Pictures from Super Chern-Simons Theory

Abstract

We study super-Chern-Simons theory on a generic supermanifold. After a self-contained review of integration on supermanifolds, the complexes of forms (superforms, pseudo-forms and integral forms) and the extended Cartan calculus are discussed. We then introduce Picture Changing Operators. We provide several examples of computation of PCO's acting on different type of forms. We illustrate also the action of the $\eta$ operator, crucial ingredient to define the interactions of super Chern-Simons theory. Then, we discuss the action for super Chern-Simons theory on any supermanifold, first in the factorized form (3-form $\times$ PCO) and then, we consider the most general expression. The latter is written in term of psuedo-forms containing an infinite number of components. We show that the free equations of motion reduce to the usual Chern-Simons equations yielding the proof of the equivalence between the formulations at different pictures of the same theory. Finally, we discuss the interaction terms. They require a suitable definition in order to take into account the picture number. That implies the construction of a 2-product which is not associative that inherits an $A_{\infty }$ algebra structure. That shares several similarities with a recent construction of a super string field theory action by Erler, Konopka and Sachs.

arXiv comment59 pages, Latex2e

References in corpus (21)

• Classical p-branes from boundary state
• N=4 Topological Strings
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Noncommutative homotopy algebras associated with open strings
• Resolving Witten's Superstring Field Theory
• Complete Action for Open Superstring Field Theory with Cyclic $A_{\infty }$ Structure
• Osp(N|4) supermultiplets as conformal superfields on \partial AdS_4 and the generic form of N=2, D=3 gauge theories
• The Geometry of Supermanifolds and New Supersymmetric Actions
• De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology
• Hodge Dualities on Supermanifolds
• Cech and de Rham Cohomology of Integral Forms
• The Integral Form of Supergravity
• Covariant Multiloop Superstring Amplitudes
• Projective Superspaces in Practice
• Supergeometry of $\Pi$-Projective Spaces
• Chern-Simons Theory on Supermanifolds
• Dual Forms on Supermanifolds and Cartan Calculus
• One-Dimensional Super Calabi-Yau Manifolds and their Mirrors
• Multimetric Supergravities
• $A_{\infty }$-Algebra from Supermanifolds
• Homotopy Algebras in String Field Theory

Cited by in corpus (11)

• Holst-MacDowell-Mansouri action for (extended) supergravity with boundaries and super Chern-Simons theory
• M5-brane in the superspace approach
• Super Chern-Simons Theory: BV-formalism and $A_{\infty }$-algebras
• Towards black hole entropy in chiral loop quantum supergravity
• Super Cartan geometry and the super Ashtekar connection
• Supersymmetric Wilson Loops via Integral Forms
• Super fiber bundles, connection forms, and parallel transport
• Supergravities and Branes from Hilbert-Poincaré Series
• On Forms, Cohomology, and BV Laplacians in Odd Symplectic Geometry
• Surface Operators in Superspace
• Geometrical framework for picture changing operators in the pure spinor formalism