a b s t r a c tWe consider algebras defined from quivers with relations that are kth order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GL n for all n, and fourdimensional Sklyanin algebras. More generally, we show that any N-Koszul, (twisted) Calabi-Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between N-Koszul twisted Calabi-Yau algebras A and algebras defined by a superpotential ω such that an associated complex is a bimodule resolution of A. Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of an extension of SL 2 (Z/4).

Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [13]. Using results of W. Crawley-Boevey and M. Holland [10], [8] and [9]we give a combinatorial description of all the relevant couples (α, λ) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness.

ABSTRACT. In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is A∞-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models. INTRODUCTIONOriginally homological mirror symmetry was developed by Kontsevich [31] as a framework to explain the similarities between the symplectic geometry and algebraic geometry of certain Calabi-Yau manifolds. More precisely its main conjecture states that for any compact Calabi-Yau manifold with a complex structure X, one can find a mirror CalabiYau manifold X ′ equipped with a symplectic structure such that the derived category of coherent sheaves over X is equivalent to the zeroth homology of the triangulated envelop of the split closure of the Fukaya category of X ′ . The latter is a category that represents the intersection theory of Lagrangian submanifolds of X ′ .Over the years it turned out that this conjecture is part of a set of equivalences which are much broader than the compact Calabi-Yau setting [27,21,1,6,7]. Removing the compactness or Calabi-Yau condition often makes the mirror a singular object, which physicists call a Landau-Ginzburg model [37,38]. A Landau-Ginzburg model (X, W ) is a pair of a smooth space X and a complex valued function W : X → C, which is called the potential. On the algebraic side we associate to it the dg-category of matrix factorizations MF(X, W ). Its objects are diagrams P 0 G G P 1 o o where P i are vector bundles and the composition of the maps results in multiplication with W . The morphisms are morphisms between these vector bundles equipped with a natural differential.On the other hand if X ′ is noncompact we need to tweak the notion of the Fukaya category, by imposing certain conditions on the behaviour of the Lagrangians near infinity and using a Hamiltonian flow to adjust the intersection theory. This gives us the notion of a wrapped Fukaya category [3].In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov proved an instance of mirror symmetry between such objects. On the symplectic side they considered a sphere with k punctures and on the algebraic side they considered a special Landau Ginzburg model on a certain toric quasiprojective noncompact Calabi Yau threefold and they proved an equivalence between the derived wrapped Fukaya category of the former and the derived category of matrix factorizations of the lat...

Abstract. Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution Y , this derived equivalence generalises the Fourier-Mukai transform relating the G-Hilbert scheme and the skew group algebra C[x, y, z] * G for a finite abelian subgroup of SL(3, C). We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of CautisLogvinenko [7] and Cautis-Craw-Logvinenko [6] to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko [25].