Wall-crossing from Lagrangian Cobordisms

Abstract

Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an extension of their theory to the pearly model of Lagrangian Floer cohomology and unobstructed Lagrangian cobordisms. In particular, we examine the suspension cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between mutant Lagrangian surfaces. In both cases, we show that these Lagrangian cobordisms can be unobstructed by bounding cochain and additionally induce an $A_\infty$ homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a non-trivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of bounding cochain from this equivalence accounts for the "instanton-corrections" considered by Auroux, Ekholm, Pascaleff, Rizell, and Tonkonog for the wall-crossing formula between Chekanov and product tori in $(\mathbb C^2)\setminus \{z_1z_2=1\}$. We additionally prove some auxiliary results that may be of independent interest. These include a weakly filtered version of the Whitehead theorem for $A_\infty$ algebras and an extension of Charest-Woodward's stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.