Logarithmic and complex constant term identities

TL;DR

This paper explores complex and logarithmic constant term identities related to vertex algebra representation theory, deriving new identities from conjectural analogues and proving specific cases for root system G_2.

Contribution

It introduces complex analogues of Dyson and Morris identities, explains their connection to logarithmic identities, and proves new identities for the G_2 root system.

Findings

Derived complex analogues of Dyson and Morris identities.

Connected logarithmic identities to derivatives of complex identities.

Proved complex and logarithmic identities for the G_2 root system.

Abstract

In recent work on the representation theory of vertex algebras related to the Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamovic and Milas arise naturally by differentiating as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove complex and logarithmic constant term identities for the root system G_2.