Contents

Contents

Graph complex

Roughly:

Consider the complex vector space $\mathcal{G}$ spanned by the isomorphism classes of oriented ribbon graphs modulo the relation $(\Gamma,-\sigma) = - (\Gamma, \sigma)$ where $\sigma$ is an orientation on the graph $\Gamma$.

A differential on this vector space is given by

$\partial(\Gamma) \coloneqq \sum_{e\in E(\Gamma)\backslash Loop(\Gamma)} \Gamma/e \,,$

where the sum is over edges $e$ which are not loops (have distinct source and target) and $\Gamma/e$ is obtained from $\Gamma$ by contraction at edge $e$ (cf. ribbon graph).

This function $\partial$ is indeed a differential, in that it satisfies $\partial^2 = 0$, because two contractions in different order produce a different orientation.

some details missing here…

The resulting chain complex $(\mathcal{G}_\bullet, \partial)$ is called the graph complex. Its chain homology $H_\bullet(\mathcal{G}, \partial)$ is called graph homology.

This was originally indicated in (Kontsevich 94, pages 11-12). For a detailed and careful account see Lambrechts-Volic 14, section 6.

Example

(the “3-term relation”)

In the graph complex the differential of the graph as shown on the left below (the vertices on the horizontal line are the external vertices, that above the line is internal) is a linear combination as shown on the right:

$\phantom{AAA}\array{ \partial \\ \phantom{A} \\ \phantom{a}}$ $\phantom{A}\array{ = \\ \phantom{A} \\ \phantom{a} }\phantom{A}$

graphics grabbed from Lambrechts-Volic 14, Figure 1 & Figure 2

Under the quasi-isomorphism from the graph complex to the de Rham complex on the Fulton-MacPherson compactification of a configuration space of points given by sending each graph to its Chern-Simons Feynman amplitude on compactified configuration spaces of points (this Prop.) this relation becomes the “3-term relation” (this Prop.):

$\left[g_{i j}\right] \wedge \left[ g_{j k} \right] + \left[g_{j k}\right] \wedge \left[ g_{k i} \right] + \left[g_{k i}\right] \wedge \left[ g_{i j} \right] \;\sim\; 0$

satisfied by the Chern-Simons propagator form

$\left( g_{i j} \;\in\; \Omega^2\left( FM_n(\mathbb{R}^d) \right) \right) \,.$

Properties

General properties

There is a canonical bigrading on the graph complex, where $\mathcal{G}_{i j}$ is generated by those graphs which have $i$ vertices and $j$ edges; the differential has bidegree $(-1,-1)$; each $\mathcal{G}_{i j}$ is finite-dimensional, while the whole complex is infinite-dimensional.

The graph complex splits into a direct sum of subcomplexes labelled by the Euler characteristics of the underlying graph.

The structure of a graph complex reflects a structure in the Chevalley-Eilenberg complex of a certain Lie algebra; and the graph homology to the relative Lie homology of that Lie algebra as shown by Kontsevich.

$L_\infty$-algebra structure

The Graph complex carries the structure of a dg-Lie algebra (L-infinity algebra) which acts on the space of choices of formal deformation quantization of Poisson manifolds. Its degree-0 chain homology is the Lie algebra of the Grothendieck-Teichmüller group.

The homology in negative degree vanishes and that in positive degree is still unknown, but computer experiements show that at least the third cohomology contains nontrivial elements.

The degree-0 homology is also isomorphic, up to one “scaling class”, to the 0th cohomology of the derivations of the E2 operad.

Relation to configuration spaces

The system of graph complexes is quasi-isomorphic to the real cohomology of configuration spaces of points (Campos-Willwacher 16, theorem 1).

Further applications

…moduli spaces

…deformation theory

…Rozansky-Witten theory

…Vassiliev invariants

…description of the classifying space $BOut(F_n)$ of the group of outer automorphisms of a free group with $n$ generators

Graph complex controls the universal $L_\infty$-deformations of the space of polyvector fields.

Generalizations

There are generalizations for $d$-algebras (algebras over little disc operad in higher dimension). The cohomological graph complex is then the case for $d=2$. There is also a “directed” version. On the other hand, graph complex

References

Various versions of the definition of the graph complex were introduced in

• Maxim Kontsevich, pages 11-12 of Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121 (pdf)

• Maxim Kontsevich, around Def. 15 and Lemma 3 in Operads and Motives in Deformation Quantization, Lett. Math. Phys. 48 35-72, 1999 (arXiv:math/9904055)

Decent review of the graph complex as a model for the de Rham cohomology of the Fulton-MacPherson compactification of configuration spaces of points (exhibiting the formality of the little n-disk operads) is in

further review:

Further discussion of the graph complex as a model for the de Rham cohomology of configuration spaces of points is in