# When does a moduli space admit a spin structure?

This is a very vague question. Is there any example of spin structures on a moduli space? References are requested. I have vaguely heard that Witten discussed when a sigma model is spin. Somehow I cannot locate it in the literature.

Thanks!

• There are of course moduli spaces which are both important and at the same time are geometrically simple, a key example being the (higher) Jacobians ncatlab.org/nlab/show/intermediate+Jacobian which are simply higher dimensional tori. Hence these even admit a framing and in particular admit whatever G-structure you want. – Urs Schreiber Sep 10 '14 at 22:40

In general in the gauge theory setting, $w_2$ of the moduli space pulls back from the space of connections mod gauge group. The tangent space to the solutions to the equations (SW or ASD Yang-Mills, or….) is defined as the kernel of the linearization of the operator defining the the equations. It follows that there is an element (the index bundle) of the K-theory of space of connections/gauge group, whose restriction to the moduli space is the tangent bundle. In particular, $w_2$ of the tangent bundle pulls back from $w_2$ of this index bundle. So in cases where $H^2$ with $Z_2$ coefficients for the space of connections/gauge vanishes, it follows that the moduli space is spin. I think this happens, eg for $M=S^4$.