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We discuss theories of gravity with independent metric (or frame field) and connection, from the point of view of effective field theory. We count the parity-even Lagrangian terms of dimension up to four and give explicit bases for the independent terms that contribute to the two-point function. We then give the decomposition of the linearized action on a complete basis of spin projectors and consider various subclasses of Metric-Affine Gravity theories (MAGs). We show that teleparallel theories can be dynamically equivalent to any metric theory of gravity and give the particle content of those whose Lagrangian contains only dimension-two terms. We point out the existence of a class of MAGs whose EOMs do not admit propagating degrees of freedoms. Finally, we construct simple MAGs that contain only a massless graviton and a state of spin/parity $2^-$ or $3^-$. As a side result, we write the relativistic wave equation for a spin/parity $2^-$ state., Comment: 96 pages

We discuss the possibility to define exact RG equations for a UV regulated Wilsonian action based on a proper time (PT) regulator function. We start from a functional mapping which shows how each particular flow equation (and RG scheme) is associated to infinitely many scale dependent field redefinitions, which are related to specific coarse-graining procedures. On specializing to a sub-family of one parameter PT regulators we briefly analyze few results for the Ising Universality class in three dimensions, obtained within a second order truncation in the derivative expansion of the Wilsonian action., Comment: 18 pages, 1 figure, revtex, few clarifying comments and references added

We construct the spin-projection operators for a theory containing a symmetric two-index tensor and a general three-index tensor. We then use them to analyse, at linearized level, the most general action for a metric-affine theory of gravity with terms up to second order in curvature, which depends on 28 parameters. In the metric case we recover known results. In the torsion-free case, we are able to determine the most general six-parameter class of theories that are projective invariant, contain only one massless spin 2 and no spin 3, and are free of ghosts and tachyons., Comment: 39 pages. Ancillary notebook with spin projectors is provided separately. Requires the package xTensor for Mathematica v2: references added, some minor corrections

We compute the one-loop divergences in a theory of gravity with Lagrangian of the general form $f(R,R_{\mu\nu}R^{\mu\nu})$, on an Einstein background. We also establish that the one-loop effective action is invariant under a duality that consists of changing certain parameters in the relation between the metric and the quantum fluctuation field. Finally, we discuss the unimodular version of such a theory and establish its equivalence at one-loop order with the general case., Comment: 14 pages, v2:minor corrections, refs. added/modified

It is shown that the one-loop effective action of unimodular gravity is the same as that of ordinary gravity, restricted to unimodular metrics. The only difference is in the treatment of the global scale degree of freedom and of the cosmological term. A constant vacuum energy does not gravitate, addressing one aspect of the cosmological constant problem., Comment: Prepared for the proceedings of the workshop "Black holes, gravitational waves and spacetime singularities", held at the Vatican Observatory, Castel Gandolfo (Rome) May 9 - 12, 2017

We compute the one-loop effective action in unimodular gravity, starting from two different classical formulations of the theory. We find that the effective action is the same in both cases, and agrees with the one of General Relativity.

We compute the one-loop divergences in a higher-derivative theory of gravity including Ricci tensor squared and Ricci scalar squared terms, in addition to the Hilbert and cosmological terms, on an (generally off-shell) Einstein background. We work with a two-parameter family of parametrizations of the graviton field, and a two-parameter family of gauges. We find that there are some choices of gauge or parametrization that reduce the dependence on the remaining parameters. The results are invariant under a recently discovered "duality" that involves the replacement of the densitized metric by a densitized inverse metric as the fundamental quantum variable., Comment: 29 pages, to appear in EPJC

We perform a general computation of the off-shell one-loop divergences in Einstein gravity, in a two-parameter family of path integral measures, corresponding to different ways of parametrizing the graviton field, and a two-parameter family of gauges. Trying to reduce the gauge- and measure-dependence selects certain classes of measures and gauges respectively. There is a choice of two parameters (corresponding to the exponential parametrization and the partial gauge condition that the quantum field be traceless) that automatically eliminates the dependence on the remaining two parameters and on the cosmological constant. We observe that the divergences are invariant under a $\mathbf{Z}_2$ "duality" transformation that (in a particularly important special case) involves the replacement of the densitized metric by a densitized inverse metric as the fundamental quantum variable. This singles out a formulation of unimodular gravity as the unique "self-dual" theory in this class., Comment: 23 pages, 16 figures, minor improvements, to appear in JHEP

Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension $D = p + 1$. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the $O(D,D)$ string T-duality, here we show in the case of the M2-brane in $D = 3$ that a generalised geometry renders manifest the $SL(3) \times SL(2)$ U-duality. In the case of M2-brane in $D=4$, with and without extra target space coordinates, we show that only the ${\rm GL}(4,R)\ltimes R^4$ subgroup of the expected $SL(5,R)$ U-duality symmetry is realised., Comment: 26 pages

There have been some speculations about the existence of critical unitary O(N)-invariant scalar field theories in dimensions 4

We discuss general theories of N scalar fields with O(N) symmetry. In addition to the standard case of linearly realized symmetry there are also examples that carry nonlinear realizations, with the topology of a cylinder $R\times S^{N-1}$ or a sphere $S^N$. We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential., Comment: 32 pages; v.2: one reference added, small corrections, to appear in PRD

We compute the one-loop beta functions of the cosmological constant, Newton's constant and the topological mass in topologically massive supergravity in three dimensions. We use a variant of the proper time method supplemented by a simple choice of cutoff function. We find that the dimensionless coefficient of the Chern-Simons term, $\nu$, has vanishing beta function. The flow of the cosmological constant and Newton's constant depends on $\nu$; we study analytically the structure of the flow and its fixed points in the limits of small and large $\nu$., Comment: 36 pages. References added

We consider matter fields conformally coupled to a background metric and dilaton and describe in detail a quantization procedure and related renormalization group flow that preserve Weyl invariance. Even though the resulting effective action is Weyl-invariant, the trace anomaly is still present, with all its physical consequences. We discuss first the case of free matter and then extend the result to interacting matter. We also consider the case when the metric and dilaton are dynamical and gravitons enter in the loops., Comment: 32 pages, 3 figures

Any theory can be made Weyl invariant by introducing a dilaton. It is shown how to construct renormalization group equations for gravity that maintain this property. Explicit calculations are given only in the simplest approximation, namely for the one loop beta functions of a dilaton conformally coupled to a dynamical metric, but the results have wider validity. This formalism could be used to define the meaning of a theory with a position-dependent cutoff: it is equivalent to a theory with a constant cutoff, but a conformally related metric., Comment: To appear in New Journal of Physics. 20 pages

I discuss the notion of asymptotic safety and possible applications to quantum field theories of gravity and matter., Comment: In the proceedings of the conference "Time and matter", Budva, Montenegro, October 2010. 18 pages

We consider a model in which Goldstone bosons, described by a SU(N) chiral nonlinear sigma model, are coupled to an N-plet of colored fermions by means of a Yukawa interaction. We study the one-loop renormalization group flow and show that the non-Gaussian UV fixed point, which is present in the purely bosonic model, is lost because of fermion loop effects unless N is sufficiently large. We then add four-fermion contact interactions to the lagrangian and show that in this case there exist several non-Gaussian fixed points. The strength of the contact interactions, predicted by the requirement that the theory flows towards a fixed point in the UV, is compared to the current experimental bounds. This toy model could provide an important building block of an asymptotically safe model of the weak interactions., Comment: 7 pages, 3 figures, 1 table

We consider the standard model without the Higgs boson, where the Goldstone modes are described by a nonlinear sigma model. We study the renormalization group flow of the sigma model coupling f and of the electroweak parameters S and T. The condition that the couplings reach a fixed point at high energy leaves the low energy values of f and T arbitrary (to be determined experimentally) and fixes S to a value compatible with electroweak precision data., Comment: 4 pages, 2 figures

We study the beta functions of the leading, two-derivative terms of the left-gauged SU(N) nonlinear sigma-model in d dimensions. In d>2, we find the usual Gaussian ultraviolet fixed point for the gauge coupling and an attractive non-Gaussian fixed point for the Goldstone boson coupling. The position of the latter fixed point controls the chiral expansion, unitarity and the strength of the tree-level Goldstone boson scattering amplitudes. Attention is paid to the gauge- and scheme-dependence of the results., Comment: REVTeX4, 11 pages, no figures, v2. Revised version accepted for publication in PRD where comments on large N have been removed

There seems to be a common prejudice that asymptotic safety is either incompatible with, or at best unrelated to, the other topics in the title. This is not the case. In fact, we show that 1) the existence of a fixed point with suitable properties is a promising way of deriving emergent properties of gravity, and 2) there is a sense in which asymptotic safety implies a minimal length. In so doing we also discuss possible signatures of asymptotic safety in scattering experiments., Comment: LaTEX, 20 pages, 2 figures; v.2: minor changes, reflecting published version

We calculate the running of the three coupling constants in cosmological, topologically massive 3d gravity. We find that \nu, the dimensionless coefficient of the Chern-Simons term, has vanishing beta function. The flow of the cosmological constant and Newton's constant depends on \nu, and for any positive \nu there exist both a trivial and a nontrivial fixed point., Comment: 44 pages, 16 figures

In these lectures I review the status of gravity from the point of view of the gauge principle and renormalization, the main tools in the toolbox of theoretical particle physics. In the first lecture I start from the old question "in what sense is gravity a gauge theory?" I will reformulate the theory of gravity in a general kinematical setting which highlights the presence of two Goldstone boson-like fields, and the occurrence of a gravitational Higgs phenomenon. The fact that in General Relativity the connection is a derived quantity appears to be a low energy consequence of this Higgs phenomenon. From here it is simple to see how to embed the group of local frame transformations and a Yang Mills group into a larger unifying group, and how the distinction between these groups, and the corresponding interactions, derives from the VEV of an order parameter. I will describe in some detail the fermionic sector of a realistic "GraviGUT" with SO(3,1)xSO(10) \subset SO(3,11). In the second lecture I will discuss the possibility that the renormalization group flow of gravity has a fixed point with a finite number of attractive directions. This would make the theory well behaved in the ultraviolet, and predictive, in spite of being perturbatively nonrenormalizable. There is by now a significant amount of evidence that this may be the case. There are thus reasons to believe that quantum field theory may eventually prove sufficient to explain the mysteries of gravity., Comment: Lectures given at the Fifth International School on Field Theory and Gravitation, Cuiaba, Brazil April 20-24 2009. To appear in PoS

There are deep analogies between Einstein's theory of gravity and the nonlinear sigma models. It is suggested that these similarities may extend also to the ultraviolet behaviour, in the sense that both theories could turn out to be asymptotically safe., Comment: Talk given at International Workshop on Continuum and Lattice Approaches to Quantum Gravity, Brighton, United Kingdom, 17-19 Sep 08. To appear in PoS

We calculate the one loop beta functions for nonlinear sigma models in four dimensions containing general two and four derivative terms. In the O(N) model there are four such terms and nontrivial fixed points exist for all N \geq 4. In the chiral SU(N) models there are in general six couplings, but only five for N=3 and four for N=2; we find fixed points only for N=2,3. In the approximation considered, the four derivative couplings are asymptotically free but the coupling in the two derivative term has a nonzero limit. These results support the hypothesis that certain sigma models may be asymptotically safe., Comment: 26 pages

We show how to obtain a single chiral family of an SO(10) GUT, starting from a Majorana-Weyl representation of a unifying ("GraviGUT") group SO(3,11), which contains the gravitational Lorentz group SO(3,1). An action is proposed, which reduces to the correct fermionic GUT action in the broken phase., Comment: 7 pages. Discussion clarified. Version matching the published one

Applying functional renormalization group methods, we describe two inequivalent ways of defining the renormalization group of matter-coupled four dimensional gravity, in the approximation where only the conformal factor is dynamical and taking the trace anomaly explicitly into account. We make contact with earlier work and briefly discuss the presence or absence of fixed points, depending on the truncation of the action and other approximations., Comment: 24 pages

We compute the gravitational corrections to the running of couplings in a scalar-fermion system, using the Wilsonian approach. Our discussion is relevant for symmetric as well as for broken scalar phases. We find that the Yukawa and quartic scalar couplings become irrelevant at the Gaussian fixed point., Comment: 5 pages, RevTex, discussion extended, typos corrected, reference added. Published online on PLB.

Using Wilsonian methods, we study the renormalization group flow of the Nonlinear Sigma Model in any dimension $d$, restricting our attention to terms with two derivatives. At one loop we always find a Ricci flow. When symmetries completely fix the internal metric, we compute the beta function of the single remaining coupling, without any further approximation. For $d>2$ and positive curvature, there is a nontrivial fixed point, which could be used to define an ultraviolet limit, in spite of the perturbative nonrenormalizability of the theory. Potential applications are briefly mentioned., Comment: 9 pages

This note addresses the question whether in a gauge theory coupled to gravity internal and spacetime transformation can be mixed. It is shown that if the VEV of the gauge field is flat, the symmetry group is always a product of internal and spacetime symmetries. On the other hand, if the VEV of the gauge field is not flat it is impossible to properly define the notion of a ``spacetime'' transformation; as a consequence, if the symmetry group is nontrivial, mixing generically occurs., Comment: Plain TEX, 6 pages

The Higgs phenomenon occurs in theories of gravity in which the connection is an independent dynamical variable. The role of order parameters is played by the soldering form and a fiber metric. The breaking of the original gauge symmetry is linked to the appearance of geometrical structures on spacetime. These facts suggest certain modifications and generalizations of the theory. We propose a Higgs-like model which provides a dynamical explanation for the nondegeneracy of the metric and a framework for the unification of gravity with the other interactions., Comment: This paper was published long ago but was not previously available in the archive. Some updates have been added in a postscript and some recent references added

Asymptotic safety is a set of conditions, based on the existence of a nontrivial fixed point for the renormalization group flow, which would make a quantum field theory consistent up to arbitrarily high energies. After introducing the basic ideas of this approach, I review the present evidence in favor of an asymptotically safe quantum field theory of gravity., Comment: In "Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter", ed. D. Oriti, Cambridge University Press A sign corrected in eq. (1.4.7), a reference updated

The coupling of chiral fermions to gravity makes use only of the selfdual SU(2) subalgebra of the (complexified) SO(3,1) algebra. It is possible to identify the antiselfdual subalgebra with the SU(2)_L isospin group that appears in the Standard Model, or with its right-handed counterpart SU(2)_R that appears in some extensions. Based on this observation, we describe a form of unification of the gravitational and weak interactions. We also discuss models with fermions of both chiralities, the inclusion strong interactions, and the way in which these unified models of gravitational and gauge interactions avoid conflict with the Coleman-Mandula theorem., Comment: 18 pages, typos corrected and improved wording

We recalculate the beta functions of higher derivative gravity in four dimensions using the one--loop approximation to an Exact Renormalization Group Equation. We reproduce the beta functions of the dimensionless couplings that were known in the literature but we find new terms for the beta functions of Newton's constant and of the cosmological constant. As a result, the theory appears to be asymptotically safe at a non--Gaussian Fixed Point, rather than perturbatively renormalizable and asymptotically free., Comment: Introduction expanded, some references added

We show that the running of gravitational couplings, together with a suitable identification of the renormalization group scale can give rise to modified dispersion relations for massive particles. This result seems to be compatible with both the frameworks of effective field theory with Lorentz invariance violation and deformed special relativity. The phenomenological consequences depend on which of the frameworks is assumed. We discuss the nature and strength of the available constraints for both cases and show that in the case of Lorentz invariance violation, the theory would be strongly constrained., Comment: revtex4, 9 pages, updated to match published version

A theory of gravity with a generic action functional and minimally coupled to N matter fields has a nontrivial fixed point in the leading large N approximation. At this fixed point, the cosmological constant and Newton's constant are nonzero and UV relevant; the curvature squared terms are asymptotically free with marginal behaviour; all higher order terms are irrelevant and can be set to zero by a suitable choice of cutoff function., Comment: LaTEX, 4 pages. Relative to the published paper, a sign has been corrected in equations (17) and (18)

We discuss the generalization to global gauge anomalies of the familiar procedure for the cancellation of local gauge anomalies in effective theories of spontaneously broken symmetries. We illustrate this mechanism in a recently proposed six-dimensional extension of the standard model., Comment: 5 pages; v2: version to appear in Phys. Rev. D

Nonlinear sigma models arise in supergravity theories with or without matter couplings in various dimensions and they are important in understanding the duality symmetries of M-theory. With this motivation in mind, we review the salient features of gauged and ungauged nonlinear sigma models with or without Wess-Zumino terms for general target spaces in a minimal as well as lifted formulation. Relevant to the question of finding interesting vacua of gauged supergravity theories is the highly constrained potential which arises naturally in these theories. Motivated by this fact, we derive a general and simple formula for a gauge invariant potential of this kind., Comment: 24 pages, latex, uses sprocl.sty, contribution to the R. Arnowitt Fest: A Symposium on Supersymmetry and Gravitation, College Station, TX, 5-7 April 1998

Using renormalization group methods we calculate the derivative expansion of the effective Lagrangian for a covariantly constant gauge field in curved spacetime. Curvature affects the vacuum, in particular it could induce phase transitions between different vacua. We also consider the effect of quantum fluctuations of the metric , in the context of a renormalizable $R^2$ theory. In this case the critical curvature depends on the gravi tational coupling constants., Comment: Latex file, 8 pages

We consider the gravitational coupling of a scalar field, in a reformulation of General Relativity exhibiting local GL(4) invariance at the classical level. We compute the one-loop contribution of the scalar to the quantum effective potential of the vierbein and find that it does not have GL(4) invariance. The minima of the effective potential occur for a vierbein which is proportional to the unit matrix., Comment: 9 pages, plain-TeX, no figures

We discuss the definition of the average effective action in terms of the heat-kernel. As an example we examine a model describing a self-interacting scalar field, both in flat and curved background, and study the renormalization group flow of some of the parameters characterizing its effective potential. Some implications of the running of these parameters for inflationary cosmology are also briefly discussed., Comment: 9 pages, plain-Tex

We study a scalar field theory coupled to gravity on a flat background, below Planck's energy. Einstein's theory is treated as an effective field theory. Within the context of Wilson's renormalization group, we compute gravitational corrections to the beta functions and the anomalous dimension of the scalar field, taking into account threshold effects., Comment: 13 pages, plainTex

We consider a scalar-metric gauge theory of gravity with independent metric, connection and dilaton. The role of the dilaton is to provide the scale of all masses, via its vacuum expectation value. In this theory, we study the renormalization group flow of the dilaton potential, taking into account threshold effects at the Planck scale. Due to the running of the VEV of the dilaton all particles that would naively seem to have masses larger than Planck's mass, may actually not propagate. This could solve the problem of unitarity in these theories., Comment: 27 pages, plain-Tex

We study the target-space duality transformations in $p$--branes as transformations which mix the worldvolume field equations with Bianchi identities. We consider an $(m+p+1)$-dimensional spacetime with $p+1$ dimensions compactified, and a particular form of the background fields. We find that while a $GL(2)=SL(2)\times R$ group is realized when $m=0$, only a two parameter group is realized when $m>0$., Comment: 12 pages, plain TEX, SISSA 44/94/EP, CTP-TAMU-15/94

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., Comment: 7 pages, plain TEX, SISSA 45/94/EP

There is a general mechanism by which certain matter fields coupled to gravity can generate a nontrivial effective potential for the conformal factor of the metric. It is based on a nonstandard regularization method, with the cutoff being defined independently of the conformal factor. This mechanism produces a coupling of the matter fields to a dilaton, and a complicated interaction between matter, dilaton and metric. When it is applied to the standard model, it gives an effective potential which can be used to predict the top and Higgs masses. If the purely gravitational contribution to the potential is added, the mass of the dilaton is of the order of Planck's mass and the large hierarchy between the Planck and Fermi scales appears to be due to the smallness of the Higgs-dilaton coupling., Comment: 7 pages, plain TEX, ITP and SISSA

In a four dimensional theory of gravity with lagrangian quadratic in curvature and torsion, we compute the effective action for metrics of the form $g_{\mu\nu}=\rho^2\delta_{\mu\nu}$, with $\rho$ constant. Using standard field-theoretic methods we find that one loop quantum effects produce a nontrivial effective potential for $\rho$. We explain this unexpected result by showing how our regularization procedure differs from the one that is usually adopted in Quantum Gravity. Using the method of the average effective potential, we compute the scale dependence of the v.e.v. of the conformal factor., Comment: 8 pages, plain TEX, SISSA 71/93-EP

We generalize to dimension $p>1$ the notion of string structure and discuss the related obstruction. We apply our results to a model of bosonic $p$-branes propagating on a principal $G$-bundle, coupled to a Yang--Mills field and an antisymmetric tensor field and in the presence of a Wess-Zumino term in the Lagrangian. We construct the quantization line bundle and discuss the action of background gauge transformations on wave functions., Comment: 14 pages, plain TEX, SISSA 36/93/EP

We consider, for $p$ odd, a $p$--brane coupled to a $(p+1)$th rank background antisymmetric tensor field and to background Yang-Mills (YM) fields {\it via} a Wess-Zumino term. We obtain the generators of antisymmetric tensor and Yang-Mills gauge transformations acting on $p$--brane wavefunctionals (functions on `$p$-loop space'). The Yang-Mills generators do not form a closed algebra by themselves; instead, the algebra of Yang-Mills and antisymmetric tensor generators is a $U(1)$ extension of the usual algebra of Yang-Mills gauge transformations. We construct the $p$-brane's Hamiltonian and thereby find gauge-covariant functional derivatives acting on $p$--brane wavefunctionals that commute with the YM and $U(1)$ generators., Comment: 13 pages, UG-5/92; CTP-TAMU-63/92; DAMTP/R-92/32; Imperial/TP/91-92/30; SISSA 157/92/EP

Using canonical methods, we study the invariance properties of a bosonic $p$--brane propagating in a curved background locally diffeomorphic to $M\times G$, where $M$ is spacetime and $G$ a group manifold. The action is that of a gauged sigma model in $p+1$ dimensions coupled to a Yang--Mills field and a $(p+1)$--form in $M$. We construct the generators of Yang-Mills and tensor gauge transformations and exhibit the role of the $(p+1)$--form in cancelling the potential Schwinger terms. We also discuss the Noether currents associated with the global symmetries of the action and the question of the existence of infinite dimensional symmetry algebras, analogous to the Kac-Moody symmetry of the string., Comment: 15 pages, CTP-TAMU-32/92, SISSA 182/92/EP

The algebra W_{1+\infty} with central charge c=0 can be identified with the algebra of quantum observables of a particle moving on a circle. Mathematically, it is the universal enveloping algebra of the Euclidean algebra in two dimensions. Similarly, the super W_\infty algebra is found to be the universal enveloping algebra of the super-Euclidean algebra in two dimensions., Comment: (8 pages)