Conformal blocks on elliptic curves and the Knizhnik--Zamolodchikov--Bernard equations

Abstract

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan--Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik--Zamolodchikov equation.

32 pages, amslatex

Cited by in corpus (23)

• Affine SL(2) conformal blocks from 4d gauge theories
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Chern-Simons theory and BCS superconductivity
• Chiral Extensions of the WZNW Phase Space, Poisson-Lie Symmetries and Groupoids
• On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras
• Classification of Isomonodromy Problems on Elliptic Curves
• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• N-point spherical functions and asymptotic boundary KZB equations
• Hitchin Systems at Low Genera
• Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
• Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model
• The Chiral WZNW Phase Space and its Poisson-Lie Groupoid
• Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras
• On the Integrability of Four Dimensional N=2 Gauge Theories in the Omega Background
• Modular differential equations for torus one-point functions
• Lectures on conformal field theory and Kac-Moody algebras
• Special functions, conformal blocks, Bethe ansatz, and SL(3,Z)
• Dynamically twisted algebra $A_{q,p;\hat{\pi }}(\widehat{g{l}_2})$ as current algebra generalizing screening currents of q-deformed Virasoro algebra
• Solitons on Noncommutative Torus as Elliptic Calogero Gaudin Models, Branes and Laughlin Wave Functions
• Classical Wakimoto Realizations of Chiral WZNW Bloch Waves
• Quasitriangular chiral WZW model in a nutshell
• Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations
• Graphical calculus for quantum vertex operators, I: The dynamical fusion operator