Communications in Mathematical Physics

, Volume 273, Issue 2, pp 317–355 | Cite as

Large N Expansion of q-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras

  • Sebastian de Haro
  • Sanjaye Ramgoolam
  • Alessandro Torrielli


We derive the q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU(N) Yang-Mills theory. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl duality of quantum groups. The appearance of Euler characters of configuration spaces of Riemann surfaces in the expansion persists. We discuss the geometrical meaning of these formulae.


Partition Function Modulus Space Riemann Surface Wilson Loop Conjugacy Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Migdal, A.A.: Recursion Equations In Gauge Field Theories. Sov. Phys. JETP 42, 413 (1975) [Zh. Eksp. Teor. Fiz. 69, 810 (1975)]Google Scholar
  2. 2.
    Gross D.J. (1993). Two-dimensional QCD as a string theory. Nucl. Phys. B 400: 161 MATHCrossRefADSGoogle Scholar
  3. 3.
    Gross D.J. and Taylor W.I. (1993). Two-dimensional QCD is a string theory. Nucl. Phys. B 400: 181 MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Gross D.J. and Taylor W.I. (1993). Twists and Wilson loops in the string theory of two-dimensional QCD. Nucl. Phys. B 403: 395 MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Cordes S., Moore G.W. and Ramgoolam S. (1995). Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories. Nucl. Phys. Proc. Suppl. 41: 184 MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Cordes S., Moore G.W. and Ramgoolam S. (1997). Large N 2-D Yang-Mills theory and topological string theory. Commun. Math. Phys 185: 543 MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Horava P.: Topological strings and QCD in two-dimensions. http://, 1993Google Scholar
  8. 8.
    Vafa C.: Two dimensional Yang-Mills, black holes and topological strings., 2004Google Scholar
  9. 9.
    Bryan J., Pandharipande R.: The local Gromov-Witten theory of curves. 0411037, 2004Google Scholar
  10. 10.
    Aganagic M., Ooguri H., Saulina N. and Vafa C. (2005). Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys. B 715: 304 MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Haro S. (2006). A note on knot invariants and q-deformed 2d Yang-Mills. Phys. Lett. B 634: 78 CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Boulatov D.V. (1993). q deformed lattice gauge theory and three manifold invariants. Int. J. Mod. Phys. A 8: 3139 MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Buffenoir E. and Roche P. (1995). Two-dimensional lattice gauge theory based on a quantum group. Commun. Math. Phys. 170: 669 MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Klimcik C. (2001). The formulae of Kontsevich and Verlinde from the perspective of the Drinfeld double. Commun. Math. Phys. 217: 203 MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Jimbo M. (1986). A q-analog of U(gl(N + 1)), Hecke algebras and the Yang-Baxter equation. Lett. Math. Phys. 11: 247–252 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    King R.C. and Wybourne B.G. (1992). Representations and traces of the Hecke algebras H n(q) of type A n-1. J. Math. Phys. 33(1): 4 MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Faddeev L.D., Reshetikhin N.Y. and Takhtajan L.A. (1990). Quantization Of Lie Groups And Lie Algebras. Lengingrad Math. J. 1: 193 MATHMathSciNetGoogle Scholar
  18. 18.
    Majid S. (1995). Foundations of quantum group theory. Cambridge Univ. Press, Cambridge MATHGoogle Scholar
  19. 19.
    Coquereaux, R., Schieber, G.E.: Action of a finite quantum group on the algebra of complex N ×  N matrices. AIP Conf. Proc. 453, Melville, NY: Amer. Inst. of Physics, 1998 pp. 9–23Google Scholar
  20. 20.
    Ram A. (1991). A Frobenius formula for characters of the Hecke algebra. Invent. Math. 106: 461–488 MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Gyoja A. (1986). A q-analogue of Young Symmetrizer. Osaka J. Math. 23: 841–852 MATHMathSciNetGoogle Scholar
  22. 22.
    Francis A. (1999). The Minimal Basis for the Centre of an Iwahori-Hecke Algebra. J. Algebra 221: 1–28 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Francis A. and Jones L. (2005). On bases of centres of Iwahori-Hecke algebras of the symmetric group. J. Algebra 289(1): 42–69 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ramgoolam S. (1996). Wilson loops in 2-D Yang-Mills: Euler characters and loop equations. Int. J. Mod. Phys. A 11: 3885 MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Dijkgraaf R., Vafa C., Verlinde E.P. and Verlinde H.L. (1989). The Operator Algebra Of Orbifold Models. Commun. Math. Phys. 123: 485 MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Dijkgraaf R. and Witten E. (1990). Topological Gauge Theories And Group Cohomology. Commun. Math. Phys. 129: 393 MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Freed D.S. and Quinn F. (1993). Chern-Simons theory with finite gauge group. Commun. Math. Phys. 156: 435 MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Haro S. (2005). Chern-Simons theory, 2d Yang-Mills and lie algebra wanderers. Nucl. Phys.B 730: 312 MATHCrossRefADSGoogle Scholar
  29. 29.
    Dijkgraaf R. and Moore G.W. (1997). Balanced topological field theories. Commun. Math. Phys. 185: 411 MATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Caporaso N., Cirafici M., Griguolo L., Pasquetti S., Seminara D. and Szabo R.J. (2006). Topological strings and large N phase transitions. I: Nonchiral expansion of q-deformed Yang-Mills theory. JHEP 0601: 035 CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Caporaso N., Cirafici M., Griguolo L., Pasquetti S., Seminara D. and Szabo R.J. (2006). Topological strings and large N phase transitions. II: Chiral expansion of q-deformed Yang-Mills theory. JHEP 0601: 036 CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Brzezinski T., Dabrowski H. and Rembielinski J. (1992). On the quantum differential calculus and the quantum holomorphicity. J. Math. Phys. 33(1): 19–24 MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Krieg A. Hecke Algebras. Memoirs of the American Mathematical Society, 87, 435, Providence, RI: Amer. Math. Soc, 1990Google Scholar
  34. 34.
    Fulton, W.: Hurwitz schemes and irreducibility of the moduli spaces of algebraic curves. Ann. of Math. (2) 90, 542–575 (1969)Google Scholar
  35. 35.
    Haro S. (2004). Chern-Simons theory in lens spaces from 2d Yang-Mills on the cylinder. JHEP 0408: 041 CrossRefGoogle Scholar
  36. 36.
    Beasley C. and Witten E. (2005). Non-abelian localization for Chern-Simons theory. J. Diff. Geom. 70: 183–323 MATHMathSciNetGoogle Scholar
  37. 37.
    Blau, M., Thompson, G.: Chern-Simons theory on S**1-bundles: Abelianisation and q-deformed Yang-Mills theory. JHEP 0605003 , 183 (2006)Google Scholar
  38. 38.
    Chung S.W., Fukuma M. and Shapere A.D. (1994). Structure of topological lattice field theories in three-dimensions. Int. J. Mod. Phys. A 9: 1305 MATHCrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Fukuma M., Hosono S. and Kawai H. (1994). Lattice topological field theory in two-dimensions. Commun. Math. Phys. 161: 157 MATHCrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Martin P.M.: On Schur-Weyl duality, A n Hecke algebras and quantum sl(N) on \(\bigotimes^{n+1}C^N\) . Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys. Vol. 16 River Edge. NJ: World Sci. Publ., 1992, pp. 645–673Google Scholar
  41. 41.
    Beĭlinson, A.A., Lusztig, G., MacPherson, R.D.: Duke Math. J. 61(2), 655–677 (1990)CrossRefMathSciNetGoogle Scholar
  42. 42.
    LeClair A., Ludwig A. and Mussardo G. (1998). Integrability of coupled conformal field theories. Nucl. Phys. B 512: 523 MATHCrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Katriel J., Abdelassam B. and Chakrabarti A. (1995). The fundamental invariant of the Hecke algebra H n(q) characterizes the representations of H n(q), S n, SU q(N) and SU(N). J. Math. Phys. 36: 5139–5158 MATHCrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Dipper R. and James G.D. (1987). Blocks and idempotents of Hecke algebras of general linear groups. Proc. Lon. Math. Soc. 3(54): 57 CrossRefMathSciNetGoogle Scholar
  45. 45.
    Ogievetsky O., Pyatov P.: Lecture on Hecke algebra. Based on lectures at the International School “Symmetries and Integrable systems”. (Dubna, 8–11 June, 1999). Dubna: JINR Publ. Dept., 2000Google Scholar
  46. 46.
    Nomura M. (1990). Representation functions \(d^j_{mk}\) of U[sl q(2) ] as wavefunctions of “Quantum symmetric tops” and Relationship to Braiding matrices J. Phys. Soc. Japan 59(12): 4260–4271CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Kirillov A.N. and Reshetikhin N.Yu. (1989). Representations of the algebra U q(sl(2)), q orthogonal polynomials and invariants of links. Adv. Series in Math. Phys. 7: 285–339 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Sebastian de Haro
    • 1
  • Sanjaye Ramgoolam
    • 2
  • Alessandro Torrielli
    • 3
  1. 1.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutGolmGermany
  2. 2.Department of Physics, Queen MaryUniversity of LondonLondonUK
  3. 3.Institut für PhysikHumboldt Universität zu BerlinBerlinGermany

Personalised recommendations