173 research outputs found
Survey of mathematical foundations of QFT and perturbative string theory
Recent years have seen noteworthy progress in the mathematical formulation of
quantum field theory and perturbative string theory. We give a brief survey of
these developments. It serves as an introduction to the more detailed
collection "Mathematical Foundations of Quantum Field Theory and Perturbative
String Theory".Comment: This is the introduction to the upcoming volume "Mathematical
Foundations of Quantum Field Theory and Perturbative String Theory", edited
by the authors and published by the American Mathematical Societ
Lie n-algebras of BPS charges
We uncover higher algebraic structures on Noether currents and BPS charges.
It is known that equivalence classes of conserved currents form a Lie algebra.
We show that at least for target space symmetries of higher parameterized
WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on
the Noether currents themselves. Applied to the Green-Schwarz-type action
functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra
refinements of the traditional BPS brane charge extensions of supersymmetry
algebras. We discuss this in the generality of higher differential geometry,
where it applies also to branes with (higher) gauge fields on their
worldvolume. Applied to the M5-brane sigma-model we recover and properly
globalize the M-theory super Lie algebra extension of 11-dimensional
superisometries by 2-brane and 5-brane charges. Passing beyond the
infinitesimal Lie theory we find cohomological corrections to these charges in
higher analogy to the familiar corrections for D-brane charges as they are
lifted from ordinary cohomology to twisted K-theory. This supports the proposal
that M-brane charges live in a twisted cohomology theory.Comment: 19 pages, v2: references added, details of the main computation
spelled ou
Connections on non-abelian Gerbes and their Holonomy
We introduce an axiomatic framework for the parallel transport of connections
on gerbes. It incorporates parallel transport along curves and along surfaces,
and is formulated in terms of gluing axioms and smoothness conditions. The
smoothness conditions are imposed with respect to a strict Lie 2-group, which
plays the role of a band, or structure 2-group. Upon choosing certain examples
of Lie 2-groups, our axiomatic framework reproduces in a systematical way
several known concepts of gerbes with connection: non-abelian differential
cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These
relationships convey a well-defined notion of surface holonomy from our
axiomatic framework to each of these concrete models. Till now, holonomy was
only known for abelian gerbes; our approach reproduces that known concept and
extends it to non-abelian gerbes. Several new features of surface holonomy are
exposed under its extension to non-abelian gerbes; for example, it carries an
action of the mapping class group of the surface.Comment: 57 pages. v1 is preliminary. v2 is completely rewritten, former
Sections 1 and 2 have been moved into a separate paper (arxiv:1303.4663), and
the discussion of non-abelian surface holonomy has been improved and
extended. v3 is the final and published version with a few minor correction
The inner automorphism 3-group of a strict 2-group
Any group gives rise to a 2-group of inner automorphisms,
. It is an old result by Segal that the nerve of this is the
universal -bundle. We discuss that, similarly, for every 2-group
there is a 3-group and a slightly smaller 3-group
of inner automorphisms. We describe these for
any strict 2-group, discuss how can be
understood as arising from the mapping cone of the identity on and
show that its underlying 2-groupoid structure fits into a short exact sequence
.
As a consequence, encodes the properties of the
universal 2-bundle.Comment: references added, relation to simplicial constructions expanded,
version to appear in JHR
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