Abstract
Starting with a choice of gauge algebras, specification of a 4d gauge theory involves additional data, namely the gauge groups and the discrete theta angles. Equivalently, one needs to specify the set of charges of allowed line operators. In this note, we study how these additional data are represented in 6d, when the 4d theory in question is an \( \mathcal{N} \) = 4 super Yang-Mills theory or an \( \mathcal{N} \) = 2 class S theory. We will see that the ℤ N symmetry of the so-called T N theory plays an important role.
As a byproduct, we will find that the superconformal index of class S theories can be refined so that it can give 2d q-deformed Yang-Mills theory with different gauge groups associated to the same gauge algebra.
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References
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, arXiv:1006.0146 [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
O. Aharony and E. Witten, Anti-de Sitter space and the center of the gauge group, JHEP 11 (1998) 018 [hep-th/9807205] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP 12 (1998) 012 [hep-th/9812012] [INSPIRE].
G.W. Moore, Anomalies, Gauss laws and Page charges in M-theory, Comptes Rendus Physique 6 (2005) 251 [hep-th/0409158] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
M. Henningson, The partition bundle of type A N −1(2, 0) theory, JHEP 04 (2011) 090 [arXiv:1012.4299] [INSPIRE].
D.S. Freed and C. Teleman, Relative quantum field theory, Commun. Math. Phys. 326 (2014) 459 [arXiv:1212.1692] [INSPIRE].
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
D. Mumford, M. Nori and P. Norman, Tata Lectures on Theta III, Birkhäuser, (1991).
S.S. Razamat and B. Willett, Global Properties of Supersymmetric Theories and the Lens Space, arXiv:1307.4381 [INSPIRE].
A. Kapustin and R. Thorngren, Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement, arXiv:1308.2926 [INSPIRE].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [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].
Y. Tachikawa and N. Watanabe, Lines in 4d class S theories and in q-deformed Yang-Mills, work in progress.
Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-Deformed YM, Nucl. Phys. B 869 (2013) 493 [arXiv:1210.2855] [INSPIRE].
D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
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Tachikawa, Y. On the 6d origin of discrete additional data of 4d gauge theories. J. High Energ. Phys. 2014, 20 (2014). https://doi.org/10.1007/JHEP05(2014)020
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DOI: https://doi.org/10.1007/JHEP05(2014)020