Multi-trace Deformations Research Articles

Universal IR holography, scalar fluctuations, and glueball spectra

We show that the d’Alembertian operator with a possible mass term in the anti–de Sitter (AdS) soliton and more general confining gravity dual backgrounds admits infinitely many different spectra. These can be interpreted as different theories in the infrared and correspond to multitrace deformations of either the Dirichlet or the Neumann theory. We prove that all these fluctuations are normalizable and provide examples of their spectra. Therefore, the AdS soliton can be interpreted as giving a holographic Renormalization Group flow between a universal UV theory at the AdS boundary and these infinitely many possibilities in the IR, obtained by deformations. The massive spectrum of the double trace deformation in AdS5 allows the matching of the large-N glueball masses of lattice QCD3; the ratio of the ground states of the 2++ and 0++ channels are in full agreement with the lattice prediction. When considering AdS7 and the four-dimensional pure glue theory, a remarkably general picture emerges, where we can write formulas for the fluctuations that are in agreement with ones from holographic high-energy scattering and from AdS/CFT with IR and UV cutoff. We point out that this log branch in the IR in D dimensions can be seen as the usual logarithmic branch of scalar fields saturating the Breitenlohner-Freedman bound in a conformally rescaled metric, with AdSD−1×S1 asymptotics. Published by the American Physical Society 2024

Effective actions and bubble nucleation from holography

We discuss the computation of the quantum effective action of strongly interacting field theories using holographic duality, and its use to determine quasi-equilibrium parameters of first order phase transitions relevant for gravitational wave production. A particularly simple holographic model is introduced, containing only the metric and a free massive scalar field. Despite the simplicity, the model contains a rich phase diagram, including first order phase transitions at non-zero temperature, due to various multi-trace deformations. We obtain the leading terms in the effective action from homogeneous black brane solutions in the gravity dual, and linearised perturbations around them. We then employ the effective action to construct bubble and domain wall solutions in the field theory side and study their properties. In particular, we show how the scaling of the effective action with the effective number of degrees of freedom of the quantum field theory determines the corresponding scaling of gravitational wave parameters.

Light scalar modes from holographic deformations

We consider different ways of modifying the mass spectrum of a strongly coupled gauge theory with confinement using the AdS/CFT correspondence. Single- and multitrace deformations are introduced, such that the resulting theory has a mode lighter than the confinement scale. The multitrace deformation is shown to be a possible way of achieving a light composite mode, unlike the single-trace deformation where the light mode is an admixture of elementary and composite states.

Log corrections to entropy of three dimensional black holes with soft hair

We calculate log corrections to the entropy of three-dimensional black holes with “soft hairy” boundary conditions. Their thermodynamics possesses some special features that preclude a naive direct evaluation of these corrections, so we follow two different approaches. The first one exploits that the BTZ black hole belongs to the spectrum of Brown-Henneaux as well as soft hairy boundary conditions, so that the respective log corrections are related through a suitable change of the thermodynamic ensemble. In the second approach the analogue of modular invariance is considered for dual theories with anisotropic scaling of Lifshitz type with dynamical exponent z at the boundary. On the gravity side such scalings arise for KdV-type boundary conditions, which provide a specific 1-parameter family of multi-trace deformations of the usual AdS3/CFT2 setup, with Brown-Henneaux corresponding to z = 1 and soft hairy boundary conditions to the limiting case z → 0+. Both approaches agree in the case of BTZ black holes for any non-negative z. Finally, for soft hairy boundary conditions we show that not only the leading term, but also the log corrections to the entropy of black flowers endowed with affine û (1) soft hair charges exclusively depend on the zero modes and hence coincide with the ones for BTZ black holes.

Black holes in ω-deformed gauged [formula omitted] supergravity

Motivated by the recently found 4-dimensional ω-deformed gauged supergravity, we investigate the black hole solutions within the single scalar field consistent truncations of this theory. We construct black hole solutions that have spherical, toroidal, and hyperbolic horizon topologies. The scalar field is regular everywhere outside the curvature singularity and the stress–energy tensor satisfies the null energy condition. When the parameter ω does not vanish, there is a degeneracy in the spectrum of black hole solutions for boundary conditions that preserve the asymptotic Anti-de Sitter symmetries. These boundary conditions correspond to multi-trace deformations in the dual field theory.

Holographic quantum criticality from multi-trace deformations

We explore the consequences of multi-trace deformations in applications of gauge-gravity duality to condensed matter physics. We find that they introduce a powerful new “knob” that can implement spontaneous symmetry breaking, and can be used to construct a new type of holographic superconductor. This knob can be tuned to drive the critical temperature to zero, leading to a new quantum critical point. We calculate nontrivial critical exponents, and show that fluctuations of the order parameter are ‘locally’ quantum critical in the disordered phase. Most notably the dynamical critical exponent is determined by the dimension of an operator at the critical point. We argue that the results are robust against quantum corrections and discuss various generalizations.

Multitrace deformations, Gamow states, and stability of AdS/CFT

We analyze the effect of multitrace deformations in conformal field theories at leading order in a large N approximation. These theories admit a description in terms of weakly coupled gravity duals. We show how the deformations can be mapped into boundary terms of the gravity theory and how to reproduce the RG equations found in field theory. In the case of doubletrace deformations, and for bulk scalars with masses in the range − d 2 /4 <m 2 < − d 2 /4 + 1, the deformed theory flows between two fixed points of the renormalization group, manifesting a resonant behavior at the scale characterizing the transition between the two CFT’s. On the gravity side the resonance is mapped into an IR non-normalizable mode (Gamow state) whose overlap with the UV region increases as the dual operator approaches the free field limit. We argue that this resonant behavior is a generic property of large N theories in the conformal window, and associate it to a remnant of the Nambu-Goldstone mode of dilatation invariance. We emphasize the role of nonminimal couplings to gravity and establish a stability theorem for scalar/gravity systems with AdS boundary conditions in the presence of arbitrary boundary potentials and nonminimal coupling.

D-brane potentials from multi-trace deformations in AdS/CFT

It is known that certain AdS boundary conditions allow smooth initial data to evolve into a big crunch. To study this type of cosmological singularity, one can use the dual quantum field theory, where the non-standard boundary conditions are reflected by the presence of a multi-trace potential unbounded below. For specific AdS_4 and AdS_5 models, we provide a D-brane (or M-brane) interpretation of the unbounded potential. Using probe brane computations, we show that the AdS boundary conditions of interest cause spherical branes to be pushed to the boundary of AdS in finite time, and that the corresponding potential agrees with the multi-trace deformation of the dual field theory. Systems with expanding spherical D3-branes are related to big crunch supergravity solutions by a phenomenon similar to geometric transition.

Marginal deformations and 3-algebra structures

We study marginal deformations of superconformal Chern–Simons matter theories that are based on 3-algebras. For this, we introduce the notion of an associated 3-product, which captures very general gauge invariant deformations of the superpotentials of the BLG and ABJM models. We also consider conformal multi-trace deformations preserving N = 2 supersymmetry. We then use N = 2 supergraph techniques to compute the two-loop beta functions of these deformations. Besides confirming conformal invariance of both the BLG and ABJM models, we also verify that the recently proposed β-deformations of the ABJM model are indeed marginal to the order we are considering.

Single Trace and Multitrace Deformations of Supersymmetric QCD

In this report we briefly discuss phenomenologically-viable metastable dynamical supersymmetry breaking in supersymmetric QCD. The construction relies on the work of Intriligator, Seiberg and Shih, complemented with single trace and multitrace deformations. This is a summary of work done in collaboration with R. Essig, K. Sinha, G. Torroba and M.J. Strassler [R. Essig, J. F. Fortin, K. Sinha, G. Torroba and M. J. Strassler, arXiv:0812.3213 [hep-th] ].

Metastable supersymmetry breaking and multitrace deformations of SQCD

Metastable vacua in supersymmetric QCD in the presence of single and multitrace deformations of the superpotential are explored, with the aim of obtaining an acceptable phenomenology. The metastable vacua appear at one loop, have a broken R-symmetry, and a magnetic gauge group that is completely Higgsed. With only a single trace deformation, the adjoint fermions from the meson superfield are approximately massless at one loop, even though they are massive at tree level and R-symmetry is broken. Consequently, if charged under the standard model, they are unacceptably light. A multitrace quadratic deformation generates fermion masses proportional to the deformation parameter. Phenomenologically viable models of direct gauge mediation can then be obtained, and some of their features are discussed.

Supersymmetric multi-trace boundary conditions in AdS

Boundary conditions for massive fermions are investigated in AdSd for d ⩾ 2. For fermion masses in the range 0 ⩽ |m| < 1/2ℓ with ℓ being the AdS length, the standard notion of normalizeability allows a choice of boundary conditions. As in the case of scalars at or slightly above the Breitenlohner–Freedman (BF) bound, such boundary conditions correspond to multi-trace deformations of any CFT dual. By constructing appropriate boundary superfields, for d = 3, 4, 5 we identify joint scalar/fermion boundary conditions which preserve either supersymmetry or superconformal symmetry on the boundary. In particular, we identify boundary conditions corresponding via AdS/CFT (at large N) to a 595-parameter family of double-trace marginal deformations of the low-energy theory of N M2-branes which preserve superconformal symmetry. We also establish that (at large N and large 't Hooft coupling λ) there are no marginal or relevant multi-trace deformations of 3+1 super Yang–Mills which preserve even supersymmetry.

Multitrace Deformations of Vector and Adjoint Theories and their Holographic Duals

We present general methods to study the effect of multitrace deformations in conformal theories admitting holographic duals in Anti de Sitter space. In particular, we analyse the case that these deformations introduce an instability both in the bulk AdS space and in the boundary CFT. We also argue that multitrace deformations of the O ( N ) linear sigma model in three dimensions correspond to nontrivial time-dependent backgrounds in certain theories of infinitely many interacting massless fields on AdS 4 , proposed years ago by Fradkin and Vasiliev. We point out that the phase diagram of a truly marginal large- N deformation has an infrared limit in which only an O ( N ) singlet field survives. We draw from this case lessons on the full string-theoretical interpretation of instabilities of the dual boundary theory and exhibit a toy model that resolves the instability of the O ( N ) model, generated by a marginal multitrace deformation. The resolution suggests that the instability may not survive in an appropriate UV completion of the CFT.

Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT

We present a general and systematic treatment of multi-trace deformations in the AdS/CFT correspondence in the large N limit, pointing out and clarifying subtleties relating to the formulation of the boundary value problem on a conformal boundary. We then apply this method to study multi-trace deformations in the presence of a scalar VEV, which requires the coupling to gravity to be taken into account. We show that supergravity solutions subject to `mixed' boundary conditions are in one-to-one correspondence with critical points of the holographic effective action of the dual theory in the presence of a multi-trace deformation, and we find a number of new exact analytic solutions involving a minimally or conformally coupled scalar field satisfying `mixed' boundary conditions. These include the generalization to any dimension of the instanton solution recently found in hep-th/0611315. Finally, we provide a systematic method for computing the holographic effective action in the presence of a multi-trace deformation in a derivative expansion away from the conformal vacuum using Hamilton-Jacobi theory. Requiring that this effective action exists and is bounded from below reproduces recent results on the stability of the AdS vacuum in the presence of `mixed' boundary conditions.

Non-supersymmetric membrane flows from fake supergravity and multi-trace deformations

We use fake supergravity as a solution generating technique to obtain a continuum of non-supersymmetric asymptotically $AdS_4\times S^7$ domain wall solutions of eleven-dimensional supergravity with non-trivial scalars in the $SL(8,\mathbb{R})/SO(8)$ coset. These solutions are continuously connected to the supersymmetric domain walls describing a uniform sector of the Coulomb branch of the $M2$-brane theory. We also provide a general argument that under certain conditions identifies the fake superpotential with the exact large-N quantum effective potential of the dual theory, describing a marginal multi-trace deformation. This identification strongly motivates further study of fake supergravity as a solution generating method and it allows us to interpret our non-supersymmetric solutions as a family of marginal triple-trace deformations of the Coulomb branch that completely break supersymmetry and to calculate the exact large-N anomalous dimensions of the operators involved. The holographic one- and two-point functions for these solutions are also computed.

Boundary conditions and dualities: vector fields in AdS/CFT

In AdS, scalar fields with masses slightly above the Breitenlohner-Freedman bound admit a variety of possible boundary conditions which are reflected in the Lagrangian of the dual field theory. Generic small changes in the AdS boundary conditions correspond to deformations of the dual field theory by multi-trace operators. Here we extend this discussion to the case of vector gauge fields in the bulk spacetime using the results of Ishibashi and Wald [hep-th/0402184]. As in the context of scalar fields, general boundary conditions for vector fields involve multi-trace deformations which lead to renormalization-group flows. Such flows originate in ultra-violet CFTs which give new gauge/gravity dualities. At least for AdS4/CFT3, the dual of the bulk photon appears to be a propagating gauge field instead of the usual R-charge current. Applying similar reasoning to tensor fields suggests the existence of a duality between string theory on AdS4 and a quantum gravity theory in three dimensions.

Multitrace deformations of vector and adjoint theories and their holographic duals

We present general methods to study the effect of multitrace deformations in conformal theories admitting holographic duals in Anti de Sitter space. In particular, we analyse the case that these deformations introduce an instability both in the bulk AdS space and in the boundary CFT. We also argue that multitrace deformations of the O(N) linear sigma model in three dimensions correspond to nontrivial time-dependent backgrounds in certain theories of infinitely many interacting massless fields on AdS_4, proposed years ago by Fradkin and Vasiliev. We point out that the phase diagram of a truly marginal large-N deformation has an infrared limit in which only an O(N) singlet field survives. We draw from this case lessons on the full string-theoretical interpretation of instabilities of the dual boundary theory and exhibit a toy model that resolves the instability of the O(N) model, generated by a marginal multitrace deformation. The resolution suggests that the instability may not survive in an appropriate UV completion of the CFT.

Non-local effects of multi-trace deformations in the AdS/CFT correspondence

The AdS/CFT correspondence relates deformations of the CFT by "multi-trace operators" to "non-local string theories". The deformed theories seem to have non-local interactions in the compact directions of space-time; in the gravity approximation the deformed theories involve modified boundary conditions on the fields which are explicitly non-local in the compact directions. In this note we exhibit a particular non-local property of the resulting space-time theory. We show that in the usual backgrounds appearing in the AdS/CFT correspondence, the commutator of two bulk scalar fields at points with a large enough distance between them in the compact directions and a small enough time-like distance between them in AdS vanishes, but this is not always true in the deformed theories. We discuss how this is consistent with causality.

Multitrace AdS/CFT and master field dynamics

We consider gauge theories with multitrace deformations in the context of certain AdS/CFT models with explicit breaking of conformal symmetry and supersymmetry. In particular, we study the standard four-dimensional confining model based on the D4-brane metric at finite temperature. We work in the self-consistent Hartree approximation, which becomes exact in the large-N limit and is equivalent to the AdS/CFT multitrace prescription that has been proposed in the literature. We show that generic multitrace perturbations have important effects on the phase structure of these models. Most notably they can induce new types of large-N first-order phase transitions.

A Note on Multi-trace Deformations and AdS/CFT

We derive the general formula, at a finite cutoff, for the change in the boundary condition of a scalar field in AdS under a Multiple-trace deformation of the dual CFT. Our analysis suggests that fluctuations around the classical solution in AdS should not be constrained by boundary conditions.