Abstract
An infinite class of analytic AdS_7 x S^3 solutions has recently been found. The S^3 is distorted into a "crescent roll" shape by the presence of D8-branes. These solutions are conjectured to be dual to a class of "linear quivers", with a large number of gauge groups coupled to (bi-)fundamental matter and tensor fields. In this paper we perform a precise quantitative check of this correspondence, showing that the a Weyl anomalies computed in field theory and gravity agree. In the holographic limit, where the number of gauge groups is large, the field theory result is a quadratic form in the gauge group ranks involving the inverse of the A_N Cartan matrix C. The agreement can be understood as a continuum limit, using the fact that C is a lattice analogue of a second derivative. The discrete data of the field theory, summarized by two partitions, become in this limit the continuous functions in the geometry. Conversely, the geometry of the internal space gets discretized at the quantum level to the discrete data of the two partitions.
Highlights
Solutions were conjectured to be dual to the CFTs described above in [6].1 Later, their analytical expression was found [9]
The S3 is distorted into a “crescent roll” shape by the presence of D8-branes. These solutions are conjectured to be dual to a class of “linear quivers”, with a large number of gauge groups coupled tofundamental matter and tensor fields
The geometry of the internal space gets discretized at the quantum level to the discrete data of the two partitions
Summary
The theories were originally inferred to exist from brane configurations involving NS5branes, D6-branes and D8-branes. (From the four scalars in each hypermultiplet one can form hyper-momentum maps for the U(1)i centers of the U(ri) gauge groups, which are equated to πi by the equations of motion.) Given all these ingredients, at generic points on the tensor branch where Φi = Φi+1 one can write the equations of motion of these theories (or equivalently a “pseudo-action” on top of which one has to impose the self-duality constraints Hi = ∗Hi by hand). The k in (2.10) turns out to be the same as the k we defined in field theory, namely the maximum rank.7 This correspondence was originally motivated by the similarity of the data of the brane diagrams and of the AdS7 solutions (see figure 3). In the language of brane diagrams, the correspondence says that a D8-brane on which μ D6-branes end (in the configuration where all the D8’s are on the outside, as in figure 3(b)) becomes in the AdS7 solution a D8 with D6-charge μ
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