A conjecture on Hubbard-Stratonovich transformations for the Pruisken-Sch\"afer parameterisations of real hyperbolic domains

TL;DR

This paper investigates the Hubbard-Stratonovich transformation within real hyperbolic domains, identifying issues with naive volume element choices and proposing a conjecture for the correct form, supported by explicit solutions and numerical methods.

Contribution

It introduces a conjecture for the correct volume element in Hubbard-Stratonovich transformations for hyperbolic domains, validated through explicit solutions and numerical analysis.

Findings

Naive volume element choice invalidates the transformation.

A conjecture for the correct volume element is proposed.

Supported by analytic solutions for specific groups and numerical evaluation.

Abstract

Rigorous justification of the Hubbard-Stratonovich transformation for the Pruisken-Sch\"afer type of parameterisations of real hyperbolic O(m,n)-invariant domains remains a challenging problem. We show that a naive choice of the volume element invalidates the transformation, and put forward a conjecture about the correct form which ensures the desired structure. The conjecture is supported by complete analytic solution of the problem for groups O(1,1) and O(2,1), and by a method combining analytical calculations with a simple numerical evaluation of a two-dimensional integral in the case of the group O(2,2).