Curved Fukaya algebras and the Dubrovin spectrum

Abstract

Under simplified axioms on moduli spaces of pseudo-holomorphic curves, we show that weakly unobstructed Fukaya algebras of Floer-nontrivial Lagrangians in a compact symplectic manifold must have curvature in the spectrum of an operator introduced by Dubrovin, which acts on the big quantum cohomology. We use the example of the complex Grassmannian $\operatorname{Gr}(2,4)$ to illustrate a decoupling phenomenon, where the eigenvalues of finite energy truncations become simple under explicit bulk-deformations.

Figures (7)

• Figure 1: Spectrum of the $\alpha=2$ truncation of $K^B$ with $B=0$ (red) and $B=t[X_d]$ (blue) for different Schubert cycles $X_d\subset\operatorname{Gr}(2,4)$ and $t\in\mathbb{C}^\times$, after specializing to $q=1$. The zero eigenvalue has algebraic multiplicity two, and it decouples whenever $X_d\neq X_{\ydiagram{2,1}}$.
• Figure 2: Some correspondences induced by moduli of stable $J$-holomorphic spheres/disks. The interior/boundary marked points denoted $\blacksquare$ are constrained to map to $B/D$.
• Figure 3: Some correspondences induced by moduli of stable $J$-holomorphic spheres/disks with geodesic constraints. The interior/boundary marked points denoted $\blacksquare$ are constrained to map to $B/D$.
• Figure 4: Some correspondences induced by moduli of stable $J$-holomorphic disks with single geodesic constraints. The interior/boundary marked points denoted $\blacksquare$ are constrained to map to $B/D$.
• Figure 5: Degenerations of disks contributing to $Q\cap_{B,D}C$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.3
• Definition 2.1
• Definition 3.1
• Proposition 3.2
• Definition 3.3
• Proposition 3.4
• Definition 4.1
• Lemma 4.2
• Definition 4.3