The higher structure of the Floer cochains of the diagonal in CP^ x CP^n

Question

I am trying to learn a little Lagrangian Floer theory and I was hoping someone could explain the following calculation. Consider CP^n x CP^n with the form (omega,-omega) and the diagonal Lagrangian L. Now the FH*(L,L) is isomorphic to the quantum cohomology of CP^n as a ring. How about the higher A-infinity structure on the Floer cochains, CH*(L,L)? Can we compute this in a reasonable way? I'd be happy to even understand this for CP^1, though I suspect this might be obvious somehow. Is there a way to extract this from the Gromov Witten invariants of CP^n?

Answer 1

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 $d\neq 2$.

Suppose inductively that we can kill the $d$-fold products $\mu^d$ for $3\leq d\leq m$. 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 $\Delta_M\subset M\times M$ 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).

Answer 2

I was browsing old symplectic questions and spotted this one. I realise that in the intervening decade you "learned a little Lagrangian Floer theory" like the question says, but maybe you'll enjoy another answer anyway.

In the paper https://arxiv.org/abs/1507.05842 by me and YankI Lekili, we give some examples of non-formality for the diagonal for toric manifolds (see corollary 7.3.5). Indeed the simplest is $CP^1$ when Floer cohomology is considered in characteristic 2 (whereas it is formal in characteristic zero, as Tim explains above). Here's how to see it in this simplest case.

The toric Hamiltonian circle action gives you a Lagrangian "moment correspondence" from $T^*S^1$ to $CP^1\times CP^1$ which gives an $A_\infty$ functor from the wrapped Fukaya category to the Fukaya category. This functor sends the cotangent fibre to the diagonal and sends the zero section to the product torus (equator times equator). In particular, it gives an $A_\infty$ map from chains on the based loopspace of $S^1$ (as a ring this is $k[z,z^{-1}]$, and higher products vanish for degree reasons) to quantum cohomology of $CP^1$ (as a ring this is $k[H]/(H^2-1)$, with the higher products to be determined below; note that I'm not working over a Novikov ring, I just set the area parameter equal to 1). Note that I'm using a minimal ($\mu_1=0$) model for the $A_\infty$ structure on the Floer complex of the diagonal, just identifying it with QH with some higher products.

In $T^*S^1$, you can write the zero section as the cone on a morphism from the cotangent fibre to itself. Under the moment correspondence functor, this means you can write the product torus as the cone on an element of quantum cohomology, considered as a morphism from the diagonal to itself. Of course it's nontrivial to compute what that element of QH is because it comes out of an $A_\infty$ functor counting pseudoholomorphic quilts (followed by an $A_\infty$ homotopy transfer to get to a minimal model). However, if we take the field $k$ to be $Z/(2)$ then there are only four elements of QH: $0$, $1$, $H$ and $1+H$.

So you try taking the cone on each of these and computing its self-homs (e.g. in the category of twisted complexes). Only one choice gives the right dimension to match with the Floer cohomology of the product torus (I think* it was $1+H$).

You can get this far only knowing the quantum product on QH, not the higher products. Now compute the ring structure on the cone. For this you need to know $\mu_4$ on QH. Since we can assume by a further homotopy that the $A_\infty$ structure is strictly unital, the only possibilities are $\mu_4(H,H,H,H)=0$ or $H$. To get the correct product structure on the Floer cohomology of the product torus you need this to be H. Therefore Floer cochains on the diagonal are non-formal in characteristic 2.

More generally, corollary 7.3.5 referred to above shows that self-Floer cochains of the diagonal is not formal if QH is not semisimple. It would be fun to explicitly compute the higher products for $CP^2$ in characteristic 3, etc.

*As I say, you can check this is the only possibility using nothing more than linear algebra. But to see why it is geometrically reasonable, remember that, given a Morse function and gradient like vector field, the zero section in $T^*M$ is built out of cotangent fibres at critical points using morphisms coming from Morse flow lines. On the circle you have two flow lines. Now shrink one flowline to bring the two critical points together. Roughly, our moment correspondence functor should be giving Seidel elements: from one flowline (the one which wraps around) you'll get the Seidel element of the $S^1$ action, namely H; from the one which shrinks you'll get the Seidel element of the trivial circle action, namely 1.