Pure Spinor Superfields: An Overview

Abstract

Maximally supersymmetric theories do not allow off-shell superspace formulations with traditional superfields containing a finite set of auxiliary fields. It has become clear that off-shell supersymmetric action formulations of such models can be achieved by the introduction of pure spinors. In this talk, an overview of this formalism is given, with emphasis on D = 10 super-Yang-Mills theory and D = 11 supergravity. This a somewhat expanded version of a talk presented at the workshop "Breaking of supersymmetry and ultraviolet divergences in extended supergravity" (BUDS), Laboratori Nazionali di Frascati, March 25-28, 2013.

Twistors and supertwistors for exceptional field theory

“…If at some point the issue is resolved, it should be possible to construct off-shell supersymmetric actions for extended supersymmetric field theory and supergeometry along the lines of refs. [76][77][78][79][80].…”

Pure spinor superspace action for D = 6, N = 1 super-Yang-Mills theory

“…This is a typical feature of the pure spinor superfield formalism, and the present behaviour mirrors that of maximally supersymmetric SYM. Even more drastic reduction of the order of the interactions are seen in the actions for BLG and ABJM models [2][3][4], in the Born-Infeld deformation of D = 10 SYM [8,11,12], and in D = 11 supergravity [5,6]. The equations of motion following from the action (4.12) are 13) where "•" is shorthand for formation of the adjoint of the gauge algebra, and [·, ·] denotes adjoint action.…”

“…Some more spinor identities are collected in appendix A. The numbering for Dynkin labels is that of figure 1, where an upper spinor index is represented by (001) (1). The symmetry properties of spinor bilinears are:…”

“…A convenient way to represent them is as two-component spinors with quaternionic entries. One then uses the isomorphism SL(2; H) ≈ Spin (1,5), and the R-symmetry SU(2) acts by quaternionic multiplication with elements of unit norm from the right. We will use this language only occasionally, but instead work with matrices (γ a ) αβ or (γ a ) αβ , a = 1, .…”

“…[1] and references therein) have been used in the construction of actions for maximally supersymmetric theories [2][3][4][5][6][7][8][9][10][11][12][13]. It is there that the formalism, originating in superstring theory [14][15][16][17] and in the deformation theory for maximally supersymmetric super-Yang-Mills theory (SYM) and supergravity [18][19][20][21][22][23][24][25], has its greatest power.…”