Fermions from Half-BPS Supergravity

TL;DR

This paper connects half-BPS supergravity geometries to free fermion phase space, revealing a noncommutative structure and providing a new approach to counting supersymmetric configurations.

Contribution

It introduces a collective coordinate quantization method for LLM geometries, linking supergravity configurations to fermion phase space and noncommutative geometry.

Findings

u function as classical fermion phase space distribution

Configuration space becomes noncommutative in half-BPS sector

Provides a new framework for counting supersymmetric states

Abstract

We discuss collective coordinate quantization of the half-BPS geometries of Lin, Lunin and Maldacena (hep-th/0409174). The LLM geometries are parameterized by a single function $u$ on a plane. We treat this function as a collective coordinate. We arrive at the collective coordinate action as well as path integral measure by considering D3 branes in an arbitrary LLM geometry. The resulting functional integral is shown, using known methods (hep-th/9309028), to be the classical limit of a functional integral for free fermions in a harmonic oscillator. The function $u$ gets identified with the classical limit of the Wigner phase space distribution of the fermion theory which satisfies u * u = u. The calculation shows how configuration space of supergravity becomes a phase space (hence noncommutative) in the half-BPS sector. Our method sheds new light on counting supersymmetric configurations in supergravity.