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Fermions from Half-BPS Supergravity

Gautam Mandal

TL;DR

This paper develops a collective-coordinate quantization of Lin–Lunin–Maldacena half-BPS geometries by promoting the LLM function $u(x_1,x_2)$ to the central collective degree of freedom and deriving its action and measure from D3-brane dynamics. The resulting $u$-path integral is shown to be the $\hbar\to 0$ limit of a functional integral for free fermions in a harmonic oscillator, with $u$ identified as the classical limit of the Wigner phase-space density, thereby turning the configuration space into a noncommutative phase space in the half-BPS sector. The work connects supergravity configuration counting to a phase-space, Kirillov-form description and discusses how this framework reproduces D3-brane dynamics for various brane configurations (giant gravitons, dual giants, and branes in arbitrary LLM geometries). It also outlines a first-principles derivation via collective coordinates and highlights implications for counting supersymmetric configurations and potential extensions to finite-$\hbar$ effects and nonperturbative corrections. Overall, the paper provides a concrete link between LLM geometries, noncommutative phase-space dynamics, and fermionic holographic descriptions, with broad implications for entropy counting and BPS state enumeration in supergravity.

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.

Fermions from Half-BPS Supergravity

TL;DR

This paper develops a collective-coordinate quantization of Lin–Lunin–Maldacena half-BPS geometries by promoting the LLM function to the central collective degree of freedom and deriving its action and measure from D3-brane dynamics. The resulting -path integral is shown to be the limit of a functional integral for free fermions in a harmonic oscillator, with identified as the classical limit of the Wigner phase-space density, thereby turning the configuration space into a noncommutative phase space in the half-BPS sector. The work connects supergravity configuration counting to a phase-space, Kirillov-form description and discusses how this framework reproduces D3-brane dynamics for various brane configurations (giant gravitons, dual giants, and branes in arbitrary LLM geometries). It also outlines a first-principles derivation via collective coordinates and highlights implications for counting supersymmetric configurations and potential extensions to finite- effects and nonperturbative corrections. Overall, the paper provides a concrete link between LLM geometries, noncommutative phase-space dynamics, and fermionic holographic descriptions, with broad implications for entropy counting and BPS state enumeration in supergravity.

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 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 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.

Paper Structure

This paper contains 16 sections, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The moduli space of 1/2-BPS Supergravity
  3. Quantization of half-BPS vacua
  4. Correspondence between checkerboard configurations and IIB geometries
  5. Recipe for the collective coordinate action
  6. Single giant graviton in $AdS_5 \times S^5$
  7. D3 brane in arbitrary LLM geometry
  8. Collective coordinate action
  9. Action
  10. Measure
  11. Equivalence to Fermion path integral
  12. Remarks on collective coordinate method with BPS constraint
  13. Conclusion
  14. Acknowledgments
  15. Phase space density action for a single cell
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Checkerboard parameterization. The white rectangles inside the circle represent the regions $H_j$ in (\ref{['checkerboard']}), while the black rectangles outside the circle denote the regions $P_i$. A small number of isolated cells represents giant gravitons in $S^5$ or in $AdS_5$. When the number of cells is large, each additional cell (black or white) can be regarded as a D3 brane in an arbitrary background LLM geometry defined by the rest of the pattern.
  • Figure 2: The fluctuations $p^+(x_1)$ and $p^-(x_1)$ extend from the original droplet to distances $O(\sqrt\hbar)$. The Lagrangian for these fluctuations evaluates to $O(g_s)$. This cancels the prefactor $1/g_s$ sitting outside the collective action (\ref{['action-for-delta-u']}), consistent with fundamental string tension which is independent of $g_s$.