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Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
, 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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Cited by 25 (3 self)
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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 p-form electrodynamics, in which (p − 1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups 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 non-Abelian symmetry groups. The main new feature is that our model involves both parallel transports along paths and generalized transports along surfaces with a non-trivial 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 non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian 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.
Positivity of relativistic spin network evaluations
, 2002
"... Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of G ×G-symmetric spin networks is non-negative whenever the edges are labeled by representations of the form V ⊗ V ∗ where V is a representation of G, and the intertwiners are generalizations of the ..."
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Cited by 3 (2 self)
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Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of G ×G-symmetric spin networks is non-negative whenever the edges are labeled by representations of the form V ⊗ V ∗ where V is a representation of G, and the intertwiners are generalizations of the Barrett–Crane intertwiner. This includes in particular the relativistic spin networks with symmetry group Spin(4) or SO(4). We also present a counterexample, using the finite group S3, to the stronger conjecture that all spin network evaluations are non-negative as long as they can be written using only group integrations and index contractions. This counterexample applies in particular to the product of five 6j-symbols which appears in the spin foam model of the S3-symmetric BFtheory on the two-complex dual to a triangulation of the sphere S 3 using five tetrahedra. We show that this product is negative real for a particular assignment of representations to the edges. PACS: 04.60.Nc key words: Spin network, spin network evaluations, spin foam model
On the causal Barrett–Crane model: measure, coupling constant, Wick rotation, symmetries and observables
, 2002
"... We discuss various features and details of two versions of the Barrett–Crane spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian model and second of the SL(2,�)-symmetric Lorentzian version in which all tetrahedra are space-like. Recently, Livine and Oriti proposed to intro ..."
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Cited by 2 (0 self)
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We discuss various features and details of two versions of the Barrett–Crane spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian model and second of the SL(2,�)-symmetric Lorentzian version in which all tetrahedra are space-like. Recently, Livine and Oriti proposed to introduce a causal structure into the Lorentzian Barrett–Crane model from which one can construct a path integral that corresponds to the causal (Feynman) propagator. We show how to obtain convergent integrals for the 10j-symbols and how a dimensionless constant can be introduced into the model. We propose a ‘Wick rotation ’ which turns the rapidly oscillating complex amplitudes of the Feynman path integral into positive real and bounded weights. This construction does not yet have the status of a theorem, but it can be used as an alternative definition of the propagator and makes the causal model accessible by standard numerical simulation algorithms. In addition, we identify the local symmetries of the models and show how their four-simplex amplitudes can be re-expressed in terms of the ordinary relativistic 10j-symbols. Finally, motivated by possible numerical simulations, we express the matrix elements that are defined by the model, in terms of the continuous connection variables and determine the most general observable in the connection picture. Everything is done on a fixed two-complex. PACS: 04.60.Nc
unknown title
, 2002
"... A spin foam model for pure gauge theory coupled to quantum gravity ..."
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DUAL COMPUTATIONS OF NON-ABELIAN YANG-MILLS ON THE LATTICE
, 705
"... Abstract. In the past several decades there have been a number of proposals for computing with dual forms of non-abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit ..."
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Abstract. In the past several decades there have been a number of proposals for computing with dual forms of non-abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit the question of whether it is practical to perform numerical computation using non-abelian dual models. Specifically, we consider three-dimensional SU(2) pure Yang-Mills as an accessible yet non-trivial case in which the gauge group is non-abelian. Using methods developed recently in the context of spin foam quantum gravity, we describe a Metropolis algorithm for sampling the dual ensemble and efficiently computing the dual amplitude. We relate our algorithms to prior work in non-abelian dual computations of Hari Dass and his collaborators, addressing several problems that have (to the best our knowledge) been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest. 1.
