相关论文: Explicit Generation of Integer Solutions via CY Mf…

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric…

A long-term project is to construct a complete Calabi-Yau metric on the complement of the anticanonical divisor in a compact K\"ahler manifold $\oM$. We focus on the case where this smooth divisor has multiplicity 2 and is itself a compact…

We define an integer-valued virtual count of embedded pseudo-holomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi--Yau $3$-folds, which can be viewed as an extension of Taubes' Gromov…

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly…

There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…

The $tt^*$ equations define a flat connection on the moduli spaces of $2d, \mathcal{N}=2$ quantum field theories. For conformal theories with $c=3d$, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat…

We develop some computational methods for the integrals over the 3-chains on the compact Calabi-Yau 3-folds that plays a prominent role in the analysis of the topological B-model in the context of the open mirror symmetry. We discuss such…

Based on Cynk-Hulek method we construct complex Calabi-Yau varieties of arbitrary dimensions using elliptic curves with automorphism of order 6. Also we give formulas for Hodge numbers of varieties obtained from that construction. We shall…

The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to…

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…

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form…

We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau…

A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated K\"ahler manifolds. The solutions to these systems arise from sets of geodesic second order non-linear differential…

This note is a survey of the enumerative geometry of rational curves on Calabi-Yau threefolds, based on a talk given by the author at the May 1991 Workshop on Mirror Symmetry at MSRI. An earlier version appeared in "Essays on Mirror…

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the…

We investigate D-branes on a noncompact singular Calabi-Yau manifold by using the boundary CFT description, and calculate the open string Witten indices between the boundary states. The B-type D-branes turn out to be characterized by the…

The B-model approach of topological string theory leads to difference equations by quantizing algebraic mirror curves. It is known that these quantum mechanical systems are solved by the refined topological strings. Recently, it was pointed…

Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

We present new invariant machine learning models that approximate the Ricci-flat metric on Calabi-Yau (CY) manifolds with discrete symmetries. We accomplish this by combining the $\phi$-model of the cymetric package with non-trainable,…