Studying the 3d Ising surface CFTs on the fuzzy sphere

Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3D Ising CFT. We propose two schemes by cutting a boundary in the orbital space or the real space to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through state-operator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary $o$, the displacement operator $\mathrm{D}$ and their conformal descendants and extract their scaling dimensions. We also study the one-point and two-point correlation functions and extract the bulk-to-surface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3D Ising surface CFTs non-perturbatively. Other conformal data obtained in this way also agrees with prior methods.