A new formulation of higher parallel transport in higher gauge theory

Soncini, Emanuele; Zucchini, Roberto

doi:10.1016/j.geomphys.2015.04.010

Cited by 25 publications

(44 citation statements)

References 30 publications

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“…A connection in higher gauge theory based on a Lie 2-group G given by the crossed module (G, H, ⊲, ∂) may be described locally in terms of a 1-form A, valued in Lie(G), the Lie algebra of G, and a 2-form B, valued in Lie(H), the Lie algebra of H. There are also Lie(H)-valued transition 1-forms and transition functions valued in G and H, and a global description in terms of parallel transport, to which we will return in our forthcoming work [16]. For further discussion of higher gauge theory from this point of view, the reader may consult a variety of works on the subject, such as [2,18,11,19].…”

Section: Effect Of Changes In Discretizationmentioning

confidence: 99%

2-group actions and moduli spaces of higher gauge theory

Morton¹,

Picken²

2020

Journal of Geometry and Physics

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A framework for higher gauge theory based on a 2-group is presented, by constructing a groupoid of connections on a manifold acted on by a 2-group of gauge transformations, following previous work by the authors where the general notion of the action of a 2-group on a category was defined. The connections are discretized, given by assignments of 2-group data to 1-and 2cells coming from a given cell structure on the manifold, and likewise the gauge transformations are given by 2-group assignments to 0-cells. The 2-cells of the manifold are endowed with a bigon structure, matching the 2-dimensional algebra of squares which is used for calculating with 2-group data. Showing that the action of the 2-group of gauge transformations on the groupoid of connections is well-defined is the central result. The effect, on the groupoid of connections, of changing the discretization is studied, and partial results and conjectures are presented around this issue. The transformation double category that arises from the action of a 2-group on a category, as defined in previous work by the authors, is described for the case at hand, where it becomes a transtormation double groupoid. Finally, examples of the construction are given for simple choices of manifold: the circle, the 2-sphere and the torus.

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Section: Effect Of Changes In Discretizationmentioning

confidence: 99%

2-group actions and moduli spaces of higher gauge theory

Morton¹,

Picken²

2020

Journal of Geometry and Physics

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“…For any point p, any curves γ, γ 1 , γ 2 and any surfaces Σ, Σ 1 , Σ 2 , relations (23)- (25) and the further relations…”

Section: Higher Parallel Transportmentioning

confidence: 99%

“…Earlier endeavours on higher parallel transport includes the work of Caetano and Picken, [14] Baez and Schreiber [15,16] Schreiber and Waldorf, [17][18][19] Faria Martins and Picken, [20,21] Chatterjee, Lahiri and Sengupta [22][23][24] Soncini and Zucchini, [25] Abbaspour and Wagemann [26] and Arias Abad and Schaetz. [27,28] Earlier results on higher holonomy were obtained by Cattaneo, Cotta-Ramusino and Rinaldi, [29] Cattaneo and Rossi [30] and Faria Martins and Picken.…”

Section: Introductionmentioning

confidence: 99%

Wilson Surfaces for Surface Knots: A Field Theoretic Route to Higher Knots

Zucchini

2019

Fortschritte der Physik

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Holonomy invariants in strict higher gauge theory have been studied in depth aiming to applications to higher Chern–Simons theory. For a flat 2–connection, the holonomy of surface knots of arbitrary genus has been defined and its covariance properties under 1–gauge transformation and change of base data have been determined. Using quandle theory, a definition of trace over a crossed module such to yield surface knot invariants upon application to 2–holonomies has been given.

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“…This latter result does not apply (in this form) to other surfaces since the mapping class group is then more complicated: in general an isotopy class of embeddings is needed to define the 2-dimensional holonomy of a 2-gauge connection along an embedded surface. For discussion see [32,63].…”

Section: Introductionmentioning

confidence: 99%

“…We note that topological phases protected by higher gauge symmetry are also proposed in [42].Higher gauge theory [3, 5] is a generalisation of ordinary gauge theory with further levels of structure and symmetry. A key feature of higher gauge theory is parallel transport along surfaces embedded in a manifold where a gauge 2-connection is present [3,5,32,63]. In higher gauge theory, instead of local gauge symmetry groups we have local gauge symmetry 2-groups.…”

mentioning

confidence: 99%

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

et al. 2019

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Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialised CWdecompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.1 Due to a lack of local observables, experimentally distinguishing different topological phases can be a difficult task from a microscopic point of view. Instead the characterising properties of topological phases are most efficiently described by their emergent behaviours. Signatures for the presence of topological order include ground state degeneracies which depend on the spatial topology of the material in question [69,43], universal negative corrections to the entanglement entropy [44,37,48] and the presence of stable topological excitations which provide non-trivial representations of their respective motion groups [47,41]. This means the braid group for point particles (anyons) in 2+1D [54] and the loop braid group for loop excitations in 3+1D [66].In 2+1D there exist several constructions for TQFTs (see for instance [64]). Path-integral models arise from Chern-Simons-Witten theory [73] and from BF theory [1], while the discrete realisation of BF-theory coincides with the Turaev-Viro [64]/Barrett-Westbury [6] state-sum (see [2]). In contrast, in 3+1D a framework general enough to capture all features of 4D topology is still lacking. Nevertheless, we have the Crane-Yetter TQFT [23,64] and its generalisations [71,25]; and the Yetter homotopy 2-type TQFT [74,59,33], derived from a strict finite 2-group [4]. All of these 3+1D TQFTs give rise to topological invariant...

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A new formulation of higher parallel transport in higher gauge theory

Cited by 25 publications

References 30 publications

2-group actions and moduli spaces of higher gauge theory

2-group actions and moduli spaces of higher gauge theory

Wilson Surfaces for Surface Knots: A Field Theoretic Route to Higher Knots

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

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