B-RNS-GSS formalism and $L_{\infty}$-actions

TL;DR

The paper develops a general framework to strictify $L_{\infty}$-actions of a Lie superalgebra on a $Q$-manifold within the BV formalism, by enlarging the space with auxiliary variables and performing a similarity transformation. It shows that any $L_{\infty}$-action is quasi-isomorphic to a strict action on a larger manifold, via a strictification that uses spectator ghosts and an explicit $F$-transformation, and that integrating out the auxiliary fields yields an effective action reproducing the original $L_{\infty}$-data. The construction is applied to the ten-dimensional SUSY algebra in the RNS formalism, and to the B-RNS-GSS model, providing explicit formulas for the similarity transformation and clarifying how the Berkovits equivalence chain can be realized at the level of BRST complexes and BV actions. These results connect homotopy-algebraic structures with string-theory formalisms, offering concrete tools to relate RNS and pure-spinor descriptions via strictified symmetry actions and homotopy transfer.

Abstract

Pure spinor formalism and RNS formalism are related by a chain of equivalences constructed by introducing and integrating-out BRST quartets. This is known as B-RNS-GSS formalism. One of the steps can be understood as adding auxiliary fields to lift a strong homotopy action of the SUSY Lie superalgebra in the large Hilbert space to a strict action. We develop a general prescription for this ``strictification'' procedure, which can be applied for any strong homotopy action of a Lie superalgebra. We explain how it is related to the B-RNS-GSS formalism.