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Alberto S. Cattaneo

Publications and source records attributed to Alberto S. Cattaneo.

At least 19 recordsLinked to original sources
arXiv2026

The reduced Dirac structure of General Relativity on manifolds with corners

In this paper, the corner Poisson structure of four-dimensional Palatini-Cartan gravity is derived. Building on the classical description of gravity on manifolds with boundary, specifically on the boundary constraint algebra, a pre-Dirac structure on the space of corner fields is obtained together with a reduction procedure that yields a maximal Dirac structure, identified as the graph of a Poisson bivector field, on the reduced space of corner fields. It is further shown that this Poisson structure admits an equivalent affine Poisson description, which naturally exhibits the reduced corner theory as a $BF$-like theory and leads to a BF$^2$V formulation. This provides the basis for a unified framework for the bulk, boundary, and corner structures of Palatini-Cartan gravity.

math-ph↗
arXiv2026

Corner Quantization of 4D $BF$ Theory

This note studies the quantized corner structure of four-dimensional $BF$ theory, classifies the associated free and physical corner algebras and constructs possible representations. In the abelian case, for arbitrary closed oriented surfaces and in the presence or absence of a cosmological term, explicit presentations of the corner algebras are obtained in terms of generators and relations, identifying them as infinite-dimensional oscillator-type Lie algebras with an abelian summand. A construction of infinite families of simple modules via bosonic Fock space representations is provided. In the non-abelian case on the torus, the corner algebras are described as quotients constructed from the central extensions of double-loop algebras over certain non-semisimple Lie algebras. A construction of infinite families of simple Fock-type modules of the free corner algebra via an induced module procedure is also provided. The resulting modules descend only trivially to the physical quotient, revealing an obstruction in the present construction in the non-abelian setting.

math-ph↗
arXiv2026

From $L_\infty$ algebroids to $L_\infty$ spaces: Part I

The notion of $L_\infty$ spaces over dg manifolds is developed. An equivalence between the category of transitive $L_\infty$ algebroids and that of $L_\infty$ spaces is established, and this equivalence detects weak equivalences. Moreover, a faithful functor from $L_\infty$ algebroids to $L_\infty$ spaces is constructed, which also detects weak equivalences.

math.DG↗
arXiv2026

Graded geometry and generalized reduction

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.

math.SG↗
arXiv2026

BV pushforward as a quasi-isomorphism

Given a BV theory on a space of fields split into two subspaces ("infrared" and "ultraviolet"), one has the BV pushforward map $P_*$, sending observables to observables of the effective theory on the infrared space. This note proves that $P_*$ is a quasi-isomorphism of BV complexes, by realizing it as a part of a strong deformation retraction constructed using the homological perturbation lemma. Two proofs are given: (i) comparing Feynman diagrams for $P_*$ with "cable diagrams" arising from homological perturbation theory and (ii) using topological quantum mechanics. This construction gives a formula for the quasi-inverse $i_\mathrm{int}$ of $P_*$ - the map lifting observables of the effective theory to the full theory. The topological quantum mechanics perspective - and its realization as an AKSZ theory - allows one to write $i_\mathrm{int}$ as a path integral (realizing cable diagrams for $i_\mathrm{int}$ as Feynman diagrams) and to study its classical limit.

math-ph↗
arXiv2026

The Reduced Phase Space of $N=1, D=4$ Supergravity in the BV-BFV formalism

This paper describes the reduced phase space of $N=1$, $D=4$ supergravity in the fully off-shell Palatini--Cartan formalism. This is achieved through the KT construction, allowing an explicit description of first-class constraints on the boundary. The corresponding BFV description is obtained, and its relation with the BV one in the bulk is described by employing the BV pushforward in the particular example of a cylindrical spacetime.

math-ph↗
arXiv2025

Surface Observables, $2$-Knot Invariants, and Nonabelian Electric Fluxes

This work introduces a surface observable for nonabelian four-dimensional $BF$ theory with a cosmological term. The surface observable yields new $2$-knot invariants that may extend beyond known examples such as the Alexander invariant. By BV pushforward, the surface observable induces an electric observable in nonabelian Yang-Mills theory, offering a concrete realization of 't~Hooft operators. An application to self-dual Yang-Mills theory is also discussed.

hep-th↗
arXiv2025

4D Palatini-Cartan Gravity in Hamiltonian Form

In this note the Hamiltonian formulation of four-dimensional gravity, in the Palatini-Cartan formalism, is recovered by elimination of an auxiliary field appearing as part of the connection.

gr-qc↗
arXiv2025

BV Pushforward of Palatini-Cartan gravity

The goal of this note is to show that the standard BV formulation of gravity in the Palatini-Cartan formalism is equivalent, using a BV pushforward, to an AKSZ-like version compatible with BFV data on the boundary.

math-ph↗
arXiv2025

Gravity Coupled with Scalar, SU$(n)$, and Spinor Fields on Manifolds with Null-Boundary

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.

math-ph↗
arXiv2025

BV description of $N = 1$, $D = 4$ Supergravity in the first order formalism

This note examines the BV formulation of $N=1$, $D=4$ supergravity in the first-order Palatini--Cartan framework. Challenges in achieving an off-shell formulation are addressed by introducing corrections to the rank 2 BV action, offering in addition a solid foundation for the study of the theory on manifolds with boundary.

math-ph↗
arXiv2025

Equivariant BV-BFV Formalism

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.

math-ph↗
arXiv2025

Teaching "Foundations of Mathematics" with the Lean Theorem Prover

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.

math.HO↗
arXiv2024

Boundary structure of gauge and matter fields coupled to gravity

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.

math-ph↗
arXiv2023

Gravity with torsion as deformed $BF$ theory

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.

math-ph↗
arXiv2023

Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

This note describes a local Poisson structure (up to homotopy) associated to corners in four-dimensional gravity in the coframe (Palatini--Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure contains in particular an Atiyah algebroid that couples the internal symmetries to diffeomorphisms. The relation with $BF$ theory is also described.

math-ph↗