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Riccardo Villa

Publications and source records attributed to Riccardo Villa.

8 recordsLinked to original sources
arXiv2026

On gauging Abelian extensions of finite and U(1) groups

We consider Abelian extensions of global symmetries of the form $A \to G \to K$, with $A$ finite. For a quantum field theory $\mathcal{T}$ with symmetry $G$, we compare gauging $G$ directly with gauging first $A$ and then $K$, and show that for finite Abelian groups and for $K \simeq U(1)$ the two procedures are equivalent as expected, $\mathcal{T}/G \simeq \mathcal{T}/A/K$. In the continuous case $K=U(1)$, after gauging the full extension, the dual symmetry $\widehat{\mathbb{Z}}_q^{(d-2)}$ fits into an extension characterizing the topological data of the magnetic $U(1)_m^{(d-3)}$ symmetry. This is better described using differential cohomology.

hep-th↗
arXiv2026

$Γ$-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations

We present several asymptotic results concerning the non-local Massari Problem for sets with prescribed mean curvature. In particular, we show that the fractional Massari functional $Γ$-converges to the classical one, and this convergence preserves minimizers in the $L^1_{\mbox{loc}}$-topology. This returns useful information about the asymptotic behavior of the solutions of the inhomogeneous Allen-Cahn equation in the forced and the mass-prescribed settings. In this context, a new geometric object, which we refer to as "non-local hybrid mean curvature", naturally appears.

math.AP↗
arXiv2026

Gauging in superconductors and other electronic systems

Ordinary, s-wave superconductors have been recognized as being topological phases of matter, in which the dynamical gauge field implies less understood global features. Using the tools of topological field theories and generalized symmetries, we provide an updated description of these systems. At very low energies, the Higgs model reduces to the BF theory, which exhibits topological order. Furthermore, the gauge field must be a spin$_c$ connection, to describe the spin of fermions forming Cooper pairs. Gauging implies that superconductors are inherently bosonic systems, yet they are endowed with a gravito-magnetic anomaly that is the remnant of their fermionic origin. We recognize that this anomaly is related to the Gaiotto-Kapustin-Thorngren bosonization, achieved via gauging fermion parity $(-1)^F$, now included in the gauge dynamics. This anomaly characterizes gauged electronic matter in great generality in three and four spacetime dimensions, forbidding trivial massive phases at low energy. It holds beyond the validity of the Higgs model, nd in other kinds of superconductors as well. It also appears in the nontrivial massless phase of three-dimensional electrodynamics, recently understood.

hep-th↗
arXiv2026

Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.

math.AP↗
arXiv2025

On non-local almost minimal sets and an application to the non-local Massari's Problem

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter. In addition, we also discuss stickiness phenomena for non-local almost minimal sets.

math.AP↗
arXiv2025

Bosonizations and dualities in 2+1 dimensions

We discuss two methods for relating bosonic and fermionic relativistic field theories in 2+1 dimensions, the $Z_2^f$ gauging and the flux attachment. The first is primarily a correspondence between topological theories. It amounts to summing over fermionic spin structures, as is familiar in two-dimensional conformal theories. Its inverse map, fermionization, shows how spin structures and $Z_2^f$ fermion parity emerge from a bosonic theory equipped with a dual $Z_2^{(1)}$ generalized symmetry. The second method,flux attachment, gives spin and statistics to charged particles by coupling them to a Chern-Simons theory, and provides the basis for the Abelian dualities. We illustrate the two bosonizations with explicit results in a solvable semiclassical conformal theory, and show their differences and interplays with particle-vortex dualities. We employ the so-called loop model, which can describe general infrared critical points in 2+1 dimensions in the semiclassical limit. We also combine the two bosonizations to obtain further duality relations. By applying $Z_2^f$ gauging to the Dirac-boson and Majorana-boson flux-attachment dualities, we find new relations between bosonic theories.

hep-th↗
arXiv2024

Bosonization of 2+1 dimensional fermions on the surface of topological insulators

Three-dimensional topological insulators can be described by an effective field theory involving two `hydrodynamic' Abelian gauge fields. The action contains a bulk topological BF term and a surface term, called loop model. This describes the massless 2+1 dimensional excitations and provides them with a semiclassical, yet non-trivial conformal invariant dynamics. Given that topological insulators are originally fermionic, this physical setting is ideal for realizing the bosonization of massless fermions in terms of gauge fields. Building on earlier analyses of the loop model, we find that fermions belong to the solitonic spectrum and can be described by Wilson lines, through the generalization of 1+1 dimensional vertex operators. Their correlation functions agree with conformal invariance. The bosonic loop model is then mapped into a fermionic theory by using the general construction of fermionic topological phases described in the literature. It requires the identification of the characteristic one-form $Z_2$ symmetry of the bosonic theory and its gauging, which originates the fermion number $(-1)^F$, the spin sectors and the time reversal symmetry obeying ${\cal T}^2=(-1)^F$. These results are detailed for the effective action and the partition function on the geometry $S^2\times S^1$.

hep-th↗