Moduli Space Of Graphs Research Articles

Schwinger, ltd: loop-tree duality in the parametric representation

We derive a variant of the loop-tree duality for Feynman integrals in the Schwinger parametric representation. This is achieved by decomposing the integration domain into a disjoint union of cells, one for each spanning tree of the graph under consideration. Each of these cells is the total space of a fiber bundle with contractible fibers over a cube. Loop-tree duality emerges then as the result of first decomposing the integration domain, then integrating along the fibers of each fiber bundle.As a byproduct we obtain a new proof that the moduli space of graphs is homotopy equivalent to its spine. In addition, we outline a potential application to Kontsevich’s graph (co-)homology.

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out (F_n) and Aut (F_n) from classes for groups \Gamma_{k,s} , where the \Gamma_{k,s} generalize Out (F_k)=\Gamma_{k,0} and Aut (F_k)=\Gamma_{k,1} . The symmetric group \mathfrak G_s acts on H_*(\Gamma_{k,s}) by permuting leaves, and for trivial rational coefficients we compute the \mathfrak G_s -module structure on H_*(\Gamma_{k,s}) completely for k \leq 2 . Assembling these classes then produces all the known nontrivial rational homology classes for Aut (F_n) and Out (F_n) with the possible exception of classes for n=7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut (F_n) and Out (F_n) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

It is not known whether or not the stable rational cohomology groups $\tilde H^*(\operatorname{Aut}(F_\infty);\mathbb{Q})$ always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of e;outer spacee: and e;auter spacee;, respectively, introduced in [3] by Culler and Vogtmann and [6] by Hatcher and Vogtmann.