Conformal embeddings and higher-spin bulk duals

It is well known that conformal embeddings can be used to construct nondiagonal modular invariants for affine Lie algebras. This idea can be extended to construct infinite series of nondiagonal modular invariants for coset conformal field theories (CFTs). In this paper, we systematically approach the problem of identifying higher-spin bulk duals for these kind of nondiagonal invariants. In particular, for a special value of the ’t Hooft coupling, there exist a class of partition functions that have enhanced supersymmetry, which should be reflected in a bulk dual. As an illustration of this, we show that a partition function of an orthogonal group coset CFT has an $\mathcal{N}=1$ supersymmetric higher-spin bulk dual, in the ’t Hooft limit. We also propose that two of the series of CFT partition functions, obtained from conformal embeddings, are equal, generalizing the well-known dual interpretation of the three-state Potts model as a $\frac{SU(2{)}_3\otimes SU(2{)}_1}{SU(2{)}_4}$ and also as a $\frac{SU(3{)}_1\otimes SU(3{)}_1}{SU(3{)}_2}$ coset model.

Figure 1

Young diagram with Frobenius coordinates given by the matrix in Eq. (a10). The shaded boxes represent the main diagonal.