arXiv subjects

Florian Girelli

Publications and source records attributed to Florian Girelli.

At least 19 recordsLinked to original sources

arXiv2026

Integrability from categorification and the 2-Kac-Moody Algebra

The theory of Lie bialgebras and the classical Yang-Baxter equation plays a major role in the study of 1+1d integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra. A categorified, higher homotopy notion of Lie algebras has been studied, which gave rise to the notion of (strict) Lie 2-bialgebras (Lie algebra crossed-modules) and the 2-graded Yang-Baxter equations. In this paper, we use these differential graded structures to generalize the construction of a Lax pair and introduce an appropriate notion of higher-dimensional integrability. Within this framework, we introduce a higher derived version of the affine Kac-Moody algebra, which underpins the 2-graded Lax integrability that we have developed here as a zero 2-curvature condition. As an explicit demonstration, we will consider a 3d field theory and show that it (i) is 2-graded Lax integrable and (ii) hosts symmetries governed by a Kac-Moody 2-algebra.

math-ph↗

arXiv2026

Gauging the Gauge and Anomaly Resolution

In this paper, we explore the algebraic and geometric structures that arise from a procedure we dub "gauging the gauge", which involves the promotion of a certain global, coordinate independent symmetry to a local one. By gauging the global 1-form shift symmetry in a gauge theory, we demonstrate that the structure of a Lie algebra crossed-module and its associated 2-gauge theory arises. Moreover, performing this procedure once again on a 2-gauge theory generates a 3-gauge theory, based on Lie algebra 2-crossed-modules. As such, we show that the physical procedure of "gauging the gauge" can be understood mathematically as a categorification. Applications of such higher-gauge structures are considered, including general relativity, high-energy physics and condensed matter theory. Of particular interest is the mechanism of anomaly resolution, in which one introduces a higher-gauge structure to absorb curvature defects. This mechanism has been shown to allow one to consistently gauge an anomalous background symmetry in QFT

hep-th↗

arXiv2026

Topological field theory plus local Lorentz symmetry is gravity

Four-dimensional gravity admits many equivalent formulations - metric, Einstein-Cartan, teleparallel, McDowell-Mansouri, among others - each offering distinct advantages, particularly, in view of quantization. We propose a new formulation based on Weyl spinor-valued 1-forms, ultimately encoding the frame-field data. Starting from a topological field theory with a global $\mathrm{SL}(2,\mathbb{C})$ symmetry, we show that promoting this symmetry to a local gauge symmetry leads to the emergence of gravity. We analyze the covariant phase space of this theory, its symmetries and charge structure and explore the role of admissible corner terms together with their impact on boundary charges and their algebra. We study several extensions of this framework, including the incorporation of a cosmological constant and a novel $ G \rightarrow 0 $ scaling limit obtained from this model. The presence of the frame field already at the topological level allows point particles to be coupled uniformly in both the topological and gravitational theories. We perform a detailed Hamiltonian analysis of the theory and clarify the implementation of the reality conditions. We argue that this formulation provides structural features that make it particularly well suited for both discretization and quantization.

gr-qc↗

arXiv2026

A Covariant Formulation of Logarithmic Supertranslations at Spatial Infinity

We investigate the asymptotic symmetries of asymptotically flat spacetimes at spatial infinity. We propose a new symplectic structure and conservative boundary conditions in a polyhomogeneous Beig-Schmidt expansion. The asymptotic symmetries extend the BMS algebra by abelian sectors, notably incorporating regular log-translations and log-supertranslations. The associated charges are finite and conserved, and we show that their algebra admits a central extension between supertranslations and log-supertranslations, and between the singular translations and regular log-translations. Our analysis is compatible with, and extends, both the work of arXiv:1106.4045 and arXiv:2211.10941 : it extends the former by incorporating log-supertranslations, and the latter by allowing both parities of the log-supertranslations in the same phase space. These newly identified symmetries at spatial infinity encode novel physical information that has not been revealed in other regions of asymptotically flat spacetimes, thereby opening the door to new observables to consider at null and timelike infinity.

hep-th↗

arXiv2026

Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory

We explore global Poisson-Lie (PL) symmetries using a Lagrangian, or "covariant phase space" approach, that manifestly preserves spacetime covariance. PL symmetries are the classical analog of quantum-group symmetries. In the Noetherian framework symmetries leave the Lagrangian invariant up to boundary terms and necessarily yield (on closed manifolds) $\mathfrak{g}^{*}$-valued conserved charges which serve as Hamiltonian generators of the symmetry itself. Non-trivial PL symmetries transcend this framework by failing to be symplectomorphisms and by admitting (conserved) non-Abelian group-valued momentum maps. In this paper we discuss various structural and conceptual challenges associated with the implementation of PL symmetries in field theory, focusing in particular on non-locality. We examine these issues through explicit examples of low-dimensional field theories with non-trivial PL symmetries: the deformed spinning top (or, the particle with curved momentum and configuration space) in 0+1D; the non-linear $σ$-model by Klimčík and Ševera (KS) in 1+1D; and gravity with a cosmological constant in 2+1D. Although these examples touch on systems of different dimensionality, they are all ultimately underpinned by 2D $σ$-models, specifically the A-model and KS model.

hep-th↗

arXiv2026

Hopf 2-algebras and Braided Monoidal 2-Categories

Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of $\mathsf{2Vect}^{hBC}$, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-$R$-matrices", which can be seen as an extension of the usual notion of $R$-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-$R$-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit.

math.QA↗

arXiv2025

Teleparallel gravity at null infinity

Four-dimensional asymptotically flat spacetimes have been central to recent developments in infrared physics. Gravitational waves reaching the asymptotic boundary reveal an infinite-dimensional symmetry group known as the Bondi-Metzner-Sachs (BMS) group. The vacuum structure breaks this symmetry, giving rise to Goldstone modes that play a pivotal role in the analysis of scattering amplitudes. However, these modes must be added to the phase space of the metric formulation. In this work, we explore an alternative formulation of general relativity, teleparallel gravity, which is dynamically equivalent to the standard metric description in the bulk. This framework relies on a tetrad field to encode the gravitational degrees of freedom and a flat connection to represent inertial effects. Leveraging this decomposition, we propose a novel encoding of the Goldstone modes within the flat connection. We examine the implications of this approach in the covariant phase space framework, focusing on the symplectic potential and its connection to the Wald-Zoupas prescription.

gr-qc↗

arXiv2025

Origin of the quantum group symmetry in 3d quantum gravity

It is well-known that quantum groups are relevant to describe the quantum regime of 3d gravity. They encode a deformation of the gauge symmetries parametrized by the value of the cosmological constant. They appear as a form of regularization either through the quantization of the Chern-Simons formulation or the state sum approach of Turaev-Viro. Such deformations are perplexing from a continuum and classical picture since the action is defined in terms of undeformed gauge invariance. We present here a novel way to derive from first principle and from the classical action such quantum group deformation. The argument relies on two main steps. First we perform a canonical transformation, which deformed the gauge invariance and the boundary symmetries, and makes them depend on the cosmological constant. Second we implement a discretization procedure relying on a truncation of the degrees of freedom from the continuum.

gr-qc↗

arXiv2022

Diffeomorphisms as quadratic charges in 4d BF theory and related TQFTs

We present a Sugawara-type construction for boundary charges in 4d BF theory and in a general family of related TQFTs. Starting from the underlying current Lie algebra of boundary symmetries, this gives rise to well-defined quadratic charges forming an algebra of vector fields. In the case of 3d BF theory (i.e. 3d gravity), it was shown in [PRD 106 (2022), arXiv:2012.05263 [hep-th]] that this construction leads to a two-dimensional family of diffeomorphism charges which satisfy a certain modular duality. Here we show that adapting this construction to 4d BF theory first requires to split the underlying gauge algebra. Surprisingly, the space of well-defined quadratic generators can then be shown to be once again two-dimensional. In the case of tangential vector fields, this canonically endows 4d BF theory with a $\mathrm{diff}(S^2)\times\mathrm{diff}(S^2)$ or $\mathrm{diff}(S^2)\ltimes\mathrm{vect}(S^2)_\mathrm{ab}$ algebra of boundary symmetries depending on the gauge algebra. The prospect is to then understand how this can be reduced to a gravitational symmetry algebra by imposing Plebański simplicity constraints.

hep-th↗

arXiv2022

Local Observables in $\operatorname{SU}_q(2)$ Lattice Gauge Theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

hep-lat↗

arXiv2022

Group field theory on quantum groups

We introduce the framework of Hopf algebra field theory (HAFT) which generalizes the notion of group field theory to the quantum group (Hopf algebra) case. We focus in particular on the 3d case and show how the HAFT we considered is topological. The highlight of the construction is the notion of plane-wave which leads, in the specific example of SUq (2) with q real, to a discretization of the Euclidian BF action with a negative cosmological constant. This work can be viewed as the generalization of the seminal work by Baratin and Oriti: we have a (non-commutative) formulation of simplicial 3d gravity in the presence of a non-zero cosmological constant.

hep-th↗

arXiv2022

Group field theory on 2-groups

Group field theories are quantum field theories built on groups. They can be seen as a tool to generate topological state-sums or quantum gravity models. For four dimensional manifolds, different arguments have pointed towards 2-groups (such as crossed modules) as the relevant symmetry structure to probe four dimensional topological features. Here, we introduce a group field theory built on crossed modules which generate a four dimensional topological model, as we prove that the Feynman diagram amplitudes can be related by Pachner moves. This model is presumably the dual version of the Yetter-Mackaay model.

hep-th↗

arXiv2022

Numerical evaluation of spin foam amplitudes beyond simplices

We present the first numerical calculation of the 4D Euclidean spin foam vertex amplitude for vertices with hypercubic combinatorics. Concretely, we compute the amplitude for coherent boundary data peaked on cuboid and frustum shapes. We present the numerical algorithms to explicitly compute the vertex amplitude and compare the results in different cases to the semi-classical approximation of the amplitude. Overall we find good qualitative agreement of the amplitudes and evidence of convergence of the asymptotic formula to the full amplitude already at fairly small spins, yet also differences in the frequency of oscillations and a phase shift absent in the 4-simplex case. However, due to rapidly growing numerical costs, we cannot reach sufficiently high spins to prove agreement of both amplitudes. Lastly, this setup allows us to explore non-uniform vertex amplitudes, where some representations are small while others are large; we find indications that scenarios might exist in which the semi-classical amplitude is a valid approximation even if some spins remain small. This suggests that the transition of the quantum to the semi-classical regime (for a single vertex amplitude) is intricate.

gr-qc↗

arXiv2022

Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity

Recently, a new choice of variables was identified to understand how the quantum group structure appeared in three-dimensional gravity [1]. These variables are introduced via a canonical transformation generated by a boundary term. We show that this boundary term can actually be taken to be the volume of the boundary and that the new variables can be defined in any dimension greater than three. In addition, we study the associated metric and teleparallel formalisms. The former is a variant of the Henneaux--Teitelboim model for unimodular gravity. The latter provides a non-Abelian generalization of the usual Abelian teleparallel formulation.

gr-qc↗

arXiv2022

(2-)Drinfel'd Double and (2-)BF Theory

The gauge symmetry and shift/translational symmetry of a 3D BF action, which are associated to a pair of dual Lie algebras, can be combined to form the Drinfel'd double. This combined symmetry is the gauge symmetry of the Chern-Simons action which is equivalent to the BF action, up to some boundary term. We show that something similar happens in 4D when considering a 2-BF action (aka BFCG action), whose symmetries are specified in terms of a pair of dual strict Lie 2-algebras (ie. crossed-modules). Combining these symmetries gives rise to a 2-Drinfel'd double which becomes the gauge symmetry structure of a 4D BF theory, up to a boundary term. Concretely, we show how using 2-gauge transformations based on dual crossed-modules, the notion of 2-Drinfel'd double defined in Ref. arXiv:1109.1344 appears. We also discuss how, similarly to the Lie algebra case, the symmetric contribution of the $r$-matrix of the 2-Drinfel'd double can be interpreted as a quadratic 2-Casimir, which allows to recover the notion of duality.

hep-th↗

arXiv2021

Semidual Kitaev lattice model and tensor network representation

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on $M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to $D(H)$. Given a finite-dimensional Hopf algebra $H$, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

math.QA↗

arXiv2021

Polyhedron phase space using 2-groups: kappa-Poincare as a Poisson 2-group

We construct a phase space for a three dimensional cellular complex with decorations on edges and faces using crossed modules (strict 2-groups) equipped with a (non-trivial) Poisson structure. We do not use the most general crossed module, but only the ones where the target map (t-map) is trivial. As a particular case, we recover that deformations of the Poincare group can be exported to deformations of the Poincare 2-group. The kappa-deformation case provides a natural candidate for describing discrete geometries with curved edge decorations (but still flat face decorations). Our construction generalizes the classical phase space defined in [5] for a 3d triangulation in terms of an un-deformed Poincare 2-group.

hep-th↗

arXiv2021

Discretization of 4d Poincaré BF theory: from groups to 2-groups

We study the discretization of a Poincaré/Euclidean BF theory. Upon the addition of a boundary term, this theory is equivalent to the BFCG theory defined in terms of the Poincaré/Euclidean 2-group. At an intermediate step in the discretization, we note that there are multiple options for how to proceed. One option brings us back to recovering the discrete variables and phase space of BF theory. Another option allows us to rediscover the phase space related to the G-networks given in [2]. Indeed, our main result is that we are now able to relate the continuum fields with the discrete variables in [2]. This relation is important to determine how to implement the simplicity constraints to recover gravity using the BFCG action. In fact, we show that such relation is not as simple as in the BF discretization: the discretized variables on the triangles actually depend on several of the continuum fields instead of solely the continuum B-field. We also compare and contrast the discretized BF and BFCG models as pairs of dual 2-groups. This work highlights (again) how the choice of boundary term influences the resulting symmetry structure of the \textit{discretized theory} -- and hence ultimately the choice of quantum states.

hep-th↗