of the [*WDVV equations*]{} $$\begin{gathered}
\label{eq:WDVV equations}
\frac{\d^3 F}{\d t^\alpha\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^3 F}{\d t^\nu\d t^\gamma\d t^\delta}=\frac{\d^3 F}{\d t^\delta\d t^\beta\d t^\mu}\eta^{\mu\nu}\frac{\d^3 F}{\d t^\nu\d t^\gamma\d t^\alpha},\quad 1\le\alpha,\beta,\gamma,\delta\le N,\end{gathered}$$ where $F = F(t^1,\dots,t^N)$ is an analytic function defined on some open subset $M\subset{\mathbb C}^N$, $\eta=(\eta_{\alpha\beta})$ is an $N\times N$ symmetric non-degenerate matrix with complex coefficients, $(\eta^{\alpha\beta}):=\eta^{-1}$ and we use the convention of sum over repeated Greek indices. The WDVV equations appear in many areas of mathematics, including singularity theory and curve counting theories in algebraic geometry. In Gromov–Witten theory the WDVV equations describe the structure of primary Gromov–Witten invariants in genus $0$ and naturally come from a certain relation in the cohomology of the moduli space of stable curves.

Suppose that a function $F$ satisfies the WDVV equations together with the additional assumption $$\begin{gathered}
\label{eq:unit condition for F}
\frac{\d^3 F}{\d t^1\d t^\alpha\d t^\beta}=\eta_{\alpha\beta}.\end{gathered}$$ The function $F$ defines a commutative product $\circ$ on each tangent space $T_p M$ by $$\frac{{{\partial}}}{{{\partial}}t^\alpha} \circ \frac{{{\partial}}}{{{\partial}}t^\beta}:=\frac{{{\partial}}^3F}{{{\partial}}t^\alpha {{\partial}}t^\beta {{\partial}}t^\gamma} \eta^{\gamma \delta} \frac{{{\partial}}}{{{\partial}}t^\delta}, \quad 1 \le \alpha,\beta \le N.$$ One can immediately see that the WDVV equations are equivalent to the associativity of this product and property  means that the vector field $\frac{\d}{\d t^1}$ is the unit. One can go 