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I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields.

For example, the first Chern class of a complex line bundle on a manifold is some element of $H^2(X,\mathbb{Z})$, which on a 2d manifold equivalent to specifying an integer, the Chern number $c_1 \in \mathbb{Z}$.

All of these concepts have local interpretations after putting a $U(1)$ connection on the bundle, in terms of the connection 1-form $A_\mu dx^\mu$ and the curvature 2-form field on the manifold, $F = (\partial_\mu A_\nu) dx^\mu \wedge \ dx^\nu$. In this case, the Chern class can be thought of as $\frac{1}{2\pi}F \in H^2(X,\mathbb{Z}) \subset H^2_\text{(de Rham)}(X,\mathbb{R})$. And the Chern number is given as the integral of this local quantity, $c_1 = \frac{1}{2\pi}\int_X F$.

I was curious whether or not there was an analogous interpretation of a spin structure and the Arf invariant.

From my (pedestrian, physics) point of view (as explained in section 2.2 of this paper), a spin structure is an element $\rho \in H_1(X,\mathbb{Z}_2)$ which specifies periodic or antiperiodic boundary conditions on each nontrivial cycle, and the Arf invariant is $ind_{\text{mod 2}}(D_\rho)$, where $D_\rho$ is the Dirac operator associated to the spin structure and $ind_{\text{mod 2}}(D_\rho) $ is the mod 2 index of the operator.

My question could also be phrased as whether this mod 2 index of a Dirac operator is some integral of some local quantity that is "canonically" associated to the spin structure.

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    $\begingroup$ The index of the Dirac operator is given by the $\hat{A}$ genus and hence is independent of the spin structure. However, the Arf invariant is not the same for all spin structures, so I am a bit confused by your claim that "the Arf invariant is $(-1)^{ind(D_{\rho})}$". $\endgroup$
    – Michael Albanese
    Jun 2, 2019 at 17:09
  • $\begingroup$ I should have said Arf = $ind(D_\rho) \text{ (mod 2)}$, which is how it's defined in the paper I linked in section 2.2. For a 2-torus, I believe the index depends on the spin structure. The Dirac operator is $\partial_x + i \partial_y$, acting on functions with either periodic or antiperiodic boundary conditions on opposite sides of the fundamental domain (each of the 2x2 choices of boundary conditions is a spin structure). This operator on the torus can only have a zero mode for constant functions, which only happens for the doubly periodic case. $\endgroup$
    – Joe
    Jun 2, 2019 at 17:34
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    $\begingroup$ @Joe that notation is a little misleading to me. The "mod 2 index" is different from "the value of the index modulo 2", so I personally wouldn't denote it by $\mathit{ind}(D_\rho) \pmod 2$. $\endgroup$
    – Arun Debray
    Jun 3, 2019 at 3:24
  • $\begingroup$ Thank you, I'll edit my post. I'm still familiarizing myself with these concepts $\endgroup$
    – Joe
    Jun 3, 2019 at 4:57
  • $\begingroup$ @ArunDebray: What is the mod 2 index? $\endgroup$
    – Michael Albanese
    Jun 6, 2019 at 18:03

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