arXiv subjects

Mauro Mantegazza

Publications and source records attributed to Mauro Mantegazza.

5 recordsLinked to original sources

arXiv2025

Higher Order Connections in Noncommutative Geometry

We prove that, in the setting of noncommutative differential geometry, a system of higher order connections is equivalent to a suitable generalization of the notion of phase space quantization (in the sense of Moyal star products on the symbol algebra). Moreover, we show that higher order connections are equivalent to (ordinary) connections on jet modules. This involves introducing the notion of natural linear differential operator, as well as an important family of examples of such operators, namely the Spencer operators, generalizing their corresponding classical analogues. Spencer operators form the building blocks of this theory by providing a method of converting between the different manifestations of higher order connections. A system of such higher order connections then gives a quantization, by which we mean a splitting of the quotient projection that defines symbols as classes of differential operators up to differential operators of lower order. This yields a notion of total symbol and of star product, the latter of which corresponds, when restricted to the classical setting, to phase space quantization in the context of quantum mechanics. In this interpretation, we allow the analogues of the position coordinates to form a possibly noncommutative algebra.

math.QA↗

arXiv2025

Symbols in Noncommutative Geometry

In this paper we prove that the classical Lie bracket of vector fields can be generalized to the noncommutative setting by antisymmetrizing (in a suitable noncommutative sense) their compositions. This construction turns out to depend on the representability of linear differential operators, as it relies on the interpretation of vector fields as differential operators. In particular we provide necessary and sufficient conditions for (noncommutative) jet modules to be representing objects for differential operators. Furthermore, the primary ingredient for guaranteeing the closure of a bracket operation is a treatment of symbols, which classically represent, in an intrinsic way, the highest-order term of a differential operator. Thus, we provide an extensive theory of symbols herein.

math.QA↗

arXiv2025

Jet Functors in Noncommutative Geometry

In this article we construct three infinite families of endofunctors $J_d^{(n)}$, $J_d^{[n]}$, and $J_d^n$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $\mathbb{k}$, equipped with an exterior algebra $Ω^\bullet_d$. We prove that these functors generalize the corresponding classical notions of nonholonomic, semiholonomic, and holonomic jet functors, respectively. Our functors come equipped with natural transformations from the identity functor to the corresponding jet functors, which play the rôles of the classical prolongation maps. This allows us to define the notion of linear differential operators with respect to $Ω^{\bullet}_d$. We show that if $Ω^1_d$ is flat as a right $A$-module, the semiholonomic jet functor satisfies the semiholonomic jet exact sequence $0 \rightarrow \bigotimes^n_A Ω^1_d \rightarrow J^{[n]}_d\rightarrow J^{[n-1]}_d \rightarrow 0$. Moreover, we construct a functor of symmetric (in a suitable noncommutative sense) forms $S^n_d$ associated to $Ω^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $δ$-complex. We give necessary and sufficient conditions under which the holonomic jet functor $J_d^n$ satisfies the (holonomic) jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n \rightarrow J_d^{n-1} \rightarrow 0$. In particular, for $n=1$ the sequence is always exact, for $n=2$ it is exact for $Ω^1_d$ flat as a right $A$-module, and for $n\ge 3$, it is sufficient to have $Ω^1_d$, $Ω^2_d$, and $Ω^3_d$ flat as right $A$-modules and the vanishing of the Spencer $δ$-cohomology $H^{\bullet,2}_{δ_d}$.

math.QA↗

arXiv2022

The c-map as a functor on certain variations of Hodge structure

We give a new manifestly natural presentation of the supergravity c-map. We achieve this by giving a more explicit description of the correspondence between projective special Kähler manifolds and variations of Hodge structure, and by demonstrating that the twist construction of Swann, for a certain kind of twist data, reduces to a quotient by a discrete group. We combine these two ideas by showing that variations of Hodge structure give rise to the aforementioned kind of twist data and by then applying the twist realisation of the c-map due to Macia and Swann. This extends previous results regarding the lifting of general isomorphisms along the undeformed c-map, and of infinitesimal automorphisms along the deformed c-map. We show in fact that general isomorphisms can be naturally lifted along the deformed c-map.

math.DG↗

arXiv2021

Construction of projective special Kähler manifolds

In this paper we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on $\mathrm{SU}(n,1)/\mathrm{S}(\mathrm{U}(n)\mathrm{U}(1))$. We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.

math.DG↗