259 research outputs found
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
The spectral shift function and spectral flow
This paper extends Krein's spectral shift function theory to the setting of
semifinite spectral triples. We define the spectral shift function under these
hypotheses via Birman-Solomyak spectral averaging formula and show that it
computes spectral flow.Comment: 47 page
Twisted K-theory and finite-dimensional approximation
We provide a finite-dimensional model of the twisted K-group twisted by any
degree three integral cohomology class of a CW complex. One key to the model is
Furuta's generalized vector bundle, and the other is a finite-dimensional
approximation of Fredholm operators.Comment: 26 pages, LaTeX 2e, Xypic; main theorem improve
Geometric Aspects of D-branes and T-duality
We explore the differential geometry of T-duality and D-branes. Because
D-branes and RR-fields are properly described via K-theory, we discuss the
(differential) K-theoretic generalization of T-duality and its application to
the coupling of D-branes to RR-fields. This leads to a puzzle involving the
transformation of the A-roof genera in the coupling.Comment: 26 pages, JHEP format, uses dcpic.sty; v2: references added, v3:
minor change
Families index theorem in supersymmetric WZW model and twisted K-theory: The SU(2) case
The construction of twisted K-theory classes on a compact Lie group is
reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The
Quillen superconnection is introduced for a family of supercharges parametrized
by a compact Lie group and the Chern character is explicitly computed in the
case of SU(2). For large euclidean time, the character form is localized on a
D-brane.Comment: Version 2: Essentially simplified proof of the main result using a
map from twisted K-theory to gerbes modulo the twisting gerbe; references
added + minor correction
Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field
In the framework of algebraic quantum field theory we analyze the anomalous
statistics exhibited by a class of automorphisms of the observable algebra of
the two-dimensional free massive Dirac field, constructed by fermionic gauge
group methods. The violation of Haag duality, the topological peculiarity of a
two-dimensional space-time and the fact that unitary implementers do not lie in
the global field algebra account for strange behaviour of statistics, which is
no longer an intrinsic property of sectors. Since automorphisms are not inner,
we exploit asymptotic abelianness of intertwiners in order to construct a
braiding for a suitable -tensor subcategory of End(). We
define two inequivalent classes of path connected bi-asymptopias, selecting
only those sets of nets which yield a true generalized statistics operator.Comment: 24 page
Cyclic cocycles on twisted convolution algebras
We give a construction of cyclic cocycles on convolution algebras twisted by
gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu
and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted
convolution algebra and twisted cohomology groups which is similar to a
construction of Mathai and Stevenson. When the groupoid is not proper, we
cannot construct an invariant connection on the gerbe; therefore to study this
algebra, we instead develop simplicial techniques to construct a simplicial
curvature 3-form representing the class of the gerbe. Then by using a JLO
formula we define a morphism from a simplicial complex twisted by this
simplicial curvature 3-form to the mixed bicomplex computing the periodic
cyclic cohomology of the twisted convolution algebras. The results in this
article were originally published in the author's Ph.D. thesis.Comment: 39 page
Exact solution of a 2D interacting fermion model
We study an exactly solvable quantum field theory (QFT) model describing
interacting fermions in 2+1 dimensions. This model is motivated by physical
arguments suggesting that it provides an effective description of spinless
fermions on a square lattice with local hopping and density-density
interactions if, close to half filling, the system develops a partial energy
gap. The necessary regularization of the QFT model is based on this proposed
relation to lattice fermions. We use bosonization methods to diagonalize the
Hamiltonian and to compute all correlation functions. We also discuss how,
after appropriate multiplicative renormalizations, all short- and long distance
cutoffs can be removed. In particular, we prove that the renormalized two-point
functions have algebraic decay with non-trivial exponents depending on the
interaction strengths, which is a hallmark of Luttinger-liquid behavior.Comment: 59 pages, 3 figures, v2: further references added; additional
subsections elaborating mathematical details; additional appendix with
details on the relation to lattice fermion
On unbounded p-summable Fredholm modules
We prove that odd unbounded p-summable Fredholm modules are also bounded
p-summable Fredholm modules (this is the odd counterpart of a result of A.
Connes for the case of even Fredholm modules)
Fractional Loop Group and Twisted K-Theory
We study the structure of abelian extensions of the group of
-differentiable loops (in the Sobolev sense), generalizing from the case of
central extension of the smooth loop group. This is motivated by the aim of
understanding the problems with current algebras in higher dimensions. Highest
weight modules are constructed for the Lie algebra. The construction is
extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An
application to the twisted K-theory on is discussed.Comment: Final version in Commun. Math. Phy
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