group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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.
(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.):
satisfied by the Chern-Simons propagator form
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.
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.
See at deformation quantization – Motivic Galois group action on the space of quantizations.
See at Grothendieck-Teichmüller group – relation to the graph complex.
The system of graph complexes is quasi-isomorphic to the real cohomology of configuration spaces of points (Campos-Willwacher 16, theorem 1).
…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.
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
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
Najib Idrissi, The Lambrechts-Stanley Model of Configuration Spaces, Invent. Math, 2018 (arXiv:1608.08054, doi:10.1007/s00222-018-0842-9)
Ricardo Campos, Thomas Willwacher, A model for configuration spaces of points (arXiv:1604.02043)
Ricardo Campos, Najib Idrissi, Pascal Lambrechts, Thomas Willwacher, Configuration Spaces of Manifolds with Boundary (arXiv:1802.00716)
Ricardo Campos, Julien Ducoulombier, Najib Idrissi, Thomas Willwacher, A model for framed configuration spaces of points (arXiv:1807.08319)
See also
Maxim Kontsevich, Rozansky–Witten invariants via formal geometry, Compositio Mathematica 115: 115–127, 1999, doi, arXiv:dg-ga/9704009
Martin Markl, Steve Shnider, James Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.
Andrey Lazarev, Operads and topological conformal field theories, pdf; and older versio: Graduate lectures on operads and topological field theories, zip file with 11 pdfs, over 5 Mb
Alastair Hamilton, A super-analogue of Kontsevich’s theorem on graph homology, Lett. Math. Phys. 76 (2006), no. 1, 37–55, math.QA/0510390
A. Lazarev, Alexander Voronov, Graph homology: Koszul and Verdier duality (math.QA/0702313)
Mikhail Movshev, A definition of graph homology and graph K-theory of algebras (math.KT/9911111)
Alberto S. Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (doi:10.1142/S0218216505004019, math.GT/0307218, MR2006g:58021)
Kiyoshi Igusa, Graph cohomology and Kontsevich cycles, Topology 43 (2004), n. 6, p. 1469-1510, MR2005d:57028, doi
Thomas Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra (arxiv/1009.1654)
Vasily Dolgushev, Christopher Rogers, Thomas Willwacher, Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields (arxiv/1211.4230)
Damien Calaque, Carlo A. Rossi, Lectures on Duflo isomorphisms in Lie algebra and complex geometry, European Math. Soc. 2011
Sergei Merkulov, Graph complexes with loops and wheels, in (Manin’s Festschrift:) Algebra, Arithmetic, and Geometry, Progress in Mathematics 270 (2009) 311-354, doi, pdf
Martin Markl, Sergei Merkulov, S. Shadrin, Wheeled PROPs, graph complexes and the master equation, J. Pure Appl. Algebra, 213(4):496–535, 2009, math.AT/0610683
The following survey has discussion of context between the graph complex and Batalin-Vilkovisky formalism:
Jian Qiu, Maxim Zabzine, Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications, arxiv/1105.2680
Jian Qiu, Maxim Zabzine, Knot weight systems from graded symplectic geometry, arxiv/1110.5234
Alastair Hamilton, Andrey Lazarev, Graph cohomology classes in the Batalin-Vilkovisky formalism, J.Geom.Phys. 59:555-575, 2009, arxiv/0701825
Last revised on February 7, 2019 at 08:53:02. See the history of this page for a list of all contributions to it.