Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems

Abstract

We work out finite-dimensional integral formulae for the scalar product of genus one states of the group $G$ Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system.

24 pages, latex file

References in corpus (10)

• Supersymmetric Yang-Mills Systems And Integrable Systems
• Gaudin Model, Bethe Ansatz and Critical Level
• Quasiclassical asymptotics of solutions to the KZ equations
• Off-Shell Bethe Ansatz Equation and N-point Correlators in the SU(2) WZNW Theory
• Chern-Simons States at Genus One
• Hitchin systems, higher Gaudin operators and $r$-matrices
• SU(2) WZW Theory at Higher Genera
• Coulomb Gas Representation of the SU(2) WZW Correlators at Higher Genera
• Elliptic Wess-Zumino-Witten Model from Elliptic Chern-Simons Theory
• Spherical functions on affine Lie groups

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• Explicit solution of the (quantum) elliptic Calogero-Sutherland model
• Self-duality of the SL_2 Hitchin integrable system at genus two
• Series Solutions of the Non-Stationary Heun Equation
• Exact solutions by integrals of the non-stationary elliptic Calogero-Sutherland equation
• Relaxation of nonlinear oscillations in BCS superconductivity
• Coherent oscillations in superconducting cold Fermi atoms and their applications