格罗滕迪克-斯普林格解析与TQFTs
Grothendieck-Springer resolutions and TQFTs
AI总结：
本文通过构造新的TQFT，将格罗滕迪克-斯普林格解析与辛流形的拉格朗日关系相结合，探索了纯李论数据与TQFT的联系。
摘要：
摩尔-塔奇克awa猜想认为，每个连通的复半单群G都决定了一个二维TQFT在一个哈密顿辛流形的范畴中。虽然证明这个猜想会很有价值，但我们的目标却截然不同。我们反而将摩尔-塔奇克awa猜想视为系统地将新的TQFT分配给纯李论数据的第一步。同时，这些新的TQFT应与摩尔和塔奇克awa所推测的TQFT有密切的关系。我们的手稿旨在整合这两种观点。设g为G的李代数。考虑一个共轭类C的抛物子代数。这个类确定了部分格罗滕迪克-斯普林格解析μ_C: g_C→g* = g和ν_C: G_C→G。我们构造了一个规范的辛群oid (T*G)_C→g_C和准辛群oid D(G)_C→G_C。通过考虑一个科斯坦特切片Kos⊆g和斯坦因伯格切片Ste⊆G，我们证明了配对(((T*G)_C)_{reg}→(g_C)_{reg}, μ_C^{-1}(Kos))和((D(G)_C)_{reg}→(G_C)_{reg}, ν_C^{-1}(Ste))在1-位移的Weinstein辛范畴中决定了新的且明确的TQFT。然后我们显示，某些来自我们新TQFT的辛流形与开放的摩尔-塔奇克awa流形有规范的拉格朗日关系。
英文摘要：
The Moore-Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional TQFT in a category of Hamiltonian symplectic varieties. While it would be worthwhile to prove this conjecture outright, our objectives are drastically different. We instead view the Moore--Tachikawa conjecture as a first step in systematically assigning new TQFTs to purely Lie-theoretic data. At the same time, one should expect these new TQFTs to bear a close relation to those conjectured by Moore and Tachikawa. Our manuscript aims to integrate these two points of view. Let $\mathfrak{g}$ be the Lie algebra of $G$. Consider a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$. This class determines partial Grothendieck--Springer resolutions $μ_{\mathcal{C}}:\mathfrak{g}_{\mathcal{C}}\longrightarrow\mathfrak{g}^*=\mathfrak{g}$ and $ν_{\mathcal{C}}:G_{\mathcal{C}}\longrightarrow G$. We construct a canonical symplectic groupoid $(T^*G)_{\mathcal{C}}\substack{\longrightarrow\\[-9pt] \longrightarrow}\mathfrak{g}_{\mathcal{C}}$ and quasi-symplectic groupoid $\mathrm{D}(G)_{\mathcal{C}}\substack{\longrightarrow\\[-9pt] \longrightarrow} G_{\mathcal{C}}$. By considering a Kostant slice $\mathrm{Kos}\subseteq\mathfrak{g}$ and Steinberg slice $\mathrm{Ste}\subseteq G$, we prove that the pairs $(((T^*G)_{\mathcal{C}})_{\text{reg}}\substack{\longrightarrow\\[-9pt] \longrightarrow}(\mathfrak{g}_{\mathcal{C}})_{\text{reg}},μ_{\mathcal{C}}^{-1}(\mathrm{Kos}))$ and $((\mathrm{D}(G)_{\mathcal{C}})_{\text{reg}}\substack{\longrightarrow\\[-9pt] \longrightarrow}(G_{\mathcal{C}})_{\text{reg}},ν_{\mathcal{C}}^{-1}(\mathrm{Ste}))$ determine new and explicit TQFTs in a $1$-shifted Weinstein symplectic category. We then show that certain symplectic varieties arising from our new TQFTs have canonical Lagrangian relations to the open Moore-Tachikawa varieties.
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