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Higher gauge theory
"... Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where t ..."
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Cited by 65 (15 self)
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Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2groups and Lie 2algebras. We describe a theory of 2connections on principal 2bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2connection allows parallel transport along paths and surfaces in a parametrizationindependent way. In terms of Breen and Messing’s framework, this requires that the ‘fake curvature ’ must vanish. In this paper we summarize the main results of our theory without proofs. 1
Bundle Gerbes for ChernSimons and WESSZUMINOWITTEN THEORIES
, 2005
"... We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invarian ..."
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Cited by 48 (9 self)
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We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invariant. We apply these notions to refine the DijkgraafWitten correspondence between three dimensional ChernSimons functionals and WessZuminoWitten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H 4 (BG, Z) to H3 (G, Z). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for nonsimply connected Lie groups. The implications for WessZuminoWitten models are also discussed.
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 48 (10 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.
Twisted differential String and Fivebrane structures
, 2009
"... Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten an ..."
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Cited by 26 (21 self)
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Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian O(n) and U(n)principal bundles underlying the tangent and the gauge bundle on target space. In this article we unify the picture by describing nonabelian differential cohomology and twisted nonabelian differential cohomology and apply it to these situations. We demonstrate that the FreedWitten mechanism for the Bfield, the GreenSchwarz mechanism for the H3field, as well as its magnetic dual version for the H7field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted Spin(n), twisted String(n) and twisted Fivebrane(n)structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of U(n). We work out the (nonabelian) L∞algebra valued connection data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the
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 26 (13 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.
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 19 (5 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
E8 Gauge Theory and Gerbes in String Theory
, 2006
"... The reduction of the E8 gauge theory to ten dimensions leads to a loop group, which in relation to twisted Ktheory has a DixmierDouady class identified with the NeveuSchwarz Hfield. We give an interpretation of the degree two part of the etaform by comparing the adiabatic limit of the eta invari ..."
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Cited by 15 (15 self)
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The reduction of the E8 gauge theory to ten dimensions leads to a loop group, which in relation to twisted Ktheory has a DixmierDouady class identified with the NeveuSchwarz Hfield. We give an interpretation of the degree two part of the etaform by comparing the adiabatic limit of the eta invariant with the one loop term in type IIA. More generally, starting with a Gbundle, the comparison for manifolds with String Structure identifies G with E8 and the representation as the adjoint, due to an interesting appearance of the dual Coxeter number. This makes possible a description in terms of a generalized WZW model at the critical level. We also discuss the relation to the index gerbe, the possibility of obtaining such bundles from loop space, and the symmetry breaking to finitedimensional bundles. We discuss the implications of this and we give several proposals.
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 15 (14 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
Division Algebras, Supersymmetry and Higher Gauge Theory
, 2012
"... A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no o ..."
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Cited by 5 (1 self)
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A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no one deserves more credit for my mathematical and professional growth during this program. “Thanks ” does not seem sufficient, but it is all I have to give. Also deserving special mention is John’s collaborator, James Dolan. I am convinced there is no subject in mathematics for which Jim does not have some deep insight, and I thank him for sharing a few of these insights with me. Together, John and Jim are an unparalleled team: there are no two better people with whom to talk about mathematics, and no two people more awake to the joy of mathematics. I would also like to thank Geoffrey Dixon, Tevian Dray, Robert Helling, Corinne Manogue, Chris Rogers, Hisham Sati, James Stasheff, and Riccardo Nicoletti for helpful conversations and correspondence. I especially thank Urs Schreiber for many discussions of higher gauge theory and L∞superalgebras, smooth ∞groups, and supergeometry.