The Yang-Mills heat flow with random distributional initial data

Cao, Sky; Chatterjee, Sourav

doi:10.48550/arxiv.2111.10652

Cited by 3 publications

(14 citation statements)

References 26 publications

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“…A major obstruction in Charalambous and Gross's program, as it turned out, was that it was not known whether the Yang-Mills heat flow exists and is well-behaved if the initial data is a distributional connection. In this paper (combined with the companion paper [8]), we show that this is indeed the case, if the initial distributional connection is the GFF, or "GFF-like" (in terms of short-distance behavior). 1.10.…”

Section: Introductionmentioning

confidence: 66%

“…We next cite the following direct consequence of [8,Proposition 3.30] (applied with d = 3 and ε = 1/2 (say)). Note that the assumptions that we made on {A n } n≥1 in Section 1.7 are exactly the Assumptions A-E which are assumed in [8,Proposition 3.30] (the fact that Assumption C is satisfied follows from [8, Lemma 3.7]). Proposition 4.2 (See Proposition 3.30 in [8]).…”

Section: Proofs Of the Second And Third Main Resultsmentioning

confidence: 99%

“…Note that the assumptions that we made on {A n } n≥1 in Section 1.7 are exactly the Assumptions A-E which are assumed in [8,Proposition 3.30] (the fact that Assumption C is satisfied follows from [8, Lemma 3.7]). Proposition 4.2 (See Proposition 3.30 in [8]). Let {A n } n≥1 be as in Theorem 1.10.…”

Section: Proofs Of the Second And Third Main Resultsmentioning

confidence: 99%

“…The proof of the next theorem is in Section 4. The main technical details in the proof are in the companion paper [8].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 (the bulk of this paper), we prove Theorem 1.4, and along the way develop the technical details needed in the proof. In Section 4, we prove Theorems 1.10 and 1.13; the main technical details behind the proofs of these theorems are in the companion paper [8].…”

Section: Introductionmentioning

confidence: 99%

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A state space for 3D Euclidean Yang-Mills theories

Cao¹,

Chatterjee²

2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 66%

Section: Proofs Of the Second And Third Main Resultsmentioning

confidence: 99%

Section: Proofs Of the Second And Third Main Resultsmentioning

confidence: 99%

“…The proof of the next theorem is in Section 4. The main technical details in the proof are in the companion paper [8].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A state space for 3D Euclidean Yang-Mills theories

Cao¹,

Chatterjee²

2021

Preprint

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

Chevyrev¹

2022

Preprint

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We consider a non-linear heat equation ∂ t u = ∆u + B(u, Du) + P (u) 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 u 0 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 in [CC21b, CCHS22]. 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∞,∞ is an endpoint since the equation is locally well-posed for B η ∞,∞ for every η > − 1 2 .

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Well-posedness of a gauge-covariant wave equation with space-time white noise forcing

Bringmann¹,

Rodnianski²

2023

Preprint

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We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove the failure of a probabilistic null-form estimate, which exposes a potential obstruction towards the probabilistic well-posedness of a stochastic Maxwell-Klein-Gordon equation.

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The Yang-Mills heat flow with random distributional initial data

Cited by 3 publications

References 26 publications

A state space for 3D Euclidean Yang-Mills theories

A state space for 3D Euclidean Yang-Mills theories

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

Well-posedness of a gauge-covariant wave equation with space-time white noise forcing

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