arXiv subjects

Claudio Meneses

Publications and source records attributed to Claudio Meneses.

15 recordsLinked to original sources

arXiv2025

Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $\lambda$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperk\"ahler metric on the hyperpolygon space $\mathcal X_\alpha$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperk\"ahler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.

math.DG↗

arXiv2025

Semi-Supervised Image-Based Narrative Extraction: A Case Study with Historical Photographic Records

This paper presents a semi-supervised approach to extracting narratives from historical photographic records using an adaptation of the narrative maps algorithm. We extend the original unsupervised text-based method to work with image data, leveraging deep learning techniques for visual feature extraction and similarity computation. Our method is applied to the ROGER dataset, a collection of photographs from the 1928 Sacambaya Expedition in Bolivia captured by Robert Gerstmann. We compare our algorithmically extracted visual narratives with expert-curated timelines of varying lengths (5 to 30 images) to evaluate the effectiveness of our approach. In particular, we use the Dynamic Time Warping (DTW) algorithm to match the extracted narratives with the expert-curated baseline. In addition, we asked an expert on the topic to qualitatively evaluate a representative example of the resulting narratives. Our findings show that the narrative maps approach generally outperforms random sampling for longer timelines (10+ images, p < 0.05), with expert evaluation confirming the historical accuracy and coherence of the extracted narratives. This research contributes to the field of computational analysis of visual cultural heritage, offering new tools for historians, archivists, and digital humanities scholars to explore and understand large-scale image collections. The method's ability to generate meaningful narratives from visual data opens up new possibilities for the study and interpretation of historical events through photographic evidence.

cs.CV↗

arXiv2020

Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.

math.AG↗

arXiv2020

Homotopy classes of gauge fields and the lattice

For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of finite-dimensional families of paths relative to $\mathscr{C}$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG fields of a given pair $(M,\mathscr{C})$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $G$-bundles on $M$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

math-ph↗

arXiv2019

Macroscopic observables from the comparison of local reference systems

Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of ``macroscopic observables'' characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a $G$-bundle over $M$ (and over submanifolds of $M$ that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.

gr-qc↗

arXiv2019

On a functional of Kobayashi for Higgs bundles

We define a functional ${\cal J}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that ${\cal J}(h)$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating ${\cal J}(h)$ and another functional ${\cal I}(h)$, closely related to the Yang-Mills-Higgs functional \cite{Bradlow-Wilkin, Wentworth}, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of ${\cal J}(h)$, which is expressed as a certain $L^{2}$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of ${\cal J}(h)$ if and only if the corresponding Hitchin--Simpson mean curvature is parallel with respect to the Hitchin--Simpson connection.

math.DG↗

arXiv2019

Thin homotopy and the holonomy approach to gauge theories

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations -- such as the so-called thin homotopy -- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group $G$ satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal $G$-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

math-ph↗

arXiv2019

Linear phase space deformations with angular momentum symmetry

Motivated by the work of Leznov--Mostovoy, we classify the linear deformations of standard $2n$-dimensional phase space that preserve the obvious symplectic $\mathfrak{o}(n)$-symmetry. As a consequence, we describe standard phase space, as well as $T^{*}S^{n}$ and $T^{*}\mathbb{H}^{n}$ with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+2}$.

math.SG↗

arXiv2018

Optimum weight chamber examples of moduli spaces of stable parabolic bundles in genus 0

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of the complexity of the actions of the corresponding groups of bundle automorphisms on stable parabolic structures. For the given choices of parabolic weights, $\mathscr{N}$ consists entirely of isomorphism classes of strictly stable parabolic bundles whose underlying Birkhoff-Grothendieck splitting coefficients are constant and minimal, is constructed as a quotient of a set of stable parabolic structures by a group of bundle automorphisms, and is a smooth, compact complex manifold biholomorphic to $\left(\mathbb{C}\mathbb{P}^{1}\right)^{n-3}$ for even degree, and $\mathbb{C}\mathbb{P}^{n-3}$ for odd degree. As an application of the construction of such explicit models, we provide an explicit characterization of the nilpotent cone locus on $T^{*}\mathscr{N}$ for Hitchin's integrable system.

math.AG↗

arXiv2017

On Shimura's isomorphism and $(Γ, G)$-bundles on the upper-half plane

For a compact real form $U$ of a complex simple Lie group $G$, and an irreducible representation $ρ:Γ\to U$ of a Fuchsian group of the first kind $Γ$, it is shown that the classical isomorphism of Shimura, for the periods of a cusp form of weight 2 with values in $\mathfrak{g}$ and the representation $\textrm{Ad}ρ:Γ\to\textrm{Aut}\mathfrak{g}$, can be interpreted as the differential at a point of the zero section, for a natural map from the cotangent bundle of the moduli space of certain $(Γ, G)$-bundles over $\mathbb{H}$ (in the sense of Seshadri) to an open set in the smooth locus of the character variety $\textrm{Hom}_{\mathbf{t}}(Γ,G)/PG$. Emphasis is put on analytic techniques.

math.CV↗

arXiv2017

Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types' automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.

math.CV↗

arXiv2017

On vector-valued Poincaré series of weight 2

Given a pair $(Γ,ρ)$ of a Fuchsian group of the first kind, and a unitary representation $ρ$ of $Γ$ of arbitrary rank, the problem of construction of vector-valued Poincaré series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.

math.CV↗

arXiv2014

Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere

Moduli spaces of stable parabolic bundles of parabolic degree $0$ over the Riemann sphere are stratified according to the Harder--Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $\mathscr{N}_{0}$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $\mathscr{S}$ is defined as the regularized critical value of the non-compact Wess--Zumino--Novikov--Witten action functional. The definition of $\mathscr{S}$ depends on a suitable notion of parabolic bundle `uniformization map' following from the Mehta--Seshadri and Birkhoff--Grothendieck theorems. It is shown that $-\mathscr{S}$ is a primitive for a (1,0)-form $\vartheta$ on $\mathscr{N}_{0}$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $-\mathscr{S}$ is a K\"ahler potential for $(\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}$, where $\Omega$ is the Narasimhan--Atiyah--Bott K\"ahler form in $\mathscr{N}$ and $\Omega_{\mathrm{T}}$ is a certain linear combination of tautological $(1,1)$-forms associated with the marked points. These results provide an explicit relation between the cohomology class $[\Omega]$ and tautological classes, which holds globally over certain open chambers of parabolic weights where $\mathscr{N}_{0} = \mathscr{N}$.

math.CV↗

arXiv2005

Geometry of C-flat connections, coarse graining and the continuum limit

A notion of effective gauge fields which does not involve a background metric is introduced. The role of scale is played by cellular decompositions of the base manifold. Once a cellular decomposition is chosen, the corresponding space of effective gauge fields is the space of flat connections with singularities on its codimension two skeleton, ${\cal A}_{C-flat} \subset \bar{\cal A}_M$. If cellular decomposition $C_2$ is finer than cellular decomposition $C_1$, there is a coarse graining map $π_{C_2 \to C_1}: {\cal A}_{C_2-flat} \to {\cal A}_{C_1-flat}$. We prove that the triple $({\cal A}_{C_2-flat}, π_{C_2 \to C_1}, {\cal A}_{C_1-flat})$ is a principal fiber bundle with a preferred global section given by the natural inclusion map $i_{C_1 \to C_2}: {\cal A}_{C_1-flat} \to {\cal A}_{C_2-flat}$. Since the spaces ${\cal A}_{C-flat}$ are partially ordered (by inclusion) and this order is directed in the direction of refinement, we can define a continuum limit, $C \to M$. We prove that, in an appropriate sense, $\lim_{C \to M} {\cal A}_{C-flat} = \bar{\cal A}_M$. We also define a construction of measures in $\bar{\cal A}_M$ as the continuum limit (not a projective limit) of effective measures.

hep-th↗

arXiv2004

Differentiation matrices for meromorphic functions

A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some singular differential problems defined in the complex domain. The potential use of these matrices is illustrated with the case of elliptic functions.

math.NA↗