论文标题
通过增强的符号类别量化双重
Towards a quantization of the double via the enhanced symplectic category
论文作者
论文摘要
本文考虑了增强的符号“类别”，目的是量化准哈米尔顿$ g $ - 空格，其中$ g $是一个紧凑的简单谎言组。我们的起点是汉密尔顿几何形状的cotangent束$ t^*g $之间的类似类比，而内部融合的double $ d（g）= g \ times g $ g \ times g $ in quasi-hamiltonian几何形状。 Guillemin和Sternberg考虑了前者，在其所谓的角色Lagrangians上进行了半浓度和相位功能，$λ_{\ Mathcal {o}} \ subseteq t^*g $。我们的准哈米尔顿同行将这些角色拉格朗日替换为通用中心化$λ_ {\ Mathcal {c}} \ longrightArrow \ Mathcal \ Mathcal {C of常规，$ \ frac {1} {1} {k} {k} {k} $ - Integral conjugacy类$ \ m artercal $ \ mathcal $ subsete $ subsete $ supsete $ supsete $ secseete c <c。我们向每个通用centralizer显示为$ d（g）$中的“准哈米顿拉格朗日”，并配备了半密度和相位功能。 同时，我们认为缺乏自然的哈密顿类似物的Dehn Twist引起的自动形态$ R：D（G）\ LongrightArrow D（G）$。每个Quasi-hamiltonian lagrangian $ r（λ_ {\ Mathcal {c}}）$显示与每$λ_ {\ Mathcal {C}'} $具有干净的交点，并配备了其本身的半密度和相位功能。这导致我们考虑了bks配对的bks配对的良好行为，准黑米顿的概念的可能性（λ_ {\ mathcal {c}}}）$和$λ_{\ Mathcal {c}'} $。我们构建了这样的配对并研究其特性。这是由$ r（λ_ {\ Mathcal {c}}）\capλ_ {\ Mathcal {c}'} $的$ r（λ_ {\ Mathcal {c}}）的良好几何恐惧和经典BKS配对的重新制作所促进的。我们的工作也许是通过增强的符号“类别”迈出$ d（g）$的级别$ k $量化的第一步。
This paper considers the enhanced symplectic "category" for purposes of quantizing quasi-Hamiltonian $G$-spaces, where $G$ is a compact simple Lie group. Our starting point is the well-acknowledged analogy between the cotangent bundle $T^*G$ in Hamiltonian geometry and the internally fused double $D(G)=G\times G$ in quasi-Hamiltonian geometry. Guillemin and Sternberg consider the former, studing half-densities and phase functions on its so-called character Lagrangians $Λ_{\mathcal{O}}\subseteq T^*G$. Our quasi-Hamiltonian counterpart replaces these character Lagrangians with the universal centralizers $Λ_{\mathcal{C}}\longrightarrow\mathcal{C}$ of regular, $\frac{1}{k}$-integral conjugacy classes $\mathcal{C}\subseteq G$. We show each universal centralizer to be a "quasi-Hamiltonian Lagrangian" in $D(G)$, and to come equipped with a half-density and phase function. At the same time, we consider a Dehn twist-induced automorphism $R:D(G)\longrightarrow D(G)$ that lacks a natural Hamiltonian analogue. Each quasi-Hamiltonian Lagrangian $R(Λ_{\mathcal{C}})$ is shown to have a clean intersection with every $Λ_{\mathcal{C}'}$, and to come equipped with a half-density and phase function of its own. This leads us to consider the possibility of a well-behaved, quasi-Hamiltonian notion of the BKS pairing between $R(Λ_{\mathcal{C}})$ and $Λ_{\mathcal{C}'}$. We construct such a pairing and study its properties. This is facilitated by the nice geometric fearures of $R(Λ_{\mathcal{C}})\capΛ_{\mathcal{C}'}$ and a reformulation of the classical BKS pairing. Our work is perhaps the first step towards a level-$k$ quantization of $D(G)$ via the enhanced symplectic "category".
