Airy Functions for Compact Lie Groups

TL;DR

This paper extends the concept of Airy functions to integrals over the Lie algebra of compact Lie groups, providing criteria for their properties and explicit evaluations, generalizing previous matrix-based results.

Contribution

It generalizes Kontsevich's Airy integral to arbitrary compact Lie groups and offers explicit formulas and criteria for the Airy property of invariant polynomials.

Findings

Established a criterion for the Airy property of polynomials on Lie algebras.

Explicitly evaluated invariant integrals on hermitian matrices.

Unified previous results as special cases within a broader framework.

Abstract

The classical Airy function has been generalised by Kontsevich to a function of a matrix argument, which is an integral over the space of (skew) hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is generalised to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases.