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The combinatorial mutation of polygons, which makes a given lattice polygon another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give the $mathbb{Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, we assign the lattice polygon called the perfect matching polygon. It is known that for each lattice polygon $P$ there exists a dimer model such that it gives $P$ as the perfect matching polygon and satisfies the consistency condition. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations that we call the deformations of consistent dimer models, and show that the deformations of consistent dimer models induce the combinatorial mutations of the associated perfect matching polygons.
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Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space algebraic curves are governed, in particular, by the dKP, WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems. Particular attention is paid to the study of two- and three-dimensional deformations of elliptic curves. Problem of an appropriate choice of Poisson structure is discussed.
Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fresan, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a description of the connected components of this graph, and use this to prove our result on boundedness of the variations.
The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian algebra $mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a $2$-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. In this paper, we show that internal perfect matchings, which correspond to toric exceptional divisors on a crepant resolution of a $3$-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Moreover, combining this result with the theorems due to Amiot-Iyama-Reiten, we show that the stable category of graded maximal Cohen-Macaulay modules admits a tilting object for any $3$-dimensional Gorenstein toric isolated singularity. We then show that all internal perfect matchings corresponding to the same toric exceptional divisor are transformed into each other using the mutations of perfect matchings, and this induces derived equivalences of $2$-representation infinite algebras.
Motivated by the BPS/CFT correspondence, we explore the similarities between the classical $beta$-deformed Hermitean matrix model and the $q$-deformed matrix models associated to 3d $mathcal{N}=2$ supersymmetric gauge theories on $D^2times_{q}S^1$ and $S_b^3$ by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the $W$-algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.
We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds $Y$ dual in F-theory to a gauge theory with gauge group $G$. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the $L$-genus, the $hat{A}$-genus, and the curvature invariant $X_8$ that appears in M-theory. We also show that certain characteristic classes are independent on the choice of the Kodaria fiber characterizing the group $G$. That is the case of $int_Y c_1^2 c_2$, the arithmetic genus, and the $hat{A}$-genus. Thus, it is enough to know $int_Y c_2^2$ and the Euler characteristic $chi(Y)$ to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of $G=$ SU($n$) for ($n=2,3,4,5,6,7$), USp($4$), Spin($7$), Spin($8$), Spin($10$), G$_2$, F$_4$, E$_6$, E$_7$, or E$_8$.
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