Uncovering conformal symmetry in the 3D Ising transition I: State-operator correspondence

The 3D Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry. Conformal symmetry is believed to be a key ingredient for finding the long sought-after exact solution of the 3D Ising transition, but its emergence at the transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions. Here, we design an innovative way to study the quantum version of 3D Ising phase transition on the sphere, using the ``fuzzy (non-commutative) sphere" regularization. We accurately calculate and analyze the energy spectra at the transition, and demonstrate the state-operator correspondence (i.e. radial quantization), an important property of conformal field theory. Our result directly elucidates the emergent conformal symmetry of the 3D Ising transition, a conjecture made by Polyakov half a century ago.