The chiral critical locus and topological structures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916148778369024 |
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| author | Bouaziz, Emile |
| author_facet | Bouaziz, Emile |
| contents | We study a differential graded VOA associated to the derived critical locus of a function $f$ on a smooth oriented $D$-dimensional variety $(X,\mathbf{vol})$. Informally, this VOA, $\mathbf{crit}^{ch}_{f}$, is just the algebra of chiral differential operators on the derived critical locus $\mathbf{crit}_{f}$. We prove, using a generalization of a physical construction of Witten, the $\mathbf{crit}^{ch}_{f}$ admits a \emph{topological structure} if $f$ is homogeneous for a $\mathbf{G}_{m}$ action on $(X,\mathbf{vol})$. If $\mathbf{vol}$ has weight $b$ and $f$ has weight $a$, we compute the rank of the topological structure in terms of the discrete invariants of the theory to be $$d=\Big(D-\frac{2b}{a}\Big).$$ We conclude with some remarks about BV quantization and a simple computation of characters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The chiral critical locus and topological structures Bouaziz, Emile Algebraic Geometry Representation Theory We study a differential graded VOA associated to the derived critical locus of a function $f$ on a smooth oriented $D$-dimensional variety $(X,\mathbf{vol})$. Informally, this VOA, $\mathbf{crit}^{ch}_{f}$, is just the algebra of chiral differential operators on the derived critical locus $\mathbf{crit}_{f}$. We prove, using a generalization of a physical construction of Witten, the $\mathbf{crit}^{ch}_{f}$ admits a \emph{topological structure} if $f$ is homogeneous for a $\mathbf{G}_{m}$ action on $(X,\mathbf{vol})$. If $\mathbf{vol}$ has weight $b$ and $f$ has weight $a$, we compute the rank of the topological structure in terms of the discrete invariants of the theory to be $$d=\Big(D-\frac{2b}{a}\Big).$$ We conclude with some remarks about BV quantization and a simple computation of characters. |
| title | The chiral critical locus and topological structures |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2403.03630 |