Abstract
We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from the viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.
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G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
A.A. Gerasimov and S.L. Shatashvili, Higgs Bundles, Gauge Theories and Quantum Groups, Commun. Math. Phys. 277 (2008) 323 [hep-th/0609024] [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW model and Bethe Ansatz for the phase model, JHEP 11 (2012) 146 [arXiv:1209.3800] [INSPIRE].
N.M. Bogoliubov, R.K. Bullough and G.D. Pang, Exact solution of a q-boson hopping model, Phys. Lett. B 47 (1993) 11495.
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nucl. Phys. B 408 (1993) 345 [hep-th/9305010] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.M. Bogoliubov, A.G. Izergin and N.A. Kitanine, Correlation functions for a strongly correlated boson system, Nucl. Phys. B 516 (1998) 501 [solv-int/9710002].
M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1989) 175 [INSPIRE].
G. Segal, The definition of conformal field theory, in proceedings of Symposium on Topology, Geometry and Quantum Field Theory (Segalfest), 24-29 Jun 2002, Oxford, England, United Kingdom [INSPIRE].
R. Dijkgraaf, Les Houches lectures on fields, strings and duality, hep-th/9703136 [INSPIRE].
R. Dijkgraaf, A Geometrical Approach to Two-Dimensional Conformal Field Theory, Ph.D. Thesis, Utrecht (1989).
C. Korff, Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra, Commun. Math. Phys. 318 (2013) 173 [arXiv:1110.6356].
C. Korff and C. Stroppel, The sl(n)-WZNW Fusion Ring: a combinatorial construction and a realisation as quotient of quantum cohomology Adv. Math. 225 (2010) 200 [arXiv:0909.2347].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge (1993).
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].
N.A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz, Theor. Math. Phys. 79 (1989) 502.
N.A. Slavnov, The algebraic Bethe ansatz and quantum integrable systems, Russian Math. Surv. 62 (2007) 727.
T. Deguchi and C. Matsui, Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry, Nucl. Phys. B 814 (2009) 405 [Erratum ibid. B 851 (2011) 238-243] [arXiv:0807.1847] [INSPIRE].
T. Deguchi and C. Matsui, Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method, Nucl. Phys. B 831 (2010) 359 [arXiv:0907.0582] [INSPIRE].
P.P. Kulish, Contraction of quantum algebras and q-oscillators, Theor. Math. Phys. 86 (1991) 108 [INSPIRE].
A. Gerasimov, Localization in GWZW and Verlinde formula, hep-th/9305090 [INSPIRE].
A. Miyake, K. Ohta and N. Sakai, Volume of Moduli Space of Vortex Equations and Localization, Prog. Theor. Phys. 126 (2011) 637 [arXiv:1105.2087] [INSPIRE].
N.S. Manton and S.M. Nasir, Volume of vortex moduli spaces, Commun. Math. Phys. 199 (1999) 591 [hep-th/9807017] [INSPIRE].
E. Witten, On Holomorphic factorization of WZW and coset models, Commun. Math. Phys. 144 (1992) 189 [INSPIRE].
J. Kock, Frobenius algebras and 2D Topological Quantum Field Theories, Cambridge, Cambridge University Press (1985).
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press (1979).
K. Ohta and Y. Yoshida, Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold, Phys. Rev. D 86 (2012) 105018 [arXiv:1205.0046] [INSPIRE].
E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, hep-th/9312104 [INSPIRE].
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].
E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].
N.J. Hitchin, The Selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
C. Teleman and C.T. Woodward, The Index Formula on the Moduli of G-bundles, Ann. Math. 170 (2009) 495 [math.AG/0312154].
C. Teleman, Loop Groups and G-bundles on curves, http://math.berkeley.edu/~teleman/lectures.html
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Okuda, S., Yoshida, Y. G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra. J. High Energ. Phys. 2014, 3 (2014). https://doi.org/10.1007/JHEP03(2014)003
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DOI: https://doi.org/10.1007/JHEP03(2014)003