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.


  • $\begingroup$ 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. $\endgroup$ – Urs Schreiber Sep 10 '14 at 22:40

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 $M=S^4$.

  • $\begingroup$ @Thank you! This is exactly the moduli space I am looking for, especially the ASD Yang-Mills moduli. Could you possibly recall a reference? $\endgroup$ – Piojo Sep 11 '14 at 0:57
  • $\begingroup$ The description of the tangent bundle (as an element in KO-theory, which suffices for computing characteristic classes) is found in Donaldson's famous paper, An application of gauge theory to four dimensional topology, JDG 18 (1983), 279-315. He used it to prove orientability. (There are some technicalities coming from non-compactness of the moduli space, but these don't matter for computing w_2). You might look in the book of Donaldson-Kronheimer. For S^4, see Atiyah-Jones, Topological Aspects of Yang-Mills theory. $\endgroup$ – Danny Ruberman Sep 11 '14 at 1:22
  • $\begingroup$ Also, I recall (from long ago) an unpublished preliminary version of a paper by Ohta that proves that a certain ASD moduli space is spin. I think the underlying 4-manifold is of the form S^1 \times Y where Y is a homology 3-sphere, and the bundle has c_2 = 1. If you email me off-line, then I might be able to send you a scan. (If the paper is still in my office.) $\endgroup$ – Danny Ruberman Sep 11 '14 at 1:36
  • $\begingroup$ The preprint of Ohta is called Spin structure on Moduli spaces of Yang-Mills connections. The main theorem says that the moduli space of ASD SU(2) connections (for generic metrics) on a 4-manifold X is spin if X has an almost complex structure, H_1(X) has no even torsion, and either the intersection form of X is odd, or it is even with c_2 of the bundle odd. Elaborating on what I said above, the idea is to show by some algebraic topology that w_2 lifts to an integral class c_1, which can be evaluated using the families index theorem. There are some examples at the end of the paper. $\endgroup$ – Danny Ruberman Sep 11 '14 at 13:46

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