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We study the critical properties of the Ising-Higgs gauge theory in $(2+1)D$along the self-dual line which have recently been a subject of debate. For thefirst time, using machine learning techniques, we determine the low energyoperator content of the associated field theory. Our approach enables us tolargely refute the existence of an emergent current operator and with it thestanding conjecture that this transition is of the $XY^*$ universality class.We contrast these results with the ones obtained for the $(2+1)D$ Ashkin-Tellertransverse field Ising model where we find the expected current operator. Ournumerical technique extends the recently proposed Real-Space Mutual Informationallowing us to extract sub-leading non-linear operators. This allows acontrolled and computationally scalable approach to target CFT spectrum anddiscern universality classes beyond $(1+1)D$ from Monte Carlo data.

Machine Learning the Operator Content of the Critical Self-Dual Ising-Higgs Gauge Model