We find a concise relation between the moduli , of a rational Narain lattice (,) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on certain complex Lie groups and their quotients. Both flat and non-flat cases are discussed in detail. This paper answers a question proposed by Andreas and Garcia-Fernandez in Comm Math Phys 332(3):13811383, 2014.

We consider the Einstein/Yang-Mills equations in 3+ 1 space time dimensions with SU (2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.

In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2) mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian manifolds.

A bstract 6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by 2 curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the A case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the D and E cases as

We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya C^{\infty} -manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with Gmez-Sharpe (arXiv: hep-th/0008150), Donagi-Katz-Sharpe (arXiv: hep-th/0309270), and Denef (arXiv: hep-th/0107152). Some string-theoretical remarks are given at the end of each section.

We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

This is the first of a series of papers in which we systematically use singularity theory to study four dimensional N= 2 superconformal field theories. Our main focus in this paper is to identify what kind of singularity is needed to define a SCFT. The constraint for a hypersurface singularity has been found by Sharpere and Vafa, and here the complete set of solutions are listed using a related mathematical result of Stephen ST Yau and Yu. We also study other type of singularities such as the complete intersection, quotient of hypersurface singularity by a finite group and non-isolated singularity. We finally conjecture that any three dimensional rational Gorenstein graded isolated singularity should define a N= 2 SCFT. We explain how to extract various interesting physical quantities such as Seiberg-Witten geometry, central charges, exact marginal deformations, BPS quiver, RG flow trajectory, etc from the properties of singularity.