Cluster algebras and monotone Lagrangian tori

Motivated by the construction of Newton–Okounkov bodies and toric degenerations via cluster algebras in [37], [27], we consider a family of Newton–Okounkov polytopes of a complex smooth Fano variety X related by a composition of tropicalized cluster mutations. According to the work of [44], the toric degeneration associated with each Newton–Okounkov polytope Δ in the family produces a completely integrable system of X over Δ. We investigate circumstances in which each completely integrable system possesses a monotone Lagrangian torus fiber. We provide a sufficient condition, based on the data of tropical integer points and exchange matrices, for the family of constructed monotone Lagrangian tori to contain infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphism. By employing this criterion and exploiting the correspondence between the tropical integer points and the dual canonical basis elements, we generate infinitely many distinct monotone Lagrangian tori on flag manifolds of arbitrary type except in a few cases.