Yang-Mills Heat Flow Research Articles

A State Space for 3D Euclidean Yang–Mills Theories

It is believed that Euclidean Yang–Mills theories behave like the massless Gaussian free field (GFF) at short distances. This makes it impossible to define the main observables for these theories—the Wilson loop observables—in dimensions greater than two, because line integrals of the GFF do not exist in such dimensions. Taking forward a proposal of Charalambous and Gross, this article shows that it is possible to define Euclidean Yang–Mills theories on the 3D unit torus as ‘random distributional gauge orbits’, provided that they indeed behave like the GFF in a certain sense. One of the main technical tools is the existence of the Yang–Mills heat flow on the 3D torus starting from GFF-like initial data, which is established in a companion paper. A key consequence of this construction is that under the GFF assumption, one can define a notion of ‘regularized Wilson loop observables’ for Euclidean Yang–Mills theories on the 3D unit torus.

Norm inflation for a non-linear heat equation with gaussian initial conditions

We consider a non-linear heat equation ∂tu=Δu+B(u,Du)+P(u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u = \\Delta u + B(u,Du)+P(u)$$\\end{document} posed on the d-dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document} is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space B∞,∞-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{-1/2}_{\\infty ,\\infty }$$\\end{document}; the space B∞,∞-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{-1/2}_{\\infty ,\\infty }$$\\end{document} is an endpoint since the equation is locally well-posed for B∞,∞η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{\\eta }_{\\infty ,\\infty }$$\\end{document} for every η>-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta >-\\frac{1}{2}$$\\end{document}.

The Yang-Mills heat flow with random distributional initial data

We construct local solutions to the Yang–Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato–Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way toward the construction of 3D Yang–Mills measures.

The Yang-Mills heat flow and the caloric gauge

The Yang-Mills heat flow and the caloric gauge

Stochastic quantization of Yang–Mills

We review two works [Chandra et al., Publ. Math. l’IHÉS (published online, 2022) and Chandra et al., arXiv:2201.03487 (2022)] that study the stochastic quantization equations of Yang–Mills on two- and three-dimensional Euclidean space with finite volume. The main result of these works is that one can renormalize the 2D and 3D stochastic Yang–Mills heat flow so that the dynamic becomes gauge covariant in law. Furthermore, there is a state space of distributional 1-forms S to which gauge equivalence approximately extends and such that the renormalized stochastic Yang–Mills heat flow projects to a Markov process on the quotient space of gauge orbits S/∼. In this Review, we give unified statements of the main results of these works, highlight differences in the methods, and point out a number of open problems.

Langevin dynamic for the 2D Yang–Mills measure

We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

Levy Laplacian on Manifold and Yang—Mills Heat Flow

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang-Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Laplacian on the infinite dimensional manifold.

Stable blowup for the supercritical Yang–Mills heat flow

In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations, 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.

Yang–Mills heat flow on gauged holomorphic maps

We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps $\mathcal{H}(P,X)$, where $P$ is a principal bundle on a Riemann surface $\Sigma$ and $X$ is a K\"ahler Hamiltonian $G$-manifold. For compact $\Sigma$, possibly with boundary, we prove long time existence of the gradient flow. The flow lines converge to critical points of the functional. So, there is a stratification on $\mathcal{H}(P,X)$ that is invariant under the action of the complexified gauge group. Symplectic vortices are the zeros of the functional we study. When $\Sigma$ has boundary, similar to Donaldson's result for the Hermitian Yang-Mills equations, we show that there is only a single stratum - any element of $\mathcal{H}(P,X)$ can be complex gauge transformed to a symplectic vortex. This is a version of Mundet's Hitchin-Kobayashi result on a surface with boundary.

Finite energy global well-posedness of the Yang–Mills equations on R 1 + 3 : An approach using the Yang–Mills heat flow

In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated \emph{Yang-Mills heat flow}. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings. Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by S. Klainerman and M. Machedon (1995) using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck's lemma (1982), and hence does \emph{not} involve localization in space-time.

GAUGE CHOICE FOR THE YANG–MILLS EQUATIONS USING THE YANG–MILLS HEAT FLOW AND LOCAL WELL-POSEDNESS IN H1

We introduce a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ℝ1+3, which uses the Yang–Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As a first application, we give an alternative proof of the local well-posedness of the Yang–Mills equations for initial data [Formula: see text], which is a classical result of Klainerman and Machedon (1995) that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space–time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang–Mills equations, also using the Yang–Mills heat flow, is established in a companion article.

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.

The Yang-Mills heat flow on the moduli space of framed bundles on a surface

We study the analog of the Yang-Mills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity.

Nonuniqueness for the Yang–Mills heat flow

We prove nonuniqueness for the Yang–Mills heat flow on bundles over manifolds of dimension m⩾5. For 5⩽ m⩽9 and any n∈ N there is an initial connection on the trivial bundle R m× SO(m) which, when evolved by the Yang–Mills heat flow, develops a point singularity in finite time, such that there are at least n different smooth continuations after the singular time. Moreover, the solution to the Yang–Mills heat flow may continue on a different bundle after the singular time, and for m∈{6,8} not even the topology of the bundle is determined uniquely.

Finite time blow-up for the Yang-Mills heat flow in higher dimensions

We consider the $L^2$ -gradient flow associated with the Yang-Mills functional, the so-called Yang-Mills heat flow. In the setting of a trivial principal SO(n)-bundle over ${\mathbb R}^n$ in dimension n greater than 4, we show blow-up in finite time for a class of SO(n)-equivariant initial connections.

Global existence of the equivariant Yang-Mills heat flow in four space dimensions

We show that the Yang-Mills heat flow of SO (4)-equivariant connections on an SU (2)-bundle over a ball in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] admits a smooth solution for all time for any initial and boundary data of finite energy.

The Yang-Mills flow in four dimensions

Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.