The heptagon-wheel cocycle in the Kontsevich graph complex

Abstract

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce -- under the orientation mapping -- infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the $(2\ell+1)$-wheel graph with a nonzero coefficient at every $\ell\in\mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell = 1$ and $\ell = 2$ of one and two graphs respectively, the cocycle condition $d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel cocycle (known to exist at $\ell = 3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.Comment: Special Issue JNMP 2017 `Local and nonlocal symmetries in Mathematical Physics'; 17 journal-style pages, 54 figures, 3 tables; v2 accepte

The defining properties of the Kontsevich unoriented graph complex

“…Denote by Gra the quotient of the vector space of graphs at hand modulo the above relation (cf. [5,6] or [2]). By definition, zero graphs represent the zero class in Gra, that is, the (formal sums of) graphs which equal minus themselves under a symmetry that induces a parity-odd permutation of edges.…”

“…The knowledge of cocycles in the Kontsevich unoriented graph complex (Gra, d) is important because under the orientation morphism O r, the d-cocycles on n vertices and 2n−2 edges are mapped to the universal deformations of Poisson brackets on finite-dimensional affine manifolds (see [5] or [2,3,4] and [1]). For example, the tetrahedron γ 3 ∈ ker d from [5], as well as the pentagon-and heptagon-wheel cocycles γ 5 , γ 7 ∈ ker d are presented in [2] (see also references therein). The Poisson structure symmetry which corresponds to γ 5 is found in [3,4].…”

“…We follow the notation and conventions which are adopted in [4] (and in [2] where examples are given with practical calculations in the Kontsevich graph complex). We refer to [1] or [6] for more details about the complex Gra (introduced in [5]).…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…Examples of this construction for known graph cocycles γ 2ℓ+1 on n vertices and 2n − 2 edges (namely, n = 2ℓ + 2 = 4, 6, and 8) have been given in [2,3], [7], and [4], respectively. Practical calculation of graph cocycles is addressed in [8,30]; the algorithms to verify the Poisson cocycle factorisation through the Jacobi identity are available from [9]. In the fundamental work [29], see also [27], Willwacher related the unoriented graph complex to generators of the Grothendieck-Teichmüller Lie algebra grt.…”

“…Are the graph cocycles truly graphs ? In the papers [21] (see also [22] and [4,8,28] for a pedagogical review), Kontsevich introduced the graph complex -one of the many -with parity-even vertices, with a wedge ordering of parity-odd edges, and the differential d = [•−•, ·] produced by the graded commutator of graph insertions into vertices. This direction of research was furthered by Willwacher et al [11,14,29]: in particular, in [30] a generating function counts the numbers of nonzero (i.e.…”