Abstract
This is a companion paper of a long work appeared in [1] discussing the super-Chern-Simons theory on supermanifolds. Here, it is emphasized that the BV formalism is naturally formulated using integral forms for any supersymmetric and supergravity models and we show how to deal with $A_\infty$-algebras emerging from supermanifold structures.
Highlights
The Batalin-Vilkovisky (BV) formalism and supergeometry have been extensively studied during the last years
It has been shown the naturalness of the BV formalism in the supergeometry approach (QP manifolds, oddsymplectic structures, BV bracket) because all fields have their own opposite-statistic partner leading to a BV-symplectic two-form corresponding to an oddsymplectic structure
The application of the BV formalism was ubiquitous in quantum field theory and string theory, but in our opinion the BV formalism for supersymmetric theories has never been deeply explored from the supergeometric point of view and this is the aim of the present work
Summary
The Batalin-Vilkovisky (BV) formalism and supergeometry have been extensively studied during the last years. When working with the theory at 0 picture number, the natural set of antifields lies into the integral forms complex instead of the usual superforms complex. We show that the BV formalism adapts to the present framework and the result leads to Chern-Simons (CS) action where the picture number 1 gauge field Að1j1Þ is replaced by a generic pseudoform A with any form degree (in particular, it can be negative, exactly as in string field theory [12]) and picture number fixed to 1. Given a ðpjqÞ-form ωðpjqÞ ∈ ΩðpjqÞ, we define the picture lowering operator ZD as. Zv∶ΩðpjqÞ → Ωðpjq−1Þ; ωðpjqÞ ↦ ZvðωðpjqÞÞ 1⁄4 1⁄2d; −iΘðιDÞωðpjqÞ; where 1⁄2·; · denotes as usual a graded commutator and the action of the operator ΘðιvÞ is defined by the Fourier-like relation of the Heaviside step function.
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