Example of ''annihilation'' of Seiberg-Witten Equation solutions

Question

The proof that the Seiberg-Witten invariants of a 4-manifold $X$ with fixed Spin$^c$ structure really are invariant wrt the metric used to define them goes roughly as follows (for simplicity let $b_2^+(X) > 1$ and assume the expected dimension of the SW moduli space is 0):

If $g_0$ and $g_1$ are metrics on $X$ then they may be joined by a path $(g_t)_{0 \le t\le 1}$ of metrics where each $g_t$ is such that the corresponding SW moduli space avoids reducible solutions. Now the collection of moduli spaces $(M_t)_{0\le t\le1}$ of the SW equations for each metric $g_t$ can be viewed as a cobordism between the 0-dimensional oriented manifolds $M_0$ and $M_1$. Hence the number of solutions with orientation accounted for is the same in each.

This allows for an "annihilation" phenomenon to occur among solutions: for example, {+,+,-} is cobordant to {-} via a cobordism consisting of one line from + to - and a U-shaped line along which + and - annihilate.

My question is whether anyone can illustrate an explicit example of two solutions to the SW equations wrt a given metric $g_0$ that have opposite orientation and "annihilate" along a path $(g_t)_{0 \le t \le 1}$ to some final metric $g_1$. I would also be interested in any qualitative discussion of this phenomenon.

Answer 1

I can get you very close to explicit, while giving a qualitative discussion, though you may find what follows as cheating. For starters, I won't be writing down local coordinates. Anyway, Taubes gave a fairly explicit construction for how SW solutions relate to $J$-holomorphic curves in symplectic manifolds (and the signs attached to curves agree with the signs attached to SW solutions). Here, deforming the metric is equivalent to deforming $J$.

The relation between curves and SW solutions: Given a curve, the corresponding SW solution will be a pair $(A,\psi)$ of connection and spinor, where $A$ is flat away from the curve while its curvature builds up over the curve (the energy of the curve is related to the norm of the curvature), and $\psi$ decomposes into a pair of sections $(\alpha,\beta)$ for which $\alpha$ vanishes along the curve (roughly speaking).

So we can look for annihilations and bifurcations of $J$-holomorphic curves, which you can then build corresponding SW solutions. In one of Taubes' papers is an explicit example with holomorphic tori. But let's make this simpler by taking the explicit example $X=S^1\times T_\phi$, where $T_\phi$ is the mapping torus of a symplectomorphism $\phi:\mathbb{T}^2\to\mathbb{T}^2$ of a torus. Now isotoping $\phi$ ultimately corresponds to deforming the metric (and deforming $J$), and periodic orbits of $\phi$ correspond to SW solutions (and $J$-holomorphic tori). And you can write down a bifurcation in which an elliptic orbit cancels a positive hyperbolic orbit of the same period; the orbits have opposite "Lefschetz sign".

So if you trace back through all of that: By writing down an explicit dynamical system where orbits bifurcate, then you can write down explicit connections and spinors which almost solve the SW equations (Taubes' construction is ultimately a perturbative approach, he gets these configurations from a curve and then uses the Implicit Function Theorem to get a nearby honest SW solution).