Limit of geometric quantizations on Kähler manifolds with T-symmetry
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914960356933632 |
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| author | Leung, Naichung Conan Wang, Dan |
| author_facet | Leung, Naichung Conan Wang, Dan |
| contents | A compact Kähler manifold $\left( M,ω,J\right) $ with $T$-symmetry admits a natural mixed polarization $\mathcal{P}_{\mathrm{mix}}$ whose real directions come from the $T$-action. In \cite{LW1}, we constructed a one-parameter family of Kähler structures $\left( ω,J_{t}\right) $'s with the same underlying Kä hler form $ω$ and $J_{0}=J$, such that (i) there is a $T$-equivariant biholomorphism between $\left( M,J_{0}\right) $ and $\left( M,J_{t}\right) $ and (ii) Kähler polarizations $\mathcal{P} _{t}$'s corresponding to $J_{t}$'s converge to $\mathcal{P}_{\mathrm{mix}}$ as $t$ goes to infinity.
In this paper, we study the quantum analog of above results. Assume $L$ is a pre-quantum line bundle on $\left( M,ω\right) $. Let $\mathcal{H}_{t}$ and $ \mathcal{H}_{\mathrm{mix}}$ be quantum spaces defined using polarizations $\mathcal{P}_{t}$ and $\mathcal{P}_{\mathrm{mix}}$ respectively. In particular, $\mathcal{H}_{t}=H_{\bar{\partial}_{t}}^{0}\left( M,L\right) $. They are both representations of $T$. We show that (i) there is a $T$-equivariant isomorphism between $\mathcal{H}_{0}$ and $\mathcal{H}_{\mathrm{mix}}$ and (ii) for regular $T$-weight $λ$, corresponding $λ$-weight spaces $ \mathcal{H}_{t,λ}$'s converge to $\mathcal{H}_{\mathrm{mix},λ}$ as $t$ goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07759 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Limit of geometric quantizations on Kähler manifolds with T-symmetry Leung, Naichung Conan Wang, Dan Symplectic Geometry Mathematical Physics Differential Geometry A compact Kähler manifold $\left( M,ω,J\right) $ with $T$-symmetry admits a natural mixed polarization $\mathcal{P}_{\mathrm{mix}}$ whose real directions come from the $T$-action. In \cite{LW1}, we constructed a one-parameter family of Kähler structures $\left( ω,J_{t}\right) $'s with the same underlying Kä hler form $ω$ and $J_{0}=J$, such that (i) there is a $T$-equivariant biholomorphism between $\left( M,J_{0}\right) $ and $\left( M,J_{t}\right) $ and (ii) Kähler polarizations $\mathcal{P} _{t}$'s corresponding to $J_{t}$'s converge to $\mathcal{P}_{\mathrm{mix}}$ as $t$ goes to infinity. In this paper, we study the quantum analog of above results. Assume $L$ is a pre-quantum line bundle on $\left( M,ω\right) $. Let $\mathcal{H}_{t}$ and $ \mathcal{H}_{\mathrm{mix}}$ be quantum spaces defined using polarizations $\mathcal{P}_{t}$ and $\mathcal{P}_{\mathrm{mix}}$ respectively. In particular, $\mathcal{H}_{t}=H_{\bar{\partial}_{t}}^{0}\left( M,L\right) $. They are both representations of $T$. We show that (i) there is a $T$-equivariant isomorphism between $\mathcal{H}_{0}$ and $\mathcal{H}_{\mathrm{mix}}$ and (ii) for regular $T$-weight $λ$, corresponding $λ$-weight spaces $ \mathcal{H}_{t,λ}$'s converge to $\mathcal{H}_{\mathrm{mix},λ}$ as $t$ goes to infinity. |
| title | Limit of geometric quantizations on Kähler manifolds with T-symmetry |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2307.07759 |