论文标题
从重力到弦拓扑
From gravity to string topology
论文作者
论文摘要
作者早些时候提出的链重力适用于任何环状$ a_ \ in_ \ infty $代数的循环hochschild，配备了标量$ -d $的标量产品。特别是，它作用于任何繁殖二元组的循环霍基柴尔德综合体，以及通过商DG Porporad $ st_ {3-d} $的功能区图的动作因素。我们表明，其共同体proporad $ h^\ bullet（st_ {3-d}）$是高度的，并且它在简化的Equivariant同源性$ \ bar {h} _ \ bullet^{s^1}（s^1}（lm）$ lm lm $ lm $ lm $ lm $ lm $ d $ d $ d $ dimensy $ $ $ $ $ $ $ $ $ m的行为上。通过其非常结构，弦拓扑均$ h^\ bullet（st_ {3-d}）$配备了重力术的形态，由模量空间$ m_ {g，n}稳定的稳定代数c $ G $ G $ g with Marked Points的稳定代数曲线的紧凑型共同体确定。该结果引起了弦拓扑的新普遍操作，并以统一的方式复制了几种已知的构造：我们表明（i）$ h^\ bullet（st_ {3-d}）$在$ 3-d $ 3-d $ 3-d $ 3-d $ 3-d $ 3-d $ 3-d $ 3-d的行动下也是$ \ bar { M. Chas和D. Sullivan的纯几何结构，（ii）$ h^\ bullet（st_ {3-d}）$在同型均匀参与的lie bialgebras下是$ 2-d $的适当的; （iii）E。Getzler的重力作业注入$ h^\ bullet（st_ {3-d}）$，这意味着C. Westerland的几何结构纯粹是代数的对应物，建立了重力作用，以$ \ bar {h}
The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic $A_\infty$ algebra equipped with a scalar product of degree $-d$. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree $d$, and that action factors through a quotient dg properad $ST_{3-d}$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $H^\bullet(ST_{3-d})$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $\bar{H}_\bullet^{S^1}(LM)$ of the loop space $LM$ of any simply connected $d$-dimensional closed manifold $M$. By its very construction, the string topology properad $H^\bullet(ST_{3-d})$ comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces $M_{g,n}$ of stable algebraic curves of genus $g$ with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) $H^\bullet(ST_{3-d})$ is also a properad under the properad of involutive Lie bialgebras in degree $3-d$ whose induced action on $\bar{H}_\bullet^{S^1}(LM)$ agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) $H^\bullet(ST_{3-d})$ is a properad under the properad of homotopy involutive Lie bialgebras in degree $2-d$; (iii) E. Getzler's gravity operad injects into $H^\bullet(ST_{3-d})$ implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on $\bar{H}_\bullet^{S^1}(LM)$.
