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Branes at angles and calibrated geometry

BS Acharya, JM Figueroa-O'Farrill, B Spence

TL;DR

This work reframes the problem of supersymmetric intersections of $\,\mathsf{M}5\,$-branes (and $\,\mathsf{M}2\,$-branes) in terms of calibrated geometry. By tying the SUSY (half-BPS) conditions to calibrations arising from spinors and reduced holonomy, it provides an invariant derivation of the two-brane results and extends them to arbitrary numbers of branes, highlighting special Lagrangian and Cayley calibrations as key structures. The approach clarifies how angle data and spinor isotropy constrain supersymmetry, yielding precise criteria for when brane intersections preserve fractions of supersymmetry and how these configurations relate to volume minimization. Overall, the paper establishes a geometric framework that connects brane dynamics to calibrated submanifolds and special holonomy, with implications for classifying supersymmetric brane configurations in M-theory.

Abstract

In a recent paper, Ohta and Townsend studied the conditions which must be satisfied for a configuration of two intersecting M5-branes at angles to be supersymmetric. In this paper we extend this result to any number of M5-branes or any number of M2-branes. This is accomplished by interpreting their results in terms of calibrated geometry, which is of independent interest.

Branes at angles and calibrated geometry

TL;DR

This work reframes the problem of supersymmetric intersections of -branes (and -branes) in terms of calibrated geometry. By tying the SUSY (half-BPS) conditions to calibrations arising from spinors and reduced holonomy, it provides an invariant derivation of the two-brane results and extends them to arbitrary numbers of branes, highlighting special Lagrangian and Cayley calibrations as key structures. The approach clarifies how angle data and spinor isotropy constrain supersymmetry, yielding precise criteria for when brane intersections preserve fractions of supersymmetry and how these configurations relate to volume minimization. Overall, the paper establishes a geometric framework that connects brane dynamics to calibrated submanifolds and special holonomy, with implications for classifying supersymmetric brane configurations in M-theory.

Abstract

In a recent paper, Ohta and Townsend studied the conditions which must be satisfied for a configuration of two intersecting M5-branes at angles to be supersymmetric. In this paper we extend this result to any number of M5-branes or any number of M2-branes. This is accomplished by interpreting their results in terms of calibrated geometry, which is of independent interest.

Paper Structure

This paper contains 14 sections, 1 theorem, 31 equations.

Table of Contents

  1. Introduction
  2. Supersymmetric pairs of $\mathsf{M}5$-branes at angles
  3. Calibrated geometry
  4. Calibrations and volume minimisation
  5. Calibrations and holonomy
  6. Calibrations and spinors
  7. The Angle Theorem and the Nance calibration
  8. The Angle Theorem
  9. The Nance calibrations
  10. Intersecting $\mathsf{M}5$-branes at angles
  11. Special lagrangian calibrations and the condition \ref{['eq:msusy']}
  12. Spinors and the generalisation to more than two branes
  13. A brief look at $\mathsf{M}2$-branes
  14. The many faces of susy

Key Result

Theorem 1

Two oriented $p$-planes in $\mathbb{E}^{2p}$ defined by the two simple unit $p$-vectors: $\zeta(0) = e_1 \wedge \cdots \wedge e_p$ and $\zeta(\theta)$, belong to the same $\varphi$-grassmannian if and only if the following inequality is satisfied:

Theorems & Definitions (1)

  • Theorem 1: Angle Theorem