Abstract
We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T 4 with flat metric using Dirac quantization. In addition to an \( \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) \) symmetry, it possesses \( \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) \) symmetry that is electromagnetic S-duality. We show explicitly how this \( \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) \) S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2, 0) tensor theory, by computing the partition function of a single fivebrane compactified on T 2 times T 4, which has \( \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right)\times \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) \) symmetry. If we identify the couplings of the abelian gauge theory \( \tau =\frac{\theta }{2\pi }+i\frac{4\pi }{e^2} \) with the complex modulus of the T 2 torus \( \tau ={\beta}^2+i\frac{R_1}{R_2} \), then in the small T 2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the \( \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) \) symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the \( \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) \) acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.
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References
C. Montonen and D.I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978) 97 [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
E. Witten, Geometric Langlands from six dimensions, arXiv:0905.2720 [INSPIRE].
E. Witten, Some comments on string dynamics, in Future perspectives in string theory, Los Angeles U.S.A. (1995), pg. 501 [hep-th/9507121] [INSPIRE].
E. Witten, Conformal field theory in four and six dimensions, arXiv:0712.0157 [INSPIRE].
E.P. Verlinde, Global aspects of electric-magnetic duality, Nucl. Phys. B 455 (1995) 211 [hep-th/9506011] [INSPIRE].
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 2, Cambridge University Press, Cambridge U.K. (1987), pg. 40 [INSPIRE].
L. Dolan and C.R. Nappi, A modular invariant partition function for the five-brane, Nucl. Phys. B 530 (1998) 683 [hep-th/9806016] [INSPIRE].
L. Dolan and C.R. Nappi, The Ramond-Ramond selfdual five form’s partition function on T 10, Mod. Phys. Lett. A 15 (2000) 1261 [hep-th/0005074] [INSPIRE].
D. Bak and A. Gustavsson, M5/D4 brane partition function on a circle bundle, JHEP 12 (2012) 099 [arXiv:1209.4391] [INSPIRE].
R. Zucchini, Abelian duality and Abelian Wilson loops, Commun. Math. Phys. 242 (2003) 473 [hep-th/0210244] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383 [hep-th/9505186] [INSPIRE].
L. Dolan and Y. Sun, Partition functions for Maxwell theory on the five-torus and for the fivebrane on S 1 × T 5, JHEP 09 (2013) 011 [arXiv:1208.5971] [INSPIRE].
M. Henningson, The quantum Hilbert space of a chiral two form in d = (5 + 1)-dimensions, JHEP 03 (2002) 021 [hep-th/0111150] [INSPIRE].
H. Kikuchi, Poincaré invariance in temporal gauge canonical quantization and theta vacua, Int. J. Mod. Phys. A 9 (1994) 2741 [hep-th/9302045] [INSPIRE].
P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science, Yeshiva University, New York U.S.A. (1964).
A. Das, Lectures on quantum field theory, World Scientific, Hackensack U.S.A. (2008) [INSPIRE].
M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover Publications, New York U.S.A. (1972).
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [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].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2∗ super Yang-Mills on S 4, JHEP 03 (2012) 017 [arXiv:1004.1222] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
G. Etesi and A. Nagy, S-duality in Abelian gauge theory revisited, J. Geom. Phys. 61 (2011) 693 [arXiv:1005.5639] [INSPIRE].
A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].
H. Coxeter and W. Moser, Generators and relations for discrete groups, Springer Verlag, New York U.S.A. (1980).
S. Trott, A pair of generators for the unimodular group, Canad. Math. Bull. 5 (1962) 245.
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Dolan, L., Sun, Y. Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on T 2 × T 4, via their partition functions. J. High Energ. Phys. 2015, 134 (2015). https://doi.org/10.1007/JHEP06(2015)134
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DOI: https://doi.org/10.1007/JHEP06(2015)134