# Relaxed highest-weight modules over affine VOAs

The (chiral) symmetry algebras of 2d conformal field theory are described as vertex operator algebras (VOAs).

Among them, C_2-cofinite ones with semisimple module categories correspond to

rational conformal field theory, whose characters span a SL_2(\Z)-invariant vector space

and whose fusion rules in module categories satisfy the celebrated Verlinde formula.

The affine VOAs are those construted by (and almost the same as) affine Kac-Moody algebras.

Among them, non-integrable affine VOAs are important examples of non-C_2-cofinite VOAs.

The Verlinde formula for non-integrable (admissible) affine VOAs proposed by T. Creutzig and D. Ridout

involves in relaxed highest-weight modules over affine Kac-Moody Lie algebras, which are modules parabolically

induced from weight modules over the associated finite-dimensional simple Lie algebras.

In this talk, we briefly review the Creutzig-Ridout theory and discuss recent progress on

character formulas and classification of relaxed highest-weight modules.

This talk is based on joint works with David Ridout.