Year of Graduation
Cluster Coordinates for the Classical XXZ Chain and Seiberg-Witten Theory
Mathematics and Mathematical Physics
In this thesis we continue recently initiated identification of objects, which appear in context of integrability in 5d N=1 theory and those, associated with the cluster integrable systems (CIS).We focus on the integrable systems corresponding to the 5d Seiberg-Witten theories with linear quivers – so-called classical XXZ spin chains. We find out that bipartite graph – main combinatorial data of CIS – defining CIS isomorphic to XXZ spin chain of rank M on N sites is N x M ‘fence net’ lattice. Properly processed Kasteleyn operator (whose determinant gives spectral curve of CIS) naturally give Lax operators of spin chain. Natural symmetry of bipartite graph under N-M permutation is proved to be a realization of so-called spectral duality, exchanging rank of spin chain and its length. Finally, we provide explicit construction for the embedding of CIS with arbitrary symmetric Newton polygon into co-extended general linear group of proper rank.