arXiv research

Generative adversarial networks (GANs) evolved into one of the most successful unsupervised techniques for generating realistic images. Even though it has recently been shown that GAN training converges, GAN models often end up in local Nash equilibria that are associated with mode collapse or otherwise fail to model t…

Smooth Yang-Mills fields proved in supercritical dimensions.

Study on Yang-Mills fields blow-up in 4D, proving certain configurations impossible.

The paper recasts Penrose-Sparling's non-Hausdorff twistor space using noncommutative geometry.

Quantized Coulomb branches linked to skein algebras.

We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quanti…

Study of Hilbert schemes and Coulomb branches of hypertoric varieties.

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…

New $G_2$-holonomy manifolds from 5d N=1 theories domain walls.

Paper analyzes convergence rates of mean-field SVGD method.

We calculate the free energy of Coulomb gas systems on Riemann surfaces.

Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.

We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.

Floer homology connects to quiver Hecke algebras in Coulomb branches.

This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586

Proposes a Coulomb-like model for international trade flows, fitting real-world data.

We develop a new method for proving regularity for small energy stationary solutions of coupled gauge field equations. Our results duplicate those of Tian--Tao [7] for the pure Yang Mills equations, but our proof is simpler, and obtains bounded curvature without the use of Coulomb gauges. It relies instead on the Weitz…

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a math…

We study solutions of the Bogomolny equation on R^2\times S^1$ with prescribed singularities. We show that Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particu…

2d GLSM connects Berry connections to Coulomb branch via difference equations.

New method uses Coulomb gases for Monte Carlo integration with reduced errors.

The paper connects $G_2$-manifolds to Coulomb and Higgs phases of gauge theories.

This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannia…

Tripod spiders' energy control analyzed for Hooke and Coulomb potentials.

This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), wri…

This is the third companion paper of arXiv:1601.03586. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme o…

We introduce a novel class of localized atomic environment representations, based upon the Coulomb matrix. By combining these functions with the Gaussian approximation potential approach, we present LC-GAP, a new system for generating atomic potentials through machine learning (ML). Tests on the QM7, QM7b and GDB9 biom…

New gauge condition fixes $L^2$ metric divergence in hyperbolic monopole spaces.

In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimension…

Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.

New theorems in 2D and 4D for metrics with curvature or singularity.

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on $\mathbb R^4$, $SU(2)$-monopole moduli spa…

Extends Coulomb gauge existence to non-associative gauge theory.

The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.

Study on Wasserstein gradient flow for MMD between Coulomb measures.

This is a supplement to [arXiv:1503.03676], where an approach towards a mathematically rigorous definition of the Coulomb branch of a $3$-dimensional $\mathcal N=4$ SUSY gauge theory was proposed. We ask questions on their expected properties, especially in relation to the corresponding Higgs branch, partly motivated b…

Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold wit…

The paper studies how adding a 'Gauge Mass' term breaks gauge symmetry in Yang-Mills-Higgs systems and analyzes the resulting behavior.

Monoidal categorifies genus zero skein algebra using K-theory.

This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake ca…

Let $A$ be a $W^{1,2}$-connection on a principle $\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\|F_A\|_{L^2(M)}\le Λ$. Our main result is the existence of a global section $σ: M\to P$ with finite singularities on $M$ such that the connection form $σ^*A$ satisfies the Coulomb…

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequalit…

We study which geometric structure can be constructed from the vierbein (frame/coframe) variables and which field models can be related to this geometry. The coframe field models, alternative to GR, are known as viable models for gravity, since they have the Schwarzschild solution. Since the local Lorentz invariance is…

In this work, we consider solutions of the Maxwell equations on the Schwarzschild-de Sitter family of black hole spacetimes. We prove that, in the static region bounded by black hole and cosmological horizons, solutions of the Maxwell equations decay to stationary Coulomb solutions at a super-polynomial rate, with deca…

Explains a property of algebras related to quantum field theories.

Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.

We count the supersymmetric bound states of many distinct BPS monopoles in N=4 Yang-Mills theories and in pure N=2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The number of purely magnetic bound states is again found to be cons…

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…