154 research outputs found
q-deformed KZB heat equation: completeness, modular properties and SL(3,Z)
We study the properties of one-dimensional hypergeometric integral solutions
of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard
equations on tori. We show that they also obey a difference KZB heat equation
in the modular parameter, give formulae for modular transformations, and prove
a completeness result, by showing that the associated Fourier transform is
invertible. These results are based on SL(3,Z) transformation properties
parallel to those of elliptic gamma functions.Comment: 39 page
Resonance Relations for Solutions of the Elliptic QKZB Equations, Fusion Rules, and Eigenvectors of Transfer Matrices of Restricted Interaction-round-a-face Models
Conformal blocks for the WZW model on tori can be represented by vector
valued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfying
vanishing conditions on root hyperplanes. We introduce a quantum version of
these vanishing conditions in the sl(2) case. They are compatible with the qKZB
equations and are obeyed by the hypergeometric solutions as well as by their
critical level counterpart, which are Bethe eigenfunctions of IRF row-to-row
transfer matrices. In the language of IRF models the vanishing conditions turn
out to be equivalent to the sl(2) fusion rules defining restricted models.Comment: 64 pages, late
A Riemann-Roch-Hirzebruch formula for traces of differential operators
Let D be a holomorphic differential operator acting on sections of a
holomorphic vector bundle on an n-dimensional compact complex manifold. We
prove a formula, conjectured by Feigin and Shoikhet, for the Lefschetz number
of D as the integral over the manifold of a differential form. The class of
this differential form is obtained via formal differential geometry from the
canonical generator of the Hochschild cohomology of the algebra of differential
operators in a formal neighbourhood of a point. If D is the identity, the
formula reduces to the Riemann--Roch--Hirzebruch formula.Comment: 31 pages, 1 figure. Misprints corrected and appendix with analytical
details added in v
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