Abstract We develop a framework for the reconstruction of the bulk theory dual to conformal field theory without any assumption by means of a flow equation. To this end we investigate a minimal extension of the free-flow equation and find that at a special parametrization the conformal transformation for a normalized smeared operator exactly becomes the isometry of anti-de Sitter space (AdS). By employing this special flow equation for O(N) vector models, we explicitly show that the AdS geometry as well as the scalar field satisfying the GKP–Witten relation concurrently emerge in this framework.

Japan

Sinya Aoki

Hungary

Janos Balog

Tetsuya Onogi

Shuichi Yokoyama

Special flow equation and the GKP–Witten relation Supersymmetric gauge theories and the AdS/CFT correspondence

Abstract We explore the construction of the dual bulk theory in the flow equation approach. We compute the vacuum expectation value of the Einstein operator at the next-to-leading order in the  expansion using a free O() vector model. We interpret the next-to-leading correction as the quantum correction to the cosmological constant of the anti-de Sitter space. We finally comment on how to generalize this computation to matrix elements of the Einstein operator for excited states.

Holographic computation of quantum corrections to the bulk cosmological constant

We study the flow equation of the O( ) model in dimensions at the next-to-leading order (NLO) in the expansion. Using the Schwinger–Dyson equation, we derive 2-pt and 4-pt functions of flowed fields. As the first application of the NLO calculations, we study the running coupling defined from the connected 4-pt function of flowed fields in -dimensional theory. We show in particular that this running coupling has not only an ultraviolet fixed point but also an infrared fixed point (Wilson–Fisher fixed point) in 3-dimensional massless scalar theory. As the second application, we calculate the NLO correction to the induced metric in dimensions with in the massless limit. While the induced metric describes a 4-dimensional Euclidean Anti-de-Sitter (AdS) space at the leading order, as shown in the previous paper, the NLO corrections make the space asymptotically AdS only in the UV and IR limits. Remarkably, while the AdS radius does not receive an NLO correction in the UV limit, the AdS radius decreases at the NLO in the IR limit, which corresponds to the Wilson–Fisher fixed point in the original scalar model in 3 dimensions.

Germany

Peter Weisz

Flow equation for the scalar model in the large expansion and its applications

We study the proposal that a ( d1 )-dimensional induced metric is constructed from a d -dimensional field theory using gradient flow. Applying the idea to the O( N ) 4 model and normalizing the flow field, we have shown in the large N limit that the induced metric is finite and universal in the sense that it does not depend on the details of the flow equation and the original field theory except for the renormalized mass, which is the only relevant quantity in this limit. We have found that the induced metric describes Euclidean anti-de Sitter (AdS) space in both ultraviolet (UV) and infrared (IR) limits of the flow direction, where the radius of the AdS is bigger in the IR than in the UV.

Flow equation for the large N scalar model and induced geometries

Erratum to: Walking in the 3-dimensional large N scalar model

The solvability of the three-dimensional O( N ) scalar field theory in the large N limit makes it an ideal toy model exhibiting “walking” behavior, expected in some SU( N ) gauge theories with a large number of fermion flavors. We study the model using lattice regularization and show that when the ratio of the particle mass to an effective 4-point coupling (with dimension mass) is small, the beta function associated to the running 4-point coupling is “walking”. We also study lattice artifacts and finite size effects, and find that while the former can be sizable at realistic correlation length, the latter are under control already at lattice sizes a few (∼3) correlation lengths. We show the robustness of the walking phenomenon by showing that it can also be observed by studying physical observables such as the scattering phase shifts and the mass gap in finite volume.