Chiral Structures on the 4d Spin Cobordism Category

We use the isomorphism Spin(4) = Spin(3) × Spin(3) to propose a quantization the four-dimensional spin cobordism category away from the prime two, as a Weyl algebra of endomorphisms of the three-dimensional spin cobordism category. 1. The 4D spin cobordism category . . . The classifying spectrum of the (symmetric monoidal) cobordism category with compact oriented d-manifolds as objects, and D = (d+1)-dimensional cobordisms between them as morphisms, has been identified, in the remarkable work of Galatius, Madsen, Tillmann and Weiss [9], as MTSO(D) ∼ BSO(D) ; where B(D) is the vector bundle associated to the basic real D-dimensional representation of the special orthogonal group. [X denotes the Thom spectrum of a (possibly virtual) vector bundle E over X.] The authors of that paper remark that their methods apply more generally, to cobordism categories defined similarly by manifolds with more elaborate structures on their tangent bundles. This note is concerned with two closely related cases, when d = 2 or 3 (hence D = 3 or 4), and the structure on the tangent bundle is defined by a lifting from the special orthogonal to the spin group. The exceptional isomorphism Spin(4) ∼= Spin(3) × Spin(3) ∼= SU(2)× SU(2) (and its associated automorphism ε(qL × qR) = qR × qL) have profound implications in quantum physics; this paper speculates about some of its consequences for the algebraic topology of these cobordism categories. Thanks to: S. Galatius, A. Givental, M. Hopkins, G. Laures, C. Teleman, S. Zelditch . . .Of course none of them bear any responsibility for the flight of fancy recorded here. I would also like to thank the 07 Spring Midwest Topology Seminar for a chance to talk about these ideas. Date: 15 July 2007. The author was supported by the NSF. 1