Abstract
We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d \( \mathcal{N} \) = 4 gauge theories. We conjecture various relations between these boundary VOA’s and properties of the (topologically twisted) bulk theories. We discuss applications to the Symplectic Duality and Geometric Langlands programs.
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References
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
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 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
D. Gaiotto, Twisted compactifications of 3d \( \mathcal{N} \) = 4 theories and conformal blocks, JHEP 02 (2019) 061 [arXiv:1611.01528] [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
D. Gaiotto, S-duality of boundary conditions and the Geometric Langlands program, Proc. Symp. Pure Math. 98 (2018) 139 [arXiv:1609.09030] [INSPIRE].
T. Creutzig and D. Gaiotto, Vertex Algebras for S-duality, arXiv:1708.00875 [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
L. Rozansky and E. Witten, HyperKähler geometry and invariants of three manifolds, Selecta Math. 3 (1997) 401 [hep-th/9612216] [INSPIRE].
A. Kapustin and K. Vyas, A-Models in Three and Four Dimensions, arXiv:1002.4241 [INSPIRE].
A. Kapustin, L. Rozansky and N. Saulina, Three-dimensional topological field theory and symplectic algebraic geometry I, Nucl. Phys. B 816 (2009) 295 [arXiv:0810.5415] [INSPIRE].
A. Kapustin and L. Rozansky, Three-dimensional topological field theory and symplectic algebraic geometry II, Commun. Num. Theor. Phys. 4 (2010) 463 [arXiv:0909.3643] [INSPIRE].
M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry and Symplectic Duality in 3d \( \mathcal{N} \) = 4 Gauge Theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
H.-J. Chung and T. Okazaki, (2, 2) and (0, 4) supersymmetric boundary conditions in 3d \( \mathcal{N} \) = 4 theories and type IIB branes, Phys. Rev. D 96 (2017) 086005 [arXiv:1608.05363] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
K. Costello, T. Dimofte and D. Gaiotto, Boundary vertex algebras and holomorphic twists, to appear.
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
D. Butson and P. Yoo, Degenerate Classical Field Theories and Boundary Theories, arXiv:1611.00311 [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, Lines and Spectral Dualities in 3d Gauge Theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
A. Beilinson and V. Drinfeld, American Mathematical Society Colloquium Publications. Vol. 51: Chiral algebras, American Mathematical Society, Providence U.S.A. (2004).
A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, II, arXiv:1601.03586 [INSPIRE].
A. Kapustin and L. Rozansky, On the relation between open and closed topological strings, Commun. Math. Phys. 252 (2004) 393 [hep-th/0405232] [INSPIRE].
K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007) 165 [math/0412149] [INSPIRE].
K. Costello, T. Creutzig and D. Gaiotto, Higgs and coulomb branches from vertex operator algebras, to appear.
B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].
D. Karabali and H.J. Schnitzer, BRST Quantization of the Gauged WZW Action and Coset Conformal Field Theories, Nucl. Phys. B 329 (1990) 649 [INSPIRE].
H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].
A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].
E. Frenkel and D. Gaiotto, Gauge theory, vertex algebras and the geometric langlands duality, to appear.
D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The theta-Angle in N = 4 Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N = 4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets, JHEP 07 (2008) 091 [arXiv:0805.3662] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
A. Kapustin and N. Saulina, Chern-Simons-Rozansky-Witten topological field theory, Nucl. Phys. B 823 (2009) 403 [arXiv:0904.1447] [INSPIRE].
D. Tong, The holographic dual of AdS 3 × S 3 × S 3 × S 1, JHEP 04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
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Costello, K., Gaiotto, D. Vertex Operator Algebras and 3d \( \mathcal{N} \) = 4 gauge theories. J. High Energ. Phys. 2019, 18 (2019). https://doi.org/10.1007/JHEP05(2019)018
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DOI: https://doi.org/10.1007/JHEP05(2019)018