- Email: [email protected]
The unification of quantum gravity
PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 88 (2000) 3-9
The unification
of quantum
www.elstier.nl/locate/npe
gravity
K.S. Stelle * The Blackett Laboratory, Imperial College Prince Consort Road, London SW7 2BW, UK
The subject of quantum gravity had its origins in the thirties, very soon after the great SUCcess with nonrelativistic quantum mechanics and the first, albeit noncovariant, steps towards QED. Gravity proved to be a problem of a different magnitude, however. The infinities associated with the dimensionful coupling constant immediately stood out as a central problem of the subject, but there was nonetheless hope that a sufficiently careful approach to quantization might resolve such difficulties. Subsequent approaches to the subject acquired different flavors, according to the individual backgrounds and predilections of the physicists involved. Relativists clearly took to geometrical methods, and readily adopted Dirac’s canonical quantization for gauge theories when this was developed in the 60s. On the other hand, quantum field theorists preferred to face the problems in a perturbative analysis about flat space. There have also been S-matrix approaches to the problem. All approaches, however, led to difficulties. Moreover these difficulties were sufficiently different in their natures that it became difficult to compare the successes and disappointments of the various approaches. As a result, there was a tendency for the different communities of researchers to lose contact, with insufficient interaction between the different programs. Meanwhile, quantum field theory itself made steady progress in other areas, with progressive understanding of the infinities problem and of the proper quantization procedure for gauge theories. Renormalization techniques, path-integral formulations, the introduction of ghost fields and their control through BRST methods: all these ele*Research No. 613
supported
in part
by PPARC
under
SPG
grant
0920-5632/00/$ - see front matter 0 2000 Elsevier Science B.\! PI1 SO920-5632(00)00747-7
ments became part of the central canon of techniques. The weak and electromagnetic interactions and the strong interactions are now successfully handled within this context, while renormalization group analysis and the study of solitons and instantons has opened the door to some nonperturbative understanding as well. On another track, in classical general relativity the understanding of physics in curved spacetime made a slow but progressive advance since the theory’s birth over 80 years ago. Many striking exact solutions were found, and techniques were developed for the analysis of global structure, the initial value problem, and the nature of spacetime singularities. Crucial but sparse observational data made it clear that general relativity was in full agreement with astronomical reality. With data from the binary pulsar, clear information about gravitational radiation, albeit indirect, is also now available. Accordingly, classical general relativity has now become one of the most securely grounded physical theories. But quantum gravity has remained elusive. In this contrasting world of approaches and developments, it is only recently that results from different directions have perhaps begun coalescing into a coherent view of the outstanding problems of quantum gravity. Within a field theoretic perspective, the exploration of supersymmetric theories began, initially with a view to applications in domains accessible to accelerator experiments. But, surprisingly, this program became entangled with the problems of gravity. This arose from the need for a realistic pattern of spontaneous supersymmetry breaking, a problem not at first obviously connected to gravity. But the constraints of supersymmetry are so strong that it turned out All rights reserved.
4
K.S. Stelle/Nuclear Physics B (Proc. Suppl.) 88 (2000) 3-9
that the only viable mechanisms for supersymmetry breaking involve coupling to supergravity. The discovery of supergravity in 1976, by Freedman, Ferrara and Van Nieuwenhuizen from a particle physics perspective but followed instantly by Deser and Zumino replying more to relativists’ concerns, was itself a major achievement. Later came understanding of its couplings and the construction of locally supersymmetric models, through the development of supersymmetric tensor calculus and superspace. This allowed the creation of particle physics models whose possible confirmation is now a main hope for the next generation of accelerators, notably the LHC at CERN. After supplying the mechanism for spontaneous supersymmetry breaking, however, supergravity has typically been retired to the sidelines, leaving the discussion to proceed in a context of flat-space renormalizable particle field theory. But this is clearly an arbitrary truncation of the full theory, perhaps understandable in the rush to understand supersymmetric extensions of the standard model, but not ultimately sustainable. Supergravity itself gave rise initially to high hopes for a solution to the infinities problem. Indeed, local supersymmetry proves to be a teaser here, for it does significantly increase the control over infinities, leading to cancellations at one and two loops in pure D = 4 supergravity theories with less than the maximal amount of supersymmetry, and even all the way through four loops (as we hear from Zvi Bern in this conference) in the maximally extended N = 8 theory. But, eventually, the infinities inevitably return, once more dashing hopes for a field-theoretic resolution of gravity’s difficulties. Out of the mysterious world of the S-matrix and the struggle to understand hadronic physics in the late 1960’s, however, another approach was brewing. The Veneziano amplitude gave a stunning description of the Regge resonances observed in hadronic data. No field theoretic derivation of this amplitude was known, but then it was realized by Koba, Nielsen, Susskind and others that a one-dimensional extended object, the relativistic string, could be the underlying dynamics for this. String theory progressed independently
from particle field theory for awhile, with a strong phenomenological bent. Although the Regge behavior of string-derived amplitudes agreed with observation, accelerator experiments at fixed angles eventually gave evidence of a hard “partonic” structure of the hadrons that was in serious disagreement with the very soft string interactions in this parameter region. With great regret, string theory fell out of favor as a theory of the strong interactions. Moreover, it had to be admitted that it had in fact a number of other embarrassments. For example, it required formulation in higher dimensional spacetimes, and also had, as a result of the quantum offset in the Regge trajectories, a strange zero mass state of spin two, which was quite unwanted in hadronic physics. Nonetheless, as happened with Yang-Mills nonabelian gauge theories, which also suffered an early disappointment in their initial intended application, a certain stubborn band of physicists continued to work on the theory anyway, despite its physical disappointments. Thus it was that Scherk, Schwarz and Green pressed forward into development of the superstring. The hadronic disappointment of the zero-mass spin two state now became the theory’s main hope: this could be the graviton. String theory, as a theory of extended objects, has a totally different ultraviolet structure from that of particle field theories. Although new types of infinities can arise in string amplitudes, thus complicating the story for awhile, it eventually became clear that superstring theory does manage to provide the sought-for cure to the infinities problem of quantum gravity. Moreover, superstring theories have correspondence limits to supergravity theories, making a link to previous ambitions for supersymmetry in the theory of gravitation. And these theories are also naturally formulated in ten dimensional spacetimes, just as many of the principal supergravities. But there is an exception: the most majestic supergravity, the D = 11 theory constructed by Cremmer, Julia and Scherk, which seemed to be left out of the string-related picture. The higher spacetime dimensions themselves were not so much of a worry: these could be dealt using the geometrically powerful techniques of Kaluza-Klein theory, compact-
K.S. Stelle/Nuclear Physics B (Proc. SuppL) 88 (2000) 3-9
ifying unobserved dimensions down to small sizes, but also providing an attractive possible explanation for "internal" symmetries. Together with a hopeful cure for the difficulties of quantum gravity, the subject was now getting more ambitious: it seemed t h a t only totally unified theories had a chance of working. This inevitably upped the stakes from a search for a theory of quantum gravity to that for a quantum theory of all matter and interactions. In 1984 it was discovered by Green and Schwarz that the successful resolution of certain difficulties with gauge anomalies restricted the viable candidate superstring theories down to a very small set with very particular (and phenomenologically interesting) gauge symmetries. After this, interest in superstrings rose to a fever pitch. A whole new generation of young physicists got into the act, bringing with them new techniques of conformal field theory and introducing sophisticated new mathematics into the theory, while leaving aside earlier field theoretic and relativistic methods. This was at last the birth of a perhaps viable approach to quantum gravity. Now that a candidate existed, one could really get down to see if anything like our physical world could be described by it. Here, however, an apparent setback was encountered. The supposed quasi-uniqueness of the viable superstring theories began to dissolve: to start with, there were in fact five candidate theories in the initial ten-dimensional spacetime; moreover the details of compactification down to D = 4 gave rise to a very large number of candidates, so this seemed hardly a unique theory after all. Enthusiasm was giving way to a certain degree of muddle. While all of this was going on, another renegade band, principally of supergravity enthusiasts, were pursuing a somewhat heretical line of enquiry. If one had to give up point particles for extended objects, then why stop at onedimensional strings? Why not consider membranes, or even higher-dimensional extended objects? Perhaps because this work was in any case rather disregarded by the majority string community, this band of practitioners felt free to make light of things and termed the more general extended objects "p-branes," so that a particle was
5
a 0-brane, a string was a 1-brane, etc. The downside was that resorting to intrinsic dimensions higher than one promptly reintroduced many of the difficulties that string theory was meant to cure, such as the dreaded infinities. But higherdimensional extended objects still held their fascination. In particular, they fitted in perfectly with the abandoned but still majestic D = 11 supergravity, which was found to interact most naturally with 2-branes and 5-branes. From the same perspective, the D = 10 theories naturally interacted with strings, to be sure, but also with a whole range of extended objects of other dimensionalities. And since string theories have correspondence limits to supergravities, should not the flora of p-branes be taken seriously in string theory as well? All of this started coming together finally through the influence of developments associated with another concept: "duality." It was known that strings propagating on compactified backgrounds with Killing symmetries could be transformed, by gauging those symmetries and then eliminating variables in a different order, into strings propagating on a different background, with a compactification radius inverse to t h a t of the original background. These so-called Tdualities also provided a link between two of the 5 consistent D = 10 string theories, i.e. between the type IIA and IIB theories. Furthermore, these T-duality transformations had the effect of transforming the familiar Neumann boundary conditions for open strings into a Dirichtet type of condition that was puzzling because it breaks Poincar~ invariance. The penny dropped when it was realized that the specially selected spacetime hyperplanes upon which the Dirichlet boundary conditions are imposed could be interpreted as solitonic higher-dimensional extended objects in the theory. In this way, the p-branes came to be identified in string theory, ... and pbrahe practitioners were invited back in out of the rain. This started to get things going again. The relations between the known duality symmetries and nonlinearly realized classical symmetries originally found by Cremmer and Julia in supergravity theories gave rise to a bold new hypothesis by Hull and Townsend and Witten: there was a
6
K.S. SteIle/Nuclear
Physics B (proc. Suppl.) 88 (2000) 3-9
whole “U-duality” group of symmetries active at the quantum level in string theory, including new strong-weak coupling dualities, which could take one far into the nonperturbative region. And duality links were also forged to connect all of the five D = 10 superstring theories. Most impressive of all was the return from exile of D = 11 supergravity, which now came to be seen as the field-theoretic description of the strong-coupling limit of type IIA string theory. Everything was becoming one again. String theory is no longer just a theory of strings, but of particles, strings, membranes and higher-dimensional pbranes as well. And perhaps we even live in a brane, a 3brane propagating in a higher-dimensional world of which we are not yet fully aware. This eruption of solitonic physics in the string program has had the effect of reawakening an appreciation of the wisdom of the ages that has been acquired in general relativity and in quantum field theory. This is illustrated2 in Figure 1. Moreover, many of the other approaches that have been followed in the long and diverse struggle for quantum gravity may now once again have a role to play. In this new perspective, many features of earlier discussions may make return visits, but in For example, the three-loop divernew guises. gences in D = 4 supergravity field theories involve a locally supersymmetric invariant containing the square of the Bel-Robinson tensor N tr(RR + *R*R)2, which exists in all D = 4 supergravities. Directly related invariants exist in higherdimensional supergravities, from which the D = 4 invariants can be obtained by dimensional reduction. The D = 10 “oxidized” analogues of this term occur with finite coefficients in superstring theories. There is also a D = 11 invariant in this oxidation sequence, which is related by supersymmetry to the term sMll A,,, A tr(R A R A R A R) that is required in the M-theory effective action in order to cancel chiral anomalies on the 5-brane worldvolume. Moreover, this term also gives rise to calculable corrections to other quantities of interest, such as the Hawking-Page thermodynamic free energy. Thus, old disasters may revisit us in ‘With
many thanks to Andrei Linde for the cartoon.
Figure
1. Sage and students
milder form, and can even play constructive roles from the new M-theory perspective. Of course, one persistent criticism of string theory has been its reliance upon formulations in particular spacetime backgrounds. For a theory of gravity, this is clearly a shortcoming, and begs to be remedied by a fully background-independent formulation of the theory. This still remains a major challenge. However, the range of specific backgrounds upon which detailed formulations of the theory have been made has been significantly extended. No longer just flat space, but now also curved backgrounds including branes or black holes have been included in the list of consistent string backgrounds. There are hints that discrete structures of spacetime, or non-commutative geometry, may be relevant here. Moreover, one of the key ideas of Kaluza-Klein theory, that the apparent dimensionality of spacetime may vary, is wholly applicable from the new viewpoint. A striking example of this has been the duality-based correspondence, proposed by Maldacena, between strings in 5-dimensional anti
KS. Stelle/Nuclear
I
Physics B (Proc. Suppl.) 88 (2000) 3-9
de Sitter spacetime and U(N) Yang-Mills gauge theories on the 4-dimensional boundary of that spacetime, in the limit of large N. This correspondence has a brane interpretation as well, in terms of the 3-branes of type IIB string theory. This brane spacetime has an asymptotic structure (AdS)s x S5 near the brane horizon, which is in this case a fully nonsingular surface both for the metric and for the scalar fields of the theory. Treating the compact S5 as a KaluzaKlein internal space, one obtains a 5-dimensional interpretation for this anti de Sitter/conformal field theory (AdS/CFT) duality relation. The 3brane’s 4-dimensional worldvolume is the arena for the U(N) gauge theory, as well as the boundary of (AdS)5. The duality relation links type IIB strings in the (AdS)s x S5 bulk spacetime with the worldvolume super-Yang-Mills theory on the boundary, taken in the ‘t Hooft limit of large N while keeping g&N fixed, where gYM is the Yang-Mills coupling constant. This relation has given rise to a hope of obtaining at last some understanding of confinement in realistic nonabelian gauge theory models. Another curious aspect of the emerging unified picture of quantum gravity and the other interactions might be termed “atavism.” We have become used to treating yesterday’s field theories as correspondence limits, or effective field theories, of the underlying microscopic dynamics. But in many cases, these effective theories “remember” some key features of the underlying microscopic theory. A classic example of this is the way in which pion theory, as described by a nonlinear sigma model, has an anomaly structure that must match that of the underlying microscopic theory through the ‘t Hooft anomaly matching condition. In a similar way, some aspects of the incomplete quantum structure of supergravity theories must reflect the structure of the underlying Mtheory. Anomalies are once again an example of this, and have been powerfully employed in the Hoiava-Witten picture of spacetime compactified on an interval from D = 11 to D = 10, yielding thereby a very beautiful duality link between M-theory and the & x ES heterotic string. Another example of “atavism” is the way in which the supergravity brane spectrum “remem-
bers” the duality symmetric charge-lattice structure required in M-theory (and since these SUpersymmetric branes are extremal in a fashion similar to the extremal Reissner-Nordstrom SO~Ution of GR, the charge lattice is also the lattice of brane tensions). D = 11 supergravity contains a characteristic three-form gauge field A,,,, with field strength F,*] = dA,,, . The bosonic part of the D = 11 action s &j(R - &(.F’[41)2) - $ sFrl] A F14, A A,,, has in its last term an interaction of Chern-Simons form, which is only invariant up to a surface term under the 6A,,, = A,,,, &‘I,,, = 0 This gives rise to a Noether gauge invariance. charge
&A le) =
Jm,,, A,31 A (*II -
$13,
is the canonical
A &j)> momen-
where nijk = & tum. When conside;ed for cohomologically non. . trivial parameters &], these charges have an algebra that represents the cohomology ring of spacetime,
[Qfil,Q$f
[Jl
]P.B.
=
@I,:)~,,~, , where 121 IX
QiGi = sM1(, AIal A F14] is the magnetic Considering to 2-branes
charge.
also an electric coupling of M-theory Ts sM, A,,, and a magnetic cou-
pling to 5-branes TGJm, &.lr the requirements of gauge invariance for cohomologically nontrivial parameters Ais and A,,, lead to the restrictions T3 sm, A,,, = 2rk, T6 sM6 &,I = 2m. Consistency of these with the algebra A,,, = A:,, A A;2,, yields the brane-tension relation T6 = &T: for the charge-lattice unit elements. Taken together with the Dirac quantization relations between conjugated electric and magnetic charges, this analysis determines the full supersymmetric brane-charge lattice. And this in turn is the principal representation of M-theory U-duality. Although the full microscopic and backgroundindependent formulation of M-theory is still not known, it is hoped that such “atavistic” clues from the effective field theory may point us in the right direction. Another striking feature of the emerging picture of gravity and the other interactions is that the dimensionality of spacetime itself has become a dynamical quantity. In Kaluza-Klein theory, determining the dynamics of a low-energy theory by the choice of a higher-dimensional “vatuum” has long been a familiar feature. For ex-
8
K.S. Stelle/Nuclear Physics B (Proc. Suppl.) 88 (2000) 3-9
ample, in the compactification of superstrings on Calabi-Yau 3-folds, the detailed choice of the internal manifold determines the gauge group and the matter representations of the low-energy effective theory. And in the limit where the size of the internal manifold tends to infinity, KaluzaKlein towers of massive states re-coalesce into a continuum, reinstating the higher-dimensional description of the theory as the most appropriate one. By contrast, another point that has been recognized recently in string theory is that the effective dimensionality of spacetime can also be less than the apparent dimensionality. This idea goes by the keyword of “holography,” by which it is meant that “bulk” amplitudes can be determined in terms of functions of boundary fields a. The best example of this is in the AdS/CFT correspondence, in which the brane horizon (i.e. the AdS horizon) has one dimension less than that of the bulk theory, but the AdS/CFT duality encodes the bulk dynamics in terms of that of the boundary fields. This is manifested in the AdS/CFT relation between Yang-Mills correlat,ion functions and superstring amplitudes: [~A]~-‘YA~[A]+~,,‘[A]J = I
,-h[@m(J)] (1)
where, on the left-hand side, 0, is a Yang-Mills composite operator with source J, while on the right-hand side, (a,(J) is the supergravity bulk field that couples to O,, which is determined by the boundary value J. This relation gives important information when read from left to right, and also from right to left. Reading from left to right, we obtain nonperturbative information on a nonabelian gauge theory from supergravity couplings. Reading from right to left, we see how an apparently higher-dimensional theory is determined by the dynamics of a lower-dimensional one. The open-minded attitude towards the effective dimensionality of spacetime extends also to phenomenology, where even the possibility of detecting consequences of a 5th spacetime dimension already in energies in the TeV range has been given considerable attention recently. Other traditional techniques employed in the long search for a quantum theory of gravity may
also be expected synthesis:
to find diverse
roles in the new
l
Canonical methods and studies of integrable models can help in understanding some very central but very uncomfortable For example, one still lacks problems. proper methods to quantize theories with self-duality conditions, such as the worldvolume dynamics of the M-theory 5-brane, which has a self-dual 3-form H,,, = *H,,,.
l
Euclidean quantum gravity is still a key tool in inflationary cosmology discussions. Inflation is mostly still discussed in a general framework without a detailed origin in fundamental theory, but there are increasingly attempts made to obtain an inflationary universe from M-theory. The first attempts in this direction using instantons have failed because the characteristic exponential potentials are too steep to give significant inflation. However, this subject has only begun to be investigated, and brane scenarios and higher-dimensional spacetimes still may give many possibilities.
l
Black-hole thermodynamics is now a core topic of M-theory. Indeed, it represents the most dramatic example to date of the power of this new synthesis to make inroads on old problems. Microscopic state analysis now fully reproduces the Bekenstein-Hawking entropy formula, and the predicted Hawking radiation includes even the non-thermal “greybody” factors.
The diversity of these different approaches and techniques is illustrated in Figure 2, superimposed upon a map of the beautiful island of Sardinia where this conference unifying the diverse approaches to quantum gravity took place. What is most needed now is a fundamental definition of just what M-theory really is.
KS. Stelle/Nuclear
Physics B (Proc. Suppl.) 88 (2000) 3-9
Quantum Gravity Country
&alitg
Non-perturbative gauge theory
Figure 2.
The unification
of quantum
www.elstier.nl/locate/npe
gravity
K.S. Stelle * The Blackett Laboratory, Imperial College Prince Consort Road, London SW7 2BW, UK
The subject of quantum gravity had its origins in the thirties, very soon after the great SUCcess with nonrelativistic quantum mechanics and the first, albeit noncovariant, steps towards QED. Gravity proved to be a problem of a different magnitude, however. The infinities associated with the dimensionful coupling constant immediately stood out as a central problem of the subject, but there was nonetheless hope that a sufficiently careful approach to quantization might resolve such difficulties. Subsequent approaches to the subject acquired different flavors, according to the individual backgrounds and predilections of the physicists involved. Relativists clearly took to geometrical methods, and readily adopted Dirac’s canonical quantization for gauge theories when this was developed in the 60s. On the other hand, quantum field theorists preferred to face the problems in a perturbative analysis about flat space. There have also been S-matrix approaches to the problem. All approaches, however, led to difficulties. Moreover these difficulties were sufficiently different in their natures that it became difficult to compare the successes and disappointments of the various approaches. As a result, there was a tendency for the different communities of researchers to lose contact, with insufficient interaction between the different programs. Meanwhile, quantum field theory itself made steady progress in other areas, with progressive understanding of the infinities problem and of the proper quantization procedure for gauge theories. Renormalization techniques, path-integral formulations, the introduction of ghost fields and their control through BRST methods: all these ele*Research No. 613
supported
in part
by PPARC
under
SPG
grant
0920-5632/00/$ - see front matter 0 2000 Elsevier Science B.\! PI1 SO920-5632(00)00747-7
ments became part of the central canon of techniques. The weak and electromagnetic interactions and the strong interactions are now successfully handled within this context, while renormalization group analysis and the study of solitons and instantons has opened the door to some nonperturbative understanding as well. On another track, in classical general relativity the understanding of physics in curved spacetime made a slow but progressive advance since the theory’s birth over 80 years ago. Many striking exact solutions were found, and techniques were developed for the analysis of global structure, the initial value problem, and the nature of spacetime singularities. Crucial but sparse observational data made it clear that general relativity was in full agreement with astronomical reality. With data from the binary pulsar, clear information about gravitational radiation, albeit indirect, is also now available. Accordingly, classical general relativity has now become one of the most securely grounded physical theories. But quantum gravity has remained elusive. In this contrasting world of approaches and developments, it is only recently that results from different directions have perhaps begun coalescing into a coherent view of the outstanding problems of quantum gravity. Within a field theoretic perspective, the exploration of supersymmetric theories began, initially with a view to applications in domains accessible to accelerator experiments. But, surprisingly, this program became entangled with the problems of gravity. This arose from the need for a realistic pattern of spontaneous supersymmetry breaking, a problem not at first obviously connected to gravity. But the constraints of supersymmetry are so strong that it turned out All rights reserved.
4
K.S. Stelle/Nuclear Physics B (Proc. Suppl.) 88 (2000) 3-9
that the only viable mechanisms for supersymmetry breaking involve coupling to supergravity. The discovery of supergravity in 1976, by Freedman, Ferrara and Van Nieuwenhuizen from a particle physics perspective but followed instantly by Deser and Zumino replying more to relativists’ concerns, was itself a major achievement. Later came understanding of its couplings and the construction of locally supersymmetric models, through the development of supersymmetric tensor calculus and superspace. This allowed the creation of particle physics models whose possible confirmation is now a main hope for the next generation of accelerators, notably the LHC at CERN. After supplying the mechanism for spontaneous supersymmetry breaking, however, supergravity has typically been retired to the sidelines, leaving the discussion to proceed in a context of flat-space renormalizable particle field theory. But this is clearly an arbitrary truncation of the full theory, perhaps understandable in the rush to understand supersymmetric extensions of the standard model, but not ultimately sustainable. Supergravity itself gave rise initially to high hopes for a solution to the infinities problem. Indeed, local supersymmetry proves to be a teaser here, for it does significantly increase the control over infinities, leading to cancellations at one and two loops in pure D = 4 supergravity theories with less than the maximal amount of supersymmetry, and even all the way through four loops (as we hear from Zvi Bern in this conference) in the maximally extended N = 8 theory. But, eventually, the infinities inevitably return, once more dashing hopes for a field-theoretic resolution of gravity’s difficulties. Out of the mysterious world of the S-matrix and the struggle to understand hadronic physics in the late 1960’s, however, another approach was brewing. The Veneziano amplitude gave a stunning description of the Regge resonances observed in hadronic data. No field theoretic derivation of this amplitude was known, but then it was realized by Koba, Nielsen, Susskind and others that a one-dimensional extended object, the relativistic string, could be the underlying dynamics for this. String theory progressed independently
from particle field theory for awhile, with a strong phenomenological bent. Although the Regge behavior of string-derived amplitudes agreed with observation, accelerator experiments at fixed angles eventually gave evidence of a hard “partonic” structure of the hadrons that was in serious disagreement with the very soft string interactions in this parameter region. With great regret, string theory fell out of favor as a theory of the strong interactions. Moreover, it had to be admitted that it had in fact a number of other embarrassments. For example, it required formulation in higher dimensional spacetimes, and also had, as a result of the quantum offset in the Regge trajectories, a strange zero mass state of spin two, which was quite unwanted in hadronic physics. Nonetheless, as happened with Yang-Mills nonabelian gauge theories, which also suffered an early disappointment in their initial intended application, a certain stubborn band of physicists continued to work on the theory anyway, despite its physical disappointments. Thus it was that Scherk, Schwarz and Green pressed forward into development of the superstring. The hadronic disappointment of the zero-mass spin two state now became the theory’s main hope: this could be the graviton. String theory, as a theory of extended objects, has a totally different ultraviolet structure from that of particle field theories. Although new types of infinities can arise in string amplitudes, thus complicating the story for awhile, it eventually became clear that superstring theory does manage to provide the sought-for cure to the infinities problem of quantum gravity. Moreover, superstring theories have correspondence limits to supergravity theories, making a link to previous ambitions for supersymmetry in the theory of gravitation. And these theories are also naturally formulated in ten dimensional spacetimes, just as many of the principal supergravities. But there is an exception: the most majestic supergravity, the D = 11 theory constructed by Cremmer, Julia and Scherk, which seemed to be left out of the string-related picture. The higher spacetime dimensions themselves were not so much of a worry: these could be dealt using the geometrically powerful techniques of Kaluza-Klein theory, compact-
K.S. Stelle/Nuclear Physics B (Proc. SuppL) 88 (2000) 3-9
ifying unobserved dimensions down to small sizes, but also providing an attractive possible explanation for "internal" symmetries. Together with a hopeful cure for the difficulties of quantum gravity, the subject was now getting more ambitious: it seemed t h a t only totally unified theories had a chance of working. This inevitably upped the stakes from a search for a theory of quantum gravity to that for a quantum theory of all matter and interactions. In 1984 it was discovered by Green and Schwarz that the successful resolution of certain difficulties with gauge anomalies restricted the viable candidate superstring theories down to a very small set with very particular (and phenomenologically interesting) gauge symmetries. After this, interest in superstrings rose to a fever pitch. A whole new generation of young physicists got into the act, bringing with them new techniques of conformal field theory and introducing sophisticated new mathematics into the theory, while leaving aside earlier field theoretic and relativistic methods. This was at last the birth of a perhaps viable approach to quantum gravity. Now that a candidate existed, one could really get down to see if anything like our physical world could be described by it. Here, however, an apparent setback was encountered. The supposed quasi-uniqueness of the viable superstring theories began to dissolve: to start with, there were in fact five candidate theories in the initial ten-dimensional spacetime; moreover the details of compactification down to D = 4 gave rise to a very large number of candidates, so this seemed hardly a unique theory after all. Enthusiasm was giving way to a certain degree of muddle. While all of this was going on, another renegade band, principally of supergravity enthusiasts, were pursuing a somewhat heretical line of enquiry. If one had to give up point particles for extended objects, then why stop at onedimensional strings? Why not consider membranes, or even higher-dimensional extended objects? Perhaps because this work was in any case rather disregarded by the majority string community, this band of practitioners felt free to make light of things and termed the more general extended objects "p-branes," so that a particle was
5
a 0-brane, a string was a 1-brane, etc. The downside was that resorting to intrinsic dimensions higher than one promptly reintroduced many of the difficulties that string theory was meant to cure, such as the dreaded infinities. But higherdimensional extended objects still held their fascination. In particular, they fitted in perfectly with the abandoned but still majestic D = 11 supergravity, which was found to interact most naturally with 2-branes and 5-branes. From the same perspective, the D = 10 theories naturally interacted with strings, to be sure, but also with a whole range of extended objects of other dimensionalities. And since string theories have correspondence limits to supergravities, should not the flora of p-branes be taken seriously in string theory as well? All of this started coming together finally through the influence of developments associated with another concept: "duality." It was known that strings propagating on compactified backgrounds with Killing symmetries could be transformed, by gauging those symmetries and then eliminating variables in a different order, into strings propagating on a different background, with a compactification radius inverse to t h a t of the original background. These so-called Tdualities also provided a link between two of the 5 consistent D = 10 string theories, i.e. between the type IIA and IIB theories. Furthermore, these T-duality transformations had the effect of transforming the familiar Neumann boundary conditions for open strings into a Dirichtet type of condition that was puzzling because it breaks Poincar~ invariance. The penny dropped when it was realized that the specially selected spacetime hyperplanes upon which the Dirichlet boundary conditions are imposed could be interpreted as solitonic higher-dimensional extended objects in the theory. In this way, the p-branes came to be identified in string theory, ... and pbrahe practitioners were invited back in out of the rain. This started to get things going again. The relations between the known duality symmetries and nonlinearly realized classical symmetries originally found by Cremmer and Julia in supergravity theories gave rise to a bold new hypothesis by Hull and Townsend and Witten: there was a
6
K.S. SteIle/Nuclear
Physics B (proc. Suppl.) 88 (2000) 3-9
whole “U-duality” group of symmetries active at the quantum level in string theory, including new strong-weak coupling dualities, which could take one far into the nonperturbative region. And duality links were also forged to connect all of the five D = 10 superstring theories. Most impressive of all was the return from exile of D = 11 supergravity, which now came to be seen as the field-theoretic description of the strong-coupling limit of type IIA string theory. Everything was becoming one again. String theory is no longer just a theory of strings, but of particles, strings, membranes and higher-dimensional pbranes as well. And perhaps we even live in a brane, a 3brane propagating in a higher-dimensional world of which we are not yet fully aware. This eruption of solitonic physics in the string program has had the effect of reawakening an appreciation of the wisdom of the ages that has been acquired in general relativity and in quantum field theory. This is illustrated2 in Figure 1. Moreover, many of the other approaches that have been followed in the long and diverse struggle for quantum gravity may now once again have a role to play. In this new perspective, many features of earlier discussions may make return visits, but in For example, the three-loop divernew guises. gences in D = 4 supergravity field theories involve a locally supersymmetric invariant containing the square of the Bel-Robinson tensor N tr(RR + *R*R)2, which exists in all D = 4 supergravities. Directly related invariants exist in higherdimensional supergravities, from which the D = 4 invariants can be obtained by dimensional reduction. The D = 10 “oxidized” analogues of this term occur with finite coefficients in superstring theories. There is also a D = 11 invariant in this oxidation sequence, which is related by supersymmetry to the term sMll A,,, A tr(R A R A R A R) that is required in the M-theory effective action in order to cancel chiral anomalies on the 5-brane worldvolume. Moreover, this term also gives rise to calculable corrections to other quantities of interest, such as the Hawking-Page thermodynamic free energy. Thus, old disasters may revisit us in ‘With
many thanks to Andrei Linde for the cartoon.
Figure
1. Sage and students
milder form, and can even play constructive roles from the new M-theory perspective. Of course, one persistent criticism of string theory has been its reliance upon formulations in particular spacetime backgrounds. For a theory of gravity, this is clearly a shortcoming, and begs to be remedied by a fully background-independent formulation of the theory. This still remains a major challenge. However, the range of specific backgrounds upon which detailed formulations of the theory have been made has been significantly extended. No longer just flat space, but now also curved backgrounds including branes or black holes have been included in the list of consistent string backgrounds. There are hints that discrete structures of spacetime, or non-commutative geometry, may be relevant here. Moreover, one of the key ideas of Kaluza-Klein theory, that the apparent dimensionality of spacetime may vary, is wholly applicable from the new viewpoint. A striking example of this has been the duality-based correspondence, proposed by Maldacena, between strings in 5-dimensional anti
KS. Stelle/Nuclear
I
Physics B (Proc. Suppl.) 88 (2000) 3-9
de Sitter spacetime and U(N) Yang-Mills gauge theories on the 4-dimensional boundary of that spacetime, in the limit of large N. This correspondence has a brane interpretation as well, in terms of the 3-branes of type IIB string theory. This brane spacetime has an asymptotic structure (AdS)s x S5 near the brane horizon, which is in this case a fully nonsingular surface both for the metric and for the scalar fields of the theory. Treating the compact S5 as a KaluzaKlein internal space, one obtains a 5-dimensional interpretation for this anti de Sitter/conformal field theory (AdS/CFT) duality relation. The 3brane’s 4-dimensional worldvolume is the arena for the U(N) gauge theory, as well as the boundary of (AdS)5. The duality relation links type IIB strings in the (AdS)s x S5 bulk spacetime with the worldvolume super-Yang-Mills theory on the boundary, taken in the ‘t Hooft limit of large N while keeping g&N fixed, where gYM is the Yang-Mills coupling constant. This relation has given rise to a hope of obtaining at last some understanding of confinement in realistic nonabelian gauge theory models. Another curious aspect of the emerging unified picture of quantum gravity and the other interactions might be termed “atavism.” We have become used to treating yesterday’s field theories as correspondence limits, or effective field theories, of the underlying microscopic dynamics. But in many cases, these effective theories “remember” some key features of the underlying microscopic theory. A classic example of this is the way in which pion theory, as described by a nonlinear sigma model, has an anomaly structure that must match that of the underlying microscopic theory through the ‘t Hooft anomaly matching condition. In a similar way, some aspects of the incomplete quantum structure of supergravity theories must reflect the structure of the underlying Mtheory. Anomalies are once again an example of this, and have been powerfully employed in the Hoiava-Witten picture of spacetime compactified on an interval from D = 11 to D = 10, yielding thereby a very beautiful duality link between M-theory and the & x ES heterotic string. Another example of “atavism” is the way in which the supergravity brane spectrum “remem-
bers” the duality symmetric charge-lattice structure required in M-theory (and since these SUpersymmetric branes are extremal in a fashion similar to the extremal Reissner-Nordstrom SO~Ution of GR, the charge lattice is also the lattice of brane tensions). D = 11 supergravity contains a characteristic three-form gauge field A,,,, with field strength F,*] = dA,,, . The bosonic part of the D = 11 action s &j(R - &(.F’[41)2) - $ sFrl] A F14, A A,,, has in its last term an interaction of Chern-Simons form, which is only invariant up to a surface term under the 6A,,, = A,,,, &‘I,,, = 0 This gives rise to a Noether gauge invariance. charge
&A le) =
Jm,,, A,31 A (*II -
$13,
is the canonical
A &j)> momen-
where nijk = & tum. When conside;ed for cohomologically non. . trivial parameters &], these charges have an algebra that represents the cohomology ring of spacetime,
[Qfil,Q$f
[Jl
]P.B.
=
@I,:)~,,~, , where 121 IX
QiGi = sM1(, AIal A F14] is the magnetic Considering to 2-branes
charge.
also an electric coupling of M-theory Ts sM, A,,, and a magnetic cou-
pling to 5-branes TGJm, &.lr the requirements of gauge invariance for cohomologically nontrivial parameters Ais and A,,, lead to the restrictions T3 sm, A,,, = 2rk, T6 sM6 &,I = 2m. Consistency of these with the algebra A,,, = A:,, A A;2,, yields the brane-tension relation T6 = &T: for the charge-lattice unit elements. Taken together with the Dirac quantization relations between conjugated electric and magnetic charges, this analysis determines the full supersymmetric brane-charge lattice. And this in turn is the principal representation of M-theory U-duality. Although the full microscopic and backgroundindependent formulation of M-theory is still not known, it is hoped that such “atavistic” clues from the effective field theory may point us in the right direction. Another striking feature of the emerging picture of gravity and the other interactions is that the dimensionality of spacetime itself has become a dynamical quantity. In Kaluza-Klein theory, determining the dynamics of a low-energy theory by the choice of a higher-dimensional “vatuum” has long been a familiar feature. For ex-
8
K.S. Stelle/Nuclear Physics B (Proc. Suppl.) 88 (2000) 3-9
ample, in the compactification of superstrings on Calabi-Yau 3-folds, the detailed choice of the internal manifold determines the gauge group and the matter representations of the low-energy effective theory. And in the limit where the size of the internal manifold tends to infinity, KaluzaKlein towers of massive states re-coalesce into a continuum, reinstating the higher-dimensional description of the theory as the most appropriate one. By contrast, another point that has been recognized recently in string theory is that the effective dimensionality of spacetime can also be less than the apparent dimensionality. This idea goes by the keyword of “holography,” by which it is meant that “bulk” amplitudes can be determined in terms of functions of boundary fields a. The best example of this is in the AdS/CFT correspondence, in which the brane horizon (i.e. the AdS horizon) has one dimension less than that of the bulk theory, but the AdS/CFT duality encodes the bulk dynamics in terms of that of the boundary fields. This is manifested in the AdS/CFT relation between Yang-Mills correlat,ion functions and superstring amplitudes: [~A]~-‘YA~[A]+~,,‘[A]J = I
,-h[@m(J)] (1)
where, on the left-hand side, 0, is a Yang-Mills composite operator with source J, while on the right-hand side, (a,(J) is the supergravity bulk field that couples to O,, which is determined by the boundary value J. This relation gives important information when read from left to right, and also from right to left. Reading from left to right, we obtain nonperturbative information on a nonabelian gauge theory from supergravity couplings. Reading from right to left, we see how an apparently higher-dimensional theory is determined by the dynamics of a lower-dimensional one. The open-minded attitude towards the effective dimensionality of spacetime extends also to phenomenology, where even the possibility of detecting consequences of a 5th spacetime dimension already in energies in the TeV range has been given considerable attention recently. Other traditional techniques employed in the long search for a quantum theory of gravity may
also be expected synthesis:
to find diverse
roles in the new
l
Canonical methods and studies of integrable models can help in understanding some very central but very uncomfortable For example, one still lacks problems. proper methods to quantize theories with self-duality conditions, such as the worldvolume dynamics of the M-theory 5-brane, which has a self-dual 3-form H,,, = *H,,,.
l
Euclidean quantum gravity is still a key tool in inflationary cosmology discussions. Inflation is mostly still discussed in a general framework without a detailed origin in fundamental theory, but there are increasingly attempts made to obtain an inflationary universe from M-theory. The first attempts in this direction using instantons have failed because the characteristic exponential potentials are too steep to give significant inflation. However, this subject has only begun to be investigated, and brane scenarios and higher-dimensional spacetimes still may give many possibilities.
l
Black-hole thermodynamics is now a core topic of M-theory. Indeed, it represents the most dramatic example to date of the power of this new synthesis to make inroads on old problems. Microscopic state analysis now fully reproduces the Bekenstein-Hawking entropy formula, and the predicted Hawking radiation includes even the non-thermal “greybody” factors.
The diversity of these different approaches and techniques is illustrated in Figure 2, superimposed upon a map of the beautiful island of Sardinia where this conference unifying the diverse approaches to quantum gravity took place. What is most needed now is a fundamental definition of just what M-theory really is.
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Physics B (Proc. Suppl.) 88 (2000) 3-9
Quantum Gravity Country
&alitg
Non-perturbative gauge theory
Figure 2.