1,957 research outputs found
Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichm\"uller Space
There exists on each Teichm\"uller space (comprising compact Riemann
surfaces of genus ), a natural sequence of determinant (of cohomology) line
bundles, , related to each other via certain ``Mumford isomorphisms''.
There is a remarkable connection, (Belavin-Knizhnik), between the Mumford
isomorphisms and the existence of the Polyakov string measure on the
Teichm\"uller space. This suggests the question of finding a genus-independent
formulation of these bundles and their isomorphisms. In this paper we combine a
Grothendieck-Riemann-Roch lemma with a new concept of bundles
to construct such an universal version. Our universal objects exist over the
universal space, , which is the direct limit of the as the
genus varies over the tower of all unbranched coverings of any base surface.
The bundles and the connecting isomorphisms are equivariant with respect to the
natural action of the universal commensurability modular group.Comment: ACTA MATHEMATICA (to appear); finalised version with a note of
clarification regarding the connection of the commensurability modular group
with the virtual automorphism group of the fundamental group of a closed
Riemann surface; 25 pages. LATE
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
Covering rough sets based on neighborhoods: An approach without using neighborhoods
Rough set theory, a mathematical tool to deal with inexact or uncertain
knowledge in information systems, has originally described the indiscernibility
of elements by equivalence relations. Covering rough sets are a natural
extension of classical rough sets by relaxing the partitions arising from
equivalence relations to coverings. Recently, some topological concepts such as
neighborhood have been applied to covering rough sets. In this paper, we
further investigate the covering rough sets based on neighborhoods by
approximation operations. We show that the upper approximation based on
neighborhoods can be defined equivalently without using neighborhoods. To
analyze the coverings themselves, we introduce unary and composition operations
on coverings. A notion of homomorphismis provided to relate two covering
approximation spaces. We also examine the properties of approximations
preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin
Dequantisation of the Dirac Monopole
Using a sheaf-theoretic extension of conventional principal bundle theory,
the Dirac monopole is formulated as a spherically symmetric model free of
singularities outside the origin such that the charge may assume arbitrary real
values. For integral charges, the construction effectively coincides with the
usual model. Spin structures and Dirac operators are also generalised by the
same technique.Comment: 22 pages. Version to appear in Proc. R. Soc. London
Large N 2D Yang-Mills Theory and Topological String Theory
We describe a topological string theory which reproduces many aspects of the
1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the
zero coupling (A=0) limit. The string theory is a modified version of
topological gravity coupled to a topological sigma model with spacetime as
target. The derivation of the string theory relies on a new interpretation of
Gross and Taylor's ``\Omega^{-1} points.'' We describe how inclusion of the
area, coupling of chiral sectors, and Wilson loop expectation values can be
incorporated in the topological string approach.Comment: 95 pages, 15 Postscript figures, uses harvmac (Please use the "large"
print option.) Extensive revisions of the sections on topological field
theory. Added a compact synopsis of topological field theory. Minor typos
corrected. References adde
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