Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
Apr, 2006Citations per year
Abstract:
The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the ``quantum spectral curve'' and argue that it takes the analogous structural and unifying role on the quantum level also. In the simplest, but essential case the ``quantum spectral curve'' is given by the formula det(L(z)-dz) [Talalaev04] (hep-th/0404153). As an easy application of our constructions we obtain the following: quite a universal receipt to define quantum commuting hamiltonians from the classical ones, in particular an explicit description of a maximal commutative subalgebra in U(gl(n)[t])/t^N and in U(\g[t^{-1}])\otimes U(t\g[t]): its relation with the center on the of the affine algebra: an explicit formula for the center generators and a conjecture on W-algebra generators: a receipt to obtain the q-deformation of these results: the simple and explicit construction of the Langlands correspondence: the relation between the ``quantum spectral curve'' and the Knizhnik-Zamolodchikov equation: new generalizations of the KZ-equation: the conjecture on rationality of the solutions of the KZ-equation for special values of level. In the simplest cases we observe the coincidence of the ``quantum spectral curve'' and the so-called Baxter equation. Connection with the KZ-equation offers a new powerful way to construct the Baxter's Q-operator.- integrability
- quantization
- spectral representation
- algebra: GL(N)
- geometry
- bibliography
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