Skip to main content
SpringerLink
Log in

Gauge theories as matrix models

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.

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

Access this article

Log in via an institution

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. N. Seiberg and E. Witten, Nucl. Phys. B, 426, 19–52 (1994); Erratum, 430, 485–486 (1994); arXiv:hep-th/9407087v1 (1994); 431, 484–550 (1994); arXiv:hep-th/9408099v1 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. P. Argyres, M. Plesser, and N. Seiberg, Nucl. Phys. B, 471, 159–194 (1996); arXiv:hep-th/9603042v1 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. A. Marshakov and A. Yung, Nucl. Phys. B, 831, 72–104 (2010); arXiv:0912.1366v2 [hep-th] (2009); A. V. Marshakov, Theor. Math. Phys., 165, 1650–1661 (2010).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. N. Nekrasov, Adv. Theor. Math. Phys., 7, 831–864 (2004); arXiv:hep-th/0206161v1 (2002).

    MathSciNet  Google Scholar 

  5. A. S. Losev, A. Marshakov, and N. Nekrasov, “Small instantons, little strings, and free fermions,” in: From Fields to Strings: Circumnavigating Theoretical Physics (M. Shifman et al., eds.), World Scientific, River Edge, N. J. (2005), p. 581–621; arXiv:hep-th/0302191v3 (2003).

    Google Scholar 

  6. N. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions,” in: The Unity of Mathematics (Progr. Math., Vol. 244), Birkhäuser, Boston, Mass. (2006), pp. 525–596; arXiv:hep-th/0306238v2 (2003).

    Chapter  Google Scholar 

  7. A. Marshakov and N. Nekrasov, JHEP, 0701, 104 (2007); arXiv:hep-th/0612019v2 (2006); A. V. Marshakov, Theor. Math. Phys., 154, 362–384 (2008); arXiv:0706.2857v2 [hep-th] (2007).

    Article  ADS  MathSciNet  Google Scholar 

  8. A. Marshakov, JHEP, 0803, 055 (2008); arXiv:0712.2802v3 [hep-th] (2007).

    Article  ADS  MathSciNet  Google Scholar 

  9. T. Nakatsu and K. Takasaki, Commun. Math. Phys., 285, 445–468 (2009); arXiv:0710.5339v2 [hep-th] (2007); T. Nakatsu, Y. Noma, and K. Takasaki, Internat. J. Mod. Phys. A, 23, 2332–2342 (2008); arXiv:0806.3675v1 [hep-th] (2008); Nucl. Phys. B, 808, 411–440 (2009); arXiv:0807.0746v2 [hep-th] (2008).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. A. Braverman, “Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors,” arXiv:math/0401409v2 (2004); A. Braverman and P. Etingof, “Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential,” arXiv:math/0409441v3 (2004).

  11. R. Flume and R. Pogossian, Internat. J. Mod. Phys. A, 18, 2541–2563 (2003); arXiv:hep-th/0208176v2 (2002); H. Nakajima and K. Yoshioka, “Instanton counting on blowup I: 4-Dimensional pure gauge theory,” arXiv:math/0306198v2 (2003); “Lectures on instanton counting,” arXiv:math/0311058v1 (2003); U. Bruzzo and F. Fucito, Nucl. Phys. B, 678, 638–655 (2004); arXiv:math-ph/0310036v1 (2003); F. Fucito, J. Morales, and R. Poghossian, JHEP, 0410, 037 (2004); arXiv:hep-th/0408090v2 (2004); S. Shadchin, SIGMA, 0602, 008 (2006); arXiv:hep-th/0601167v3 (2006).

    Article  MATH  ADS  Google Scholar 

  12. L. Alday, D. Gaiotto, and Y. Tachikawa, Lett. Math. Phys., 91, 167–197 (2010); arXiv:0906.3219v2 [hep-th] (2009).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. N. Wyllard, JHEP, 0911, 002 (2009); arXiv:0907.2189v2 [hep-th] (2009); A. Mironov and A. Morozov, Nucl. Phys. B, 825, 1-37 (2009); arXiv:0908.2569v2 [hep-th] (2009); Phys. Lett. B, 682, 118-124 (2009); arXiv:0909.3531v1 [hep-th] (2009); N. Drukker, J. Gomis, T. Okuda, and J. Teschner, JHEP, 1002, 057 (2010); arXiv:0909.1105v3 [hep-th] (2009); G. Bonelli and A. Tanzini, Phys. Lett. B, 691, 111-115 (2010); arXiv:0909.4031v3 [hep-th] (2009); L. Alday, F. Benini, and Y. Tachikawa, Phys. Rev. Lett., 105, 141601 (2010); arXiv:0909.4776v1 [hep-th] (2009); R. Poghossian, JHEP, 0912, 038 (2009); arXiv:0909.3412v2 [hepth] (2009); L. Hadasz, Z. Jaskólski, and P. Suchanek, JHEP, 1001, 063 (2010); “Proving the AGT relation for N f = 0, 1, 2 antifundamentals,” arXiv:1004.1841v2 [hep-th] (2010); V. Alba and A. Morozov, Nucl. Phys. B, 840, 441-468 (2010); arXiv:0912.2535v2 [hep-th] (2009); M. Taki, JHEP, 1005, 038 (2010); arXiv:0912.4789v1 [hep-th] (2009); JHEP, 1107, 047 (2011); arXiv:1007.2524v2 [hep-th] (2010); S. Yanagida, J. Algebra, 333, 273-294 (2011); arXiv:1003.1049v2 [math.QA] (2010); “Norms of logarithmic primaries of Virasoro algebra,” arXiv:1010.0528v2 [math.QA] (2010); N. Drukker, D. Gaiotto, and J. Gomis, “The virtue of defects in 4D gauge theories and 2D CFTs,” arXiv:1003.1112v2 [hep-th] (2010); H. Itoyama and T. Oota, Nucl. Phys. B, 838, 298–330 (2010); arXiv:1003.2929v2 [hep-th] (2010); C. Kozçaz, S. Pasquetti, F. Passerini, and N. Wyllard, JHEP, 1101, 045 (2011); arXiv:1008.1412v2 [hep-th] (2010); H. Itoyama, T. Oota, and N. Yonezawa, Phys. Rev. D, 82, 085031 (2010); arXiv:1008.1861v2 [hep-th] (2010); K. Maruyoshi and F. Yagi, JHEP, 1101, 042 (2011); arXiv:1009.5553v3 [hep-th] (2010); G. Bonelli, K. Maruyoshi, A. Tanzini, and F. Yagi, JHEP, 1107, 055 (2011); arXiv:1011.5417v2 [hep-th] (2010); A. Mironov, A. Morozov, and Sh. Shakirov, J. Phys. A, 44, 085401 (2011); arXiv:1010.1734v1 [hep-th] (2010); JHEP, 1103, 102 (2011); arXiv:1011.3481v2 [hep-th] (2010).

    Article  ADS  MathSciNet  Google Scholar 

  14. B. Eynard, N. Orantin, and M. Mariño, JHEP, 0706, 058 (2007); arXiv:hep-th/0702110v3 (2007); B. Eynard, J. Stat. Mech. Theory Exp., 10, 10011 (2009); arXiv:0905.0535v3 [math-ph] (2009); L. Chekhov, B. Eynard, and O. Marchal, Theor. Math. Phys., 166, 141-185 (2011); arXiv:1009.6007v1 [math-ph] (2010).

    Article  ADS  Google Scholar 

  15. A. Mironov, A. Morozov, and Sh. Shakirov, JHEP, 1002, 030 (2010); arXiv:0911.5721v2 [hep-th] (2009); “Towards a proof of AGT conjecture by methods of matrix models,” arXiv:1011.5629v1 [hep-th] (2010).

    Article  ADS  MathSciNet  Google Scholar 

  16. T. Eguchi and S.-K. Yang, Modern Phys. Lett. A, 9, 2893–2902 (1994); arXiv:hep-th/9407134v1 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. A. Klemm and P. Sulkowski, Nucl. Phys. B, 819, 400–430 (2009); arXiv:0810.4944v2 [hep-th] (2008).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. R. Dijkgraaf and C. Vafa, “Toda theories, matrix models, topological strings, and N=2 gauge systems,” arXiv:0909.2453v1 [hep-th] (2009).

  19. A. Marshakov, “Strings, integrable systems, geometry, and statistical models,” in: Lie Theory and Its Applications in Physics V (H.-D. Doebner and V. K. Dobrev, eds.), World Scientific, River Edge, N. J. (2004), pp. 231–240; arXiv:hep-th/0401199v2 (2004).

    Google Scholar 

  20. A. Migdal, Phys. Rept., 102, 199–290 (1983).

    Article  ADS  Google Scholar 

  21. R. Dijkgraaf and C. Vafa, Nucl. Phys. B, 644, 3–20, 21–39 (2002); arXiv:hep-th/0206255v2 (2002); arXiv:hepth/0207106v1 (2002); “A perturbative window into non-perturbative physics,” arXiv:hep-th/0208048v1 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. L. Chekhov and A. Mironov, Phys. Lett. B, 552, 293–302 (2003); arXiv:hep-th/0209085v2 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. L. Chekhov, A. Marshakov, A. Mironov, and D. Vasiliev, Phys. Lett. B, 562, 323–338 (2003); arXiv:hep-th/0301071v1 (2003); L. O. Chekhov, A. V. Marshakov, A. D. Mironov, and D. Vasiliev, Proc. Steklov Inst. Math., 251, 254-292 (2005); arXiv:hep-th/0506075v1 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. V. Kazakov and A. Marshakov, J. Phys. A, 36, 3107–3136 (2003); arXiv:hep-th/0211236v4 (2002).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. A. V. Marshakov, Theor. Math. Phys., 147, 583–636, 777-820 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, Nucl. Phys. B, 357, 565–618 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  27. A. Hanany and Y. Oz, Nucl. Phys. B, 452, 283–312 (1995); arXiv:hep-th/9505075v1 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 380, 75–80 (1996); arXiv:hep-th/9603140v1 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  29. A. Marshakov, A. Mironov, and A. Morozov, JHEP, 0911, 048 (2009); arXiv:0909.3338v3 [hep-th] (2009).

    Article  ADS  MathSciNet  Google Scholar 

  30. D. Gaiotto, “N=2 dualities,” arXiv:0904.2715v1 [hep-th] (2009).

  31. T. Eguchi and K. Maruyoshi, JHEP, 1002, 022 (2010); arXiv:0911.4797v3 [hep-th] (2009); 1007, 081 (2010); arXiv:1006.0828v2 [hep-th] (2010).

    Article  ADS  MathSciNet  Google Scholar 

  32. A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B, 477, 577–605 (1996); arXiv:hep-th/9506136v2 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. D. Gaiotto, “Asymptotically free N=2 theories and irregular conformal blocks,” arXiv:0908.0307v1 [hep-th] (2009).

  34. A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B, 682, 125–129 (2009); arXiv:0909.2052v1 [hep-th] (2009).

    Article  ADS  MathSciNet  Google Scholar 

  35. E. Barnes, Proc. London Math. Soc., 31, 358–381 (1899); Phil. Trans. Roy. Soc. London A, 196, 265–387 (1901); Trans. Cambridge Phil. Soc., 19, 374–425 (1904).

    Article  MATH  Google Scholar 

  36. A. Gerasimov, A. Levin, and A. Marshakov, Nucl. Phys. B, 360, 537–558 (1991); A. Bilal, V. V. Fock, and I. Kogan, Nucl. Phys. B, 359, 635–672 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  37. I. M. Krichever, Commun. Pure. Appl. Math., 47, 437–475 (1994); arXiv:hep-th/9205110v1 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  38. I. Krichever, A. Marshakov, and A. Zabrodin, Commun. Math. Phys., 259, 1–44 (2005); arXiv:hep-th/0309010v3 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. L. Faddeev and R. Kashaev, Modern Phys. Lett. A, 9, 427–434 (1994); arXiv:hep-th/9310070v1 (1993); R. Kashaev, “Quantization of Teichmuller spaces and the quantum dilogarithm,” arXiv:q-alg/9705021v1 (1997); L. O. Chekhov and V. V. Fock, Theor. Math. Phys., 120, 1245–1259 (1999); arXiv:math/9908165v2 (1999); V. V. Fock and A. Goncharov, “Moduli spaces of local systems and higher Teichmüller theory,” arXiv:math/0311149v4 (2003); J. Teschner, Internat. J. Mod. Phys. A, 19, 459–477 (2004); arXiv:hep-th/0303149v2 (2003); “Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I,” arXiv:1005.2846v4 [hep-th] (2010).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Al. Zamolodchikov, Internat. J. Mod. Phys. A, 19, 510–523 (2004); arXiv:hep-th/0312279v1 (2003); “On the three-point function in minimal Liouville gravity,” arXiv:hep-th/0505063v1 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, Commun. Math. Phys., 225, 573–609 (2002); arXiv:hep-th/0102180v2 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. N. A. Nekrasov and S. L. Shatashvili, Prog. Theoret. Phys. (Suppl.), 177, 105–119 (2009); arXiv:0901.4748v2 [hep-th] (2009).

    Article  MATH  ADS  Google Scholar 

  43. K. Kozlowski and J. Teschner, “TBA for the Toda chain,” arXiv:1006.2906v1 [math-ph] (2010).

  44. M. Spreafico, J. Number Theory, 129, 2035–2063 (2009).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tamm Theory Department, Lebedev Physical Institute, RAS, Moscow, Russia

    A. V. Marshakov

  2. Institute of Theoretical and Experimental Physics, Moscow, Russia

    A. V. Marshakov

Authors
  1. A. V. Marshakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Marshakov.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 169, No. 3, pp. 391–412, December, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marshakov, A.V. Gauge theories as matrix models. Theor Math Phys 169, 1704–1723 (2011). https://doi.org/10.1007/s11232-011-0146-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0146-3

Keywords

Search

Navigation