TQFT’s and gerbes

Picken, Roger

doi:10.2140/agt.2004.4.243

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…This extends earlier results in this direction in [Gaw88,BTW04,Pic04]. Let us illustrate some aspects of the results in Theorem 3.12.…”

Section: Transgression Of Gerbes and Holonomy On Surfaces With Boundarysupporting

confidence: 84%

Gerbes in Geometry, Field Theory, and Quantisation

Bunk

2021

Preprint

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This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1-and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

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“…This extends earlier results in this direction in [Gaw88,BTW04,Pic04]. Let us illustrate some aspects of the results in Theorem 3.12.…”

Section: Transgression Of Gerbes and Holonomy On Surfaces With Boundarysupporting

confidence: 84%

Gerbes in Geometry, Field Theory, and Quantisation

Bunk

2021

Preprint

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“…As in the case of 2-bundles, the 1-, 2-and 3-gauge transformations would be better understood by passing to the notion of a Gray triple groupoid, which (to keep the size of this paper within limits) we will analyse in a future article, where we will also address the general definition of a Gray 3-bundle, and describe the corresponding 3-dimensional parallel transport, categorifying the results of [SW3,FMP2,MP,P].…”

Section: Introductionmentioning

confidence: 99%

The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

Martins

Picken²

2011

Differential Geometry and its Applications

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“…In fact, it has only recently been achieved for the 1-dimensional case by Berwick-Evans and Pavlov [BEP15]. Earlier work on abelian gerbes indicates this should be the case in general [Pic04] though this has not been fully worked out. Part of the reason is due to the fact that the representation theory for higher groups is a rather young subject [BBFW12].…”

Section: Two-dimensional Algebra and Surface Transportmentioning

confidence: 99%

Two-dimensional algebra in lattice gauge theory

Parzygnat

2019

Journal of Mathematical Physics

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We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1-and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor networks, to compute the parallel transport along a surface using approximations on a lattice. Although this work is mainly intended as expository, we prove a convergence theorem for the surface transport in the continuum limit. Locality is used to define infinitesimal parallel transport and twodimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data. The correct surface ordering is dictated by two-dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice. The surface ordering is inherently complicated, but we prove a simplification theorem confirming earlier results of Schreiber and Waldorf. Assuming little background, we present a simple way to understand some abstract concepts of higher category theory. In doing so, we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory.

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TQFT’s and gerbes

Cited by 10 publications

References 22 publications

Gerbes in Geometry, Field Theory, and Quantisation

Gerbes in Geometry, Field Theory, and Quantisation

The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

Two-dimensional algebra in lattice gauge theory

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