The higher structure of the Floer cochains of the diagonal in CP^ x CP^n - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:54:21Z http://mathoverflow.net/feeds/question/27087 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27087/the-higher-structure-of-the-floer-cochains-of-the-diagonal-in-cp-x-cpn The higher structure of the Floer cochains of the diagonal in CP^ x CP^n Dan 2010-06-04T19:21:44Z 2010-06-05T02:58:08Z <p>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? </p> http://mathoverflow.net/questions/27087/the-higher-structure-of-the-floer-cochains-of-the-diagonal-in-cp-x-cpn/27122#27122 Answer by Tim Perutz for The higher structure of the Floer cochains of the diagonal in CP^ x CP^n Tim Perutz 2010-06-05T00:52:27Z 2010-06-05T02:58:08Z <p>Here's an argument that the diagonal Lagrangian correspondence $\Delta$ in $\mathbb{C}P^n \times \mathbb{C}P^n$ is <i>formal</i>. 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 $\mu^2$.</p> <p>Be critical; I might have slipped up!</p> <p>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 <i>intrinsically formal</i>, 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 $d\neq 2$.</p> <p>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. </p> <hr> <p>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).</p>