Coherent states over Grassmann manifolds and the WKB-exactness in path integral

TL;DR

This paper formulates N coherent states on Grassmann manifolds to analyze WKB-exactness in path integrals, revealing localization mechanisms via Schwinger bosons, with implications for semiclassical approximations in representation theory.

Contribution

It introduces a coherent state formulation on Grassmann manifolds to demonstrate WKB-exactness using localization and Schwinger boson techniques.

Findings

Demonstrates WKB-exactness in path integrals over Grassmann manifolds.

Uncovers the localization mechanism via Schwinger boson representation.

Shows the semiclassical approximation becomes exact as the label k ightarrow \u221e.

Abstract

\(\Un{N}\) coherent states over Grassmann manifold, \(\grsmn{N}{n}\simeq\Un{N}/ (\Un{n}\times \Un{N-n})\), are formulated to be able to argue the WKB-exactness, so called the localization of Duistermaat-Heckman, in the path integral representation of a character formula. The exponent in the path integral formula is proportional to an integer \(k\) that labels the \(\Un{N}\) representation. Thus when \(k \rightarrow\infty\) a usual semiclassical approximation, by regarding \(k \sim 1 / \hbar\), can be performed yielding to a desired conclusion. The mechanism of the localization is uncovered by means of a view from an extended space realized by the Schwinger boson technique.