Results 1  10
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38
From loop groups to 2groups
 HHA
"... We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2gr ..."
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Cited by 23 (11 self)
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We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2group is a categorified version of a Lie group. If G is a simplyconnected compact simple Lie group, there is a 1parameter family of Lie 2algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3form on G. There appears to be no Lie 2group having gk as its Lie 2algebra, except when k = 0. Here, however, we construct for integral k an infinitedimensional Lie 2group PkG whose Lie 2algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the levelk Kac– Moody central extension of the loop group ΩG. This 2group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group PkG  that is an extension of G by K(Z, 2). When k = ±1, PkG  can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), PkG  is none other than String(n). 1 1
Parallel transport and functors
"... Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. Th ..."
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Cited by 15 (5 self)
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Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport. Table of Contents
Categorified symplectic geometry and the classical string
, 2008
"... A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poi ..."
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Cited by 10 (5 self)
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A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an ndimensional field theory using a phase space that is an ‘nplectic manifold’: a finitedimensional manifold equipped with a closed nondegenerate (n + 1)form. Here we consider the case n = 2. For any 2plectic manifold, we construct a Lie 2algebra of observables. We then explain how this Lie 2algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2plectic structure for the string.
Higher gauge theory
"... I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to ce ..."
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Cited by 9 (0 self)
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I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to certain 2categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2category of 2bundles over a given 2space under a given 2group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2space is the 2space corresponding to a given space and the 2group is the automorphism 2group of a given group, then this 2category is equivalent to the 2category of gerbes over that space under that group (being described by the same cohomological data).
L∞algebra connections and applications to String and ChernSimons ntransport
, 2008
"... We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspect ..."
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Cited by 9 (5 self)
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We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a Dbrane the KalbRamond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)bundle on the Dbrane to a U(H)bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the Stringextension BU(1) → String(G) → G. Here the obstruction to the lift is a 3bundle with connection (a 2gerbe): the ChernSimons 3bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a Stringstructure. We discuss how to describe this obstruction problem in terms of Lie nalgebras and their corresponding categorified CartanEhresmann connections. Generalizations even beyond Stringextensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized ChernSimons 7bundles classified by the second Pontrjagin class.
A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE OVER POSETS
"... Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theorie ..."
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Cited by 5 (3 self)
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Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general nonAbelian, of a poset with values in a group G. Interpreting a 1–cocycle as a principal bundle, a connection turns out to be a 1–cochain associated in a suitable way with this 1–cocycle; the curvature of a connection turns out to be its 2–coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the AmbroseSinger theorem. 1.
Lattice pform electromagnetism and chain field theory
 LOOPS ’05, ALBERT EINSTEIN INSTITUT, MAX PLANCK GESELLSCHAFT, GOLM
, 2005
"... Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in str ..."
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Cited by 4 (2 self)
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Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as pform electromagnetism, including the KalbRamond field in string theory, and its nonabelian generalizations. It is desirable to discretize such ‘higher gauge theories ’ in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of pform electromagnetism. We show that discrete pform electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for pform electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of pform electromagnetism as a ‘chain field theory’ — a theory analogous to topological quantum field theory, but with chain complexes replacing
Fivebrane Structures
, 2008
"... We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the oneloop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a ..."
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Cited by 4 (4 self)
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We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the oneloop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the GreenSchwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual ” version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3but even 7connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under
HOMOTOPY TRANSITION COCYCLES
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 1(1), 2006, PP.273–283
, 2006
"... For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that gαβgβγ is homotopic to gαγ as a map of Uα ∩ Uβ ∩ Uγ into H. Moreover, on mul ..."
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Cited by 3 (0 self)
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For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that gαβgβγ is homotopic to gαγ as a map of Uα ∩ Uβ ∩ Uγ into H. Moreover, on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis [17]. Here we present it using language that has been developed in the interim. We also show how this points a direction ‘on beyond gerbes’.