论文标题
绘制班级组的量子表示的同源结构
Homological Construction of Quantum Representations of Mapping Class Groups
论文作者
论文摘要
我们为绘制由非偏smisimple TQFT（拓扑量子场理论）产生的量子群的量子表示家族提供了同源模型。我们的方法给出了关于这些表示形式的新几何观点，并将其聚集在一个理论的两个理论中，这是研究映射类群体线性的最有前途的结构。更确切地说，如果$ \ varsigma_ {g，1} $是$ g $的表面，$ 1 $ bounce consem零件，我们考虑其映射类组$ \ mathrm {modrm {mathrm {\ varsigma_ {g，1}）$的（交叉）动作，其配置空间$ \ mathrm {conf} _n（\ varsigma_ {g，1}）$在其表面编织组的$π_1（\ mathrm {conf} _n（\ conf} _n（\ varsigma _ _ = g，1}））的Heisenberg商$ \ mathbb {h} _g $中的扭曲系数。我们表明，这种动作将$ \ mathfrak {sl} _2 $的量子组的动作交织在一起，我们通过纯粹的同源方式定义了该动作。对于$ \ mathbb {h} _g $的有限维线性表示（取决于Unity $ζ$），我们调整构造以获得$ \ mathrm {modrm {mod}的投影表示形式（\ varsigma_ {g，1}）$。最后，通过明确的同构，我们确定了$ \ mathrm {mod}（\ varsigma_ {g，1}）$的子代表，这与量子表示相同，与非semisimple tqft产生的量子表示与与量子$ \ m mathfrak {sl} sl} $ qual相关。在此过程中，我们为所有标准生成器提供了具体的基础和明确的公式，以$ \ mathrm {mod}（\ varsigma_ {g，1}）和量子$ \ mathfrak {slfrak {sl} _2 _2 $在等价问题上的问题上，以及回答Crivelli，felelder，felderder，felderderder，ekkekk的问题。 We also make sure that the restriction of these representations to the Torelli group $\mathcal{I}(\varSigma_{g,1})$ are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers $\mathbb{Z}[ζ]$, when expressed in these bases.
We provide a homological model for a family of quantum representations of mapping class groups arising from non-semisimple TQFTs (Topological Quantum Field Theories). Our approach gives a new geometric point of view on these representations, and it gathers into one theory two of the most promising constructions for investigating linearity of mapping class groups. More precisely, if $\varSigma_{g,1}$ is a surface of genus $g$ with $1$ boundary component, we consider a (crossed) action of its mapping class group $\mathrm{Mod}(\varSigma_{g,1})$ on the homology of its configuration space $\mathrm{Conf}_n(\varSigma_{g,1})$ with twisted coefficients in the Heisenberg quotient $\mathbb{H}_g$ of its surface braid group $π_1(\mathrm{Conf}_n(\varSigma_{g,1}))$. We show that this action intertwines an action of the quantum group of $\mathfrak{sl}_2$, that we define by purely homological means. For a finite-dimensional linear representation of $\mathbb{H}_g$ (depending on a root of unity $ζ$), we tweak the construction to obtain a projective representation of $\mathrm{Mod}(\varSigma_{g,1})$. Finally, we identify, by an explicit isomorphism, a subrepresentation of $\mathrm{Mod}(\varSigma_{g,1})$ that is equivalent to the quantum representation arising from the non-semisimple TQFT associated with quantum $\mathfrak{sl}_2$ at $ζ$. In the process, we provide concrete bases and explicit formulas for the actions of all the standard generators of $\mathrm{Mod}(\varSigma_{g,1})$ and of quantum $\mathfrak{sl}_2$ on both sides of the equivalence, and answer a question by Crivelli, Felder, and Wieczerkowski. We also make sure that the restriction of these representations to the Torelli group $\mathcal{I}(\varSigma_{g,1})$ are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers $\mathbb{Z}[ζ]$, when expressed in these bases.
