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

Abstract

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.

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

“…However, in order to apply bosonization one has to restrict the elements g ∈ N G (h) ⊂ G to the Cartan normalizer, and this will be the key object in our definition of the twist fields. In particular, bosonization means that one considers the space H˙g as a sum of twisted representations of the Heisenberg algebra h. These representations depend not only on the elements g ∈ N G (h), but also on additional data: eigenvalues of the zero modes of the ginvariant part of h. This extra data, below to be called r-charges following [16], has discrete freedom, since only the exponents of such eigenvalues are specified by g. We also denote below the most refined data asg = (g, r).…”

“…These "averaged over a cycle" parameters have been called as r-charges in [16]. Hence, all elements of g ∈ N GL(N ) (h) can be conjugated to the products over the cycles…”

“…For a single cycle K = 1 this gives a product formula for the lattice A N −1 -theta function (A. 16), which for N = 2…”

“…Even in such situation, in case of generic monodromies one cannot write explicit formula for the conformal block. Below, following [16], we are going to restrict ourselves to the case of so-called twist fields [17], corresponding to the quasi-permutation monodromies, when the representations of W-algebras become related to the twisted representations of the corresponding KM algebras 2 [18,19].…”

“…Finally, we propose a construction of the exact conformal blocks of the twist fields for W-algebras of D-series, generalizing approach of [17,16], and obtain explicit formula, expressing the multipoint blocks in terms of algebro-geometric objects on the branched cover of the original curve with extra involution. Some important technical details are collected in Appendices.…”