# Vertex algebras (Bourbaki seminar) by Frenkel.

By Frenkel.

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RADUL and W. WANG – W1+∞ and WN with central charge N , Comm. Math. Phys. 170 (1995) 337–357. [FR] E. FRENKEL and N. RESHETIKHIN – Towards deformed chiral algebras, Preprint q-alg/9706023. [FK] I. FRENKEL and V. KAC – Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23–66. [FGZ] I. FRENKEL, H. GARLAND and G. ZUCKERMAN – Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. A. 83 (1986) 8442–8446. [FLM] I. FRENKEL, J. LEPOWSKY and A. MEURMAN – Vertex Operator Algebras and the Monster.

182–191, Springer–Verlag 1991. [Se] G. SEGAL – The Definition of Conformal Field Theory, unpublished manuscript. [So] C. SORGER – La formule de Verlinde, S´em. Bourbaki, Exp. 793, Asterisque 237 (1996) 87–114. [TK] A. TSUCHIYA and Y. KANIE – Vertex operators in conformal field theory on P1 and monodromy represenations of the braid group, in Adv. Stud. Pure Math 16, pp. 297–372, Academic Press 1988. [TUY] A. TSUCHIYA, K. UENO and Y. YAMADA – Conformal field theory on universal family of stable curves with gauge symmetries, Adv.

875-31 Since LG is ample, we can recover the moduli space of semi-stable G–bundles on X as the Proj of the graded ring ⊕k≥0 ΓN k = ⊕k≥0 C(X, x, LN k (g)) for large N (the ring of “non-abelian theta functions”). Thus, if we could define the product on conformal blocks in a natural way, we would obtain a description of the moduli space. Using the correlation function description of conformal blocks Feigin and Stoyanovsky [FS] have identified the space C(X, x, Lk (g)) with the space of sections of a line bundle on the power of X satisfying certain conditions.