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Mirror and Global Symmetry
Three-dimensional N = 4 theories are characterized by Mirror symmetry [78] [45]. This is a particular symmetry which exchanges the Higgs with the Coulomb branch of two theories, exchanges the Fayet-Iliopoulos and mass parameters and finally the SU(2)R and SU(2)R0 of the SO(4) R-symmetry. Regarding the above brief discussion regarding the moduli space of vacua, we see that the Coulomb branch of a theory, which is in pronciple difficult to study due to the fact that receives quantum corrections, can be studied from the classical Higgs branch of the mirror theory.
Regarding the linear quivers studied in this work, mirror symmetry exchanges the integer partitions and ˆ. Of course this implies that we have another IR flow condition which should be satisfied in order for the mirror symmetric quiver to have a non trivial fixed point in the IR: T ˆ SU(N) $ T ˆ SU(N) , T > ˆ $ ˆT > (1.3.2) Specifically, both theories of the mirror pair flow in the IR to the same fixed point. Apart from mirror symmetry, these theories are characterized by additional global symmetries, which are determined by and ˆ. From the previous discussion on the general linear quivers, these global symmetries correspond to the ones rotating the fundamental hypermultiplets of the theory: H = Q i U(Mi) and Hˆ = Q i U( ˆMi) for the mirror theory. Therefore at the fixed point, the total global symmetry is expected to be: H × Hˆ.
The holographic viewpoint
The holographic duality described in the first part of this work, is instrumental for the treatment of the problem of the AdS4 graviton Higgsing, the details of which have been introduced in the corresponding part of the general introduction. Although the main analysis is carried out in the gravitational side of the holographic correspondence, starting the study of the problem from the point of view of the dual sCFT3 is instructive regarding the construction of the massive AdS4 supergravity solutions.
The question of the presence of a bulk massive graviton is related to the energy-momentum conservation in the dual boudary CFT. Holographically the AdS4 graviton is dual to the stress tensor of the dual sCFT3, the scaling dimension of which is related to the graviton mass [6]: m2L24 = ( − 3) (3.1.1) , with L4 being the AdS4 radius and the scaling dimension of the stress-tensor. The representation of the three-dimensional conformal algebra so(2, 3) which contains T and its conformal descendants, is actually short, due to the conservation of the stress-tensor which results to three null states (null descendants). The scaling dimension of the conserved stress-tensor does not receive quantum corrections and hence it is canonical: = 3. From the above holographic expression, it becomes obvious that a conserved stress tensor of the boundary three-dimensional CFT corresponds to a massless AdS4 graviton. Therefore, a slightly massive graviton would correnspond to a non-conserved stress-tensor.
In this case, its scaling dimension would receive quantum corrections in the form of a small anomalous dimension [5], 1, which would then correspond to a small graviton mass: m2L24 O().
Higgsing in Representation theory
Before moving to geometry, let us discuss the Higgsing from the point of view of representation theory. The notation and the conventions, along with selected aspects of representation theory for three-dimensional N = 2 and N = 4 theories are given in the Appendix A Let D(, s) denote a unitary highest-weight representation of so(2, 3) with conformal primary of spin s and scaling dimension . Massive gravitons belong to long representations of the algebra.
The decomposition of a long spin-s representation at the unitarity threshold reads [101] D(s + 1 + ; s) !0 −−−−! D(s + 1; s) D(s + 2; s − 1) . (3.2.1).
Thus the AdS4 graviton (s = 2) acquires a mass by eating a massive Goldstone vector. In the 10d supergravity this vector must be the combination of off-diagonal components of the metric and tensor fields that is dual to the CFT operator Ta4.
Since we will here deal with N = 4 backgrounds, fields and dual operators fit in representations of the larger superconformal algebra osp(4|4). These have been all classified under mild assumptions [59][39]. In the notation of [39] (slightly retouched in [19]) the supersymmetric extension of the above decomposition reads L[0](0;0) 1+ !0 −−−−!.
Partitions for good quivers
The field theories of our holographic setup are three-dimensional N = 4 gauge theories that can be engineered with D3-branes suspended between D5-branes and NS5-branes [73]. Let A,N, ˆN be respectively the number of these three types of brane. To define the gauge theory one must give two ordered partitions of A in N or ˆN positive integers A = l1 + l2 + · · · + lN = ˆl1 +ˆl2 + · · ·ˆl ˆN , (4.2.1) where li li+1 andˆl ˆi ˆl ˆi +1. These describe the distribution of the D3-branes among NS5-branes on the left and D5-branes on the right. Equivalently, the partitions define two Young diagrams, and ˆ, both with the same number A of boxes. The diagram has li boxes in the ith row, and ˆ has ˆl ˆj boxes in theˆj th row. We label the rows of the transposed Young diagram T (i.e. the columns of ) by hatted Latin letters, and the rows of the transposed Young diagrams ˆT by unhatted letters. The reason for this notation will soon be clear. The length of theˆj th row in T is l T ˆj , and the length of the jth row in ˆT isˆl T j .
Quivers whose gauge symmetry can be entirely Higgsed correspond to pairs obeying the ordering condition T > ˆ. It was conjectured by Gaiotto and Witten [65] that at the origin of their Higgs branch such ‘good theories’ flow to strongly-coupled supersymmetric CFTs that are irreducible with no free-field factors. We can put the ordering condition in compact form by introducing the integrated row lengths.
Table of contents :
Introduction
I 3dN = 4 Superconformal Theories and type IIB Supergravity Duals
1 3d N = 4 Superconformal Theories
1.1 N = 4 supersymmetric gauge theories in three dimensions
1.2 Linear quivers and their Brane Realizations
1.3 Moduli Space and Symmetries
2 Holographic Duals: IIB Supergravity on AdS4 × S2 × ˆ S2 n (2)
2.1 The supergravity solutions
2.2 Holographic Dictionary
II String Theory embeddings of Massive AdS4 Gravity and Bimetric Models
3 Massive AdS4 gravity from String Theory
3.1 The holographic viewpoint
3.2 Higgsing in Representation theory
3.3 Massive spin-2 on AdS4 ×M6
3.4 Conclusions and perspectives
4 Stringy AdS4 Bigravity 45
4.1 Introduction
4.2 Partitions for good quivers
4.3 Quantum gates as box moves
4.4 Geometry of the gates
4.5 Mixing of the gravitons
4.6 Bimetric and Massive AdS4 gravity
4.7 Concluding Remarks
III T ˆ [SU(N)] Superconformal Manifolds
5 Exactly marginal Deformations
5.1 Introduction
5.2 Superconformal index of T ˆ [SU(N)]
5.3 Characters of OSp(4|4) and Hilbert series
5.4 Calculation of the index
5.5 Counting the N = 2 moduli
Appendix
A Elements of representation theory for 3d N = 2 and N = 4 theories
B The supersymmetric Janus solution
C Combinatorics of linear quivers
D Index and plethystic exponentials
E Superconformal index of T[SU(2)]
E.1 Analytical computation of the index
E.2 T[SU(2)] index as holomorphic blocks
Bibliography
