Abstract
We calculate the gravitational corrections to the effective action of N=2 SU(2) Seiberg-Witten theory with matter using modularity, the holomorphic anomaly equation and expected behavior at the boundaries of the moduli space. As in pure gauge theory we show that the gap condition at the dyon singularities completely fixes the gravitational corrections. We discuss the behavior of the gravitational corrections at the conformal points. We compare our results with the recursive solution of the loop equation in the matrix model approach, which provides in addition open amplitudes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.-x. Huang and A. Klemm, Holomorphic anomaly in gauge theories and matrix models, JHEP 09 (2007) 054 [hep-th/0605195] [SPIRES].
E. Witten, Quantum background independence in string theory, hep-th/9306122 [SPIRES].
M. Aganagic, V. Bouchard and A. Klemm, Topological strings and (almost) modular forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [SPIRES].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [SPIRES].
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] [SPIRES].
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] [SPIRES].
S. Kachru, A. Klemm, W. Lerche, P. Mayr and C. Vafa, Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications, Nucl. Phys. B 459 (1996) 537 [hep-th/9508155] [SPIRES].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Self-dual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [SPIRES].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [SPIRES].
R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].
R. Dijkgraaf and C. Vafa, On geometry and matrix models, Nucl. Phys. B 644 (2002) 21 [hep-th/0207106] [SPIRES].
R. Dijkgraaf, A. Sinkovics and M. Temurhan, Matrix models and gravitational corrections, Adv. Theor. Math. Phys. 7 (2004) 1155 [hep-th/0211241] [SPIRES].
B. Eynard, All orders asymptotic expansion of large partitions, J. Stat. Mech. (2008) P07023 [arXiv:0804.0381] [SPIRES].
A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [SPIRES].
M. Mariño, Open string amplitudes and large order behavior in topological string theory, JHEP 03 (2008) 060 [hep-th/0612127] [SPIRES].
B. Haghighat, A. Klemm and M. Rauch, Integrability of the holomorphic anomaly equations, JHEP 10 (2008) 097 [arXiv:0809.1674] [SPIRES].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].
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] [SPIRES].
P.C. Argyres, M. Ronen Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [SPIRES].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Topological open strings on orbifolds, Commun. Math. Phys. 296 (2010) 589 [arXiv:0807.0597] [SPIRES].
S. Yamaguchi and S.-T. Yau, Topological string partition functions as polynomials, JHEP 07 (2004) 047 [hep-th/0406078] [SPIRES].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [SPIRES].
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] [SPIRES].
M. Alim and J .D. Lange, Polynomial structure of the (open) topological string partition function, JHEP 10 (2007) 045 [arXiv:0708.2886] [SPIRES].
S. Hosono, BCOV ring and holomorphic anomaly equation, arXiv:0810.4795.
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048] [SPIRES].
E. D’Hoker, I.M. Krichever and D.H. Phong, The effective prepotential of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 489 (1997) 179 [hep-th/9609041] [SPIRES].
W. Nahm, On the Seiberg-Witten approach to electric-magnetic duality, hep-th/9608121 [SPIRES].
K. Ito and S.-K. Yang, Prepotentials in N = 2SU(2) supersymmetric Yang-Mills theory with massless hypermultiplets, Phys. Lett. B 366 (1996) 165 [hep-th/9507144] [SPIRES].
D. Zagier, Elliptic modular forms and their applications, in The 1-2-3 of Modular Forms, Springer, Berlin Germany (2008).
C. Vafa, A stringy test of the fate of the conifold, Nucl. Phys. B 447 (1995) 252 [hep-th/9505023] [SPIRES].
D. Ghoshal and C. Vafa, C = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121 [hep-th/9506122] [SPIRES].
M.-x. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [SPIRES].
A. Klemm, W. Lerche and S. Theisen, Nonperturbative effective actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 1929 [hep-th/9505150] [SPIRES].
Y. Ohta, Prepotential of N = 2SU(2) Yang-Mills gauge theory coupled with a massive matter multiplet, J. Math. Phys. 37 (1996) 6074 [hep-th/9604051] [SPIRES].
Y. Ohta, Prepotentials of N = 2SU(2) Yang-Mills theories coupled with massive matter multiplets, J. Math. Phys. 38 (1997) 682 [hep-th/9604059] [SPIRES].
R. Fricke, Die modularen Funktionen und ihre Anwendungen, Bd 1 Teubner Leibzig (1916).
A. Brandhuber and S. Stieberger, Periods, coupling constants and modular functions in N = 2SU(2) SYM with massive matter, Int. J. Mod. Phys. A 13 (1998) 1329 [hep-th/9609130] [SPIRES].
T. Eguchi and H. Kanno, Topological strings and Nekrasov’s formulas, JHEP 12 (2003) 006 [hep-th/0310235] [SPIRES].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [SPIRES].
A. Klemm, On the geometry behind N = 2 supersymmetric effective actions in four dimensions, hep-th/9705131 [SPIRES].
M. Alim, J .D. Lange and P. Mayr, Global properties of topological string amplitudes and orbifold invariants, JHEP 03 (2010) 113 [arXiv:0809.4253] [SPIRES].
A. Klemm and E. Zaslow, Local mirror symmetry at higher genus, hep-th/9906046 [SPIRES].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP 02 (2004) 010 [hep-th/0211098] [SPIRES].
B. Eynard, M. Mariño and N. Orantin, Holomorphic anomaly and matrix models, JHEP 06 (2007) 058 [hep-th/0702110] [SPIRES].
J . Walcher, Extended holomorphic anomaly and loop amplitudes in open topological string, Nucl. Phys. B 817 (2009) 167 [arXiv:0705.4098] [SPIRES].
D. Krefl and J . Walcher, The real topological string on a local Calabi-Yau, arXiv:0902.0616 [SPIRES].
M. Mariño, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009) 052301 [arXiv:0809.2619] [SPIRES].
B. Eynard and M. Mariño, A holomorphic and background independent partition function for matrix models and topological strings, arXiv:0810.4273 [SPIRES].
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] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0902.1325
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Huang, Mx., Klemm, A. Holomorphicity and modularity in Seiberg-Witten theories with matter. J. High Energ. Phys. 2010, 83 (2010). https://doi.org/10.1007/JHEP07(2010)083
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2010)083