Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

TL;DR

This paper provides an exact method to compute conformal blocks for W-algebras with extended symmetry, linking free field theory, twist fields, and isomonodromic tau-functions, advancing understanding in CFT and gauge theory correspondence.

Contribution

It introduces a novel exact computation of conformal blocks for W-algebras using free field theory on covering surfaces, even with non-abelian monodromy groups.

Findings

Exact conformal blocks computed via free field theory on Riemann surfaces.

Established relation between conformal blocks and isomonodromic tau-functions.

Proposed method for calculating W-charges of twist fields.

Abstract

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.