1,763 research outputs found
Configuration spaces and Vassiliev classes in any dimension
The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is
studied by using configuration space integrals. Nontrivial classes are
explicitly constructed. As a by-product, we prove the nontriviality of certain
cycles of imbeddings obtained by blowing up transversal double points in
immersions. These cohomology classes generalize in a nontrivial way the
Vassiliev knot invariants. Other nontrivial classes are constructed by
considering the restriction of classes defined on the corresponding spaces of
immersions.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm
Geometric Engineering of N=2 CFT_{4}s based on Indefinite Singularities: Hyperbolic Case
Using Katz, Klemm and Vafa geometric engineering method of
supersymmetric QFTs and results on the classification of generalized
Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of
CFTs based on \textit{indefinite} singularities. We show
that the vanishing condition for the general expression of holomorphic beta
function of quiver gauge QFTs coincides exactly with the
fundamental classification theorem of KM algebras. Explicit solutions are
derived for mirror geometries of CY threefolds with \textit{% hyperbolic}
singularities.Comment: 23 pages, 4 figures, minor change
Modified 6j-symbols and 3-manifold invariants
37 pages, 16 figuresInternational audienceWe show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects
Counting generalized Jenkins-Strebel differentials
We study the combinatorial geometry of "lattice" Jenkins--Strebel
differentials with simple zeroes and simple poles on and of the
corresponding counting functions. Developing the results of M. Kontsevich we
evaluate the leading term of the symmetric polynomial counting the number of
such "lattice" Jenkins-Strebel differentials having all zeroes on a single
singular layer. This allows us to express the number of general "lattice"
Jenkins-Strebel differentials as an appropriate weighted sum over decorated
trees.
The problem of counting Jenkins-Strebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for the
volume of any stratum of meromorphic quadratic differentials recently obtained
by the authors leads to an interesting combinatorial identity for our sums over
trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with
arXiv:1212.166
- …