On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.

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On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks. (1995)

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by Alexander A. Kirillov , Jr.

Venue:	I & II, math.QA/9508017 and q-alg/9611008
Citations:	43 - 0 self

BibTeX

@ARTICLE{Kirillov95oninner,
    author = {Alexander A. Kirillov and Jr.},
    title = {On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.},
    journal = {I & II, math.QA/9508017 and q-alg/9611008},
    year = {1995},
    volume = {9508017},
    pages = {1135--1169}
}

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Abstract

this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from Wess-ZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and later refined by Kazhdan and Lusztig ([KL1--4]) and Finkelberg ([F]) in Section 9. In particular, spaces of homomorphisms in this category are the spaces of conformal blocks of WZW model. Thus, the general theory developed in Section 2 of [K] gives us an inner product on the space of conformal blocks, and so defined inner product is modular invariant. This definition is constructive: we show how it can be rewritten so that it only involves Drinfeld associator, or, equivalently, asymptotics of solutions of Knizhnik-Zamolodchikov equations. Since there are integral formulas for the solutions of KZ equations, this shows that the inner product on the space of conformal blocks can be written explicitly in terms of certain integrals. In the case g = sl 2 these integrals can be calculated (see [V]), using Selberg integral, and the answer is written in terms of \Gamma-functions. Thus, in this case we can write explicit formulas for inner product on the space of conformal blocks. These expressions are 1991 Mathematics Subject Classification. Primary 81R50, 05E35, 18D10; Secondary 57M99

Keyphrases

conformal block inner product ii inner product modular tensor category recall construction wzw model general theory mathematics subject classification drinfeld associator explicit formula defined inner product certain integral kz equation knizhnik-zamolodchikov equation integrable representation affine lie algebra conformal field theory wess-zuminowitten model integral formula modular invariant

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