Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level

TL;DR

This paper investigates fusion rules, crossing symmetry, and monodromy in conformal field theories with $SL(2)$ current algebra at fractional levels, confirming previous fusion rules and analyzing conformal blocks and operator algebra coefficients.

Contribution

It provides a detailed analysis of fusion rules and conformal blocks for $SL(2)$ current algebra with fractional levels, confirming earlier results and establishing equivalence with other formulations.

Findings

Fusion rules consistent with previous literature are realized.

Explicit crossing matrices between s- and t-channel conformal blocks are computed.

Operator algebra coefficients are derived from monodromy invariants.

Abstract

Based on our earlier work on free field realizations of conformal blocks for conformal field theories with $SL(2)$ current algebra and with fractional level and spins, we discuss in some detail the fusion rules which arise. By a careful analysis of the 4-point functions, we find that both the fusion rules previously found in the literature are realized in our formulation. Since this is somewhat contrary to our expectations in our first work based on 3-point functions, we reanalyse the 3-point functions and come to the same conclusion. We compare our results on 4-point conformal blocks in particular with a different realization of these found by O. Andreev, and we argue for the equivalence. We describe in detail how integration contours have to be chosen to obtain convenient bases for conformal blocks, both in his and in our own formulation. We then carry out the rather lengthy calculation to obtain the crossing matrix between s- and t-channel blocks, and we use that to determine the monodromy invariant 4-point greens functions. We use the monodromy coefficients to obtain the operator algebra coefficients for theories based on admissible representations.