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ESI Senior Research Fellow Program, spring term 2005
K-theory and Physics
Course of advanced graduate lectures by
Professor Mathai Varghese
(University of Adelaide, Australia)
Tuesday and Thursday, 16:15-17:45, Schrödinger lecture hall
starting on March 7, 2006
The course will focus mainly on operator K-theory and some of the
applications to mathematical physics. C*-algebras, smooth
subalgebras, K-theory, index theory, cyclic cohomology,
Poincaré duality, classical and noncommutative differential
geometry will be covered with a view to applications. The two main
applications will be as follows.
Quantum Hall effect
I will survey some models of the integer and fractional quantum Hall
effect based on noncommutative geometry. The classical geometry of
electrons in solids and the passage to noncommutative geometry
produced by the presence of a magnetic field, enables one to model the
quantum Hall effect. The quantum Hall effect is 2 dimensional: the
Euclidean case simulates the single electron model of the integer
quantum Hall effect as studied by Bellissard and coauthors, whereas
the hyperbolic case simulating the multi-electron interactions and
models the fractional quantum Hall effect, . The fractional values of
the Hall conductance are derived as integer multiples of orbifold
Euler characteristics.
- Geometry of the quantum Hall effect
- Continuous model: magnetic Schrödinger operators in 2 dimensions - basic
properties
- Discrete model: Harper operators in 2 dimensions - basic properties
- Magnetic translations and the noncommutative geometry of observables
- Kubo conductance cyclic cocycle and the Chern character cyclic cocycle
on the noncommutative geometry of observables
- The range of the Kubo conductance cyclic cocycle and the twisted
Baum-Connes conjecture
T-duality in string theory via noncommutative geometry
In the presence of a background H-flux, spacetime becomes a (mildly)
noncommutative geometry, which is locally Morita equivalent to smooth
functions on usual spacetime. Such noncommutative geometries are
classified by the Dixmier-Douady class and are geometrically
constructed via bundle gerbes. The D-brane charges and Ramond-Ramond
fields in type II string theory are classified by the K-theory of this
noncommutative geometry, called twisted K-theory. We will give
geometric realizations of twisted K-theory. I will give an axiomatic
definition of T-duality for noncommutative geometries. The examples
of noncommutative geometries arising from the presence of a background
flux, on spacetimes that are compactifed as principal torus bundles
will be analysed, and will be shown to satisfy these axioms.
- Dixmier-Douady theory and bundle gerbes
- Twisted K-theory and geometric realizations
- Twisted Chern character and twisted cohomology
- T-duality in the presence of background H-fluxes I
- T-duality in the presence of background H-fluxes II
ESI Senior Research Fellow Program coordinated by Prof. Joachim
Schwermer, Fakultät für Mathematik, Universität Wien,
Nordbergstraße 15, A-1090 Wien (Joachim.Schwermer@univie.ac.at).
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