Seminar talk, 27 November 2024

Speaker: Ekaterina Shemyakova

Title: On differential operators generating higher brackets

Abstract:
On supermanifolds, a Poisson structure can be either even, corresponding to a Poisson  bivector, or odd, corresponding to an odd Hamiltonian quadratic in momenta. An odd  Poisson bracket can also be defined by an odd second-order differential operator that  squares to zero, known as a "BV-type" operator.

A higher analog, $P_{\infty }$ or $S_{\infty }$, is a series of brackets of alternating  parities or all odd, respectively, that satisfy relations that are higher homotopy  analogs of the Jacobi identity. These brackets are generated by arbitrary multivector  fields or Hamiltonians. However, generating an $S_{\infty }$-structure by a higher-order  differential operator is not straightforward, as this would violate the Leibniz  identities. Kravchenko and others studied these structures, and Voronov addressed  the Leibniz identity issue by introducing formal $\hbar$-differential operators. 

In this talk, we revisit the construction of an $\hbar$-differential operator that  generates higher Koszul brackets on differential forms on a $P_{\infty }$-manifold. 

It is well known that a chain map between the de Rham and Poisson complexes on a  Poisson manifold at the same time maps the Koszul bracket of differential forms to the  Schouten bracket of multivector fields. In the $P_{\infty }$-case, however, the chain map  is also known, but it does not connect the corresponding bracket structures. An  $L_{\infty }$-morphism from the higher Koszul brackets to the Schouten bracket has been  constructed recently, using Voronov's thick morphism technique. In this talk, we will  show how to lift this morphism to the level of operators.

The talk is partly based on joint work with Yagmur Yilmaz.