TITLE: ubiquitous quantum cohomology
QUESTION [12 upvotes]: Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which I have read are projective, I am curious about the behaviour of quantum cohomology under monodromy.
Edit: A coming seminar: "Quantum motives: realizations, detection, applications", incl. a lecture "Quantum motives: review (of) the classical idea of how to linearize algebraic geometry with an eye to utilizing it in the quantum setup." and a minicourse "Geometric Langlands and quantum motives: a link".  
Edit: The slides on Manin's talk on the concept of classical motives and it's relation to  quantum cohomology are here. 
Edit: Manin's and Smirnov's interesting computations and thoughts on e.g. a "membrane quantum cohomology" (+ russian videos of a talk in june 2011 on the program "... elucidation of this "self-referentiality" of quantum cohomology has just begun. The talk tries to outline the contours of this huge program and the first steps of it.": part 1, part 2). BTW, has anyone the text by Kapranov which is mentioned in the paper above and in Hacking's introduction?

REPLY [6 votes]: About monodromy and quantum cohomology: while everybody knows that Gromov-Witten invariants are invariant under deformations, the immediate consequence that they are invariant under the monodromy action for any smooth family of varieties is sometimes forgotten. In other words, they are invariant under the mapping class group. This can be extremely useful in computations when the mapping class group is big.
To elaborate a bit: when we are saying "GW invariants are invariant under small deformations", then we are implicitly already using the Gauss-Manin connection: If $X \to T$ is a family, say over the disc, then it doesn't make sense to identify Gromov-Witten invariants of $X_{t_1}$ and $X_{t_2}$ unless you know which GW-invariants to compare; in other words, given cohomology classes $\gamma_1, \dots, \gamma_n$ and a homology class $\beta$ on $X_{t_1}$, we need to find corresponding classes on $X_{t_2}$. Well, fortunately, this is not a problem, as $X_{t_1} \cong X_{t_2}$ as smooth manifolds, and thus $H^*(X_{t_1}) \cong H^*(X_{t_2})$.
This identification is nothing but the Gauss-Manin connection. In particular, when the bases $T$ is more complicated, you need to choose a path from $t_1$ to $t_2$ to obtain the identification; and it will depend on the homotopy class in the path. In particular, when $T$ is not simply-connected, we get a representation of $\pi_1(T)$ on $H^*(X_0)$, i.e. the monodromy action. Since it is pieced together out of identifications $H^*(X_0) \cong H^*(X_{t_1}) \cong H^*(X_{t_2}) \cong \dots$ as above, GW-invariants are invariant under this group action:
$
\langle \gamma_1, \dots, \gamma_n \rangle_{\beta}^{g,n} =
\langle \Phi(\gamma_1), \dots, \Phi(\gamma_n) \rangle_{\Phi(\beta)}^{g,n} 
$
for any $\Phi$ in the image of $\pi_1(T) \to \mathrm{Aut} H^*(X, \mathbb{Q})$.
(The mapping class group is basically the biggest possible group $\pi_1(T)$ you can obtain this way, i.e. the fundamental group of the moduli space of varieties diffeomorphic to $X$.)