11,949 research outputs found
Dixmier traces and some applications to noncommutative geometry
This is a survey of some recent advances in the theory of singular traces in
which the authors have played some part and which were inspired by questions
raised by the book of Alain Connes (Noncommutative Geometry, Academic Press
1994). There are some original proofs and ideas but most of the results have
appeared elsewhere. Detailed information on the contents is contained in the
Introduction.Comment: To appear in Russian Mathematical Surveys (in Russian). New version
corrects Latex problems, minor errors and reference
Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)
In [CPR2], we presented a K-theoretic approach to finding invariants of
algebras with no non-trivial traces. This paper presents a new example that is
more typical of the generic situation. This is the case of an algebra that
admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main
results are index theorems (which calculate spectral flow), one using ordinary
cyclic cohomology and the other using twisted cyclic cohomology, where the
twisting comes from the generator of the modular group of the Haar state. In
contrast to the Cuntz algebras studied in [CPR2], the computations are
considerably more complex and interesting, because there are nontrivial `eta'
contributions to this index.Comment: 25 pages, 1 figur
Topological Quantum Field Theory and Seiberg-Witten Monopoles
A topological quantum field theory is introduced which reproduces the
Seiberg-Witten invariants of four-manifolds. Dimensional reduction of this
topological field theory leads to a new one in three dimensions. Its partition
function yields a three-manifold invariant, which can be regarded as the
Seiberg-Witten version of Casson's invariant. A Geometrical interpretation of
the three dimensional quantum field theory is also given.Comment: 15 pages, Latex file, no figure
A Simple Algebraic Derivation of the Covariant Anomaly and Schwinger Term
An expression for the curvature of the "covariant" determinant line bundle is
given in even dimensional space-time. The usefulness is guaranteed by its
prediction of the covariant anomaly and Schwinger term. It allows a parallel
derivation of the consistent anomaly and Schwinger term, and their covariant
counterparts, which clarifies the similarities and differences between them. In
particular, it becomes clear that in contrary to the case for anomalies, the
difference between the consistent and covariant Schwinger term can not be
extended to a local form on the space of gauge potentials.Comment: 16 page
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