  Causally ordered correlation functions of local operators in near-thermal
quantum systems computed using the Schwinger-Keldysh formalism obey a set of
Ward identities. These can be understood rather simply as the consequence of a
topological (BRST) algebra, called the universal Schwinger-Keldysh
superalgebra, as explained in our companion paper arXiv:1610.01940. In the
present paper we provide a mathematical discussion of this topological algebra.
In particular, we argue that the structures can be understood in the language
of extended equivariant cohomology. To keep the discussion self-contained, we
provide a basic review of the algebraic construction of equivariant cohomology
and explain how it can be understood in familiar terms as a superspace gauge
algebra. We demonstrate how the Schwinger-Keldysh construction can be
succinctly encoded in terms a thermal equivariant cohomology algebra which
naturally acts on the operator (super)-algebra of the quantum system. The main
rationale behind this exploration is to extract symmetry statements which are
robust under renormalization group flow and can hence be used to understand
low-energy effective field theory of near-thermal physics. To illustrate the
general principles, we focus on Langevin dynamics of a Brownian particle,
rephrasing some known results in terms of thermal equivariant cohomology. As
described elsewhere, the general framework enables construction of effective
actions for dissipative hydrodynamics and could potentially illumine our
understanding of black holes.
