My research revolves around the path integral and various mathematical formulations of it. In particular, I am studying its perturbative formulation in the Batalin-Vilkovisky formalism and looking for connections with non-perturbative definitions, such as functorial quantum field theories.
Below is a list of my research papers in approximately chronological order (newer ones first).
14. Pavel Mnev, Konstantin Wernli: Gluing formulae for heat kernels, arXiv:2404.00156
We state and prove two gluing formulae for the heat kernel of the Laplacian on a Riemannian manifold of the form M_1 \cup_\gamma M_2. We present several examples.
Publication: Journal of Geometry and Physics, Volume 216, October 2025. https://doi.org/10.1016/j.geomphys.2025.105594
13. Ivan Contreras, Santosh Kandel, Pavel Mnev, Konstantin Wernli: Combinatorial QFT on graphs: first quantization formalism, arXiv:2308.07801
We study a combinatorial model of the quantum scalar field with polynomial potential on a graph. In the first quantization formalism, the value of a Feynman graph is given by a sum over maps from the Feynman graph to the spacetime graph (mapping edges to paths). This picture interacts naturally with Atiyah-Segal-like cutting-gluing of spacetime graphs. In particular, one has combinatorial counterparts of the known gluing formulae for Green's functions and (zeta-regularized) determinants of Laplacians.
Publication: . Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024), published online first. https://doi.org/10.4171/aihpd/194
12. Olga Chekeres, Santosh Kandel, Andrey Losev, Pavel Mnev, Konstantin Wernli, Donald Youmans: On enumerative problems for maps and quasimaps: freckles and scars, arXiv:2308.06844
We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the "smooth conjecture" that may lead to computation of the number of maps by an integral over the space of quasimaps.
11. Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli: Quantum Chern-Simons theories on cylinders: BV-BFV partition functions, arXiv:2012.13983
We compute partition functions of Chern-Simons type theories for cylindrical spacetimes I×Σ, with I an interval and dimΣ=4l+2, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case dimΣ=0 is considered as a toy example. We show that one can identify - for certain choices of residual fields - the "physical part" (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper [arXiv:2012.13270], without any quantum corrections. This Hamilton-Jacobi action is the action functional of a conformal field theory on Σ. For dimΣ=2, this implies a version of the CS-WZW correspondence. For dimΣ=6, using a particular polarization on one end of the cylinder, the Chern-Simons partition function is related to Kodaira-Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.
Publication: Commun. Math. Phys. 398, 133-218 (2023), https://doi.org/10.1007/s00220-022-04513-8
10. Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli: Constrained systems, generalized Hamilton-Jacobi actions, and quantization, arXiv:2012.13270
Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton-Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern-Simons theory, where the HJ action turns out to be the gauged Wess-Zumino-Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin-Vilkovisky (BV) formalism in the bulk and of the Batalin-Fradkin-Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern-Simons theory and the toy model for 7D Chern-Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [arXiv:2012.13983]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.
Publication: Journal of Geometric Mechanics 14,179-272 (2022), https://doi.org/10.3934/jgm.2022010
9. Ivan Contreras, Nima Moshayedi, Konstantin Wernli: Convolution algebras for Relational Groupoids and Reduction, arXiv:2008.05281
We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group G and a given normal subgroup H. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.
Publication: Pacific J. Math. 313 (2021) 75-102, https://doi.org/10.2140/pjm.2021.313.75
8. Santosh Kandel, Pavel Mnev, Konstantin Wernli: Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing, arXiv:1912.11202
We study the perturbative quantization of 2-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorial quantum field theory framework of Atiyah-Segal. In particular, we prove that the perturbative partition function defined in terms of integrals over configuration spaces of points on the surface satisfies an Atiyah-Segal type gluing formula. Tadpoles (short loops) behave nontrivially under gluing and play a crucial role in the result.
Publication: Advances in Theoretical and Mathematical Physics 25, no. 7 (2021): 1847-1952, https://doi.org/10.4310/ATMP.2021.v25.n7.a5
7. Pavel Mnev, Michele Schiavina, Konstantin Wernli: Towards holography in the BV-BFV setting, arXiv:1905.00952
We show how the BV-BFV formalism provides natural solutions to descent equations, and discuss how it relates to the emergence of holographic counterparts of given gauge theories. Furthermore, by means of an AKSZ-type construction we reproduce the Chern-Simons to Wess-Zumino-Witten correspondence from infinitesimal local data, and show an analogous correspondence for BF theory. We discuss how holographic correspondences relate to choices of polarisation relevant for quantisation, proposing a semi-classical interpretation of the quantum holographic principle.
Publication: Ann. Henri Poincaré 21, 993–1044 (2020). https://doi.org/10.1007/s00023-019-00862-8
6. Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli: Theta Invariants of lens spaces via the BV-BFV formalism, arXiv:1810.06663
The goal of this paper is to investigate the Theta invariant --- an invariant of framed 3-manifolds associated with the lowest order contribution to the Chern-Simons partition function --- in the context of the quantum BV-BFV formalism. Namely, we compute the state on the solid torus to low degree in ℏ, and apply the gluing procedure to compute the Theta invariant of lens spaces. We use a distributional propagator which does not extend to a compactified configuration space, so to compute loop diagrams we have to define a regularization of the product of the distributional propagators, which is done in an \emph{ad hoc} fashion. Also, a polarization has to be chosen for the quantization process. Our results agree with results in the literature for one type of polarization, but for another type of polarization there are extra terms.
Publication: In: Alekseev, A., Frenkel, E., Rosso, M., Webster, B., Yakimov, M. (eds) Representation Theory, Mathematical Physics, and Integrable Systems. Progress in Mathematics, vol 340. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-78148-4_3
5. Cattaneo, A.S., Moshayedi, N. & Wernli, K. On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism, arXiv:1808.01832
We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum GBFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology.
Publication: Commun. Math. Phys. 375, 41–103 (2020). https://doi.org/10.1007/s00220-020-03726-z
4. Alberto S. Cattaneo, Nima Moshayedi, Konstantin Wernli: Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary, arXiv:1807.11782
We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. This is done by using concepts of formal geometry. Moreover, we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero. This condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we call the modified "differential" Quantum Master Equation.
Publication: Commun. Math. Phys. 372, 213–260 (2019). https://doi.org/10.1007/s00220-019-03591-5
3. Alberto S. Cattaneo, Nima Moshayedi, Konstantin Wernli: Relational Symplectic Groupoid Quantization for Constant Poisson Structures, arXiv:1611.05617
As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization constructed in this paper induces Kontsevich's deformation quantization on the underlying Poisson manifold, i.e., the Moyal product, which is known in full details. This allows focussing on the BV-BFV technology and testing it. For the unexperienced reader, this is also a practical and reasonably simple way to learn it.
Publication: Lett Math Phys 107, 1649–1688 (2017). https://doi.org/10.1007/s11005-017-0959-6
2. Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli: Split Chern-Simons theory in the BV-BFV formalism, arXiv:1512.00588
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.
Publication: In: Cardona A., Morales P., Ocampo H., Paycha S., Reyes Lega A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-65427-0_9
1.Konstantin Wernli: Computing Entanglement Polytopes (Master Thesis), arXiv:1808.03382
In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the entanglement polytopes. Furthermore we present a numerical method which, in theory, can compute the entanglement polytope of any given SLOCC class given a representative. Using it we classify the entanglement polytopes of 2×3×N systems.