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CLUSTER ALGEBRAS VIA CLUSTER CATEGORIES WITH INFINITEDIMENSIONAL MORPHISM SPACES
"... Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an e ..."
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Cited by 49 (3 self)
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Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the Einvariant and show that an arbitrary decorated representation with vanishing Einvariant is characterized by its gvector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid
DILOGARITHM IDENTITIES FOR CONFORMAL FIELD THEORIES AND CLUSTER ALGEBRAS: Simply Laced Case
, 2010
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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.
2Frieze Patterns and the Cluster Structure of the Space of Polygons
"... We study the space of 2frieze patterns generalizing that of the classical CoxeterConway frieze patterns. The geometric realization of this space is the space of ngons (in the projective plane and in 3dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n ..."
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Cited by 13 (7 self)
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We study the space of 2frieze patterns generalizing that of the classical CoxeterConway frieze patterns. The geometric realization of this space is the space of ngons (in the projective plane and in 3dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.
Categorical tinkertoys for N = 2 gauge theories
"... In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abe ..."
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Cited by 6 (5 self)
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In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite–dimensional) representations of the Jacobian algebra CQ/(∂W) should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal ‘generic ’ subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. More precisely, there is a family of ‘light ’ subcategories Lλ ⊂ rep(Q,W), indexed by points λ ∈ N, where N is a projective variety whose irreducible components are copies of P1 in one–to–one correspondence with the simple factors of G. If λ is the generic point of the i–th irreducible component, Lλ is the universal subcategory corresponding to the i–th simple factor of G. Matter, on the contrary, is encoded in the subcategories Lλa where {λa} is a finite set of closed points in N. In particular, for a Gaiotto theory there is one such family of subcategories, Lλ∈N, for each maximal degeneration of the corresponding surface Σ, and the index variety N may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed–point subcategories to ‘fixtures ’ (spheres with three punctures of various kinds) and higher–order generalizations. The rules for ‘gluing ’ categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N = 2 theories which are not of the Gaiotto class. We include several examples and some amusing consequence, as the characterization in terms of quiver combinatorics of asymptotically free theories.