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.