Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center

TL;DR

This paper establishes a deep connection between the modular group representations in logarithmic conformal field theories and the quantum group center, revealing their algebraic structures and fusion rules.

Contribution

It demonstrates the equivalence of SL(2,Z) representations in quantum groups and logarithmic CFTs, and links the Grothendieck ring to the fusion algebra with new q-binomial identities.

Findings

SL(2,Z) representation on quantum group center matches that on extended characters

Grothendieck ring coincides with the fusion algebra of the (1,p) model

Derived new q-binomial identities from fusion algebra structure

Abstract

The SL(2,Z) representation $\pi$ on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL(2,Z) representation on the extended characters of the logarithmic (1,p) conformal field theory model. The multiplicative Jordan decomposition of the U_{q}sl(2) ribbon element determines the decomposition of $\pi$ into a ``pointwise'' product of two commuting SL(2,Z) representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2,Z) representation on the (1,p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of U_{q}sl(2) at the primitive 2p-th root of unity is shown to coincide with the fusion algebra of the (1,p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of~U_{q}sl(2).