998 research outputs found
Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
We construct noncommutative `Riemannian manifold' structures on dual
quasitriangular Hopf algebras such as with its standard bicovariant
differential calculus, using the quantum frame bundle formalism introduced
previously. The metric is provided by the braided-Killing form on the
braided-Lie algebra on the tangent space and the -bein by the Maurer-Cartan
form. We also apply the theory to finite sets and in particular to finite group
function algebras with differential calculi and Killing forms determined
by a conjugacy class. The case of the permutation group is worked out
in full detail and a unique torsion free and cotorsion free or `Levi-Civita'
connection is obtained with noncommutative Ricci curvature essentially
proportional to the metric (an Einstein space). We also construct Dirac
operators in the metric background, including on finite groups such as .
In the process we clarify the construction of connections from gauge fields
with nonuniversal calculi on quantum principal bundles of tensor product form.Comment: 43 pages, 1 figure. Revised August 2001 to cut page length (eg
deleted appendix) for publication in CMP. Also fleshed out ex. of q-Killing
metric for q-SU_2 previously mentioned (no significant additions
Gauge theories on noncommutative euclidean spaces
We consider gauge theories on noncommutative euclidean space . In particular,
we discuss the structure of gauge group following standard mathematical
definitions and using the ideas of hep-th/0102182.Comment: Corrections made, references adde
Compact Kac algebras and commuting squares
We consider commuting squares of finite dimensional von Neumann algebras
having the algebra of complex numbers in the lower left corner. Examples
include the vertex models, the spin models (in the sense of subfactor theory)
and the commuting squares associated to finite dimensional Kac algebras. To any
such commuting square we associate a compact Kac algebra and we compute the
corresponding subfactor and its standard invariant in terms of it.Comment: 14 pages, some minor change
Infinite random matrices and ergodic measures
We introduce and study a 2-parameter family of unitarily invariant
probability measures on the space of infinite Hermitian matrices. We show that
the decomposition of a measure from this family on ergodic components is
described by a determinantal point process on the real line. The correlation
kernel for this process is explicitly computed.
At certain values of parameters the kernel turns into the well-known sine
kernel which describes the local correlation in Circular and Gaussian Unitary
Ensembles. Thus, the random point configuration of the sine process is
interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices
distributed according to the corresponding measure.Comment: 36 page
On Superpotentials for D-Branes in Gepner Models
A large class of D-branes in Calabi-Yau spaces can be constructed at the
Gepner points using the techniques of boundary conformal field theory. In this
note we develop methods that allow to compute open string amplitudes for such
D-branes. In particular, we present explicit formulas for the products of open
string vertex operators of untwisted A-type branes. As an application we show
that the boundary theories of the quintic associated with the special
Lagrangian submanifolds Im \omega_i z_i = 0 where \omega_i^5=1 possess no
continuous moduli.Comment: 33 pages, 2 figure
Deformation Theory of Infinity Algebras
This work explores the deformation theory of algebraic structures in a very
general setting. These structures include commutative, associative algebras,
Lie algebras, and the infinity versions of these structures, the strongly
homotopy associative and Lie algebras. In all these cases the algebra structure
is determined by an element of a certain graded Lie algebra which plays the
role of a differential on this algebra. We work out the deformation theory in
terms of the Lie algebra of coderivations of an appropriate coalgebra structure
and construct a universal infinitesimal deformation as well as a miniversal
formal deformation. By working at this level of generality, the main ideas
involved in deformation theory stand out more clearly.Comment: 31 pages, LaTeX2
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