arXiv subjects

Jeffrey Giansiracusa

Publications and source records attributed to Jeffrey Giansiracusa.

At least 19 recordsLinked to original sources

arXiv2026

Simplicity of confinement in SU(3) Yang-Mills theory

We introduce a novel observable associated to Abelian monopole currents defined in the Maximal Abelian Projection of SU(3) Yang-Mills theory that captures the topology of the current loop. This observable, referred to as the $\textit{simplicity}$, is defined as the ratio of the zeroth over the first Betti number of the current graph for a given field configuration. A numerical study of the expectation value of the simplicity performed in the framework of Lattice Gauge Theories enables us to determine the deconfinement temperature to a higher degree of accuracy than that reached by conventional methods at a comparable computational effort. Our results suggest that Abelian current loops are strongly correlated with the degrees of freedoms of the theory that determine confinement. Our investigation opens new perspectives for the definition of an order parameter for deconfinement in Quantum Chromodynamics able to expose the potentially rich phase structure of the theory.

hep-lat↗

arXiv2026

Three lectures on tropical algebra

This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical polynomials, ideals, congruences, and how the connection with tropical geometry is made via congruences of bend relations. Tropical geometry and matroid theory are telling us that we should focus attention on a narrow slice of the world of tropical algebra, and this leads to the theory of tropical ideals (as developed by Maclagan and Rinc\'{o}n) and an abundance of interesting open questions. In the second lecture we examine the relationship between Berkovich analytification and tropicalization from the perspective of bend relations, giving a refinement of Payne's influential limit theorem. In the third lecture we set aside geometry and focus on tropicalization via bend relations as a construction in commutative and non-commutative algebra. Constructions such as symmetric algebras, exterior algebras, matrix algebras, and Clifford algebras can be tropicalized. In the case of exterior algebras, the resulting tropical notion beautifully completes the picture of the Pl\"{u}cker embedding and gives a new perspective on the tropical Pl\"{u}cker relations. For matrix algebras and Clifford algebras, Morita theory becomes an interesting topic.

math.CO↗

arXiv2025

Emergent photons and mechanisms of confinement

We numerically study $\mathbb{Z}_N$ lattice gauge theories in 4D as prototypical models of systems with $\mathbb{Z}_N$ 1-$\textit{form symmetry}$. For $N \geq 3$, we provide evidence that such systems exhibit not only the expected phases with spontaneously broken/restored symmetry but also a third photon phase. When present, the 1-form symmetry provides a precise notion of confinement, and it is commonly believed that confinement ensues due to the proliferation of extended, string-like objects known as $\textit{center vortices}$, which carry a $\mathbb{Z}_N$ flux. However, this picture is challenged by the three-phase scenario investigated here. We show that both the confined and the photon phases are associated with the proliferation of center vortices and that the key difference between them lies in whether or not vortex-junctions - the $\textit{monopoles}$ - proliferate.

hep-lat↗

arXiv2025

Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theories

Motivated by recent literature on the possible existence of a second higher-temperature phase transition in Quantum Chromodynamics, we revisit the proposal that colour confinement is related to the dynamics of magnetic monopoles using methods of Topological Data Analysis, which provides a mathematically rigorous characterisation of topological properties of quantities defined on a lattice. After introducing persistent homology, one of the main tools in Topological Data Analysis, we shall discuss how this concept can be used to quantitatively analyse the behaviour of monopoles across the deconfinement phase transition. Our approach is first demonstrated for Compact $U(1)$ Lattice Gauge Theory, which is known to have a zero-temperature deconfinement phase transition driven by the restoration of the symmetry associated with the conservation of the magnetic charge. For this system, we perform a finite-size scaling analysis of observables capturing the homology of magnetic current loops, showing that the expected value of the deconfinement critical coupling is reproduced by our analysis. We then extend our method to $SU(3)$ gauge theory, in which Abelian magnetic monopoles are identified after projection in the Maximal Abelian Gauge. A finite-size scaling of our homological observables of Abelian magnetic current loops at temporal size $N_t = 4$ provides the expected value of the critical coupling with an accuracy that is generally higher than that obtained with conventional thermodynamic approaches at comparable statistics, hinting towards the relevance of topological properties of monopole currents for confinement.

hep-lat↗

arXiv2024

Topological Data Analysis of Monopole Current Networks in $U(1)$ Lattice Gauge Theory

In $4$-dimensional pure compact $U(1)$ lattice gauge theory, we analyse topological aspects of the dynamics of monopoles across the deconfinement phase transition. We do this using tools from Topological Data Analysis (TDA). We demonstrate that observables constructed from the zeroth and first homology groups of monopole current networks may be used to quantitatively and robustly locate the critical inverse coupling $β_{c}$ through finite-size scaling. Our method provides a mathematically robust framework for the characterisation of topological invariants related to monopole currents, putting on firmer ground earlier investigations. Moreover, our approach can be generalised to the study of Abelian monopoles in non-Abelian gauge theories.

hep-lat↗

arXiv2024

Log-concavity for independent sets of valuated matroids

Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials, developed by Brändén and Huh, to prove a generalization of the Mason-Welsh conjecture to the context of valuated matroids. In fact, we provide a log-concavity result in the more general setting of valuated discrete polymatroids, or equivalently, M-convex functions. Our approach is via the construction of a generic extension of a valuated matroid or M-convex function, so that the bases of the extension are related to the independent sets of the original matroid. We also provide a similar log-concavity result for valuated bimatroids, which, we believe, might be of independent interest.

math.CO↗

arXiv2024

Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each $n\geq 2$ and $d\geq 1$ our construction yields a non-realizable degree $d$ hypersurface scheme in $\mathbb{P}^n$. Maclagan-Rincón produced a non-realizable line in $\mathbb{P}^n$ for each $n$, and for $(d,n)=(1,2)$ the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

math.AG↗

arXiv2023

Algebraic Dynamical Systems in Machine Learning

We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including 'hybrid symbolic-numeric' models.

math.CT↗

arXiv2023

Ladder Decomposition for Morphisms of Persistence Modules

The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we investigate when a morphism $Φ\colon V \to W$ of persistence modules admits an analogous direct sum decomposition. Jacquard et al. showed that a ladder decomposition can be obtained whenever the barcodes of $V$ and $W$ do not have any strictly nested bars. We refine this result and show that even in the presence of nested bars, a ladder decomposition exists when the morphism is sufficiently close to being invertible relative to the scale of the nested bars.

math.AT↗

arXiv2023

A general framework for tropical differential equations

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. We then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

math.AG↗

arXiv2023

An $E_\infty$ structure on the matroid grassmannian

In analogy with the origin of the additive structure of $K$-theory, we construct an $E_\infty$ structure on the matroid Grassmannian (the space of oriented matroids) for which the underlying binary operation is the direct sum of matroids. The proof involves lifting the polyhedral fan structure of the Dressian to a polyhedral model for the matroid Grassmannian, and introducing a novel $E_\infty$ operad made from the space of infinite subsets of $\mathbb{N}$.

math.AT↗

arXiv2023

Probing center vortices and deconfinement in $\mathrm{SU}(2)$ lattice gauge theory with persistent homology

We investigate the use of persistent homology, a tool from topological data analysis, as a means to detect and quantitatively describe center vortices in $\mathrm{SU}(2)$ lattice gauge theory in a gauge-invariant manner. We provide evidence for the sensitivity of our method to vortices by detecting a vortex explicitly inserted using twisted boundary conditions in the deconfined phase. This inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a simple $k$-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical $β$ and critical exponent of correlation length $ν$ of the deconfinement phase transition.

hep-lat↗

arXiv2022

Persistent homology as a probe for center vortices and deconfinement in SU(2) lattice gauge theory

Topological Data Analysis (TDA) is a field that leverages tools and ideas from algebraic topology to provide robust methods for analysing geometric and topological aspects of data. One of the principal tools of TDA, persistent homology, produces a quantitative description of how the connectivity and structure of data changes when viewed over a sequence of scales. We propose that this presents a means to directly probe topological objects in gauge theories. We present recent work on using persistent homology to detect center vortices in SU(2) lattice gauge theory configurations in a gauge-invariant manner. We introduce the basics of persistence, describe our construction, and demonstrate that the result is sensitive to vortices. Moreover we discuss how, with simple machine learning, one can use the resulting persistence to quantitatively analyse the deconfinement transition via finite-size scaling, providing evidence on the role of vortices in relation to confinement in Yang-Mills theories.

hep-lat↗

arXiv2022

The universal tropicalization and the Berkovich analytification

Given an integral scheme X over a non-archimedean valued field k , we construct acuniversal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by earlier work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification of X. Moreover, using the scheme-theoretic tropicalization, we obtain a tropical scheme $Trop_{univ}(X)$ whose T-points give the analytification and which canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X = spec A is affine, we show that $Trop_{univ}(X)$ is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $Trop_{univ}(X)$ represents the moduli functor of semivaluations on X, and when X = spec A is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.

math.AG↗

arXiv2022

Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

We use persistent homology and persistence images as an observable of three different variants of the two-dimensional XY model in order to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbours models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

cond-mat.stat-mech↗

arXiv2019

Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)\to π_0 Diff(K)$ does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.

math.AT↗

arXiv2018

The circle transfer and cobordism categories

The circle transfer $QΣ(LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let $C_1(X)$ denote the 1-dimensional cobordism category and let $Circ(X) \subset C_1(X)$ denote the subcategory whose objects are disjoint unions of unparametrised circles in $\mathbb{R}^\infty$. Multiplication in $S^1$ induces a functor $Circ(X) \to Circ(LX)$, and the composition of this functor with the inclusion of $Circ(LX)$ into $C_1(LX)$ is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the 2-dimensional cobordism category $C_2(X)$ and find that it is null-homotopic when $X$ is a point.

math.AT↗

arXiv2017

The diffeomorphism group of a K3 surface and Nielsen realization

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.

math.GT↗