Disk potential functions for quadrics

Abstract

We compute the disk potential of Gelfand-Zeitlin torus fiber in a quadric hypersurface. Contents 1. Introduction 1 2. Preliminaries on quadrics 3 3. Lagrangian Floer theory and monotone Fukaya category 7 4. Disk potential functions for quadrics 11 5. Effective disk classes 14 6. Computing disk potential functions 18 7. Final remarks 23 References 24

Chekanov torus and Gelfand--Zeitlin torus in $S^2 \times S^2$

“…The image of the system on the quadric hypersurface of complex dimension two agrees with the moment polytope of CP (1, 1, 2), which is one of the starting points of this work. Second, the disk potential of the GZ torus fiber T GZ computed in [Kim21] agrees with that of T CS in [Aur07, CS10] and that of T FOOO in [FOOO12] up to some coordinate changes. Theorem 4 claims a stronger relationship between T GZ and T CS .…”

“…Employing the technique of the gradient Hamiltonian disks in [CK19], the torus T GZ was shown to be monotone. Lemma 3 (Proposition 3.7 in [Kim21]). The above Gelfand-Zeitlin torus fiber T GZ is monotone.…”