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On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined
On twodimensional holonomy
 Trans. Amer. Math. Soc
"... We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rankn homotop ..."
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Cited by 5 (4 self)
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We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rankn homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E → G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M, ∗) → C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1form on P, a principal G bundle over M, and m is a 2form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context. Key words and phrases nonabelian gerbe; 2bundle, twodimensional holonomy; crossed module; categorical group; Wilson sphere
Infinitedimensional representations of 2groups. Available as arXiv:0812.4969
"... A ‘2group ’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2groups have representations on ‘2vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2groups typically have few r ..."
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Cited by 3 (3 self)
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A ‘2group ’ is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2groups have representations on ‘2vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2groups typically have few representations on the finitedimensional 2vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinitedimensional 2vector spaces called ‘measurable categories ’ (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinitedimensional representations of certain Lie 2groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2intertwiners for any skeletal measurable 2group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and subintertwiners—features not seen in ordinary group representation theory. We classify irreducible and indecomposable representations and intertwiners. We also classify ‘irretractable ’ representations—another feature not seen in ordinary
An Invitation to Higher Gauge Theory
, 2010
"... In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie gr ..."
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Cited by 3 (2 self)
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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2group’, which serves as a gauge 2group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2group’. We also touch upon higher structures such as the ‘gravity 3group’, and the Lie 3superalgebra that governs 11dimensional supergravity. 1
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic JahnTeller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, GrassmannHopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
Contents
, 2008
"... We define the fundamental strict categorical group P2(M, ∗) of a based smooth manifold (M, ∗) and construct categorical holonomies, being smooth morphisms P2(M, ∗) → C(G), where C(G) is a Lie categorical group, by using a notion of categorical connections, which we define. As a result, we are able ..."
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We define the fundamental strict categorical group P2(M, ∗) of a based smooth manifold (M, ∗) and construct categorical holonomies, being smooth morphisms P2(M, ∗) → C(G), where C(G) is a Lie categorical group, by using a notion of categorical connections, which we define. As a result, we are able to define Wilson spheres in this context.
DAMTP200327 Higher Gauge Theory and a nonAbelian generalization
, 2003
"... In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of ..."
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In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G = U(1), there exists a generalization, known as pform electrodynamics, in which (p − 1)dimensional charged objects can be propagated along psurfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed psurfaces. In this article, we use Lie 2groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p = 2 to possibly nonAbelian symmetry groups. The main new feature is that our model involves both parallel transports along paths and generalized transports along surfaces with a nontrivial interplay of these two types of variables. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be nonAbelian and which others are always Abelian. A discrete version of connections on nonAbelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.
Cubical 2Bundles with Connection and Wilson Spheres
, 2009
"... We define a cubical version of categorical group 2bundles with connection and consider their twodimensional parallel transport with the aim of defining Wilson surface functionals. Key words and phrases cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; crossed module; categorical g ..."
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We define a cubical version of categorical group 2bundles with connection and consider their twodimensional parallel transport with the aim of defining Wilson surface functionals. Key words and phrases cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; crossed module; categorical group; double groupoid; Wilson sphere; Wilson surface; Higher gauge theory 2000 Mathematics Subject Classification 53C29 (primary); 18D05, 70S15 (secondary) 1
ON THE REGULAR REPRESENTATION OF AN (ESSENTIALLY) FINITE
, 907
"... Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard ..."
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Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers ” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2functor ω: Rep2Vectk G�2Vectk is representable with the regular representation as representing object. As a consequence we obtain a klinear equivalence between the 2vector space Vect G k of functors from the underlying groupoid of G to Vectk, on the one hand, and the klinear category End ω of pseudonatural endomorphisms of ω, on the other hand. We conclude that End ω is a 2vector space, and we (partially) describe a basis of it. 1.