Double-trace Deformations Research Articles

Time singularities of correlators from Dirichlet conditions in AdS/CFT

Within AdS/CFT, we establish a general procedure for obtaining the leading singularity of two-point correlators involving operator insertions at different times. The procedure obtained is applied to operators dual to a scalar field which satisfies Dirichlet boundary conditions on an arbitrary time-like surface in the bulk. We determine how the Dirichlet boundary conditions influence the singularity structure of the field theory correlation functions. New singularities appear at boundary points connected by null geodesics bouncing between the Dirichlet surface and the boundary. We propose that their appearance can be interpreted as due to a non-local double trace deformation of the dual field theory, in which the two insertions of the operator are separated in time. The procedure developed in this paper provides a technical tool which may prove useful in view of describing holographic thermalization using gravitational collapse in AdS space.

Effective AdS/renormalized CFT

For an effective AdS theory, we present a simple prescription to compute the renormalization of its dual boundary field theory at \( \mathcal{O}\left( {{{1} \left/ {{{N^2}}} \right.}} \right) \). In particular, we define anomalous dimension holographically as the dependence of the wave-function renormalization factor on the radial cutoff in the Poincare patch of AdS. With this definition, the anomalous dimensions of both single- and double-trace operators are calculated. Three different dualities are considered with the field theory being CFT, CFT with a double-trace deformation and spontaneously broken CFT. For the second dual pair, we compute scaling corrections at the UV and IR fixed points of the RG flow triggered by the double-trace deformation. For the last case, we discuss whether our prescription is sensitive to the AdS interior or equivalently, the IR physics of the dual field theory.

Orbifold equivalence for finite density QCD and effective field theory

In the large N c limit, some apparently different gauge theories turn out to be equivalent due to large N c orbifold equivalence. We use effective field theory techniques to explore orbifold equivalence, focusing on the specific case of a recently discovered relation between an SO(2N c ) gauge theory and QCD. The equivalence to QCD has been argued to hold at finite baryon chemical potential, μ B , so long as one deforms the SO(2N c ) theory by certain “double-trace” terms. The deformed SO(2N c ) theory can be studied without a sign problem in the chiral limit, in contrast to SU(N c ) QCD at finite μ B . The purpose of the double-trace deformation in the SO(2N c ) theory is to prevent baryon number symmetry from breaking spontaneously at finite density, which is necessary for the equivalence to large N c QCD to be valid. The effective field theory analysis presented here clarifies the physical significance of double-trace deformations, and strongly supports the proposed equivalence between the deformed SO(2N c ) theory and large N c QCD at finite density.

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.

Adventures in holographic dimer models

We abstract the essential features of holographic dimer models, and develop several new applications of these models. Firstly, semi-holographically coupling free band fermions to holographic dimers, we uncover novel phase transitions between conventional Fermi liquids and non-Fermi liquids, accompanied by a change in the structure of the Fermi surface. Secondly, we make dimer vibrations propagate through the whole crystal by way of double trace deformations, obtaining nontrivial band structure. In a simple toy model, the topology of the band structure experiences an interesting reorganization as we vary the strength of the double trace deformations. Finally, we develop tools that would allow one to build, in a bottom-up fashion, a holographic avatar of the Hubbard model.

Double-trace deformations, holography and the c-conjecture

A double-trace deformation is the simplest perturbation of a conformal field theory that has a gravity dual. In this paper we review the existing results for the case in which the deformation is composed from a scalar operator, and extend them to the case of a spinor operator. In particular we check the validity of the c-conjecture along the RG flow induced by the deformation, using both Cardy's c-function and the recent proposal by Myers and Sinha of a c-function from entanglement entropy.

The conformal window of deformed conformal field theories in the planar limit

We discuss in the planar approximation the effect of double-trace deformations on CFT's. We show that this large class of models posses a conformal window describing a non-trivial flow between two fixed points of the renormalization group, and reveal the presence of a resonance which we associate to the remnant of a dilaton pole. As the conformal window shrinks to zero measure the theory undergoes a conformal phase transition separating a symmetric from a nonsymmetric phase. The recently conjectured strongly coupled branch of non-supersymmetric, non-abelian gauge theories with a large number of flavors is analyzed in light of these results, and a model for the strong branch is proposed. Some phenomenological implications in the context of unparticle physics are also emphasized.

The three dimensional dual of 4D chirality

Chiral gauge theories can be defined in four-dimensional Anti de Sitter space, but AdS boundary conditions explicitly break the chiral symmetry in a specific, well defined manner, which in turns results in an anomalous Ward identity. When the 4D theory admits a dual description in terms of a 3D CFT, the 3D dual of the broken chiral symmetry is a certain double-trace deformation of the CFT, which produces the same anomalous chiral Ward identities that obtains in the 4D bulk theory.

Yang-Mills theories with chiral matter at strong coupling

Strong coupling dynamics of Yang-Mills theories with chiral fermion content remained largely elusive despite much effort over the years. In this work, we propose a dynamical framework in which we can address non-perturbative properties of chiral, non-supersymmetric gauge theories, in particular, chiral quiver theories on S{sub 1} x R{sub 3}. Double-trace deformations are used to stabilize the center-symmetric vacuum. This allows one to smoothly connect smaller(S{sub 1}) to larger(S{sub 1}) physics (R{sub 4} is the limiting case) where the double-trace deformations are switched off. In particular, occurrence of the mass gap in the gauge sector and linear confinement due to bions are analytically demonstrated. We find the pattern of the chiral symmetry realization which depends on the structure of the ring operators, a novel class of topological excitations. The deformed chiral theory, unlike the undeformed one, satisfies volume independence down to arbitrarily small volumes (a working Eguchi-Kawai reduction) in the large N limit. This equivalence, may open new perspectives on strong coupling chiral gauge theories on R{sub 4}.

Weak-interacting holographic QCD

We propose a simple prescription for including low-energy weak-interactions into the framework of holographic QCD, based on the standard AdS/CFT dictionary of double-trace deformations. As our proposal enables us to calculate various electro-weak observables involving strongly coupled QCD, it opens a new perspective on phenomenological applications of holographic QCD. We illustrate efficiency and usefulness of our method by performing a few exemplar calculations; neutron beta decay, charged pion weak decay, and meson–nucleon parity non-conserving (PNC) couplings. The idea is general enough to be implemented in both Sakai–Sugimoto as well as Hard/Soft Wall models.

Center-stabilized Yang-Mills theory: Confinement and largeNvolume independence

We examine a double trace deformation of $SU(N)$ Yang-Mills theory which, for large $N$ and large volume, is equivalent to unmodified Yang-Mills theory up to $O(1/{N}^{2})$ corrections. In contrast to the unmodified theory, large $N$ volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large $N$ volume independence in small volumes. For small values of $N$, if the theory is formulated on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ with a sufficiently small compactification size $L$, then an analytic treatment of the nonperturbative dynamics of the deformed theory is possible. In this regime, we show that the deformed Yang-Mills theory has a mass gap and exhibits linear confinement. Increasing the circumference $L$ or number of colors $N$ decreases the separation of scales on which the analytic treatment relies. However, there are no order parameters which distinguish the small and large radius regimes. Consequently, for small $N$ the deformed theory provides a novel example of a locally four-dimensional pure-gauge theory in which one has analytic control over confinement, while for large $N$ it provides a simple fully reduced model for Yang-Mills theory. The construction is easily generalized to QCD and other QCD-like theories.

QCD-like theories onR3×S1: A smooth journey from small to larger(S1)with double-trace deformations

We consider QCD-like theories with one massless fermion in various representations of the gauge group $\mathrm{SU}(N)$. The theories are formulated on ${R}_{3}\ifmmode\times\else\texttimes\fi{}{S}_{1}$. In the decompactification limit of large $r({S}_{1})$ all these theories are characterized by confinement, mass gap, and spontaneous breaking of a (discrete) chiral symmetry ($\ensuremath{\chi}\mathrm{SB}$). At small $r({S}_{1})$, in order to stabilize the vacua of these theories at a center-symmetric point, we suggest to perform a double-trace deformation. With this deformation, the theories at hand are at weak coupling at small $r({S}_{1})$ and yet exhibit basic features of the large $r({S}_{1})$ limit: confinement and $\ensuremath{\chi}\mathrm{SB}$. We calculate the string tension, mass gap, bifermion condensates, and $\ensuremath{\theta}$ dependence. The double-trace deformation becomes dynamically irrelevant at large $r({S}_{1})$. Despite the fact that at small $r({S}_{1})$ confinement is Abelian, while it is expected to be non-Abelian at large $r({S}_{1})$, we argue that small and large $r({S}_{1})$ physics are continuously connected. If so, one can use small $r({S}_{1})$ laboratory to extract lessons about QCD and QCD-like theories on ${R}_{4}$.

Polyakov formulas for GJMS operators from AdS/CFT

We argue that the AdS/CFT calculational prescription for double-trace deformations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a quantum 1-loop correction to the SUGRA action and the boundary counterpart accounts for a sub-leading term in the large-N limit. The sequence of GJMS conformal Laplacians shows up in the two-point function of the CFT operator dual to a bulk scalar field at certain values of its scaling dimension. The restriction to conformally flat boundary metrics reduces the bulk computation to that of volume renormalization which renders the universal type A anomaly. In this way, we directly connect two chief roles of the Q-curvature: the main term in Polyakov formulas on one hand, and its relation to the Poincare metrics of the Fefferman-Graham construction, on the other hand. We find agreement with previously conjectured patterns including a generic and simple formula for the type A anomaly coefficient that matches all reported values in the literature concerning GJMS operators, to our knowledge.

Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT

According to the AdS/CFT dictionary, perturbing the large N boundary theory by a relevant double-trace deformation of the form f O^2 corresponds in the bulk to imposing ``mixed'' boundary conditions for the field dual to O. In this note we address various aspects of this correspondence. The change c_{UV} - c_{IR} of the central charge between the UV ad IR fixed points is known from explicit calculations (hep-th/0210093, hep-th/0212138) to be exactly the same in the bulk and in the boundary theories. By comparing the appropriate bulk and boundary functional determinants, we give a simple ``kinematic'' explanation for this universal agreement. We also clarify the prescription for computing AdS/CFT correlators with Delta_- boundary conditions.

Partition functions and double-trace deformations in AdS/CFT

We study the effect of a relevant double-trace deformation on the partition function (and conformal anomaly) of a CFT at large N and its dual picture in AdS. Three complementary previous results are brought into full agreement with each other: bulk and boundary computations, as well as their formal identity. We show the exact equality between the dimensionally regularized partition functions or, equivalently, fluctuational determinants involved. A series of results then follows: (i) equality between the renormalized partition functions for all d; (ii) for all even d, correction to the conformal anomaly; (iii) for even d, the mapping entails a mixing of UV and IR effects on the same side (bulk) of the duality, with no precedent in the leading order computations; and finally, (iv) a subtle relation between overall coefficients, volume renormalization and IR-UV connection. All in all, we get a clean test of the AdS/CFT correspondence beyond the classical SUGRA approximation in the bulk and at subleading O(1) order in the large-N expansion on the boundary.

On the AdS higher spin / O(N) vector model correspondence: degeneracy of the holographic image

We explore the conjectured duality between the critical O(N) vector model and minimal bosonic massless higher spin (HS) theory in AdS. In the boundary free theory, the conformal partial wave expansion (CPWE) of the four-point function of the scalar singlet bilinear is reorganized to make it explicitly crossing-symmetric and closed in the singlet sector, dual to the bulk HS gauge fields. We are able to analytically establish the factorized form of the fusion coefficients as well as the two-point function coefficient of the HS currents. We insist in directly computing the free correlators from bulk graphs with the unconventional branch. The three-point function of the scalar bilinear turns out to be an "extremal" one at d=3. The four-leg bulk exchange graph can be precisely related to the CPWs of the boundary dual scalar and its shadow. The flow in the IR by Legendre transforming at leading 1/N, following the pattern of double-trace deformations, and the assumption of degeneracy of the hologram lead to the CPWE of the scalar four-point function at IR. Here we confirm some previous results, obtained from more involved computations of skeleton graphs, as well as extend some of them from d=3 to generic dimension 2<d<4.

A note on causality in the bulk and stability on the boundary

By carefully analyzing the radial part of the wave-equation for a scalar field in AdSd+1, we show that for a particular range of boundary conditions on the scalar field, the radial spectrum contains a bound state. Using the AdS/CFT correspondence, we interpret this peculiar phenomenon as being dual to an unstable double trace deformation of the boundary conformal field theory. We thus show how the bulk theory holographically detects whether a boundary perturbation is stable.

3D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS

We study a duality, recently conjectured by Klebanov and Polyakov, between higher-spin theories on AdS4 and O(N) vector models in 3D. These theories are free in the UV and interacting in the IR. At the UV fixed point, the O(N) model has an infinite number of higher-spin conserved currents. In the IR, these currents are no longer conserved for spin s>2. In this Letter, we show that the dual interpretation of this fact is that all fields of spin s>2 in AdS4 become massive by a Higgs mechanism, that leaves the spin-2 field massless. We identify the Higgs field and show how it relates to the RG flow connecting the two CFTs, which is induced by a double trace deformation.

A universal result on central charges in the presence of double-trace deformations

We study large N conformal field theories perturbed by relevant double-trace deformations. Using the auxiliary field trick, or Hubbard–Stratonovich transformation, we show that in the infrared the theory flows to another CFT. The generating functionals of planar correlators in the ultraviolet and infrared CFTs are shown to be related by a Legendre transform. Our main result is a universal expression for the difference of the scale anomalies between the ultraviolet and infrared fixed points, which is of order 1 in the large N expansion. Our computations are entirely field theoretic, and the results are shown to agree with predictions from AdS/CFT. We also remark that a certain two-point function can be computed for all energy scales on both sides of the duality, with full agreement between the two and no scheme dependence.

`Double-trace' deformations, boundary conditions and spacetime singularities

Double-trace deformations of the AdS/CFT duality result in a new perturbation expansion for string theory, based on a non-local worldsheet. We discuss some aspects of the deformation in the low energy gravity approximation, where it appears as a change in the boundary condition of fields. We relate unique features of the boundary of AdS to the worldsheet becoming non-local, and conjecture that non-local worldsheet actions may be generic in other classes of backgrounds.