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. Contents

Abstract. 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.

Abstract. 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 G T,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 G T,γ .

We present a unified mathematical framework that elegantly describes minimally SUSY gauge theories in even dimension, ranging from 6d to 0d, and their dualities. This approach combines recent developments on graded quiver with potentials, higher Ginzburg algebras and higher cluster categories (also known as m-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order (m+1) dualities of the gauge theories. Our work suggests that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities, whose physical interpretation is yet to be understood. arXiv:1711.01270v1 [hep-th] 3 Nov 2017 Contents 42 12 Conclusions and Outlook 45 A Allowable Potential Terms and Mutations 47 B Mutation of Differentials and Relation to Oppermann's Work 48 C Background on Cluster Categories 53 D Silting 57 1 IntroductionRecently, it was realized that minimally supersymmetric gauge theories in 6 − 2m dimensions exhibit order (m + 1) dualities, generalizing the well known case of Seiberg duality for 4d N = 1 theories [1]. 1 The first hint in this direction was the discovery that 2d N = (0, 2) gauge theories enjoy an order 3 duality named triality [2]. This was soon followed by the proposal of quadrality, an order 4 duality, for 0d N = 1 gauge theories [3]. There has also been significant progress in the brane engineering of 2d N = (0, 2) and 0d N = 1 theories. These constructions include D-brane probes of toric Calabi-Yau (CY) singularities [4], T-dual brane configurations generalizing brane tilings [5-8] and D-branes in the mirror geometries [3,9]. 2 These brane configurations have been useful for both understanding and postulating some of these dualities.In parallel, there have been interesting mathematical developments concerning graded quivers with potentials [14,15], higher Ginzburg algebras [15,16] and higher cluster categories [14]. While these topics are closely related to each other, its presentation in the literature has not been fully integrated. In this paper, we will show that they can be combined into a unified mathematical framework that elegantly describes minimally SUSY gauge theories in even dimension, ranging from 6d to 0d. Moreover, quiver mutations studied in the mathematical context precisely correspond to the order 1 By "order (m + 1) duality" we mean a generalization of duality relating (m + 1) different theories. Furthermore, in this case, (m + 1) consecutive applications of an elementary duality transformation amounts to the identity.2 See [10-13] for alternative constructions of 2d N = (0, 2) theories.-1 -(m + 1) dualities of the gauge theories. Higher Ginzburg algebras thus provide an algebraic unification of gauge theories in different dimensions and their dualities, which is similar to the geometric unification attained in [3,9,17] using mirror symmetry. Interestingly, this realization suggests that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities, whose physical int...

Abstract. This paper concerns cluster algebras with principal coefficients A•(S, M ) associated to bordered surfaces (S, M ), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -with or without self-intersections -we associate an element of (the fraction field of) A•(S, M ), using products of elements of P SL2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW,MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [FG1,FG2,FG3], who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to show that certain collections of arcs and loops comprise a vector-space basis for A•(S, M ).