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1. Stochastic quantisation of Yang–Mills–Higgs in 3D

   https://doi.org/10.1007/s00222-024-01264-2  

   Chandra, Ajay; Chevyrev, Ilya; Hairer, Martin; Shen, Hao (August 2024, Inventiones mathematicae)

   Abstract We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space$$\mathcal{S}$$ $S$is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to$$\mathcal{S}$$ $S$and thus define a quotient space of “gauge orbits”$$\mathfrak {O}$$ $O$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on$$\mathfrak {O}$$ $O$(up to a potential finite time blow-up) associated to the stochastic YMH flow. 

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2. The Yang-Mills heat flow and the caloric gauge

   Sung-Jin Oh; Daniel Tataru (January 2022, Asterisque)

   This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs. Scattering Dichotomy for the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space-time. The primary subject of this paper, however, is another PDE, namely the energy critical Yang-Mills heat flow on the 4-dimensional Euclidean space. Our first goal is to establish sharp criteria for global existence and asymptotic convergence to a flat connection for this system in H1, including the Dichotomy Theorem (i.e., either the above properties hold or a harmonic Yang-Mills connection bubbles off) and the Threshold Theorem (i.e., if the initial energy is less than twice that of the ground state, then the above properties hold). Our second goal is to use the Yang-Mills heat flow in order to define the caloric gauge, which will play a major role in the analysis of the hyperbolic Yang-Mills equation in the subsequent papers. 

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3. A Puzzle About General Covariance and Gauge

   https://doi.org/10.1086/736478  

   March, Eleanor; Weatherall, James (May 2025, The British Journal for the Philosophy of Science)

   We consider two simple criteria for when a physical theory should be said to be ``generally covariant'', and we argue that these criteria are not met by Yang-Mills theory, even on geometric formulations of that theory. The reason, we show, is that the bundles encountered in Yang-Mills theory are not natural bundles; instead, they are gauge-natural. We then show how these observations relate to previous arguments about the significance of solder forms in assessing disanalogies between general relativity and Yang-Mills theory. We conclude by suggesting that general covariance is really about functoriality. 

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4. On Sugawara construction on celestial sphere

   https://doi.org/10.1007/JHEP09(2020)139  

   Fan, Wei; Fotopoulos, Angelos; Stieberger, Stephan; Taylor, Tomasz R. (September 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract Conformally soft gluons are conserved currents of the Celestial Conformal Field Theory (CCFT) and generate a Kac-Moody algebra. We study celestial amplitudes of Yang-Mills theory, which are Mellin transforms of gluon amplitudes and take the double soft limit of a pair of gluons. In this manner we construct the Sugawara energy-momentum tensor of the CCFT. We verify that conformally soft gauge bosons are Virasoro primaries of the CCFT under the Sugawara energy-momentum tensor. The Sugawara tensor though does not generate the correct conformal transformations for hard states. In Einstein-Yang- Mills theory, we consider an alternative construction of the energy-momentum tensor, similar to the double copy construction which relates gauge theory amplitudes with gravity ones. This energy momentum tensor has the correct properties to generate conformal transformations for both soft and hard states. We extend this construction to supertranslations. 

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5. Convergence of the self‐dual U (1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

   https://doi.org/10.1002/cpa.22150  

   Parise, Davide; Pigati, Alessandro; Stern, Daniel (January 2024, Communications on Pure and Applied Mathematics)

   Abstract Given a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionalsdefined for couples consisting of a section  and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of . 

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