Growth of spinors in the generalized Seiberg-Witten equations on $\mathbb R^4$ and $\mathbb R^3$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918107411382272 |
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| author | Bera, Gorapada |
| author_facet | Bera, Gorapada |
| contents | The classical Seiberg-Witten equations in dimensions three and four admit a natural generalization within a unified framework known as the generalized Seiberg-Witten (GSW) equations, which encompasses many important equations in gauge theory. This article proves that the averaged $L^2$-norm of any spinor with non-constant pointwise norm in the GSW equations on $\mathbb R^4$ and $\mathbb R^3$, measured over large-radius spheres, grows faster than a power of the radius, under a suitable curvature decay assumption. Separately, it is shown that if the Yang-Mills-Higgs energy of any solution of these equations is finite, then the pointwise norm of the spinor in it must converge to a non-negative constant at infinity. These two behaviors cannot occur simultaneously unless the spinor has constant pointwise norm. This work may be seen as partial generalization of results obtained by Taubes[Tau17a], and Nagy and Oliveira [NO19] for the Kapustin-Witten equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Growth of spinors in the generalized Seiberg-Witten equations on $\mathbb R^4$ and $\mathbb R^3$ Bera, Gorapada Differential Geometry The classical Seiberg-Witten equations in dimensions three and four admit a natural generalization within a unified framework known as the generalized Seiberg-Witten (GSW) equations, which encompasses many important equations in gauge theory. This article proves that the averaged $L^2$-norm of any spinor with non-constant pointwise norm in the GSW equations on $\mathbb R^4$ and $\mathbb R^3$, measured over large-radius spheres, grows faster than a power of the radius, under a suitable curvature decay assumption. Separately, it is shown that if the Yang-Mills-Higgs energy of any solution of these equations is finite, then the pointwise norm of the spinor in it must converge to a non-negative constant at infinity. These two behaviors cannot occur simultaneously unless the spinor has constant pointwise norm. This work may be seen as partial generalization of results obtained by Taubes[Tau17a], and Nagy and Oliveira [NO19] for the Kapustin-Witten equations. |
| title | Growth of spinors in the generalized Seiberg-Witten equations on $\mathbb R^4$ and $\mathbb R^3$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2401.00100 |