$SU_3$ coherent state operators and invariant correlation functions and their quantum group counterparts

TL;DR

This paper develops coherent state operators for SL(n,C), explores their application to invariant correlation functions in SU(3), and extends the framework to quantum groups, introducing a polynomial basis for non-commutative coset spaces.

Contribution

It introduces a new class of coherent state operators for SL(n,C) and their quantum group counterparts, enabling the explicit construction of invariant correlation functions.

Findings

Constructed coherent state operators acting on holomorphic representations.

Derived explicit formulas for SU(3) invariant 2- and 3-point functions.

Developed a polynomial basis for quantum coset spaces related to Lusztig bases.

Abstract

Coherent state operators (CSO) are defined as operator valued functions on G=SL(n,C), homogeneous with respect to right multiplication by lower triangular matrices. They act on a model space containing all holomorphic finite dimensional representations of G with multiplicity 1. CSO provide an analytic tool for studying G invariant 2- and 3-point functions, which are written down in the case of $SU_3$. The quantum group deformation of the construction gives rise to a non-commutative coset space. We introduce a "standard" polynomial basis in this space (related to but not identical with the Lusztig canonical basis) which is appropriate for writing down $U_q(sl_3)$ invariant 2-point functions for representaions of the type $(\lambda,0)$ and $(0,\lambda)$. General invariant 2-point functions are written down in a mixed Poincar\'e-Birkhoff-Witt type basis.