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

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.

43 pages, LaTeX, minor change to correct a misprint

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• Coherent states, Path integral, and Semiclassical approximation
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• Symplectic Manifolds, Coherent States and Semiclassical Approximation
• More about Path Integral for Spin
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