We apply the functional renormalization group equation to a massive Fierz-Pauli action in curved space and find that, even though a massive term is a modification in the infrared sector, the mass term modifies the value of the non-gaussian fixed point in the UV sector.We obtained the beta function for the scale dependent mass parameter and found that the massive Fierz-Pauli case still seems to be an asymptotically safe theory.
We construct a gravity dual for scale invariant but non-conformal field theories with a cyclic renormalization group flow. A slight modification of our construction gives a gravity dual of discretely scale invariant field theories. The underlying gravitational theory breaks the null energy condition.
This is a copy of the 2009 Princeton University thesis which examined various aspects of gauge/gravity duality, including renormalization group flows, phase transitions of the holographic entanglement entropy, and instabilities associated with the breaking of supersymmetry. Chapter 5 contains new unpublished material on various instabilities of the weakly curved non-supersymmetric $AdS_4$ backgrounds of M-theory.
A general model of dialton-Maxwell gravity in two dimensions is investigated. The corresponding one-loop effective action and the generalized $β$-functions are obtained. A set of models that are fixed points of the renormalization group equations are presented.
We discuss D-dimensional scalar and electromagnetic fields interacting with a quantized metric. The gravitons depend solely on twodimensional coordinates. We consider a conformal field theory model for the metric tensor. We show that an interaction with gravity improves the short distance behaviour. As a result there is no charge renormalization in the fourdimensional Higgs model.
We argue that the torus partition sum in $2d$ (super) gravity, which counts physical states in the theory, is a decreasing function of the renormalization group scale. As an application we chart the space of $(\hat c\leq1)$ $c\leq1$ models coupled to (super) gravity, confirming and extending ideas due to A. Zamolodchikov, and discuss briefly string theory, where our results imply that the number of degrees of freedom decreases with time.
We study the one loop renormalization in the most general metric-dilaton theory with second derivative only. In constant background dilaton theory, there are two types of gravity background which enable the theory renormalizable at one-loop level. We show this concretely to discuss on the spherical symmetric background.
Lorentzian simplicial quantum gravity is a non-perturbatively defined theory of quantum gravity which predicts a positive cosmological constant. Since the approach is based on a sum over space-time histories, it is perturbatively non-renormalizable even in three dimensions. By mapping the three-dimensional theory to a two-matrix model with ABAB interaction we show that both the cosmological and the (perturbatively) non-renormalizable gravitational coupling constant undergo additive renormalizations consistent with canonical quantization.
We look into the $AdS$ black holes from two-dimensional gravity perspective. In this work, we extend the previous results of holographic renormalization group flows to dimensions two. By introducing a superpotential, we derive the flow equations in two-dimensional dilaton gravity. We find a quantity which monotonically decreases along flows and give some comments on holographic $c$-theorem. As examples, we show that recently studied supersymmetric $AdS$ black hole solutions generically dimensionally reduce to two-dimensional dilaton gravity, and obtain the flow equations for black hole solutio…
In the Brans-Dicke model we treat the scalar field exactly and expand the gravitational field in a power series. A comparison with 2D sigma models and ϕ^{4} perturbation theory in four dimensions suggests that the perturbation series in 4D Brans-Dicke model is renormalizable.
The quantum properties of two-dimensional matter-dilaton gravity ---which includes a large family of actions for two-dimensional gravity (in particular, string-inspired models)--- are investigated. The one-loop divergences in linear covariant gauges are calculated and the structure of the one-loop renormalization is studied. The explicit forms of the dilaton potential, dilaton-Maxwell, and dilaton-scalar couplings for which the theory is one-loop multiplicatively renormalizable are found. A comparison with the one-loop renormalization structure of four-dimensional gravity-matter theory is give…
Quantum gravitational effects on the renormalization group equation are studied in the $(2+ε)$-dimensional approach. Divergences in a matter one-loop effective action do not receive gravitational radiative corrections. The renormalization factor for beta functions recently found by Klebanov, Kogan and Polyakov is obtained by using the renormalized cosmological constant to define the physical scale transformation.
We discuss exact renormalization group (RG) in $R^2$-gravity using effective average action formalism. The truncated evolution equation for such a theory on De Sitter background leads to the system of nonperturbative RG equations for cosmological and gravitational coupling constants. Approximate solution of these RG equations shows the appearence of antiscreening and screening behaviour of Newtonian coupling what depends on higher derivative coupling constants.
We study the effective theory of the conformal factor near its infrared stable fixed point.The renormalization group equations for the effective coupling constants are found and their solutions near the critical point are obtained, providing the logarithmic corrections to scaling relations.Some cosmological applications of the running of coupling constants are briefly discussed.
These lecture notes provide a pedagogical introduction to a specific continuum implementation of the Wilsonian renormalization group, the effective average action. Its general properties and, in particular, its functional renormalization group equation are explained in a simple scalar setting. The approach is then applied to Quantum Einstein Gravity (QEG). The possibility of constructing a fundamental theory of quantum gravity in the framework of Asymptotic Safety is discussed and the supporting evidence is summarized.
We consider renormalization groups of transformations composed of a Gaussian convolution and a field dilatation. As an example, we consider perturbations of a single component real Euclidean free field $ϕ$ with covariance $(-\bigtriangleup)^{-1+\fracε{2}}$. We show that the renormalization group admits two equivalent formulations called massless picture and massive picture respectively. We then show in the massive picture that the renormalization group has a symmetry. The symmetry consists of global scale transformations composed with certain Gaussian convolutions. We translate the symmetry ba…
We study the question of the gauge dependence of the quantum gravity contribution to the running gauge coupling constant for electromagnetism. The calculations are performed using dimensional regularization in a manifestly gauge invariant and gauge condition independent formulation of the effective action. It is shown that there is no quantum gravity contribution to the running charge, and hence there is no alteration to asymptotic freedom at high energies as predicted by Robinson and Wilczek.
We study two--loop renormalization in $(2+ε)$--dimensional quantum gravity. As a first step towards the full calculation, we concentrate on the divergences which are proportional to the number of matter fields. We calculate the $β$ functions and show how the nonlocal divergences as well as the infrared divergences cancel among the diagrams. Although the formalism includes a subtlety concerning the general covariance due to the dynamics of the conformal mode, we find that the renormalization group allows the existence of a fixed point which possesses the general covariance. Our results strongly…
We consider a bosonic $\s$--model coupled to two--dimensional gravity. In the semiclassical limit, $c\rightarrow -\infty$, we compute the gravity dressing of the $\b$--functions at two--loop order in the matter fields. We find that the corrections due to the presence of dynamical gravity are {\em not} expressible simply in terms of a multiplicative factor as previously obtained at the one--loop level. Our result indicates that the critical points of the theory are nontrivially influenced and modified by the induced gravity.
We discuss the ``gravitationally dressed'' beta functions in the Gross--Neveu model interacting with 2d Liouville theory and in $SU(N)$ gauge theory interacting with the conformal sector of 4d quantum gravity. Among the effects that we suggest may feel the gravitational dressing are the minimum of the effective potential and the running of the gauge coupling.
Using the method of renormalization group, we improve the two-loop effective potential of the massive $ϕ^4$ theory to obtain the next-next-to-leading logarithm correction in the $\bar{MS}$ scheme. Our result well reproduces the next-next-to-leading logarithm parts of the ordinary loop expansion result known up to the four-loop order.
It has been proposed in \cite{Park:2014tia} that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization. The analysis in \cite{Park:2014tia} was carried out at the classical/operator level. We review the proposal and make a transition to the path integral account. We then set the stage for explicitly carrying out the two-loop renormalization procedure of the resulting 3D action. We also address a potentially subtle issue in the gravity context concerning whether renormalizability d…
We compute the renormalization group running of the Newton constant and the parameter $λ$ in $(3+1)$-dimensional projectable Horava gravity. We use the background field method expanding around configurations with flat spatial metric, but non-vanishing shift. This allows us to reduce the number of interaction vertices and thereby drastically simplify the calculations. The gauge invariant $β$-function of $λ$ has two families of zeros, attractive in the infrared and ultraviolet respectively. They are candidates for the fixed points of the full renormalization group flow of the theory, once the $β…
We extend the relation between stochstic quantization and Holographic Wilsonian Renormalization Group to the massive fermion case in Euclidean AdS space. The original suggestion about the explicit relation between the two theories is in arXiv:1209.2242. To give a fermion bulk action the Dirichlet boundary condition and the Neumann boundary condition, there are some additional terms to the bulk action. We have confirmed that the relation also hold for massive fermion case.
We study holographic renormalization for three dimensional new massive gravity (NMG). By studying the general fall off conditions for the metric allowed by the model at infinity, we show that at the critical point where the central charges of the dual CFT are zero it contains a leading logarithmic behavior. In the context of AdS/CFT correspondence it can be identified as a source for an irrelevant operator in the dual CFT. The presence of the logarithmic fall off may be interpreted as the fact that the dual CFT would be a LCFT.
We explore the implications of recent work by Brézin and Zinn-Justin, applying the renormalization group techniques from critical phenomena to the scaling limit of matrix models in two-dimensional quantum gravity. They endeavor to get the lowest order fixed points of the theory giving insight upon the critical points of the theory. We show that at leading order the perturbative result is equal to the saddle-point approximation result. We calculate the next-to-leading order in the perturbative expansion exploring the goodness of the approach.
In three spacetime dimensions, where no graviton propagates, pure gravity is known to be finite. It is natural to inquire whether finiteness survives the coupling with matter. Standard arguments ensure that there exists a subtraction scheme where no Lorentz-Chern-Simons term is generated by radiative corrections, but are not sufficiently powerful to ensure finiteness. Therefore, it is necessary to perform an explicit (two-loop) computation in a specific model. I consider quantum gravity coupled with Chern-Simons U(1) gauge theory and massless fermions and show that renormalization originates f…
We summarize our renormalization group approach for the vector model as well as the matrix model which are the discretized quantum gravity in one- and two-dimensional spacetime. A difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space is formulated by means of the loop equation. The reparametrization identities reduce the flow in the infinite dimensional coupling constant space to that in finite dimensions. The matrix model gives a nonlinear differential equation as an effective renormalization group equation. The…
We study quantum gravity in $2+ε$ dimensions in such a way to preserve the volume preserving diffeomorphism invariance. In such a formulation, we prove the following trinity: the general covariance, the conformal invariance and the renormalization group flow to Einstein theory at long distance. We emphasize that the consistent and macroscopic universes like our own can only exist for matter central charge $0<c<25$. We show that the spacetime singularity at the big bang is resolved by the renormalization effect and universes are found to bounce back from the big crunch. Our formulation may be v…
We present a comprehensive analysis of the prescription we recently put forward for the computation of real-time correlation functions using gauge/gravity duality. The prescription is valid for any holographic supergravity background and it naturally maps initial and final data in the bulk to initial and final states or density matrices in the field theory. We show in detail how the technique of holographic renormalization can be applied in this setting and we provide numerous illustrative examples, including the computation of time-ordered, Wightman and retarded 2-point functions in Poincare …
We obtain holographically renormalized boundary stress tensors with the emphasis on a special point in the parameter space of three dimensional new massive gravity, using the so-called Fefferman-Graham coordinates with relevant counter terms. Through the linearized equations of motion with a standard prescription, we also obtain correlators among these stress tensors. We argue that the self-consistency of holographic renormalization determines counter terms up to unphysical ambiguities. Using these renormalized stress tensors in Fefferman-Graham coordinates, we obtain the central charges of du…
We perform a holographic renormalization of gravity duals to little string theories. In particular, we construct counterterms which yield a well-defined type II action for NS-sector linear dilaton backgrounds. Our methods are based on a similar recent construction in asymptotically flat spacetimes, and our work demonstrates in detail the parallels between asymptotically flat and linear dilaton boundary conditions. The counterterms guarantee that (i) the on-shell action is finite and (ii) asymptotically linear dilaton solutions are stationary points of the action under all boundary condition pr…
A theory of higher-derivative 2D dilaton gravity which has its roots in the massive higher-spin mode dynamics of string theory is suggested. The divergences of the effective action to one-loop are calculated, both in the covariant and in the conformal gauge. Some technical problems which appear in the calculations are discussed. An interpretation of the theory as a particular D=2 higher-derivative $σ$-model is given. For a specific case of higher-derivative 2D dilaton gravity, which is one loop multiplicatively renormalizable, static configurations corresponding to black holes are shown to exi…
Motivated by Weinberg's asymptotic safety scenario, we investigate the gravitational renormalization group flow in the Einstein-Hilbert truncation supplemented by the wave-function renormalization of the ghost fields. The latter induces non-trivial corrections to the beta-functions for Newton's constant and the cosmological constant. The resulting ghost-improved phase diagram is investigated in detail. In particular, we find a non-trivial ultraviolet fixed point in agreement with the asymptotic safety conjecture, which also survives in the presence of extra dimensions. In four dimensions the g…
We study the Casimir effect for a massive bosonic string terminating on D-brains, and living in a flat space with an antisymmetric background B-field. We find the Casimir energy and Casimir force as functions of the mass and length of the string and show the force does not depend on B-field. We also find the divergent part of the vacuum state energy, as a function of the mass and length of the string and offer an idea for renormalization of the mass of the string.
The asymptotic behavior of new massive gravity (NMG) is analyzed for all values of the mass parameter satisfying the Breitenlohner-Freedman bound. The traditional Fefferman-Graham expansion fails to capture the dynamics of NMG, and new terms in the asymptotic expansion are needed to include the massive graviton modes. New boundary conditions are discovered for a range of values $-1<2m^2l^2<1$ at which non-Einstein modes decay more slowly than the Brown-Henneaux boundary conditions. The holographically renormalized stress tensor is computed for these modes, and the relevant counterterms are ide…
In this paper we explicitly carry out the perturbative renormalization of the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions up to second order in the coupling constant. This is done by computing the two-to-two $S$-matrix using the LSZ reduction formula and canceling out the divergences by introducing counterterms. We demonstrate that the renormalized Lagrangian is unambiguously determined by demanding that it gives the correct $S$-matrix of a $T\bar{T}$-deformed integrable field theory. Remarkably, the renormalized Lagrangian is qualitatively very different from its classica…
As a check of the first massive integrated vertex operator in the pure spinor formalism constructed in arXiv:1802.04486, we compute the one loop 2-point function of the stable non-BPS massive states in SO(32) heterotic string theory. This allows us to compute the one loop renormalized mass of these states using the pure spinor formalism. Our results are in agreement with the corresponding results obtained by Sen using the RNS formalism. This provides an instance of the equivalence between the RNS and the pure spinor formalism for the massive states at loop level.
Using as an example the Einstein gravity with the cosmological constant, we discuss the calculation of renormalization group functions off shell. We found, that gauge dependent terms should be absorbed by the nonlinear renormalization of metric. Nevertheless, some terms can be included in the renormalization of Newton's constant. This ambiguity in the renormalization prescription is discussed.
Two dimensional dilaton gravity interacting with a four-fermion model and scalars is investigated, all the coefficients of the Lagrangian being arbitrary functions of the dilaton field. The one-loop covariant effective action for 2D dilaton gravity with Majorana spinors (including the four-fermion interaction) is obtained, and the technical problems which appear in an attempt at generalizing such calculations to the case of the most general four-fermion model described by Dirac fermions are discussed. A solution to these problems is found, based on its reduction to the Majorana spinor case. Th…
The coefficient of the dimensionally regularized two-loop R^3 divergence of (nonsupersymmetric) gravity theories has recently been shown to change when non-dynamical three forms are added to the theory, or when a pseudo-scalar is replaced by the anti-symmetric two-form field to which it is dual. This phenomenon involves evanescent operators, whose matrix elements vanish in four dimensions, including the Gauss-Bonnet operator which is also connected to the trace anomaly. On the other hand, these effects appear to have no physical consequences in renormalized scattering processes. In particular,…
The fourth derivative models for two dimensional gravity are shown to be equivalent to the special version of the nonlinear sigma models coupled to 2d quantum gravity. The reduction consists in the introduction of the auxiliary scalar fields and can be performed in an explicit way for both metric and general metric-dilaton cases. In view of this we can evaluate the structure of possible counterterms and show that they contains second derivative structures only. The calculations in the theory with an auxiliary fields require some special procedure to be applied. We perform the explicit calculat…
I prove that classical gravity coupled with quantized matter can be renormalized with a finite number of independent couplings, plus field redefinitions, without introducing higher-derivative kinetic terms in the gravitational sector, but adding vertices that couple the matter stress-tensor with the Ricci tensor. The theory is called "acausal gravity", because it predicts the violation of causality at high energies. Renormalizability is proved by means of a map M that relates acausal gravity with higher-derivative gravity. The causality violations are governed by two parameters, a and b, that …
We give an introductory account to the renormalization of models without metric background. We sketch the application to certain discrete models of quantum gravity such as spin foam models.
Black hole no-hair theorems are proven using inequalities that govern the radial dependence of spherically symmetric configurations of matter fields. In this paper, we analyze the analogous inequalities for geometries dual to renormalization group flows via the AdS/CFT correspondence. These inequalities give much useful information about the qualitative properties of such flows. For Poincare invariant flows, we show that generic flows of relevant or irrelevant operators lead to singular geometries. For the case of irrelevant operators, this leads to an apparent conflict with Polchinski's decou…
We use the functional renormalization group equation for quantum gravity to construct a non-perturbative flow equation for modified gravity theories of the form $S = \int d^dx \sqrt{g} f(R)$. Based on this equation we show that certain gravitational interactions monomials can be consistently decoupled from the renormalization group (RG) flow and reproduce recent results on the asymptotic safety conjecture. The non-perturbative RG flow of non-local extensions of the Einstein-Hilbert truncation including $\int d^dx \sqrt{g} \ln(R)$ and $\int d^dx \sqrt{g} R^{-n}$ interactions is investigated in …
Quantum gravity is analyzed from the viewpoint of the renormalization group. The analysis is based on methods introduced by J. Polchinski concerning the perturbative renormalization with flow equations. In the first part of this work, the program of renormalization with flow equations is reviewed and then extended to effective field theories that have a finite UV cutoff. This is done for a scalar field theory by imposing additional renormalization conditions for some of the nonrenormalizable couplings. It turns out that one so obtains a statement on the predictivity of the effective theory at …
In linearized quantum gravity, a shift of the average energy-momentum can be compensated by a shift of the average gravitational field. This allows a renormalization scheme that naturally removes the contribution of quantum vacuum fluctuations to the cosmological constant, solving the old cosmological-constant problem for weak gravitational fields.
We calculate the beta-functions of the general massive (p,q) supersymmetric sigma model to two loop order using (1,0) superfields. The conditions for finiteness are discussed in relation to (p,q) supersymmetry. We also calculate the effective potential using component fields to one loop order and consider the possibility of perturbative breaking of supersymmetry. The effect of one loop finite local counter terms and the ultra-violet behaviour of the off-shell (p,q) models to all orders in perturbation theory are also addressed.
Classical higher-derivative gravity is investigated in the context of the holographic renormalization group (RG). We parametrize the Euclidean time such that one step of time evolution in (d+1)-dimensional bulk gravity can be directly interpreted as that of block spin transformation of the d-dimensional boundary field theory. This parametrization simplifies the analysis of the holographic RG structure in gravity systems, and conformal fixed points are always described by AdS geometry. We find that higher-derivative gravity generically induces extra degrees of freedom which acquire huge mass ar…
A variational principle is constructed for gravity coupled to an asymptotically linear dilaton and a p-form field strength. This requires the introduction of appropriate surface terms -- also known as `boundary counterterms' -- in the action. The variation of the action with respect to the boundary metric yields a boundary stress tensor, which is used to construct conserved charges that generate the asymptotic symmetries of the theory. In most cases a minimal set of assumptions leads to a unique set of counterterms. However, for certain examples we find families of actions that depend on one o…
We investigate the phase structure and the infrared properties of higher-derivative quantum gravity (QG) with matter, in $4-\varepsilon$ dimensions. The renormalization group (RG) equations in $4-\varepsilon$ dimensions are analysed for the following types of matter: the $fφ^4$-theory, the $O(N) φ^4$-theory, scalar electrodynamics, and the $SU(2)$ model with scalars. New fixed points for the scalar coupling appear ---one of which is IR stable--- being some of them induced by QG. The IR stable fixed point perturbed by QG leads to a second-order phase transition for the theory at non-zero temper…
We investigate the gauge symmetry and gauge fixing dependence properties of the effective average action for quantum gravity models of general form. Using the background field formalism and the standard BRST-based arguments, one can establish the special class of regulator functions that preserves the background field symmetry of the effective average action. Unfortunately, regardless the gauge symmetry is preserved at the quantum level, the non-invariance of the regulator action under the global BRST transformations leads to the gauge fixing dependence even under the use of the on-shell condi…
A new way how to calculate the off-shell renormalization functions within the $R^2$-gravity has been proposed. The one-loop renormalization group equations in the approach suggested have been constructed. The behaviour of effective potential for an massless scalar field interacting with the quantum gravitational field has been analyzed in this approach.
The renormalization group method in $R^2$-gravity without matter fields is discussed. A criterion for the existence of the renormalization constant for the metric has been found, two-loop higher order poles have been calculated, a relation which allows us to find the one-loop renormalization constant of the Newtonian constant has been suggested.
This work explores an alternative solution to the problem of renormalizability in Einstein gravity. In the proposed approach, Einstein gravity is transformed into the renormalizable theory of four-derivative gravity by applying a field redefinition containing an infinite number of higher derivatives. It is also shown that the current-current amplitude is invariant with the field redefinition, and thus the unitarity of Einstein gravity is preserved.
We re-examine the OSp(N,4) invariant interacting model of massless chiral and gauge superfields, whose superconformal invariance was instrumental, both in proving the all-order no-renormalization of the mass and chiral self-interaction lagrangians, and in determining the linear superfield renormalization needed. We show that the renormalization of the gravitational action modifies only the cosmological term, without affecting higher-order tensors. This could explain why the effect of the cosmological constant is shadowed by the effects of newtonian gravity.
In our previously published papers, it was proved that the chromodynamics with massive gluons can well be set up on the gauge-invariance principle. The quantization of the chromodynamics was perfectly performed in the both of Hamiltonian and Lagrangian path-integral formalisms by using the Lagrangian undetermined multiplier method. In this paper, It is shown that the quantum theory is invariant with respect to a kind of BRST-transformations. From the BRST-invariance of the theory, the Ward-Takahashi identities satisfied by the generating functionals of full Green functions, connected Green fun…
Stability of dilatonic AdS spaces due to quantum effects of dilaton coupled conformal matter is considered. When such spaces do not exist on classical level, their dynamical generation occurs. Explicit examples corresponding to quantum creation of d4 dilatonic AdS Universe and of d2 dilatonic AdS Black Hole (BH) are presented. Motivated by holographic RG, in the similar approach the complete d5 effective action is discussed. The intermediate region where it is the sum of two parts: bulk (classical gravity) and boundary quantum action is investigated. The effective equations solution representi…
We study the renormalization group flow of higher derivative gravity, utilizing the functional renormalization group equation for the average action. Employing a recently proposed algorithm, termed the universal renormalization group machine, for solving the flow equation, all the universal features of the one-loop beta-functions are recovered. While the universal part of the beta-functions admits two fixed points, we explicitly show that the existence of one of them depends on the choice of regularization scheme, indicating that it is most probably unphysical.
