Related papers: Lagrangians, SO(3)-instantons and the Atiyah-Floer…

Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric…

This paper classifies separated bounding pairs for Lagrangian submanifolds that are homologically trivial inside the ambient space, under the assumption that restriction on cohomology from the ambient space to the Lagrangian is surjective.…

D-instantons are used to probe the near-horizon geometry of D3-branes systems on orbifold spaces. For fractional D3-branes, D-instanton calculus correctly reproduces the gauge beta-function and U(1)_R anomaly of the corresponding N=2…

We prove an explicit form of the Crepant Transformation Conjecture for Grassmannian flops. Our approach uses abelianization to first relate the restrictions of the Lagrangian cones to degree-2 classes, and then deduces the general result…

We introduce the concepts of branched coarse coverings and transfers between coarse homology theories along them. We show that various versions of coarse $K$-homology theories admit the additional structure of transfers. We show versions of…

In 2001 Sir M. F. Atiyah formulated a conjecture (C1) and later with P. Sutcliffe two stronger conjectures (C2) and (C3). These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for…

The axion Weak Gravity Conjecture implies that when parametrically increasing the axion decay constants, instanton corrections become increasingly important. We provide strong evidence for the validity of this conjecture by studying the…

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on…

N=1^* gauge theories are believed to have fractional instanton contributions in the confining vacua. D3 brane probe computations in gravitation dual of large-N N=2^* gauge theories point to the absence of such contributions in the low…

We examine 4-dimensional string backgrounds compactified over a two torus. There exist two alternative effective Lagrangians containing each two $SL(2)/U(1)$ sigma-models. Two of these sigma-models are the complex and the K\"ahler…

Using earlier work of S\'a Earp and the author [SEW13] we construct an irreducible unobstructed $G_2$-instanton on an $\mathrm{SO}(3)$-bundle over a twisted connected sum recently discovered by Crowley-Nordstr\"om [CN14].

Lattice artefacts are used, through modified lattice actions, as a tool to find the largest instantons in a toroidal geometry [0,L]^3X[0,T] for T to infinity. It is conjectured that the largest instanton is associated with tunnelling…

We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold…

The geometric Lagrangian theory (of arbitrary order) is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc.…

The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of…

The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different…

We provide a proof of a variant of the Landau-Siegel Zeros conjecture.

A plane curve C defined by a homogeneous polynomial satisfying Laplace's equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two…

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the…

In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to an $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by…