Related papers: Birational Transformations on Dimer Integrable Sys…

Brane tilings are bipartite periodic graphs on the 2-torus and realize a large family of 4d N=1 supersymmetric gauge theories corresponding to toric Calabi-Yau 3-folds. We present a complete classification of dimer integrable systems…

Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system…

We consider examples of extremal transitions between families of Calabi-Yau complete intersection threefolds in toric varieties, which are induced by toric embeddings of one toric variety into the other. We show that the toric map induced…

We show that a family of birational transformations that relate toric Fano 3-folds defined by reflexive lattice polytopes can be identified with mass deformations of corresponding 2d (0,2) supersymmetric quiver gauge theories. These…

The interplay between toric Calabi-Yau 3-folds, dimer integrable systems, and 5-dimensional quantum field theories has proved fruitful. We extend this framework to generalized toric polygons (GTPs) and show that their integrable systems…

We introduce the notion of birational complexity of a log Calabi-Yau pair. This invariant measures how far the log Calabi-Yau pair is to being birational to a toric pair. We study fundamental properties of the new invariant, with a…

This paper continues the study of two examples of extremal transitions between families of Calabi-Yau threefolds. In a previous paper we suggested that the "mirror transition" between mirror families predicted by Morrison could be achieved…

R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the…

P. Berglund, T. H\"ubsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries…

We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the…

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli…

In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and…

Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anti-canonical hypersurfaces in G/B. Recently Alexeev and…

Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair $a,b$ of positive integers, and geometrically related to $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifolds of local…

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to…

We propose exact quantization conditions for the quantum integrable systems of Goncharov and Kenyon, based on the enumerative geometry of the corresponding toric Calabi-Yau manifolds. Our conjecture builds upon recent results on the…

We investigate the equivalence theorem for integrable systems using two formulations of the Alday-Arutyunov-Frolov model. We show that the S-matrix is invariant under the field transformation which reduces the non-linear Dirac brackets of…

For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge…

We prove that the class of cluster integrable systems constructed by Goncharov and Kenyon out of the dimer model on a torus coincides with the one defined by Gekhtman, Shapiro, Tabachnikov, and Vainshtein using Postnikov's perfect networks.…

Localization methods have produced explicit expressions for the sphere partition functions of (2,2) superconformal field theories. The mirror symmetry conjecture predicts an IR duality between pairs of Abelian gauged linear sigma models, a…