Abstract
A notable class of superconformal theories (SCFTs) in six dimensions is parameterized by an integer N , an ADE group G, and two nilpotent elements μL,R in G. Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs. In this paper we test this conjecture for G = SU(k), where AdS7 duals exist in IIA. We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two SU(k) non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS7 vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS7 solutions. The BPS equations for domain walls connecting two such vacua can be solved analytically, up to a Nahm equation with certain boundary conditions. The latter admit a solution connecting two vacua if and only if the corresponding nilpotent elements are related by the natural partial ordering, in agreement with the field theory conjecture.
Highlights
IIA supergravity, we will work with an effective seven-dimensional description that contains all the expected vacua with a given k
Nilpotent elements have a natural partial ordering, which has been conjectured to coincide with the hierarchy of renormalization-group flows among the SCFTs
We work with a seven-dimensional gauged supergravity, consisting of the gravity multiplet and two SU(k) non-Abelian vector multiplets. We show that this theory has many supersymmetric AdS7 vacua, determined by two nilpotent elements, which are naturally interpreted as IIA AdS7 solutions
Summary
We will begin with a quick review of the six-dimensional SCFTs that we are going to investigate holographically. We see that on the left we have the vertical Young diagram , corresponding to the partition [16] and to μ = 0, which belongs to the smallest possible orbit; this is depicted in the sketch of figure 1 as the tip of the cone. There may be D6-branes at the endpoints of I The correspondence between these AdS7 solutions and the SCFTs is easy to write down: the μta, which we defined earlier as the number of boxes in each column of the Young diagram associated to μ, give the slope of the piecewise-linear function α (see [7] for more details, and especially figure 2 there).
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