Geometry of canonical bases and mirror symmetry
Geometry of canonical bases and mirror symmetry
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正则基几何和镜像对称
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Li
中科院分区：
文献类型：
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作者：
A. Goncharov;Li
修饰曲面$$S$$ S是一个有方向的曲面，具有边界和边界上的一组有限的、可能是空的特殊点，被认为是模同位素。设$$mathrm{G}$$ G是$${mathbb Q}$$ q上的一个分裂约化群。对$$(mathrm{G}, S)$$ （G，S）得到一个模空间$${mathcal A}_{mathrm{G}, S}$$ AG，S，它与$$S$$ S上的$$mathrm{G}$$ G局部系统的模空间密切相关，并具有一个正结构（Fock and Goncharov, Publ Math IHES 103:1-212, 2006）。因此定义了一个集$${mathcal A}_{mathrm{G}, S}({mathbb Z}^t)$$ AG，S（Zt）的积分热带点。我们在空间$${mathcal A}_{mathrm{G}, S}$$ AG，S上引入一个有理正函数$${mathcal W}$$ W，称为势。它的热带化是一个函数$${mathcal W}^t: {mathcal A}_{mathrm{G}, S}({mathbb Z}^t)
ightarrow {mathbb Z}$$ Wt:AG，S（Zt）→Z。条件$${mathcal W}^tge 0$$ Wt≥0定义了正积分热带点$${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$ AG,S+(Zt)的子集。对于$$mathrm{G=SL}_2$$ G=SL2，我们从Fock和Goncharov （Publ Math IHES 103:1-212, 2006）中恢复了$$S$$ S上的正积分$${mathcal A}$$ a - lamations集。我们证明了当$$S$$ S是边界上有$$n$$ n个特殊点的圆盘时，集合$${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$ AG,S+(Zt)参数化了卷积映射纤维的顶维分量。因此，通过几何Satake对应(Lusztig, ast<s:1> risque 101-102:208-229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1): 95-143, 2007；Beilinson和Drinfeld，手性代数。他们提供了Langlands对偶群的不可约模的张量积不变量的正则基础$$mathrm{G}^L$$ GL: 1 $$egin{aligned} (V_{lambda _1}otimes ldots otimes V_{lambda _n})^{mathrm{G}^L}. end{aligned}$$ (Vλ1⊗…⊗Vλn)GL。当$$mathrm{G=GL}_m$$ G=GLm, $$n=3$$ n=3时，在$${mathcal A}_{mathrm{G}, S}$$ AG，S上存在一个特殊的坐标系（Fock and Goncharov, Publ Math IHES 103:1-212, 2006）。我们证明了它识别集$${mathcal A}^+_{mathrm{GL_m}, S}({mathbb Z}^t)$$ AGLm,S+(Zt)与Knutson - Tao的蜂箱（Knutson和Tao，蜂窝模型的GL(n)张量积I：饱和猜想的证明，1998）。我们的结果推广了Kamnitzer的一个定理（Hives and the fibers of the convolution morphism, 2007），他使用Hives对$$mathrm{G=GL}_m$$ G=GLm, $$n=3$$ n=3的卷积变种的顶成分进行参数化。对于$$mathrm{G=GL}_m$$ G=GLm, $$n>3$$ n>3，我们证明了Kamnitzer的猜想（Kamnitzer, Hives and the fibers of the convolution morphism, 2012）。我们的参数化是自然循环不变的。我们表明，对于任何$$mathrm{G}$$ G和$$n=3$$ n=3，它符合Berenstein - Zelevinsky的参数化（Berenstein和Zelevinsky， Invent Math 143(1): 77-128, 2001），其循环不变性是模糊的。我们定义了具有势能$$({mathcal A}, {mathcal W})$$ （A，W）的更一般的正空间，参数化了标志的混合构型。利用它们，我们定义了Mirković-Vilonen循环的泛化(Mirkovic and Vilonen, Ann Math（2) 166(1):95 - 143,2007）和$$V_{lambda _1}otimes ldots otimes V_{lambda _n}$$ Vλ1⊗…⊗Vλn的正则基，推广了$$V_{lambda }$$ Vλ中的Mirković-Vilonen基。我们的构造自然带有广义MV周期的参数化。对于经典的MV循环，它相当于Kamnitzer （Mirkovich-Vilonen cycles and polytopes, 2005）所发现的。我们证明了集合$${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$ AG,S+(Zt)参数化了一个新的模空间的顶维分量，曲面仿射Grasmannian，推广了卷积映射的纤维。这些分量通常是无限大的。我们将它们的维度定义为$${mathbb Z}$$ z轴的一个元素，而不是一个整数。我们定义了一个新的模空间$$mathrm{Loc}_{G^L, S}$$ LocGL，S，当$$S$$ S没有特殊点时，它简化为$$S$$ S上的$$G^L$$ gl局部系统的模空间。集合$${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$ AG,S+(Zt)参数化了$$mathrm{Loc}_{G^L, S}$$ LocGL，S上正则函数线性空间中的一组基。我们认为潜在的$${mathcal W}$$ W本身，而不仅仅是它的热带化，是重要的——它应该被视为$${mathcal A}_{mathrm{G}, S}$$ AG，S上的朗道-金兹堡模型的潜力。我们推测对$$({mathcal A}_{mathrm{G}, S}, {mathcal W})$$ （AG，S，W）是$$mathrm{Loc}_{G^L, S}$$ LocGL，S的镜像对偶。在一种特殊情况下，我们恢复了gimental对flag变体的量子上同态连接及其推广的描述（Gerasimov等人，经典李群的Whittaker函数的新积分表示，2012;Rietsch，量子Toda格的镜像对称解，2012）。我们给出了与正则基的参数化平行的等变同调镜像对称猜想。
A decorated surface$$S$$S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let $$mathrm{G}$$G be a split reductive group over $${mathbb Q}$$Q. A pair $$(mathrm{G}, S)$$(G,S) gives rise to a moduli space $${mathcal A}_{mathrm{G}, S}$$AG,S, closely related to the moduli space of $$mathrm{G}$$G-local systems on $$S$$S. It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set $${mathcal A}_{mathrm{G}, S}({mathbb Z}^t)$$AG,S(Zt) of its integral tropical points is defined. We introduce a rational positive function $${mathcal W}$$W on the space $${mathcal A}_{mathrm{G}, S}$$AG,S, called the potential. Its tropicalisation is a function $${mathcal W}^t: {mathcal A}_{mathrm{G}, S}({mathbb Z}^t)
ightarrow {mathbb Z}$$Wt:AG,S(Zt)→Z. The condition $${mathcal W}^tge 0$$Wt≥0 defines a subset of positive integral tropical points$${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$AG,S+(Zt). For $$mathrm{G=SL}_2$$G=SL2, we recover the set of positive integral $${mathcal A}$$A-laminations on $$S$$S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when $$S$$S is a disc with $$n$$n special points on the boundary, the set $${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$AG,S+(Zt) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Astérisque 101–102:208–229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group $$mathrm{G}^L$$GL: 1$$egin{aligned} (V_{lambda _1}otimes ldots otimes V_{lambda _n})^{mathrm{G}^L}. end{aligned}$$(Vλ1⊗…⊗Vλn)GL.When $$mathrm{G=GL}_m$$G=GLm, $$n=3$$n=3, there is a special coordinate system on $${mathcal A}_{mathrm{G}, S}$$AG,S (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). We show that it identifies the set $${mathcal A}^+_{mathrm{GL_m}, S}({mathbb Z}^t)$$AGLm,S+(Zt) with Knutson–Tao’s hives (Knutson and Tao, The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, 1998). Our result generalises a theorem of Kamnitzer (Hives and the fibres of the convolution morphism, 2007), who used hives to parametrise top components of convolution varieties for $$mathrm{G=GL}_m$$G=GLm, $$n=3$$n=3. For $$mathrm{G=GL}_m$$G=GLm, $$n>3$$n>3, we prove Kamnitzer’s conjecture (Kamnitzer, Hives and the fibres of the convolution morphism, 2012). Our parametrisation is naturally cyclic invariant. We show that for any $$mathrm{G}$$G and $$n=3$$n=3 it agrees with Berenstein–Zelevinsky’s parametrisation (Berenstein and Zelevinsky, Invent Math 143(1):77–128, 2001), whose cyclic invariance is obscure. We define more general positive spaces with potentials $$({mathcal A}, {mathcal W})$$(A,W), parametrising mixed configurations of flags. Using them, we define a generalization of Mirković–Vilonen cycles (Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007), and a canonical basis in $$V_{lambda _1}otimes ldots otimes V_{lambda _n}$$Vλ1⊗…⊗Vλn, generalizing the Mirković–Vilonen basis in $$V_{lambda }$$Vλ. Our construction comes naturally with a parametrisation of the generalised MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer (Mirkovich–Vilonen cycles and polytopes, 2005). We prove that the set $${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$AG,S+(Zt) parametrises top dimensional components of a new moduli space, surface affine Grasmannian, generalising the fibers of the convolution maps. These components are usually infinite dimensional. We define their dimension being an element of a $${mathbb Z}$$Z-torsor, rather then an integer. We define a new moduli space $$mathrm{Loc}_{G^L, S}$$LocGL,S, which reduces to the moduli spaces of $$G^L$$GL-local systems on $$S$$S if $$S$$S has no special points. The set $${mathcal A}^+_{mathrm{G}, S}({mathbb Z}^t)$$AG,S+(Zt) parametrises a basis in the linear space of regular functions on $$mathrm{Loc}_{G^L, S}$$LocGL,S. We suggest that the potential $${mathcal W}$$W itself, not only its tropicalization, is important—it should be viewed as the potential for a Landau–Ginzburg model on $${mathcal A}_{mathrm{G}, S}$$AG,S. We conjecture that the pair $$({mathcal A}_{mathrm{G}, S}, {mathcal W})$$(AG,S,W) is the mirror dual to $$mathrm{Loc}_{G^L, S}$$LocGL,S. In a special case, we recover Givental’s description of the quantum cohomology connection for flag varieties and its generalisation (Gerasimov et al., New integral representations of Whittaker functions for classical Lie groups, 2012; Rietsch, A mirror symmetric solution to the quantum Toda lattice, 2012). We formulate equivariant homological mirror symmetry conjectures parallel to our parametrisations of canonical bases.
量子Toda晶格的镜像对称解
影响因子：
2.4
作者：
Rietsch K
通讯作者：
Rietsch K
DOI：
10.4171/jems/791
发表时间：
2018
期刊：
arXiv: Algebraic Geometry
影响因子：
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作者：
T. Dyckerhoff;M. Kapranov
通讯作者：
M. Kapranov