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!
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Sign up to join this communityThis 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!
You didn't specify what kind of moduli space you meant, but there certainly are examples coming from gauge theory. The paper Spin structures on the Seiberg-Witten moduli spaces, by H. Sasahira shows that under some topological hypotheses on a 4-manifold M, the Seiberg-Witten moduli space has a spin structure. There are similar results about the anti-self-dual Yang-Mills moduli space, but I would have to dig around to get references.
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 $M=S^4$.