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The electrostatic view on M-theory LLM geometries

Aristomenis Donos, Joan Simon

TL;DR

The work provides a concrete electrostatic reformulation of the LLM sector in M-theory, recasting the non-linear Toda system as Laplace’s equation under a U(1) isometry and formulating a boundary-value problem for both regular and superstar-like geometries. It identifies the building blocks of regular vacua as semi-infinite line charges plus conducting disks (or planes) and derives explicit flux quantisation conditions that fix AdS radii and M2/M5 charges, yielding finite-$L$ masses consistent with BPS bounds. The paper then extends to excited states via multi-disk configurations, derives mass formulas $M_4 = \frac{1}{2L_4}\sum_{i}\sum_{j=1}^{i} N_2^{j} N_5^{i}$ and $M_7 = \frac{2}{L_7}\sum_{i}\sum_{j=1}^{i} N_2^{j} N_5^{i}$, and shows a clear connection between continuous disk distributions and superstar singularities. Overall, the results provide a precise dictionary between electrostatic data, flux quantisation, and gravitational masses for half-BPS M-theory geometries, clarifying the microstate interpretation of both regular and singular LLM configurations.

Abstract

We describe the geometry of the R x SO(3) x SO(6) x U(1) invariant half-BPS M-theory configurations considered in LLM in terms of their electrostatic variables. We discuss both regular configurations, such as AdS_4 x S^7 and AdS_7 x S^4 vacua or simple excited solutions, and singular ones such as the superstar geometries. This allows us to identify the appropriate boundary conditions describing the most general smooth and superstar-like singular configurations. We also compute their masses, matching the expected result from their microscopic interpretation, but now at finite radius of curvature.

The electrostatic view on M-theory LLM geometries

TL;DR

The work provides a concrete electrostatic reformulation of the LLM sector in M-theory, recasting the non-linear Toda system as Laplace’s equation under a U(1) isometry and formulating a boundary-value problem for both regular and superstar-like geometries. It identifies the building blocks of regular vacua as semi-infinite line charges plus conducting disks (or planes) and derives explicit flux quantisation conditions that fix AdS radii and M2/M5 charges, yielding finite- masses consistent with BPS bounds. The paper then extends to excited states via multi-disk configurations, derives mass formulas and , and shows a clear connection between continuous disk distributions and superstar singularities. Overall, the results provide a precise dictionary between electrostatic data, flux quantisation, and gravitational masses for half-BPS M-theory geometries, clarifying the microstate interpretation of both regular and singular LLM configurations.

Abstract

We describe the geometry of the R x SO(3) x SO(6) x U(1) invariant half-BPS M-theory configurations considered in LLM in terms of their electrostatic variables. We discuss both regular configurations, such as AdS_4 x S^7 and AdS_7 x S^4 vacua or simple excited solutions, and singular ones such as the superstar geometries. This allows us to identify the appropriate boundary conditions describing the most general smooth and superstar-like singular configurations. We also compute their masses, matching the expected result from their microscopic interpretation, but now at finite radius of curvature.

Paper Structure

This paper contains 27 sections, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Regular configurations
  3. The $AdS_{4}\times S^{7}$ ground state
  4. The $AdS_4\times S^7$ electrostatics :
  5. First principles derivation:
  6. The $AdS_7\times S^4$ ground state
  7. First principle derivation:
  8. Conformal symmetry action in the electrostatic variables
  9. AdS${}_{7}\times$S${}^{4}$ as a limit of AdS${}_{4}\times$S${}^{7}$ :
  10. Regularity
  11. Regularity close to the linear charge distribution ($U(1)$ shrinking):
  12. Regularity at $\rho=0$, away from the linear charge distribution ($S^{5}$ shrinking):
  13. Regularity on the disk surfaces ($S^{2}$ shrinking):
  14. Regularity close to the rim of a conducting disk:
  15. Flux quantisation
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Surface which when fibered over the round $S^{2}$ gives the unique four-cycle of the $AdS_{7}\times S^{4}$ background.
  • Figure 2: Surfaces describing 4-cycles and 7-cycles giving rise to non-trivial fluxes.
  • Figure 3: Deformed surface used to construct four- and seven-cycles. The seven-form integral receives non-zero contribution only from the blue segments while the four-form integral receives non-zero contribution from the red line segments.