Abstract
We examine the partition function of \( \mathcal{N}={2}^{\ast } \) supersymmetric SU(N) Yang-Mills theory on the four-sphere in the large radius limit. We point out that the large radius partition function, at fixed N, is computed by saddle-points lying on walls of marginal stability on the Coulomb branch of the theory on \( {\mathrm{\mathbb{R}}}^4 \). For N an even (odd) integer and θ YM = 0(π), these include a point of maximal degeneration of the Donagi-Witten curve to a torus where BPS dyons with electric charge \( \left[\frac{N}{2}\right] \) become massless. We argue that the dyon singularity is the lone saddle-point in the SU(2) theory, while for SU(N) with N > 2, we characterize potentially competing saddle-points by obtaining the relations between the Seiberg-Witten periods at such points. Using Nekrasov’s instanton partition function, we solve for the maximally degenerate saddle-point and obtain its free energy as a function of g YM and N, and show that the results are “large-N exact”. In the large-N theory our results provide analytical expressions for the periods/eigenvalues at the maximally degenerate saddle-point, precisely matching previously known formulae following from the correspondence between \( \mathcal{N}={2}^{\ast } \) theory and the elliptic Calogero-Moser integrable model. The maximally singular point ceases to be a saddle-point of the partition function above a critical value of the coupling, in agreement with the recent findings of Russo and Zarembo.
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References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
F. Passerini and K. Zarembo, Wilson loops in N = 2 Super-Yang-Mills from matrix model, JHEP 09 (2011) 102 [Erratum ibid. 10 (2011) 065] [arXiv:1106.5763] [INSPIRE].
B. Fraser and S.P. Kumar, Large rank Wilson loops in N = 2 superconformal QCD at strong coupling, JHEP 03 (2012) 077 [arXiv:1112.5182] [INSPIRE].
J.G. Russo and K. Zarembo, Large-N limit of N = 2 SU(N) gauge theories from localization, JHEP 10 (2012) 082 [arXiv:1207.3806] [INSPIRE].
J.G. Russo and K. Zarembo, Evidence for large-N phase transitions in N = 2* theory, JHEP 04 (2013) 065 [arXiv:1302.6968] [INSPIRE].
J.G. Russo and K. Zarembo, Massive N = 2 gauge theories at large-N, JHEP 11 (2013) 130 [arXiv:1309.1004] [INSPIRE].
N. Dorey, An elliptic superpotential for softly broken N = 4 supersymmetric Yang-Mills theory, JHEP 07 (1999) 021 [hep-th/9906011] [INSPIRE].
N. Dorey and A. Sinkovics, N = 1* vacua, fuzzy spheres and integrable systems, JHEP 07 (2002) 032 [hep-th/0205151] [INSPIRE].
N. Dorey, T.J. Hollowood, S.P. Kumar and A. Sinkovics, Exact superpotentials from matrix models, JHEP 11 (2002) 039 [hep-th/0209089] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
N. Dorey and S.P. Kumar, Softly broken N = 4 supersymmetry in the large-N limit, JHEP 02 (2000) 006 [hep-th/0001103] [INSPIRE].
O. Aharony, N. Dorey and S.P. Kumar, New modular invariance in the N = 1* theory, operator mixings and supergravity singularities, JHEP 06 (2000) 026 [hep-th/0006008] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
J.G. Russo, \( \mathcal{N}=2 \) gauge theories and quantum phases, JHEP 12 (2014) 169 [arXiv:1411.2602] [INSPIRE].
J.G. Russo, Large-N c from Seiberg-Witten curve and localization, Phys. Lett. B 748 (2015) 19 [arXiv:1504.02958] [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, hep-th/0306238 [INSPIRE].
R. Dijkgraaf and C. Vafa, A perturbative window into nonperturbative physics, hep-th/0208048 [INSPIRE].
V.A. Kazakov, I.K. Kostov and N.A. Nekrasov, D particles, matrix integrals and KP hierarchy, Nucl. Phys. B 557 (1999) 413 [hep-th/9810035] [INSPIRE].
C. Fraser and T.J. Hollowood, On the weak coupling spectrum of N = 2 supersymmetric SU(N) gauge theory, Nucl. Phys. B 490 (1997) 217 [hep-th/9610142] [INSPIRE].
M. Billó et al., Modular anomaly equations in \( \mathcal{N}={2}^{\ast } \) theories and their large-N limit, JHEP 10 (2014) 131 [arXiv:1406.7255] [INSPIRE].
F. Ferrari, Charge fractionization in N = 2 supersymmetric QCD, Phys. Rev. Lett. 78 (1997) 795 [hep-th/9609101] [INSPIRE].
K. Konishi and H. Terao, CP, charge fractionalizations and low-energy effective actions in the SU(2) Seiberg-Witten theories with quarks, Nucl. Phys. B 511 (1998) 264 [hep-th/9707005] [INSPIRE].
N. Dorey, V.V. Khoze and M.P. Mattis, On mass deformed N = 4 supersymmetric Yang-Mills theory, Phys. Lett. B 396 (1997) 141 [hep-th/9612231] [INSPIRE].
J.A. Minahan, D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 super Yang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE].
A. Ritz and A.I. Vainshtein, Long range forces and supersymmetric bound states, Nucl. Phys. B 617 (2001) 43 [hep-th/0102121] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
K. Zarembo, Strong-coupling phases of planar N = 2* Super-Yang-Mills theory, Theor. Math. Phys. 181 (2014) 1522 [arXiv:1410.6114] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrability in N = 2 gauge theory: a proof, hep-th/9511052 [INSPIRE].
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
M.R. Douglas and S.H. Shenker, Dynamics of SU(N) supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 271 [hep-th/9503163] [INSPIRE].
E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge University Press, Cambridge U.K. (1927).
N. Koblitz, Introduction to elliptic curves and modular forms, Springer, Germany (1984).
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ArXiv ePrint: 1509.00716
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Hollowood, T.J., Kumar, S.P. Partition function of \( \mathcal{N}={2}^{\ast } \) SYM on a large four-sphere. J. High Energ. Phys. 2015, 1–42 (2015). https://doi.org/10.1007/JHEP12(2015)016
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DOI: https://doi.org/10.1007/JHEP12(2015)016