Ph.D in Mathematical Physics, Russian Academy of Sciences. 2008
M.S. in Physics, St. Petersburg State University. 2005
B.S. in Physics, St. Petersburg State University. 2002
Mathematical Physics, Topology
My research is in mathematical physics, more precisely I am interested in the interactions of quantum field theory with topology, homological/homotopical algebra and supergeometry.
One important homological algebra technique that allows one to construct mathematically the path integral quantization of gauge theories (and, in particular, topological field theories) is the Batalin-Vilkovisky formalism. E.g. for a class of topological field theories (the so-called Alexandrov-Kontsevich-Schwarz-Zaboronsky sigma models) it leads to quantum partition functions expressed in terms of (finite-dimensional) integrals over Fulton-MacPherson compactified configuration spaces of points on a manifold which constitute interesting manifold invariants compatible with gluing-cutting.
Some questions I am interested in:
- Constructing exact discretizations of topological field theories on manifolds endowed with CW decompositions.
- Comparison of perturbative results in quantum field theory with non-perturbative ones, via the globalization procedure (relying on techniques of formal geometry) on the moduli space of solutions to Euler-Lagrange equations.
- Extending the perturbative path integral construction of quantum field theory to manifolds with corners of codimension 2 and higher, and comparing with Baez-Dolan-Lurie framework of (fully) extended topological quantum field theory.
- “Secondary" renormalization flow in topological field theory and Igusa-Klein higher torsions.
- How does renormalization in non-topological field theories interact with gluing/cutting?
- P. Mnev, “Discrete BF theory," arXiv:0809.1160.
- A. Alekseev, P. Mnev, "One-dimensional Chern-Simons theory,” Commun. Math. Phys. 307.1 (2011) 185-227.
- A. S. Cattaneo, P. Mnev, N. Reshetikhin, “Classical BV theories on manifolds with boundary,” Commun. Math. Phys. 332.2 (2014) 535-603.
- P. Mnev, "A construction of observables for AKSZ sigma models,” Lett. Math. Phys. 105.12 (2015) 1735-1783.
- A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Perturbative quantum gauge theories on manifolds with boundary," arXiv:1507.01221 (2015), to appear in Commun. Math. Phys.