Toda Conformal Blocks, Quantum Groups, and Flat Connections

Abstract

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the W-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra sl 3 on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat SL(3)-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel-Nielsen coordinates from Teichmüller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.

Higher-rank isomonodromic deformations and W-algebras

“…The elements of the basis of vertex operators are labeled in this approach by a finite number of moduli parameterizing the monodromy data [32,54,31,18]. For generic central charge, analogous definition is not available so far, but it is expected to be consistent with an action of the algebra of Verlinde loop operators on the space of 3-point conformal blocks, see recent work [19].…”

Bootstrapping the S5 partition function

“…While our Higgs branch result formally follows from the residue calculation of the matrix integral, we do not have a rigorous proof of its equivalence with the Coulomb branch expression due to some assumption on the contributing poles. However, we support our proposal by showing that it has all the expected properties, both from the the gauge theory and 5d AGT perspectives: i) the partition function has poles at special values of the mass parameters due to the pinching mechanism of the integration contour, and taking the residues can be interpreted as RG flows to defect theories; our proposal indeed assumes the form of a discrete sum whose summands capture the partition functions of codimension 2 and 4 defects which are consistent with the Higgs phase from the brane picture; ii) our proposal looks like the compact space version of a similar result in the 5d Ω-background [58], which can be proven combinatorially; iii) the defect partition functions can manifestly be constructed through the q-Virasoro modular triple [59], in agreement with [35,36]; quite interestingly, it turns out that the free boson correlators form a natural basis for solutions to q-deformed Ward identities or q-conformal blocks, and a similar approach has been recently used to study W 3 conformal blocks of Toda 2d CFT [60].…”

Multi-centered higher spin solutions from $$ {\mathcal{W}}_N $$ conformal blocks

“…The explicit coefficients 8 can be found in [34]. Upon inserting the last equation into the correlation function (4.6), we obtain a shortening relation of the form…”

“…As for the Virasoro algebra, the W N blocks are defined to be the building blocks of correlators which capture the purely kinematical information which is fixed by the W N Ward identities. The W N -blocks are much less studied objects than their Virasoro counterparts, see however [31][32][33][34].…”

“…As is the case for Virasoro blocks (see e.g. [4]) the W N blocks in a certain large-c limit can be determined by solving a monodromy problem [34,35]. To derive this, one considers the amplitude with an insertion of an additional degenerate primary and makes use of the constraints imposed by the decoupling of its null descendants.…”