Fixed-Functionals of three-dimensional Quantum Einstein Gravity

A bstractWe study the non-perturbative renormalization group flow of f (R)-gravity in three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the conformally reduced approximation, we derive an exact partial differential equation governing the RG-scale dependence of the function f (R). This equation is shown to possess two isolated and one continuous one-parameter family of scale-independent, regular solutions which constitute the natural generalization of RG fixed points to the realm of infinite-dimensional theory spaces. All solutions are bounded from below and give rise to positive definite kinetic terms. Moreover, they admit either one or two UV-relevant deformations, indicating that the corresponding UV-critical hypersurfaces remain finite dimensional despite the inclusion of an infinite number of coupling constants. The impact of our findings on the gravitational Asymptotic Safety program and its connection to new massive gravity is briefly discussed.

[1]  Frank Saueressig,et al.  Ghost wavefunction renormalization in asymptotically safe quantum gravity , 2010, 1001.5032.

[2]  P. Townsend,et al.  Massive gravity in three dimensions. , 2009, Physical review letters.

[3]  M. Niedermaier,et al.  The Asymptotic Safety Scenario in Quantum Gravity , 2006, Living reviews in relativity.

[4]  M. Reuter,et al.  Is quantum Einstein gravity nonperturbatively renormalizable , 2002 .

[5]  Frank Saueressig,et al.  The R^2 phase-diagram of QEG and its spectral dimension , 2012, 1206.0657.

[6]  Christoph Rahmede,et al.  Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation , 2008, 0805.2909.

[7]  Christoph Rahmede,et al.  ULTRAVIOLET PROPERTIES OF f(R)-GRAVITY , 2007, 0705.1769.

[8]  O. Zanusso,et al.  Off-diagonal heat-kernel expansion and its application to fields with differential constraints , 2011, 1112.4856.

[9]  Martin Reuter,et al.  Conformal sector of quantum Einstein gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance , 2008, 0804.1475.

[10]  M. Reuter,et al.  The role of background independence for asymptotic safety in Quantum Einstein Gravity , 2009, 0903.2971.

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Frank Saueressig,et al.  The universal RG machine , 2010, 1012.3081.

[13]  A. Codello.  Large N quantum gravity , 2011, 1108.1908.

[14]  J. Braun.  Fermion interactions and universal behavior in strongly interacting theories , 2011, 1108.4449.

[15]  Tim R. Morris.  Derivative expansion of the exact renormalization group , 1994 .

[16]  C. Wetterich,et al.  Non-perturbative renormalization flow in quantum field theory and statistical physics , 2002 .

[17]  C. Wetterich,et al.  Exact evolution equation for the effective potential , 1993, 1710.05815.

[18]  Astrid Eichhorn,et al.  Ghost anomalous dimension in asymptotically safe quantum gravity , 2010, 1001.5033.

[19]  J. Vidal,et al.  Nonperturbative renormalization group approach to the Ising model: A derivative expansion at order ∂4 , 2003 .

[20]  P. Townsend,et al.  More on Massive 3D Gravity , 2009, 0905.1259.

[21]  Oliver J. Rosten.  Fundamentals of the Exact Renormalization Group , 2010, 1003.1366.

[22]  S. Weinberg.  Ultraviolet divergences in quantum theories of gravitation. , 1980 .

[23]  C. Bervillier,et al.  Exact renormalization group equations. An Introductory review , 2000 .

[24]  Andreas Nink,et al.  On the physical mechanism underlying asymptotic safety , 2012, 1208.0031.

[25]  A. Wipf,et al.  Functional renormalization group of the nonlinear sigma model and theO(N)universality class , 2012, 1207.4499.

[26]  M. Reuter,et al.  Ultraviolet fixed point and generalized flow equation of quantum gravity , 2001 .

[27]  Frank Saueressig,et al.  Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity , 2007, 0708.1317.

[28]  S. Nagy,et al.  Infrared fixed point in quantum Einstein gravity , 2012, 1203.6564.

[29]  F. Saueressig,et al.  A Class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior , 2002 .

[30]  Frank Saueressig,et al.  Quantum Einstein gravity , 2012, 1202.2274.

[31]  A. Bonanno,et al.  Inflationary solutions in asymptotically safe f(R) theories , 2010, 1006.0192.

[32]  Holger Gies.  Introduction to the Functional RG and Applications to Gauge Theories , 2006 .

[33]  Steven Weinberg,et al.  What is Quantum Field Theory, and What Did We Think It Is? , 1996, hep-th/9702027.

[34]  D. Litim,et al.  Limit cycles and quantum gravity , 2012, 1205.4218.

[35]  O. Zanusso,et al.  On the non-local heat kernel expansion , 2012, 1203.2034.

[36]  F. Saueressig,et al.  Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation , 2002 .

[37]  H. Gies,et al.  Asymptotically free scalar curvature-ghost coupling in quantum Einstein gravity , 2009, 0907.1828.

[38]  Christoph Rahmede,et al.  Renormalization group flow in scalar-tensor theories: II , 2009, 0911.0394.

[39]  Frank Saueressig,et al.  Taming perturbative divergences in asymptotically safe gravity , 2009, 0902.4630.

[40]  S. Hawking,et al.  General Relativity; an Einstein Centenary Survey , 1979 .

[41]  Norman M. Dott.  An Introductory Review , 1962 .

[42]  R. Percacci,et al.  Conformally reduced quantum gravity revisited , 2009, 0904.2510.

[43]  Francesco Caravelli,et al.  The local potential approximation in quantum gravity , 2012, 1204.3541.

[44]  Frank Saueressig,et al.  ASYMPTOTIC SAFETY IN HIGHER-DERIVATIVE GRAVITY , 2009, 0901.2984.

[45]  J. S. Dowker,et al.  Effective Lagrangian and energy-momentum tensor in de Sitter space , 1976 .

[46]  M. Reuter,et al.  Running boundary actions, Asymptotic Safety, and black hole thermodynamics , 2012, 1205.3583.

[47]  Frank Saueressig,et al.  On the Renormalization Group Flow of Gravity , 2007, 0712.0445.

[48]  Howard Georgi,et al.  Effective Field Theory , 1993 .

[49]  Frank Saueressig,et al.  Matter Induced Bimetric Actions for Gravity , 2010, 1003.5129.

[50]  Jan M. Pawlowski,et al.  The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows , 2012, 1203.4207.

[51]  S. Paycha,et al.  Geometric and Topological Methods for Quantum Field Theory , 2007 .

[52]  Martin Reuter,et al.  Nonperturbative evolution equation for quantum gravity , 1998 .

[53]  N. Ohta.  Beta function and asymptotic safety in three-dimensional higher derivative gravity , 2012, 1205.0476.

[54]  O. Zanusso,et al.  Higher Derivative Gravity from the Universal Renormalization Group Machine , 2011, 1111.1743.

[55]  A. Codello,et al.  Low energy quantum gravity from the effective average action , 2010, 1006.3808.

[56]  C. Rahmede.  Old and new results from the Wilsonian approach to gravity , 2011 .

[57]  Martin Reuter,et al.  Background Independence and Asymptotic Safety in Conformally Reduced Gravity , 2008, 0801.3287.

[58]  D. Litim.  Optimized renormalization group flows , 2001, hep-th/0103195.

[59]  A. Codello.  Scaling solutions in a continuous dimension , 2012, 1204.3877.

[60]  S. Weinberg.  Effective Field Theory, Past and Future , 2009, 0908.1964.

[61]  Frank Saueressig,et al.  Bimetric renormalization group flows in quantum Einstein gravity , 2010, 1006.0099.

[62]  G. Hooft.  The Conformal Constraint in Canonical Quantum Gravity , 2010, 1011.0061.

[63]  E. Elizalde,et al.  One-loop f(R) gravity in de Sitter universe , 2005, hep-th/0501096.

[64]  D. Litim.  Fixed points of quantum gravity and the renormalisation group , 2008, 0810.3675.

[65]  Wataru Souma,et al.  Non-Trivial Ultraviolet Fixed Point in Quantum Gravity , 1999, hep-th/9907027.

[66]  Carlo Rovelli.  Quantum gravity , 2008, Scholarpedia.

[67]  Roberto Percacci,et al.  The running gravitational couplings , 1998 .

[68]  Bertrand Delamotte,et al.  An Introduction to the Nonperturbative Renormalization Group , 2007, cond-mat/0702365.

[69]  D. Litim.  Fixed points of quantum gravity , 2003, hep-th/0312114.

[70]  Elements of the Continuous Renormalization Group , 1998, hep-th/9802039.

[71]  D. Litim,et al.  Ising exponents from the functional renormalisation group , 2010, 1009.1948.

[72]  A. Bonanno,et al.  Universality and Symmetry Breaking in Conformally Reduced Quantum Gravity , 2012, 1206.6531.

[73]  Jan M. Pawlowski.  Aspects of the functional renormalisation group , 2007 .

[74]  A. Codello,et al.  Polyakov effective action from functional renormalization group equation , 2010, 1004.2171.

[75]  M. Reuter,et al.  Flow equation of quantum Einstein gravity in a higher derivative truncation , 2002 .

[76]  I. Avramidi.  Heat Kernel and Quantum Gravity , 2000 .