Modular and conformal invariance constraints in representation theory of affine algebras (Q1112945)

The basic theme of this long and detailed paper is the study of the restriction of an integrable highest weight representation \({\mathcal L}(\Lambda)\) of an affine Lie algebra \({\mathfrak g}\) to an affine subalgebra \({\mathfrak p}\). More precisely, by the coset construction [\textit{P. Goddard}, \textit{A. Kent} and \textit{D. Olive}, Phys. Lett. B 152, 88-92 (1985; see the preceding review Zbl 0661.17015)], \({\mathcal L}(\Lambda)\) is a representation of the direct sum \({\mathfrak p}\oplus Vir\) and so decomposes as a sum \(L(\Lambda)=\oplus_{V}V\otimes M(V,\Lambda)\) over the integrable highest weight representations V of \({\mathfrak p}\), where M(V,\(\Lambda)\) is a representation of the Virasoro algebra Vir. Giving this decomposition is equivalent to giving the branching functions \(b^ V_{\Lambda}(\tau)=q^{-\pi /24}\chi_{M(V,\lambda)}(q)\) where \(q=e^{2\pi i\tau}\), z is the central charge of the Virasoro on L(\(\Lambda)\), and \(\chi_ M\) is the character of M. It is shown that the \(b^ V_{\Lambda}\) are holomorphic functions on the upper half-plane, and enjoy simple transformation properties under \(SL_ 3(O)\). In particular, they are modular functions with respect to a congruence subgroup. The authors calculate the branching functions in several cases explicitly and study their asymptotic behaviour at the boundary. Along the way, a great deal of background information is given which makes the paper a valuable reference.