sg.symplectic geometry - The higher structure of the Floer cochains of the diagonal in CP^ x CP^n

Here's an argument that the diagonal Lagrangian correspondence $Delta$ in $mathbb{C}P^n times mathbb{C}P^n$ is formal. That is, its Floer cochains $CF^ast(Delta,Delta)$, as an $A_infty$-algebra over the rational Novikov field $Lambda=Lambda_mathbb{Q}$ (say), are quasi-isomorphic to the underlying cohomology algebra $HF^ast(Delta, Delta)cong QH^ast(mathbb{C}P^n; Lambda)$ with trivial $A_infty$ operations $mu^d$ except for the product $mu^2$.

Be critical; I might have slipped up!

Write $A$ for $QH^ast(mathbb{C}P^n; Lambda)=Lambda[t]/(t^{n+1}=q)$. Here $q$ is the Novikov parameter. I claim that $A$ is intrinsically formal, meaning that every $A_infty$-structure on $A$, with $mu^1=0$ and $mu^2$ the product on $A$, can be modified by a change of variable so that $mu^d=0$ for $dneq 2$.

Suppose inductively that we can kill the $d$-fold products $mu^d$ for $3leq dleq m$. Then $mu^{m+1}$ is a cycle for the Hochschild (cyclic bar) complex $C^{m+1}(A,A)$. The obstruction to killing it by a change of variable (leaving the lower order terms untouched) is its class in $HH^{m+1}(A,A)$. But $A$ is a finite extension field of $Lambda$ (and, to be safe, we're in char zero). So, as proved in Weibel's homological algebra book, $HH^ast(A,A)=0$ in positive degrees, and therefore the induction works. Taking a little care over what "change of variable" actually means in terms of powers of $q$, one concludes intrinsic formality.

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You made a much more geometric suggestion - to invoke GW invariants. If you want to handle $Delta_Msubset Mtimes M$ more generally, I think this is a good idea, though I can't immediately think of a suitable reference. One can show using open-closed TQFT arguments that $HF(Delta_M,Delta_M)$ is isomorphic to Hamiltonian Floer cohomology $HF(M)$. One could do this at cochain level and thereby show that the $A_infty$ product $mu^d$ of $HF(Delta_M,Delta_M)$ corresponds to the operation in the closed-string TCFT of Hamiltonian Floer cochains arising from a genus zero surface with $d$ incoming punctures and one outgoing puncture (and varying conformal structure). Via a "PSS" isomorphism with $QH(M)$, these operations should then be computable as genus-zero GW invariants (or at any rate, the cohomology-level Massey products derived from the $A_infty$-structure should be GW invariants).