Resonance Relations, Holomorphic Trace Functions and Hypergeometric Solutions to qKZB and Macdonald-Ruijsenaars Equations

The resonance relations are identities between coordinates of functions Ψ(λ) with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group Uq(sl2), and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weyl anti-invariant hypergeometric qKZB solutions with values in a tensor product of finite-dimensional Uq(sl2)-modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of Uq(sl2)-invariants, and when q is a root of unity is computed by the Verlinde algebra

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of $SL_q(2)$ and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.Comment: 50 pages, 7 figures, minor revisions, typos corrected, Communications in Mathematical Physics, in pres