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Superbranes and Superembeddings

Dmitri Sorokin

TL;DR

Sorokin’s review articulates the superembedding (doubly supersymmetric) framework as a geometric unifier for brane dynamics, recasting κ-symmetry as worldvolume supersymmetry on a superspace embedded into a target superspace. The central notion, the superembedding condition E_{ ext{α}}^{ ext{a}}=0 (and its curved-space generalizations), generates Virasoro-like constraints and, in many cases, the full brane equations of motion from integrability, while allowing worldvolume actions to be formulated when the embedding is off-shell. The article develops both foundational (particles, strings) and advanced (M2, M5) applications, showing how momentum, twistors, Lorentz harmonics, and self-dual tensor fields arise naturally within this geometric picture. It also surveys couplings to supersymmetric backgrounds (Maxwell, supergravity) and the tension-generation mechanism for strings, highlighting the framework’s capacity to encode background constraints and gauge dynamics without relying solely on component actions. Collectively, the work positions superembedding as a powerful, covariant tool for understanding brane dynamics in M-theory and its reductions, with broad implications for quantization and background consistency.

Abstract

We review the geometrical approach to the description of the dynamics of super-p-branes, Dirichlet branes and the M5-brane, which is based on a generalization of the elements of surface theory to the description of the embedding of supersurfaces into target superspaces. Being manifestly supersymmetric in both, the superworldvolume of the brane and the target superspace, this approach unifies the Neveu-Schwarz-Ramond and the Green-Schwarz formulation and provides the fermionic kappa-symmetry of the Green-Schwarz-type superbrane actions with a clear geometrical meaning of standard worldvolume local supersymmetry. We describe the properties of doubly supersymmetric (superembedding) brane actions and show how they are related to the standard Green-Schwarz formulation. In the second part of the article basic geometrical grounds of the (super)embedding approach are considered and applied to the description of the M2-brane and the M5-brane. Various applications of the superembedding approach are reviewed.

Superbranes and Superembeddings

TL;DR

Sorokin’s review articulates the superembedding (doubly supersymmetric) framework as a geometric unifier for brane dynamics, recasting κ-symmetry as worldvolume supersymmetry on a superspace embedded into a target superspace. The central notion, the superembedding condition E_{ ext{α}}^{ ext{a}}=0 (and its curved-space generalizations), generates Virasoro-like constraints and, in many cases, the full brane equations of motion from integrability, while allowing worldvolume actions to be formulated when the embedding is off-shell. The article develops both foundational (particles, strings) and advanced (M2, M5) applications, showing how momentum, twistors, Lorentz harmonics, and self-dual tensor fields arise naturally within this geometric picture. It also surveys couplings to supersymmetric backgrounds (Maxwell, supergravity) and the tension-generation mechanism for strings, highlighting the framework’s capacity to encode background constraints and gauge dynamics without relying solely on component actions. Collectively, the work positions superembedding as a powerful, covariant tool for understanding brane dynamics in M-theory and its reductions, with broad implications for quantization and background consistency.

Abstract

We review the geometrical approach to the description of the dynamics of super-p-branes, Dirichlet branes and the M5-brane, which is based on a generalization of the elements of surface theory to the description of the embedding of supersurfaces into target superspaces. Being manifestly supersymmetric in both, the superworldvolume of the brane and the target superspace, this approach unifies the Neveu-Schwarz-Ramond and the Green-Schwarz formulation and provides the fermionic kappa-symmetry of the Green-Schwarz-type superbrane actions with a clear geometrical meaning of standard worldvolume local supersymmetry. We describe the properties of doubly supersymmetric (superembedding) brane actions and show how they are related to the standard Green-Schwarz formulation. In the second part of the article basic geometrical grounds of the (super)embedding approach are considered and applied to the description of the M2-brane and the M5-brane. Various applications of the superembedding approach are reviewed.

Paper Structure

This paper contains 43 sections, 513 equations.

Table of Contents

  1. Introduction
  2. Doubly supersymmetric particles
  3. The bosonic particle
  4. Spinning particles
  5. Superparticles, $\kappa$-symmetry
  6. Spinning superparticles
  7. Worldline superfield actions for standard superparticles. The superembedding condition.
  8. $N=1, D=3$ superparticle and twistors
  9. On--shell relationship between superparticles and spinning particles
  10. The superembedding condition
  11. $N=1$, $D=4,6,10$ superparticles
  12. Coupling to supersymmetric Maxwell fields
  13. Superembedding into curved superspaces
  14. Superstrings
  15. Green--Schwarz formulation
  16. ...and 28 more sections