Burge multipartitions and the AGT correspondence

Burge multipartitions are tuples of partitions that satisfy a cyclic embedding condition. When uncoloured, Burge m-multipartitions give a combinatorial model for characters of the W_m algebras (W₂ is the Virasoro algebra). When n-coloured, they generalise the "cylindric partitions" that provide a combinatorial model (and a crystal graph) for integrable characters of the affine Lie algebra $\hat{{\mathfrak{s}}{\mathfrak{l}}}(n)$.

Here, we show that the n-coloured Burge m-multipartitions yield the characters of the CFT cosets $\hat{{\mathfrak{g}}{\mathfrak{l}}}(d{)}_{m}/\hat{{\mathfrak{g}}{\mathfrak{l}}}(d-n{)}_{m}$. Having previously shown that the same combinatorial objects give the SU(m) instanton partition functions in = 2 supersymmetric gauge theories on ²/_n, we have thus established a wide-ranging extension of the AGT correspondence.

This talk is based on a collaboration with N.Macleod (Melbourne).