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Non-invertible twisted compactification of class $\mathcal S$ theory and $(B,B,B)$ branes

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Pith's one-line read Twisting class S theory on S1 yields (B,B,B) brane targets

desk verdict A plausible extension of non-invertible twisted compactification to class S, with explicit genus-2 affine equations, but the key fixed-locus step is asserted and the global-structure quotient is dropped. read the letter →

arxiv 2412.06729 v2 pith:G3JV7356 submitted 2024-12-09 hep-th

Yankun Ma This is my paper · ORCID
classification hep-th
keywords non-invertiblesymmetryclassStheorytwistedcompactificationHitchinmodulispace(BBB)branemappinggroupaffinevarietycharacter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that compactifying a class S theory on a circle with a non-invertible self-duality defect inserted, instead of compactifying directly, produces a three-dimensional $\mathcal{N}=4$ $\sigma$ model whose target is a hyperkähler submanifold of the Hitchin moduli space. The submanifold is shown to be the fixed point set of a finite subgroup of the mapping class group of the Riemann surface that defines the class S theory, and such fixed point sets are exactly the $(B,B,B)$ branes studied in gauge theory. The paper then gives a concrete algebraic description of these branes as affine varieties and computes explicit examples for type $A_1$ genus 2 class S theory, including zero loci of explicit polynomial equations in loop coordinates. If correct, this turns a symmetry principle—non-invertible duality—into a systematic way of generating new three-dimensional theories with hyperkähler target spaces.

What carries the argument

The carrying object is the fixed-point locus $M' = \{p \in M \mid N\cdot p = p\}$ inside the Hitchin moduli space $M$—the moduli space of Higgs bundles, equivalently flat connections, on the Riemann surface—together with the identification of its character variety description in complex structure $J$ as an affine variety generated by loop coordinates. The mechanism is that a non-invertible self-duality defect $N = \sigma \circ F$ composes a duality action $F$ with a topological manipulation $\sigma$; because $\sigma$ only changes the global structure and, at a self-dual point, $F$ is a finite-order mapping class element, the fixed locus is exactly the fixed point set of a finite group of holomorphic automorphisms of the Riemann surface, and a standard theorem on such finite group actions guarantees that this fixed point set is a $(B,B,B)$ brane. For computations, the paper uses traces of words in the fundamental group as coordinates, where the mapping class group acts by Poisson automorphisms; the fixed locus is therefore described by adjoining linear fixed-point equations to the polynomial defining equations of the character variety, and the paper shows concretely that these equations cut out affine varieties.

What would settle it

One could settle it by computing, for the order-6 generator $I$ of the genus-2 type $A_1$ example, the Jacobian rank of the system consisting of the 19 polynomial relations (4.21) and the fixed-point equations (4.31): if the fixed locus is singular or has a dimension incompatible with a smooth hyperkähler submanifold, the $(B,B,B)$ identification fails.

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Extended reading notes

Core claim

The central claim is that the vacuum moduli space of the non-invertible twisted compactification is the fixed locus $M' = \{p \in M \mid N\cdot p = p\}$ in the Hitchin moduli space $M = M_H(\hat G, \Sigma_g)/L$. Since the non-invertible defect $N$ combines a mapping class group duality $F$ with a topological manipulation that cancels the change of global structure, the fixed locus is equivalently the set of points fixed by a finite-order element of the mapping class group; by the classical realization theorem for finite subgroups of the mapping class group, this is the fixed point set of a finite group of holomorphic automorphisms of the underlying Riemann surface, and a known theorem on such finite group actions guarantees that this fixed point set is a $(B,B,B)$ brane, i.e. a hyperkähler submanifold. The paper makes this concrete for type $A_1$ genus 2: using the loop-coordinate description of the $SL(2,\mathbb{C})$ character variety as an affine variety in $\mathbb{C}^{15}$ defined by 19 polynomial relations, it identifies the fixed loci of finite mapping class subgroups as the zero loci of those polynomials together with the fixed-point equations, and works out examples such as the order-6 generator $I$ acting by cyclic permutation of the loop coordinates.

Load-bearing premise

The whole argument rests on assuming that the only effect of the non-invertible defect on vacuum configurations is to demand that the configuration be fixed by the defect's action, so the new vacuum space is literally the fixed-point set of that action.

Editorial extensions

If this is right

  • Non-invertible twisted compactification provides a general recipe: every self-dual point of a class S conformal manifold yields a 3d $\mathcal{N}=4$ sigma model whose target is the fixed locus of the corresponding finite mapping class subgroup.
  • For type $A_1$ genus 2, the self-duality fixed points of $Sp(4,\mathbb{Z})$ give a catalogue of $(B,B,B)$ branes, each described as an affine variety cut out by the 19 polynomial relations of the character variety plus linear fixed-point equations.
  • The order-6 generator $I$ of the genus-2 mapping class group acts by cyclic permutations on the loop coordinates, so its fixed locus is the intersection of the polynomial relations (4.21) with $z_1=\cdots=z_6$, $z_{12}=\cdots=z_{61}$, and $z_{123}=z_{234}=z_{345}$.
  • Because the mapping class group acts by automorphisms of the Poisson algebra of loop coordinates, and of its quantum deformation, the fixed loci carry an algebraic structure that can be studied independently of the physical construction.
  • The one-punctured torus example shows the mechanism also reproduces zero-dimensional branes: the $S$-transformation fixed locus is finite, the roots of $3x_1^4+8x_1^2-8-4m=0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the identification of the vacuum space with the fixed locus is correct, the same construction should work for punctured class S theories and for type $A_{N-1}$ with $N>2$, using traces of $SL(N,\mathbb{C})$ words as loop coordinates.
  • A subtle point the paper leaves open is the quotient by the global-structure lattice $L$; one could test whether the fixed locus of the non-invertible defect on $M_H/L$ differs from the fixed locus on $M_H$, since a difference would require modifying Eq. (3.17).
  • The computed affine varieties could be fed into 3d mirror symmetry: the mirror of a non-invertibly twisted compactification may be an orbifold of the known star-shaped quiver mirror, and comparing Hilbert series of the two would be a concrete check.
  • The algebraic structure coming from the q,t deformation of the coordinate ring suggests the fixed loci might be realizable as moduli spaces of equivariant Higgs bundles, which would give an independent mathematical construction of the same branes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes that inserting a non-invertible self-duality defect in a class S theory on S^1 (non-invertible twisted compactification) yields a 3d N=4 sigma model whose target space is a (B,B,B) brane of Hitchin moduli space. The main claim is that this brane is the fixed point set of a finite subgroup of the mapping class group of the underlying Riemann surface, and the paper gives an affine-variety description of such fixed loci for type A1 genus 2. The derivation proceeds by identifying the vacuum moduli space of the twisted compactification with the fixed locus of the defect action, then invoking Nielsen realization and a theorem of Heller and Schaposnik to show this locus is hyperkähler. Concrete computations are presented using loop coordinates on the SL(2,C) character variety.

Significance. If the central identification is correct, the paper gives a new physical construction of (B,B,B) branes and provides the first explicit affine-variety descriptions of these fixed loci, which are of interest to both the class S and Higgs bundle communities. The paper correctly imports independent theorems (Heller–Schaposnik, Nielsen realization) and the coordinate computations are concrete and reproducible from the cited literature. However, the main result is only as strong as the unproven step that identifies the moduli space of the non-invertible twisted compactification with the pointwise fixed locus, and the treatment of the global-structure quotient raises additional issues that affect the validity of the computed examples as physical target spaces.

major comments (2)
  1. [§3.2, Eq. (3.17)] The identification of the vacuum moduli space with the pointwise fixed locus M' = {p in M | N·p = p} is asserted without derivation. A non-invertible defect is not a group element, and its action in the S^1-compactified theory is not simply a classical map on field configurations: the topological manipulation σ entering N = σF involves gauging and SPT stacking, and the path integral over S^1 bundles can produce twisted sectors not captured by the fixed-point condition. The author should derive (3.17) from the path integral, or support it by a concrete example such as the N=4 SYM S-duality twist of [9] where the target space can be computed independently, before using it as the foundation for the (B,B,B) brane claim.
  2. [§3.2, Eq. (3.20), footnote 5] The quotient by the global-structure lattice L is dropped too quickly. The physical moduli space is M = MH(Ĝ,Σ)/L, and the duality F acts on L through its action on H^1(Σ,Z_N). Thus the correct fixed-point condition in M is φ(p) = l·p for some l in L, which is generally a larger set than the image of MH^φ in M. The affine computations in §4 (e.g., Eq. (4.31)) are performed in the unquotiented χ2,0 and therefore describe only the l = 0 part of the target; the (B,B,B) property of any l-twisted components is not addressed by the theorem of [18]. The paper should compute the action of F on L for the cases in Table 1 and identify the full fixed locus in M before claiming that the computed affine varieties are the target spaces.
minor comments (6)
  1. [§2, paragraph before Eq. (2.9)] The phrase "dosen’t" should be "doesn’t", and "A straight computation" should be "A straightforward computation".
  2. [§3.2, paragraph after Theorem] The word "hyerKähler" should be "hyperKähler".
  3. [§5, first paragraph] The word "invertble" should be "invertible".
  4. [§4.1, paragraph on Poisson algebra] The term "Poission" should be "Poisson".
  5. [Eq. (4.21)] The third displayed equation contains the term "+ z_{i+5} z_{i,i+1}" three times, which is likely a typographical duplication; the formula should be cross-checked against reference [50].
  6. [§4.1, first paragraph] The sentence "For type An+1 the Lie group is G = SU(n)" is confusing: for type A1 the group is SU(2), so the notation should be adjusted (e.g., type A_{n-1} for SU(n)) to avoid a mismatch between the index and the group.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the fixed-point target space is an explicit physical modeling assumption, and the (B,B,B) brane claim and affine variety examples follow from external theorems and prior coordinate computations.

full rationale

The paper's derivation chain is: (i) review of non-invertible duality defects in class S (Section 2, based on [14-16]); (ii) straight circle compactification giving Hitchin moduli space (Section 3.1, [17,19]); (iii) proposal that inserting the defect restricts the vacuum moduli space to the fixed-point locus M' = {p | N·p = p} (Eq. 3.17); (iv) identification of N's action with a finite subgroup H of the mapping class group, using the claim that the topological manipulation G has no action on the Hitchin moduli space; (v) proof that M' is a (B,B,B) brane by the external Heller-Schaposnik theorem [18]; and (vi) description as an affine variety using loop coordinates and polynomial relations from Arthamonov [50]. No step fits a parameter and then re-predicts it, no definition is chosen so that the conclusion is true by fiat, and there are no self-citations. Equation (3.17) is a physical assumption about how a non-invertible topological defect acts on the vacuum manifold, not a hidden equivalence, and footnote 5 explicitly concedes the omission of the L-quotient; these are correctness/rigor limitations rather than circularity. The concrete fixed-locus computations in Section 4 are independent uses of external coordinate data, so the paper's added content does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper does not introduce free parameters or new entities. It relies on a chain of established results: Nielsen realization, the Heller-Schaposnik theorem, the identification of straight circle compactification with Hitchin moduli, and the loop-coordinate presentation of the genus 2 character variety from [50]. The main non-standard input is the fixed-locus identification Eq. (3.17), which functions as an ad hoc domain assumption rather than a derived statement.

assumptions (7)
  • standard math Nielsen realization theorem: a finite subgroup of the mapping class group that fixes a point on Teichmüller space is realized as a group of holomorphic automorphisms of the Riemann surface (cited [40,41]).
    Used in Section 3.2 to pass from fixed points of the duality group on the conformal manifold to actual surface automorphisms acting on Hitchin moduli space.
  • standard math Heller-Schaposnik theorem: for a finite group acting holomorphically on a Riemann surface of genus g at least 2, the fixed point set on the Hitchin moduli space M_H(G,Sigma_g) is a (B,B,B) brane (cited [18]).
    Used as the core input that identifies the target of non-invertible twisted compactification as a hyperkähler submanifold.
  • domain assumption The mapping class group action on the character variety M_H(G,Sigma_g) is holomorphic and preserves the holomorphic symplectic form Omega_J when the complex structure of Sigma_g is fixed by the subgroup.
    Invoked in Sections 3.2 and 4.2 to ensure fixed loci are holomorphic symplectic submanifolds; depends on the standard character-variety description of Hitchin moduli in complex structure J.
  • domain assumption The non-invertible defect N = sigma o F acts on the moduli space of vacua M = M_H(G,Sigma)/L only through the duality action F, with the topological manipulation sigma acting trivially on M and the combined action preserving the global structure.
    Used in Section 3.2 to reduce the fixed-point condition to the mapping class group action; this is asserted rather than derived.
  • ad hoc to paper The vacuum moduli space of the twisted compactification is exactly the fixed point set M' = {p in M | N·p = p}, Eq. (3.17).
    This is the key step of the paper; it is presented as a natural consequence of the defect imposing conditions on field configurations, but no derivation is given.
  • standard math The genus 2 SL(2,C) character variety is the zero locus of the 19 polynomial relations (4.21) in the 15 loop coordinates, as computed in [50] with related results in [51,52].
    Taken from prior literature; the paper uses these relations to describe the fixed locus as an affine variety without re-deriving them.
  • standard math The traces of all words in the fundamental group are generated by 2^(2g)-1 loop coordinates (Fricke-Magnus theorem, cited [45-47]), so invariant subrings of the coordinate ring can be described by polynomial conditions on these coordinates.
    Used in Section 4 to justify working with loop coordinates and to claim the affine-variety description.

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Pith. "Pith review of Non-invertible twisted compactification of class $\mathcal S$ theory and $(B,B,B)$ branes." pith.science (2026). https://pith.science/paper/G3JV7356

@misc{pith2026241206729,
  author       = {Pith},
  title        = {Pith review of: Non-invertible twisted compactification of class $\mathcal S$ theory and $(B,B,B)$ branes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3JV7356}},
  note         = {Machine review of arXiv:2412.06729}
}
abstract

We study non-invertible twisted compactification of class $\mathcal S$ theories on $S^1$: we insert a non-invertible symmetry defect at $S^1$ extending along remaining directions and then compactify on $S^1$. We show that the resulting 3d theory is 3d $\mathcal N=4$ sigma model whose target space is a hyperK\"ahler submanifold of Hitchin moduli space, i.e. a $(B,B,B)$ brane. The $(B,B,B)$ brane is the fixed point set on Hitchin moduli space of a finite subgroup of mapping class group of underlying Riemann surface. We describe the $(B,B,B)$ branes as affine varieties and calculate concrete examples of these $(B,B,B)$ branes for type $A_1$, genus $2$ class $\mathcal S$ theory.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

58 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [9]

    Kaidi, G

    J. Kaidi, G. Zafrir, and Y. Zheng, Non-invertible symmetries of N = 4 SYM and twisted compactification, JHEP 08 (2022) 053, [ arXiv:2205.01104]

    arXiv 2022

  2. [18]

    Heller and L

    S. Heller and L. P. Schaposnik, Branes through finite group actions , J. Geom. Phys. 129 (2018) 279–293, [ arXiv:1611.00391]

    arXiv 2018

  3. [1]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [ arXiv:1412.5148]

    arXiv 2015

  4. [2]

    E. P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory , Nucl. Phys. B 300 (1988) 360–376

    1988

  5. [3]

    Frohlich, J

    J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354–430, [ hep-th/0607247]

    arXiv 2007

  6. [4]

    Chang, Y.-H

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions , JHEP 01 (2019) 026, [arXiv:1802.04445]

    arXiv 2019

  7. [5]

    Frohlich, J

    J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601, [ cond-mat/0404051]

    arXiv 2004

  8. [6]

    Kaidi, K

    J. Kaidi, K. Ohmori, and Y. Zheng, Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories, Phys. Rev. Lett. 128 (2022), no. 11 111601, [ arXiv:2111.01141]

    arXiv 2022

Show all 58 references
  1. [7]

    Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Noninvertible duality defects in 3+1 dimensions , Phys. Rev. D 105 (2022), no. 12 125016, [ arXiv:2111.01139]

    2022 arXiv

  2. [8]

    Roumpedakis, S

    K. Roumpedakis, S. Seifnashri, and S.-H. Shao, Higher Gauging and Non-invertible Condensation Defects, Commun. Math. Phys. 401 (2023), no. 3 3043–3107, [arXiv:2204.02407]

    2023 arXiv

  3. [10]

    O. J. Ganor and Y. P. Hong, Selfduality and Chern-Simons Theory , arXiv:0812.1213

    arXiv

  4. [11]

    O. J. Ganor, Y. P. Hong, and H. S. Tan, Ground States of S-duality Twisted N=4 Super Yang-Mills Theory, JHEP 03 (2011) 099, [ arXiv:1007.3749]

    2011 arXiv

  5. [12]

    O. J. Ganor, Y. P. Hong, R. Markov, and H. S. Tan, Static Charges in the Low-Energy Theory of the S-Duality Twist , JHEP 04 (2012) 041, [ arXiv:1201.2679]

    2012 arXiv

  6. [13]

    Gaiotto, N=2 dualities , JHEP 08 (2012) 034, [ arXiv:0904.2715]

    D. Gaiotto, N=2 dualities , JHEP 08 (2012) 034, [ arXiv:0904.2715]

    2012 arXiv

  7. [14]

    Bashmakov, M

    V. Bashmakov, M. Del Zotto, and A. Hasan, On the 6d origin of non-invertible symmetries in 4d , JHEP 09 (2023) 161, [ arXiv:2206.07073]

    2023 arXiv

  8. [15]

    Bashmakov, M

    V. Bashmakov, M. Del Zotto, A. Hasan, and J. Kaidi, Non-invertible symmetries of class S theories, JHEP 05 (2023) 225, [ arXiv:2211.05138]

    2023 arXiv

  9. [16]

    Antinucci, C

    A. Antinucci, C. Copetti, G. Galati, and G. Rizi, “Zoology” of non-invertible duality defects: the view from class S, JHEP 04 (2024) 036, [ arXiv:2212.09549]

    2024 arXiv

  10. [17]

    Gaiotto, G

    D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013) 239–403, [ arXiv:0907.3987]

    2013 arXiv

  11. [19]

    Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories , JHEP 05 (2014) 020, [ arXiv:1309.0697]

    Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories , JHEP 05 (2014) 020, [ arXiv:1309.0697]

    2014 arXiv

  12. [20]

    Gukov, P.-S

    S. Gukov, P.-S. Hsin, and D. Pei, Generalized global symmetries of T [M ] theories. Part I , JHEP 04 (2021) 232, [ arXiv:2010.15890]. – 27 –

    2021 arXiv

  13. [21]

    Lawrie, X

    C. Lawrie, X. Yu, and H. Y. Zhang, Intermediate defect groups, polarization pairs, and noninvertible duality defects , Phys. Rev. D 109 (2024), no. 2 026005, [ arXiv:2306.11783]

    2024 arXiv

  14. [22]

    Gaiotto and J

    D. Gaiotto and J. Kulp, Orbifold groupoids, JHEP 02 (2021) 132, [ arXiv:2008.05960]

    2021 arXiv

  15. [23]

    Tachikawa, On gauging finite subgroups , SciPost Phys

    Y. Tachikawa, On gauging finite subgroups , SciPost Phys. 8 (2020), no. 1 015, [arXiv:1712.09542]

    2020 arXiv

  16. [24]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions , JHEP 03 (2018) 189, [ arXiv:1704.02330]

    2018 arXiv

  17. [25]

    Aharony, N

    O. Aharony, N. Seiberg, and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115, [ arXiv:1305.0318]

    2013 arXiv

  18. [26]

    Gottschling, ¨Uber die fixpunkte der siegelschen modulgruppe , Mathematische Annalen 143 (Apr, 1961) 111–149

    E. Gottschling, ¨Uber die fixpunkte der siegelschen modulgruppe , Mathematische Annalen 143 (Apr, 1961) 111–149

    1961

  19. [27]

    Gottschling, ¨Uber die fixpunktuntergruppen der siegelschen modulgruppe , Mathematische Annalen 143 (Oct, 1961) 399–430

    E. Gottschling, ¨Uber die fixpunktuntergruppen der siegelschen modulgruppe , Mathematische Annalen 143 (Oct, 1961) 399–430

    1961

  20. [28]

    Gottschling, Die uniformisierbarkeit der fixpunkte eigentlich diskontinuierlicher gruppen von biholomorphen abbildungen , Mathematische Annalen 169 (Mar, 1967) 26–54

    E. Gottschling, Die uniformisierbarkeit der fixpunkte eigentlich diskontinuierlicher gruppen von biholomorphen abbildungen , Mathematische Annalen 169 (Mar, 1967) 26–54

    1967

  21. [29]

    Kuribayashi and A

    I. Kuribayashi and A. Kuribayashi, Automorphism groups of compact riemann surfaces of genera three and four , Journal of Pure and Applied Algebra 65 (1990), no. 3 277–292

    1990

  22. [30]

    Kuribayashi and H

    A. Kuribayashi and H. Kimura, On automorphism groups of compact riemann surfaces of genus 5, 1986

    1986

  23. [31]

    H. P. Nilles, S. Ramos-Sanchez, A. Trautner, and P. K. S. Vaudrevange, Orbifolds from Sp(4,Z) and their modular symmetries , Nucl. Phys. B 971 (2021) 115534, [arXiv:2105.08078]

    2021 arXiv

  24. [32]

    Gaiotto, G

    D. Gaiotto, G. W. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163–224, [arXiv:0807.4723]

    2010 arXiv

  25. [33]

    Seiberg and E

    N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory , Nucl. Phys. B 426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl.Phys.B 430, 485–486 (1994)]

    1994 arXiv

  26. [34]

    Seiberg and E

    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484–550, [ hep-th/9408099]

    1994 arXiv

  27. [35]

    Gomez and R

    C. Gomez and R. Hernandez, Electric - magnetic duality and effective field theories , in Advanced School on Effective Theories , 6, 1995. hep-th/9510023

    1995 arXiv

  28. [36]

    N. J. Hitchin, The Selfduality equations on a Riemann surface , Proc. Lond. Math. Soc. 55 (1987) 59–131

    1987

  29. [37]

    Kapustin and E

    A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1–236, [ hep-th/0604151]

    2007 arXiv

  30. [38]

    Biswas, S

    I. Biswas, S. Heller, and L. P. Schaposnik, Branes and moduli spaces of Higgs bundles on smooth projective varieties., Res. Math. Sci. 8 (2021), no. 3 52, [ arXiv:2005.02564]

    2021 arXiv

  31. [39]

    L. P. Schaposnik, Advanced topics in gauge theory: mathematics and physics of Higgs bundles, arXiv:1907.09800. – 28 –

    1907

  32. [40]

    S. P. Kerckhoff, The nielsen realization problem , Bulletin of the American Mathematical Society 2 (1980) 452–454

    1980

  33. [41]

    Nakamura and T

    G. Nakamura and T. Nakanishi, Generation of finite subgroups of the mapping class group of genus 2 surface by dehn twists , Journal of Pure and Applied Algebra 222 (2018), no. 11 3585–3594

    2018

  34. [42]

    V. G. Turaev, Skein quantization of poisson algebras of loops on surfaces , Annales scientifiques de l’ ´Ecole Normale Sup´ erieure24 (1991), no. 6 635–704

    1991

  35. [43]

    Kabiraj, Center of poisson and skein algebras associated to loops on surfaces , arXiv:2011.08283

    A. Kabiraj, Center of poisson and skein algebras associated to loops on surfaces , arXiv:2011.08283

    2011 arXiv

  36. [44]

    Dimofte and S

    T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality , JHEP 05 (2013) 109, [arXiv:1106.4550]

    2013 arXiv

  37. [45]

    Magnus, Rings of fricke characters and automorphism groups of free groups , Mathematische Zeitschrift 170 (Feb, 1980) 91–103

    W. Magnus, Rings of fricke characters and automorphism groups of free groups , Mathematische Zeitschrift 170 (Feb, 1980) 91–103

    1980

  38. [46]

    Vorlesungen ¨ uber die theorie der automorphen funktionen, Monatshefte f¨ ur Mathematik und Physik 25 A34

  39. [47]

    R. D. Horowitz, Characters of free groups represented in the two-dimensional special linear group*, Communications on Pure and Applied Mathematics 25 (1972) 635–649

    1972

  40. [48]

    S. P. Tan, Y. L. Wong, and Y. Zhang, The SL(2,C) character variety of the one-holed torus , math/0509033

    arXiv

  41. [49]

    Nekrasov, A

    N. Nekrasov, A. Rosly, and S. Shatashvili, Darboux coordinates, Yang-Yang functional, and gauge theory, Nucl. Phys. B Proc. Suppl. 216 (2011) 69–93, [ arXiv:1103.3919]

    2011 arXiv

  42. [50]

    Arthamonov, Classical Limit of genus two DAHA , arXiv:2309.01011

    S. Arthamonov, Classical Limit of genus two DAHA , arXiv:2309.01011

    arXiv

  43. [51]

    Arthamonov and S

    S. Arthamonov and S. Shakirov, Genus two generalization of A1 spherical DAHA, Selecta Mathematica 25 (2019), no. 2 [ arXiv:1704.02947]

    2019 arXiv

  44. [52]

    Cooke and P

    J. Cooke and P. Samuelson, On the genus two skein algebra , Journal of the London Mathematical Society 104 (July, 2021) 2260–2298, [arXiv:2008.12695]

    2021 arXiv

  45. [53]

    J. S. Birman and H. M. Hilden, On the mapping class group of closed surfaces as covering spaces, 1971

    1971

  46. [54]

    K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories , Phys. Lett. B 387 (1996) 513–519, [ hep-th/9607207]

    1996 arXiv

  47. [55]

    Benini, Y

    F. Benini, Y. Tachikawa, and D. Xie, Mirrors of 3d Sicilian theories , JHEP 09 (2010) 063, [arXiv:1007.0992]

    2010 arXiv

  48. [56]

    Rozansky and E

    L. Rozansky and E. Witten, HyperKahler geometry and invariants of three manifolds , Selecta Math. 3 (1997) 401–458, [ hep-th/9612216]

    1997 arXiv

  49. [57]

    Gukov, P.-S

    S. Gukov, P.-S. Hsin, H. Nakajima, S. Park, D. Pei, and N. Sopenko, Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants , J. Geom. Phys. 168 (2021) 104311, [arXiv:2005.05347]

    2021 arXiv

  50. [58]

    Gukov, B

    S. Gukov, B. Haghighat, and N. Reshetikhin, Foams and KZ-equations in Rozansky-Witten theories, arXiv:2407.19757. – 29 –

    arXiv

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Reviewed August 11, 2026 · model on record in the stance chip above.