Quantum Group as Semi-infinite Cohomology

Details Quantum Group as Semi-infinite Cohomology Quantum Group as Semi-infinite Cohomology

2008-12-09

arxiv.org/abs/0812.1620v2

Commun. Math. Phys. 297 (2010) 687-732

Mathematics

Representation Theory

50 pages, 7 figures, minor revisions, typos corrected, Communications in Mathematical Physics, in press

Scientific paper

10.1007/s00220-010-1055-2

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.

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