22 research outputs found
Cap Products in String Topology
Chas and Sullivan showed that the homology of the free loop space
LM of an oriented closed smooth finite dimensional manifold M admits the
structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative
product called the loop product and a Lie bracket called the loop bracket. We
show that the cap product is compatible with the above two products in the loop
homology. Namely, the cap product with cohomology classes coming from M via the
circle action acts as derivations on loop products as well as on loop brackets.
We show that Poisson identities and Jacobi identities hold for the cap product
action, extending the BV structure in the loop homology to the one including
the cohomology of M. Finally, we describe the cap product in terms of the BV
algebra structure in the loop homology.Comment: 19 pages. Revised version 2 with added references, improved
exposition, and simplified sign
Infinite Product Decomposition of Orbifold Mapping Spaces
Physicists showed that the generating function of orbifold elliptic genera of
symmetric orbifolds can be written as an infinite product. We show that there
exists a geometric factorization on space level behind this infinite product
formula in much more general framework, where factors in the infinite product
correspond to isomorphism classes of connected finite covering spaces of
manifolds involved. From this formula, a concept of geometric Hecke operators
for functors emerges. We show that these Hecke operators indeed satisfy the
usual identity of Hecke operators for the case of 2-dimensional tori.Comment: Version 3: Two more references added and minor revision of
introduction was mad
Generalized Orbifold Euler Characteristic of Symmetric Products and Equivariant Morava K-Theory
We introduce the notion of generalized orbifold Euler characteristic
associated to an arbitrary group, and study its properties. We then calculate
generating functions of higher order (p-primary) orbifold Euler characteristic
of symmetric products of a G-manifold M. As a corollary, we obtain a formula
for the number of conjugacy classes of d-tuples of mutually commuting elements
(of order powers of p) in the wreath product G wreath S_n in terms of
corresponding numbers of G. As a topological application, we present generating
functions of Euler characteristic of equivariant Morava K-theories of symmetric
products of a G-manifold M.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-6.abs.htm
Elliptic genera and vertex operator super-algebras
This monograph deals with two aspects of the theory of elliptic genus: its topological aspect involving elliptic functions, and its representation theoretic aspect involving vertex operator super-algebras. For the second aspect, elliptic genera are shown to have the structure of modules over certain vertex operator super-algebras. The vertex operators corresponding to parallel tensor fields on closed Riemannian Spin Kähler manifolds such as Riemannian tensors and Kähler forms are shown to give rise to Virasoro algebras and affine Lie algebras. This monograph is chiefly intended for topologists and it includes accounts on topics outside of topology such as vertex operator algebras