Quantization of the space of conformal blocks

Abstract

We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated to $gl(N)$, defined in terms of rational R-matrices. We prove that under certain resonance conditions, the qKZ connection has a non-trivial invariant subbundle which we call the subbundle of quantized conformal blocks. The subbundle is given explicitly by algebraic equations in terms of the Yangian $Y(gl(N))$ action. The subbundle is a deformation of the subbundle of conformal blocks in CFT. The proof is based on an identity in the algebra with two generators $x,y$ and defining relation $xy=yx+yy$.