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Line defect half-indices of $SU(N)$ Chern-Simons theories (2403.03439v3)

Published 6 Mar 2024 in hep-th, math.CO, and math.NT

Abstract: We study the Wilson line defect half-indices of 3d $\mathcal{N}=2$ supersymmetric $SU(N)$ Chern-Simons theories of level $k\le -N$ with Neumann boundary conditions for the gauge fields, together with 2d Fermi multiplets and fundamental 3d chiral multiplets to cancel the gauge anomaly. We derive some exact results and also make some conjectures based on expansions of the $q$-series. We find several interesting connections with special functions known in the literature, including Rogers-Ramanujan functions for which we conjecture integral representations, and the appearance of Appell-Lerch sums for certain Wilson line half-index grand canonical ensembles which reveal an unexpected appearance of mock modular functions. We also find intriguing $q$-difference equations relating half-indices to Wilson line half-indices. Some of these results also have a description in terms of a dual theory with Dirichlet boundary conditions for the vector multiplet in the dual theory.

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References (51)
  1. A. Gadde, S. Gukov, and P. Putrov, “Walls, Lines, and Spectral Dualities in 3d Gauge Theories,” JHEP 1405 (2014) 047, arXiv:1302.0015 [hep-th].
  2. T. Okazaki and S. Yamaguchi, “Supersymmetric boundary conditions in three-dimensional N=2 theories,” Phys.Rev. D87 no. 12, (2013) 125005, arXiv:1302.6593 [hep-th].
  3. A. Gadde, S. Gukov, and P. Putrov, “Fivebranes and 4-manifolds,” Prog. Math. 319 (2016) 155–245, arXiv:1306.4320 [hep-th].
  4. Y. Yoshida and K. Sugiyama, “Localization of three-dimensional 𝒩=2𝒩2\mathcal{N}=2caligraphic_N = 2 supersymmetric theories on S1×D2superscript𝑆1superscript𝐷2S^{1}\times D^{2}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT,” PTEP 2020 no. 11, (2020) 113B02, arXiv:1409.6713 [hep-th].
  5. T. Dimofte, D. Gaiotto, and N. M. Paquette, “Dual boundary conditions in 3d SCFT’s,” JHEP 05 (2018) 060, arXiv:1712.07654 [hep-th].
  6. I. Brunner, J. Schulz, and A. Tabler, “Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models,” JHEP 06 (2019) 046, arXiv:1904.07258 [hep-th].
  7. K. Costello, T. Dimofte, and D. Gaiotto, “Boundary Chiral Algebras and Holomorphic Twists,” Commun. Math. Phys. 399 no. 2, (2023) 1203–1290, arXiv:2005.00083 [hep-th].
  8. K. Sugiyama and Y. Yoshida, “Supersymmetric indices on I×T2𝐼superscript𝑇2I\times T^{2}italic_I × italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, elliptic genera and dualities with boundaries,” Nucl. Phys. B 960 (2020) 115168, arXiv:2007.07664 [hep-th].
  9. H. Jockers, P. Mayr, U. Ninad, and A. Tabler, “BPS indices, modularity and perturbations in quantum K-theory,” JHEP 02 (2022) 044, arXiv:2106.07670 [hep-th].
  10. M. Dedushenko and N. Nekrasov, “Interfaces and Quantum Algebras, I: Stable Envelopes,” arXiv:2109.10941 [hep-th].
  11. K. Zeng, “Monopole operators and bulk-boundary relation in holomorphic topological theories,” SciPost Phys. 14 no. 6, (2023) 153, arXiv:2111.00955 [hep-th].
  12. T. Okazaki and D. J. Smith, “Seiberg-like dualities for orthogonal and symplectic 3d 𝒩𝒩\mathcal{N}caligraphic_N = 2 gauge theories with boundaries,” JHEP 07 (2021) 231, arXiv:2105.07450 [hep-th].
  13. T. Okazaki and D. J. Smith, “Web of Seiberg-like dualities for 3D N=2 quivers,” Phys. Rev. D 105 no. 8, (2022) 086023, arXiv:2112.07347 [hep-th].
  14. T. Okazaki and D. J. Smith, “Boundary confining dualities and Askey-Wilson type q-beta integrals,” JHEP 08 (2023) 048, arXiv:2305.00247 [hep-th].
  15. T. Okazaki and D. J. Smith, “3d exceptional gauge theories and boundary confinement,” JHEP 11 (2023) 199, arXiv:2308.14428 [hep-th].
  16. F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups,” Lett. Math. Phys. 104 (2014) 465–493, arXiv:1305.0533 [hep-th].
  17. F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic Genera of 2d 𝒩𝒩{\mathcal{N}}caligraphic_N = 2 Gauge Theories,” Commun. Math. Phys. 333 no. 3, (2015) 1241–1286, arXiv:1308.4896 [hep-th].
  18. A. Gadde and S. Gukov, “2d Index and Surface operators,” JHEP 03 (2014) 080, arXiv:1305.0266 [hep-th].
  19. T. Dimofte, D. Gaiotto, and S. Gukov, “Gauge Theories Labelled by Three-Manifolds,” Commun.Math.Phys. 325 (2014) 367–419, arXiv:1108.4389 [hep-th].
  20. C. Beem, T. Dimofte, and S. Pasquetti, “Holomorphic Blocks in Three Dimensions,” JHEP 12 (2014) 177, arXiv:1211.1986 [hep-th].
  21. T. Dimofte, D. Gaiotto, and S. Gukov, “3-Manifolds and 3d Indices,” Adv. Theor. Math. Phys. 17 no. 5, (2013) 975–1076, arXiv:1112.5179 [hep-th].
  22. B. C. Berndt, Ramanujan’s notebooks. Part III. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2.
  23. T. Okazaki, “Abelian dualities of 𝒩=(0,4)𝒩04\mathcal{N}=(0,4)caligraphic_N = ( 0 , 4 ) boundary conditions,” JHEP 08 (2019) 170, arXiv:1905.07425 [hep-th].
  24. T. Okazaki, “Abelian mirror symmetry of 𝒩𝒩\mathcal{N}caligraphic_N = (2, 2) boundary conditions,” JHEP 03 (2021) 163, arXiv:2010.13177 [hep-th].
  25. R. Askey and J. Wilson, “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials,” Mem. Amer. Math. Soc. 54 no. 319, (1985) iv+55. https://doi.org/10.1090/memo/0319.
  26. B. Nassrallah and M. Rahman, “Projection formulas, a reproducing kernel and a generating function for q𝑞qitalic_q-Wilson polynomials,” SIAM J. Math. Anal. 16 no. 1, (1985) 186–197. https://doi.org/10.1137/0516014.
  27. M. Rahman, “An integral representation of a φ910subscriptsubscript𝜑910{}_{10}\varphi_{9}start_FLOATSUBSCRIPT 10 end_FLOATSUBSCRIPT italic_φ start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT and continuous bi-orthogonal φ910subscriptsubscript𝜑910{}_{10}\varphi_{9}start_FLOATSUBSCRIPT 10 end_FLOATSUBSCRIPT italic_φ start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT rational functions,” Canad. J. Math. 38 no. 3, (1986) 605–618. https://doi.org/10.4153/CJM-1986-030-6.
  28. R. A. Gustafson, “Some q𝑞qitalic_q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras,” Trans. Amer. Math. Soc. 341 no. 1, (1994) 69–119. https://doi.org/10.2307/2154615.
  29. R. A. Gustafson, “Some q𝑞qitalic_q-beta integrals on SU⁢(n)SU𝑛{\rm SU}(n)roman_SU ( italic_n ) and Sp⁢(n)Sp𝑛{\rm Sp}(n)roman_Sp ( italic_n ) that generalize the Askey-Wilson and Nasrallah-Rahman integrals,” SIAM J. Math. Anal. 25 no. 2, (1994) 441–449. https://doi.org/10.1137/S0036141092248614.
  30. M. Ito, “Askey-Wilson type integrals associated with root systems,” Ramanujan J. 12 no. 1, (2006) 131–151. https://doi.org/10.1007/s11139-006-9579-y.
  31. M. G. Alford, K. Benson, S. R. Coleman, J. March-Russell, and F. Wilczek, “Zero modes of nonabelian vortices,” Nucl. Phys. B 349 (1991) 414–438.
  32. J. Preskill and L. M. Krauss, “Local Discrete Symmetry and Quantum Mechanical Hair,” Nucl. Phys. B 341 (1990) 50–100.
  33. M. Bucher, K.-M. Lee, and J. Preskill, “On detecting discrete Cheshire charge,” Nucl. Phys. B386 (1992) 27–42, arXiv:hep-th/9112040 [hep-th].
  34. M. G. Alford, K.-M. Lee, J. March-Russell, and J. Preskill, “Quantum field theory of nonAbelian strings and vortices,” Nucl. Phys. B384 (1992) 251–317, arXiv:hep-th/9112038 [hep-th].
  35. L. Brekke, T. D. Imbo, H. Dykstra, and A. F. Falk, “Novel spin and statistical properties of nonAbelian vortices,” Phys. Lett. B 304 (1993) 127–133, arXiv:hep-th/9210131.
  36. G. E. Andrews, The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original.
  37. L. J. Rogers, “On two theorems of combinatory analysis and some allied identities,” Proceedings of the London Mathematical Society 2 no. 1, (1917) 315–336.
  38. A. Rocha-Caridi, “Vacuum vector representations of the Virasoro algebra,” in Vertex operators in mathematics and physics (Berkeley, Calif., 1983), vol. 3 of Math. Sci. Res. Inst. Publ., pp. 451–473. Springer, New York, 1985. https://doi.org/10.1007/978-1-4613-9550-8_22.
  39. P. Paule and S. Radu, “Rogers-Ramanujan functions, modular functions, and computer algebra,” in Advances in computer algebra, vol. 226 of Springer Proc. Math. Stat., pp. 229–280. Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-73232-9_10.
  40. G. E. Andrews and D. Hickerson, “Ramanujan’s “lost” notebook vii: The sixth order mock theta functions,” Advances in Mathematics 89 no. 1, (1991) 60–105. https://www.sciencedirect.com/science/article/pii/000187089190083J.
  41. D. R. Hickerson and E. T. Mortenson, “Hecke-type double sums, appell–lerch sums, and mock theta functions, i,” Proceedings of the London Mathematical Society 109 no. 2, (2014) 382–422, https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/pdu007. https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/pdu007.
  42. S. Zwegers, Mock Theta Functions. PhD thesis, 2008. arXiv:0807.4834 [math.NT]. http://inspirehep.net/record/1288143/files/arXiv:0807.4834.pdf.
  43. A. Dabholkar, S. Murthy, and D. Zagier, “Quantum Black Holes, Wall Crossing, and Mock Modular Forms,” arXiv:1208.4074 [hep-th].
  44. S. K. Ashok, E. Dell’Aquila, and J. Troost, “Higher Poles and Crossing Phenomena from Twisted Genera,” JHEP 08 (2014) 087, arXiv:1404.7396 [hep-th].
  45. T. Okazaki and D. J. Smith, “Mock Modular Index of M2-M5 Brane System,” Phys. Rev. D96 no. 2, (2017) 026017, arXiv:1612.07565 [hep-th].
  46. T. Okazaki and D. J. Smith, “Topological M-Strings and Supergroup WZW Models,” Phys. Rev. D94 (2016) 065016, arXiv:1512.06646 [hep-th].
  47. W. N. Bailey, “Identities of the Rogers-Ramanujan type,” Proc. London Math. Soc. (2) 50 (1948) 1–10. https://doi.org/10.1112/plms/s2-50.1.1.
  48. L. J. Slater, “Further identities of the Rogers-Ramanujan type,” Proc. London Math. Soc. (2) 54 (1952) 147–167. https://doi.org/10.1112/plms/s2-54.2.147.
  49. J. Mc Laughlin, A. V. Sills, and P. Zimmer, “Rogers-Ramanujan-Slater type identities,” Electron. J. Combin. DS15 no. Dynamic Surveys, (2008) 59.
  50. G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook. Part I. Springer, New York, 2005.
  51. M. Hajij, “The tail of a quantum spin network,” Ramanujan J. 40 no. 1, (2016) 135–176. https://doi.org/10.1007/s11139-015-9705-9.

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