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Cluster Coordinates for the Classical XXZ Chain and Seiberg-Witten Theory

Student: Semenyakin Mykola

Supervisor: Andrey Marshakov

Faculty: Faculty of Mathematics

Educational Programme: Mathematics and Mathematical Physics (Master)

Final Grade: 10

Year of Graduation: 2018

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.

Full text (added June 1, 2018)

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