# Manchester Geometry Seminar 2015/2016

**Thursday 25 February 2016. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
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Weak Poisson Brackets for Gauge Systems

Matthew Peddie (University of Manchester)

`matthew.peddie@manchester.ac.uk`

In a generic gauge system, the existence of a classical BRST operator allows one to identify the equations of motion and the gauge symmetries independently of any Lagrangian or Hamiltonian function. In order to deal with non-Lagrangian/Hamiltonian gauge systems, an arbitrary geometric construction was given which admits an embedding into the BRST framework. The construction was equipped with a Poisson type bracket in order to consistently quantise these theories, and the geometric set-up together with the bracket is called a weak Poisson system.

In this talk I will give details of this construction and define a lift of the weak Poisson structure to an $L_\infty$-structure in the algebra of differential forms. This $L_\infty$-structure reduces to an odd Poisson bracket, the Koszul bracket, on the subalgebra of differential forms invariant over the gauge orbits. Such a bracket can be viewed as an extension of the even Poisson bracket constructed on the space of co-exact forms for generalised Hamiltonian mechanics to the case of dynamical systems with constraints and gauge symmetries.