arXiv subjects

Adrià Marín-Salvador

Publications and source records attributed to Adrià Marín-Salvador.

8 recordsLinked to original sources

arXiv2026

Continuous Tambara-Yamagami tensor categories

We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group $G$. A continuous Tambara-Yamagami tensor category associated to a locally compact group $G$ is a continuous tensor category that has a single non-invertible simple object $τ$ such that $τ\otimes τ$ decomposes as a direct integral indexed over $G$, meaning $τ\otimesτ\cong L^2(G)$. We show that continuous Tambara-Yamagami tensor categories for $G$ are classified by a continuous symmetric nondegenerate bicharacter $χ: G\times G\to U(1)$ and a sign $ξ\in\{\pm 1\}$. We also prove that, if a $\mathrm{W}^*$-tensor category $\mathcal{C}$ obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that $\mathcal{C}$ is obtained from a continuous tensor category by forgetting its topology.

math.QA↗

arXiv2026

Twisted representations of conformal nets and crossed balanced tensor categories

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with an action of a discrete group $G$. We show that the category $\text{Rep}^G(\mathcal{A})$ of $G$-twisted representations of $\mathcal{A}$ is canonically a $G$-crossed balanced $\mathrm{W}^*$-tensor category. This extends the results of Müger arXiv:math/0403322, in the language of localized endomorphisms, that $\text{Rep}^G(\mathcal{A})$ is a $G$-crossed braided tensor category.

math.QA↗

arXiv2026

Balanced tensor categories of representations of fixed-points conformal nets

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let $\text{Rep}^G(\mathcal{A})$ be the $G$-crossed balanced $\mathrm{W}^*$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced $\mathrm{W}^*$-tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ between the $G$-equivariantization of $\text{Rep}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net $\mathcal{A}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

math.QA↗

arXiv2026

The crossed braided tensor category of twisted and untwisted representations of the Heisenberg conformal net

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}/2$-crossed braiding. By taking a $\mathbb{Z}/2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net $\text{Heis}^{\mathbb{Z}/2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.

math.QA↗

arXiv2026

The balanced structure on the category of representations of a conformal net

Let $\mathcal{A}$ be a (not necessarily rational) conformal net. We show that the braided $\mathrm{W}^*$-tensor category $\text{Rep}(\mathcal{A})$ of representations of $\mathcal{A}$ is canonically a balanced $\mathrm{W}^*$-tensor category. The balance is given by the action of $e^{-2πi L_0}$, where $L_0$ denotes the generator of rotations on $S^1$. In arXiv:2606.03623, we generalize this result to the larger context of a group acting on $\mathcal{A}$. We provide here a more accessible proof for the case where no group is present.

math.QA↗

arXiv2023

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes $\{(\mathbb{S}^2\times\mathbb{S}^1, g_\circ-\frac{d^2}{c^2}dt^2)\}_{d,c\in\mathbb{N}^+\text{ coprime}}$, with $g_\circ$ the round metric on $\mathbb{S}^2$ and $t$ the $\mathbb{S}^1$-coordinate. We find that these are the lens spaces $L(2c,1)$ together with the pushforward of the canonical contact structure on $ST\mathbb{S}^2\cong L(2,1)$ under the natural projection $L(2,1)\to L(2c,1)$. We extend this computation to $Z\times \mathbb{S}^1$ for $Z$ a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.

math.DG↗

arXiv2021

Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation

This work applies the contact formalism of classical mechanics and classical field theory, introduced by Herglotz and later developed in the context of contact geometry, to describe electromagnetic systems with dissipation. In particular, we study an electron in a non-perfect conductor and a variation of the cyclotron radiation. In order to apply the contact formalism to a system governed by the Lorentz force, it is necessary to generalize the classical electromagnetic gauge and add a term in the Lagrangian. We also apply the k-contact theory for classical fields to model the behaviour of electromagnetic fields themselves under external damping. In particular, we show how the theory describes the evolution of electromagnetic fields in media under some circumstances. The corresponding Poynting theorem is derived. We discuss its applicability to the Lorentz dipole model and to a highly resistive dielectric.

physics.class-ph↗

arXiv2021

On the Canonical Contact Structure of the Space of Null Geodesics of a Spacetime

The space of null geodesics of a spacetime carries a canonical contact structure which has proved to be key in the discussion of causality in spacetimes. However, not much progress has been made on its nature and not many explicit calculations for specific spacetimes can be found over the literature. We compute the spaces of null geodesics and their canonical contact structures for the manifold $\mathbb{S}^2\times\mathbb{S}^1$ equipped with the family of metrics $\{g_c = g_\circ-\frac{1}{c^2}dt^2 \}_{c\in\mathbb{N}^+}$. We obtain that these are the lens spaces $L(2c,1)$ and that the contact structures are the pushforward of the canonical contact structure on $ST\mathbb{S}^2\cong L(2,1)$ under the projection map. We also study the applicability of Engel geometry on the discussion of three-dimensional spacetimes. We show that, for a particular type of three-dimensional spacetimes, one can obtain the space of null geodesics and its contact structure solely from the information of the Lorentz prolongation of the spacetime. We present an approach that makes use of this result to recover the spacetime from its space of null geodesics and skies.

math.DG↗