Quantization commutes with reduction again: the quantum GIT conjecture I
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913371041824768 |
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| author | Pomerleano, Daniel Teleman, Constantin |
| author_facet | Pomerleano, Daniel Teleman, Constantin |
| contents | For a compact monotone symplectic manifold $X$ with Hamiltonian action of a compact Lie group $G$ and smooth symplectic reduction, we relate its gauged $2$-dimensional $A$-model to the $A$-model of $X/\!/G$. This (long conjectured) result is parallel to the ($B$-model!) \emph{quantization commutes with reduction} theorem of Guillemin and Sternberg in quantum mechanics. Here, we spell out some of the precise statements, and outline the proof of equality for the spaces of states (quantum cohomology). We also indicate the way to some related results in the non-monotone case. Additional Floer theory details will be included in a follow-up paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantization commutes with reduction again: the quantum GIT conjecture I Pomerleano, Daniel Teleman, Constantin Symplectic Geometry 53D20, 53D37, 53D40 For a compact monotone symplectic manifold $X$ with Hamiltonian action of a compact Lie group $G$ and smooth symplectic reduction, we relate its gauged $2$-dimensional $A$-model to the $A$-model of $X/\!/G$. This (long conjectured) result is parallel to the ($B$-model!) \emph{quantization commutes with reduction} theorem of Guillemin and Sternberg in quantum mechanics. Here, we spell out some of the precise statements, and outline the proof of equality for the spaces of states (quantum cohomology). We also indicate the way to some related results in the non-monotone case. Additional Floer theory details will be included in a follow-up paper. |
| title | Quantization commutes with reduction again: the quantum GIT conjecture I |
| topic | Symplectic Geometry 53D20, 53D37, 53D40 |
| url | https://arxiv.org/abs/2405.20301 |