Ezra Getzler: The Variational Bicomplex and BV Cohomology (mini-course), lecture 3

Abstract

In the study of perturbative field theories, one of the most important invariants is the BV cohomology of the theory: \(H^{-1}\) parametrizes symmetries of the theory, \(H^0\) parametrizes deformations (and in particular, the vector field whose integral is the renormalization group takes values in this space), and \(H^1\) parametrizes obstructions to the integration of deformations.

In this mini-course, we will introduce an important tool in the study of functionals, the variational bicomplex \(\Omega_\infty^{p,q}\), which is the de Rham complex of the jet-space of a bundle \(E\) over the \(d\)-dimensional world-sheet; sections of \(E\) are the fields of the theory. The differential \(d\) splits into the sum of a horizontal differential \(d^{1,0}\) and a vertical differential \(d^{0,1}\). The cokernel of the horizontal differential \(d^{1,0} : \Omega_\infty^{d-1,q} \to \Omega_\infty^{d,q}\) is the space \(F^q\) of variational \(q\)-forms of the field theory; for this reason, the variational bicomplex is a powerful tool for the study of the variational de Rham complex.

For example, the vertical differential \(d^{0,1} : F = F^0 \to \Omega_\infty^{d,1}\) is the Euler-Lagrange operator, or variational derivative.

Soloviev established a canonical lift of Poisson brackets of Gelfand-Dikii type on \(F\) to the complex \((\Omega_\infty^{*,0}, d^{1,0})\). His formula may be applied to the BV antibracket, and gives a resolution of the differential graded (dg) Lie algebra \(F\) in BV theory by a dg Lie algebra structure on \(\Omega_\infty^{*, 0}\).

In these lectures, we will introduce these structures and show how their properties are established. As our main application, we will discuss a refinement of the theorem of Barnich and Grigoriev: the Batalin-Vilkovisky dg Lie algebra of an AKSZ field theory is quasi-isomorphic to the dg Poisson algebra of its target symplectic dg manifold. Time permitting, we will discuss the special cases of a spinning particle (whose quantization is the Dirac operator) and Chern-Simons theory.