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Mass Deformations and Sheets in Logahoric Higgs Moduli

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Abstract

We use the framework of the logahoric Hitchin integrable system introduced in our earlier work and the sheets of the Lie algebra in order to describe the tame defects in a six dimensional $\mathcal{N}=(2,0)$ superconformal theory under mass deformations. We show that varying leaves of fixed dimension in the Hitchin system amounts to varying residue orbits of fixed dimension. Moreover, we verify the period and dimension conditions in mass-like deformations, as proposed by Balasubramanian–Distler, thus realizing the sheets of the Lie algebra on the Poisson moduli space of logahoric Higgs torsors.

Introduction

One of the remarkable properties of Hitchin systems is that they arise as the Coulomb branch of class $S$ $4d$ $\mathcal{N}=2$ quantum field theories compactified on $\mathbb{R}^{3}\times S^{1}$ [11]. A class $S$ theory is specified by the data of a punctured Riemann surface, a compact simple Lie algebra of type $A$ , $D$ or $E$ and certain constraints at the punctures of the surface called defects. Such theories are obtained by compactifying the $6d$ $\mathcal{N}=(2,0)$ superconformal field theory on the Riemann surface with a partial holomorphic twist, and, indeed, class $S$ theories can be uniformly described in terms of these $6d$ theories (see [13]). A class $S$ theory corresponds to the $4d$ $\mathcal{N}=2$ theory obtained by compactifying the $6d$ theory, with codimension-2 defects inserted at the punctures. The codimension-2 defects of the theory are then exhibited as tame or wild singularities of the Hitchin system.

Chacaltana, Distler and Tachikawa in [7] studied the behavior of codimension-2 defects of $6d$ $\mathcal{N}=(2,0)$ theories via dimensional reductions. One first such reduction is by formulating the theory on a circle transverse to the defect, and this associates a nilpotent orbit $\mathcal{O}_{N}$ in the Lie algebra $\mathfrak{g}$ , called the Nahm label, to the defect. Another dimensional reduction is by compactification to five dimensions on a circle that the defect wraps, and so a nilpotent orbit $\mathcal{O}_{H}$ in the Langlands dual Lie algebra $\mathfrak{g}^{\vee}$ , called the Hitchin label can be associated to the defect [12]. In both cases, a five-dimensional gauge theory emerges: for group $G$ on the Nahm side and for group $G^{\vee}$ on the Hitchin side. The pair of nilpotent orbits $(\mathcal{O}_{N},\mathcal{O}_{H})$ is then describing the defect.

The study of mass deformations of $T^{\rho}[\mathfrak{g}]$ theories already appears in [7]. Balasubramanian and Distler consider in [1] a tamely ramified Hitchin system, meaning a meromorphic Hitchin system in which the Higgs field has simple poles and the residues at these poles are fixed adjoint orbits. In their work, they classify all mass-like deformations of this system and then identify those mass-like deformations that indeed correspond to actual mass deformations of $T^{\rho}[\mathfrak{g}]$ theories. In particular, they use sheets of the Hitchin Lie algebra to organize deformations of these local residue orbits. For a complex Lie algebra $\mathfrak{g}$ and for $U_{d}$ the union of adjoint orbits in $\mathfrak{g}$ of complex dimension $d$ , then the irreducible components of $U_{d}$ are called sheets of the Lie algebra. Sheets were first studied by Borho and Kraft [2], [3]; see also [1 Section 4.1]. A sheet always contains exactly one nilpotent orbit but it may also contain non-nilpotent orbits. One calls a sheet Dixmier, when it contains orbits that are semisimple. Sheets are parameterized by sheet labels, that is, pairs of certain Levi subalgebras and nilpotent orbits in the Levi subalgebras. In [1], the Kirillov–Kostant–Souriau (KKS) holomorphic symplectic form $\Omega_{\mathrm{KKS}}$ on the symplectic leaves of

Key words and phrases. mass deformation, parahoric Higgs torsor, Hamiltonian reduction, symplectic leaf, sheet, Hitchin fibration, affine Springer fiber, Duistermaat–Heckman variation. the Hitchin system varies linearly with the hypermultiplet mass parameters. These mass-like deformations are labeled by sheet labels.

The aim of this paper is to place the study of mass-deformations in the framework of the logahoric Hitchin integrable system that was formally introduced in our previous work [19]. This was established as an algebraically completely integrable Hamiltonian system over a smooth Riemann surface with a finite number of distinct punctures, for a general complex reductive Lie algebra, and involves a (tame) moduli space of parahoric torsors over the surface with Higgs fields having logarithmic singularities at the punctures. We use this new mathematical framework in order to describe the tame defects of the $6d$ theory under mass deformations, considering at the same time the approach of Balasubramanian and Distler [1] who used the sheets of the Lie algebra to study these tame defects. We realize how the symplectic geometry of the logahoric Hitchin system varies with the residues by constructing families through cotangent reduction and computing their symplectic periods. A logarithmic Higgs field is locally a Lie algebra valued one-form $A(z)\,dz/z$ , with residue $A(0)$ . Changing the trivialization conjugates this residue, so its orbit is intrinsic, and these orbits determine the symplectic leaves in the Poisson geometry of the moduli space of logahoric Higgs torsors. A more detailed account of our main results is provided below.

Let $X$ be a smooth complex projective curve with one marked point, $G$ a connected complex semisimple group, and $G_\theta$ a parahoric group scheme with reductive quotient $L_\theta$ . Let $\mathcal{B}$ denote a fixed nonempty connected component of the moduli space of regularly stable parahoric torsors $G_\theta$ over $X$ ; the automorphism group of these torsors is the center $Z(G)$ . A level structure can be seen as a choice of framing for the $L_\theta$ -torsor at the marked point. Modulo these central automorphisms, levels therefore form a principal $G_D=L_\theta/Z(G)$ -bundle $\mathcal{U}\to\mathcal{B}$ , and $T^*\mathcal{U}$ parametrizes Higgs fields with their levels.

Let now $\mathfrak{g}_D=\operatorname{Lie}(G_D)$ denote the Lie algebra of this level group $G_D$ . An invariant nondegenerate form $B_{\mathfrak{g}_D}$ converts a residue $y\in\mathfrak{g}_D$ into its coresidue $B_{\mathfrak{g}_D}(y,-)$ . This coresidue then provides a cotangent moment map $\mu:T^*\mathcal{U}\to\mathfrak{g}_D^*$ (Theorem 4.3), as established in [19]. Since the Poisson moduli space $\mathcal{P}=T^*\mathcal{U}/G_D$ forgets the level, fixing the residue orbit $\mathcal{O}$ gives $\mathcal{L}_{\mathcal{O}}=\mu^{-1}(\mathcal{O})/G_D$ , with adjoint and coadjoint orbits identified by $B_{\mathfrak{g}_D}$ and we show that cotangent reduction provides precisely the leaves of the Poisson foliation and their dimensions:

Theorem A (Theorem 4.6). Every $\mathcal{L}_{\mathcal{O}}$ is nonempty, smooth, and connected, of dimension $2\dim\mathcal{B}+\dim\mathcal{O}$ . These reductions are exactly the symplectic leaves of the Poisson space $\mathcal{P}$ .

Thus varying leaves of a fixed dimension amounts to varying residue orbits of a fixed dimension. Balasubramanian–Distler identify this dimension condition and linear symplectic periods in mass-like deformations of tame Hitchin systems [1 Section 2.4.1]. A local example in $\mathfrak{sl}_2$ shows why retaining a parabolic reduction helps at the nilpotent limit. A residue preserving a chosen line has the form

y(a,b)=\begin{pmatrix} a & b \\ 0 & -a \end{pmatrix}.

For $a\ne0$ , it is conjugate to $\operatorname{diag}(a,-a)$ ; for $a=0$ and $b\ne0$ , it is nonzero nilpotent. We observe that both orbits have dimension two. Retaining the line includes $y=0$ in a smooth space: at $a=0$ , compatible pairs satisfy $\operatorname{im} y\subset\ell\operatorname{ker} y$ , so the operators over $\ell$ form $\operatorname{Hom}(\mathbb{C}^2/\ell,\ell)=T^*_\ell\mathbb{P}^1$ . The zero fiber is therefore $T^*\mathbb{P}^1$ , whose complement of the zero section maps isomorphically to the nonzero nilpotent orbit.

In higher rank, the data consist of a parabolic $P_M=MN\subset G_D$ , with Levi factor $M$ and unipotent radical $N$ , its Levi algebra $\mathfrak{m}=\operatorname{Lie}(M)$ and center $\mathfrak{z}=\mathfrak{z}(\mathfrak{m})$ , and a fixed nilpotent $M$ -orbit $\mathcal{O}\subset[\mathfrak{m},\mathfrak{m}]$ . A parameter $m\in\mathfrak{z}$ prescribes the Levi residue $m+e$ , $e\in\mathcal{O}$ . When the centralizer of $m$ is $M$ , the stabilizer of $m+e$ is the stabilizer of $e$ in $M$ ; hence its orbit dimension stays constant as $m$ varies. At zero, Lusztig–Spaltenstein induction [20] supplies an orbit of the same dimension: $\mathcal{O}_0=\operatorname{Ind}_{\mathfrak{m}}^{\mathfrak{g}_D}(\mathcal{O})$ meets $\mathcal{O}+\operatorname{Lie}(N)$ in an open dense subset. Retaining the parabolic reduction connects these orbits through the generalized Springer construction studied by Borho–MacPherson [4] and Broer [5], and its deformations parametrized by the Levi center developed by Namikawa [21, 22, 23].

For Higgs moduli, the quotient spaces are $\mathcal{B}_{P_M}=\mathcal{U}/P_M$ and $\mathcal{V}=\mathcal{U}/N$ , so $\mathcal{V}\to\mathcal{B}_{P_M}$ is an $M$ -bundle. Since $\operatorname{Lie}(N)^\perp=\operatorname{Lie}(P_M)$ under $B_{\mathfrak{g}_D}$ , reduction by $N$ at zero imposes the parabolic residue condition and gives $T^*\mathcal{V}$ . Its remaining moment map $\mu_M$ records the Levi residue. Note that $M$ acts trivially on $\mathfrak{z}$ , so the central parameter survives the subsequent quotient by $M$ .

Theorem B (Proposition 5.3 and Theorem 5.7). The projection

q:\widetilde{\mathcal{L}}=\mu_M^{-1}(\mathfrak{z}+\mathcal{O})/M\longrightarrow\mathfrak{z}

is smooth and surjective, with symplectic fibers of dimension $2\dim B+\dim\mathfrak{g}_D-\dim\mathfrak{m}+\dim\mathcal{O}$ . For $m\in\mathfrak{z}^{\mathrm{reg}}=\{m:Z_{G_D}(m)=M\}$ , its fiber is symplectomorphic to $\mathcal{L}_{G_D\cdot(m+e)}$ , where $e\in\mathcal{O}$ . The open part of the zero fiber with residue in $\mathcal{O}_0$ is a finite étale symplectic cover of $\mathcal{L}_{\mathcal{O}_0}$ , of generalized Springer degree.

Here $Z_{G_D}(m)$ denotes the centralizer of $m$ . For fixed $m\in\mathfrak{z}^{\mathrm{reg}}$ , the residue determines the compatible parabolic reduction uniquely, whereas an induced nilpotent residue can admit several; the generalized Springer degree counts them over a general point of $\mathcal{O}_0$ . The closure of $\mathcal{O}$ includes limits of the nilpotent Levi residue. Local cotangent product charts give flatness and reduced fibers, and the projectivity of $G_D/P_M$ makes forgetting the reduction proper (Propositions 5.10 and 5.11).

Comparison of the symplectic classes requires a differentiable identification of the fibers. A connection on $\mathcal{V}\to\mathcal{B}_{P_M}$ separates cotangent vectors into base and fiber directions; subtracting $m$ from the Levi residue then gives the common manifold

Y=T^*\mathcal{B}_{P_M}\mathbin{\times_{\mathcal{B}_{P_M}}}(\mathcal{V}\times_M\mathcal{O}).

The reduced forms on $Y$ are denoted by $\Omega_m$ . Sternberg’s coupling formula [28] expresses the $m$ -dependent term by pairing $m$ with the connection curvature; hence the classes vary affinely, as in Duistermaat–Heckman [10]. Nilpotent scaling makes $\Omega_0$ exact, so the constant term vanishes. Such periods also describe formal symplectic deformations under the cohomological hypotheses of Kaledin–Verbitsky [15].

Theorem C (Theorem 6.9). There is a connection-independent class $c\in H^2(Y,\mathbb{C})\otimes\mathfrak{z}^*$ , determined by the characteristic classes of the associated principal $M/[M,M]$ -bundle, such that

[\Omega_m]=\langle m,c\rangle.

Therefore, we conclude that, together, Theorem C and the equality of orbit dimensions verify the period and dimension conditions of Balasubramanian–Distler. Theorem A also turns fixed orbit dimension into fixed Poisson rank. Consequently, sheets, the irreducible components of loci of fixed orbit dimension, organize the corresponding rank loci in $\mathcal{P}$ . The sheet theory of Borho–Kraft [3], Borho [2], and Broer [6] applies through the local cotangent product charts, which carry these irreducible components to one another (see Theorem 4.10).

Finally, the parabolic reduction remains part of the data after fixing the Hitchin value. In Yun’s global Springer theory, such additional structure supports Springer-type actions on fiber cohomology [30]. For our family, the Hitchin map records the conjugacy invariants of the central residue. Retaining $m$ itself therefore requires adjoining it to the Hitchin target; Appendix B constructs this target and establishes compatibility with the regular- and zero-fiber comparisons.

The paper is organized as follows. Section 2 reviews nilpotent induction and studies residue orbits with a compatible parabolic reduction. Section 3 relates these residue families to sheets and formulates the dimension and period conditions. In Section 4 we describe level structures, the coresidue moment map, and the symplectic leaves of the logahoric quotient. In Section 5 we construct the simultaneous parabolic reduction and compare its fibers with the residue-orbit reductions, while in Section 6 we compute the variation of the symplectic class. In the final Section 7, we illustrate the construction with a few explicit examples in types $A$ , $D$ , and $E$ .

Levi Residues and Induced Nilpotent Orbits

The Levi decomposition and the central parameter

2.1. The Levi decomposition and the central parameter. Consider a group $G_D$ which is connected, complex, and reductive, with Lie algebra $\mathfrak{g}_D$ and a fixed nondegenerate invariant symmetric form $B_{\mathfrak{g}_D}$ . A closed subgroup $P_M\subset G_D$ is parabolic if $G_D/P_M$ is projective. The Levi decomposition is $P_M = MN$ , where $N$ is its largest connected normal unipotent subgroup and $M$ is a reductive subgroup mapping isomorphically to $P_M/N$ . The subgroup $M$ is a Levi factor. The corresponding Lie algebras are $\mathfrak{m} = \operatorname{Lie}(M)$ , $\mathfrak{p} = \operatorname{Lie}(P_M)$ , and $\mathfrak{n} = \operatorname{Lie}(N)$ ; the last is the nilradical of $\mathfrak{p}$ . The Levi projection $\operatorname{pr}_{\mathfrak{m}} : \mathfrak{p} \to\mathfrak{m}$ gives the vector-space decompositions

References

[1]A. Balasubramanian, and J. Distler, Masses, sheets and rigid SCFTs. Adv. Theor. Math. Phys. 25 (2021), no. 8, 1953–2054.DOI
[2]W. Borho, Über schichten halbeinfacher Lie-algebren. Invent. Math. 65 (1981), no. 2, 283–317.DOI
[3]W. Borho, and H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54 (1979), no. 1, 61–104.DOI
[4]W. Borho, and R. MacPherson, Partial resolutions of nilpotent varieties. Astérisque 101–102 (1983), 23–74.
[5]A. Broer, Decomposition varieties in semisimple Lie algebras. Canad. J. Math. 50 (1998), no. 5, 929–971.DOI
[6]A. Broer, Lectures on decomposition classes. In Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad. Publ., Dordrecht, 1998, 39–83.DOI
[7]O. Chacaltana, J. Distler, and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2, 0) theories. Int. J. Mod. Phys. A 28 (2013), 1340006.DOI
[8]D. H. Collingwood, and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold, New York, 1993.
[9]W. A. de Graaf, and A. G. Elashvili, Induced nilpotent orbits of the simple Lie algebras of exceptional type. Georgian Math. J. 16 (2009), no. 2, 257–278.
[10]J. J. Duistermaat, and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), no. 2, 259–268.DOI
[11]D. Gaiotto, 𝒩 = 2 dualities. JHEP 08 (2012), p. 034.
[12]D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234 (2013), 239–403.arxiv.org/abs/0907.3987
[13]D. Gaiotto, and E. Witten, S-Duality of boundary conditions in 𝒩 = 4 super Yang–Mills Theory. Adv. Theor. Math. Phys. 13.3 (2009), 721–896.DOI
[14]M. Goresky, R. Kottwitz, and R. MacPherson, Purity of equivalued affine Springer fibers. Represent. Theory 10 (2006), 130–146.DOI
[15]D. Kaledin, and M. Verbitsky, Period map for non-compact holomorphically symplectic manifolds. Geom. Funct. Anal. 12 (2002), 1265–1295.DOI
[16]D. Kazhdan, and G. Lusztig, Fixed point varieties on affine flag manifolds. Israel J. Math. 62 (1988), no. 2, 129–168.DOI
[17]A. A. Kirillov, Elements of the Theory of Representations. Grundlehren der mathematischen Wissenschaften 220, Springer-Verlag, Berlin–New York, 1976.DOI
[18]G. Kydonakis, H. Sun, and L. Zhao, Logahoric Higgs torsors for a complex reductive group. Math. Ann. 388 (2024), no. 3, 3183–3228.arxiv.org/abs/2107.01977
[19]G. Kydonakis, and L. Zhao, Level structures on parahoric torsors and complete integrability. arXiv preprint, arXiv: 2506.12302v1 (2025).arxiv.org/abs/2506.12302
[20]G. Lusztig, and N. Spaltenstein, Induced unipotent classes. J. London Math. Soc. (2) 19 (1979), no. 1, 41–52.DOI
[21]Y. Namikawa, Birational geometry and deformations of nilpotent orbits. Duke Math. J. 143 (2008), 375–405.DOI
[22]Y. Namikawa, Induced nilpotent orbits and birational geometry. Adv. Math. 222 (2009), 547–564.DOI
[23]Y. Namikawa, Birational geometry for the covering of a nilpotent orbit closure II. J. Algebra 600 (2022), 152–194.DOI
[24]B. C. Ngô, Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169.
[25]G. Pappas, and M. Rapoport, Twisted loop groups and their affine flag varieties. Adv. Math. 219 (2008), no. 1, 118–198.arxiv.org/abs/math/0607130
[26]A. J. Parameswaran, and Y. Pandey, The étale fundamental group of moduli of parahoric group scheme torsors over a curve. Proc. Indian Acad. Sci. Math. Sci. 133 (2023), article 19.DOI
[27]The Stacks Project Authors, The Stacks Project. Lemma 29.26.16, Tag 081I, https://stacks.math.columbia.edu/tag/081I.
[28]S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field. Proc. Natl. Acad. Sci. USA 74 (1977), 5253–5254.pubmed.ncbi.nlm.nih.gov/16592472
[29]J. Summerfield, Lie algebra decomposition classes for reductive algebraic groups in arbitrary characteristic. arXiv preprint, arXiv: 2511.01440v2 (2026).arxiv.org/abs/2511.01440
[30]Z. Yun, Global Springer theory. Adv. Math. 228 (2011), 266–328.DOI

Mass Deformations and Sheets in Logahoric Higgs Moduli

Abstract

Introduction

Levi Residues and Induced Nilpotent Orbits

The Levi decomposition and the central parameter

References

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