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The periodicity conjecture for pairs of Dynkin diagrams
, 2010
"... We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories. ..."
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We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.
Classical and Quantum Dilogarithm Identities
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2011
"... Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturall ..."
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Cited by 16 (4 self)
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Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.
Periodicities Of T and YSystems, DILOGARITHM IDENTITIES, AND CLUSTER ALGEBRAS I: Type Br
, 2010
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Dilogarithm Identities for SineGordon and Reduced SineGordon YSystems
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... We study the family of Ysystems and Tsystems associated with the sineGordon models and the reduced sineGordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and t ..."
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We study the family of Ysystems and Tsystems associated with the sineGordon models and the reduced sineGordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.
J H E P12(2011)027
, 2011
"... Abstract: When formulated in twistor space, the Dinstanton corrected hypermultiplet moduli space in N = 2 string vacua and the Coulomb branch of rigid N = 2 gauge theories on R3 × S1 are strikingly similar and, to a large extent, dictated by consistency with wallcrossing. We elucidate this similar ..."
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Abstract: When formulated in twistor space, the Dinstanton corrected hypermultiplet moduli space in N = 2 string vacua and the Coulomb branch of rigid N = 2 gauge theories on R3 × S1 are strikingly similar and, to a large extent, dictated by consistency with wallcrossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternionKähler manifolds with a quaternionic isometry and, on the other hand, hyperkähler manifolds with a rotational isometry, equipped with a canonical hyperholomorphic circle bundle and a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wallcrossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wallcrossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wallcrossing and cluster algebras.