Abstract
We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of \( \mathcal{N}={2^{*}} \) gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasimodular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.
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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, A N −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
V. Fateev and A. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [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].
M.-X. Huang, On gauge theory and topological string in Nekrasov-Shatashvili limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].
M.-X. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [INSPIRE].
M.-X. Huang and A. Klemm, Direct integration for general Ω backgrounds, arXiv:1009.1126 [INSPIRE].
M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid N = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
M.C. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].
A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys. B 855 (2012) 128 [arXiv:1105.0948] [INSPIRE].
A. Zamolodchikov, Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419.
A. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.
A. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
J. Minahan, D. Nemeschansky and N. Warner, Instanton expansions for mass deformed N =4 super Yang-Mills theories, Nucl. Phys. B 528 (1998) 109[hep-th/9710146] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2004) 457 [hep-th/0212279] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
T. Dimofte and S. Gukov, Refined, motivic and quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
E.W. Barnes, The theory of the double gamma function, Phil. Trans. Roy. Soc. London A 196 (1901) 265.
D. Krefl and J. Walcher, Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. 95 (2011) 67 [arXiv:1007.0263] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
T. Eguchi and H. Ooguri, Conformal and current algebras on general Riemann surface, Nucl. Phys. B 282 (1987) 308 [INSPIRE].
J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011) 471 [arXiv:1005.2846] [INSPIRE].
M. Aganagic, M.C. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, Correlators of primary fields in the SU(2) W ZW theory on Riemann surfaces, Nucl. Phys. B 305 (1988) 219 [INSPIRE].
P. Di Francesco, H. Saleur and J. Zuber, Critical Ising correlation functions in the plane and on the torus, Nucl. Phys. B 290 (1987) 527 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].
E. D’Hoker and D. Phong, Calogero-Moser systems in SU(N ) Seiberg-Witten theory, Nucl. Phys. B 513 (1998) 405 [hep-th/9709053] [INSPIRE].
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) 237.
J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, hep-th/0308031 [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett. B 685 (2010) 79 [arXiv:0911.4296] [INSPIRE].
B. Feigin and D. Fuks, Verma modules over the Virasoro algebra, Funct. Anal. Appl. 17 (1983)241 [INSPIRE].
J. Teschner, Operator product expansion and factorization in the \( H_3^{+} \) WZNW model, Nucl. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE].
A. Hanany, N. Prezas and J. Troost, The partition function of the two-dimensional black hole conformal field theory, JHEP 04 (2002) 014 [hep-th/0202129] [INSPIRE].
J. Troost, The non-compact elliptic genus: mock or modular, JHEP 06 (2010) 104 [arXiv:1004.3649] [INSPIRE].
G.H. Halphen, Traité des fonctions elliptiques et de leurs applications (in French), Gauthier-Villars, Paris France (1886).
M. Grosset and A.P. Veselov, Elliptic Faulhaber polynomials and Lamé densities of states, math-ph/0508066.
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ArXiv ePrint: 1212.0722
Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie 6, UMR 8549.
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Kashani-Poor, AK., Troost, J. The toroidal block and the genus expansion. J. High Energ. Phys. 2013, 133 (2013). https://doi.org/10.1007/JHEP03(2013)133
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DOI: https://doi.org/10.1007/JHEP03(2013)133