dorsal/arxiv
View SchemaOn inner product in modular tensor categories. II. Inner product on conformal blocks and affine inner product identities
| Authors | Alexander Kirillov Jr |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9611008 |
| URL | https://arxiv.org/abs/q-alg/9611008 |
| Journal | Adv.Theor.Math.Phys.2:155-180,1998 |
Abstract
This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.
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"abstract": "This is the second part of the paper (the first part is published in Jour. of\nAMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every\nmodular tensor category (MTC) inner products on the spaces of morphisms and\nproved that the inner product on the space $\\Hom (\\bigoplus X_i\\otimes X^*_i,\nU)$ is modular invariant. Also, we have shown that in the case of the MTC\narising from the representations of the quantum group $U_q \\sln$ at roots of\nunity and $U$ being a symmetric power of the fundamental representation, this\ninner product coincides with so-called Macdonald\u0027s inner product on symmetric\npolynomials.\n In this paper, we apply the same construction to the MTC coming from the\nintegrable representations of affine Lie algebras. In this case our\nconstruction immediately gives a hermitian form on the spaces of conformal\nblocks, and this form is modular invariant (Warning: we cannot prove that it is\npositive definite). We show that this form can be rewritten in terms of\nasymptotics of KZ equations, and calculate it for $sl_2$, in which case the\nformula is a natural affine analogue of Macdonald\u0027s inner product identities.\nWe also formulate as a conjecture similar formula for $sl_n$.",
"arxiv_id": "q-alg/9611008",
"authors": [
"Alexander Kirillov Jr"
],
"categories": [
"q-alg",
"hep-th",
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],
"journal_ref": "Adv.Theor.Math.Phys.2:155-180,1998",
"title": "On inner product in modular tensor categories. II. Inner product on conformal blocks and affine inner product identities",
"url": "https://arxiv.org/abs/q-alg/9611008"
},
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