SpringerLink
Log in

Extended Holomorphic Anomaly in Gauge Theory

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) ar**v:hep-th/9408099

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. 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). ar**v:hep-th/9407087

    Google Scholar 

  3. Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B 534, 549 (1998) ar**v:hep-th/9711108

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97 (2000) ar**v:hep-th/9712241

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Losev, A., Nekrasov, N., Shatashvili, S.L.: Testing Seiberg-Witten solution. ar**v:hep-th/9801061

  6. Moore G.W., Nekrasov N., Shatashvili S.: D-particle bound states and generalized instantons. Commun. Math. Phys. 209, 77 (2000) ar**v:hep-th/9803265

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831 (2004) ar**v:hep-th/0206161

    MathSciNet  Google Scholar 

  8. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. ar**v:hep-th/0306238

  9. Nakajima, H., Yoshioka, K.: Instanton counting on blowup. Part I: 4-dimensional pure gauge theory. ar**v:math.ag/0306198

  10. Moore G.W., Witten E.: Integration over the u-plane in Donaldson theory. Adv. Theor. Math. Phys. 1, 298 (1998) ar**v:hep-th/9709193

    MathSciNet  Google Scholar 

  11. Antoniadis I., Gava E., Narain K.S., Taylor T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994) ar**v:hep-th/9307158

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. 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) ar**v:hep-th/9309140

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. 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) ar**v:hep-th/9604034

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Katz S.H., Klemm A., Vafa C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997) ar**v:hep-th/9609239

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Klemm A., Marino M., Theisen S.: Gravitational corrections in supersymmetric gauge theory and matrix models. JHEP 0303, 051 (2003) ar**v:hep-th/0211216

    Article  MathSciNet  ADS  Google Scholar 

  16. Iqbal A., Kashani-Poor A.K.: Instanton counting and Chern-Simons theory. Adv. Theor. Math. Phys. 7, 457 (2004) ar**v:hep-th/0212279

    MathSciNet  Google Scholar 

  17. Eguchi T., Kanno H.: Topological strings and Nekrasov’s formulas. JHEP 0312, 006 (2003) ar**v:hep-th/0310235

    Article  MathSciNet  ADS  Google Scholar 

  18. Huang M.X., Klemm A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 0709, 054 (2007) ar**v:hep-th/0605195

    Article  MathSciNet  ADS  Google Scholar 

  19. Huang, M.X., Klemm, A.: Holomorphicity and modularity in Seiberg-Witten theories with matter. ar**v:0902.1325[hep-th]

  20. Ghoshal D., Vafa C.: C = 1 String as the topological theory of the conifold. Nucl. Phys. B 453, 121 (1995) ar**v:hep-th/9506122

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Antoniadis, I., Hohenegger, S., Narain, K.S., Taylor, T.R.: Deformed topological partition function and Nekrasov backgrounds. ar**v:1003.2832[hep-th]

  22. Walcher J.: Extended holomorphic anomaly and loop amplitudes in open topological string. Nucl. Phys. B 817, 167 (2009) ar**v:0705.4098[hep-th]

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Bouchard V., Florea B., Marino M.: Counting higher genus curves with crosscaps in Calabi-Yau orientifolds. JHEP 0412, 035 (2004) ar**v:hep-th/0405083

    Article  MathSciNet  ADS  Google Scholar 

  24. Walcher J.: Evidence for tadpole cancellation in the topological string. Comm. Number Theor. Phys. 3, 111–172 (2009) ar**v:0712.2775[hep-th]

    MATH  MathSciNet  Google Scholar 

  25. Krefl, D., Walcher, J.: The real topological string on a local Calabi-Yau. ar**v:0902. 0616[hep-th]

  26. Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167 (2010) ar**v:0906.3219 [hep-th]

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Dijkgraaf, R., Vafa, C.: Toda theories, matrix models, topological strings, and N = 2 gauge systems. ar**v:0909.2453[hep-th]

  28. Yamaguchi S., Yau S.T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004) ar**v:hep-th/0406078

    Article  MathSciNet  ADS  Google Scholar 

  29. Alim M., Lange J.D.: Polynomial structure of the (open) topological string partition function. JHEP 0710, 045 (2007) ar**v:0708.2886[hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  30. Konishi, Y., Minabe, S.: On solutions to Walcher’s extended holomorphic anomaly equation. ar**v:0708.2898[math.AG]

  31. Grimm T.W., Klemm A., Marino M., Weiss M.: Direct integration of the topological string. JHEP 0708, 058 (2007) ar**v:hep-th/0702187

    Article  MathSciNet  ADS  Google Scholar 

  32. Gopakumar, R., Vafa, C.: M-theory and topological strings. Part I, II. ar**v:hep-th/9809187, ar**v:hep-th/9812127

  33. Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  34. Hollowood T.J., Iqbal A., Vafa C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008) ar**v:hep-th/0310272

    Article  MathSciNet  ADS  Google Scholar 

  35. Iqbal A., Kozcaz C., Vafa C.: The refined topological vertex. JHEP 0910, 069 (2009) ar**v:hep-th/0701156

    Article  MathSciNet  ADS  Google Scholar 

  36. Morrison D.R., Walcher J.: D-branes and Normal Functions. Adv. Theor. Math. Phys. 13, 553–598 (2009) ar**v:0709.4028[hep-th]

    MATH  MathSciNet  Google Scholar 

  37. 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) ar**v:0909.0945[hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  38. Kozcaz, C., Pasquetti, S., Wyllard, N.: A & B model approaches to surface operators and Toda theories. ar**v:1004.2025[hep-th]

  39. Dimofte, T., Gukov, S., Hollands, L.: Vortex counting and Lagrangian 3-manifolds. ar**v:1006.0977[hep-th]

  40. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. ar**v:0908.4052[hep-th]

  41. Dijkgraaf R., Verlinde E.P., Verlinde H.L.: C = 1 Conformal field theories on Riemann surfaces. Commun. Math. Phys. 115, 649 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. Bouchard V., Florea B., Marino M.: Topological open string amplitudes on orientifolds. JHEP 0502, 002 (2005) ar**v:hep-th/0411227

    Article  MathSciNet  ADS  Google Scholar 

  43. Krefl D., Pasquetti S., Walcher J.: The real topological vertex at work. Nucl. Phys. B 833, 153 (2010) ar**v:0909.1324[hep-th]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Aganagic M., Klemm A., Marino M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) ar**v:hep-th/0305132

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. 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) ar**v: hep-th/0208176

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. Bruzzo U., Fucito F., Morales J.F., Tanzini A.: Multi-instanton calculus and equivariant cohomology. JHEP 0305, 054 (2003) ar**v:hep-th/0211108

    Article  MathSciNet  ADS  Google Scholar 

  47. Nakajima, H., Yoshioka, K.: Lectures on instanton counting. ar**v:math/0311058

Download references

Author information

Authors and Affiliations

  1. IPMU, The University of Tokyo, Kashiwa, Japan

    Daniel Krefl

  2. PH-TH Division, CERN, Geneva, Switzerland

    Johannes Walcher

Authors
  1. Daniel Krefl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johannes Walcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johannes Walcher.

Rights and permissions

Reprints 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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-010-0432-2

Mathematics Subject Classification (2000)

Keywords

Search

Navigation