Towards a quantization of the double via the enhanced symplectic category
| العنوان: | Towards a quantization of the double via the enhanced symplectic category |
|---|---|
| المؤلفون: | Peter Crooks, Jonathan Weitsman |
| سنة النشر: | 2020 |
| مصطلحات موضوعية: | Applied Mathematics, Simple Lie group, 010102 general mathematics, Lambda, Automorphism, 01 natural sciences, Centralizer and normalizer, Theoretical Computer Science, 010101 applied mathematics, Combinatorics, Computational Mathematics, Mathematics (miscellaneous), Conjugacy class, Mathematics - Symplectic Geometry, Symplectic category, FOS: Mathematics, Symplectic Geometry (math.SG), Cotangent bundle, 0101 mathematics, Mathematics::Symplectic Geometry, Symplectic geometry, Mathematics |
| الوصف: | 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 $\Lambda_{\mathcal{O}}\subseteq T^*G$. Our quasi-Hamiltonian counterpart replaces these character Lagrangians with the universal centralizers $\Lambda_{\mathcal{C}}\longrightarrow\mathcal{C}$ 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 $R(\Lambda_{\mathcal{C}})$ is shown to have a clean intersection with every $\Lambda_{\mathcal{C}'}$, 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 $R(\Lambda_{\mathcal{C}})$ and $\Lambda_{\mathcal{C}'}$. We construct such a pairing and study its properties. This is facilitated by the nice geometric fearures of $R(\Lambda_{\mathcal{C}})\cap\Lambda_{\mathcal{C}'}$ 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". Comment: 47 pages |
| اللغة: | English |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::a1002450ec557f3cbfa9466e9c7710a6 http://arxiv.org/abs/2012.11383 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....a1002450ec557f3cbfa9466e9c7710a6 |
| قاعدة البيانات: | OpenAIRE |
| الوصف غير متاح. |