To the memory of José Guadalupe Ramírez-Rocha. Abstract. This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brüstle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation. Contents

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type Ã.

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ.

Abstract not found

Abstract not found

Abstract not found

Abstract not found

Abstract. We introduce a new class of finite dimensional gentle alge-bras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invari-ant of Avella-Alaminos and Geiss for surface algebras and we provide a

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.