# Twist-field representations of W-algebras, exact conformal blocks and character identities

## TL;DR

This paper explores twist-field representations of W-algebras, deriving character identities and proposing methods to compute exact conformal blocks using geometric data from Prym varieties, extending to D- and B-series.

## Contribution

It generalizes the construction of vertex operators for twist-fields in W-algebras to D- and B-series and links character computations to lattice theta-function identities.

## Key findings

- Derived new character identities involving lattice theta-functions.
- Proposed a geometric approach to compute conformal blocks for D-series.
- Extended twist-field representations to D- and B-series W-algebras.

## Abstract

We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.