To understand some physics problem, I want to know if there is (non-$L^2$ or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$ \begin{equation} {D}_A \psi = 0\\ F_A^+ = i\zeta\omega + \sigma(\psi) \end{equation} where $\zeta$ is a real number, and $\omega$ is the standard symplectic form on $\mathbb{R}^4$.

I know that when $\zeta = 0$, a square-integrable solution would imply $\psi = 0$ everywhere, because by Weitzenbock formula \begin{equation} D_A^*{D_A}\psi = \nabla _A^*{\nabla _A}\psi + \frac{s}{4}\psi + \frac{1}{2}F_A^ + \psi \end{equation} and then using the S-W equation itself, one gets \begin{equation} \int {{{\left| {{\nabla _A}\psi } \right|}^2} + \frac{1}{4}{{\left| \psi \right|}^4} + \left( {i\zeta\omega \cdot \psi ,\psi } \right)} = - \frac{1}{4}\int {s{{\left| \psi \right|}^2}} \end{equation} which forces $\psi = 0$ when $s = \zeta = 0$

But I wonder what happens if $\zeta \ne 0$?

Is there known solution (non-$L^2$ is also good) to the equation? Any reference would be great.

  • $\begingroup$ See the answer below. $\endgroup$ – Liviu Nicolaescu Mar 24 '14 at 11:27

The noncompactness is bound to create trouble. For what to expect in the case of a compact symplectic manifold check page 272-275 of these notes Kronheimer and Mrowka have investigated the noncompact case in a beautiful paper going back to the mid 90s. Their main result seems to say that the answer to your question is positive. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$

Comment 1. If we think $\bR^4=\bC^2$, then we have

$$\Psi=\alpha+\beta,\;\;\alpha\in \Omega^0(\bC^2),\;\; \beta\in \Omega^{0,2}(\bC^2). $$

Closing our eyes and pretending that $\bR^4$ is compact we deduce from the SW equations $\newcommand{\bpar}{\bar{\partial}}$

$$ F_A^{0,2}=0,\;\;\bpar_A \alpha=0=\bpar_A^*\beta, $$

$$\alpha\beta=0, $$

$$ F_A\wedge \omega= \sqrt{-1}(|\alpha|^2-|\beta|^2-\zeta) dV. $$

One of $\alpha$ or $\beta$ is identically zero. If $F_A\wedge \omega \to 0$ at $\infty$ then either $|\alpha|^2\to \zeta$ or $|\beta|^2\to-\zeta$ at $\infty$ vilating the $L^2$-assumption on $\Psi$. As in the paper of Kronheimer and Mrowka, you need to impose a different condition at $\infty$ to get nontrivial solutions.

  • $\begingroup$ Thanks for your answer (But the way, your link to notes contains some invalid character after ".pdf"). It is intriguing to see "contact structure" appearing in the title, since my question also has a K-contact structure origin, but in different setting: near some points on a 5d K-contact $M$, certain physics equations reduce to SW equation on $\mathbb{R}^4$ plus an equation along the fifth direction. $\endgroup$ – Lelouch Mar 24 '14 at 14:55
  • $\begingroup$ I fixed the broken link. The paper of Kronheimer Mrowka refers to $4$-manifolds bounding contact $3$-manifolds. As a curiosity, about 15 years ago I looked at SW equations on K-contact 3-manifolds and reduce them to vortices on the base. $\endgroup$ – Liviu Nicolaescu Mar 24 '14 at 15:00
  • $\begingroup$ I guess what we've been mentioning are along the same line (and do you mean your paper <Three-dimensional Seiberg-Witten theory >?). In physics, your situation should correspond to Higgs branch localization (see arXiv:1312.6078), where "BPS equation" reduces to vortex eq. in neighborhoods of closed Reeb orbits ($\sim S^1\times \mathbb{R}^2$), and the "partition function" receives contributions from these vortices. I focus on 5d, so my situation is: 5d K-contact $M$ and SW equation. $\endgroup$ – Lelouch Mar 24 '14 at 15:13
  • $\begingroup$ I meant my papers on SW theory on Seifert manifolds. As for $\mathbb{R}^4$ its noncompactness worries me and that is why I am reluctant to make definite statements. A good guide on dealing with noncompactness could be Jaffe & Taubes' book on vortices and monopoles. Its precise title escapes me at the moment. $\endgroup$ – Liviu Nicolaescu Mar 24 '14 at 16:44
  • $\begingroup$ Perhaps you mean <Vortices and Monopoles: Structure of Static Gauge Theories>. Well, for kink, vortex, monopole and their physics applications, non-compactness doesn't seem like much a problem (solutions on $\mathbb{R}^{1, 2, 3}$ are standard and taught in our QFT class). So I'm not sure what they could tell me about how to deal with spinorial SW-eq. on $\mathbb{R}^4$. $\endgroup$ – Lelouch Mar 24 '14 at 17:10

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