Triangle Seminars
Wednesday, 3 Sep 2025
Spontaneous quantization of the Yang-Mills gradient flow
📍 London
Alexander Migdal
(IAS Princeton)
Abstract:
We formulate a nonsingular loop-space calculus for the Yang-Mills (YM) gradient flow, in which all variations act within the manifold of smooth loops via ``dot derivatives'' that are finite, parametrization-invariant, and free of cusp or backtracking singularities. This yields an exact momentum-loop representation and a universal trilinear loop-space diffusion equation, valid for any non-Abelian gauge group. We identify two distinct classes of exact solutions. The first is a self-dual (Hodge-dual) matrix-valued minimal surface whose area functional, when exponentiated, solves the fixed-point loop equation exactly, without contact terms or ambiguities; for planar loops the dual area equals \(2\sqrt{2}\) times the Euclidean minimal area, providing a geometrically grounded confinement mechanism. We also prove that the ordinary minimal surface in ℝ4 fails to satisfy the fixed-point loop equation, due to a singular nonvanishing contribution from the loop operator. The second is a decaying-flow solution in which the momentum loop performs a periodic random walk on regular star polygons – the ``Euler ensemble'' previously found in Navier-Stokes turbulence – realizing a form of spontaneous quantization in the YM gradient flow. We discuss the emergence of quantum-like Wilson-loop statistics from deterministic classical dynamics, potential implications for confinement in QCD, and the role of these fixed points as attractors in the space of YM gradient-flow trajectories.
We formulate a nonsingular loop-space calculus for the Yang-Mills (YM) gradient flow, in which all variations act within the manifold of smooth loops via ``dot derivatives'' that are finite, parametrization-invariant, and free of cusp or backtracking singularities. This yields an exact momentum-loop representation and a universal trilinear loop-space diffusion equation, valid for any non-Abelian gauge group. We identify two distinct classes of exact solutions. The first is a self-dual (Hodge-dual) matrix-valued minimal surface whose area functional, when exponentiated, solves the fixed-point loop equation exactly, without contact terms or ambiguities; for planar loops the dual area equals \(2\sqrt{2}\) times the Euclidean minimal area, providing a geometrically grounded confinement mechanism. We also prove that the ordinary minimal surface in ℝ4 fails to satisfy the fixed-point loop equation, due to a singular nonvanishing contribution from the loop operator. The second is a decaying-flow solution in which the momentum loop performs a periodic random walk on regular star polygons – the ``Euler ensemble'' previously found in Navier-Stokes turbulence – realizing a form of spontaneous quantization in the YM gradient flow. We discuss the emergence of quantum-like Wilson-loop statistics from deterministic classical dynamics, potential implications for confinement in QCD, and the role of these fixed points as attractors in the space of YM gradient-flow trajectories.
Posted by: Evgeny Sobko