Where are the $U(1)$ topological gauge theory zero modes? In this 1995 paper, Witten claims the following:

The zero modes (i.e. of the gauge field) do not give factors of $\Im \tau$; they are tangent to the space of classical minima, which is a torus of dimension $b_1(X)$ and has a volume independent of $\Im \tau$.

The idea is that the photon partition function for an abelian gauge theory in a curved compact manifold $X$ includes an integration over the $b_1$ zero modes of flat connections. A flat connection $A$ is one such that its curvature vanishes. Also, for the following questions to make sense we have to assume that $b_1 > 0$, i.e. the fundamental group of $X$ is not trivial.
Why these zero modes span a torus $T^{b_1}=H^2(X,\mathbb{R})/H^2(X,\mathbb{Z})$?
The latter is found in a paper by Marino and Moore (page 2). These zero modes live on $T^{b_1}$ or in its tangent space as this paper says?
And most importantly these zero modes are the zero modes of some Laplace operator. What is this Laplace operator then?
 A: The Laplace operator is just the general Laplace-deRham operator $\Delta = \mathrm{d}\delta + \delta \mathrm{d}$, for $\mathrm{d}$ the exterior derivative and $\delta$ the codifferential. Forms $\omega$ with $\Delta \omega = 0$ are called harmonic forms and by Hodge's theorem, the space of harmonic $p$-forms is isomorphic to the $p$-th deRham cohomology, therefore the number of harmonic 1-forms (which is what Witten means by "possible zero modes of $A$") is the dimension of the first cohomology, which, for real coefficients, is the dual of first homology and therefore has dimension $b_1$. Note that it is crucial for this correspondence that the gauge group is $\mathrm{U}(1)$, since then $A$ is just a $\mathbb{R}$-valued form.
The flat connections have $\mathrm{d}A = F = 0$ and are therefore closed forms. The gauge symmetry corresponds exactly to quotienting out exact 1-forms, so the space of gauge equivalence classes of flat connections is $H^1(X,\mathbb{R})$. The harmonic forms span this vector space, and quotienting out $H^1(X,\mathbb{Z})$ makes this into a torus-like space which still has the original vector space as tangent space at every point, hence the tangent space is spanned by the zero modes/harmonic forms.
I suspect that the quotient by $H^1(X,\mathbb{Z})$ on the space of classical minima is related to the $\theta$-symmetry of the action, but I don't actually know much Seiberg-Witten theory, so I can't say for certain.
