ln Z r o t β ) = 2 π κ ( κ + 1 ) V ( − 1 ) κ ∫ D Υ (208) \mathcal{G}\circ\left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=\frac{2\pi^\kappa}{(\kappa+1)\mathcal{V}}(-1)^\kappa\int_{\mathcal{D}}\Upsilon \tag*{(208)} G ∘ ( β ln Z rot ) = ( κ + 1 ) V 2 π κ ( − 1 ) κ ∫ D Υ (208) where the integration domain D \mathcal{D} D
D = D 0 − D 1 + D 2 − … + ( − 1 ) κ D κ . (209) \mathcal{D}=\mathcal{D}_0-\mathcal{D}_1+\mathcal{D}_2-\ldots+(-1)^\kappa\mathcal{D}_\kappa. \tag*{(209)} D = D 0 − D 1 + D 2 − … + ( − 1 ) κ D κ . (209) Here D a \mathcal{D}_a D a κ − 1 \kappa-1 κ − 1 κ \kappa κ p a b p_{ab} p ab ( b = 0 … κ , b ≠ a ) (b=0\ldots\kappa,\ b\ne a) ( b = 0 … κ , b = a ) κ \kappa κ p a b p_{ab} p ab b ≠ a b\ne a b = a κ = 2 \kappa=2 κ = 2 κ = 3 \kappa=3 κ = 3 Figure 1 ) and tetrahedra when κ = 4 \kappa=4 κ = 4 ( κ − 1 ) (\kappa-1) ( κ − 1 ) κ ≥ 5 \kappa\ge5 κ ≥ 5
(S150) tells us that the integration domain is the union of κ + 1 \kappa+1 κ + 1 d = 6 d=6 d = 6 κ − 1 \kappa-1 κ − 1 p a b p_{ab} p ab ( m 2 , … , m κ ) (m_2,\ldots,m_\kappa) ( m 2 , … , m κ ) ( κ + 1 ) κ 2 \frac{(\kappa+1)\kappa}{2} 2 ( κ + 1 ) κ ( κ − 1 ) (\kappa-1) ( κ − 1 ) κ + 1 \kappa+1 κ + 1 c a c_a c a (192) .
In our d = 6 d=6 d = 6 Figure 1 ) that the integration domain for generic p a b p_{ab} p ab d = 2 κ d=2\kappa d = 2 κ κ ≥ 3 \kappa\ge3 κ ≥ 3 κ + 1 \kappa+1 κ + 1 2 ( κ + 1 ) 2(\kappa+1) 2 ( κ + 1 ) κ \kappa κ Δ ( 2 , κ + 1 ) \Delta(2,\kappa+1) Δ ( 2 , κ + 1 )
Consider a Cartesian R κ + 1 \mathbb{R}^{\kappa+1} R κ + 1 Δ ( 2 , κ + 1 ) \Delta(2,\kappa+1) Δ ( 2 , κ + 1 ) ∑ a t a = 2 \sum_a t_a=2 ∑ a t a = 2
Δ ( 2 , κ + 1 ) : = { ( t 0 , … , t κ ) | 0 ≤ t a ≤ 1 , ∑ a = 0 κ t a = 2 } . (210) \Delta(2,\kappa+1):=\left\{(t_0,\ldots,t_\kappa)\,\middle|\,0\le t_a\le1,\ \sum_{a=0}^{\kappa}t_a=2\right\}. \tag*{(210)} Δ ( 2 , κ + 1 ) := { ( t 0 , … , t κ ) 0 ≤ t a ≤ 1 , a = 0 ∑ κ t a = 2 } . (210) This shape is easily visualized in the case κ = 2 \kappa=2 κ = 2 Figure 3 where it consists of the interior of a triangle. The boundaries of this triangle are the three integration domains D 0 \mathcal{D}_0 D 0 D 1 \mathcal{D}_1 D 1 D 2 \mathcal{D}_2 D 2 κ ≥ 3 \kappa\ge3 κ ≥ 3 κ \kappa κ κ \kappa κ (210) ) has two different kinds of flat κ − 1 \kappa-1 κ − 1 κ + 1 \kappa+1 κ + 1 t a = 1 t_a=1 t a = 1 ( a = 0 … κ ) (a=0\ldots\kappa) ( a = 0 … κ ) κ + 1 \kappa+1 κ + 1 t a = 0 t_a=0 t a = 0 ( a = 0 … κ ) (a=0\ldots\kappa) ( a = 0 … κ ) a t h a^{\mathrm{th}} a th t a = 1 t^a=1 t a = 1 D a \mathcal{D}_a D a a t h a^{\mathrm{th}} a th t a = 0 t_a=0 t a = 0
The identification spelled out above can be justified as follows. Notice that the surface element t a = 1 t^a=1 t a = 1 ∑ b ≠ a t b = 1 \sum_{b\ne a}t^b=1 ∑ b = a t b = 1 b ≠ a b\ne a b = a t b > 0 t^b>0 t b > 0 ( κ − 1 ) (\kappa-1) ( κ − 1 ) a t h a^{\mathrm{th}} a th b t h b^{\mathrm{th}} b th t a = t b = 1 t_a=t_b=1 t a = t b = 1 D a \mathcal{D}_a D a t a = t b = 1 t_a=t_b=1 t a = t b = 1 b = 0 … κ b=0\ldots\kappa b = 0 … κ b ≠ a b\ne a b = a p a b p_{ab} p ab
The full boundary of Δ ( 2 , κ + 1 ) \Delta(2,\kappa+1) Δ ( 2 , κ + 1 ) ( κ + 1 ) (\kappa+1) ( κ + 1 ) t a = 0 t^a=0 t a = 0 31
Δ ( 2 , κ ) \Delta(2,\kappa) Δ ( 2 , κ ) 32 t a = 0 t^a=0 t a = 0 t b = 1 t^b=1 t b = 1 b ≠ a b\ne a b = a κ − 2 \kappa-2 κ − 2 κ − 1 \kappa-1 κ − 1
p c b , c = 0 … κ , c ≠ b , c ≠ a . (211) p_{cb},\quad c=0\ldots\kappa,\ c\ne b,\ c\ne a. \tag*{(211)} p c b , c = 0 … κ , c = b , c = a . (211) In addition, the hole t a = 0 t^a=0 t a = 0 t b = 0 t^b=0 t b = 0 b ≠ a b\ne a b = a Δ ( 2 , κ − 1 ) \Delta(2,\kappa-1) Δ ( 2 , κ − 1 )
We reemphasize that the discussion of the previous paragraphs describes our integration domain only in a topological sense. The R κ + 1 \mathbb{R}^{\kappa+1} R κ + 1 m i m_i m i R κ − 1 \mathbb{R}^{\kappa-1} R κ − 1 m 2 … m κ m_2\ldots m_\kappa m 2 … m κ t a = 1 t_a=1 t a = 1 D a \mathcal{D}_a D a a a a D a \mathcal{D}_a D a t a = 0 t_a=0 t a = 0
In the physical problem of interest, our integration surfaces D a \mathcal{D}_a D a H a \mathcal{H}_a H a D \mathcal{D} D κ − 2 \kappa-2 κ − 2 κ \kappa κ D b \mathcal{D}_b D b b ≠ a b\ne a b = a κ − 2 \kappa-2 κ − 2 κ − 1 \kappa-1 κ − 1 p b c p_{bc} p b c c ≠ a c\ne a c = a c ≠ b c\ne b c = b D b \mathcal{D}_b D b b ≠ a b\ne a b = a
We denote this κ − 1 \kappa-1 κ − 1 R κ − 1 \mathbb{R}^{\kappa-1} R κ − 1 m i m_i m i H a \mathcal{H}_a H a 33
In the discussion above we have characterized the ‘shaded integration surfaces’ and the ‘white holes’ in every κ \kappa κ κ − 1 \kappa-1 κ − 1 Δ ( 2 , κ ) \Delta(2,\kappa) Δ ( 2 , κ )
Table 1.
As an aside for the interested reader, we pause to describe the case d = 8 d = 8 d = 8
polytope is Δ ( 2 , 5 ) \Delta(2,5) Δ ( 2 , 5 ) Δ ( 2 , 5 ) \Delta(2,5) Δ ( 2 , 5 ) t a = 1 t_a = 1 t a = 1 t a = 0 t_a = 0 t a = 0 Figure 4 displays the two kinds of three-dimensional boundary cells.
Figure 4. The boundary cells of Δ ( 2 , 5 ) \Delta(2,5) Δ ( 2 , 5 ) d = 8 d = 8 d = 8 t a = 1 t_a = 1 t a = 1 t a = 0 t_a = 0 t a = 0
The remaining discussion proceeds as in d = 6 d = 6 d = 6 κ − 1 \kappa- 1 κ − 1 m i m_i m i κ − 1 \kappa- 1 κ − 1 Δ ( 2 , κ + 1 ) \Delta(2,\kappa+ 1) Δ ( 2 , κ + 1 ) D \mathcal{D} D D + H \mathcal{D} + \mathcal{H} D + H
H = ∑ a = 0 κ H a . (212) \mathcal{H} = \sum_{a=0}^{\kappa} \mathcal{H}_a . \tag*{(212)} H = a = 0 ∑ κ H a . (212) In equations
∫ D + H Υ = 0. (213) \int_{\mathcal{D}+\mathcal{H}} \Upsilon= 0 . \tag*{(213)} ∫ D + H Υ = 0. (213) As in d = 6 d = 6 d = 6 p a b p_{ab} p ab d = 6 d = 6 d = 6 p a b p_{ab} p ab (192) . This has the following crucial consequence. Recall that we are free to choose the shapes of the holes H a \mathcal{H}_a H a κ \kappa κ H a \mathcal{H}_a H a D b \mathcal{D}_b D b (211) , (for the κ \kappa κ b = 0 , … κ b = 0, \ldots\kappa b = 0 , … κ b ≠ a b \ne a b = a H a \mathcal{H}_a H a H b \mathcal{H}_b H b κ \kappa κ b b b b ≠ a b \ne a b = a H b \mathcal{H}_b H b H a \mathcal{H}_a H a
In the subsequent paragraph, we will demonstrate that each of the vectors in Eq. (211) actually lie in a particular κ − 2 \kappa- 2 κ − 2 R κ − 1 \mathbb{R}^{\kappa-1} R κ − 1 b b b b ≠ a b \ne a b = a 34 35 H a \mathcal{H}_a H a D b \mathcal{D}_b D b κ − 2 \kappa-2 κ − 2 H a \mathcal{H}_a H a κ − 2 \kappa-2 κ − 2 36 a a a H \mathcal{H} H
∫ D R = 0 (214) \int_{\mathcal{D}}\mathfrak{R}=0 \tag*{(214)} ∫ D R = 0 (214) and it follows from (S149) that
G ∘ ( ln Z r o t β ) = 0 \mathcal{G}\circ\left(\frac{\ln Z^{\mathrm{rot}}}{\beta}\right)=0 G ∘ ( β ln Z rot ) = 0 as we set out to prove.
As promised above, we now demonstrate that the vectors p b d p_{bd} p b d b ≠ a b\ne a b = a d ≠ a d\ne a d = a κ − 2 \kappa-2 κ − 2
q a ( t ) : = ∏ m ≠ a ( t − c m ) = t κ − e 1 ( a ) t κ − 1 + e 2 ( a ) t κ − 2 − ⋯ + ( − 1 ) κ e κ ( a ) , (215) q_a(t):=\prod_{m\ne a}(t-c_m)=t^\kappa-e_1^{(a)}t^{\kappa-1}+e_2^{(a)}t^{\kappa-2}-\cdots+(-1)^\kappa e_\kappa^{(a)}, \tag*{(215)} q a ( t ) := m = a ∏ ( t − c m ) = t κ − e 1 ( a ) t κ − 1 + e 2 ( a ) t κ − 2 − ⋯ + ( − 1 ) κ e κ ( a ) , (215) (notice the product above runs over all m ≠ a m\ne a m = a d ≠ a d\ne a d = a b ≠ a b\ne a b = a q a ( c d ) = q a ( c b ) = 0 q_a(c_d)=q_a(c_b)=0 q a ( c d ) = q a ( c b ) = 0 q a ( c d ) − q a ( c b ) = 0 q_a(c_d)-q_a(c_b)=0 q a ( c d ) − q a ( c b ) = 0 c d − c b c_d-c_b c d − c b
m κ − e 1 ( a ) m κ − 1 + e 2 ( a ) m κ − 2 − ⋯ + ( − 1 ) κ − 1 e κ − 1 ( a ) = 0 , (216) m_\kappa-e_1^{(a)}m_{\kappa-1}+e_2^{(a)}m_{\kappa-2}-\cdots+(-1)^{\kappa-1}e_{\kappa-1}^{(a)}=0, \tag*{(216)} m κ − e 1 ( a ) m κ − 1 + e 2 ( a ) m κ − 2 − ⋯ + ( − 1 ) κ − 1 e κ − 1 ( a ) = 0 , (216) where m j m_j m j p d b p_{db} p d b p d b p_{db} p d b d , b ≠ a d,b\ne a d , b = a R κ − 1 \mathbb{R}^{\kappa-1} R κ − 1 m j m_j m j a t h a^{\mathrm{th}} a th
Evaluation of cyclic sum of the Kerr-AdS2 κ + 1 _{2\kappa+1} 2 κ + 1
In this small section, we will derive the cyclic identity for the Kerr-AdS2 κ + 1 _{2\kappa+1} 2 κ + 1 ln Z B H β \frac{\ln Z_{\mathrm{BH}}}{\beta} β l n Z BH Ω a \Omega^a Ω a Ω \Omega Ω Ω b h \Omega_{\mathrm{bh}} Ω bh
Ω a ∘ [ ln Z B H β ] = π κ − 1 8 ( κ − 1 ) ! G 2 κ + 1 ( 1 − q a 2 ) ∏ i = 0 i ≠ a κ 1 + q i 2 q a 2 − q i 2 \Omega^a\circ\left[\frac{\ln Z_{\mathrm{BH}}}{\beta}\right]
=\frac{\pi^{\kappa-1}}{8(\kappa-1)!G_{2\kappa+1}}(1-q_a^2)\prod_{\substack{i=0\\i\ne a}}^\kappa\frac{1+q_i^2}{q_a^2-q_i^2} Ω a ∘ [ β ln Z BH ] = 8 ( κ − 1 )! G 2 κ + 1 π κ − 1 ( 1 − q a 2 ) i = 0 i = a ∏ κ q a 2 − q i 2 1 + q i 2 = π κ − 1 8 ( κ − 1 ) ! G 2 κ + 1 [ ∏ j = 0 κ ( 1 + q j 2 ) ] 1 − q a 2 1 + q a 2 ∏ i = 0 i ≠ a κ 1 q a 2 − q i 2 , for a = 0 , 1 , ⋯ , κ . (217) \qquad=\frac{\pi^{\kappa-1}}{8(\kappa-1)!G_{2\kappa+1}}
\left[\prod_{j=0}^\kappa(1+q_j^2)\right]\frac{1-q_a^2}{1+q_a^2}\prod_{\substack{i=0\\i\ne a}}^\kappa\frac{1}{q_a^2-q_i^2},
\qquad\text{for }a=0,1,\cdots,\kappa. \tag*{(217)} = 8 ( κ − 1 )! G 2 κ + 1 π κ − 1 [ j = 0 ∏ κ ( 1 + q j 2 ) ] 1 + q a 2 1 − q a 2 i = 0 i = a ∏ κ q a 2 − q i 2 1 , for a = 0 , 1 , ⋯ , κ . (217) Summing over a a a κ \kappa κ
( 1 + Ω + ⋯ + Ω κ ) ∘ [ ln Z B H β ] = π κ − 1 8 ( κ − 1 ) ! G 2 κ + 1 [ ∏ j = 0 κ ( 1 + q j 2 ) ] ∑ a = 0 κ 1 − q a 2 1 + q a 2 ∏ i = 0 i ≠ a κ 1 q a 2 − q i 2 . (218) (1+\Omega+\cdots+\Omega^\kappa)\circ\left[\frac{\ln Z_{\mathrm{BH}}}{\beta}\right]
=\frac{\pi^{\kappa-1}}{8(\kappa-1)!G_{2\kappa+1}}
\left[\prod_{j=0}^\kappa(1+q_j^2)\right]\sum_{a=0}^\kappa\frac{1-q_a^2}{1+q_a^2}\prod_{\substack{i=0\\i\ne a}}^\kappa\frac{1}{q_a^2-q_i^2}. \tag*{(218)} ( 1 + Ω + ⋯ + Ω κ ) ∘ [ β ln Z BH ] = 8 ( κ − 1 )! G 2 κ + 1 π κ − 1 [ j = 0 ∏ κ ( 1 + q j 2 ) ] a = 0 ∑ κ 1 + q a 2 1 − q a 2 i = 0 i = a ∏ κ q a 2 − q i 2 1 . (218) We will evaluate the a a a f ( z ) f(z) f ( z )
f ( z ) : = 1 − z 1 + z ∏ i = 0 κ 1 z − q i 2 . (219) f(z):=\frac{1-z}{1+z}\prod_{i=0}^\kappa\frac{1}{z-q_i^2}. \tag*{(219)} f ( z ) := 1 + z 1 − z i = 0 ∏ κ z − q i 2 1 . (219) We can see that f ( z ) f(z) f ( z ) z = q a 2 z=q_a^2 z = q a 2 a = 0 , 1 , ⋯ , κ a=0,1,\cdots,\kappa a = 0 , 1 , ⋯ , κ z = − 1 z=-1 z = − 1 ∣ z ∣ |z| ∣ z ∣ f ( z ) ∼ − 1 z κ + 1 f(z)\sim-\frac{1}{z^{\kappa+1}} f ( z ) ∼ − z κ + 1 1 f ( z ) f(z) f ( z ) 1 z 2 \frac{1}{z^2} z 2 1 κ ≥ 1 \kappa\geq1 κ ≥ 1 f ( z ) f(z) f ( z ) z z z
∑ a = 0 κ Res z = q a 2 f ( z ) + Res z = − 1 f ( z ) = 0. (220) \sum_{a=0}^{\kappa}\operatorname{Res}_{z=q_a^2}f(z)+\operatorname{Res}_{z=-1}f(z)=0. \tag*{(220)} a = 0 ∑ κ Res z = q a 2 f ( z ) + Res z = − 1 f ( z ) = 0. (220) The residues of f ( z ) f(z) f ( z ) z = q a 2 z=q_a^2 z = q a 2
Res z = q a 2 f ( z ) = 1 − q a 2 1 + q a 2 ∏ i = 0 i ≠ a κ 1 q a 2 − q i 2 , (221) \operatorname{Res}_{z=q_a^2}f(z)=\frac{1-q_a^2}{1+q_a^2}\prod_{\substack{i=0\\i\ne a}}^{\kappa}\frac{1}{q_a^2-q_i^2}, \tag*{(221)} Res z = q a 2 f ( z ) = 1 + q a 2 1 − q a 2 i = 0 i = a ∏ κ q a 2 − q i 2 1 , (221) and the residue of f ( z ) f(z) f ( z ) z = − 1 z=-1 z = − 1
Res z = − 1 f ( z ) = 2 ∏ j = 0 κ 1 ( − 1 − q j 2 ) = − 2 ( − 1 ) κ ∏ j = 0 κ 1 ( 1 + q j 2 ) . (222) \operatorname{Res}_{z=-1}f(z)=2\prod_{j=0}^{\kappa}\frac{1}{(-1-q_j^2)}=-2(-1)^\kappa\prod_{j=0}^{\kappa}\frac{1}{(1+q_j^2)}. \tag*{(222)} Res z = − 1 f ( z ) = 2 j = 0 ∏ κ ( − 1 − q j 2 ) 1 = − 2 ( − 1 ) κ j = 0 ∏ κ ( 1 + q j 2 ) 1 . (222) Substituting the residues into (220) , we get
∑ a = 0 κ 1 − q a 2 1 + q a 2 ∏ i = 0 i ≠ a κ 1 q a 2 − q i 2 = 2 ( − 1 ) κ ∏ j = 0 κ 1 ( 1 + q j 2 ) . (223) \sum_{a=0}^{\kappa}\frac{1-q_a^2}{1+q_a^2}\prod_{\substack{i=0\\i\ne a}}^{\kappa}\frac{1}{q_a^2-q_i^2}=2(-1)^\kappa\prod_{j=0}^{\kappa}\frac{1}{(1+q_j^2)}. \tag*{(223)} a = 0 ∑ κ 1 + q a 2 1 − q a 2 i = 0 i = a ∏ κ q a 2 − q i 2 1 = 2 ( − 1 ) κ j = 0 ∏ κ ( 1 + q j 2 ) 1 . (223) Substituting the above identity in (218) , we have obtained the (224)
( 1 + Ω + ⋯ + Ω κ ) ∘ [ ln Z B H β ] = ( − 1 ) κ π κ − 1 4 ( κ − 1 ) ! G 2 κ + 1 = ( − 1 ) κ κ ! a 2 κ (224) \left(1+\Omega+\cdots+\Omega^\kappa\right)\circ\left[\frac{\ln Z_{\mathrm{BH}}}{\beta}\right]=(-1)^\kappa\frac{\pi^{\kappa-1}}{4(\kappa-1)!G_{2\kappa+1}}=(-1)^\kappa\kappa!a_{2\kappa} \tag*{(224)} ( 1 + Ω + ⋯ + Ω κ ) ∘ [ β ln Z BH ] = ( − 1 ) κ 4 ( κ − 1 )! G 2 κ + 1 π κ − 1 = ( − 1 ) κ κ ! a 2 κ (224) where the a a a a 2 κ a_{2\kappa} a 2 κ d = 2 κ d=2\kappa d = 2 κ a 2 κ = π κ − 1 4 κ κ ! ( κ − 1 ) ! G 2 κ + 1 a_{2\kappa}=\frac{\pi^{\kappa-1}}{4^\kappa\kappa!(\kappa-1)!G_{2\kappa+1}} a 2 κ = 4 κ κ ! ( κ − 1 )! G 2 κ + 1 π κ − 1
Spectral equivalence between S 3 × R S^3\times\mathbb{R} S 3 × R A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1
We consider a conformally coupled scalar on S 3 × R S^3\times\mathbb{R} S 3 × R
( □ − ξ R S 3 ) Φ ( x ) = 0 (225) (\Box-\xi R_{S^3})\Phi(x)=0 \tag*{(225)} ( □ − ξ R S 3 ) Φ ( x ) = 0 (225) The metric of S 3 × R S^3\times\mathbb{R} S 3 × R
d s S 3 × R 2 = − d t 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 , (226) ds^2_{S^3\times\mathbb{R}}=-dt^2+d\theta^2+\sin^2\theta\,d\phi_1^2+\cos^2\theta\,d\phi_2^2, \tag*{(226)} d s S 3 × R 2 = − d t 2 + d θ 2 + sin 2 θ d ϕ 1 2 + cos 2 θ d ϕ 2 2 , (226) where the scalar modes can be written as
Φ = e − i E t Y n , m , p ( θ , ϕ 1 , ϕ 2 ) , n , m ∈ Z . (227) \Phi= e^{-iEt}Y_{n,m,p}(\theta,\phi_1,\phi_2), \quad n,m \in\mathbb{Z}. \tag*{(227)} Φ = e − i E t Y n , m , p ( θ , ϕ 1 , ϕ 2 ) , n , m ∈ Z . (227) where the scalar harmonics Y n , m , p ( θ , ϕ 1 , ϕ 2 ) Y_{n,m,p}(\theta,\phi_1,\phi_2) Y n , m , p ( θ , ϕ 1 , ϕ 2 ) S 3 S^3 S 3
Y n , m , p ( θ , ϕ 1 , ϕ 2 ) ∝ e i n ϕ 1 + i m ϕ 2 ( sin θ ) ∣ n ∣ ( cos θ ) ∣ m ∣ P p ( ∣ n ∣ , ∣ m ∣ ) ( cos 2 θ ) , p = 0 , 1 , 2 , … , (228) Y_{n,m,p}(\theta,\phi_1,\phi_2) \propto e^{in\phi_1+im\phi_2}(\sin\theta)^{|n|}(\cos\theta)^{|m|}P_p^{(|n|,|m|)}(\cos2\theta), \quad p=0,1,2,\ldots, \tag*{(228)} Y n , m , p ( θ , ϕ 1 , ϕ 2 ) ∝ e in ϕ 1 + im ϕ 2 ( sin θ ) ∣ n ∣ ( cos θ ) ∣ m ∣ P p ( ∣ n ∣ , ∣ m ∣ ) ( cos 2 θ ) , p = 0 , 1 , 2 , … , (228) where, P p ( ∣ n ∣ , ∣ m ∣ ) ( z ) P_p^{(|n|,|m|)}(z) P p ( ∣ n ∣ , ∣ m ∣ ) ( z ) p p p
Scalar harmonics Y n , m , p Y_{n,m,p} Y n , m , p S 3 S^3 S 3
− ∇ S 3 2 Y l = l ( l + 2 ) Y l l = ∣ n ∣ + ∣ m ∣ + 2 p . (229) -\nabla^2_{S^3}Y_l=l(l+2)Y_l \qquad l=|n|+|m|+2p. \tag*{(229)} − ∇ S 3 2 Y l = l ( l + 2 ) Y l l = ∣ n ∣ + ∣ m ∣ + 2 p . (229) Using Eq. (229) and including the contribution of the conformal coupling ξ R S 3 = 1 6 × 6 = 1 \xi R_{S^3}=\frac{1}{6}\times6=1 ξ R S 3 = 6 1 × 6 = 1 (225) , we obtain
E 2 = l ( l + 2 ) + 1 = ( l + 1 ) 2 . (230) E^2=l(l+2)+1=(l+1)^2. \tag*{(230)} E 2 = l ( l + 2 ) + 1 = ( l + 1 ) 2 . (230) Finally, we obtain the energy as
E S 3 × R = ∣ n ∣ + ∣ m ∣ + 2 p + 1. (231) E_{S^3\times\mathbb{R}}=|n|+|m|+2p+1. \tag*{(231)} E S 3 × R = ∣ n ∣ + ∣ m ∣ + 2 p + 1. (231) Next, we perform the following coordinate change,
l = sin 2 θ = 1 cosh 2 ρ , (232) l=\sin^2\theta=\frac{1}{\cosh^2\rho}, \tag*{(232)} l = sin 2 θ = cosh 2 ρ 1 , (232) and rewrite the metric of S 3 × R S^3\times\mathbb{R} S 3 × R A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1
d s S 3 × R 2 = W 2 [ d ϕ 1 2 + d ρ 2 − cosh 2 ρ d t 2 + sinh 2 ρ d ϕ 2 2 ] = 1 cosh 2 ρ d s A d S 3 × S 1 2 ; W 2 = 1 cosh 2 ρ (233) ds^2_{S^3\times\mathbb{R}}=W^2\left[d\phi_1^2+d\rho^2-\cosh^2\rho\,dt^2+\sinh^2\rho\,d\phi_2^2\right]=\frac{1}{\cosh^2\rho}ds^2_{AdS_3\times S^1};\quad W^2=\frac{1}{\cosh^2\rho} \tag*{(233)} d s S 3 × R 2 = W 2 [ d ϕ 1 2 + d ρ 2 − cosh 2 ρ d t 2 + sinh 2 ρ d ϕ 2 2 ] = cosh 2 ρ 1 d s A d S 3 × S 1 2 ; W 2 = cosh 2 ρ 1 (233) where, d s A d S 3 × S 1 2 ds^2_{AdS_3\times S^1} d s A d S 3 × S 1 2 A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1 S 1 S^1 S 1 ϕ 1 \phi_1 ϕ 1 ϕ 1 \phi_1 ϕ 1
θ = 0 , l = sin 2 θ → 0 , (234) \theta=0,\quad l=\sin^2\theta\to0, \tag*{(234)} θ = 0 , l = sin 2 θ → 0 , (234) and locally, the metric takes the form
d s 2 ≈ d r 2 + r 2 d ϕ 1 2 + ⋯ , r ≈ sin θ = l . (235) ds^2\approx dr^2+r^2d\phi_1^2+\cdots,\qquad r\approx\sin\theta=\sqrt{l}. \tag*{(235)} d s 2 ≈ d r 2 + r 2 d ϕ 1 2 + ⋯ , r ≈ sin θ = l . (235) For a mode with angular momentum n n n
Φ ∼ e i n ϕ 1 \Phi\sim e^{in\phi_1} Φ ∼ e in ϕ 1 Hence, the smoothness at the origin implies
Φ S 3 ∼ r ∣ n ∣ ∼ l ∣ n ∣ / 2 . (236) \Phi_{S^3}\sim r^{|n|}\sim l^{|n|/2}. \tag*{(236)} Φ S 3 ∼ r ∣ n ∣ ∼ l ∣ n ∣/2 . (236) Upon using Eq. (232) , this gives
Φ S 3 ∼ ( cosh ρ ) − ∣ n ∣ . (237) \Phi_{S^3}\sim(\cosh\rho)^{-|n|}. \tag*{(237)} Φ S 3 ∼ ( cosh ρ ) − ∣ n ∣ . (237) Since the scalar has Weyl weight one, we obtain
Φ A d S 3 × S 1 = W Φ S 3 ∼ ( cosh ρ ) − ( ∣ n ∣ + 1 ) , ρ → ∞ . (238) \Phi_{AdS_3\times S^1}=W\Phi_{S^3}\sim(\cosh\rho)^{-(|n|+1)},\qquad\rho\to\infty. \tag*{(238)} Φ A d S 3 × S 1 = W Φ S 3 ∼ ( cosh ρ ) − ( ∣ n ∣ + 1 ) , ρ → ∞. (238) Thus, the smoothness boundary condition at the shrinking ϕ i \phi_i ϕ i A d S 3 AdS_3 A d S 3
For a conformally coupled scalar on A d S 3 × S 1 AdS_3\times S^1 A d S 3 × S 1
Φ A d S 3 × S 1 = e i n ϕ 1 φ n , (239) \Phi_{AdS_3\times S^1}=e^{in\phi_1}\varphi_n, \tag*{(239)} Φ A d S 3 × S 1 = e in ϕ 1 φ n , (239) the Kaluza–Klein momentum contributes n 2 n^2 n 2 − 1 -1 − 1 A d S 3 AdS_3 A d S 3
m n 2 = n 2 − 1. (240) m_n^2 = n^2 - 1. \tag*{(240)} m n 2 = n 2 − 1. (240) Using the AdS-CFT map, we obtain
m n 2 = Δ n ( Δ n − 2 ) . ⟹ Δ n = 1 ± ∣ n ∣ (241) m_n^2 = \Delta_n(\Delta_n - 2). \quad\Longrightarrow\quad\Delta_n = 1 \pm|n| \tag*{(241)} m n 2 = Δ n ( Δ n − 2 ) . ⟹ Δ n = 1 ± ∣ n ∣ (241) Next, using the falloff condition in Eq. (238) we obtain,
Δ n = ∣ n ∣ + 1. (242) \Delta_n = |n| + 1. \tag*{(242)} Δ n = ∣ n ∣ + 1. (242) A scalar of dimension Δ n \Delta_n Δ n A d S 3 AdS_3 A d S 3 m m m p p p
E A d S 3 = Δ n + ∣ m ∣ + 2 p . (243) E_{AdS_3} = \Delta_n + |m| + 2p. \tag*{(243)} E A d S 3 = Δ n + ∣ m ∣ + 2 p . (243) Upon using Eq. (241) , we finally obtain the expression for the energy as
E A d S 3 × S 1 = ∣ n ∣ + ∣ m ∣ + 2 p + 1. (244) E_{AdS_3 \times S^1} = |n| + |m| + 2p + 1. \tag*{(244)} E A d S 3 × S 1 = ∣ n ∣ + ∣ m ∣ + 2 p + 1. (244) which exactly agrees with E S 3 × R E_{S^3 \times\mathbb{R}} E S 3 × R (231) .
