arXiv subjects
Publications and source records attributed to Pavel Mnev.
We study a one-dimensional toy version of the Chern-Simons theory. We construct its simplicial version which comprises features of a low-energy effective gauge theory and of a topological quantum field theory in the sense of Atiyah.
The perturbative Chern-Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties (as is the case, e.g., with the propagator defined by Axelrod and Singer). It turns out that the effective BV action is a function on cohomology (with shifted degrees) that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely. Out of it one obtains invariants.
We describe a canonical reduction of AKSZ-BV theories to the cohomology of the source manifold. We get a finite dimensional BV theory that describes the contribution of the zero modes to the full QFT. Integration can be defined and correlators can be computed. As an illustration of the general construction we consider two dimensional Poisson sigma model and three dimensional Courant sigma model. When the source manifold is compact, the reduced theory is a generalization of the AKSZ construction where we take as source the cohomology ring. We present the possible generalizations of the AKSZ theory.
In this work we discuss the simplicial program for topological field theories for the case of non-abelian BF theory. Discrete BF theory with finite-dimensional space of fields is constructed for a triangulated manifold (or for a manifold equipped with cubical cell decomposition), that is in a sense equivalent to the topological BF theory on manifold. This discrete version allows one to calculate interesting quantities from the BF theory, like the effective action on cohomology, in terms of finite-dimensional integrals instead of functional integrals, as demonstrated in a series of explicit examples. We also discuss the interpretation of discrete BF action as the generating function for $qL_\infty$ structure (certain "one-loop version" of ordinary $L_\infty$ algebra) on the cell cochains of triangulation, related to the de Rham algebra of the underlying manifold by homotopy transfer procedure. This work is a refinement of older text hep-th/0610326.
In this work we discuss the construction of "simplicial BF theory", the field theory with finite-dimensional space of fields, associated to a triangulated manifold, that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of simplicial program - program of constructing discrete topological field theories. We also discuss the relation of these constructions to homotopy algebra.
We give a new proof of the trace formula for regular graphs. Our approach is inspired by path integral approach in quantum mechanics, and calculations are mostly combinatorial.