On universal quantum dimensions

R.L Mkrtchyan

doi:10.1016/j.nuclphysb.2017.05.021

Abstract

We represent in the universal form restricted one-instanton partition function of supersymmetric Yang–Mills theory. It is based on the derivation of universal expressions for quantum dimensions (universal characters) of Cartan powers of adjoint and some other series of irreps of simple Lie algebras. These formulae also provide a proof of formulae for universal quantum dimensions for low-dimensional representations, needed in derivation of universal knot polynomials (i.e. colored Wilson averages of Chern–Simons theory on 3d sphere). As a check of the (complicated) formulae for universal quantum dimensions we prove numerically Deligne's hypothesis on universal characters for symmetric cube of adjoint representation.

Highlights

This paper is the step in realization of program of representation of partition functions and observables in gauge theories in universal form
This will be the first appearance of universal formula in four-dimensional Yang-Mills theory. Another aim is derivation of formulae for quantum dimensions, used in [3] for calculation of universal expressions for some knot polynomials (Wilson averages in Chern-Simons theory). Both results are based on the derivation of new formulae for universal quantum dimensions of some series of irreps, appearing in decomposition of powers of adjoint representation
This program is successful for Chern-Simons theory on the 3d sphere [1, 2], and has led to establishment of exact Chern-Simons/topological strings duality for SU(N) gauge groups [6], similar results in refined cases [6], gauge/string duality conjecture for exceptional groups [23], universal knot polynomials [3], and other achievements

Summary

Introduction

This paper is the step in realization of program of representation of partition functions and observables in gauge theories (and other simple-Lie-algebrasbased theories) in universal form. Universality approach reveals [9] details of behavior of SU (N ) Chern-Simons partition function under N → −N duality, and, more generally, under permutations of universal parameters (Vogel’s symmetry) Both results are based on the derivation (see Sections 2, 3) of new formulae for universal quantum dimensions (sometimes called universal characters) of some series of irreps, appearing in decomposition of powers of adjoint representation. It is not possible to define simple modules with ring which is not integral domain, one can still have universal formulae In that case they will have an arbitrariness of adding polynomials or other functions of universal parameters which are zero for all simple Lie algebras, and even on the entire lines of classical algebras, exceptional line and line t = 0, which corresponds to the superalgebra D2,1,λ [15].

Quantum dimensions of Cartan powers of adjoint

Deligne’s hypothesis on quantum dimensions

One-instanton partition function

Conclusion

A Examples of decompositions of symmetric cube of adjoint