Deformation Theory and the Batalin-Vilkovisky Master Equation
Jim Stasheff
TL;DR
The paper shows that both classical and quantum Batalin–Vilkovisky master equations are precisely the integrability conditions for deforming (differential) graded algebras: the classical equation corresponds to integrability in differential graded commutative algebras, while the quantum equation arises in the BV–algebra framework with a second-order operator $\Delta$ leading to a Maurer–Cartan-type structure for an $L_\infty$-algebra; by embedding the BV anti-field/ghost machinery into a deformation-theoretic setting, it unifies gauge symmetry, higher homotopy structures, and field-theoretic quantization—exemplified by Zwiebach’s closed string field theory, which furnishes a concrete $L_\infty$-structure and yields the quantum Master Equation $(S,S)=\Delta S$ from moduli-space geometry. The work clarifies that the master equations are not mere analogies but essential integrability conditions for deformations, with important implications for higher-spin interactions and string field theory. It also highlights Tate resolutions and Chevalley–Eilenberg-type structures as concrete realizations of the underlying homotopy algebra governing gauge symmetries and their deformations.
Abstract
The Batalin-Vilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the Batalin-Vilkovisky approach is here translated into the language of deformation theory.