arXiv subjects

Emanuele Latini

Publications and source records attributed to Emanuele Latini.

At least 19 recordsLinked to original sources

arXiv2026

On Braided Differential Calculi and Quantum G-structures

We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford--Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer--Cartan form is shown to decompose into a direct sum of the Maurer--Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer--Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter--Drinfeld module of $O_q(\mathrm{GL}_2)$.

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arXiv2026

Noncommutative vector field calculi

We discuss noncommutative differential geometry from a vector field centric point of view. This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair. We define the universal FOVC and construct an adjunction between the categories of FOVC and that of first order differential calculi, showing that the vector field approach is dual, though not equivalent, to the differential form one. This correspondence is then extended to covariant vector field and differential calculi. On Hopf algebras, (bi)covariant FOVC are in bijection with (bicovariant) quantum tangent spaces. For Hopf--Galois extensions, quantum tangent spaces give rise to vertical vector fields and in this setup we further describe base vector fields and horizontal vector fields and show that they are related via a noncommutative Atiyah sequence. Multiple examples, based on braided derivations, finite groups and the quantum Hopf fibration, are given. The vector field approach is further enriched by a sheaf-theoretic treatment, which recovers the former as a local, or affine, picture.

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arXiv2025

Gauge transformations on quantum principal bundles

We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a compatible differential structure, one requires the coaction to extend as a morphism of differential graded algebras. This leads to an exact noncommutative Atiyah sequence, a graded Hopf--Galois extension of differential forms and a canonical braiding on total space forms such that the latter are graded-braided commutative. We recall this approach to noncommutative differential geometry and further discuss the extension of quantum gauge transformations, in the sense of Brzeziński, to differential forms. In this way we obtain an action of quantum gauge transformations on connections of the quantum principal bundle and their curvature. Explicit examples, such as the noncommutative 2-torus, the quantum Hopf fibration and smash product algebras are discussed.

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arXiv2024

On the Durdevic approach to quantum principal bundles

We revisit and extend the Durdevic theory of complete calculi on quantum principal bundles. In this setting one naturally obtains a graded Hopf-Galois extension of the higher order calculus and an intrinsic decomposition of degree 1-forms into horizontal and vertical forms. This proposal is appealing, since it is consistently equipped with a canonical braiding and exactness of the Atiyah sequence is guaranteed. Moreover, we provide examples of complete calculi, including the noncommutative 2-torus, the quantum Hopf fibration and differential calculi on crossed product algebras.

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arXiv2024

Conformally compact and higher conformal Yang-Mills equations

On conformally compact manifolds we study Yang-Mills equations, their boundary conditions, formal asymptotics, and Dirichlet-to-Neumann maps. We find that smooth solutions with "magnetic" Dirichlet boundary data are obstructed by a conformally invariant, higher order boundary current. We study the asymptotics of the interior Yang-Mills energy functional and show that the obstructing current is the variation of the conformally invariant coefficient of the first log term in this expansion which is a higher derivative conformally invariant analog of the Yang-Mills energy. The invariant energy is the anomaly for the renormalized interior Yang-Mills functional and its variation gives higher conformal Yang-Mills equations. Global solutions to the magnetic boundary problem determine higher order "electric" Neumann data. This yields the Dirichlet-to-Neumann map. We also construct conformally invariant, higher transverse derivative boundary operators. Acting on interior connections, they give obstructions to solving the Yang-Mills boundary problem, determine the asymptotics of formal solutions, and yield conformally invariant tensors capturing the (non-local) electric Neumann data. We also characterize a renormalized Yang-Mills action functional that encodes global features analogously to the renormalized volume for Poincaré-Einstein structures.

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arXiv2024

Reduction of Quantum Principal Bundles over non affine bases

In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle $\mathrm{GL}(n) \to \mathrm{GL}(n)/P= \mathbf{P}^{n-1}(\mathbb{C})$ to the Levi subgroup $G_0$ inside the maximal parabolic subgroup $P$ of $\mathrm{GL}(n)$. We characterize reductions in the sheaf theoretic setting.

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arXiv2021

One-loop quantum gravity from the N=4 spinning particle

We construct a spinning particle that reproduces the propagation of the graviton on those curved backgrounds which solve the Einstein equations, with or without cosmological constant, i.e. Einstein manifolds. It is obtained by modifying the N=4 supersymmetric spinning particle by relaxing the gauging of the full SO(4) R-symmetry group to a parabolic subgroup, and selecting suitable Chern-Simons couplings on the worldline. We test it by computing the correct one-loop divergencies of quantum gravity in D=4.

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arXiv2021

The Quantum Darboux Theorem,

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.

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arXiv2020

Quantum principal bundles on projective bases

The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study the quantization of principal bundles G -> G/P, where G is a semisimple group and P a parabolic subgroup.

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arXiv2018

Quantum Klein Space and Superspace

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.

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arXiv2017

Klein and Conformal Superspaces, Split Algebras and Spinor Orbits

We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra $\mathbb{C}_{s}$. We exploit the observation that certain split form of orthogonal groups can be realized in terms of matrix groups over split composition algebras; this leads to a natural interpretation of the the sections of the spinor bundle in the critical split dimensions $D=4$, $6$ and $10$ as $\mathbb{C}_{s}^{2}$, $\mathbb{H}_{s}^{2}$ and $\mathbb{O}_{s}^{2}$, respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-Conformal superspace structure.

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arXiv2015

The symplectic origin of conformal and Minkowski superspaces

Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.

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arXiv2015

Spinning particles and higher spin field equations

Relativistic particles with higher spin can be described in first quantization using actions with local supersymmetry on the worldline. First, we present a brief review of these actions and their use in first quantization. In a Dirac quantization scheme the field equations emerge as Dirac constraints on the Hilbert space, and we outline how they lead to the description of higher spin fields in terms of the more standard Fronsdal-Labastida equations. Then, we describe how these actions can be extended so that the propagating particle is allowed to take different values of the spin, i.e. carry a reducible representation of the Poincaré group. This way one may identify a four dimensional model that carries the same degrees of freedom of the minimal Vasiliev's interacting higher spin field theory. Extensions to massive particles and to propagation on (A)dS spaces are also briefly commented upon.

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arXiv2015

A worldline approach to colored particles

Relativistic particle actions are a useful tool to describe quantum field theory effective actions using a string-inspired first-quantized approach. Here we describe how to employ suitable particle actions in the computation of the scalar contribution to the one-loop gluon effective action. We use the well-known method of introducing auxiliary variables that create the color degrees of freedom. In a path integral they implement automatically the path ordering needed to ensure gauge invariance. It is known that the color degrees of freedom introduced this way form a reducible representation of the gauge group. We describe a method of projecting onto the fundamental representation (or any other chosen irrep, if desired) of the gauge group. Previously, we have discussed the case of anticommuting auxiliary variables. Choosing them to be in the fundamental representation allows to obtain, without any extra effort, also the situation in which the color is given by any antisymmetric tensor product of the fundamental. Here, we describe the novel case of bosonic auxiliary variables. They can be used equivalently for creating the color charges in the fundamental representation. In addition one gets, as a byproduct, the cases where the particle can have the color sitting in any symmetric tensor product of the fundamental. This is obtained by tuning to a different value a Chern Simons coupling, present in the model, which controls how the projection is achieved.

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arXiv2015

Effective action for higher spin fields on (A)dS backgrounds

We study the one loop effective action for a class of higher spin fields by using a first-quantized description. The latter is obtained by considering spinning particles, characterized by an extended local supersymmetry on the worldline, that can propagate consistently on conformally flat spaces. The gauge fixing procedure for calculating the worldline path integral on a loop is delicate, as the gauge algebra contains nontrivial structure functions. Restricting the analysis on (A)dS backgrounds simplifies the gauge fixing procedure, and allows us to produce a useful representation of the one loop effective action. In particular, we extract the first few heat kernel coefficients for arbitrary even spacetime dimension D and for spin S identified by a curvature tensor with the symmetries of a rectangular Young tableau of D/2 rows and [S] columns.

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arXiv2014

Massive and massless higher spinning particles in odd dimensions

We study actions for massive bosonic particles of higher spins by dimensionally reducing an action for massless particles. For the latter we take a model with a SO(N) extended local supersymmetry on the worldline, that is known to describe massless (conformal) particles of higher spins in flat spacetimes of even dimensions. Dimensional reduction produces an action for massive spinning particles in odd dimensions. The field equations that emerge in a quantization a la Dirac are shown to be equivalent to the Fierz-Pauli ones. The massless limit generates a multiplet of massless states with higher spins, whose first quantized field equations have a geometric form with fields belonging to various types of Young tableaux. These geometric equations can be partially integrated to show their equivalence with the standard Fronsdal-Labastida equations. We covariantize our model to check whether an extension to curved spacetimes can be achieved. Restricting to (A)dS spaces, we find that the worldline gauge algebra becomes nonlinear, but remains first class. This guarantees consistency on such backgrounds. A light cone analysis confirms the presence of the expected propagating degrees of freedom. A covariant analysis is worked out explicitly for the massive case, which is seen to give rise to the Fierz-Pauli equations extended to (A)dS spaces. It is worth noting that in D=3 the massless limit of our model when N goes to infinity has the same field content of the Vasiliev's theory that accommodates each spin exactly once.

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arXiv2013

Particles with non abelian charges

Efficient methods for describing non abelian charges in worldline approaches to QFT are useful to simplify calculations and address structural properties, as for example color/kinematics relations. Here we analyze in detail a method for treating arbitrary non abelian charges. We use Grassmann variables to take into account color degrees of freedom, which however are known to produce reducible representations of the color group. Then we couple them to a U(1) gauge field defined on the worldline, together with a Chern-Simons term, to achieve projection on an irreducible representation. Upon gauge fixing there remains a modulus, an angle parametrizing the U(1) Wilson loop, whose dependence is taken into account exactly in the propagator of the Grassmann variables. We test the method in simple examples, the scalar and spin 1/2 contribution to the gluon self energy, and suggest that it might simplify the analysis of more involved amplitudes.

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arXiv2013

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of [31] to a conformally invariant, Poincare--Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulae for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of [9]. This complements the results of Aubry and Guillarmou [3] where associated conformal harmonic spaces parametrise smooth solutions.

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