Abstract
The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.
Similar content being viewed by others
References
Moore G., Nekrasov N. and Shatashvili S. (2000). Integrating over Higgs Branches. Commun. Math. Phys. 209: 97
Hitchin N.J. (1987). The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55: 59–126
Gaudin M. (1983). La Fonction d’Onde de Bethe. Masson, Paris
Bogoliubov N., Izergin A. and Korepin V. (1993). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge
Faddeev, L.D., Sklyanin, E.K., Takhtajan, L.A.: The Quantum Inverse Problem Method. 1. Theor. Math. Phys. 40, 688–706 (1980); Teor. Mat. Fiz. 40, 194–220 (1979)
Gerasimov A., Kharchev S., Lebedev D. and Oblezin S. (2005). On a class of representations of the Yangian and moduli space of monopoles. Commun. Math. Phys. 260: 511
Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Class of Representations of Quantum Groups. http://arxiv.org/math/0501473, 2005
Neumann W.D. and Zagier D. (1985). Volumes of hyperbolic 3-manifolds. Topology 24: 307–332
Nahm, W.: Conformal field theory, dilogarithms, and three dimensional manifolds. In: Interface between physics and mathematics, Proceedings, Hangzhou 1993, W. Nahm, J.M. Shen, eds., Singapore: World Scientific, 1994
Gliozzi F. and Tateo R. (1996). Thermodynamic Bethe Ansatz and Threefold Triangulations. Int. J. of Mod. Phys. A 11: 4051
Nahm, W.: Conformal Field Theory and Torsion Elements of the Bloch group. [arXiv:hepth/0404120]. Springer
Migdal, A.: Recursion Equations in Gauge Theories. Zh. Eksp. Teor. Fiz 69, 810 (1975) (Sov. Phys. Jetp. 42, 413 (1975))
Kazakov V. and Kostov I. (1980). Nonlinear Strings in Two-Dimensional U(Infinity) Gauge Theory. Nucl. Phys. B176: 199
Kazakov V. (1981). Wilson Loop Average For An Arbitrary Contour In Two-Dimensional U(N) Gauge Theory. Nucl. Phys. B 179: 283
Rusakov B. (1990). Loop Averages And Partition Functions In U(N) Gauge Theory On Two-Dimensional Manifolds. Mod. Phys. Lett. A5: 693
Witten E. (1991). On quantum gauge theories in two-dimensions. Commun. Math. Phys. 141: 153
Witten E. (1992). Two-dimensional gauge theories revisited. J. Geom. Phys. 9: 303
Gawedzki K. and Kupiainen A. (1988). G/H Conformal Field Theory form Gauged WZW Model. Phys. Lett. B 215: 119
Spiegelglas M. and Yankielowicz S. (1993). G/G Topological Field Theories By Cosetting G(K). Nucl. Phys. B 393: 301
Gerasimov, A.: Localization in GWZW and Verlinde formula. http://arXiv.org/list/hepth/9305090, 1993
Blau M. and Thompson G. (1993). Derivation of the Verlinde formula from Chern-Simons theory and the G/G model. Nucl. Phys. B 408: 345
Varadarajan, V.S.: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Math. 576. Berlin, Heidelberg, New York: Springer-Verlag, 1977
Zelobenko D.P. and Stern A.I. (1983). Predstavlenija grupp Li. Moskow, Nauka
Lieb E.H. and Liniger W. (1963). Exact analysis of an interacting Bose gas I. The general solution and the ground state. Phys. Rev. (2) 130: 1605
Berezin, F.A., Pohil, G.P., Finkelberg, V.M.: Vestnik MGU 1, 1, 21 (1964)
Yang C.N. (1967). Some exact results for the many-body problems in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19: 1312
Yang C.N. and Yang C.P. (1969). Thermodinamics of a one-dimensional system of bosons with repulsive delta-function interaction. J. Math. Phys. 10: 1115
Kulish P.P., Manakov S.V. and Faddeev L.D. (1976). Comparison of the exact quantum and quasiclassical results for the nonlinear Schrödinger equation. Theor. Math. Phys. 28: 615
Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Berlin, Heidelberg, New York: Springer-Verlag, 1980
Faddeev, L.D.: How Algebraic Bethe Ansatz works for integrable model. In: Relativistic gravitation and gravitational radiation, Proceedings of Les Houches School of Physics 1995, Cambridge: Cambridge Univ. Press, 1997
Murakami S. and Wadati M. (1993). Connection between Yangian symmetry and the quantum inverse scattering method. J. Phys. A: Math. Gen. 29: 7903
Hikami K. (1998). Notes on the structure of the -function interacting gas. Intertwining operator in the degenerate affine Hecke algebra. J. Phys. A: Math. Gen 31(4): L85
Heckman G.J. and Opdam E.M. (1997). Yang’s system of particles and Hecke algebras. Ann. Math. 145: 139
Emsiz E., Opdam E.M. and Stokman J.V. (2006). Periodic integrable systems with delta-potentials. Commun. Math. Phys. 269: 191–225
Drinfeld V.G. (1986). Degenerate affine Hecke algebras and Yangians. Funct. Anal. Appl. 20: 58
Lusztig G. (1989). Affine Hecke algebras and their graded version. J. AMS 2(3): 599
Dunkl T.C. (1989). Differential-Difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311(1): 167
Cherednik I. (1991). A unification of Knizhnik-Zamolodchikov and Dunkle operators via affine Hecke algebras. Invent. Math. J. 106: 411
Gerasimov A.A. and Shatashvili S.L. (2000). On Exact tachyon Potential in Open String Field Theory. JHEP 0010: 034
Macdonald I. (1970). Harmonic Analysis on Semi-simple Groups. Actes, Congès Intern. Math. 2: 331
Macdonald I.G. (1979). Symmetric Functions and Hall Polynomials. Oxford University Press, London
Heckman G.J. and Schlichtkrull H. (1994). Harmonic Analysis Functions on Symmetric Spaces. Academic Press, London-New York
Alekseev A.Yu., Faddeev L.D. and Shatashvili S.L. (1988). Quantization of symplectic orbits of compact Lie groups by means of the functional integral. J. Geom. Phys. 5: 391–406
Witten E. (1985). Global gravitational anomalies. Commun. Math. Phys. 100: 197–226
Atiyah M. (1987). The logarithm of the Dedekind η-function. Math. Ann. 278: 335–380
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral Asymmetry and Riemannian Geometry. Math. Proc. Camb. Philos. Soc. 77, 43 (1975), 78, 403 (1975), 79, 71 (1976)
Lusztig G. (1985). Equivariant K-theory and Representations of Hecke Algebras. Proc. Am. Math. Soc. 94: 337
Kazhdan D. and Lusztig G. (1987). Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent. Math. 87: 153
Ginzburg V. and Chriss N. (1997). Representation theory and complex geometry. Birkhauser, Boston
Ginzburg, V.: Geometric methods in representation theory of Hecke algebras and quantum groups. Notes by Vladimir Baranovsky. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Dordrecht: Kluwer Acad. Publ., 1998, p. 14
Kalkman J. (1993). BRST Model for Equivariant Cohomology and Representative for Equivariant Thom Class. Commun. Math. Phys. 153: 477
Freed, D., Hopkins, M., Teleman, C.: Twisted K-theory and loop group representations. http://arXiv.org.list/math/0312155, 2003
Crnkovic, C., Witten, E.: Covariant description of canonical formalism in geometric theories. Newton’s tercentenary volume, edit. S. Hawking, W. Israel, Cambridge: Cambridge University Press, 1987
Narashimhan M.S. and Seshadri C.S. (1965). Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. (2) 82: 540
Ramanathan M.S. (1975). Stable bundles on a compact Riemann surface. Math. Ann. 213: 129
Nahm, W.: The construction of all self-dual multimonopoles by the ADHM method. In: Monopoles in quantum field theory, eds. N. Craigie, Singapore: World Scientific, 1982
Nahm, W.: Self-dual monopoles and calorons. Lect. Notes in Physics. 201, eds. G. Denardo, Berlin, Heidelberg, New York: Springer, 1984 p. 189
Atiyah M.F. (1988). Topological quantum field theory. Publications Mathmatiques de l’IHÉS 68: 175–186
Gorsky A. and Nekrasov N. (1994). Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory. Nucl. Phys. B 414: 213–238
Gross D.J. and Taylor W. (1993). Two Dimensional QCD is a String Theory. Nucl.Phys. B400: 181–210
Cordes S., Moore G. and Ramgoolam S. (1997). Large N 2D Yang-Mills Theory and Topological String Theory. Commun. Math. Phys. 185: 543–619
Cordes S., Moore G. and Ramgoolam S. (1995). Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories. Nucl. Phys. Proc. Suppl. 41: 184–244
Aganagic M., Ooguri H., Saulina N. and Vafa C. (2005). Black Holes, q-Deformed 2d Yang-Mills, Non-perturbative Topological Strings. Nucl. Phys. B 715: 304–348
Smirnov, F.: Quasi-classical Study of Form Factors in Finite Volume. http://arXiv.org/list/hepth/9802132, 1998
Atiyah, M., Bielawski, R.: Nahm’s equations, configuration spaces and flag manifolds. http://arXiv.org/list/math/0110112, 2001
Bump D. (1998). Automorphic Forms and Representations. Cambridge Univ. Press, Cambridge
An Introduction to the Langlands Program. ed. J. Bernstein, S. Gelbart, Basel-Boston: Birkhauser, 2003
Kapustin, A., Witten, E.: Electric-Magnetic Duality And The Geometric Langlands Program. http://arXiv.org/list/hepth/0604151, 2006
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan
Rights and permissions
About this article
Cite this article
Gerasimov, A.A., Shatashvili, S.L. Higgs Bundles, Gauge Theories and Quantum Groups. Commun. Math. Phys. 277, 323–367 (2008). https://doi.org/10.1007/s00220-007-0369-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0369-1