Superbranes and superembeddings

Abstract

We review the geometrical approach to the description of the dynamics of superparticles, superstrings and, in general, 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 κ-symmetry of the Green-Schwarz-type superbrane actions with a clear geometrical meaning of standard worldvolume local supersymmetry.The dynamics of superbranes is encoded in a generic superembedding condition. Depending on the superbrane and the target-space dimension, the superembedding condition produces either only off-shell constraints (as in the case of N = 1 superparticles and N = 1 superstrings), or also results in the full set of the superbrane equations of motion (as, for example, in the case of the M-theory branes). In the first case worldvolume superspace actions for the superbranes can be constructed, while in the second case only component or generalized superfield actions are known.We describe the properties of the doubly supersymmetric 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. † Alexander

Kappa-symmetry of Green–Schwarz actions in coset superspaces

“…The closure of the algebra of worldvolume diffeomorphisms, κ-symmetry transformations and Lorentz transformations corresponds to an integrability condition on certain vector fields, and this usually has a dual formulation in terms of differential forms. The results of this paper are presumably also closely related to the geometric interpretation of κ-symmetry transformations in terms of superembeddings [10,9,11].…”

“…Progress has been made in achieving a geometric understanding of kappa symmetry using the formalism of super-embeddings [9], based on doubly supersymmetric approaches to superstrings which incorporate both manifest worldvolume and spacetime supersymmetries [10] (for a recent review of superembeddings including a comprehensive list of references, see [11]). The worldvolume of the p-brane is extended to a supermanifold which is embedded into the target superspace in such a way that the odd subspace of the worldvolume tangent space is a subset of the odd subspace of tangent space to the target superspace.…”

SU(1, 1|N) superconformal mechanics with fermionic gauge symmetry

“…As is known [40], the κ-symmetry reduces the number of fermionic dynamical degrees of freedom by half. It proves convenient to choose the gauge fixing condition in the form 3…”

Covariant map between Ramond-Neveu-Schwarz and pure spinor formalisms for the superstring

“…However, because of the complicated bosonization formula of (1.3) used in the hybrid formalism, the map was not manifestly SO(10) covariant. Although there exists a relation at the classical level between the hybrid formalism and the manifestly covariant "superembedding" formalisms [5][6][7], this relation has not yet been understood at the quantum level.…”

“…So after adding non-minimal spinor variables to the RNS formalism, dynamically twisting as in (1.5), and performing various similarity transformations, the RNS BRST operator is covariantly mapped into the pure spinor BRST operator plus a set of "topological" variables which decouple from the cohomology. Furthermore, the non-minimal RNS formalism containing both the RNS ψ m variables and the GS θ α variables provides a natural bridge between the RNS and pure spinor formalisms which resembles the "superembedding" formalisms reviewed in [7]. Vertex operators in the GSO(+) sector can be expressed in d=10 superspace using the non-minimal RNS formalism, and in the gauge u m = Γ m = 0, they reduce to the usual pure spinor vertex operators.…”