A-D-E Polynomial and Rogers--Ramanujan Identities

Abstract

We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\mathcal{G}}^{(1)}{)}_{\ell -1}\otimes ({\mathcal{G}}^{(1)}{)}_1/({\mathcal{G}}^{(1)}{)}_{\ell }$, with $\mathcal{G}$=A${}_{n-1}$ \mbox{$(\ell \ge 2)$}, D${}_{n-1}$ $(\ell \ge 2)$, E${}_{6,7,8}$ $(\ell =2)$. In support of our conjectures we establish the correct behaviour under level-rank duality for $\mathcal{G}$=A${}_{n-1}$ and show that the A-D-E Rogers--Ramanujan identities have the expected $q\to 1^{-}$ asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary cosets are also discussed briefly.

19 pages, Latex, 1 Postscript figure