Quantum geometry of moduli spaces of local systems and representation theory
Quantum geometry of moduli spaces of local systems and representation theory
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局域系统模空间的量子几何与表示论
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发表时间：
2019
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通讯作者：
Li
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作者：
A. Goncharov;Li
设G是分裂半单伴随群，S是具有穿孔和特殊边界点的定向曲面。我们在S上引入了一个模空间P(G，S)的参数化群-局部系，并证明了它在群模群M(G，S)的作用下具有同变的群泊松结构，它包含S的映射类群，G的外自同构群，以及穿孔/边界分支上的Weyl/辫子群的乘积.证明了对偶模空间A(G，S)具有M(G，S)-等变簇结构，对(A(G，S)，P(G，S))是簇系综.这些结果推广了第一作者V.Fock和I.Le的作品。对于任意的普朗克常数h，我们量子化簇泊松变数X。H>；0或|h|=1.首先，我们在X上函数代数的朗兰兹模偶A(h；X)上定义了一个*-代数结构.我们构造了*-代数A(h；X)在簇模群M(X)的么正射影表示下等变的主列表示.这扩展了V.Fock和第一作者h>；0时的作品。结合这一结果，我们得到了由*-代数A(h；P(G，S))及其主级数表示所给出的模空间P(G，S)的M(G，S)等变量子化。我们构造了主级数*-表示的实现。特别地，当S是具有两个特殊点的穿透圆盘时，我们得到了量子群Uq(G)的朗兰兹模偶的主级数*-表示。我们猜想在W-代数的振荡表示的局部协不变量系统与S的映射类群的射影表示所提供的局部不变量系统之间存在非退化对。
Let G be a split semi-simple adjoint group, and S an oriented surface with punctures and special boundary points. We introduce a moduli space P(G,S) parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S)) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le. We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0. Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g). We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.