33,996 research outputs found
Quantum lattice gauge fields and groupoid C*-algebras
We present an operator-algebraic approach to the quantization and reduction
of lattice field theories. Our approach uses groupoid C*-algebras to describe
the observables and exploits Rieffel induction to implement the quantum gauge
symmetries. We introduce direct systems of Hilbert spaces and direct systems of
(observable) C*-algebras, and, dually, corresponding inverse systems of
configuration spaces and (pair) groupoids. The continuum and thermodynamic
limit of the theory can then be described by taking the corresponding limits,
thereby keeping the duality between the Hilbert space and observable C*-algebra
on the one hand, and the configuration space and the pair groupoid on the
other. Since all constructions are equivariant with respect to the gauge group,
the reduction procedure applies in the limit as well.Comment: 23 pages, 6 figure
Thurston boundary of Teichm\"uller spaces and the commensurability modular group
If is an unramified covering map between two compact oriented
surfaces of genus at least two, then it is proved that the embedding map,
corresponding to , from the Teichm\"uller space , for , to
actually extends to an embedding between the Thurston
compactification of the two Teichm\"uller spaces. Using this result, an
inductive limit of Thurston compactified Teichm\"uller spaces has been
constructed, where the index for the inductive limit runs over all possible
finite unramified coverings of a fixed compact oriented surface of genus at
least two. This inductive limit contains the inductive limit of Teichm\"uller
spaces, constructed in \cite{BNS}, as a subset. The universal commensurability
modular group, which was constructed in \cite{BNS}, has a natural action on the
inductive limit of Teichm\"uller spaces. It is proved here that this action of
the universal commensurability modular group extends continuously to the
inductive limit of Thurston compactified Teichm\"uller spaces.Comment: AMSLaTex file. To appear in Conformal Geometry and Dynamic
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
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