# Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

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.

• 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 $(-1)^{ind(D_{\rho})}$". – Michael Albanese Jun 2 '19 at 17:09
• 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. – Joe Jun 2 '19 at 17:34
• @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 $\mathit{ind}(D_\rho) \pmod 2$. – Arun Debray Jun 3 '19 at 3:24
• Thank you, I'll edit my post. I'm still familiarizing myself with these concepts – Joe Jun 3 '19 at 4:57
• @ArunDebray: What is the mod 2 index? – Michael Albanese Jun 6 '19 at 18:03