Abstract
The partition function of an \({\mathcal {N}=2}\) gauge theory in the Ω-background satisfies, for generic value of the parameter \({\beta=-{\epsilon_1}/{\epsilon_2}}\) , the, in general extended, but otherwise β-independent, holomorphic anomaly equation of special geometry. Modularity together with the (β-dependent) gap structure at the various singular loci in the moduli space completely fixes the holomorphic ambiguity, also when the extension is non-trivial. In some cases, the theory at the orbifold radius, corresponding to β = 2, can be identified with an “orientifold” of the theory at β = 1. The various connections give hints for embedding the structure into the topological string.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) arXiv:hep-th/9408099
Seiberg, N., Witten, E.: Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19 (1994). Erratum-ibid. B 430, 485 (1994). arXiv:hep-th/9407087
Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B 534, 549 (1998) arXiv:hep-th/9711108
Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97 (2000) arXiv:hep-th/9712241
Losev, A., Nekrasov, N., Shatashvili, S.L.: Testing Seiberg-Witten solution. arXiv:hep-th/9801061
Moore G.W., Nekrasov N., Shatashvili S.: D-particle bound states and generalized instantons. Commun. Math. Phys. 209, 77 (2000) arXiv:hep-th/9803265
Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831 (2004) arXiv:hep-th/0206161
Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. arXiv:hep-th/0306238
Nakajima, H., Yoshioka, K.: Instanton counting on blowup. Part I: 4-dimensional pure gauge theory. arXiv:math.ag/0306198
Moore G.W., Witten E.: Integration over the u-plane in Donaldson theory. Adv. Theor. Math. Phys. 1, 298 (1998) arXiv:hep-th/9709193
Antoniadis I., Gava E., Narain K.S., Taylor T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994) arXiv:hep-th/9307158
Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) arXiv:hep-th/9309140
Klemm A., Lerche W., Mayr P., Vafa C., Warner N.P.: Self-dual strings and N = 2 supersymmetric field theory. Nucl. Phys. B 477, 746 (1996) arXiv:hep-th/9604034
Katz S.H., Klemm A., Vafa C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997) arXiv:hep-th/9609239
Klemm A., Marino M., Theisen S.: Gravitational corrections in supersymmetric gauge theory and matrix models. JHEP 0303, 051 (2003) arXiv:hep-th/0211216
Iqbal A., Kashani-Poor A.K.: Instanton counting and Chern-Simons theory. Adv. Theor. Math. Phys. 7, 457 (2004) arXiv:hep-th/0212279
Eguchi T., Kanno H.: Topological strings and Nekrasov’s formulas. JHEP 0312, 006 (2003) arXiv:hep-th/0310235
Huang M.X., Klemm A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 0709, 054 (2007) arXiv:hep-th/0605195
Huang, M.X., Klemm, A.: Holomorphicity and modularity in Seiberg-Witten theories with matter. arXiv:0902.1325[hep-th]
Ghoshal D., Vafa C.: C = 1 String as the topological theory of the conifold. Nucl. Phys. B 453, 121 (1995) arXiv:hep-th/9506122
Antoniadis, I., Hohenegger, S., Narain, K.S., Taylor, T.R.: Deformed topological partition function and Nekrasov backgrounds. arXiv:1003.2832[hep-th]
Walcher J.: Extended holomorphic anomaly and loop amplitudes in open topological string. Nucl. Phys. B 817, 167 (2009) arXiv:0705.4098[hep-th]
Bouchard V., Florea B., Marino M.: Counting higher genus curves with crosscaps in Calabi-Yau orientifolds. JHEP 0412, 035 (2004) arXiv:hep-th/0405083
Walcher J.: Evidence for tadpole cancellation in the topological string. Comm. Number Theor. Phys. 3, 111–172 (2009) arXiv:0712.2775[hep-th]
Krefl, D., Walcher, J.: The real topological string on a local Calabi-Yau. arXiv:0902. 0616[hep-th]
Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167 (2010) arXiv:0906.3219 [hep-th]
Dijkgraaf, R., Vafa, C.: Toda theories, matrix models, topological strings, and N = 2 gauge systems. arXiv:0909.2453[hep-th]
Yamaguchi S., Yau S.T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004) arXiv:hep-th/0406078
Alim M., Lange J.D.: Polynomial structure of the (open) topological string partition function. JHEP 0710, 045 (2007) arXiv:0708.2886[hep-th]
Konishi, Y., Minabe, S.: On solutions to Walcher’s extended holomorphic anomaly equation. arXiv:0708.2898[math.AG]
Grimm T.W., Klemm A., Marino M., Weiss M.: Direct integration of the topological string. JHEP 0708, 058 (2007) arXiv:hep-th/0702187
Gopakumar, R., Vafa, C.: M-theory and topological strings. Part I, II. arXiv:hep-th/9809187, arXiv:hep-th/9812127
Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475 (1990)
Hollowood T.J., Iqbal A., Vafa C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008) arXiv:hep-th/0310272
Iqbal A., Kozcaz C., Vafa C.: The refined topological vertex. JHEP 0910, 069 (2009) arXiv:hep-th/0701156
Morrison D.R., Walcher J.: D-branes and Normal Functions. Adv. Theor. Math. Phys. 13, 553–598 (2009) arXiv:0709.4028[hep-th]
Alday L.F., Gaiotto D., Gukov S., Tachikawa Y., Verlinde H.: Loop and surface operators in N = 2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010) arXiv:0909.0945[hep-th]
Kozcaz, C., Pasquetti, S., Wyllard, N.: A & B model approaches to surface operators and Toda theories. arXiv:1004.2025[hep-th]
Dimofte, T., Gukov, S., Hollands, L.: Vortex counting and Lagrangian 3-manifolds. arXiv:1006.0977[hep-th]
Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052[hep-th]
Dijkgraaf R., Verlinde E.P., Verlinde H.L.: C = 1 Conformal field theories on Riemann surfaces. Commun. Math. Phys. 115, 649 (1988)
Bouchard V., Florea B., Marino M.: Topological open string amplitudes on orientifolds. JHEP 0502, 002 (2005) arXiv:hep-th/0411227
Krefl D., Pasquetti S., Walcher J.: The real topological vertex at work. Nucl. Phys. B 833, 153 (2010) arXiv:0909.1324[hep-th]
Aganagic M., Klemm A., Marino M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) arXiv:hep-th/0305132
Flume R., Poghossian R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A 18, 2541 (2003) arXiv: hep-th/0208176
Bruzzo U., Fucito F., Morales J.F., Tanzini A.: Multi-instanton calculus and equivariant cohomology. JHEP 0305, 054 (2003) arXiv:hep-th/0211108
Nakajima, H., Yoshioka, K.: Lectures on instanton counting. arXiv:math/0311058
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krefl, D., Walcher, J. Extended Holomorphic Anomaly in Gauge Theory. Lett Math Phys 95, 67–88 (2011). https://doi.org/10.1007/s11005-010-0432-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0432-2