A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate 2 categories: A wrapped Fukaya category F(Q) and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q). We show that there is a duality on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual, F(D(Q)). We also discuss the connection with classical mirror symmetry.
We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed by mutation from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of (derived) monodromy groups of associated families of Calabi-Yau varieties. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new and more informative proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. This is joint work in progress with Hugh Thomas.
In recent joint work with Benjamin Nill, we introduced a natural generalisation of the notion of a reflexive polytope. These "l-reflexive polytopes" also appear as dual pairs, and in two-dimensions they arise from reflexive polygons via a change of the underlying lattice. Furthermore, any reflexive polygon of arbitrary index satisfies the famous “number 12” property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances.
In the spirit of Arcara & Bertram, we investigate wall-crossing phenomena in the stability manifold of an irreducible principally polarized abelian surface for objects with the same invariants as (twists of) ideal sheaves of points. In particular, we construct a sequence of fine moduli spaces which are related by Mukai flops and observe that the stability of these objects is completely determined by the configuration of points. Finally, we use Fourier-Mukai theory to show that these moduli are projective.
Let G be a finite subgroup of SL(3,C) and N be a normal subgroup of G. Then G-Hilb and G/N-Hilb(N-Hilb) are both crepant resolutions of C^3/G. In the talk I will explain how can we construct them as moduli spaces of representations of the McKay quiver, also known as moduli of G-constellations, and how can we calculate them explicitly in several cases. I will also address the problem of whether they are isomorphic (or not) as moduli spaces and as algebraic varieties. This is a joint work with A. Ishii and Y. Ito.
We study the stability spaces Stab(Q) and Stab(Γ_N Q), in the sense of Bridgeland, for the bounded derived category of a Dynkin quiver Q and the finite-dimensional derived category of the Calabi-Yau-N Ginzburg algebra Γ_N Q associated to Q. We will review the result of King-Qiu about the exchange graphs of hearts in such derived categories and cluster exchange graphs. Then we prove the simply connectedness of such spaces (which provides a topological realization of almost complete cluster tilting objects). A point of view is that, the quotient space of Stab(Γ_N Q) by the Seidel-Thomas braid group should be the stability space for the higher cluster category of Q.
This talk is on joint work with Robert Marsh. We describe a Landau-Ginzburg model for an arbitrary Grassmannian X in terms of Pluecker coordinates on a dual Grassmannian. This LG-model gives rise to a vector bundle with Gauss-Manin connection, which we relate explicitly with a bundle on the A-model side of X, the trivial bundle with fibre H*(X) and Dubrovin-Givental connection. The latter connection is defined in terms of the quantum cohomology of X. Our work makes use of the cluster structure of the homogeneous coordinate ring of the dual Grassmannian and involves some beautiful Postnikov diagrams.
I shall describe the polarised moduli of one of the four known types of compact irreducible symplectic variety, the 10-dimensional examples of O'Grady. The moduli space has dimension 21 and its birational geometry may be studied using modular forms, along the lines previously used for moduli of K3 surfaces and of deformations of their Hilbert schemes. This is joint work with V. Gritsenko and K. Hulek.