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We derive equivariant localization formulas of Atiyah--Bott and cohomological field theory types in the Batalin-Vilkovisky formalism and discuss their applications in Poisson geometry and quantum field theory., Comment: 13 pages

This study aims to observe if the theorem prover Lean positively influences students' understanding of mathematical proving. To this end, we perform a pilot study concerning freshmen students at the University of Zurich (UZH). While doing so, we apply certain teaching methods and gather data from the volunteer students enrolled in the ``Foundations of Mathematics'' course. After eleven weeks of study covering some exercise questions implemented with Lean, we measure Lean students' performances in proving mathematical statements, compared to other students who are not engaged with Lean. For this measurement, we interview five Lean and four Non-Lean students and we analyze the scores of all students in the final exam. Finally, we check significance by performing a $t$-test for independent samples and the Mann-Whitney $U$-test.

Three-dimensional supergravity in the Batalin-Vilkovisky formalism is constructed by showing that the theory including the Rarita-Schwinger term is equivalent to an AKSZ theory., Comment: 24 pages

The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example., Comment: 26 pages, made some small corrections

In this paper, we present a theory for gravity coupled with scalar, SU$(n)$ and spinor fields on manifolds with null-boundary. We perform the symplectic reduction of the space of boundary fields and give the constraints of the theory in terms of local functionals of boundary vielbein and connection. For the three different couplings, the analysis of the constraint algebra shows that the set of constraints does not form a first class system.

We study a family of (possibly non topological) deformations of $BF$ theory for the Lie algebra obtained by quadratic extension of $\mathfrak{so}(3,1)$ by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin--Vilkovisky formalism. In particular, both Palatini--Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin--Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields., Comment: 61 pages + bibliography. v2: removed leftover review comments

This note gives an overview of the BV formalism in its various incarnations and applications., Comment: 16 pages. Article commissioned by the Encyclopedia of Mathematical Physics

This explanatory note, based on the geometrical method by Kijovski and Tulczyjew, describes the construction of the reduced phase space of Lagrangian field theories, i.e., the correct space of initial conditions with its symplectic structure. Several examples and, in particular, the case of four-dimensional gravity in the coframe formalism (Palatini--Cartan theory) are analyzed., Comment: 23 pages. References updated. Article commissioned by the Encyclopedia of Mathematical Physics

We present general reduction procedures for Courant, Dirac and generalized complex structures, in particular when a group of symmetries is acting. We do so by taking the graded symplectic viewpoint on Courant algebroids and carrying out graded symplectic reduction, both in the coisotropic and hamiltonian settings. Specializing the latter to the exact case, we recover in a systematic way the reduction schemes of Bursztyn-Cavalcanti-Gualtieri., Comment: 80 pages. v2: Added Lemma 7.1. Added Example 6.15 and consequences thereof (Examples 7.6, 7.7, 7.12, 8.20, 8.21, 8.22)

This is a didactical review on the canonical BV Laplacian on half-densities., Comment: 20 pages. Minor style improvements

We study four dimensional gauge theories in the context of an equivariant extension of the Batalin-Vilkovisky (BV) formalism. We discuss the embedding of BV Yang-Mills (YM) theory into a larger BV theory and their relation. Partial integration in the equivariant BV setting (BV push-forward map) is performed explicitly for the abelian case. As result, we obtain a non-local homological generalization of the Cartan calculus and a non-local extension of the abelian YM BV action which satisfies the equivariant master equation., Comment: 27 pages, refs added, published version

In classical field theory, gluing spacetime manifolds along boundary corresponds to taking a fiber product of the corresponding spaces of fields (as differential graded Fr\'echet manifolds) up to homotopy. We construct this homotopy explicitly in several examples in the setting of BV-BFV formalism (Batalin--Vilkovisky formalism with cutting--gluing).

The boundary structure of $3+1$-dimensional gravity (in the Palatini-Cartan formalism) coupled to to gauge (Yang-Mills) and matter (scalar and spinorial) fields is described through the use of the Kijowski-Tulczijew construction. In particular, the reduced phase space is obtained as the reduction of a symplectic space by some first class constraints and a cohomological description (BFV) of it is presented., Comment: 59 pages. Some minor corrections. Close to published version

An extension of the notion of classical equivalence of equivalence in the Batalin--(Fradkin)--Vilkovisky (BV) and (BFV) framework for local Lagrangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of non-Abelian Yang--Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV-BFV description, are shown to be pairwise equivalent as strict BV-BFV theories. This in particular implies that their BV-complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically-equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV-BFV formulation. They are shown to be equivalent as lax BV-BFV theories and to have isomorphic BV cohomologies. This shows that strict BV-BFV equivalence is a strictly finer notion of equivalence of theories., Comment: Published version

We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $I \times \Sigma$, with $I$ an interval and $\dim \Sigma = 4l+2$, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $\dim \Sigma = 0$ 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 $\Sigma$. For $\dim \Sigma = 2$, this implies a version of the CS-WZW correspondence. For $\dim \Sigma = 6$, 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., Comment: 81 pages, 11 figures. v.2: updated references. v.3: small expository changes

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., Comment: 96 pages, 9 figures. Updated references; some historical comments added. In v3: Section 4.3 has been rewritten

We analyse the boundary structure of General Relativity in the coframe formalism in the case of a lightlike boundary, i.e., when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1 dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case., Comment: 42 pages, accepted for publication in Ann. Henri Poincar\'e. Improved version, added section about independence of the structure from the choices

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schr\"odinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds., Comment: 47 pages

In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein-Hilbert and of the Palatini--Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein--Hilbert theory, in the latter a BV theory for Palatini--Cartan theory with a partial implementation of the torsion-free condition already on the space of fields. All theories described here are BV versions of the same classical system on cylinders. The AKSZ implementations we present have the advantage of yielding a compatible BV-BFV description, which is the required starting point for a quantization in presence of a boundary., Comment: Improved version with new results about Palatini-Cartan theory. 41 Pages

An explicit, geometric description of the first-class constraints and their Poisson brackets for gravity in the Palatini-Cartan formalism (in space-time dimension greater than three) is given. The corresponding Batalin- Fradkin-Vilkovisky (BFV) formulation is also developed., Comment: Accepted manuscript. Few exposition and stylistic improvements. 32 pages

We study an equivariant extension of the Batalin-Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang-Mills in 2d and of Donaldson-Witten theory., Comment: 22 pages, refs added, typos corrected, the final published version in J.Geom.Phys

These notes give an introduction to the mathematical framework of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms. Some of the presented content was given as a mini course by the first author at the 2018 QSPACE conference in Benasque., Comment: 63 pages, 10 figures, minor changes made (added some references, corrected some typos and some small mistakes)

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $\Theta$-invariant that we can define even if the cohomology group is not vanishing., Comment: 13 pages

A Lagrangian subspace $L$ of a weak symplectic vector space is called \emph{split Lagrangian} if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for $L$ to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace $C$ of a weak symplectic space $V$ which imply that the induced canonical relation $L_C$ from $V$ to $C/C^{\omega}$ is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations., Comment: 33 pages, 2 figures

This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the contexts of super differentiable and of derived algebraic geometry., Comment: 13 pages. Some corrections. Some references added and some updated

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 $\hbar$, 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., Comment: 25 pages, 16 figures, comments are welcome

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., Comment: 56 pages, 22 figures, glossary added, presentation improved for non experts. Section 2 is very similar to Section 2 of our companion paper arXiv:1807.11782. Also Appendix A.1-3, on the compactification of configuration spaces of manifolds with boundary, are taken from there but Section A.4, on the case of manifolds with corners, is added. To appear in Commun. Math. Phys

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., Comment: 41 pages, 9 figures, typos fixed, presentation improved for non experts, to appear in Commun. Math. Phys

We discuss observables of an equivariant extension of the A-model in the framework of the AKSZ construction. We introduce the A-model observables, a class of observables that are homotopically equivalent to the canonical AKSZ observables but are better behaved in the gauge fixing. We discuss them for two different choices of gauge fixing: the first one is conjectured to compute the correlators of the A-model with target the Marsden-Weinstein reduced space; in the second one we recover the topological Yang-Mills action coupled with A-model so that the A-model observables are closed under supersymmetry., Comment: 16 pages; minor corrections

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., Comment: 22 pages, 12 figures

The triadic description of General Relativity in three dimensions is known to be a BF theory. Diffeomorphisms, as symmetries, are easily re- covered on shell from the symmetries of BF theory. This note describes an explicit off-shell BV symplectomorphism between the BV versions of the two theories, each endowed with their natural symmetries.

We show that the Palatini--Cartan--Holst formulation of General Relativity in tetrad variables must be complemented with additional requirements on the fields when boundaries are taken into account for the associated BV theory to induce a compatible BFV theory on the boundary., Comment: 22 pages. Corrected typos in some formulae. Minor aesthetic fixes

General relativity in four dimensions can be reformulated as a gauge theory, referred to as Palatini-Cartan-Holst theory. This paper describes its reduced phase space using a geometric method due to Kijowski and Tulczyjew and its relation to that of the Einstein-Hilbert approach., Comment: Revised version comprising new results, a correction of Th 4.22 and the arguments leading to it. Manuscript accepted for publication in AHP. 31 pages

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., Comment: Ver. 2: numerous expository improvements

We show that the non-Abelian nature of geometric fluxes---the corner-stone in the definition of quantum geometry in the framework of loop quantum gravity (LQG)---follows directly form the continuum canonical commutations relations of gravity in connection variables and the validity of the Gauss law. The present treatment simplifies previous formulations and thus identifies more clearly the root of the discreteness of geometric operators in LQG. Our statement generalizes to arbitrary gauge theories and relies only on the validity of the Gauss law., Comment: 5 pages, no figures (journal version)

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., Comment: 32 pages, 15 figures, some changes (including the correction of a minor mistake) included; new appendix with the explicit computations added; to appear in Lett. Math. Phys

This note describes the functional-integral quantization of two-dimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the Batalin-Vilkovisky formalism is included., Comment: 26 pages. This is my contribution to ICM 2006 which I never uploaded on the arXiv

We discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the discussion in [arXiv:0706.3164], where a gauge fixing defined by a compatible complex structure was introduced, by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss the off-shell supersymmetry of the A-model as a residual BV symmetry of the gauge-fixed PSM action., Comment: 15 pages, one missing reference added

This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely the classically-equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method., Comment: 36 pages. Improved exposition. To appear in Lett. Math. Phys

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., Comment: 8 pages, 1 figure. (Ver. 2 has a figure added and some typos corrected.)

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., Comment: 19 pages, 2 figures. Based on the contribution presented at "Geometric, Algebraic and Topological Methods for Quantum Field Theory", July 20-31, Villa de Leyva, Colombia v2: First author's name changed to include middle name

The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any $d + 1 \not= 2$ (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered in a straightforward way., Comment: 16 pages

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., Comment: Ver. 2: Improved presentation, especially we added a detailed discussion of the space of states. To appear in Commun. Math. Phys

This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed., Comment: 36 pages, 1 figure