Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday

This special feature, dedicated to Michael J. Duff FRS on the occasion of his 70th birthday, concerns topics in 'Quantum gravity, branes and M-theory'. These three intertwining subjects have been central to Duff's work; indeed many of his contributions have come to define significant aspects of what we actually mean by these terms. From the discovery of Weyl anomalies to recognising superstrings in 10 dimensions as a special case of membranes in an 11-dimensional M-theory, Duff's insights have shaped major developments across these themes. So it is an apposite setting for such a celebration and we are delighted to be able to include in this collection contributions from many of the pioneers of quantum gravity, branes and M-theory. The breadth of these topics has placed little constraint on the multiplicity of ideas appearing in these pages, from astrophysical black holes to chaotic condensed matter. Again, this is fitting as Duff's scientific remit spans a remarkable diversity of motifs, from the fundamentals of M-theory to entanglement in quantum information.

including spacetime background fields the vanishing of the world-sheet Weyl anomaly implies the spacetime Einstein equations of (super)gravity [28,29], a remarkable result sitting at the foundations of string theory.
Crossing the pond to Brandeis, Waltham, MA, USA, in 1977, Mike joined forces with Steven M. Christensen at Harvard to compute Weyl and axial anomalies in the then recently discovered theory of supergravity. In particular, they were to show that the superpartner to the graviton, the gravitino, contributes an axial anomaly -21 times that of a Dirac spinor [30]. This was again met with some disbelief, but perhaps most interesting was their approach, generalising the classical index theorems, such as Atiyah-Singer, to arbitrary spin [31]. Such calculations revealed some unexpected subtleties. Together with Peter van Nieuwenhuizen, Mike demonstrated that the partition function and Weyl anomaly of a given field may not coincide with those of its electromagnetic dual [32]. Here the anomaly is given by tr⟨T ⟩reg − ⟨trT ⟩reg, where tr denotes the trace, ⟨−⟩reg is the regularized expectation value and T is the stress-energy tensor [27]. They used this observation to argue that theories, classically equivalent under electromagnetic duality, may fail to be so quantum mechanically [32], which is by now a well-recognised property of quantum field theory on topologically non-trivial manifolds [33][34][35][36]. This anomaly should not be confused with tr⟨T ⟩reg alone which yields equivalent results [37][38][39]. Fast-forward some 42 years, Mike demonstrated that the Weyl anomaly of (the massless sector of) type IIA string theory compactified on a 6-manifold is given by a product of Euler characteristics χ(M × X) = χ(M )χ(X), where M is the (Euclidean) spacetime 4-manifold and X is the internal 6-manifold. Moreover, for (the massless sector of) M-theory compactified on a 7-manifold Y , the Weyl anomaly is given by the product is a topological invariant reminiscent of the Ray-Singer torsion [36]. If you like, ρ is to M-theory what χ is to strings.
This early foray into supergravity marked the beginning of Mike's next major movement: Kaluza-Klein theory. In the early 1980s Mike was to return to Imperial College London and also spend time at CERN, Meyrin, Switzerland, two institutes that played an important role in the development of Kaluza-Klein supergravity. At this time supergravity offered much promise as a unified theory, necessarily including gravity. First, it was hoped that supersymmetry might ameliorate the UV divergences plaguing perturbative quantum gravity. Second, supergravity is unique and particularly elegant in D = 11 spacetime dimensions, the maximum allowed by supersymmetry. Thus, when combined with Kaluza-Klein compactification, supergravity stood out as an approach to unification 5 . In this context, Mike and his colleagues made several key advances. With Christopher N. Pope, Mike showed that D = 4, SO(8) gauged N = 8 supergravity theory could be derived as a spontaneous Kaluza-Klein compactification of D = 11 supergravity on AdS 4 × S 7 [40]. Besides its importance for unification at that time, this particular compactification has been a cornerstone of many of the subsequent advances in supergravity and string/M-theory. With Mike's PhD student, Moustafa A. Awada, they further showed that by preserving the S 7 topology while deforming its geometry one could break the N = 8 supersymmetry down to N = 1 [41]. This entailed two important insights that would shape much future work on string/M-theory compactifications. First, the holomony of the internal manifold dictates the degree of supersymmetry preserved. In the context of heterotic superstring compactifications with vanishing fluxes this famously picks out Calabi-Yau 3-folds as the internal manifolds of choice for model building. Second, Mike, Pope and Bengt E. W. Nilsson subsequently showed that the supersymmetry breaking induced by the squashed S 7 corresponded to a Higgs mechanism from the D = 4 perspective [42]. Not long after, the same trio performed the first K3 compactification [43]. This was motivated, in part, by its special SU(2) holonomy, a prelude to the all important SU(3) holonomy Calabi-Yau 3-fold superstring compactifications that would be initiated shortly after [44]. Moreover, the SU(2) holonomy implies that K3 compactifications preserve one half of the supersymmetries, opening the door to type IIA on K3 and heterotic on T 4 string/string dualities. More on that later. These developments, along with manifold pioneering 5  contributions made by many others (some of whom can be found in this very collection), were pulled together by Mike, Nilsson and Pope in what has become a standard reference for Kaluza-Klein supergravity [45].
The sharp crescendo of excitement surrounding supergravity was just as quickly muffled 6 . It had started to seem unlikely that supergravity could ultimately stave off the divergences inherent to a perturbative quantum field theory of gravity (almost 50 years on this chapter is still not quite closed, however) and Edward Witten had demonstrated that D = 11 supergravity compactified on a manifold could not accommodate the chirality needed to make contact with the Standard Model [46]. By the end of 1985 the groundbreaking discoveries of the Green-Schwarz mechanism, heterotic superstrings and Calabi-Yau 3-fold compactifications had firmly, and rightly, cemented themselves as the most promising route to superunification.
Yet, Mike and many like-minded folk had not yet given up on D = 11. On the one hand, superstrings were not an open and shut case and in his 1987 'Not the standard superstring review' [47] Mike erred on the side of caution, "In order not to be misunderstood, let me say straight away that I share the conviction that superstrings are the most exciting development in theoretical physics for many years, and that they offer the best promise to date of achieving the twin goals of a consistent quantum gravity and a unification of all the forces and particles of Nature. Where I differ is the degree of emphasis that I would place on the unresolved problems of superstrings, and the likely time scales involved before superstrings (or something like superstrings) make contact with experimental reality." He emphasised, in particular, the challenges (and opportunities) posed by the landscape problem and non-perturbative phenomena, such as black holes. On the other hand, the tension between 10 and 11 raised its own questions. Why did supersymmetry allow for 11, while superstrings only 10? If supergravity was the low-energy effective field theory of superstrings, where did that leave D = 11 supergravity? Mike vigorously maintained that 11 should be taken seriously.
Indeed, various clues that D = 11 might yet play a role had been amassing. While there are no superstrings in D = 11, there are supermembranes that couple to D = 11 supergravity [48]. It turns out that this is one of the key bridges between D = 10 string theory and D = 11 M-theory. In 1987 Mike, Paul S. Howe, Takeo Inami and Kellogg Stelle showed [49] by compactifying the D = 11 spacetime manifold on S 1 and simultaneously wrapping the supermembrane around the circle ones finds precisely the type IIA superstring in D = 10! This result pre-empted 7 important facets of the M-theory revolution of 1995 by connecting strings and membranes, along with 10 and 11 dimensions. In the same year, Mike and Miles P. Blencowe, again inspired by the discovery of supermembranes in D = 11, conjectured the existence of super p-branes on the S 1 × S p boundary of AdS p+2 and presented the corresponding (free) superconformal field theories [51]. The maximal p = 2 case corresponded to the supermembrane on AdS 4 × S 7 with the superconformal group OSp(8|4). The maximal p = 3 and p = 5 cases corresponded to the yet to be discovered D3-brane and M5-brane on AdS 5 × S 5 and AdS 7 × S 4 with superconformal groups SU(2, 2|5) and OSp(8 * |4), respectively.
Further telling clues on the road to M-theory arose in the context of branes and dualities, themselves closely related. By the mid-eighties five a priori independent consistent superstring theories had been established. However, they were not islands; for instance, the IIA and IIB theories could be connected by T-duality or mirror symmetry. What was to emerge over the next decade or so was a web of dualities, suggesting that each string theory was but a corner of a larger framework. During this period of intense activity (in 10  to Texas A&M, just in time for it to host the inaugural 'Strings 89' conference. Aptly, that year Mike addressed the question of manifest T-duality [52]. By considering two dual string theories, he introduced the notion of a doubled spacetime with a generalised O(D, D) metric H(g, B), built from the standard metric g and the Kalb-Ramond two-form B. This is, today, a key ingredient in the thriving domain of double field theory. The following year 'Strings' would return to Texas A&M and this time around Mike and his then PhD student, Jian Xin Lu, generalised these notions to membranes, where the role B is replaced by the three-form C of D = 11 supergravity [53]. The goal here was to make manifest, from the membrane's perspective, the global symmetries of D = 11 supergravity compactified on an n-torus, which would later be recognised as shadows of the U-dualities of M-theory. This time the spacetime is not merely doubled, but extended by C n 2 coordinates corresponding to the possible ways one can wrap a membrane on an ntorus. There is a generalised metric H(g, C) manifesting the appropriate symmetry group; for example, SL(5, R) for n = 4. It is interesting to note that Mike and Lu puzzled over the cases n > 4, which does not naïvely work out as expected. They resolved this question, quite naturally, by introducing additional coordinates corresponding to the Hodge dual of C and so recovered the symmetries of D = 11 supergravity on an n-torus, for 1 ≤ n ≤ 8. Of course, we now understand these coordinates as corresponding to the possible wrappings of the M5-brane that kick in at n = 5. The extended spacetimes and their generalised metrics H(g, C) are, today, central to the developments of exceptional field theory.
Another central theme of Mike's time as a Texan was the role of solitonic supersymmetric pbrane solutions that carry topological magnetic charge, and their dual relationship to elementary singular (D − p − 4)-brane solutions carrying electric Noether charge [54][55][56][57]. For example, in 1991 Mike and Stelle [58] discovered the elementary multiple membrane solutions of D = 11 supergravity, shortly followed by the dual solitonic superfivebrane solution of Gueven [59]. An other idea introduced by Mike, with Ramzi R. Khuri, Ruben Minasian, and Joachim Rahmfeld, during this period was the identification of solitonic magnetic string states as extremal black holes [60]. Then applying S-duality led Mike and Rahmfeld to relate supersymmetric massive string states with elementary black holes [61]. These ideas generalise to the black and super pbranes solutions in various dimensions [56] and have become a key concept in the understanding of black holes in string/M-theory. The profound contributions unravelling this web of ideas, by Mike and many others, are far too numerous to do justice to here. Fortunately, Mike, Lu and Khuri put together an influential review [62] of these developments up to 1994 that we can defer to. An important consequence of the p/(D − p − 4)-brane dualities [63] is the implied equivalences among string compactifications; for example, the D = 10 heterotic string compactified on a 4-torus is quantum equivalent to the D = 10 type IIA string on K3.
Another related idea introduced by Mike and his colleagues at Texas A&M, including Lu, Khuri, Minasian as well as the newer arrival James T. Liu, was that p-brane dualities could be used to explain electromagnetic duality in lower dimensions [64], as described by Witten [65]: "Mike Duff and Ramzi Khuri in 1993 had written a paper on what they called string/string duality. They had said there should be a self-dual string theory in six dimensions that, looked at in two different ways, would give electric-magnetic duality of gauge theory in four dimensions. It was actually a brilliant idea. The only trouble was they didn't have an example in which it worked." Mike and his colleagues rapidly developed an intricate web of dualities amongst strings and p-branes, and their implications for strong/weak coupling dualities, over the following years [66,66,67,67]. Note, the particular case of the self-dual string in D = 6 relates to the role of the (2, 0) theory in the geometric Langlands programme. In particular, Mike, Liu and Minasian gave evidence that membrane/fivebrane duality provides an eleven dimensional origin of string/string duality, which in turn bolsters the S-duality conjecture [63]. The original hope of Mike and Khuri was also realised together with Minasian and Witten in the context of a heterotic/heterotic duality [68]. These observations contributed (along with crucial insights of a great many others that we are shamefully unable to pay due homage to here) to the 1995 Mtheory revolution led by Witten, which marked a new phase in the development of strings and branes. The supermembrane and fivebrane were duly promoted to the M2-and M5-brane and D = 11 found its place, after all, as the low-energy limit of M-theory. Mike's conviction that D = 11 should be canon was vindicated. This period was followed by an explosion of ideas in string/M-theory, the anti-de Sitter/conformal field theory correspondence (AdS/CFT) and Randall-Sundrum brane world models to name but two examples. Mike's past work, such as the AdS compactifications and brane-scans [69], fed into many aspects of the renewed research avenues. In fact, jumping ahead a little, his 1973 work on loop corrections to the Schwarzschild solution proved important to (AdS/CFT)/Randall-Sundrum complementarity, as shown by Mike and Liu [70]. Who in 1973 saw that one coming! In particular, Mike and colleagues developed asymptotically flat and AdS black hole and p-brane solutions to M-theory [71][72][73][74]  To a degree it was time to take stock. Mike, David Gross and Witten solicited 'big questions' from the attendees and selected the ten best. Some transcended any particular approach to physics beyond the standard model or quantum gravity, for example 'Why does the universe appear to have one time and three space dimensions?'. But one was squarely in the domain of M-theory: "What are the fundamental degrees of freedom of M-theory (the theory whose lowenergy limit is eleven-dimensional supergravity and which subsumes the five consistent superstring theories) and does the theory describe Nature?" This is perhaps the question with which Mike himself has since been most preoccupied: elucidating what M-theory is. Although we have collectively uncovered a patchwork understanding, its ultimate formulation requires new ideas and insights, an endeavour Mike has constantly championed.
In 2005 Mike would come full circle, returning to Imperial College London, now as the Abdus Salam Professor of Theoretical Physics. Here he would embark on several new research journeys (but always touching on M-theory), such as black holes and qubits [75], quantum optics and Hawking radiation [76] and gravity as the 'square' of Yang-Mills theory [77].
For instance, not long after arriving, Mike and Sergio Ferrara, with various of their colleagues and students, would build a dictionary between string/M-theory black holes and various concepts from quantum information theory, qubits and entanglement measures [75,[78][79][80][81][82][83] programme grew out of the observation that the entropy of the ST U black hole 8 and the entanglement shared by three qubits are both described by Cayley's hyperdeterminant [78]. One can only assume that Cayley had anticipated both M-theory and quantum computing.
In completely separate developments, Mike initiated a programme to understand 'Einstein as the square of Yang-Mills' at the level of off-shell field theories. The notion of gravity as the 'product' of two gauge theories has a long history, but was in particular made concrete through the tree-level Kawai-Lewellen-Tye 'closed = open × open' string scattering relations. This idea has witnessed a recent renaissance driven by the 2008 Bern-Carrasco-Johansson colour/kinematics duality conjecture, which allows one to build graviton scattering amplitudes from the 'double copy' of gluon amplitudes to all orders in perturbation theory. 9 Inspired, in part, by the relationship between the symmetries of supergravity and those of super Yang-Mills theory, Mike took an off-shell field theory approach to 'gravity = gauge × gauge'. This yielded remarkable and unexpected insights such as the appearance of the Freudenthal magic square of U-dualities [85,86] and the Yang-Mills origin of (super)diffeomorphisms [87,88]. Today scattering amplitudes and the 'double copy' are being used to understand classical gravity, in particular black hole collisions. We would imagine that Salam (or a 20-year-old Mike) would have been surprised and delighted at this.