Abstract
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
N. Wyllard, AN−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability, and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 06 (2018) 46 [arXiv:1709.09993] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, II, ICCM Not. 06 (2018) 120 [arXiv:1802.01579] [INSPIRE].
M. Ashwinkumar and M.-C. Tan, Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory, Adv. Theor. Math. Phys. 24 (2020) 1681 [arXiv:1910.01134] [INSPIRE].
B. Geyer and D. Mülsch, Higher dimensional analog of the Blau-Thompson model and NT = 8, D = 2 Hodge type cohomological gauge theories, Nucl. Phys. B 662 (2003) 531 [hep-th/0211061] [INSPIRE].
C. Cordova and D.L. Jafferis, Five-dimensional maximally supersymmetric Yang-Mills in supergravity backgrounds, JHEP 10 (2017) 003 [arXiv:1305.2886] [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].
C. Elliott and V. Pestun, Multiplicative Hitchin Systems and Supersymmetric Gauge Theory, arXiv:1812.05516 [INSPIRE].
V. Mikhaylov, Teichmüller TQFT vs. Chern-Simons theory, JHEP 04 (2018) 085 [arXiv:1710.04354] [INSPIRE].
D. Gaiotto and E. Witten, Knot Invariants from Four-Dimensional Gauge Theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE].
M. Ashwinkumar, Integrable Lattice Models and Holography, JHEP 02 (2021) 227 [arXiv:2003.08931] [INSPIRE].
G. Jorjadze and G. Weigt, Poisson structure and Moyal quantization of the Liouville theory, Nucl. Phys. B 619 (2001) 232 [hep-th/0105306] [INSPIRE].
P. Bouwknegt and K. Schoutens, W-symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4 − D SYM to 2 − D sigma models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
E. Witten, Non-Abelian bosonization in two-dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.06053
Rights and permissions
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.
About this article
Cite this article
Ashwinkumar, M., Png, KS. & Tan, MC. 4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondence. J. High Energ. Phys. 2021, 57 (2021). https://doi.org/10.1007/JHEP09(2021)057
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2021)057