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Mirror symmetry for 4D A1 class-S theories: Modular properties, defects, and the Coulomb branch

Yiwen Pan

Wenbin Yan

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Phys. Rev. D 114, 085008 – Published 8 October, 2026

DOI: https://doi.org/10.1103/x51p-wdy5

Abstract

This is the companion paper to Pan and Yan [arXiv:2410.15695 (2024)], containing all the details and series of examples on a 4D mirror symmetry for the class-S theories which relates the representation theory of the chiral quantization of the Higgs branch and the geometry of the Coulomb branch. We study the representation theory by using the 4D/vertex operator algebra correspondence, (defect) Schur indices, and (flavor) modular differential equations, and match the data with the fixed manifolds of the Hitchin moduli spaces. This correspondence extends the connection between the Higgs and Coulomb branch of Argyres-Douglas theories, and can provide systematic guidance for the study of the representation theory of vertex operator algebras by exploiting results from Hitchin systems.

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References (58)

  1. K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387, 513 (1996).
  2. T. Braden, N. Proudfoot, and B. Webster, Quantizations of conical symplectic resolutions I: Local and global structure, arXiv:1208.3863.
  3. T. Braden, A. Licata, N. Proudfoot, and B. Webster, Quantizations of conical symplectic resolutions II: Category O and symplectic duality, arXiv:1407.0964.
  4. C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336, 1359 (2015).
  5. C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, J. High Energy Phys. 08 (2018) 114.
  6. J. Song, D. Xie, and W. Yan, Vertex operator algebras of Argyres-Douglas theories from M5-branes, J. High Energy Phys. 12 (2017) 123.
  7. T. Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties, arXiv:1811.01577.
  8. P. Shan, D. Xie, and W. Yan, Mirror symmetry for circle compactified 4D N=2 SCFTs, arXiv:2306.15214.
  9. P. Shan, D. Xie, and W. Yan, Modularity for W-algebras and affine Springer fibres, arXiv:2404.00760.
  10. L. Fredrickson, D. Pei, W. Yan, and K. Ye, Argyres-Douglas theories, chiral algebras and wild Hitchin characters, J. High Energy Phys. 01 (2018) 150.
  11. L. Fredrickson and A. Neitzke, From S1-fixed points to W-algebra representations, arXiv:1709.06142.
  12. Y. Pan and W. Yan, companion paper, Mirror symmetry for circle compactified 4d A1 class-S theories, arXiv:2410:15695.
  13. D. Gaiotto, N=2 dualities, J. High Energy Phys. 08 (2012) 034.
  14. D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234, 239 (2013).
  15. G. W. Moore and Y. Tachikawa, On 2D TQFTs whose values are holomorphic symplectic varieties, Proc. Symp. Pure Math. 85, 191 (2012).
  16. H. Zheng, Y. Pan, and Y. Wang, Surface defects, flavored modular differential equations, and modularity, Phys. Rev. D 106, 105020 (2022).
  17. Y. Pan and Y. Wang, Flavored modular differential equations, Phys. Rev. D 108, 085027 (2023).
  18. T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865.
  19. C. Cordova, D. Gaiotto, and S.-H. Shao, Surface defects and chiral algebras, J. High Energy Phys. 05 (2017) 140.
  20. C. Cordova, D. Gaiotto, and S.-H. Shao, Infrared computations of defect Schur indices, J. High Energy Phys. 11 (2016) 106.
  21. Y. Pan and W. Peelaers, Exact Schur index in closed form, Phys. Rev. D 106, 045017 (2022).
  22. Z. Guo, Y. Li, Y. Pan, and Y. Wang, N=2 Schur index and line operators, Phys. Rev. D 108, 106002 (2023).
  23. Y. Pan and P. Yang, Modularity of Schur index, modular differential equations, and high-temperature asymptotics, Phys. Rev. D 110, 065019 (2024).
  24. N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 3, 59 (1987).
  25. H. U. Boden and K. Yokogawa, Moduli spaces of parabolic Higgs bundles and parabolic k (d) pairs over smooth curves: I, Int. J. Math. 7, 573 (1996).
  26. B. Nasatyr and B. Steer, Orbifold Riemann surfaces and the Yang-Mills-Higgs equations, Ann. Della Scuola Normale Superiore di Pisa-Classe di Scienze 22, 595 (1995).
  27. T. Hausel and F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz, Inventiones Mathematicae 174, 555 (2008).
  28. A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, The 4d superconformal index from q-deformed 2D Yang-Mills, Phys. Rev. Lett. 106, 241602 (2011).
  29. A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, Gauge theories and macdonald polynomials, Commun. Math. Phys. 319, 147 (2013).
  30. D. Gaiotto, L. Rastelli, and S. S. Razamat, Bootstrapping the superconformal index with surface defects, J. High Energy Phys. 01 (2013) 022.
  31. L. F. Alday, M. Bullimore, M. Fluder, and L. Hollands, Surface defects, the superconformal index and q-deformed Yang-Mills, J. High Energy Phys. 10 (2013) 018.
  32. T. Nishinaka, S. Sasa, and R.-D. Zhu, On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra, J. High Energy Phys. 03 (2019) 091.

  33. There may be multiple MDEs annihilating χ0. We only concern ourselves with the one of the lowest order.

  34. As a meromorphic function of the complex structure moduli τ, ϕn(τ) may contain poles.

  35. The modular group Γ in the flavored case may be different from the one relevant in the unflavored case of the same theory: in general, flavor refinement results in a subgroup of the latter.

  36. Γ is SL(2,Z) when n is even, is Γ0(2)⊂SL(2,Z) when n odd.

  37. O. García-Prada, P. B. Gothen, and V. Muñoz, Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, Mem. Am. Math. Soc. 187, viii+80 (2007).
  38. P. Deligne, Théorie de hodge: Ii, Publ. Mathé. de l’IHÉS 40, 5 (1971).
  39. P. Deligne, Théorie de hodge: Iii, Publ. Math. de l’IHÉS 44, 5 (1974).
  40. T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J. 160, 323 (2011).
  41. A. M. Garsia and M. Haiman, A remarkable q, t-catalan sequence and q-Lagrange inversion, J. Algebraic Combinatorics 5, 191 (1996).
  42. I. G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford University Press, New York, 1998).

  43. There is an U(1)f flavor symmetry discussed in [4] that is not visible in the class-S construction. Additionally, there is an additional accidental USp(4) symmetry at the level of associated VOA [44, 45]. However, in this paper we will only consider flavor symmetry from the punctures in the class-S construction. It would be interesting to explore how the additional symmetry further refines the our proposal.

  44. K. Kiyoshige and T. Nishinaka, The chiral algebra of genus two class S theory, J. High Energy Phys. 02 (2021) 199.
  45. C. Beem and C. Meneghelli, Geometric free field realization for the genus-two class S theory of Type A1, Phys. Rev. D 104, 065015 (2021).

  46. For Argyres-Douglas theories, these vortex type defects do correspond to irreducible modules of the associated VOA. However, we are unable to show the modules here to be irreducible. Nevertheless, we proceed by assuming irreducibility.

  47. C. Beem, S. S. Razamat, and P. Singh, Schur indices of class S and quasimodular forms, Phys. Rev. D 105, 085009 (2022).
  48. O. Perse, A note on representations of some affine vertex algebras of type D, Glasnik Matematicki 48, 81 (2013).
  49. T. Arakawa and A. Moreau, Joseph ideals and Lisse minimal W-algebras, J. Inst. Math. Jussieu 17, 397 (2018).
  50. B. Li, H. Li, and W. Yan, Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras, arXiv:2304.09681.
  51. D. Adamovic, A realization of certain modules for the n=4 superconformal algebra and the affine lie algebra A2(1), Transform. Groups 21, 299 (2016).
  52. F. Bonetti, C. Meneghelli, and L. Rastelli, VOAs labelled by complex reflection groups and 4D SCFTs, J. High Energy Phys. 05 (2019) 155.
  53. T. Arakawa, T. Kuwabara, and S. Möller, Hilbert schemes of points in the plane and quasi-lisse vertex algebras with N=4 symmetry, arXiv:2309.17308.
  54. C. Beem and S. Nair, Twisted chiral algebras of class S and mixed Feigin–Frenkel gluing, Commun. Math. Phys. 399, 295 (2023).
  55. C. Beem and S. Nair, Free field realisation of the chiral universal centraliser, Ann. Henri Poincare 24, 4343 (2023).
  56. C. Beem, A. Deb, M. Martone, C. Meneghelli, and L. Rastelli, Free field realizations for rank-one SCFTs, J. High Energy Phys. 12 (2024) 004.
  57. K. Costello, T. Creutzig, and D. Gaiotto, Higgs and Coulomb branches from vertex operator algebras, J. High Energy Phys. 03 (2019) 066.
  58. C. Beem and A. E. V. Ferrari, Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions, Commun. Math. Phys. 406, 117 (2025).

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Phys. Rev. D 114, 085008– Published 8 October, 2026
  • Received 6 December 2024
  • Accepted 8 January 2026
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