nLab Baum-Douglas geometric cycle

Idea

A Baum-Douglas geometric cycle on a given manifold $X$ is a representative for K-homology classes on $X$. It is given by a submanifold $Q \hookrightarrow X$ 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

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

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 $Q$ represents the worldvolume of the D-brane and the complex vector bundle it carries the Chan-Paton gauge field.

Related concepts

• motivic function

References

The original articles:

• Paul Baum, R. Douglas, K-homology and index theory, in: R. Kadison (ed.), Operator Algebras and Applications, Proceedings of Symposia in Pure Math. 38 AMS (1982) 117-173 [ams:pspum-38-1]

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

A generalization to twisted homology is discussed in section of

• Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology, Journal of Noncommutative Geometry (arXiv:0710.1625)

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

• Alain Connes, Georges Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20,

  no. 6, 1139–1183 (1984) (pdf)

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

• Rui Reis?, Richard Szabo, Geometric K-Homology of Flat D-Branes ,Commun.Math.Phys. 266 (2006) 71-122 (arXiv:hep-th/0507043)

• Richard Szabo, D-branes and bivariant K-theory, Noncommutative Geometry and Physics 3 1 (2013): 131. (arXiv:0809.3029)