Speaker
Ezra Getzler, Northwestern University & Uppsala University
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^(p,q)_infinity, 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^(d-1,q)_infinity à Omega^(d,q)_infinity
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 à Omega^(0,1)_infinity 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^(*,0)_infinity , 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^(*,0)_infinity.
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.