nLab
Baum-Douglas geometric cycle

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A Baum-Douglas geometric cycle on a given manifold XX is a representative for K-homology classes on XX. It is given by a submanifold QXQ \hookrightarrow X equipped with spin^c structure and with a complex vector bundle. The equivalence relation identifying such data that represents the same K-homology class includes a compatible bordism relation.

Viewed as a correspondence of the form

X(Q,E)* X \stackrel{}{\leftarrow} (Q,E) \to \ast

a Baum-Douglas geometric cycle is a special case of the spans that represent classes in KK-theory (between manifolds) according to (Connes-Skandalis 84, section 3).

In string theory Baum-Douglas cycles constitute one formalization of the concept of D-brane carrying a Chan-Paton gauge field (Reis-Szabo 05, Szabo 08): the submanifold QQ represents the worldvolume of the D-brane and the complex vector bundle it carries the Chan-Paton gauge field.

References

The original articles are

  • Paul Baum, R. Douglas, K-homology and index theory: Operator Algebras and Applications (R. Kadison editor), volume 38 of Proceedings of Symposia in Pure Math., 117-173, Providence RI, 1982. AMS.

  • Paul Baum, R. Douglas. Index theory, bordism, and K-homology, Contemp. Math. 10: 1-31 1982.

A generalization to twisted homology is discussed in section of

A generalization to geometric (co)-cocycles for KK-theory is in section 3 of

More generally, a construction of general homology theories in a similar fashion is discussed in

  • S. Buoncristiano, C. P. Rourke and B. J. Sanderson, A geometric approach to homology theory, Cambridge Univ. Press, Cambridge, Mass. (1976)

and a construction of bivariant cohomology theories in this spirit is in

  • Martin Jakob, Bivariant theories for smooth manifolds, Applied Categorical Structures 10 no. 3 (2002)

The interpretation as a formalization of D-branes in string theory is highlighted in

Last revised on December 28, 2016 at 12:53:53. See the history of this page for a list of all contributions to it.