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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.21057 |
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| _version_ | 1866913996856098816 |
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| author | Doan, Aleksander Walpuski, Thomas |
| author_facet | Doan, Aleksander Walpuski, Thomas |
| contents | Motivated by counting pseudo-holomorphic curves in symplectic Calabi-Yau $3$-folds, this article studies a chamber structure in the space of real Cauchy-Riemann operators on a Riemann surface, and constructs three chambered invariants associated with such operators: $n_{\mathrm{Bl}}$, $n_{1,2}$, $n_{2,1}$. The first of these invariants is defined by counting pseudo-holomorphic sections of bundles whose fibres are modeled on the blow-up of $\mathbf{C}^2/\{\pm 1\}$. The other two are defined by counting solutions to the ADHM vortex equations. We conjecture that $n_{1,2}$ and $n_{2,1}$ are related to putative symplectic invariants generalizing the Pandharipande-Thomas and rank $2$ Donaldson-Thomas invariants in algebraic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chambered invariants of real Cauchy-Riemann operators Doan, Aleksander Walpuski, Thomas Differential Geometry Motivated by counting pseudo-holomorphic curves in symplectic Calabi-Yau $3$-folds, this article studies a chamber structure in the space of real Cauchy-Riemann operators on a Riemann surface, and constructs three chambered invariants associated with such operators: $n_{\mathrm{Bl}}$, $n_{1,2}$, $n_{2,1}$. The first of these invariants is defined by counting pseudo-holomorphic sections of bundles whose fibres are modeled on the blow-up of $\mathbf{C}^2/\{\pm 1\}$. The other two are defined by counting solutions to the ADHM vortex equations. We conjecture that $n_{1,2}$ and $n_{2,1}$ are related to putative symplectic invariants generalizing the Pandharipande-Thomas and rank $2$ Donaldson-Thomas invariants in algebraic geometry. |
| title | Chambered invariants of real Cauchy-Riemann operators |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.21057 |