Conformal blocks, Verlinde formula and diagram automorphisms

TL;DR

This paper extends the Verlinde formula to compute the trace of diagram automorphisms on conformal blocks for simple Lie algebras, providing new formulas and relations for these automorphisms.

Contribution

It introduces an analogue of the Verlinde formula for the trace of diagram automorphisms and a twining formula relating conformal blocks of specific Lie algebra pairs.

Findings

Derived an analogue of the Verlinde formula for automorphism traces

Established an analogue of the Kac-Walton formula for automorphism traces

Presented a twining formula between conformal blocks of sl(2n+1) and sp(2n)

Abstract

The Verlinde formula computes the dimension of conformal blocks associated to simple Lie algebras and stable pointed curves. If a simply-laced simple Lie algebra admits a nontrivial diagram automorphism, then this automorphism acts on the space of conformal blocks naturally. We prove an analogue of Verlinde formula for the trace of the diagram automorphism on the space of conformal blocks. Along the way, we get an analogue of Kac-Walton formula for the trace of the diagram automorphism. We also get a twining type formula between the conformal blocks for the pair $(sl(2n+1),sp(2n))$.