Pavel Mnev

At MPIM Bonn

Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556

Office: 124 Hayes-Healy Hall
+1 574 631 6288

I am an assistant professor at the math department of the University of Notre Dame. Here is my CV.

Research Interests:
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?

Publications and preprints:
  1. P. Mnev, "Towards simplicial Chern-Simons theory, I" an unpublished draft (2005), here.
    We consider the problem of discretizing topological field theories of Chern- Simons type. Our main instruments are Batalin-Vilkovysky formalism and standard Feynman diagram technique for functional integral in QFT. We solve the 1-dimensional case of the problem completely and check that the result does not exhibit nontrivial renormalization. Then we face the 2-dimensional case of discretizing theory on a triangle. We use Dupont construction for choosing the gauge and find the action of the discretized theory in lowest nontrivial order (containing higher classical and quantum operations). Next we check (by direct calculations, in lowest nontrivial order) that this action does possess nontrivial topological (BV-exact) renormalization flow for quantum operations under barycentric gluing, while classical operations are recovered.

  2. P. Mnev, "On the simplicial BF model," J. Math. Sci. 141, 4 (2007) 1429-1431.

  3. P. Mnev, "Discrete path integral approach to the trace formula for regular graphs", Commun. Math. Phys. 274.1 (2007) 233-241, arXiv:math-ph/0609028.
    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.

  4. P. Mnev, "Notes on simplicial BF theory", Moscow Math. J 9.2 (2009) 371-410, arXiv: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.

  5. P. Mnev, "On simplicial BF theory," Russian Academy of Sciences, Doklady Mathematics 77.1 (2008) 59-63. English translation. Original Russian version.
    In this note we outline the construction of simplicial BF theory on a triangulated manifold as a special case of the construction of effective action for the abstract BF theory by means of a Batalin-Vilkovisky integral. We present our results for the simplicial action as closed formulae in dimensions 0 and 1 and as perturbative answers in higher dimension. We explain the action of simplicial BF theory in terms of homotopy transfer of algebraic structures.

  6. P. Mnev, "Discrete BF theory," arXiv:0809.1160 [hep-th] (the English translation of my Ph.D. thesis).
    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 a cubical cell decomposition), that is in a sense equivalent to the topological BF theory on the manifold. This discrete version allows one to calculate certain 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 the qL-infinity structure (certain “one-loop version” of the ordinary L-infinity algebra) on the cell cochains of triangulation, related to the de Rham algebra of the underlying manifold by the homotopy transfer procedure. This work is a refinement and build-up on our previous work [arXiv:hep-th/0610326].

  7. A. S. Cattaneo, P. Mnev, "Remarks on Chern-Simons invariants", Commun. Math. Phys. 293.3 (2010) 803-836, arXiv:0811.2045 [math.QA].
    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.

  8. F. Bonechi, P. Mnev, M. Zabzine, "Finite dimensional AKSZ-BV theories", Lett. Math. Phys. 94.2 (2010) 197-228, arXiv:0903.0995 [hep-th].
    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.

  9. A. Alekseev, P. Mnev, "One-dimensional Chern-Simons theory,” Commun. Math. Phys. 307.1 (2011) 185-227, arXiv:1005.2111 [hep-th].
    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.

  10. A. S. Cattaneo, F. Bonechi, P. Mnev, "The Poisson sigma model on closed surfaces," JHEP 1 (2012), 1-27, arXiv:1110.4850 [hep-th].
    Using methods of formal geometry, the Poisson sigma model on a closed surface is studied in perturbation theory. The effective action, as a function on vacua, is shown to have no quantum corrections if the surface is a torus or if the Poisson structure is regular and unimodular (e.g., symplectic). In the case of a Kahler structure or of a trivial Poisson structure, the partition function on the torus is shown to be the Euler characteristic of the target; some evidence is given for this to happen more generally. The methods of formal geometry introduced in this paper might be applicable to other sigma models, at least of the AKSZ type.

  11. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Classical BV theories on manifolds with boundary," Commun. Math. Phys. 332.2 (2014) 535-603, arXiv:1201.0290 [math-ph].
    In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the BF theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.

  12. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Classical and quantum Lagrangian field theories with boundary,"arXiv:1207.0239 [math-ph].
    This note gives an introduction to Lagrangian field theories in the presence of boundaries. After an overview of the classical aspects, the cohomological formalisms to resolve singularities in the bulk and in the boundary theories (the BV and the BFV formalisms, respectively) are recalled. One of the goals here (and in [arXiv:1201.0290]) is to show how the latter two formalisms can be put together in a consistent way, also in view of perturbative quantization.

  13. P. Mnev, "A construction of observables for AKSZ sigma models,” Lett. Math. Phys. 105.12 (2015) 1735-1783, arXiv:1212.5751 [math-ph].
    A construction of gauge-invariant observables is suggested for a class of topological field theories, the AKSZ sigma-models. The observables are associated to extensions of the target Q-manifold of the sigma model to a Q-bundle over it with additional Hamiltonian structure in fibers.

  14. A. Alekseev, Y. Barmaz, P. Mnev, "Chern-Simons theory with Wilson lines and boundary in the BV-BFV Formalism," J. Geom. and Phys. 67 (2013) 1-15, arXiv:1212.6256 [math-ph].
    We consider the Chern-Simons theory with Wilson lines in 3D and in 1D in the BV-BFV formalism of Cattaneo-Mnev-Reshetikhin. In particular, we allow for Wilson lines to end on the boundary of the space-time manifold. In the toy model of 1D Chern-Simons theory, the quantized BFV boundary action coincides with the Kostant cubic Dirac operator which plays an important role in representation theory. In the case of 3D Chern-Simons theory, the boundary action turns out to be the odd (degree 1) version of the BF model with source terms for the B field at the points where the Wilson lines meet the boundary. The boundary space of states arising as the cohomology of the quantized BFV action coincides with the space of conformal blocks of the corresponding WZW model.

  15. A. S. Cattaneo, P. Mnev, "Wave relations," Commun. Math. Phys. 332.3 (2014) 1083-1111, arXiv:1308.5592 [math-ph].
    The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly there is still something to say on this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian evolution are not problematic from the viewpoint of relations. In the two-dimensional Minkowski case, these relations are shown to be Lagrangian. This result is then extended to a wide class of metrics and is conjectured to be true also in higher dimensions for nice enough metrics. A counterexample where the relation is not Lagrangian is provided by the Misner space.

  16. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Semiclassical quantization of classical field theories," Mathematical Aspects of Quantum Field Theories, Springer (2015) 275-324,  arXiv:1311.2490 [math-ph].
    These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline formal semiclassical quantization in the finite dimensional case. Towards the end we give an example of such a quantization in the case of Abelian Chern-Simons theory.

  17. P. Mnev, "Lecture notes on torsions," arXiv:1406.3705 [math.AT].
    These are the lecture notes for the introductory course on Whitehead, Reidemeister and Ray-Singer torsions, given by the author at the University of Zurich in Spring semester 2014.

  18. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Perturbative quantum gauge theories on manifolds with boundary," Commun. Math. Phys. 357.2 (2018) 631-730, arXiv:1507.01221 [math-ph].
    This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.

  19. A. Alekseev, O. Chekeres, P. Mnev, "Wilson surface observables from equivariant cohomology," JHEP 93.11 (2015), arXiv:1507.06343 [hep-th].
    Wilson lines in gauge theories admit several path integral descriptions. The first one (due to Alekseev-Faddeev-Shatashvili) uses path integrals over coadjoint orbits. The second one (due to Diakonov-Petrov) replaces a 1-dimensional path integral with a 2-dimensional topological σ-model. We show that this σ-model is defined by the equivariant extension of the Kirillov symplectic form on the coadjoint orbit. This allows to define the corresponding observable on arbitrary 2-dimensional surfaces, including closed surfaces. We give a new path integral presentation of Wilson lines in terms of Poisson σ-models, and we test this presentation in the framework of the 2-dimensional Yang-Mills theory. On a closed surface, our Wilson surface observable turns out to be nontrivial for G non-simply connected (and trivial for G simply connected), in particular we study in detail the cases G=U(1) and G=SO(3).

  20. A. S. Cattaneo, P. Mnev, K. Wernli, "Split Chern-Simons theory in the BV-BFV formalism," Quantization, Geometry and Noncommutative Structures in Mathematics and Physics, Springer, Cham (2017) 293–324, arXiv:1512.00588 [math.GT].
    The goal of this note is to give a brief overview of the BV-BFV formalism developed by the first two authors and Reshetikhin in [arXiv:1201.0290], [arXiv:1507.01221] in order to perform perturbative quantisation of Lagrangian field theories on manifolds with boundary, and present a special case of Chern-Simons theory as a new example.

  21. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Perturbative BV theories with Segal-like gluing," arXiv:1602.00741 [math-ph].
    This is a survey of our program of perturbative quantization of gauge theories on manifolds with boundary compatible with cutting/pasting and with gauge symmetry treated by means of a cohomological resolution (Batalin-Vilkovisky) formalism. We also give two explicit quantum examples -- abelian BF theory and the Poisson sigma model. This exposition is based on a talk by P.M. at the ICMP 2015 in Santiago de Chile.

  22. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "A cellular topological field theory," arXiv:1701.05874 [math.AT].
    We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms.

  23. P. Mnev, "Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory," arXiv:1707.08096 [math-ph].
    Lecture notes for the course "Batalin-Vilkovisky formalism and applications in topological quantum field theory" given at the University of Notre Dame in the Fall 2016 for a mathematical audience. In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.
  24. A. S. Losev, P. Mnev, D. R. Youmans, "Two-dimensional abelian BF theory in Lorenz gauge as a twisted N=(2,2) superconformal field theory," J. Geom. Phys. 131C (2018) 122-137, arXiv:1712.01186 [hep-th].
    We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N = (2, 2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The BRST charge of the gauge-fixed theory equals the total Q of type B twisted theory. This unexpected identification of two different theories opens a way for nontrivial deformations of both of these theories.
  25. A. S. Cattaneo, P. Mnev, N. Reshetikhin, "Poisson sigma model and semiclassical quantization of integrable systems,"  in Ludwig Faddeev Memorial Volume: A Life in Mathematical Physics, World Scientific (2018), arXiv:1803.07723 [math-ph].
    In this paper we outline the construction of semiclassical eigenfunctions of integrable models in terms of the semiclassical path integral for the Poisson sigma model with the target space being the phase space of the integrable system. The semiclassical path integral is defined as a formal power series with coefficients being Feynman diagrams. We also argue that in a similar way one can obtain irreducible semiclassical representations of Kontsevich's star product.

  26. R. Iraso, P. Mnev, "Two-dimensional Yang-Mills theory on surfaces with corners in Batalin-Vilkovisky formalism," arXiv:1806.04172 [math-ph].
    In this paper we recover the non-perturbative partition function of 2D Yang-Mills theory from the perturbative path integral. To achieve this goal, we study the perturbative path integral quantization for 2D Yang-Mills theory on surfaces with boundaries and corners in the Batalin-Vilkovisky formalism (or, more precisely, in its adaptation to the setting with boundaries, compatible with gluing and cutting - the BV-BFV formalism). We prove that cutting a surface (e.g. a closed one) into simple enough pieces - building blocks - and choosing a convenient gauge-fixing on the pieces, and assembling back the partition function on the surface, one recovers the known non-perturbative answers for 2D Yang-Mills theory.

Some talk slides and recordings:

Expository material:

Fall 2018: MATH 20550-04, 20550-06 Calculus III.

Spring 2018: MATH 20580-03, 20580-05 Introduction to linear algebra and differential equations.
Link to the course webpage.
Notes: 1/17, 1/19, 1/24, 1/26, 1/29, 1/31, 2/2, 2/5, 2/7, 2/9, 2/12, 2/14, 2/16, 2/19, 2/21, 2/23, 2/26, 2/28, 3/2, 3/5, 3/7, 3/9, 3/19, 3/21, 3/23, 3/26, 3/28, 4/4, 4/6, 4/9, 4/11, 4/13, 4/16, 4/18, 4/20, 4/23, 4/25, 4/27, 4/30, 5/2.

Fall 2017: MATH 70330 Intermediate Geometry and Topology.
A Kan style graduate student seminar. Here are the syllabus and the detailed plan.

Spring 2017: MATH 20580-04 Introduction to linear algebra and differential equations.
Link to the course webpage.

Fall 2016: MATH 80430 Topics in topology I: "Batalin-Vilkovisky formalism and applications in topological quantum field theory."
Link to the course webpage.

Some materials from courses and mini-courses I have taught in the past:

Quantum BV theories on manifolds with boundary (a mini-course at the conference Higher Structures in Geometry and Physics 2015, MPIM Bonn, October 2015).
Notes: part 1, part 2.

Introduction to quantum electrodynamics for mathematicians (a mini-course at MPIM Bonn, December 2014).
Notes: 1 2.

Batalin-Vilkovisky formalism in topological quantum field theory (MPIM Bonn/University of Bonn, Fall 2014).
Program:  here.
Notes: part 1, part 2, part 3.

Torsions (University of Zurich, Spring 2014).
Poster: here.
Notes: arXiv:1406.3705 [math.AT].

Moduli space of flat connections (a mini-course at the Galileo Galilei Insitute, Florence, October 2013).
Video recording of the lectures: 1 2 3.
Notes (in Russian!) from another instance of this mini-course (Chebyshev Laboratory, St. Petersburg, May 2013): 1-3.

Topological quantum field theory (University of Zurich, Spring 2013).
Notes: here.

Conformal field theory, part II (University of Zurich, Fall 2011).
Notes: 2 3 4,5 6 7,8 9 10 11 12 13 14.
Course webpage: link.

Conformal field theory, part I (University of Zurich, Spring 2011).
Notes: 1 2 3 4 5 6 7 8 9 10 11 12 13 14.
Course webpage: link.


My father is a mathematician: webpage.

Link to my old webpage.

Also, I like to play Irish and Breton music on flute.