论文标题
仿射opers and Conformal offine toda
Affine opers and conformal affine Toda
论文作者
论文摘要
对于$ \ mathfrak g $ a affine类型的kac-moody代数，我们表明有一个$ \ text {aut} \，\ mathcal o $ $ -equivariant识别$ \ \ text {fun} \，\ text {feen} \，\ text {op} {op} _ {op} _ {\ nath \ nathfrak g}（\ mathfrak g}（d） g} $ - 圆盘上的opers和$ w \ subsetπ_0$，真空卵形模块内部筛选内核的相交$π_0$。此内核$ W $由两个状态生成：一个共形向量和一个状态$δ_ { - 1} \ left | 0 \ right> $。我们表明，后者赋予$π_0$具有翻译的规范概念$ t^{\ text {（aff）}} $，并使用它来定义$π_0$π_0的密度，这是经典的共同仿射toda toda field理论的积分。 $ \ text {aut} \，\ mathcal o $ action定义了$ \ mathbb p^1 $的捆绑$π$，带有纤维$π_0$。我们表明，该产品捆绑$π\ otimesω^j $，其中$ω^j $是规范捆绑包的张量，并具有单一参数的全态连接家族，$ \ nabla^{\ nabla^{\ text {（aft）}}}} - αt} - αt^^{ C $。在H^1（\ Mathbb p^1，π\ outimemimemimemimemimemω^J，\ nabla^j，\ nabla^{\ nabla^{\ nabla^{\ nabla^{\ nabla^{\ nabla^{\ text {（aff）}）$ $ \ nabla^{\ mathrm {（aff）}} $。 $ {\ mathfrak g} $ - miura oper $χ$的任何选择给出了$ \ nabla^{\ mathrm {（aff）}} _χ$ on $ω^j $。使用Coinvariants，我们从$π\ otimesω^J $的部分定义了一个地图$ \ mathsff_χ$，向$ω^j $的部分定义。我们表明，$ \ mathsff_χ\ \ nabla^{\ text {（（aff）}} = \ nabla^{\ text {\ text {（aff）} _χ\ mthsff_χ$，以便$ \ m athsf f_ f_ f_ f_f_χ$下降到了良好定义的COLOMOLODIES图。在此地图下，$ [\ mathbf v_j dt^{j+1}] $的类将以$ h^1（\ Mathbb p^1，ω^j，\ nabla^j，\ nabla^{\ text {（aff）}} _ q {（aff）}} _取）$ {
For $\mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $\text{Aut}\, \mathcal O$-equivariant identification between $\text{Fun}\,\text{Op}_{\mathfrak g}(D)$, the algebra of functions on the space of ${\mathfrak g}$-opers on the disc, and $W\subset π_0$, the intersection of kernels of screenings inside a vacuum Fock module $π_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $δ_{-1}\left|0\right>$. We show that the latter endows $π_0$ with a canonical notion of translation $T^{\text{(aff)}}$, and use it to define the densities in $π_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $\text{Aut}\,\mathcal O$-action defines a bundle $Π$ over $\mathbb P^1$ with fibre $π_0$. We show that the product bundles $Π\otimes Ω^j$, where $Ω^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $\nabla^{\text{(aff)}} - αT^{\text{(aff)}}$, $α\in \mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[\mathbf v_j dt^{j+1} ] \in H^1(\mathbb P^1, Π\otimes Ω^j,\nabla^{\text{(aff)}})$ of the de Rham cohomology of $\nabla^{\mathrm{(aff)}}$. Any choice of ${\mathfrak g}$-Miura oper $χ$ gives a connection $\nabla^{\mathrm{(aff)}}_χ$ on $Ω^j$. Using coinvariants, we define a map $\mathsf F_χ$ from sections of $Π\otimes Ω^j$ to sections of $Ω^j$. We show that $\mathsf F_χ\nabla^{\text{(aff)}} = \nabla^{\text{(aff)}}_χ\mathsf F_χ$, so that $\mathsf F_χ$ descends to a well-defined map of cohomologies. Under this map, the classes $[\mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(\mathbb P^1, Ω^j,\nabla^{\text{(aff)}}_χ)$ defined by the ${\mathfrak g}$-oper underlying $χ$.
