HABILITATIONSSCHRIFT Gravity in 2D and 3D

Gravity in 2D and 3D

A romance of lower dimensions

Presented to the Faculty of Physics Vienna University of Technology

By

Daniel Grumiller

Institute for Theoretical Physics E-mail: grumil@hep.itp.tuwien.ac.at

Vienna, February 24, 2010

Gravity in lower dimensions provides a useful expedient for testing ideas about quantum gravity in higher dimensions. Technical simplifications in lower dimensions often lead to exact results, and this helps to address some of the conceptual problems posed by classical and quantum gravity. At the same time, these simplifications could remove some of the features that make gravity interesting. Striking a balance between models that are tractable and models that seem relevant is an art in its own right.

Two is the lowest dimension that admits gravity models containing black hole solutions. In particular, 2D dilaton gravity exhibits a wide range of in- teresting phenomena, for instance a rich structure of thermodynamic observ- ables like black hole entropy or free energy. Conceptual issues of holographic renormalization with arbitrary spacetime asymptotics can be resolved com- paratively easily. 2D gravity thus provides valuable insights for corresponding higher-dimensional theories of gravity.

Three is the lowest dimension that admits gravity models containing black hole solutions and gravitons. In particular, Cosmological Topologically Mas- sive Gravity exhibits a wide range of interesting features: it contains different black hole solutions, asymptotically Anti-deSitter solutions as well as solu- tions with different asymptotics and massive graviton excitations. In the past three years 3D gravity has been studied vigorously, and my recent research results have contributed significantly to this field. One of the main goals is to obtain a useful and soluble model of quantum gravity. Another pertinent goal is to apply the gauge/gravity duality to 3D gravity in order to describe certain condensed matter systems. Both is work in progress.

In this Habilitationsschrift I collect a selection of my papers on 2D and 3D gravity and provide a brief guideline to these papers as well as to current research of my START group at the Vienna University of Technology.

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I am grateful to my collaborators, colleagues and students for scores of hours of discussions. I thank in particular my co-authors on the papers that constitute this Habilitationsschrift: Wolfgang Kummerfor numerous valuable advises during the early stages of my career and for being such a good person; Dima Vassilevich for an enjoyable long-term collaboration on 2D gravity and the hospitality in Leipzig; Luzi Bergamin for being so good-natured, reliable and fun to work with; Ren´e Meyer for being such an excellent, agreeable and successful student; Roman Jackiw for more than two fruitful and pleasurable years of collaboration at the Massachusetts Institute of Technology; Robert McNees for his excellent introduction to holographic renormalization and his hospitality at the Perimeter Institute;

Niklas Johansson for being extremely brilliant and pleasant at the same time, and for being my first post-doc; Peter van Nieuwenhuizen for in- venting and explaining supergravity, as well as for all the wonderful stories;

Ivo Sachs for his permanently good spirit in the face of difficult tasks; and Olaf Hohm for inventing and explaining new massive gravity. Moreover, I thank Toni Rebhan, head of the Fundamental Interactions group at the Institute for Theoretical Physics of the Vienna University of Technology, for doing the right things at the right time in the right way. Finally, I thank Laurenz Widhalm, not just for splendid collaboration on outreach issues atteilchen.atand in schools, but also for his friendship and for introducing us to the world of geocaching.

Privately I would like to thank so many people — family and friends

— that this acknowledgment would grow without bound. Thus, I’ll restrict myself to three highlights in print and let everyone else know my gratefulness in private. Wiltraud. What can I say? Life without you wouldn’t bemylife, and I am happy that you took all the moving within Europe and between Europe and the US with good humour. Moving back to Vienna was the hardest part, but we did it! Laurin and Armin. You are the best boys a father can wish for: curious, cheeky, playful and reliable when it counts — it was phantastic to live with you in Leipzig, in Boston and it is phantastic to live with you in Vienna. In addition I am grateful for each and every private visit during the past six years, so to all our family members and friends who visited us in Leipzig or Boston, who kept contact with us, who helped us moving, who brought us poppy seeds or Topfen, who gave me an excellent excuse for serious cooking, who made trips with us within Leipzig or to New York or to the Blue Hills Reservoir or in Burgenland or in Zeutschach, who helped us survive the first 1.5 years back in Vienna: thank you. Thanks.

This Habilitationsschrift was supported by the START project Y435-N16 of the Austrian Science Foundation (FWF).

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And H. C. IN PARTICULAR This Work is Dedicated By a Humble Native of Flatland

In the Hope that

Even as he was Initiated into the Mysteries Of THREE Dimensions

Having been previously conversant With ONLY TWO

So the Citizens of that Celestial Region May aspire yet higher and higher

To the Secrets of FOUR FIVE OR EVEN SIX Dimensions Thereby contributing

To the Enlargement of THE IMAGINATION And the possible Development

Of that most rare and excellent Gift of MODESTY Among the Superior Races

Of SOLID HUMANITY

Edwin A. Abbot alias A. Square, “Flatland — A romance of many dimensions” (1884)

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I Guide to the Papers 1

1 Gravity in two dimensions (2002-2007) 3

2 Gravity in three dimensions (2008-2010) 7

Bibliography 11

II Papers on 2D Gravity 13

1 Dilaton gravity in two dimensions (2002) 15 2 The classical solutions of the dimensionally reduced gravita-

tional Chern-Simons theory (2003) 159

3 Supersymmetric black holes are extremal and bald in 2D

dilaton supergravity (2004) 171

4 Virtual black holes and the S-matrix (2004) 201 5 An action for the exact string black hole (2005) 229

6 Ramifications of lineland (2006) 271

7 Duality in 2-dimensional dilaton gravity (2006) 317 8 Thermodynamics of black holes in two (and higher) dimen-

sions (2007) 331

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1 Instability in cosmological topologically massive gravity at

the chiral point (2008) 397

2 Canonical analysis of cosmological topologically massive grav-

ity at the chiral point (2008) 417

3 Consistent boundary conditions for cosmological topologi- cally massive gravity at the chiral point (2008) 429 4 Holographic counterterms from local supersymmetry with-

out boundary conditions (2009) 435

5 AdS3/LCFT2 – Correlators in Cosmological Topologically Mas-

sive Gravity (2009) 445

6 AdS3/LCFT2 – Correlators in New Massive Gravity (2009) 509 7 Gravity duals for logarithmic conformal field theories (2010)519

CV 535

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Guide to the Papers

1

Gravity in two dimensions (2002-2007)

Two’s company; three’s a crowd

Proverb Part II of this Habilitationsschrift contains eight papers on 2D dilaton gravity [1–8]. The theory considered in most papers is 2D dilaton gravity, the action of which is given by

S2D =− Z

d²xp

|g|h

XR−U(X)(∇X)²−2V(X)i

+ boundary terms (1.1) where g is a 2D metric, X the dilaton field and U, V arbitrary functions thereof. Here is a brief description and list of abstracts of these papers.

Ref. [1] is an invited review article on 2D dilaton gravity. It sum- marizes my earliest research achievements until 2002 and provides a compre- hensive overview on 2D dilaton gravity. Here is its abstract:

The study of general two dimensional models of gravity allows to tackle ba- sic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade new insights regarding the exact quantization of the geometric part of such theories have been ob- tained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a

3

Ref. [2] constructs all classical solutions globally of a specific 3D theory reduced to 2D. This paper led to several invitations and to closer contact with the Massachusetts Institute of Technology, in particular with Alfredo Iorio, Roman Jackiw and Carlos Nu˜nez. Here is its abstract:

The Kaluza-Klein reduction of the 3d gravitational Chern-Simons term to a 2d theory is equivalent to a Poisson-sigma model with fourdimensional target space and degenerate Poisson tensor of rank 2. Thus two constants of motion (Casimir functions) exist, namely charge and energy. The application of well-known methods developed in the framework of first order gravity allows to construct all classical solutions straightforwardly and to discuss their global structure. For a certain fine tuning of the values of the constants of motion the solutions ofhep-th/0305117 are reproduced. Possible generalizations are pointed out.

Ref. [3] discusses all supersymmetric solutions of 2D dilaton su- pergravity. This paper considerably extends the discussion on 2D dilaton supergravity as compared to the review [1]. Here is its abstract:

We present a systematic discussion of supersymmetric solutions of 2D dilaton supergravity. In particular those solutions which retain at least half of the su- persymmetries are ground states with respect to the bosonic Casimir function (essentially the ADM mass). Nevertheless, by tuning the prepotential appro- priately, black hole solutions may emerge with an arbitrary number of Killing horizons. The absence of dilatino and gravitino hair is proven. Moreover, the impossibility of supersymmetric dS ground states and of nonextremal black holes is confirmed, even in the presence of a dilaton. In these derivations the knowledge of the general analytic solution of 2D dilaton supergravity plays an important rˆole. The latter result is addressed in the more general context of gPSMs which have no supergravity interpretation. Finally it is demonstrated that the inclusion of non-minimally coupled matter, a step which is already nontrivial by itself, does not change these features in an essential way.

Ref. [4] is an invited review article on virtual black holes. Here is its abstract:

A brief review on virtual black holes is presented, with special emphasis on phenomenologically relevant issues like their influence on scattering or on the specific heat of (real) black holes. Regarding theoretical topics results im- portant for (avoidance of ) information loss are summarized. After recalling

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for scattering experiments is addressed. Among the key features is that vir- tual black holes tend to regularize divergences of quantum field theory and that a unitary S-matrix may be constructed. Also the thermodynamical behavior of real evaporating black holes may be ameliorated by interactions with vir- tual black holes. Open experimental and theoretical challenges are mentioned briefly.

Ref. [5] constructed for the first time a local action for a specific family of string theoretic black holes. This paper circumvented an earlier no-go result by relaxing one of its premises and led to numerous in- vitations as well as to the collaboration with Robert McNees. Here is its abstract:

A local action is constructed describing the exact string black hole discovered by Dijkgraaf, Verlinde and Verlinde in 1992. It turns out to be a special 2D Maxwell-dilaton gravity theory, linear in curvature and field strength. Two constants of motion exist: massM ≥1, determined by the levelk, andU(1)- charge Q ≥ 0, determined by the value of the dilaton at the origin. ADM mass, Hawking temperature TH ∝ p

1−1/M and Bekenstein–Hawking en- tropy are derived and studied in detail. Winding/momentum mode duality implies the existence of a similar action, arising from a branch ambiguity, which describes the exact string naked singularity. In the strong coupling limit the solution dual to AdS2 is found to be the 5D Schwarzschild black hole. Some applications to black hole thermodynamics and 2D string theory are discussed and generalizations — supersymmetric extension, coupling to matter and critical collapse, quantization — are pointed out.

Ref. [6] is another invited review on 2D dilaton gravity. Its focus is particularly on the more recent developments between 2002-2006. This paper was written with my undergraduate student Ren´e Meyer in Leipzig.

Here is its abstract:

A non-technical overview on gravity in two dimensions is provided. Appli- cations discussed in this work comprise 2D type 0A/0B string theory, Black Hole evaporation/thermodynamics, toy models for quantum gravity, for nu- merical General Relativity in the context of critical collapse and for solid state analogues of Black Holes. Mathematical relations to integrable models, non-linear gauge theories, Poisson-sigma models, KdV surfaces and non- commutative geometry are presented.

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of a longer work with Roman Jackiw. Here is its abstract:

We descry and discuss a duality in 2-dimensional dilaton gravity.

Ref. [8] provides a first principle derivation of all boundary terms in 2D dilaton gravity. In particular, the holographic counterterms were constructed and exploited to derive the free energy of a large class of black holes, as well as other thermodynamic properties. Here is its abstract:

A comprehensive treatment of black hole thermodynamics in two-dimensional dilaton gravity is presented. We derive an improved action for these theories and construct the Euclidean path integral. An essentially unique boundary counterterm renders the improved action finite on-shell, and its variational properties guarantee that the path integral has a well-defined semi-classical limit. We give a detailed discussion of the canonical ensemble described by the Euclidean partition function, and examine various issues related to sta- bility. Numerous examples are provided, including black hole backgrounds that appear in two dimensional solutions of string theory. We show that the Exact String Black Hole is one of the rare cases that admits a consistent thermodynamics without the need for an external thermal reservoir. Our ap- proach can also be applied to certain higher-dimensional black holes, such as Schwarzschild-AdS, Reissner-Nordstr¨om, and BTZ.

Outlook on 2D gravity While currently my research focus is mostly on 3D gravity or higher-dimensional gravity, I still keep working in this field where I am one of the leading experts world-wide. I pursue this topic in collaboration with students at the Vienna University of Technology, and with my collaborators Luzi Bergamin, Roman Jackiw, Robert McNees, Ren´e Meyer, Dimitri Vassilevich and others.

Main goal of this line of research: thorough understanding of classi- cal, semi-classical and quantum gravitational effects and black hole properties. Since many higher-dimensional black holes can be described by 2D dilaton gravity (Schwarzschild, Reissner-Nordstr¨om, BTZ, etc.) results obtained in two dimensions can provide valuable insights also for higher di- mensions, where the “holy grail” — a consistent and comprehensive quantum theory of gravity — is still out of reach at present.

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Gravity in three dimensions (2008-2010)

Professor, you should be commended On your theory so geniusly splendid.

But some say it’s luck, And you really just suck,

’Cause your theory’s not what you intended!

Physics Limericks (Harvard University), http://www.physics.harvard.edu/academics/undergrad/limericks.html

Part III of this Habilitationsschrift contains seven papers on 3D gravity [9–

15]. The theory considered in most papers below is cosmological topologically massive gravity, whose action (up to boundary terms) is given by

S³D = 1 16πG

Z

d³xp

|g|h R+2

ℓ²+ 1

2µε^αβγΓ^ρασ ∂βΓ^σγρ+2

3Γ^σβτΓ^τγρ

i (2.1) where G is Newton’s constant, g is a 3D metric, ℓ is the AdS radius and µ the Chern–Simons coupling. Here is a brief description and list of abstracts of these papers.

Ref. [9] discovered graviton excitations in a specific 3D gravity theory that were thought to be absent by others. Moreover, it con- jectured a logarithmic conformal field theory as gauge theory dual. This paper is my best-known recent work, and most of the papers below are based on it. It was written with the graduate student Niklas Johansson, who is now my post-doc in Vienna. Here is its abstract:

We consider cosmological topologically massive gravity at the chiral point with 7

physical reasons to discard this mode, this theory is unstable. To address this issue we prove that the mode is not pure gauge and that its negative energy is time-independent and finite. The isometry generators L0 and L¯0

have non-unitary matrix representations like in logarithmic CFT. While the new mode obeys boundary conditions that are slightly weaker than the ones by Brown and Henneaux, its fall-off behavior is compatible with spacetime being asymptotically AdS³. We employ holographic renormalization to show that the variational principle is well-defined. The corresponding Brown–York stress tensor is finite, traceless and conserved. Finally we address possibilities to eliminate the instability and prospects for chiral gravity.

Ref. [10] showed that the graviton mode discovered above persists non-perturbatively. In addition we provided a novel reformulation of cos- mological topologically massive gravity in this work. Here is its abstract:

Wolfgang Kummer was a pioneer of two-dimensional gravity and a strong advocate of the first order formulation in terms of Cartan variables. In the present work we apply Wolfgang Kummer’s philosophy, the “Vienna School approach”, to a specific three-dimensional model of gravity, cosmological topo- logically massive gravity at the chiral point. Exploiting a new Chern–Simons representation we perform a canonical analysis. The dimension of the phys- ical phase space is two per point, and thus the theory exhibits a local physical degree of freedom, the topologically massive graviton.

Ref. [11] showed that there are boundary conditions consistent with asymptotic AdS behavior that encompass the graviton excitations discovered above. These boundary conditions relax the famous Brown–

Henneaux boundary conditions, but the asymptotic symmetry algebra con- sists of two copies of the Virasoro algebra, just like in 3D Einstein gravity with Brown–Henneaux boundary conditions. Here is its abstract:

We show that cosmological topologically massive gravity at the chiral point al- lows not only Brown–Henneaux boundary conditions as consistent boundary conditions, but slightly more general ones which encompass the logarithmic primary found in 0805.2610 as well as all its descendants.

Ref. [12] provided a novel way to derive holographic counterterms, namely from supersymmetry. This paper was written together with the co-inventor of supergravity and provides the basis of current collabora- tion with Anton Rebhan and Peter van Nieuwenhuizen. Here is its abstract:

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theory finite, coincide with the counterterms that are needed to make the ac- tion supersymmetric without imposing any boundary conditions on the fields.

Ref. [13] corroborates the logarithmic CFT conjecture made above by calculating 2- and 3-point correlators. Besides these slightly lengthy calculations the paper contains also a lot of material that will be useful for fu- ture purposes, like the construction of all normalizable and non-normalizable modes in cosmological topologically massive gravity on an AdS background.

Here is its abstract:

For cosmological topologically massive gravity at the chiral point we calcu- late momentum space 2- and 3-point correlators of operators in the postu- lated dual CFT on the cylinder. These operators are sourced by the bulk and boundary gravitons. Our correlators are fully consistent with the proposal that cosmological topologically massive gravity at the chiral point is dual to a logarithmic CFT. In the process we give a complete classification of nor- malizable and non-normalizeable left, right and logarithmic solutions to the linearized equations of motion in global AdS³.

Ref. [14] shows that also a related 3D gravity theory, New Massive Gravity, can be dual to a logarithmic CFT for a certain tuning of parameters. This paper was written with one of the inventors of New Massive Gravity, Olaf Hohm. Here is its abstract:

We calculate 2-point correlators for New Massive Gravity at the chiral point and find that they behave precisely as those of a logarithmic conformal field theory, which is characterized in addition to the central charges cL =cR= 0 by “new anomalies” bL = bR = −σ_G^12ℓ

N, where σ is the sign of the Einstein–

Hilbert term, ℓ the AdS radius and GN Newton’s constant.

Ref. [15] is an invited proceedings contribution. It summarizes the recent evidence that led to the proposals for specific gravity duals of logarith- mic CFTs and provides an outlook towards condensed matter applications.

Here is its abstract:

Logarithmic conformal field theories with vanishing central charge describe systems with quenched disorder, percolation or dilute self-avoiding polymers.

In these theories the energy momentum tensor acquires a logarithmic part- ner. In this talk we address the construction of possible gravity duals for these logarithmic conformal field theories and present two viable candidates for such duals, namely theories of massive gravity in three dimensions at a chiral point.

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Ertl, my post-doc Niklas Johansson, my visitors Branislav Cvetkovic, Olaf Hohm, Roman Jackiw, Ivo Sachs and Dima Vassilevich, as well as with ad- ditional international collaborators. In addition to support from my START project Y435-N16 my research group on 3D gravity is supported by the FWF project P21927-N16.

Main goal of this line of research: constructing a consistent 3D theory of quantum gravity and applying the AdS/CFT correspon- dence to provide novel gravity duals for certain 2D conformal field theories. The elusive theory of quantum gravity is often called “the holy grail of theoretical physics”. Of all the attempts to quantize gravity, string theory is the best developed theory, but it is still poorly understood and poorly supported by experiment. Thus, it is prudent to consider simpler approaches to quantum gravity, where results can be expected on a rea- sonable time-scale. Lower-dimensional models of gravity provide such an approach. In the past 20 years studies of 2-dimensional gravity led to nu- merous exciting results for classical and quantum black holes. However, the simplicity of these black hole models eliminates two important features of higher-dimensional gravity: they do not contain gravitons, and they lack a good analog for the horizon area of black holes. Both these deficiencies can be remedied by considering a suitable model of gravity in three dimensions, like cosmological topologically massive gravity. Even if this ambitious goal might be out of reach, the study of 3D gravity can lead to useful applications in unexpected fields, such as condensed matter physics: the AdS/CFT cor- respondence relates gravity theories on AdS to conformal field theories that live on the boundary of AdS. Currently we continue to unravel novel features of purported CFT duals to cosmological topologically massive gravity as well as possible condensed matter physics applications, e.g. in the description of systems with quenched disorder.

10

Selected literature on 2D gravity

[1] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253;

invited review article. p. 15 ff.

[2] D. Grumiller and W. Kummer, “The classical solutions of the dimensionally reduced gravitational Chern-Simons theory,” Annals Phys. 308 (2003) 211 hep-th/0306036. p. 159 ff.

[3] L. Bergamin, D. Grumiller and W. Kummer, “Supersymmetric black holes are extremal and bald in 2-D dilaton supergravity,” J. Phys.A37 (2004) 3881 hep-th/0310006. p. 171 ff.

[4] D. Grumiller, “Virtual black holes and the S-matrix,” Int. J. Mod.

Phys. D13(2004) 1973 hep-th/0409231; invited review article. p. 201 ff.

[5] D. Grumiller, “An action for the exact string black hole,” JHEP 0505 (2005) 028 hep-th/0501208. p. 229 ff.

[6] D. Grumiller and R. Meyer, “Ramifications of lineland,” Turk. J. Phys.

30 (2006) 349–378, hep-th/0604049; invited review article. p. 271 ff.

[7] D. Grumiller and R. Jackiw, “Duality in 2-dimensional dilaton

gravity,” Phys. Lett. B642(2006) 530–534, hep-th/0609197. p. 317 ff.

[8] D. Grumiller and R. McNees, “Thermodynamics of black holes in two (and higher) dimensions,” JHEP 04 (2007) 074, hep-th/0703230.

p. 331 ff.

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[9] D. Grumiller and N. Johansson, “Instability in cosmological

topologically massive gravity at the chiral point,” JHEP 07 (2008) 134; arXiv:0805.2610 [hep-th]. p. 397 ff.

[10] D. Grumiller, R. Jackiw and N. Johansson, “Canonical analysis of cosmological topologically massive gravity at the chiral point,”

contribution toWolfgang Kummer Memorial Volume, World Scientific;

arXiv:0806.4185 [hep-th]. p. 417 ff.

[11] D. Grumiller and N. Johansson, “Consistent boundary conditions for cosmological topologically massive gravity at the chiral point,”Int. J.

Mod. Phys. D17 (2009) 2367;arXiv:0808.2575 [hep-th]. p. 429 ff.

[12] D. Grumiller and P. van Nieuwenhuizen, “Holographic counterterms from local supersymmetry without boundary conditions,”Phys. Lett.

B682(2010) 462; arXiv:0908.3486 [hep-th]. p. 435 ff.

[13] D. Grumiller and I. Sachs, “AdS³/LCFT² – Correlators in Cosmological Topologically Massive Gravity,” arXiv:0910.5241 [hep-th], to be published in JHEP (2010). p. 445 ff.

[14] D. Grumiller and O. Hohm, “AdS³/LCFT² – Correlators in New Massive Gravity,” arXiv:0911.4274 [hep-th], submitted toPhys.

Lett. B (2010). p. 509 ff.

[15] D. Grumiller and N. Johansson, “Gravity duals for logarithmic conformal field theories,”arXiv:1001.0002 [hep-th], submitted to J. Phys. Conf. Ser. (2010). p. 519 ff.

Comments on the papers constituting this Habilitationsschrift:

• In all publications above I have contributed as main author.

• Papers [1–5,7–9,11–12] are published in peer reviewed international journals. Paper [6] is published as a peer reviewed proceedings

contribution. Paper [10] published as a book contribution. Paper [13]

is accepted for publication inJHEP. Paper [14] was submitted for publication to PLB. Paper [15] is an invited proceedings contribution.

• Page numbers at the end of each reference direct to the starting page of the corresponding paper in this

Habilitationsschrift.

12

Papers on 2D Gravity

D. Grumiller

,W. Kummer

,D.V. Vassilevich

aInstitut f¨ur Theoretische Physik, TU Wien, Wiedner Hauptstr. 8–10, A-1040 Wien, Austria

bInstitut f¨ur Theoretische Physik, Universit¨at Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany

cV.A. Fock Insitute of Physics, St. Petersburg University, 198904 St. Petersburg, Russia

Abstract

The study of general two dimensional models of gravity allows to tackle basic ques- tions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quan- tum case. In the last decade new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quan- tum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject.

Key words: dilaton gravity, quantum gravity, black holes, two dimensional models PACS:04.60.-w, 04.60.Ds, 04.60.Gw, 04.60.Kz, 04.70.-s, 04.70.Bw, 04.70.Dy, 11.10.Lm, 97.60.Lf

Email addresses: grumil@hep.itp.tuwien.ac.at(D. Grumiller),

wkummer@tph.tuwien.ac.at(W. Kummer),vassil@itp.uni-leipzig.de(D.V.

Vassilevich).

1.1 Structure of this review . . . 23 1.2 Differential geometry . . . 24 1.2.1 Short primer for general dimensions . . . 24 1.2.2 Two dimensions . . . 30

2 Models in 1 + 1 Dimensions 33

2.1 Generalized Dilaton Theories . . . 33 2.1.1 Spherically reduced gravity . . . 33 2.1.2 Dilaton gravity from strings . . . 34 2.1.3 Generalized dilaton theories – the action . . . 35 2.1.4 Conformally related theories . . . 37 2.2 Equivalence to first-order formalism . . . 39 2.3 Relation to Poisson-Sigma models . . . 42

3 General classical treatment 46

3.1 All classical solutions . . . 47 3.2 Global structure . . . 52 3.2.1 Schwarzschild metric . . . 54 3.2.2 More general cases . . . 57 3.3 Black hole in Minkowski, Rindler or de Sitter space . . . 59

4 Additional fields 63

4.1 Dilaton-Yang-Mills Theory . . . 63 4.2 Dilaton Supergravity . . . 64 4.3 Dilaton gravity with matter . . . 71 4.3.1 Scalar and fermionic matter, quintessence . . . 71 4.3.2 Exact solutions – conservation law for geometry and matter . 72

5 Energy considerations 76

5.1 ADM mass and quasilocal energy . . . 76 5.2 Conservation laws . . . 80 5.3 Symmetries . . . 81

6 Hawking radiation 84

6.1 Minimally coupled scalars . . . 84 6.2 Non-minimally coupled scalars . . . 89 7 Nonperturbative path integral quantization 94 7.1 Constraint algebra . . . 95 7.2 Path integral quantization . . . 99 7.3 Path integral without matter . . . 101 7.4 Path integral with matter . . . 105 7.4.1 General formalism . . . 105 7.4.2 Perturbation theory . . . 105 7.4.3 Exact path integral with matter . . . 107

8.2 Effective line element . . . 112 8.3 Virtual black hole . . . 113 8.4 Non-local φ⁴ vertices . . . 114 8.5 Scattering amplitude . . . 115 8.6 Implications for the information paradox . . . 117

9 Canonical quantization 119

10 Conclusions and discussion 122

Acknowledgement 125

A Spherical reduction of the curvature 2-form 126

B Heat kernel expansion 127

References 131

List of Figures

2.1 A selection of dilaton theories . . . 42 3.1 Killing norm for Schwarzschild metric . . . 55 3.2 Derivative of the second null direction . . . 55 3.3 Second null direction . . . 55 3.4 Conformal coordinates with “compression factor” . . . 56 3.5 Reorientation of Fig. 3.4: patch A . . . 56 3.6 Mirror image of Fig. 3.5: patch B . . . 57 3.7 Further flips: patchesC and D . . . 57 3.8 CP diagram for the Schwarzschild solution . . . 57 3.9 Basic patch of Reissner-Nordstr¨om metric . . . 58 3.10 Penrose diagram for Reissner-Nordstr¨om metric . . . 58 3.11 A possible RN-kink . . . 59 3.12 “Phase” diagram of CP diagrams . . . 61 8.1 CP diagram of the VBH . . . 113 8.2 TotalV⁽⁴⁾-vertex with outer legs . . . 114 8.3 Kinematic plot of s-wave cross-section dσ/dα . . . 116

1 Introduction

The fundamental difficulties encountered in the numerous attempts to merge quantum theory with General Relativity by now are well-known even far outside the narrow circle of specialists in these fields. Despite many valiant efforts and new approaches like loop quantum gravity [371] or string theory¹ a final solution is not in sight. However, even many special questions search an answer².

Of course, at energies which will be accessible experimentally in the fore- seeable future, due to the smallness of Newton’s constant, respectively the large value of the Planck mass, an effective quantum theory of gravity can be constructed [129] in a standard way which in its infrared asymptotical regime as an effective quantum theory may well describe our low energy world. Its extremely small corrections to classical General Relativity (GR) are in full agreement with experimental limits [436]. However, the fact that Newton’s constant carries a dimension, inevitably makes perturbative quantum gravity inconsistent at energies of the order of the Planck mass.

In a more technical language, starting from a fixed classical background, already a long time ago perturbation theory has shown that although pure gravity is one-loop renormalizable [404] this renormalizability breaks down at two loops [188], but already at one-loop when matter interactions are taken into account. Supergravity was only able to push the onset of non- renormalizability to higher loop order (cf. e.g. [224,38,119]). It is often argued that a full treatment of the metric, including non-perturbative effects from the backreaction of matter, may solve the problem but to this day this remains a conjecture³. A basic conceptual problem of a theory like gravity is the double role of geometric variables which are not only fields but also determine the (dynamical) background upon which the physical variables live. This is e.g. of special importance for the uncertainty relation at energies above the Planck scale leading to Wheeler’s notion of “space-time-foam” [434].

Another question which has baffled theorists is the problem of time. In ordinary quantum mechanics the time variable is set apart from the “observ- ables”, whereas in the straightforward quantum formulation of gravity (the so-called Wheeler-deWitt equation [435, 121]) a variable like time must be in- troduced more or less by hand through “time-slicing”, a multi-fingered time etc. [232]. Already at the classical level of GR “time” and “space” change their roles when passing through a horizon which leads again to considerable complications in a Hamiltonian approach [10, 272].

Measuring the “observables” of usual quantum mechanics one realizes that the genuine measurement process is related always to a determination of

1 The recent book [360] can be recommended.

2 A brief history of quantum gravity can be found in ref. [371].

3 For a recent argument in favor of this conjecture using Weinberg’s argument of

“asymptotic safety” cf. e.g. [296].

the matrix element of some scattering operator with asymptotically defined ingoing and outgoing states. For a gauge theory like gravity, existing proofs of gauge-independence for the S-matrix [279] may be applicable for asymptoti- cally flat quantum gravity systems. But the problem of other experimentally accessible (gauge independent!) genuine observables is open, when the dynam- ics of the geometry comes into play in a nontrivial manner, affecting e.g. the notion what is meant by asymptotics.

The quantum properties of black holes (BH) still pose many questions.

Because of the emission of Hawking radiation [211,412], a semi-classical effect, a BH should successively lose energy. If there is no remnant of its previous existence at the end of its lifetime, the information of pure states swallowed by it will have only turned into the mixed state of Hawking radiation, violating basic notions of quantum mechanics. Thus, of special interest (and outside the range of methods based upon the fixed background of a large BH) are the last stages of BH evaporation.

Other open problems – related to BH physics and more generally to quan- tum gravity – have been the virtual BH appearing as an intermediate stage in scattering processes, the (non-)existence of a well-defined S-matrix andCP T (non-)invariance. When the metric of the BH is quantized its fluctuations may include “negative” volumes. Should those fluctuations be allowed or excluded?

The intuitive notion of “space-time foam” seems to suggest quantum gravity induced topology fluctuations. Is it possible to extract such processes from a model without ad hoc assumptions? From experience of quantum field theory in Minkowski space one may hope that a classical singularity like the one in the Schwarzschild BH may be eliminated by quantum effects – possibly at the price of a necessary renormalization procedure. Of course, the latter may just reflect the fact that interactions with further fields (e.g. other modes in string theory) are not taken into account properly. Can this hope be fulfilled?

In attempts to find answers to these questions it seems very reasonable to always try to proceed as far as possible with the known laws of quantum mechanics applied to GR. This is extremely difficult⁴ inD= 4. Therefore, for many years a rich literature developed on lower dimensional models of gravity.

The 2D Einstein-Hilbert action is just the Gauss-Bonnet term. Therefore, intrinsically 2Dmodels are locally trivial and a further structure is introduced.

This is provided by the dilaton field which naturally arises in all sorts of compactifications from higher dimensions. Such models, the most prominent being the one of Jackiw and Teitelboim (JT), were thoroughly investigated during the 1980-s [22, 123, 405, 122, 124, 238, 250, 251, 312, 388]. An excellent summary (containing also a more comprehensive list of references on literature before 1988) is contained in the textbook of Brown [59]. Among those models spherically reduced gravity (SRG), the truncation of D = 4 gravity to its s-wave part, possesses perhaps the most direct physical motivation. One can either treat this system directly in D = 4 and impose spherical symmetry in

4 A recent survey of the present situation is the one of Carlip [79].

the equations of motion (e.o.m.-s) [276] or impose spherical symmetry already in the action [36, 412, 33, 409, 205, 324, 407, 244, 276, 295, 195], thus obtaining a dilaton theory⁵. Classically, both approaches are equivalent.

The rekindled interest in generalized dilaton theories in D = 2 (hence- forth GDTs) started in the early 1990-s, triggered by the string inspired [310, 137, 443, 127, 316, 233, 117, 254] dilaton black hole model⁶, studied in the influential paper of Callan, Giddings, Harvey and Strominger (CGHS) [71].

At approximately the same time it was realized that 2D dilaton gravity can be treated as a non-linear gauge-theory [426, 230].

As already suggested by earlier work, all GDTs considered so far could be extracted from the dilaton action [373, 349]

L^(dil) =

Z

d²x√

−g

"

XR

2 − U(X)

2 (∇X)²+V(X)

#

+L^(m), (1.1) whereRis the Ricci-scalar,Xthe dilaton,U(X) andV(X) arbitrary functions thereof, g is the determinant of the metric g_µν, and L^(m) contains eventual matter fields.

When U(X) = 0 the e.o.m. for the dilaton from (1.1) is algebraic. For invertible V^′(X) the dilaton field can be eliminated altogether, and the La- grangian density is given by an arbitrary function of the Ricci-scalar. A recent review on the classical solution of such models is ref. [381]. In comparison with that, the literature on such models generalized to depend also⁷ on torsionT^a is relatively scarce. It mainly consists of elaborations based upon a theory pro- posed by Katanaev and Volovich (KV) which is quadratic in curvature and torsion [250, 251], also known as “Poincar´e gauge gravity” [322].

A common feature of these classical treatments of models with and with- out torsion is the almost exclusive use⁸ of the gauge-fixing for the D = 2 metric familiar from string theory, namely the conformal gauge. Then the e.o.m.-s become complicated partial differential equations. The determination of the solutions, which turns out to be always possible in the matterless case (L^(m) = 0 in (1.1)), for nontrivial dilaton field dependence usually requires considerable mathematical effort. The same had been true for the first papers on theories with torsion [250, 251]. However, in that context it was realized soon that gauge-fixing is not necessary, because the invariant quantitiesRand T^aTathemselves may be taken as variables in the KV-model [390,389,391,392].

This approach has been extended to general theories with torsion⁹.

5 The dilaton appears due to the “warped product” structure of the metric. For details of the spherical reduction procedure we refer to appendix A.

6 A textbook-like discussion of this model can be found in refs. [183, 399].

7 For the definition of the Lorentz scalar formed by torsion and of the curvature scalar, both expressed in terms of Cartan variables zweibeinee^a_µand spin connection ω_µ^ab we refer to sect. 1.2 below.

8 A notable exception is Polyakov [362].

9 A recent review of this approach is provided by Obukhov and Hehl [348].

As a matter of fact, in GR many other gauge-fixings for the metric have been well-known for a long time: the Eddington-Finkelstein (EF) gauge, the Painlev´e-Gullstrand gauge, the Lemaıtre gauge etc. . As compared to the “di- agonal” gauges like the conformal and the Schwarzschild type gauge, they possess the advantage that coordinate singularities can be avoided, i.e. the singularities in those metrics are essentially related to the “physical” ones in the curvature. It was shown for the first time in [291] that the use of a temporal gauge for the Cartan variables (cf. eq. (3.3) below) in the (matterless) KV- model made the solution extremely simple. This gauge corresponds to the EF gauge for the metric. Soon afterwards it was realized that the solution could be obtained even without previous gauge-fixing, either by guessing the Darboux coordinates [377] or by direct solution of the e.o.m.-s [290] (cf. sect. 3.1). Then the temporal gauge of [291] merely represents the most natural gauge fixing within this gauge-independent setting. The basis of these results had been a first order formulation of D = 2 covariant theories by means of a covariant Hamiltonian action in terms of the Cartan variables and further auxiliary fields X^a which (beside the dilaton field X) take the role of canonical momenta (cf.

eq. (2.17) below). They cover a very general class of theories comprising not only the KV-model, but also more general theories with torsion¹⁰. The most attractive feature of theories of type (2.17) is that an important subclass of them is in a one-to-one correspondence with the GDT-s (1.1). This dynamical equivalence, including the essential feature that also the global properties are exactly identical, seems to have been noticed first in [248] and used extensively in studies of the corresponding quantum theory [281, 285, 284].

Generalizing the formulation (2.17) to the much more comprehensive class of “Poisson-Sigma models” [379, 396] on the one hand helped to explain the deeper reasons of the advantages from the use of the first oder version, on the other hand led to very interesting applications in other fields [3], including especially also string theory [382, 387]. Recently this approach was shown to represent a very direct route to 2Ddilaton supergravity [140] without auxiliary fields.

Apart from the dilaton BH [71] where an exact (classical) solution is possible also when matter is included, general solutions for generic D = 2 gravity theories with matter cannot be obtained. This has been possible only in restricted cases, namely when fermionic matter is chiral¹¹ [278] or when the interaction with (anti)selfdual scalar matter is considered [356].

Semi-classical treatments of GDT-s take the one loop correction from matter into account when the classical e.o.m.-s are solved. They have been used mainly in the CGHS-model and its generalizations [41, 117, 374, 44, 115, 147, 256, 446, 445, 209, 210, 423]. In our present report we concentrate only upon Hawking radiation as a quantum effect of matter on a fixed (classical)

10In that case there is the restriction that it must be possible to eliminate all auxiliary fieldsX^a and X (see sect. 2.1.3).

11This solution was rediscovered in ref. [393].

geometrical background, because just during the last years interesting insight has been obtained there, although by no means all problems have been settled.

Finally we turn to the full quantization of GDTs. It was believed by several authors (cf. e.g. [373, 349, 242, 139, 138]) that even in the absence of interactions with matter nontrivial quantum corrections exist and can be com- puted by a perturbative path integral on some fixed background. Again the evaluation in the temporal gauge [291], at first for the KV-model showed that the use of other gauges just obscures a very simple mechanism. Actually all divergent counter-terms can be absorbed into one compact expression. After subtracting that in the absence of matter the solution of the classical theory represents an exact “quantum” result. Later this perturbative argument has been reformulated as an exact path integral, first again for the KV-model [204]

and then for general theories of gravity in D= 2 [281, 285, 284, 196, 157, 199].

In our present review we concentrate on the path integral approach, with Dirac quantization only referred to for sake of comparison. In any case, the common starting point is the Hamiltonian analysis which in a theory for- mulated in terms of Cartan variables in D = 2 possesses substantial techni- cal advantages. The constraints, even in the presence of matter interactions, form an algebra with momentum-dependent structure constants. Despite that nonlinearity the simplest version of the Batalin-Vilkovisky procedure [27] suf- fices, namely the one also applicable to ordinary nonabelian gauge theories in Minkowski space. With a temporal gauge fixing for the Cartan variables also used in the quantized theory, the geometric part of the action yields the exact path integral. Possible background geometries appear naturally as ho- mogeneous solutions of differential equations which coincide with the classical ones, reflecting “local quantum triviality” of 2D gravity theories in the ab- sence of matter, a property which had been observed as well before in the Dirac quantization of the KV-model [377].

These features are very difficult to locate in the GDT-formulation (1.1), but become evident in the equivalent first order version with a “Hamiltonian”

action.

Of course, non-renormalizability persists in the perturbation expansion when the matter fields are integrated out. But as an effective theory in cases like spherically reduced gravity, specific processes can be calculated, relying on the (gauge-independent) concept of S-matrix elements. With this method, scattering of s-waves in spherically reduced gravity has provided a very di- rect way to create a “virtual” BH as an intermediate state without further assumptions [157].

The structure of our present report is determined essentially by the ap- proach described in the last paragraphs. One reason is the fact that a very com- prehensive overview of very general classical and quantum theories in D = 2 is made possible in this manner. Also a presentation seems to be overdue in which results, scattered now among many different original papers can be inte- grated into a coherent picture. Parallel developments and differences to other approaches will be included in the appropriate places.

1.1 Structure of this review

This review is organized as follows:

• Section 1 in its remaining part contains a short primer on differential geom- etry (with special emphasis on D = 2). En passant most of our notations are fixed in that subsection.

• Section 2 motivates the study of GDTs and introduces its action in the three most frequently used forms (dilaton action, first order action, and Poisson-Sigma action) and describes the relations between them.

• Section 3 gives all classical solutions of GDTs in the absence of matter. The global structure of such theories is discussed using Schwarzschild space- time as a simple example. As a further illustration we consider a family of dilaton models describing a single black hole in Minkowski, Rindler or de Sitter space-time.

• Section 4 extends the discussion to additional gauge-fields, supergravity and (bosonic or fermionic) matter fields.

• Section 5 considers the role of energy in GDTs. In particular, the ADM mass, quasilocal energy, an absolute conservation law and its corresponding N¨other symmetry are discussed.

• Section 6 leaves the classical realm providing a concise treatment of (semi- classical) Hawking radiation for minimally and non-minimally coupled mat- ter.

• Section 7 is devoted to non-perturbative path integral quantization of the geometric sector of GDTs with (scalar) matter, giving rise to a non-local and non-polynomial effective action depending solely on the matter fields and external sources. The matter sector is treated perturbatively.

• Section 8 shows some consequences of the previously developed perturbation theory: the virtual black hole phenomenon, the appearance of non-local vertices, andS-matrix elements fors-wave gravitational scattering.

• Section 9 describes the status of Dirac quantization for a typical example of that approach.

• Section 10 concludes with a brief summary and an outlook regarding open questions.

• Appendix A recalls the spherical reduction procedure in the Cartan formal- ism.

• Appendix B collects some basic properties of the heat kernel expansion needed in Section 6.

Several topics are closely related to the subject of this review, but are not included:

(1) Various calculations and explanations of the BH entropy [169, 355] be- came a large and rather independent field of research which shows, how- ever, overlaps [165,171] with the general treatment of the dilaton theories presented in this review. We do not cover approaches which imply fur- ther physical assumptions which transgress the orthodox application of quantum theory to gravity [34, 35, 43, 24, 31].

(2) The ideas of the holographic principle [403,400] and of the AdS/CFT cor- respondence [309, 200, 444] are now being actively applied to BH physics (see, e.g. [375] and references therein).

(3) There exist different approaches to integrability of gravity models in two dimensions [338, 269, 268, 339]. In particular, a rather sophisticated tech- nique has been applied to solve the effective 2D models emerging after toroidal reduction (instead of the spherical reduction considered in this re- view) of the four-dimensional Einstein equations [39,417]. Recently again interesting developments should be noted in Liouville gravity [151, 406].

Some relations between 2D dilaton gravity and the theory of solitons were discussed in [70, 336].

Each of these topics deserves a separate review, and in some cases such reviews exist. Therefore, we have restricted ourselves in those fields to just a few (somewhat randomly selected) references which hopefully will permit further orientation.

1.2 Differential geometry

1.2.1 Short primer for general dimensions

In the comprehensive approach advocated for D = 2 gravity the use of Cartan variables (zweibeine, spin-connection) plays a pivotal role. As an introduction and in order to fix our notations we shall review briefly this formalism. For details we refer to the mathematical literature (cf. e.g. [334]).

On a manifold with D dimensions in each point one introduces viel- beine e^µ_a(x), where Greek indices refer to the (holonomic) coordinates x^µ = (x⁰, x¹, . . . , x^D−1) and Latin indices denote the ones related to a (local) Lorentz frame with metric η= diag (1,−1, . . . ,−1). The dual vector space is spanned by the inverse vielbeine¹² e^a_µ(x):

e^µae^b_µ =η^ab (1.2)

SO(1, D−1) matrices L^ab(x) of the (local) Lorentz transformations obey

L^acLbc = δ_b^a. (1.3)

A Lorentz vector V^a=e^a_µV^µ transforms under local Lorentz transformations as

V^′a(x) =L^ab(x)V^b(x) (1.4) This implies a covariant derivative

(Dµ)^a_b = δ_b^a∂µ+ωµa

b , (1.5)

12For simplicity we shall use indiscriminately the term “vielbein” for the vielbein, the inverse vielbein and the dual basis of 1-forms (the components of which are given by the inverse vielbein) whenever the meaning is clear either from the context or from the position of indices.

if the spin-connection ωµa

b is introduced as the appropriate gauge field with transformation

ω^′_µ^ab =−Lbd (∂µL^ad) +L^acωµc

dLbd. (1.6)

The infinitesimal version of (1.6) follows from L^ab = δ^ab +l^ab +O(l²) where l^ab =−lba .

Formally also diffeomorphisms

¯

x^µ(x) =x^µ−ξ^µ(x) +O(ξ²) (1.7) can be interpreted, at least locally, as gauge transformations, when the Lie variation is employed which implies a transformation referring to the same point. In

∂x¯^µ

∂x^ν =δ^µ_ν −ξ^µ_{, ν} , ∂x^ν

∂x¯^µ =δ^ν_µ+ξ_{, µ}^ν (1.8) partial derivatives with respect tox^ν have been abbreviated by the index after a comma.

For instance, for the Lie variation of a tensor of first order ¯Vµ(¯x) =

∂x^ν

∂¯x^µVν(x) one obtains

δξVµ(x) = ¯Vµ(x)−Vµ(x) =ξ^ν,µVν +ξ^νVµ,ν . (1.9) For the dual to the tangential space, e.g. V^µ∂µ = V^µ(∂µx¯^ν) ¯∂ν = ¯V^µ∂¯µ one derives the analogous transformation

δξV^µ= ¯V^µ(x)−V^µ(x) =−ξ^µ,νV^ν+ξ^νV^µ,ν . (1.10) The metric gµν in the line element is a quadratic expression of the viel- beine

(ds)²=g_µνdx^µdx^ν =e^a_µe^b_νη_abdx^µdx^ν, (1.11) and, therefore, a less elementary variable. Also the reparametrization invariant volume element

q(−)^D−1g d^Dx=^q(−)^D−1detgµνd^Dx=

=^q(−)^D−1(dete^a_µ)² detη d^Dx=|dete^a_µ|d^Dx =|e|d^Dx (1.12) is of polynomial form if expressed in vielbein components.

The advantage of the form calculus [334] is that diffeomorphism invariance is automatically implied, when the Cartan variables are converted into one forms

e^a_µ → e^a =e^a_µdx^µ, ωµa

b → ω^ab =ωµa

bdx^µ (1.13) which are special cases of p-forms

Ωp = 1

p!Ωµ1... µpdx^µ1 ∧dx^µ2 ∧ · · · ∧dx^µp . (1.14) Due to the antisymmetry of the wedge productdx^µ∧dx^ν =dx^µ⊗dx^ν−dx^ν⊗ dx^µ = −dx^ν ∧dx^µ all totally antisymmetric tensors Ωµ1...µp are described in

this way. Clearly Ωp = 0 forp > D. The action of the (p+q)-form Ωq∧Ωq on p+q vectors is defined by

Ωp∧Ξq(V1, . . . , Vp+q) = 1 p!q!

X

π

δπΩ(Vπ(1), . . . , Vπ(p))Ξ(Vπ(p+1), . . . , Vπ(p+q)), (1.15) where the sum is taken over all permutations π of 1, . . . , p+q and δπ is +1 for an even number of transpositions and −1 for an odd number of transpo- sitions. It is convenient at this point to introduce the condensed notation for (anti)symmetrization:

α[µ1...µp]:= 1 p!

X

π

δπαa_π(1)...a_π(p), σ(µ1...µp) := 1 p!

X

π

σa_π(1)...a_π(p), (1.16) where the sum is taken over all permutations π of 1, . . . , p and δπ is defined as before. In the volume form

Ωp=D = 1

D!a[µ1...µD]˜ǫ^{µ1... µD}d^Dx= 1

D!a[µ1... µD]|e|ǫ^{µ1... µD}d^Dx (1.17) the product of differentials must be proportional to the totally antisymmetric Levi-Civit´a symbol ˜ǫ^01...(D−1) =−1 or, alternatively, to the tensor ǫ =|e|⁻¹˜ǫ (cf. (1.12)). The integral of the volume form^R_MDΩD on the manifoldMD con- tains the scalar a =a_{[µ1... µD]}ǫ^{µ1... µD} which is the starting point to construct diffeomorphism invariant Lagrangians.

By means of the metric (1.11) a mixed ǫ-tensor

ǫ_{µ1... µp}^{µp+1... µp+q} = g_µ1ν1g_µ2ν2 . . . g_µpνpǫ^{ν1... νpµp+1...µp+q} (1.18) can be defined which allows the introduction of the Hodge dual of Ωp as a D−p form

∗Ωp = Ω^′D−p = 1

p!(D−p)!ǫµ1... µD−p

ν1... νp Ων1... νp dx^µ1 ∧ . . . ∧dx^µD−p . (1.19) In D= even and for Lorentzian signature we obtain for a p-form

∗ ∗Ωp = (−1)^p+1Ωp. (1.20) The exterior differential one form d = dx^µ∂µ with d² = 0 increases the form degree by one:

dΩp = 1

p!∂µΩµ1...µpdx^µ∧dx^µ1 ∧ · · · ∧dx^µp (1.21) Onto a product of formsd acts as

d(Ωp∧Ωq) =dΩp∧Ωq+ (−1)^pΩp ∧dΩq . (1.22)

We shall need little else from the form calculus [334] except the Poincar´e Lemma which says that for a closed form, obeying dΩp = 0, in a certain (“star-shaped”) neighborhood of a pointx^µ on a manifoldM, Ωp is exact, i.e.

can be written as Ω_p =dΩ^′_p−1.

In order to simplify our notation we shall drop the ∧ symbol whenever the meaning is clear from the context.

The Cartan variables expressed as one forms (1.13) in view of their Lorentz-tensor properties are examples of algebra valued forms. This is also the case for the covariant derivative (1.5), now written as

D^ab = δ_b^ad+ω^ab , (1.23) when it acts on a Lorentz vector.

From (1.13) and (1.23) the two natural quantities to be defined on a manifold are the torsion two-form

T^a = D^abe^b (1.24)

(“First Cartan’s structure equation”) and the curvature two-form

R^ab =D^acω^cb (1.25)

(“Second Cartan’s structure equation”). From (1.23) immediately follows (D²)^ab =D^acD^cb =R^ab, (1.26) Bianchi’s first identity. Using (1.26)D³ can be written in two equivalent ways, D^a_bR^b_c−R^a_bD^b_c = 0, (1.27) corresponding to Bianchi’s second identity

(dR^ab) +ω^acR^cb+ω^bcR^ac =: (DR)^ab = 0 . (1.28) The l.h.s. defines the action of the covariant derivative (1.23) onR^ab, a Lorentz tensor with two indices. The brackets indicate that those derivatives only act upon the quantity R and not further to the right. The structure equations together with the Bianchi identities show that the covariant action for any gravity action inD dimensions depending one^a, ω^ab can be constructed as a volume form depending solely onR^ab, T^aande^a. The most prominent example is Einstein gravity in D = 4 [136, 135] which in the Palatini formulation reads [352]

LHEP ∝

Z

M4

R^abe^ce^dǫabcd , (1.29) having used the definition ǫ_abcd = ǫ_µνστe^µ_ae^ν_be^σ_ce^τ_d. The condition of vanishing torsionT^a = 0 for this special case already follows from varying ω^ab indepen- dently in (1.29).