arXiv subjects

Ricardo Buring

Publications and source records attributed to Ricardo Buring.

14 recordsLinked to original sources

arXiv2023

Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$

The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

math.QA↗

arXiv2022

The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^{d}$: dimension-specific micro-graph calculus

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\smash{\mathbb{R}^3}$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\smash{\mathbb{R}^2}$.

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arXiv2022

Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?

The Kontsevich star-product admits a well-defined restriction to the class of affine -- in particular, linear -- Poisson brackets; its graph expansion consists only of Kontsevich's graphs with in-degree $\leqslant 1$ for aerial vertices. We obtain the formula $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ with harmonic propagators for the graph weights (over $n\leqslant 7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet--Felder--Willwacher, that they match the computations using the $\textsf{kontsevint}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}(\hbar^7)$. We discover that the Riemann zeta value $\zeta(3)^2/\pi^6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ \textit{because} all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $\zeta(3)^2/\pi^6$ represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula $\star_{\text{aff}}^{\text{red}}$ mod~$\bar{o}(\hbar^7)$ with only rational coefficients.

math.QA↗

arXiv2021

The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets

Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds -- infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![ [\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$ on $\mathbb{R}^4\ni\boldsymbol{y}$, including the general case $\varrho \not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P} = Q_\gamma(P^{\otimes^{\# Vert(\gamma)}})$ is trivial in the second Poisson cohomology, $Q_\gamma = [\![ P, \vec{X}([\varrho],[a]) ]\!]$, for the Nambu-determinant bi-vectors $P(\varrho,[a])$ on $\mathbb{R}^3$. For the global Casimirs $\mathbf{a} = (a_1,\ldots,a_{d-2})$ and inverse density $\varrho$ on $\mathbb{R}^d$, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely $\dot{\varrho} = \dot{\varrho}([\varrho], [\mathbf{a}])$ and $\dot{\mathbf{a}} = \dot{\mathbf{a}}([\varrho],[\mathbf{a}])$ with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron $\gamma_3$ and five for the pentagon-wheel graph cocycle $\gamma_5$), as dictated by the behaviour $\varrho(\mathbf{x}') = \varrho(\mathbf{x}) \cdot \det \| \partial \mathbf{x}' / \partial \mathbf{x} \|$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x} \rightleftarrows \mathbf{x}'$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.

math.SG↗

arXiv2019

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(\gamma)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $\gamma$ on $n$ vertices and $2n-2$ edges in each term, then the $1$-vector $\vec{X}=Or(\gamma)(\vec{V}\otimes P^{\otimes^{n-1}})$ is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors $P$ satisfying the above assumptions, on all finite-dimensional affine manifolds $M$. Still, if the bi-vector $Q\not\equiv 0$ is exact in the respective Poisson cohomology, so there exists a vector field $\vec{Y}$ such that $Q(P)=[\![\vec{Y},P]\!]$, then the universal cocycle $\vec{X}$ does not belong to the coset of $\vec{Y}$ mod $\ker[\![P,\cdot]\!]$. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the $R$-matrices for the Lie algebra $\mathfrak{gl}(2)$. Keywords: Graph complex, Poisson bracket, deformation cohomology, affine manifold, tetrahedral flow, diffeomorphism.

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arXiv2019

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.

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arXiv2019

The Kontsevich graph orientation morphism revisited

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ with ordered sets of edges and without multiple edges and one-cycles. It is known that $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ with respect to the vertex-expanding differential $d=[{\bullet}\!\!{-}\!{-}\!\!{\bullet},\cdot]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in\ker\,[\![ P,{\cdot}]\!]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $\Pi T^* N^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $\Pi T^* N^d$.

math.CO↗

arXiv2018

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.

math.CO↗

arXiv2017

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 $[\![\mathcal{P},\mathcal{P}]\!]=0$.

math-ph↗

arXiv2017

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 $k=8$.

math.CO↗

arXiv2017

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.

math.CO↗

arXiv2017

The expansion $\star$ mod $\bar{o}(\hbar^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative $\star$-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich $\star$-product up to order 4 in the deformation parameter $\hbar$. Already at this stage, the $\star$-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo $\bar{o}(\hbar^4)$, for the newly built $\star$-product expansion.

math.CO↗

arXiv2016

The Kontsevich tetrahedral flow revisited

We prove that the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = \mathcal{Q}_{a:b} (\mathcal{P})$, the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector $\mathcal{P}$ on an affine real Poisson manifold $N^n$, does infinitesimally preserve the space of Poisson bi-vectors on $N^n$ if and only if the two monomials in $\mathcal{Q}_{a:b} (\mathcal{P})$ are balanced by the ratio $a:b=1:6$. The proof is explicit; it is written in the language of Kontsevich graphs.

math.QA↗

arXiv2016

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.-m

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