CERN-PH-TH-2012-94 RUNHETC-2012-05 TIFR/TH/12-06 NI-12-013 Multiple Membranes in M-theory Jonathan Baggera, Neil Lambertb,c,f, Sunil Mukhid,f, Constantinos Papageorgakise,f aDepartment of Physics and Astronomy, Johns Hopkins University 3400 North Charles Street, Baltimore, MD 21218, USA bTheory Division, CERN 1211 Geneva 23, Switzerland cDepartment of Mathematics, King’s College London London WC2R 2LS, UK dTata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400 005, India eNHETC and Department of Physics and Astronomy, Rutgers University 126 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA fIsaac Newton Institute for Mathematical Sciences 20 Clarkson Road, Cambridge, CB3 OEH, UK Abstract We review developments in the theory of multiple, parallel membranes in M-theory. After discussing the inherent difficulties with constructing a maximally supersymmetric lagrangian with the appropriate field content and symmetries, we introduce 3-algebras and show how they allow for such a description. Dif- ferent choices of 3-algebras lead to distinct classes of 2+1 dimensional theories with varying degrees of arXiv:1203.3546v2 [hep-th] 14 Jul 2012 supersymmetry. We then demonstrate that these theories are equivalent to conventional superconformal Chern-Simons gauge theories at level k, but with bifundamental matter. Analysing the physical properties of these theories leads to the identification of a certain subclass of models with configurations of M2-branes on Zk orbifolds. These models give rise to a whole new gauge/gravity duality in the form of an AdS4/CFT3 correspondence. We also discuss mass deformations, higher derivative corrections, and the possibility of extracting information about M5-brane physics. Key words: String theory, M-theory, Branes Preprint submitted to Physics Reports July 17, 2012 Contents 1 M-theory and M-branes: a brief review 3 1.1 Eleven-dimensionalsupergravity . ................... 6 1.2 Relationtostringtheory . ............... 9 1.3 Motivationstostudyextendedobjects . ................... 12 1.4 WorldvolumeactionsforM-theorybranes . ................... 13 1.5 M-branesassolitons .............................. ............. 20 1.6 TheM2andM5-branetension . ............. 21 1.7 Relationtobranesinstringtheory . .................. 26 2 Multiple membranes: background and early attempts 33 2.1 M2-branesasstronglycoupledD2-branes . ................... 33 2.2 Branefunnels .................................... ........... 36 2.3 Supersymmetric CS theories with 3................................. 43 N ≤ 3 Three-dimensional CS gauge theories based on 3-algebras 48 3.1 = 83-algebratheories:BLG. ......... 49 3.2 N = 63-algebratheories ................................. ........ 56 3.3 FromN = 63-algebrastoCS-mattertheories . ......... 60 3.4 Somemathematicsof3-algebrasN . ................ 65 4 The effective action of multiple M2-branes 70 4.1 Branederivation................................. ............. 70 4.2 The ABJM proposalfor AdS4/CFT3 ................................... 78 4.3 ABJanddiscretetorsion . .............. 81 5 Analysis of the theory I: basics 83 5.1 Vacuummodulispace............................... ............ 83 5.2 AnovelHiggsmechanism .... ... .... .... .... ... .... .. ............ 88 5.3 RelationofABJMtoBLG ............................. ........... 101 6 Analysis of the theory II: advanced topics 107 6.1 11Dmomentum,fluxesand’tHooftoperators . .................. 107 6.2 Hidden symmetries at k = 1, 2 ...................................... 118 6.3 Backgroundfields................................. ............ 119 6.4 Dielectricmembranes. .............. 127 7 Superconformal CS theories with reduced supersymmetry 135 7.1 Superconformal CS theories with = 5................................. 135 7.2 = 5gaugegroups .......................................N .... 139 7.3 LiftingN = 5 to = 6.......................................... 141 7.4 SuperconformalN CSN theories with = 4................................. 144 N 8 Further 3-algebra directions 149 8.1 Lorentzian3-algebras. ............... 149 8.2 Higher-derivativecorrections . ................... 152 8.3 ApplicationstoM5-branes . ............... 159 9 Closing remarks 166 Email addresses: [email protected] (Jonathan Bagger), [email protected] (Neil Lambert), [email protected] (Sunil Mukhi), [email protected] (Constantinos Papageorgakis) 2 1. M-theory and M-branes: a brief review M-theory is a proposed interacting quantum theory involving fields and extended objects (“branes”) propagating in 11 spacetime dimensions. Its existence has been inferred from the properties of its massless fields, including the spacetime metric, which couple to each other via a specific classical lagrangian known as “11d supergravity,” to be described in more detail below. In addition to the massless fields, M-theory possesses two kinds of stable branes, namely 2-branes (equivalently referred to as “membranes”), and 5- branes. These are dynamical objects that extend in two and five spatial dimensions respectively (as well as time) and possess a characteristic tension and charge. 11d supergravity [1] is a locally supersymmetric lagrangian field theory involving massless bosons and fermions. It is special in that eleven is the highest spacetime dimension in which a consistent supersymmet- ric theory can be written down that has spins 2 [2]. The theory has one 32-component spinor supercharge ≤ in eleven dimensions and, if we allow no more than two-derivative interactions, its lagrangian is unique. However, given the non-renormalisability of gravity in any dimension greater than or equal to four, it is not obvious how to extend this lagrangian to an ultraviolet-complete quantum theory. For this reason, the role of the 11d supergravity lagrangian in quantum physics remained unclear for many years. The situation for supergravity theories in 10 spacetime dimensions is superficially similar. Type II super- gravities have two spacetime supersymmetry charges, which in turn singles out ten as the maximum allowed spacetime dimension. With only two-derivative interactions and this amount of supersymmetry there is not one unique lagrangian, but rather two possible lagrangians with different field contents. These are referred to as type IIA and type IIB supergravity. Again, because of non-renormalisability, these lagrangians by themselves do not define an ultraviolet-complete quantum theory. However, in these cases, an ultraviolet completion is known. Type IIA and IIB supergravities are the low-energy limits of corresponding superstring theories. In particular, this is how type IIB supergravity was originally discovered [3] and subsequently constructed [4, 5]. As shown in Ref. [3], there are two superstring theories in 10 dimensions, type IIA and IIB. Quantisation of these strings reveals, in particular, a spectrum of massless particles. Computations of string scattering amplitudes for these modes can be used to read off their low-energy lagrangians, and the resulting theories are type IIA and type IIB supergravity.1 This fact, together with considerable evidence that string theory is ultraviolet finite, encourages us to think of superstring theory as the ultraviolet completion of type II supergravity. The full theory includes not just the massless modes of supergravity but also extended objects, specifically strings. It was later understood that the spectrum also includes extended branes. These have been studied from a variety of complementary points of view: in terms of worldvolume field theories of the degrees of freedom bound to them, as charged 1Type IIA supergravity also arises by compactifying 11d supergravity on a circle [6–8], as we will discuss in more detail later. 3 extended soliton-like solutions in supergravity, and as one-dimensional matrix models. This relation between superstring theory and 10d supergravity provides a basis to conjecture the ex- istence of a theory that similarly completes 11d supergravity in the ultraviolet (UV). Indeed, it was long expected that fundamental membranes play a role analogous to the one that strings play in completing ten- dimensional supergravities (see for example Refs.[9, 10]). This idea was further stimulated by the discovery that when compactifying 11d supergravity, wrapped membranes naturally turn into the fundamental strings of type IIA superstring theory [11]. While it has not actually proved possible to quantise fundamental mem- branes and derive 11d supergravity from them, it was argued via duality [12–14] that there is a consistent UV completion of 11d supergravity and that stable membranes are an important part of this theory. The details of this conjectured theory, called “M-theory,” will be described below. In addition, for previous reviews on M-theory, its duality properties, compactifications, as well as complementary aspects of membrane dynamics, we refer the reader to [15–20]. As we will see, 11d supergravity has no scalar fields and no dimensionless couplings. Therefore in particular it has no tunable coupling constant. The same must therefore be true of the hypothetical M-theory whose low-energy action is postulated to be 11d supergravity. It follows that unlike string theory, M-theory has no perturbative expansion. This makes it considerably harder to study than string theory. We believe in the existence of M-theory only because of many different properties and relationships that have been uncovered in the last three decades. These together provide convincing evidence for the existence of an elegant and internally consistent