Abstract
We investigate an alternative approach to the correspondence of four dimensional \( \mathcal{N} \) = 2 superconformal theories and two-dimensional vertex operator algebras, in the framework of the Ω-deformation of supersymmetric gauge theories. The twodimensional Ω-deformation of the holomorphic-topological theory on the product four manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the \( \mathcal{N} \) = 2 superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our Ω-deformation point of view on the correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-'t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
N. Nekrasov, Four dimensional holomorphic theories, Ph.D. thesis, Princeton University, Princeton, NJ, U.S.A. (1996).
L. Baulieu, A. Losev and N. Nekrasov, Chern-Simons and twisted supersymmetry in various dimensions, Nucl. Phys.B 522 (1998) 82 [hep-th/9707174] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in Proceedings, 16thInternational Congress on Mathematical Physics ( ICMP09 ), Prague, Czech Republic, 3–8 August 2009, World Scientific, Singapore (2010), pg. 265 [arXiv:0908.4052] [INSPIRE].
N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
N. Nekrasov, Tying up instantons with anti-instantons, World Scientific, Singapore (2018), pg. 351 [arXiv:1802.04202] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, W symmetry in six dimensions, JHEP05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
M. Lemos and P. Liendo, N = 2 central charge bounds from 2d chiral algebras, JHEP04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys.319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches and modular differential equations, JHEP08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
J. Song, Macdonald index and chiral algebra, JHEP08 (2017) 044 [arXiv:1612.08956] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett.118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, JHEP02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, part II: non-principal deformations, JHEP12 (2016) 103 [Addendum ibid.04 (2017) 113] [arXiv:1610.05311] [INSPIRE].
P. Agarwal, A. Sciarappa and J. Song, N = 1 Lagrangians for generalized Argyres-Douglas theories, JHEP10 (2017) 211 [arXiv:1707.04751] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, A “Lagrangian” for the E7superconformal theory, JHEP05 (2018) 193 [arXiv:1802.05268] [INSPIRE].
E. Witten, Analytic continuation of Chern-Simons theory, AMS/ IP Stud. Adv. Math.50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
J. Yagi, Ω-deformation and quantization, JHEP08 (2014) 112 [arXiv:1405.6714] [INSPIRE].
Y. Luo, M.-C. Tan, J. Yagi and Q. Zhao, Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence, JHEP02 (2015) 047 [arXiv:1410.1538] [INSPIRE].
K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories, arXiv:1810.01970 [INSPIRE].
J. Oh and J. Yagi, Chiral algebras from Ω-deformation, JHEP08 (2019) 143 [arXiv:1903.11123] [INSPIRE].
Y. Pan and W. Peelaers, Schur correlation functions on S3 × S1, JHEP07 (2019) 013 [arXiv:1903.03623] [INSPIRE].
M. Dedushenko and M. Fluder, Chiral algebra, localization, modularity, surface defects, and all that, arXiv:1904.02704 [INSPIRE].
M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb branch of 3d N = 4 theories, Commun. Math. Phys.354 (2017) 671 [arXiv:1503.04817] [INSPIRE].
C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte and A. Neitzke, Secondary products in supersymmetric field theory, arXiv:1809.00009 [INSPIRE].
A. Johansen, Infinite conformal algebras in supersymmetric theories on four manifolds, Nucl. Phys.B 436 (1995) 291 [hep-th/9407109] [INSPIRE].
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of Argyres-Douglas theory from M5 brane, arXiv:1604.02155 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1904.00927
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jeong, S. SCFT/VOA correspondence via Ω-deformation. J. High Energ. Phys. 2019, 171 (2019). https://doi.org/10.1007/JHEP10(2019)171
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)171