arXiv subjects

Francesco Bonechi

Publications and source records attributed to Francesco Bonechi.

At least 19 recordsLinked to original sources

arXiv2026

3D gravity and double copy theory

We introduce a novel reformulation of three-dimensional gravity in terms of divergenceless vector frames, inspired by the double copy for Chern-Simons theory. This formulation is on-shell equivalent to conventional 3D gravity and provides a transparent geometric interpretation of the double-copy construction. We relate the resulting theory to a Chern-Simons-like action, propose a higher-dimensional origin, and explore extensions that give rise to $AdS_3$

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arXiv2025

Off-shell double copy theories in BV

We present a construction of the double copy for gauge theories that exhibit off-shell color-kinematics duality within the Batalin-Vilkovitsky (BV) formalism. As illustrative examples, we consider the double copies of Chern-Simons theory, four-dimensional BF theory, and two-dimensional Yang-Mills theory, and we discuss possible gravity interpretations for these cases. We formalize the construction and demonstrate that Kodaira-Spencer gravity, K\"ahler gravity, and their generalizations, fit naturally within this framework. In particular, Kodaira-Spencer gravity emerges as a gauge theory describing deformations of generalized complex structures, while the double copy of Chern-Simons theory captures the deformation of the Courant bracket.

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arXiv2024

Off-shell color-kinematics duality from codifferentials

We examine the color-kinematics duality within the BV formalism, highlighting its emergence as a feature of specific gauge-fixed actions. Our goal is to establish a general framework for studying the duality while investigating straightforward examples of off-shell color-kinematics duality. In this context, we revisit Chern-Simons theory as well as introduce new examples, including BF theory and 2D Yang-Mills theory, which are shown to exhibit the duality off-shell. We emphasize that the geometric structures responsible for flat-space color-kinematics duality appear for general curved spaces as well.

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arXiv2022

Towards equivariant Yang-Mills theory

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.

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arXiv2021

Nijenhuis tensor and invariant polynomials

We discuss the diagonalization problem of the Nijenhuis tensor in a class of Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We study its action on the ring of invariant polynomials of a Thimm chain of subalgebras. The existence of $ϕ$-minimal representations defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such representations exist in the classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We discuss a second general construction that in these two cases computes partially the spectrum and hints at a different behavior with respect to the classical cases.

math.SG↗

arXiv2021

On Curvature and Torsion in Courant Algebroids

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.

math.DG↗

arXiv2020

Shifted Poisson structures on differentiable stacks

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a $\mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack ${\mathfrak X}$. It turns out that shifted $(+1)$ Poisson structures on ${\mathfrak X}$ correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla. (2) We introduce the notion of tangent complex $T_{\mathfrak X}$ and cotangent complex $L_{\mathfrak X}$ of a differentiable stack ${\mathfrak X}$ in terms of any Lie groupoid $Γ{\rightrightarrows} M$ representing ${\mathfrak X}$. They correspond to homotopy class of 2-term homotopy $Γ$-modules $A[1]\rightarrow TM$ and $T^\vee M\rightarrow A^\vee[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak X}$, defines a morphism ${L_{\mathfrak X}}[1]\to {T_{\mathfrak X}}$.

math.DG↗

arXiv2020

Equivariant Batalin-Vilkovisky formalism

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.

hep-th↗

arXiv2019

Generalised Kähler Structure on $\mathbb{C}P^2$ and Elliptic Functions

We construct a toric generalised Kähler structure on $\mathbb{C}P^2$ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.

math.DG↗

arXiv2016

Comparing Poisson Sigma Model with A-model

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.

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arXiv2015

Multiplicative integrable models from Poisson-Nijenhuis structures

We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].

math.SG↗

arXiv2015

Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.

math.SG↗

arXiv2012

AKSZ construction from reduction data

We discuss a general procedure to encode the reduction of the target space geometry into AKSZ sigma models. This is done by considering the AKSZ construction with target the BFV model for constrained graded symplectic manifolds. We investigate the relation between this sigma model and the one with the reduced structure. We also discuss several examples in dimension two and three when the symmetries come from Lie group actions and systematically recover models already proposed in the literature.

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arXiv2012

The Poisson sigma model on closed surfaces

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.

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arXiv2011

Wilson Lines from Representations of NQ-Manifolds

An NQ-manifold is a non-negatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual differential geometry, we analyze the subtleties in the generalization to the NQ-setting and we also sketch some possible applications of our construction.

math.DG↗

arXiv2009

Finite dimensional AKSZ-BV theories

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.

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arXiv2009

Poisson sigma model on the sphere

We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.

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arXiv2006

Geometric quantization and non-perturbative Poisson sigma model

In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class $[α]$. The same condition was considered before by Crainic and Zhu but in a different context. In the case when $[α]$ is in the image of the sharp map we reproduce the Vaisman's condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show, with a different procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end we discuss the case of a generic coisotropic D-brane.

math.SG↗