Associativity from genus-zero WDVV under a splitting axiom

The stack ${\bar{M}}_{0,4}$ is isomorphic to ${\mathbb{P}}^1$: the cross-ratio identifies its smooth locus with ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$, and the three omitted values are the three nodal two-component curves. Hence the boundary divisors $D_{12\mathrm{\mid }34}$ and $D_{14\mathrm{\mid }23}$ are rationally equivalent.

Extend the pairing to $H\otimes \mathrm{\Lambda }$ and put $C(a,b,c)={\sum }_{\beta }{Q}^{\beta }⟨a,b,c{⟩}_{0,3,\beta }=(a\ast b,c)$. Permutation symmetry of primary Gromov–Witten classes makes $C$ graded symmetric. The pairing and curve variables are even, so $\mathrm{\mid }a\ast b\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }$ for parity-homogeneous inputs. These are the graded tensor conventions in the assumed Gromov–Witten data.

Integrate the pullback of the equal boundary classes against the four-point virtual class with homogeneous insertions $a,b,c,d$. The assumed cohomological splitting axiom expands the $12\mathrm{\mid }34$ boundary, coefficientwise in $Q$, as $(a\ast b,c\ast d)$; the $14\mathrm{\mid }23$ boundary gives $(-1{)}^{\mathrm{\mid }d\mathrm{\mid }(\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid })}(a\ast d,b\ast c)$, since moving $d$ past $b,c$ is the permutation grouping the ordered insertions into those two pairs. These contractions use the graded inverse pairing, including its Koszul signs. Graded symmetry of $C$ gives $(a\ast b,c\ast d)=C(a\ast b,c,d)=((a\ast b)\ast c,d)$. For the other contraction, cyclic symmetry and graded symmetry of the pairing give $C(a,b\ast c,d)=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }(\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid })}(b\ast c,a\ast d)=(-1{)}^{\mathrm{\mid }d\mathrm{\mid }(\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid })}(a\ast d,b\ast c)$. Hence equality of the boundary integrals is precisely $((a\ast b)\ast c,d)=(a\ast (b\ast c),d)$. The splitting here is the cohomological assumption in the Statement; [F1] records the distinct scope of the algebraic supplier.

By the definition of $\ast$, each contracted sum in WDVV is the pairing of the corresponding iterated quantum products with the remaining input. Thus WDVV gives $((a\ast b)\ast c,d)=(a\ast (b\ast c),d)$ coefficientwise in $\mathrm{\Lambda }$ for every homogeneous $d$. The Novikov completion makes these coefficientwise sums well-defined, and nondegeneracy of the Poincaré pairing implies $(a\ast b)\ast c=a\ast (b\ast c)$. Bilinearity extends the identity to all classes.