2023/12/30 by Yunhyung Cho, Myungho Kim, Cho, Yunhyung +5
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2401.00252
openalex publication_date 2023/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the construction of Newton--Okounkov bodies and toric degenerations via cluster algebras in [GHKK18, FO25], 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 [HK15], 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.