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Modular and conformal invariance constraints in representation theory of affine algebras. (English) Zbl 0661.17016
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 [P. Goddard, A. Kent and 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.
Reviewer: A.N.Pressley

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 11F03 Modular and automorphic functions
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