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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2311.10032 |
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| _version_ | 1866914863687663616 |
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| author | Franco, Emilio Hanson, Robert |
| author_facet | Franco, Emilio Hanson, Robert |
| contents | Let $\mathcal{M}_{\mathrm{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang-Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of $\mathcal{M}_{\mathrm{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne-Hitchin twistor space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$.
Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann-Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10032 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Dirac-Higgs complex and categorification of (BBB)-branes Franco, Emilio Hanson, Robert Algebraic Geometry Let $\mathcal{M}_{\mathrm{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang-Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of $\mathcal{M}_{\mathrm{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne-Hitchin twistor space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$. Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann-Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence. |
| title | The Dirac-Higgs complex and categorification of (BBB)-branes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2311.10032 |