The orientation morphism: from graph cocycles to deformations of Poisson structures

We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[\![\mathcal{P},{\rm O\vec{r}}(\gamma)(\mathcal{P})]\!] \doteq 0$ through the differential consequences of the Jacobi identity $[\![\mathcal{P},\mathcal{P}]\!]=0$ for Poisson bi-vectors $\mathcal{P}$. To illustrate the reasoning, we use the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma_3)(\mathcal{P})$, as well as the flow produced from the Kontsevich--Willwacher pentagon-wheel cocycle $\gamma_5$ and the new flow obtained from the heptagon-wheel cocycle $\gamma_7$ in the unoriented graph complex.Comment: 12 pages. Talk given by R.B. at Group32 (Jul 9--13, 2018; CVUT Prague, Czech Republic). Big formula in Appendix A retained from the (unpublished) Appendix in arXiv:1712.05259 [math-ph]. Signs corrected in v

On the Kontsevich ⋆\star-product associativity mechanism

The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$in the algebra of formal power series in$\hbar$on a given finite-dimensional affine Poisson manifold: here$\times$is the usual multiplication,$\{\ ,\ \}_{P}\neq0$is the Poisson bracket, and$\hbar$is the deformation parameter. The product$\star$is assembled at all powers$\hbar^{k\geq0}$via summation over a certain set of weighted graphs with$k+2$vertices; for each$k>0$, every such graph connects the two co-multiples of$\star$using$k$copies of$\{\ ,\ \}_{P}$. Cattaneo and Felder [ arXiv:math/9902090 [math.QA] ] interpreted these topological portraits as the genuine Feynman diagrams in the Ikeda-Izawa model [arXiv:hep-th/9312059] for quantum gravity. By expanding the star-product up to$\bar{o}(\hbar^3)$, i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = Operator(Poisson) that converts the Jacobi identity for the bracket$\{\ ,\ \}_{P}$into the associativity of$\star$. Key words: Deformation quantization, associative algebra, Poisson bracket, graph complex, star-product PACS: 02.40.Sf, 02.10.Ox, 02.40.Gh, also 04.60.-mComment: Proc. Internaional workshop SQS'15 on Supersymmetry and Quantum Symmetries (3-8 August 2015, JINR Dubna, Russia), 4 page

The heptagon-wheel cocycle in the Kontsevich graph complex

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

Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}} = \mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $\gamma$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{\mathcal{P}} = \text{Or}(\gamma)(\mathcal{P})$ preserves the space of Poisson structures if $\gamma$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles $\boldsymbol{\gamma}_{2\ell+1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell+1)$-gon wheel with a nonzero coefficient. At $\ell=1$ the tetrahedron $\boldsymbol{\gamma}_3$ itself is a cocycle; at $\ell=2$ the Kontsevich--Willwacher pentagon-wheel cocycle $\boldsymbol{\gamma}_5$ consists of two graphs. We reconstruct the symmetry $\mathcal{Q}_5(\mathcal{P}) = \text{Or}(\boldsymbol{\gamma}_5)(\mathcal{P})$ and verify that $\mathcal{Q}_5$ is a Poisson cocycle indeed: $[\![\mathcal{P},\mathcal{Q}_5(\mathcal{P})]\!]\doteq 0$ via $[\![\mathcal{P},\mathcal{P}]\!]=0$.Comment: Int. workshop "Supersymmetries and quantum symmetries -- SQS'17" (July 31 -- August 5, 2017 at JINR Dubna, Russia), 4+v pages, 2 figures, 1 tabl

Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Let $P$ be a Poisson structure on a finite-dimensional affine real manifold. Can $P$ be deformed in such a way that it stays Poisson? The language of Kontsevich graphs provides a universal approach -- with respect to all affine Poisson manifolds -- to finding a class of solutions to this deformation problem. For that reasoning, several types of graphs are needed. In this paper we outline the algorithms to generate those graphs. The graphs that encode deformations are classified by the number of internal vertices $k$; for $k \leqslant 4$ we present all solutions of the deformation problem. For $k \geqslant 5$, first reproducing the pentagon-wheel picture suggested at $k=6$ by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a new unique solution without $2$-loops and tadpoles at $k=8$.Comment: International conference ISQS'25 on integrable systems and quantum symmetries (6-10 June 2017 in CVUT Prague, Czech Republic). Introductory paragraph I.1 follows p.3 in arXiv:1710.00658 [math.CO]; 13 pages, 3 figures, 2 table

Migraine, headache, and mortality in women: a cohort study

Background: Migraine carries a high global burden, disproportionately affects women, and has been implicated as a risk factor for cardiovascular disease. Migraine with aura has been consistently associated with increased risk of cardiovascular mortality. However, published evidence on relationships between migraine or non-migraine headache and all-cause mortality is inconclusive. Therefore, we aimed to estimate the effect of non-migraine headache and migraine as well as migraine subtypes on all-cause and cause-specific mortality in women. Methods: In total, 27,844 Women’s Health Study participants, aged 45 years or older at baseline, were followed up for a median of 22.7 years. We included participants who provided information on migraine (past history, migraine without aura, or migraine with aura) or headache status and a blood sample at study start. An endpoints committee of physicians evaluated reports of incident deaths and used medical records to confirm deaths due to cardiovascular, cancer, or female-specific cancer causes. We used multivariable Cox proportional hazards models to estimate the effect of migraine or headache status on both all-cause and cause-specific mortality. Results: Compared to individuals without any headache, no differences in all-cause mortality for individuals suffering from non-migraine headache or any migraine were observed after adjustment for confounding (HR = 1.01, 95%CI, 0.93–1.10 and HR = 0.96, 95% CI: 0.89–1.04). No differences were observed for the migraine subtypes and all-cause death. Women having the migraine with aura subtype had a higher mortality due to cardiovascular disease (adjusted HR = 1.64, 95%CI: 1.06–2.54). As an explanation for the lack of overall association with all-cause mortality, we observed slightly protective signals for any cancer and female-specific cancers in this group. Conclusions: In this large prospective study of women, we found no association between non-migraine headache or migraine and all-cause mortality. Women suffering from migraine with aura had an increased risk of cardiovascular death. Future studies should investigate the reasons for the increased risk of cardiovascular mortality and evaluate whether changes in migraine patterns across the life course have differential effects on mortality

Formality morphism as the mechanism of ⋆\star-product associativity:how it works

The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-dimensional affine manifolds. Not a Lie algebra morphism by its term $\mathcal{F}_1$ alone, the entire set $\boldsymbol{\mathcal{F}}$ is an $L_\infty$-morphism instead. It induces a map of the Maurer-Cartan elements, taking Poisson bi-vectors to deformations $\mu_A\mapsto\star_{A[[\hbar]]}$ of the usual multiplication of functions into associative noncommutative $\star$-products of power series in $\hbar$. The associativity of $\star$-products is then realized, in terms of the Kontsevich graphs which encode polydifferential operators, by differential consequences of the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mechanism for the Kontsevich $\star$-products (in particular, with harmonic propagators). We inspect how the Kontsevich weights are correlated for the orgraphs which occur in the associator for $\star$ and in its expansion using Leibniz graphs with the Jacobi identity at a vertex.Comment: Symmetries & integrability of equations of mathematical physics (22-24 December 2018, IM NASU Kiev, Ukraine), 16 page