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, pseudoforms and integral forms) and the extended Cartan calculus are discussed. We then introduce Picture Changing Operators and their mathematical properties. We show that the free equations of motion reduce to the usual Chern-Simons equations proving on-shell 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. This leads to the construction of a series of non-associative products which yield an A∞ algebra structure, sharing several similarities with the super string field theory action by Erler, Konopka and Sachs.
Highlights
Our main motivation is to provide a general method for constructing classical actions for quantum field theories on supermanifolds using the powerful methods of supergeometry
For the construction of quantum field theories on supermanifolds, one needs to fix the total picture of the action, but that does not select a given picture for the fields involved
We pointed out that the interaction term has to be built in terms of a non-associative product leading to a tower of interactions organized into a A∞ algebra
Summary
Our main motivation is to provide a general method for constructing classical actions for quantum field theories on supermanifolds using the powerful methods of supergeometry. For the construction of quantum field theories on supermanifolds (we recall that the picture in string theory is related to the superghosts zero modes which are in relations with the supermoduli space of the underlying super-Riemann surface), one needs to fix the total picture of the action, but that does not select a given picture for the fields involved This means that one can choose different set of fields, defined as forms in the full complex, and construct the corresponding action (See [2, 3]). We consider an action for super Chern-Simons theory ( SCS) built in terms of the A(1|1) gauge fields, namely those at picture one. Their expansions in term of component fields are infinite dimensional, the kinetic term is obtained by using repeated distributional properties and integrating over the supermanifold. We collect some review material on A∞ algebras and their automorphisms and some explicit computations omitted in main text
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