Exactly marginal deformations from exceptional generalised geometry

Abstract

We apply exceptional generalised geometry to the study of exactly marginal deformations of N = 1 SCFTs that are dual to generic AdS 5 flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries.Our analysis holds for any N = 2 AdS 5 flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.

AdS5 vacua from type IIB supergravity on T 1,1

“…The moduli space of the full ten-dimensional solution AdS 5 × T 1,1 is known to be complex five-dimensional [4,5,[21][22][23]. However, only two of those moduli transform as singlets under the SU(2) × SU(2) factor in the isometry group and are thus accessible via consistent truncations [25]; those are the Axion-Dilaton τ and the complex modulus z related to the topology of T 1,1 , i.e.…”

“…full ten-dimensional theory is known to be complex five-dimensional [4,5,[21][22][23] and the moduli have been identified. There is always one modulus corresponding to the AxionDilaton τ and another coming from the vacuum expectation value (VEV) of the complex B-field of type IIB supergravity integrated over the nontrivial two-cycle of T 1,1 .…”

“…Moreover we define the space of Goldstone bosons, given by G = span R { k I }. 5 If the gauge group is spontaneously broken, the corresponding Goldstone bosons are always among the invariant deformations of (2.8), G ⊂ D. However, they should not be counted as physical moduli. Thus we define the moduli space of the AdS background to be the quotient M = D/G.…”

“…Moreover, reference [20] also computes moment maps and Killing vectors for the so-called NS truncation, which is done by removing the whole RR sector of type IIB supergravity before reducing to five dimensions. However, this means that the NS truncations does not contain the Axion-Dilaton and thus is not expected to have supersymmetric AdS vacua [5]. We checked this explicitly by inserting the data for the Killing vectors and moment maps from [20] into equation (2.8).…”

“…Supersymmetric solutions of type IIB supergravity with AdS factors preserving 8 real supercharges have been extensively studied in the past [1][2][3][4][5][6]. They are related to gauge theories on the boundary of AdS by the AdS/CFT correspondence.…”