Conformal embeddings and higher-spin bulk duals

Abstract

It is well-known that conformal embeddings can be used to construct non-diagonal modular invariants for affine lie algebras. This idea can be extended to construct infinite series of non-diagonal modular invariants for coset CFTs. In this paper, we systematically approach the problem of identifying higher-spin bulk duals for these kind of non-diagonal 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 a illustration of this, we show that a partition function of a orthogonal group coset CFT has a $\mathcal N=1$ 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, generalising the well-known dual interpretation of the 3-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.Comment: 40 pages, 1 figure, Version to appear in Physical Review

Higher spin currents in the orthogonal coset theory

“…According to the field contents in [40] where k 2 is fixed as k 2 = 1, the above OPE should not contain the N = 1 higher spin integer multiplets. See also [41].…”

Even spin N = 4 $$ \mathcal{N}=4 $$ holography

“…The chiral algebra of the coset in the 't Hooft limit, and correspondingly the gauge sector of the dual higher spin theory, is spanned by one superprimary field per even conformal weight, thus extending the previous bosonic even spin construction [19,20,38], the N = 2 case [39], and a recent N = 1 proposal [21]. In the k → ∞ limit, the coset (1.2) is described by an Sp(2N ) vector model.…”

“…Nevertheless, holography gives important results, starting with a series of higher spin/CFT dualities [8] relating Vasiliev higher spin theories on AdS 4 to O(N ) vector models in three dimensions. Further work on such dualities include [9][10][11][12][13], generalised to any number of dimensions in [14,15], as well as the subsequent cases of [16,17], the 3d/2d cases of [18][19][20][21][22][23][24], and the interesting dS 6 /CFT 5 case of [25].…”

The higher spin rectangle

“…This algebra, its supersymmetric extensions and holographic duals have been explored in [26][27][28][29]. In the other direction, W ∞ [λ], as also W N algebras, can be extended by adding a spin one field: the resulting algebra is known as W 1+∞ [λ].…”