3,607 research outputs found
Boston University School of Medicine Alumni News
Newsletter for Boston University School of Medicine alumni
Higher bundle gerbes and cohomology classes in gauge theories
The notion of a higher bundle gerbe is introduced to give a geometric
realization of the higher degree integral cohomology of certain manifolds. We
consider examples using the infinite dimensional spaces arising in gauge
theories.Comment: 16 pages, LaTe
Bundle gerbes and moduli spaces
In this paper, we construct the index bundle gerbe of a family of
self-adjoint Dirac-type operators, refining a construction of Segal. In a
special case, we construct a geometric bundle gerbe called the caloron bundle
gerbe, which comes with a natural connection and curving, and show that it is
isomorphic to the analytically constructed index bundle gerbe. We apply these
constructions to certain moduli spaces associated to compact Riemann surfaces,
constructing on these moduli spaces, natural bundle gerbes with connection and
curving, whose 3-curvature represent Dixmier-Douady classes that are generators
of the third de Rham cohomology groups of these moduli spaces.Comment: 19 pages. Latex2e, typos corrected, a reference adde
Type I D-branes in an H-flux and twisted KO-theory
Witten has argued that charges of Type I D-branes in the presence of an
H-flux, take values in twisted KO-theory. We begin with the study of real
bundle gerbes and their holonomy. We then introduce the notion of real bundle
gerbe KO-theory which we establish is a geometric realization of twisted
KO-theory. We examine the relation with twisted K-theory, the Chern character
and provide some examples. We conclude with some open problems.Comment: 23 pages, Latex2e, 2 new references adde
Non-Geometric Magnetic Flux and Crossed Modules
It is shown that the BRST operator of twisted N=4 Yang-Mills theory in four
dimensions is locally the same as the BRST operator of a fully decomposed
non-Abelian gerbe. Using locally defined Yang-Mills theories we describe
non-perturbative backgrounds that carry a novel magnetic flux. Given by
elements of the crossed module G x Aut G, these non-geometric fluxes can be
classified in terms of the cohomology class of the underlying non-Abelian
gerbe, and generalise the centre ZG valued magnetic flux found by 't Hooft.
These results shed light also on the description of non-local dynamics of the
chiral five-brane in terms of non-Abelian gerbes.Comment: 26 pages, LaTeX; v2: expanded, typos corrected; v3: typos corrected,
version to apper in Nucl. Phys.
L-infinity algebra connections and applications to String- and Chern-Simons n-transport
We give a generalization of the notion of a Cartan-Ehresmann connection from
Lie algebras to L-infinity algebras and use it to study the obstruction theory
of lifts through higher String-like extensions of Lie algebras. We find
(generalized) Chern-Simons and BF-theory functionals this way and describe
aspects of their parallel transport and quantization.
It is known that over a D-brane the Kalb-Ramond background field of the
string restricts to a 2-bundle with connection (a gerbe) which can be seen as
the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We
discuss how this phenomenon generalizes from the ordinary central extension
U(1) -> U(H) -> PU(H) to higher categorical central extensions, like the
String-extension BU(1) -> String(G) -> G. Here the obstruction to the lift is a
3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by
the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a
String-structure. We discuss how to describe this obstruction problem in terms
of Lie n-algebras and their corresponding categorified Cartan-Ehresmann
connections. Generalizations even beyond String-extensions are then
straightforward. For G = Spin(n) the next step is "Fivebrane structures" whose
existence is obstructed by certain generalized Chern-Simons 7-bundles
classified by the second Pontrjagin class.Comment: 100 pages, references and clarifications added; correction to section
5.1 and further example to 9.3.1 adde
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
Abelian gerbes and twisted bundles describe the topology of the NS-NS 3-form
gauge field strength H. We review how they have been usefully applied to study
and resolve global anomalies in open string theory. Abelian 2-gerbes and
twisted nonabelian gerbes describe the topology of the 4-form field strength G
of M-theory. We show that twisted nonabelian gerbes are relevant in the study
and resolution of global anomalies of multiple coinciding M5-branes. Global
anomalies for one M5-brane have been studied by Witten and by Diaconescu, Freed
and Moore. The structure and the differential geometry of twisted nonabelian
gerbes (i.e. modules for 2-gerbes) is defined and studied. The nonabelian
2-form gauge potential living on multiple coinciding M5-branes arises as
curving (curvature) of twisted nonabelian gerbes. The nonabelian group is in
general , the central extension of the E_8 loop group. The
twist is in general necessary to cancel global anomalies due to the
nontriviality of the 11-dimensional 4-form G field strength and due to the
possible torsion present in the cycles the M5-branes wrap. Our description of
M5-branes global anomalies leads to the D4-branes one upon compactification of
M-theory to Type IIA theory.Comment: 19 page
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An Invitation to Higher Gauge Theory
In this easy introduction to higher gauge theory, we describe parallel
transport for particles and strings in terms of 2-connections on 2-bundles.
Just as ordinary gauge theory involves a gauge group, this generalization
involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie
group gives a Lie 2-group; 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 2-group; the representation of
the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, 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
2-group', which serves as a gauge 2-group in 4d BF theory, which has
topological gravity as a special case. Fourth, every Lie group has an 'inner
automorphism 2-group', which serves as the gauge group in 4d BF theory with
cosmological constant term. Fifth, every Lie group has an 'automorphism
2-group', which plays an important role in the theory of nonabelian gerbes. And
sixth, every compact simple Lie group gives a 'string 2-group'. We also touch
upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra
that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum
Gravity and Quantum Geometry at the 2009 Corfu Summer Institut
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