Homotopy classes of gauge fields and the lattice

Meneses, Claudio; Zapata, José A.

doi:10.4310/atmp.2019.v23.n8.a7

Cited by 3 publications

(16 citation statements)

References 12 publications

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“…See also the work of Lewandowski for a related study [11]. A more mathematically rigorous presentation of what is described in this section but lacks a physical interpretation can be found in [12].…”

Section: Reference Systems From the Gauge Fieldmentioning

confidence: 97%

“…In [12] we use a smaller embedded lattice that does not record the evaluation of the gluing map g τ ν at p τ , but only records the evaluation at vertices of cτ . Here we use this version of L C because it makes the study presented in Sections 5 and 7 simpler.…”

Section: Effective Gauge Theories and Macroscopic Observablesmentioning

confidence: 99%

“…• Together with the usual lattice gauge theory data, which can be inter-1 Some explanation about the gluing operation will be given in Section 7 and a detailed description is given in [12].…”

Section: Effective Gauge Theories and Macroscopic Observablesmentioning

confidence: 99%

“…There is a simplex of paths in the continuum for each simplex in the triangulation of cτ induced by N C (for each cell c ν such that cν cτ ); in some sense what we have is a cell of paths in the continuum corresponding to the pair of nested closed cells cν cτ . In [12] these cells are not subdivided into subsimplices.…”

Section: Simplicial Families Of Combinatorial Paths From Multiparamet...mentioning

confidence: 99%

“…While the space M LC is connected, the set of connected components of M NC is in one-to-one correspondence with the set of equivalence classes of G-bundles over M . For a more detailed description, see [12].…”

Section: The Space Of Extended Lattice Gauge Fieldsmentioning

confidence: 99%

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Macroscopic observables from the comparison of local reference systems

Meneses¹,

Zapata²

2019

Class. Quantum Grav.

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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. *

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Section: Reference Systems From the Gauge Fieldmentioning

confidence: 97%

Section: Effective Gauge Theories and Macroscopic Observablesmentioning

confidence: 99%

Section: Effective Gauge Theories and Macroscopic Observablesmentioning

confidence: 99%

Section: Simplicial Families Of Combinatorial Paths From Multiparamet...mentioning

confidence: 99%

Section: The Space Of Extended Lattice Gauge Fieldsmentioning

confidence: 99%

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Macroscopic observables from the comparison of local reference systems

Meneses¹,

Zapata²

2019

Class. Quantum Grav.

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Thin homotopy and the holonomy approach to gauge theories

Meneses¹

2021

Mexican Mathematicians in the World

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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 G satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal G 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.

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Homotopy data as part of the lattice field: A first study

Dall’Olio

Zapata

2021

Int. J. Mod. Phys. C

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Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those properties. In this work, we show that by augmenting the usual lattice fields to include extra variables describing local topological information (more precisely, regarding homotopy), the topology of the space of fields in the continuum is faithfully reproduced in the lattice. We apply this extended lattice formulation to some simple models with nontrivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.

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Homotopy classes of gauge fields and the lattice

Cited by 3 publications

References 12 publications

Macroscopic observables from the comparison of local reference systems

Macroscopic observables from the comparison of local reference systems

Thin homotopy and the holonomy approach to gauge theories

Homotopy data as part of the lattice field: A first study

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