arXiv subjects

Juan Orendain

Publications and source records attributed to Juan Orendain.

13 recordsLinked to original sources

arXiv2026

Homotopy lattice gauge fields 1: The fields and their properties

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.

math-ph↗

arXiv2024

A better space of generalized connections

Given a base manifold $M$ and a Lie group $G$, we define $\bar{\cal A}^H_M$ a space of generalized $G$-connections on $M$ with the following properties: - The space of smooth connections ${\cal A}^\infty_M = \sqcup_π{\cal A}^\infty_π$ is densely embedded in $\bar{\cal A}^H_M = \sqcup_π\bar{\cal A}^H_{cc(π)}$; moreover, in contrast with the usual space of generalized connections, the embedding preserves topological sectors. - It is a homogeneous covering space for the standard space of generalized connections of loop quantization $\bar{\cal A}_M$. - It is a measurable space constructed as an inverse limit of of spaces of connections with a cutoff, much like $\bar{\cal A}_M$. At each level of the cutoff, a Haar measure, a BF measure and heat kernel measures can be defined. - The topological charge of generalized connections on closed manifolds $Q= \int Tr(F)$ in 2d, $Q= \int Tr(F \wedge F)$ in 4d, etc, is defined. - On a subdivided manifold, it can be calculated in terms of the spaces of generalized connections associated to its pieces. Thus, spaces of boundary connections can be computed from spaces associated to faces. - The soul of our generalized connections is a notion of higher homotopy parallel transport defined for smooth connections. We recover standard generalized connections by forgetting its higher levels. - The kth level of our higher gauge fields is trivial if and only if $π_{k-1} G$ is trivial. Then $\bar{\cal A}^H_Σ\neq \bar{\cal A}_Σ$ if the gauge group is not simply connected and $d \geq 2$. For $G=SL(2, {\mathbb C})$ or $G=SU(2)$ and $\dim Σ= 3$, however, we get $\bar{\cal A}^H_Σ= \bar{\cal A}_Σ$: Boundary data for loop quantum gravity is consistent with our space of generalized connections, but a path integral for quantum gravity would be sensitive to homotopy data.

gr-qc↗

arXiv2024

Internalizations of decorated bicategories via $π_2$-indexings

We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures $π_2$-indexings. We present a construction associating, to every $π_2$-indexing on a decorated 2-category, a length 1 double internalization.

math.CT↗

arXiv2024

Length of fully faithful framed bicategories

The length of a double category is a numerical invariant measuring the 'work' it takes to reconstruct the double category from its globular data. The smallest possible length of a double category is 1. It is conjectured that framed bicategories are of length 1. In this paper we prove this conjecture for a particular class of framed bicategories, namely for those double categories for which all their unit squares are cartesian/opcartesian. We call these framed bicategories fully faithful/absolutely dense.

math.CT↗

arXiv2023

Higher homotopy and lattice gauge fields

We present a general formalism for higher dimensional versions of lattice gauge fields based on higher strict homotopy groupoids. First, using the language of nonabelian Algebraic Topology, we define local lattice higher gauge fields. Then, we provide local-to-global principles for lattice higher gauge fields based on the HHSvK theorem. We prove that, under the correct assumptions, lattice higher gauge field as presented here, subsumes both the notion of extended lattice gauge field and other notions of higher lattice gauge field present in the literature.

math.CT↗

arXiv2021

Free Globularly Generated Double Categories II: The Canonical Double Projection

This is the second installment of a two part series of papers studying free globularly generated double categories. We introduce the canonical double projection construction. The canonical double projection translates information from free globularly generated double categories to double categories defined through the same set of globular and vertical data. We use the canonical double projection to define compatible formal linear functorial extensions of the Haagerup standard form and the Connes fusion operation to possibly-infinite index morphisms between factors. We use the canonical double projection to prove that the free globularly generated double category construction is left adjoint to decorated horizontalization. We thus interpret free globularly generated double categories as formal decorated analogs of double categories of quintets and as generators for internalizations.

math.CT↗

arXiv2019

Free globularily generated double categories

This is the first part of a two paper series studying free globularily generated double categories. In this first installment we introduce the free globularily generated double category construction. The free globularily generated double category construction canonically associates to every bicategory together with a possible category of vertical morphisms, a double category fixing this set of initial data in a free and minimal way. We use the free globularily generated double category to study length, free products, and problems of internalization. We use the free globularily generated double category construction to provide formal functorial extensions of the Haagerup standard form construction and the Connes fusion operation to inclusions of factors of not-necessarily finite Jones index.

math.CT↗

arXiv2019

Lifting bicategories through the Grothendieck construction

We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We make use of a specific instance of the Grothendieck construction to provide, for every bicategory equipped with a possible vertical category, together with a suitable monoidal pre-cosheaf relating these two structures, a double category lifting the decorated bicategory along the category of vertical morphisms provided as set of initial conditions. We prove in particular that every decorated bicategory admits a lift to a double category. We study relations of instances of our construction to foldings, cofoldings, framed bicategories and globularily generated double categories.

math.CT↗

arXiv2018

Cylinder topological quantum field theory: A categorical presentation of classical field theory and its symmetries

We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms with corners to a certain symmetric monoidal double category. We call symmetric monoidal double functors so constructed cylinder topological quantum field theories. Our initial formulation of cylinder topological quantum field theory is presented in a purely topological context. We present appropriate refinements in order to accommodate information relevant to classical field theory.

math.CT↗

arXiv2017

Internalizing decorated bicategories: The globularily generated condition

This is the first part of a series of papers studying the problem of existence of double categories for which horizontal bicategory and object category are given. We refer to this problem as the problem of existence of internalizations for decorated bicategories. We establish a formal framework within which the problem of existence of internalizations can be correctly formulated. Further, we introduce the condition of a double category being globularily generated. We prove that the problem of existence of internalizations for a decorated bicategory admits a solution if and only if it admits a globularily generated solution, and we prove that the condition of a double category being globularily generated is precisely the condition of a solution to the problem of existence of internalizations for a decorated bicategory being minimal. The study of the condition of a double category being globularily generated will thus be pivotal in our study of the problem of existence of internalizations.

math.CT↗

arXiv2014

A Note on the Existence of Indecomposable Essential Submodules of the Ring of Quotients of Ore Domains

We study the problem of existence of essential indecomposable submodules of direct sums of copies of the ring of quotients of Ore domains. We provide, for each Ore domain $D$, with at least three non-associate irreducibles, a lower bound for the supremum of all cardinals $κ$ such that the direct sum of $κ$ copies of the ring of quotients of $D$ contains indecomposable essential $D$-submodules, and a lower bound for the number of times this bound is attained up to isomorphisms. We provide examples illustrating these results.

math.RA↗

arXiv2014

Combinatorial Dimensions: Indecomposability on Certain Local Finite Dimensional Trivial Extension Algebras

We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorifed versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

math.RA↗

arXiv2013

On Direct Sum Decompositions of Krull-Schmidt Artinian Modules

We study direct sum decompositions of modules satisfying the descending chain condition on direct summands. We call modules satisfying this condition Krull-Schmidt artinian. We prove that all direct sum decompositions of Krull-Schmidt artinian modules refine into finite indecomposable direct sum decompositions and we prove that this condition is strictly stronger than the condition of a module admitting finite indecomposable direct sum decompositions. We also study the problem of existence and uniqueness of direct sum decompositions of Krull-Schmidt artinian modules in terms of given classes of modules. We present also brief studies of direct sum decompositions of modules with deviation on direct summands and of modules with finite Krull-Schmidt length.

math.RA↗