Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

Q-manifolds are supermanifolds equipped with an odd vector field that self-commute, such vector field are called homological. Such objects are found in mathematical physics behind the BV-BRST and BFV formalism of gauge theory, as well as appearing in Poisson geometry in the guise of Lie algebroids. In this talk we revisit the notion of the modular class of a Q-manifold understood as the obstruction to the existence of a Berezin volume that is invariant with respect to the Lie derivative of the homological vector field. In this way we in fact construct a characteristic class of theQ-manifold. Although these notions seem to be known to experts little has appeared in the literature. We will look at some nice examples including L_\infty-algebroids and double Lie algebroids.