# Topological Quantum Field Theory Seminar

### Coherent states for quantum groups

Quantum groups at roots of unity appear as hidden symmetries in some conformal field theories. For this reason I. Todorov has (in 1990s) used coherent state operators for quantum groups to covariantly build the field operators in Hamiltonian formalism. I tried to mathematically found his coherent states by an analogy with the Perelomov coherent states for Lie groups. For this, I use noncommutative localization theory to define and construct the noncommutative homogeneous spaces, and principal and associated bundles over them. Then, in geometric terms, I axiomatize the covariant family of coherent states which enjoy a resolution of unity formula, crucial for physical applications. Even the simplest case of quantum $$\operatorname{SL2}$$ is rather involved and the corresponding resolution of unity formula involves the Ramanujan's $$q$$-beta integral. The correct covariant family differs from ad hoc proposed formulas in several published papers by earlier authors.