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 η 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 × 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 ∞ algebra structure. That shares several similarities with a recent construction of a super string field theory action by

Supersymmetric Wilson loops via integral forms

“…We mainly restrict to definitions and properties that will be useful in the following sections. For a more extensive introduction to this topic we refer the reader for example to [1,2,3,4,11,33].…”

“…Here we follow the strategy pioneered by Belopolsky [2], where integral forms are distributional-like forms on which a suitable Cartan calculus can be developed. We clarify the basic ingredients and rules, and refer to the literature [1,11,34,35] for a complete description.…”

“…Integral forms have been already used to develop a geometric formulation of some simple topological theories such as super Chern-Simons theory [10,11,12] towards d = 3 N = 1 supergravity. In this paper we apply this technique to reformulate supersymmetric Wilson loops in terms of integral forms.…”

Super Chern-Simons theory: Batalin-Vilkovisky formalism and A∞ algebras

“…Indeed, in order to write the interactions, one needs to insert some PCOs (lowering the number of picture) which are potential sources of ambiguities and difficulties. In [10], it is shown how to deal with those issues by introducing a suitable set of multiproducts leading to an A ∞ algebra.…”

Geometrical framework for picture changing operators in the pure spinor formalism

“…Picture changing operators were previously discussed in the context of pure spinor formalism in[7,8].More generally, the target space picture changing operators were essential in constructing the Chern-Simons actions in[9].…”