The Batalin-Vilkovisky formalism extends Noether's approach to classical field theories, which is restricted to the Euler-Lagrange locus (or "on-shell", as physicists say), off-shell. This is of course important in the study of quantization of field theories, since the quantized theory is not restricted to the Euler-Lagrange locus.

In the variational calculus, the action functional is the integral of a local expression in the fields and their derivatives. The symmetries of the action may be expressed by the classical Batalin-Vilkovisky master equation, which is a Maurer-Cartan equation for functionals of the classical fields, ghost fields expressing the symmetries of the theory, and certain auxilliary fields known as antifields.

The Batalin-Vilkovisky formalism has a natural extension in functionals are lifted to densities. In the first part of today's talk, I explain this extension. which relies on the Soloviev bracket in the variational calculus, originally introduced in the study of general relativity.

Symmetries of a field theory involving diffeomorphisms of the world sheet do not really fit into the formalism of the variational calculus. In my article "Covariance in the Batalin-Vilkovisky formalism",  I explain how to take into account such symmetries of the world sheet by incorporating a curvature term into the Batalin-Vilkovisky master equation, associated to a differential graded Lie algebra with curvature. This construction is the subject of the second part of the talk.

I will finish with a few words on the work of Bonechi, Cattaneo, Qiu and Zabzine, who studied the extension of our formalism to quantum field theory.

Ezra Getzler
University of Northwestern