arXiv 2408.015222024-08math.DG

关于超流体1群Seiberg-Witten方程在3流形上的注记

Remarks on $\mathrm{Sp}(1)$-Seiberg-Witten equation over $3$-manifolds

Gorapada Bera

备注：: Final version, to appear in Mathematical Research Letters

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AI总结：

研究证明在闭合双曲3流形上，超流体1群Seiberg-Witten方程总有一个由双曲度量诱导的canonical不可约解，并探讨其模空间切空间与局部等距扁结构的关联，以及在特定条件下解的刚性。

摘要：

我们证明在闭合双曲3流形${\mathbb H}^3/Γ$上，超流体1群Seiberg-Witten方程总是有一个由双曲度量诱导的canonical不可约解。我们还证明该模空间在该canonical解处的Zariski切空间与该双曲度量下局部等距扁结构模空间的切空间相同。此空间再次等于trace-free Codazzi张量空间，并映射到Γ-模$\mathbb R^{1,3}$的第一组上同调$H^1(Γ,\mathbb R^{1,3})$。特别地，如果$H^1(Γ,\mathbb R^{1,3})=0$，则该canonical不可约解是infinitesimally刚性的。我们还证明在$S^1\times Σ$上，超流体1群Seiberg-Witten方程没有不可约解，而可约解的模空间等于平坦的SU(2)连接模空间。

英文摘要：

We prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over a closed hyperbolic $3$-manifold ${\mathbb H}^3/Γ$ always admits a canonical irreducible solution induced by the hyperbolic metric. We also prove that the Zariski tangent space of the moduli space at this canonical solution is same as the Zariski tangent space of the moduli space of locally conformally flat structures at the hyperbolic metric. This space is again same as the space of trace-free Codazzi tensors and carries an injection to $H^1(Γ,\mathbb R^{1,3})$, the first group cohomology of the $Γ$-module $\mathbb R^{1,3}$. In particular, if $H^1(Γ,\mathbb R^{1,3})=0$ then the canonical irreducible solution is infinitesimally rigid. We also prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over $S^1\times Σ$ has no irreducible solutions and the moduli space of reducible solutions is same as the moduli space of flat $\mathrm{SU}(2)$-connections.

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