aa r X i v : . [ phy s i c s . h i s t - ph ] J a n Supergravity and its Legacy
S. Ferrara a,b,ca
Theoretical Physics Department, CERNCH - 1211 Geneva 23, SWITZERLAND b INFN - Laboratori Nazionali di FrascatiVia Enrich Fermi 40, I-00044 Frascati, ITALY c Department of Physics and Astronomy and Mani L. Bhaumik Institute for Theoretical Physics,U.C.L.A., Los Angeles CA 90095-1547, USA
Abstract
A personal recollection of events that preceded the construction of Supergravity and of somesubsequent developments.
Ettore Majorana Medal addressContribution to the Proceedings of the International School of Subnuclear Physics, 54th Course:THE NEW PHYSICS FRONTIERS IN THE LHC-2 ERA, Erice, 14 June - 23 June 2016
The Prelude
In the early 1970s I was a staff member at the Frascati National Laboratories of CNEN(then the National Nuclear Energy Agency), and with my colleagues Aurelio Grillo andGiorgio Parisi we were investigating, under the leadership of Raoul Gatto (later Professor atthe University of Geneva) the consequences of the application of “Conformal Invariance” toQuantum Field Theory (QFT), stimulated by the ongoing experiments at SLAC where an un-expected Bjorken Scaling was observed in inclusive electron-proton cross sections, which wassuggesting a larger space-time symmetry in processes dominated by short distance physics.In parallel with Alexander Polyakov, at the time in the Soviet Union, we formulated inthose days Conformally Invariant Operator Product Expansions (OPE) and proposed the“Conformal Bootstrap” as a non-perturbative approach to QFT.Conformal Invariance, OPEs and the Conformal Bootstrap have become again fashionablesubjects in recent times, because of the introduction of efficient new methods to solve the“Bootstrap Equations” (Riccardo Rattazzi, Slava Rychkov, Eric Tonni and Alessandro Vichi),and mostly because of their role in the AdS/CFT correspondence.The latter, pioneered by Juan Maldacena, Edward Witten, Steve Gubser, Igor Klebanovand Polyakov, can be regarded, to some extent, as one of the great legacies of higher dimen-sional Supergravity. It can be used to gain information on strongly coupled gauge theories,and affords a variety of applications to Particle Physics, String Theory and Condensed Mat-ter Physics.In 1973 I was awarded a Fellowship at the CERN Theory Division, which was consideredthe most prestigious place where a young European physicist could go. Subsequent eventsrevealed that going to Geneva was a truly timely choice, since when I started my Fellow-ship, in the fall of 1973, Julius Wess and Bruno Zumino had just formulated supergaugeinvariant (now called supersymmetric) QFT in four dimensions, and then found that milderdivergences occur in these theories with respect to standard renormalizable QFT.The original name was inherited from fermionic strings, where a local (conformal) fermionicsymmetry on the world sheet was introduced by André Neveu and John H. Schwarz, PierreRamond, and Jean-Loup Gervais and Benji Sakita.During my two-year period as a CERN Fellow I wrote four papers with Zumino, two of2hich also with Wess and Jean Iliopoulos (at the time a visiting scientist at CERN). WithZumino we formulated four dimensional Supersymmetric Yang Mills Theories (the new namefor supergauges introduced by Abdus Salam and John Strathdee), thus opening the way toconsider supersymmetric extensions of the Standard Model, the ISM and alike.The merging of supersymmetry with Yang-Mills gauge symmetry was a non-trivial step,and the potential link with the Standard Model was motivated by the so-called “HierarchyProblem” and “Naturalness of Scales”, which in these new theories were alleviated thanks toso-called “Non-Renormalization Theorems”.Many people contributed to these developments. Aside from the previously mentionedcollaborators of Zumino, I can surely mention Kelly Stelle, Peter West, Warren Siegel, JimGates, Marc Grisaru and Martin Roček.In retrospect, the most influential of my joint papers with Zumino, written during myperiod as a Fellow, was the one on the “Supercurrent multiplet”, a multiplet which extendedNoether’s theorem to Supersymmetry and whose components include the energy momentumtensor and a vector-spinor fermionic current, as well as a vector, a scalar and a pseudo scalar.The considerations that follow were among the starting points which led Daniel Freedman,Peter van Nieuwenhuizen and myself to formulate Supergravity in four dimensions as thesupersymmetric extension of General Relativity (GR).Since in theories with local (gauge) symmetries Noether currents couple, to first order,to gauge fields, the fact that the stress tensor couples to the metric tensor in GR indicatedthat the spinor current should have coupled to a spin-3/2 massless gauge field, of the typeintroduced by William Rarita and Julian Schwinger (RS) in 1941, and promptly used byShuichi Kusaka as a hypothetical spin-3/2 neutrino.The structure of the super-current multiplet indicated two facts that were later exploited,namely that a supersymmetric extension of GR should have involved only two particles ofhelicities (2,3/2), and that the field corresponding to the latter should have been a Rarita-Schwinger field, to be coupled to GR. This seemed in contrast with a superspace formulationof a Super Riemannian Geometry, which was originally pursued by Pran Nath, RichardArnowitt and Zumino, where many ordinary fields with different spins were needed.In the fall of 1975, even if I had offers from the US, following a suggestion of JacquesPrentki, at the time CERN-TH Division leader, I accepted a CNRS position at the École3ormale Supérieure in Paris.There I had the chance to meet Daniel Freedman, who was also trying to work out aconsistent theory for spin 3/2 coupled to GR.Peter van Nieuwenhuizen was then his colleague at Stony Brook. He was also collaboratingwith us, since he was investigating the quantum properties of fields of different spins coupledto GR, and the possibility of ultraviolet cancellations similar to those encountered in super-Yang-Mills theories.In the spring of 1976 the breakthrough was achieved. Using the Noether procedure, Fierzrearrangements and Riemann tensor identities (and a computer at the Brookhaven NationalLaboratory), we proved that there is a unique Lagrangian, with only spin 2 and 3/2, and aunique set of supersymmetry transformation rules, which defines pure Supergravity in fourdimensions, the supersymmetric extension of GR.General coordinate transformations, when combined with Supersymmetry, promoted thelatter to a gauge symmetry as well, since spinors in curved space must necessarily dependon the space time point due to the local Lorentz symmetry. The Play
The construction of Supergravity as the gauge theory of local supersymmetry promptedStanley Deser and Zumino to formulate this theory as the Einstein-Cartan version of GRminimally coupled to a (spinor valued) one-form field, the RS gravitino (word coined bySidney Coleman).In their work, a quartic spin-3/2 coupling with gravitational strength, required in our for-mulation by local supersymmetry, originated as a torsion contribution to the spin-connectionwhen treated as an independent field. This extends the so-called first-order or Palatini for-mulation of GR.A subsequent superspace formulation of Supergravity by Wess and Zumino in 1978, wherean appropriate super torsion constraint was introduced to reproduce the flat super geome-try of the super-Poincaré algebra, revealed that our second-order approach, rather than thePalatini-Einstein-Cartan formulation, is equivalent to the Wess-Zumino curved Superspace.In particular, it reproduces the so called “auxiliary fields”, needed to have an off-shell formu-4ation and thus a tensor calculus as in standard GR, introduced by Peter van Nieuwhenuizenand myself and Stelle and West.In the months following the Spring of 1976 several important results were obtained, bothduring the Summer Institute organized by the École Normale Supérieure in Paris and duringsubsequent visits of some of us to Stony Brook and to CERN.In particular, also in collaborations with Eugene Cremmer, Joel Scherk, Bernard Julia,Ferdinando Gliozzi and Peter Breitenlohner, the first matter couplings were obtained and,with Peter van Nieuwnhuizen, the first extended N=2 Supergravity formulated.In a fundamental parallel work, Gliozzi, David Olive and Scherk discovered that the dualspinor model, after a GSO projection, becomes supersymmetric in space-time and not onlyon the string world-sheet. Moreover, they found that N=4 Supergravity Coupled to N=4Yang-Mills is the low energy effective theory of the 10D dual spinor model. MeanwhileScherk, Lars Brink and Schwarz classified super Yang-Mills theories in all dimensions.In these milestones of Superstring Theory, Supergravity in diverse dimensions started toemerge as a low-energy limit.The years 1977 and 1978 were the most performing for Supergravity because the discoveryof off-shell formulations allowed to write general matter-coupled Lagrangians and to begina systematic study of the SuperHiggs effect, the mass generation for the gravitino as aconsequence of the spontaneous breaking of local Supersymmetry. This includes a jointwork with Cremmer, Julia, Scherk, Girardello and van Nieuwenhuizen.Moreover, the ultraviolet properties of Supergravity theories started to be investigatedand pure supergravity was shown to be finite at one and two loops. This is to be contrastedwith the later, fundamental work of Marc Goroff and Augusto Sagnotti, where GR wasshown to diverge at two loops. Three-loop counter terms in different forms of Supergravitywere however exhibited by different authors, including Renata Kallosh, Grisaru and Deser,Jason Kay and Stelle.In a pivotal step, in those years examples of all extended supergravities in four dimensionswere derived and the appearance of electric magnetic (e.m) duality symmetries was noticed,with duality groups that act in a non-linear way on the scalar fields of the theory.The N=8 supergravity in 4D was constructed by Cremmer and Julia, and shown to possess5 very large duality group, the exceptional group E7(7).Gauged N=8 Supergravity in 4D was constructed by Bernard de Wit and Hermann Nico-lai, and gauged Supergravity in diverse dimensions was studied by many people, includingMichael Duff, Murat Günaydin, Chris Hull, Nicholas Warner, Paul Townsend, LeonardoCastellani, Riccardo D’Auria and Pietro Fré. E.m. dualities were later promoted to stringU-dualities by Hull and Townsend.Cremmer, Julia and Scherk constructed Supergravity in the highest possible number ofdimensions, eleven, following a previous classification of Super-Poincaré algebras by WernerNahm.This theory led Witten to formulate, in the Mid 1990s, the remarkable M-theory hy-pothesis, thus identifying an elusive theory in eleven space-time dimensions via the strongcoupling limit of the Type-IIA Superstring. Through webs of dualities, this step allowedto unify the different form of 10D Superstrings, thus also connecting different types of ex-tended objects, called generically branes, which emerge as solitonic solutions from higherdimensional Supergravity.Duff had been the first, in the 1980s, to advocate the central role of generic extendedobjects in an overall picture.D-branes in string theories first appeared in the classic works of Massimo Bianchi, Gian-franco Pradisi and Sagnotti, and of Joseph Polchinski.The advent of M theory marked the “second string revolution”. The 1980s witnessedextensive investigations of Kaluza-Klein (KK) theories, largely aimed at classifying possibleKK consistent reductions (spontaneous compactifications).General Lagrangians including soft breaking terms of gravitational and/or gauge originwere derived by Cremmer, Luciano Girardello, Antoine van Proeyen and myself, and wereapplied to supersymmetric models of Particle Physics.No-scale Supergravity, a theory with a naturally vanishing cosmological constant, wasbuilt by Cremmer, Costas Kounnas, Dimitri Nanopoulos and myself, and was readily appliedto Physics beyond the Standard Model with John Ellis. Twenty years later these modelswere linked to string vacua in flux compactifications by Steve Giddings, Shamit Kachru andPolchinski. 6ll these developments made it possible to derive parameter spaces of masses and cou-plings for supersymmetric extensions of the Standard Model, which can be explored in ex-perimental searches for new physics, as was done at LEP experiments in the past and iscurrently done at the LHC collider.Supersymmetric extensions of the Standard Model were introduced and widely investi-gated by Pierre Fayet and by Savas Dimopoulos and Howard Georgi, relying heavily thesoft-breaking terms proposed by Girardello and Grisaru and derived in Supergravity byRiccardo Barbieri, Carlos Savoy and myself, and independently by Lawrence Hall, JosephLykken and Steven Weinberg, and by Arnowitt, Ali Chamseddine and Nath.“Split Supersymmetry”, a mechanism advocated in order to get large splitting amongMSSM super partners while keeping the nice predictions of gauge coupling unification waslater introduced by James Wells, Nima Arkani-Hamed, Dimopoulos, Gian Giudice, AndreaRomanino, and was further developed by the same authors.A new wave of activity in Supergravity was spurred in 1984, when Michael Green andSchwarz ignited the “first superstring revolution” with the discovery of their cancellationmechanism (for gauge groups E8xE8 or/and SO(32)), making Heterotic and Type-I super-strings (which can only afford the latter option) and their point limit, 10D (1,0) supergravity,free of potential gauge anomalies and also of the gravitational anomalies previously studiedby Luis Alvarez-Gaumé and Witten.The anomaly cancellation mechanism placed restrictions on compactification manifolds,uncovering enticing connections between the supergravity analysis and Algebraic Geometry.The Calabi-Yau compactifications investigated by Philip Candelas, Gary Horowitz, An-drew Strominger and Witten resulted in a paradigmatic example of N=2 supergravity anal-ysis in terms of Special Geometry and Mirror Symmetry.All this machinery of compactifications was thus reconsidered, taking into account re-strictions coming from the Green-Schwarz mechanism. Important implications exist forcompactifications to six dimensions preserving eight supercharges, where some anomaly can-celing terms modify the low energy couplings of tensor multiplets to gauge fields.In the 1990s, Supergravity evolved into a major tool for the study of Black Holes (BH),and in particular of their BPS properties, given their emergence as supersymmetric solitonsin four-dimensional N-extended Supergravity.7n interesting interplay between charged black-holes and the moduli-space geometry ofthe underlying Supergravity emerged through the “Attractor Mechanism”, a property noticedby Kallosh, Strominger and myself when we met at an Aspen Summer Institute, in Colorado,in 1995.In N=2 Supergravity supersymmetric black holes enjoy a universal formula for theirBekenstein-Hawking Entropy, in terms of the (square modulus) of the central charge com-puted at its extremum in moduli space (see also work with Gary Gibbons).The name attractors was coined since these solitons follow trajectories in the moduli spaceof scalar fields which, independently of the initial conditions, reach the same extremal point,which coincides with the value of field at the BH Horizon. This explains why the modulidependent ADM mass is a continuous parameter while the entropy is quantized in terms ofthe BH charges.These results were then extended to theories with N-extended supersymmetry and inhigher dimensions in work including my collaborators and friends Kallosh, Paolo Aschieri,Bianca Cerchiai, Laura Andrianopoli, Stefano Bellucci, Anna Ceresole, Chamseddine, Gi-anguido Dall’Agata, D’Auria, Fré, Murat Günaydin, Maria Lledó, Alessio Marrani, MarioTrigiante, Armen Yeranyan and Zumino.Partial supersymmetry breaking in N=2 supergravity was studied in collaboration withGirardello and Massimo Porrati, with inspiration from the work of Ignatios Antoniadis,Hervé Partouche and Tomasz Taylor, and a minimal model with flat potential was explored.In 1997 Supergravity found itself again at center stage in the work of Juan Maldacena,amusingly entitled “The Large-N Limit of Superconformal Field Theories and Supergravity”,where the famous AdS/CFT duality was formulated, which was soon followed by importantadditions by Witten, Gubser, Klebanov and Polyakov. This might be called the “third stringrevolution”, and would have been impossible without a particular compactification of typeIIB Supergravity on AdS5 x S5 derived in 1985 by Hyiung-Jin Kim, Larry Romans and Petervan Nieuwenhuizen.This completes my personal recollection of many past years devoted to Supergravity. Therecent years 2013-2016 witnessed a revival of spontaneous breaking of local Supersymmetry,boosted by applications to Cosmology, and in particular to the inflationary Universe with ade Sitter phase, with an eye to observational results of the Planck Mission.8 prominent role in this work was played by Andrei Linde, a proponent of InflationaryCosmology, and indeed Daniel Freedman and I collaborated independently with Kallosh andLinde. I also collaborated with a number of colleagues, including Antoniadis, Dall’Agata,Emilian Dudas, Alex Kehagias, Porrati, Sagnotti, Jesse Thaler, Van Proeyen, Timm Wraseand Fabio Zwirner.Supergravity has also found applications in higher-spin field theories, via the AdS/CFTcorrespondence. Supersymmetric versions of OPEs and of the Conformal Bootstrap for N-extended superconformal field theories are currently under investigation. In the quantumdomain N=8 Supergravity in four dimensions was proved to be ultraviolet finite at very highloop order, following the pioneering work of Zvi Bern, Lance Dixon and David Kosower.These properties seem to descend from some amazing relations with the rigid N=4 Yang-Mills theory, which possesses exceptional conformal and integrability properties. It is notyet excluded that N=8 Supergravity be a perturbatively finite theory of Quantum Gravity.In any case this search is going to teach us new lessons on the ultraviolet cancellations whichcan occur in theories including General Relativity.