135 research outputs found
Towards an Effectivisation of the Riemann Theorem
The Riemann Theorem states, that for any nontrivial connected and simply
connected domain on the Riemann sphere there exists some its conformal
bijection to the exterior of the unit disk. In this paper we find an explicit
form of this map for a broad class of domains with analytic boundaries.Comment: 22 pages, AmsTex, Corrected typo
Extention cohomological fields theories and noncommutative Frobenius manifolds
We construct some extension ({\it Stable Field Theory}) of Cohomological
Field Theory. The Stable Field Theory is a system of homomorphisms to some
vector spaces generated by spheres and disks with punctures. It is described by
a formal tensor series, satisfying to some system of "differential equations".
In points of convergence the tensor series generate special noncommutative
analogues of Frobenius algebras, describing 'Open-Closed' Topological Field
Theories.Comment: 19 pages, LaTe
Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
We explicitly construct the series expansion for a certain class of solutions
to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric
solutions. We express the Taylor coefficients through some universal
combinatorial constants and find recurrence relations for them. These results
are used to obtain new formulas for the genus 0 double Hurwitz numbers. They
can also serve as a starting point for a constructive approach to the Riemann
mapping problem and the inverse potential problem in 2D.Comment: 26 page
Open string theory and planar algebras
In this note we show that abstract planar algebras are algebras over the
topological operad of moduli spaces of stable maps with Lagrangian boundary
conditions, which in the case of the projective line are described in terms of
real rational functions. These moduli spaces appear naturally in the
formulation of open string theory on the projective line. We also show two
geometric ways to obtain planar algebras from real algebraic geometry, one
based on string topology and one on Gromov-Witten theory. In particular,
through the well known relation between planar algebras and subfactors, these
results establish a connection between open string theory, real algebraic
geometry, and subfactors of von Neumann algebras.Comment: 13 pages, LaTeX, 7 eps figure
- …