Gerbes on orbifolds and exotic smooth R^4

Abstract

By using the relation between foliations and exotic R^4, orbifold $K$-theory deformed by a gerbe can be interpreted as coming from the change in the smoothness of R^4. We give various interpretations of integral 3-rd cohomology classes on S^3 and discuss the difference between large and small exotic R^4. Then we show that $K$-theories deformed by gerbes of the Leray orbifold of S^3 are in one-to-one correspondence with some exotic smooth R^4's. The equivalence can be understood in the sense that stable isomorphisms classes of bundle gerbes on S^{3} whose codimension-1 foliations generates the foliations of the boundary of the Akbulut cork, correspond uniquely to these exotic R^{4}'s. Given the orbifold $SU(2)\times SU(2)\rightrightarrows SU(2)$ where SU(2) acts on itself by conjugation, the deformations of the equivariant $K$-theory on this orbifold by the elements of $H_{SU(2)}^3(SU(2),\mathbb{Z})$, correspond to the changes of suitable exotic smooth structures on R^4.

now 22 pages, the construction of the foliation in the appendix was added, no figures, adjustment of the changes in arXiv:0904.1276, subm. to Comm. Math. Phys

References in corpus (1)

• An Introduction to Gerbes on Orbifolds

Cited by in corpus (4)

• Exotic smooth R^4, noncommutative algebras and quantization
• Quantum D-branes and exotic smooth R^4
• Exotic smooth R^4, geometry of string backgrounds and quantum D-branes
• Abelian gerbes, generalized geometries and foliations of small exotic R^4