
## 1. Integrable systems

1.     Let $\theta(i)=\begin{cases}1& ,i\geq 0\\ -1 & ,i<0.\end{cases}$ Consider

\begin{align*} & \text{span}\left{ X_{n}=z^{n+1}\frac{\partial}{\partial z}\,\, \Big |\,\, n\in \mathbb{Z} \right}\\ & X_i\cdot X_j = \frac{1}{2}\left[ \theta (i)+\theta(j)+\theta(-i-j-2)+1\right] X_{i+j+1}\\ &\langle X_i,X_j\rangle=\delta_{i+j,-1}\\ &\text{deg}(X_i)=i+1. \end{align*}

#### Problem 1.1.

Is there a topological problem (e.g. SFT(?)) giving rise to this?
More about this example can be found in Infinite-Dimensional Frobenius Manifolds for 2 + 1 Integrable Systems by Carlet, Dubrovin, and Mertens, [MR2753798].
• #### Problem 1.2.

Let $V$ be an odd dimensional manifold equipped with a stable Hamiltonian structure. Find conditions on $V$ for the rational SFT homology algebra $(\mathcal{P},\{h,\cdot\})$ to have computable homology e.g. get a natural splitting. Study the meaning of the higher operations on $H_*(\mathcal{P})$.
• #### Problem 1.3.

Interpretation of higher powers of descendant classes in terms of tangency conditions to contact hyperplanes, generalizing work of Okounkov-Pandharipande for $V=S^1$.
• #### Problem 1.4.

Is there an explicit description of the $\psi$ classes only in terms of the geometry of the source curve, which is coherent?

Cite this as: AimPL: Integrable systems in Gromov-Witten and symplectic field theory, available at http://aimpl.org/gwsymplectic.