Wall-crossing from Lagrangian cobordisms

Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known nontrivial 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 a bounding cochain, and additionally induce an A_∞ homomorphism between the Floer cohomology of the ends. This gives a first example of a two-ended Lagrangian cobordism giving a nontrivial equivalence of Lagrangian Floer cohomology. A brief computation is also included which shows that the incorporation of a bounding cochain from this equivalence accounts for the “instanton-corrections” considered by Auroux (2007), Pascaleff and Tonkonog (2020) and Rizell, Ekholm and Tonkonog (2022) for the wall-crossing formula between Chekanov and product tori in ℂ² \ {z₁z₂₌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_∞ algebras and an extension of Charest and Woodward’s stabilizing divisor model of Lagrangian Floer cohomology to Lagrangian cobordisms.