Abstract
We develop a flow-based sampling algorithm for $SU(N)$ lattice gauge theories that is gauge-invariant by construction. Our key contribution is constructing a class of flows on an $SU(N)$ variable (or on a $U(N)$ variable by a simple alternative) that respect matrix conjugation symmetry. We apply this technique to sample distributions of single $SU(N)$ variables and to construct flow-based samplers for $SU(2)$ and $SU(3)$ lattice gauge theory in two dimensions.
Highlights
Gauge theories based on SUðNÞ or UðNÞ groups describe many aspects of nature
The Standard Model of nuclear and particle physics is a nonAbelian gauge theory with the symmetry group Uð1Þ× SUð2Þ × SUð3Þ, candidate theories for physics beyond the Standard Model can be defined based on strongly interacting SUðNÞ gauge theories [1,2], SUðNÞ gauge symmetries emerge in various condensed matter systems [3,4,5,6,7], and SUðNÞ and UðNÞ gauge symmetries feature in the low energy limit of certain string-theory vacua [8]
The updated links, must be transformed to guarantee equivariance. This can be achieved by making the context function [i.e., the analog of Wξ acting on frozen links in Eq (9)] invariant to symmetry transformations, and defining how the function is applied to the remaining links such that the operation commutes with symmetry transformations
Summary
Gauge theories based on SUðNÞ or UðNÞ groups describe many aspects of nature. We demonstrate how SUðNÞ gauge symmetries can be incorporated into flow-based models [15]. These models use a parametrized invertible transformation (a “flow”) to construct a variational ansatz for a target probability distribution that can be optimized via machine learning techniques to enable efficient sampling. The application of flow-based models to lattice field theory is reviewed briefly in Sec. II A.
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