Manchester Geometry Seminar 2002/2003

Given a differential operator Δ acting on functions on a manifold M. Which geometrical structures are naturally associated with it? It is well known that the top order terms define a function on the cotangent bundle, called the principal symbol. For operators of order ≤ 2 the polarization of the principal symbol is a symmetric binary operation on functions on M (a "bracket") satisfying the Leibniz rule with respect to each argument. For operators of higher order there is a hierarchy of "higher brackets".

As for the lower order terms, it turns out that for an operator acting on functions, its subprincipal symbol, introduced by Hörmander, can be interpreted as an "upper connection" in the bundle of volume forms. This prompts to consider simultaneously operators Δ acting on densities of arbitrary weights. The space V^*(M) of all densities on a manifold M is a commutative associative algebra with a unit and an invariant scalar product. For operators acting in V^*(M), we prove that (unlike the operators on functions alone) they can be uniquely recovered from the corresponding bracket. The theory is valid for supermanifolds, as well as for ordinary manifolds. In the super case, odd operators of the second order are (potentially) generating operators for the so-called Batalin-Vilkovisky algebras, important for quantum field theory. Our approach allows to give a complete analysis of this situation, linking together the conditions of "flatness", Jacobi identity, and conditions for Δ².

(This is a joint work with H. Khudaverdian.)