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Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems
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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.
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Exact solutions by integrals of the non-stationary elliptic Calogero-Sutherland equation
Integral representations are constructed for solutions of the non-stationary elliptic Calogero-Sutherland equation, yielding elliptic analogues of Jack polynomials for non-integer coupling values.
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