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Holonomy and parallel transport for Abelian gerbes
, 2001
"... In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence ..."
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In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the nonsimply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the nonabelian case via the theory of double Lie groupoids.
Mackaay: Categorical representations of categorical groups
, 2004
"... The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1 ..."
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The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1
A strict totally coordinatized version of Kapranov and Voevodsky 2vector spaces
"... The purpose of this paper is to give a concrete description of a strict totally coordinatized version of Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces. In particular, we give explicit formulas for the composition of 1morphisms and the two compositions between 2morphisms ..."
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The purpose of this paper is to give a concrete description of a strict totally coordinatized version of Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces. In particular, we give explicit formulas for the composition of 1morphisms and the two compositions between 2morphisms.
State Sum Models for Quantum Gravity
"... Abstract. This review gives a history of the construction of quantum field theory on fourdimensional spacetime using combinatorial techniques, and recent developments of the theory towards a combinatorial construction of quantum gravity. 1. State sum models In this short review I give a brief surve ..."
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Abstract. This review gives a history of the construction of quantum field theory on fourdimensional spacetime using combinatorial techniques, and recent developments of the theory towards a combinatorial construction of quantum gravity. 1. State sum models In this short review I give a brief survey of the history of state sum invariants of fourmanifolds and the attempts to modify them to give models for quantum gravity. I emphasise at the outset that these are at present just models; we do not yet know how far they incorporate all the desirable features of a quantum theory of gravity. For brevity, the review will ignore the long and distinguished lowerdimensional history of these ideas. 1.1. States and weights. The general framework is as follows. Let σn be a standard nsimplex, with vertices 0, 1, 2,..., n. The state sum model requires a set of states S to be given for each simplex. These states can be thought of either as the states of a system in statistical mechanics, or as a basis set of states in quantum mechanics. This set of states is the same for any simplex of the same dimension, so one just has to specify the set of states S(σn) for each n, up to the dimension of the spacetime, n = 4. The idea is that a state on a simplex specifies a state on any one of its faces uniquely; hence there are maps ∂i: S(σn) → S(σn−1), for each i = 0,...,n, the ith map corresponding to the ith (n − 1)dimensional face (opposite the ith vertex). These satisfy some obvious relations, and the whole setup is called a simplicial set. A weight is a complex number which gives an amplitude (or Boltzmann weight) to a state. w: S(σn) → C. A state sum model uses the states and weights as the information for constructing a functional integral on a triangulated 4manifold M. A configuration on M is an assignment of states to all the simplexes in the triangulation such that the states on the faces of any simplex are given by the boundary maps ∂i. This means that the states on intersecting simplexes are related, because the common boundary data must match.
The holonomy of gerbes with connection
, 2008
"... In this paper we study the holonomy of gerbes with connections. If the manifold, M, on which the gerbe is defined is 1connected, then the holonomy defines a group homomorphism. Furthermore we show that all information about the gerbe and its connections is contained in the holonomy by proving an ex ..."
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In this paper we study the holonomy of gerbes with connections. If the manifold, M, on which the gerbe is defined is 1connected, then the holonomy defines a group homomorphism. Furthermore we show that all information about the gerbe and its connections is contained in the holonomy by proving an explicit reconstruction theorem. We comment on the general case in which M is not 1connected, but there remains a conjecture to be proved in order to make things rigorous.
Yetter: A more sensitive Lorentzian state sum
, 2003
"... ABSTRACT We give the construction modulo normalization of a new state sum model for lorentzian quantum general relativity, using the construction of Dirac’s expansors to include quantum operators corresponding to edge lengths as well as the quantum bivectors of the Barrett Crane model, and discuss t ..."
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ABSTRACT We give the construction modulo normalization of a new state sum model for lorentzian quantum general relativity, using the construction of Dirac’s expansors to include quantum operators corresponding to edge lengths as well as the quantum bivectors of the Barrett Crane model, and discuss the problem of its normalization. The new model gives rise to a new picture of quantum geometry in which lengths come in a discrete spectrum, while areas have a continuum of values. I.
FourDimensional Spin Foam Perturbation Theory
"... Abstract. We define a fourdimensional spinfoam perturbation theory for the BFtheory with a B ∧ B potential term defined for a compact semisimple Lie group G on a compact orientable 4manifold M. This is done by using the formal spin foam perturbative series coming from the spinfoam generating f ..."
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Abstract. We define a fourdimensional spinfoam perturbation theory for the BFtheory with a B ∧ B potential term defined for a compact semisimple Lie group G on a compact orientable 4manifold M. This is done by using the formal spin foam perturbative series coming from the spinfoam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain–Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ ⊗ Λ → A, where A is the adjoint representation of g, is 1dimensional for each irrep Λ. We calculate the partition function Z in the dilutegas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4manifold M. We prove that the firstorder perturbative contribution vanishes for finite triangulations, so that we define a dilutegas limit by using the secondorder contribution. We show that Z is an analytic continuation of the Crane–Yetter partition function. Furthermore, we relate Z to the partition function for the F ∧ F theory. Key words: spin foam models; BFtheory; spin networks; dilutegas limit; Crane–Yetter invariant; spinfoam perturbation theory 2010 Mathematics Subject Classification: 81T25; 81T45; 57R56 1