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Planes, branes and automorphisms: II. Branes in motion

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

TL;DR

This work completes the supersymmetric classification of two intersecting M5-branes with relative motion by showing moving configurations arise from null-rotated Cayley planes in eight dimensions, parameterized by subgroups of Spin10,1 whose isotropy preserves spinors. The authors introduce a robust normal-form analysis of Lorentz transformations, isolate null-rotation effects, and connect the results to an exotic Spin7⋉R9 isotropy, yielding a detailed group-theoretical classification and explicit examples. Extending to multiple branes, the problem reduces to eight-dimensional Cayley-plane geometry with null rotations, predicting generic fractions of SUSY as low as 1/32 and possible enhancements up to 1/4 via sub-orbits; this is quantified through a nuanced invariant-dimension analysis yielding diverse fractions such as 1/32, 1/16, 3/32, 1/8, and 1/4. The study thus unifies moving-brane SUSY with Cayley-calibrated eight-dimensional geometry and exotic spinor isotropy, offering a pathway to broader M-theory configurations and dualities.

Abstract

We complete the classification of supersymmetric configurations of two M5-branes, started by Ohta and Townsend. The novel configurations not considered before are those in which the two branes are moving relative to one another. These configurations are obtained by starting with two coincident branes and Lorentz-transforming one of them while preserving some supersymmetry. We completely classify the supersymmetric configurations involving two M5-branes, and interpret them group-theoretically. We also present some partial results on supersymmetric configurations involving an arbitrary number of M5-branes. We show that these configurations correspond to Cayley planes in eight-dimensions which are null-rotated relative to each other in the remaining (2+1) dimensions. The generic configuration preserves 1/32 of the supersymmetry, but other fractions (up to 1/4) are possible by restricting the planes to certain subsets of the Cayley grassmannian. We discuss some examples with fractions 1/32, 1/16, 3/32, 1/8, 1/4 involving an arbitrary number of branes, as well as their associated geometries.

Planes, branes and automorphisms: II. Branes in motion

TL;DR

This work completes the supersymmetric classification of two intersecting M5-branes with relative motion by showing moving configurations arise from null-rotated Cayley planes in eight dimensions, parameterized by subgroups of Spin10,1 whose isotropy preserves spinors. The authors introduce a robust normal-form analysis of Lorentz transformations, isolate null-rotation effects, and connect the results to an exotic Spin7⋉R9 isotropy, yielding a detailed group-theoretical classification and explicit examples. Extending to multiple branes, the problem reduces to eight-dimensional Cayley-plane geometry with null rotations, predicting generic fractions of SUSY as low as 1/32 and possible enhancements up to 1/4 via sub-orbits; this is quantified through a nuanced invariant-dimension analysis yielding diverse fractions such as 1/32, 1/16, 3/32, 1/8, and 1/4. The study thus unifies moving-brane SUSY with Cayley-calibrated eight-dimensional geometry and exotic spinor isotropy, offering a pathway to broader M-theory configurations and dualities.

Abstract

We complete the classification of supersymmetric configurations of two M5-branes, started by Ohta and Townsend. The novel configurations not considered before are those in which the two branes are moving relative to one another. These configurations are obtained by starting with two coincident branes and Lorentz-transforming one of them while preserving some supersymmetry. We completely classify the supersymmetric configurations involving two M5-branes, and interpret them group-theoretically. We also present some partial results on supersymmetric configurations involving an arbitrary number of M5-branes. We show that these configurations correspond to Cayley planes in eight-dimensions which are null-rotated relative to each other in the remaining (2+1) dimensions. The generic configuration preserves 1/32 of the supersymmetry, but other fractions (up to 1/4) are possible by restricting the planes to certain subsets of the Cayley grassmannian. We discuss some examples with fractions 1/32, 1/16, 3/32, 1/8, 1/4 involving an arbitrary number of branes, as well as their associated geometries.

Paper Structure

This paper contains 19 sections, 2 theorems, 71 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. The statement of the problem
  3. Supersymmetric intersections of two $\mathsf{M}5$-branes
  4. A convenient normal form for Lorentz transformations
  5. Conditions for supersymmetry
  6. Supersymmetry and null rotations
  7. Group-theoretical analysis
  8. Null rotations and an exotic spinor isotropy group
  9. An explicit real realisation of $\mathrm{C}\ell_{1,10}$
  10. Spinor isotropies
  11. An exotic spinor isotropy group
  12. Invariants
  13. More on vectors and spinors
  14. Multiple intersections
  15. $G$-relatedness
  16. ...and 4 more sections

Key Result

Lemma 1

If a spinor in $\Delta$ is annihilated by a (nontrivial) null rotation, then it is annihilated by all null rotations.

Figures (1)

  • Figure 1: Subgroups of $\mathrm{Spin}_8$ associated with intersecting brane configurations which can be null-rotated while preserving some supersymmetry. Arrows represent embeddings, and those adorned with a $\bullet$ are such that the maximal tori agree.

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Definition 1