Infinite hierarchies of nonlocal symmetries of the Chen--Kontsevich--Schwarz type for the oriented associativity equations

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…

The Singularity of Kontsevich's Solution for $QH^{*}(CP^2)$

We study the singularity of the Kontsevich's solution of the WDVV equations of associativity. We prove that it corresponds to a singularity in the change of two coordinates systems of the Frobenius manifold given by the quantum cohomology of $CP^2$.

Kontsevich 星积 $\star$ 模 $\bar{o}(\hbar^k)$ 的结合性证书：$k\leqslant 6$ 与 $k\geqslant7$ 不同

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

本文揭示 Kontsevich 星积在 $\bar{o}(\hbar^6)$ 以内与 $\hbar^7$ 及以上阶的结合性机制不同：低阶时雅可比恒等式推论可直接由结合子得到，高阶时则需经 Kontsevich 图多步推导。

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 $μ_A\mapsto\star_{A[[\hbar]]}$ of the usual multiplica…