Hurwitz numbers for regular coverings of surfaces by seamed surfaces and CardyFrobenius algebras of finite groups
Abstract
Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfaces introduced earlier are not actually surfaces. A simple example of seamed surface is booklike seamed surface: several rectangles glued by edges like sheets in a book. We prove that Hurwitz numbers for a class of regular coverings with action of fixed finite group $G$ on cover space such that stabilizers of generic points are conjugated to a fixed subgroup $K\subset G$ defines a new example of Klein Topological Field Theory (KTFT). It is known that KTFTs are in onetoone correspondence with certain class of algebras, called in the work CardyFrobenius algebras. We constructed a wide class of CardyFrobenius algebras, including particularly all Hecke algebras for finite groups. CardyFrobenius algebras corresponding to regular coverings of surfaces by seamed surfaces are described in terms of group $G$ and its subgroups. As a result, we give an algebraic formula for introduced Hurwitz numbers.
 Publication:

arXiv eprints
 Pub Date:
 September 2007
 arXiv:
 arXiv:0709.3601
 Bibcode:
 2007arXiv0709.3601A
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Rings and Algebras;
 57 M;
 16 W
 EPrint:
 23 pages