arXiv:1903.02860v1 [hep-th] 7 Mar 2019 auaKenApoc oDul n Exceptional and Double to Approach Kaluza–Klein A eeaietecalneo iwn l h ed nsu- in fields the all viewing of challenge the examine We hti aall h oengop[ group Korean the parallel, in what ol-he hoywt manifest with theory world-sheet hster skona obefil hoy(F) Some- (DFT). theory field double as known is theory This hsppra hl olw eynnhsoial ac- non-historically very a follows whole a as paper This et fasrn naduldsaetm ihmnfs T- [ manifest Siegel Warren with before space-time duality doubled a in string a of pects i n cek[ Scherk and lia formalism. semi-covariant the called now is a produced coordinates. of and number field the twice string with background the string of modes winding momentum the and only keeping theory field string closed of il Theory Field oebgnigti hi flgcltu rtbifl de- years [ 30 briefly Duff Almost Michael subject. first the ago us of history let the logic of some of scribe Be- chain supergravity. this to beginning approach fore Kaluza–Klein theory a field on exceptional based and double to approach curate history small Some 1.1 Introduction 1 notion the and quantization. theory of the M-theory and and string space to phase relationship generalised ex- a novel as the cen- of geometry the description tended a of give reinterpretation further We a charges. and tral states fol- charged identifica on the of placed to tion leading is intuition emphasis dou- Kaluza–Klein or the particular lowing theory A field theory. exceptional field as ble known is what gives to This theory. rise higher-dimensional some of dimensional reduction like Kaluza–Klein a from arising as pergravity r hr -ult perda aietsymmetry. manifest a [ Tseytlin as Arkady Subsequently appeared T-duality where ory Berman S. David M-Theor in Structures Higher on Symposium Durham LMS/EPSRC e ra fsm er,i 09 uladZwiebach and Hull 2009, in years, some [ of break a ter 5 , 6 nprle eeomns ntewr fCemr Ju- Cremmer, of work the in developments, parallel In n hnwt om[ Hohm with then and ] 11 ,eee-iesoa uegaiyre- supergravity eleven-dimensional ], 1 eeoe tigwrdsetthe- world-sheet string a developed ] a, ∗ 7 4 , ecie sophisticated a described ] 2 8 , xmndatruncation a examined ] 3 ecie ifrn as- different described ] O 9 , ( 10 d , eeoe what developed ] d ymty Af- symmetry. ) - ok hnatmtdt eomlt uegaiythe- supergravity reformulate to attempted then works aeit aietsmer n n[ in and symmetry manifest a into made E xedn h ubro iesos h group the dimensions, of number the extending a ∗ [ developed in been has scheme this of in nu- role geometry generalised other the notably been most have perhaps developments, There merous dimensions. of number the SL a on duced h deto -hoy[ M-theory of advent the aietsmer fteter,ntbyi h numer- [ the West of in works notably ous theory, a the become of would symmetry symmetry manifest exceptional that U- such ories as known Various S-duality. theories with string T-duality combined of that duality duality a to extended tp htw ileuaeltrfraltebsncfields of bosonic series the all a for for later prescription emulate a will give we that can steps we so is the- This Kaluza–Klein ory. traditional of review a with begin us Let theory Kaluza–Klein 1.2 true. isome- not is requires this and form background) the usual of tries its in (T-duality without isometries backgrounds the- for redundant a theories would as these so make do To (EFT) manifest. should theory symmetries duality one field make to exceptional and ory or theories DFT these see this of not here by-product described a approach wish only the We is in theories. that these emphasize the for to at developments were the M-theory of and heart string in symmetries duality approach. our be not will this so and coordinates extra d odn 2 ieEdRa,Lno 14S ntdKingdom United 4NS, E1 London Road, End of Mile University 327 Mary London, Queen Astronomy, and Physics of School e-mail: author Corresponding 5 a iial aemnfs gi yextending by again manifest made similarly was (5) 18 rcal,a ecnsefo h bv artv,the narrative, above the from see can we as Crucially, xetoa ru fgoa ymtis ae with Later symmetries. global of group exceptional , 19 .Temi huto hsppri ofcso the on focus to is paper this of thrust main The ]. d dmninltrswssont xii an exhibit to shown was torus -dimensional 14 ,Ncli[ Nicolai ], 12 [email protected] , 13 y 15 this ] n tes n[ In others. and ] E d smer was -symmetry 17 h group the ] E 16 7 ,by ], was

1 Proceedings Proceedings where nte a otiko hsi htteoiia oa trans- local original the that is this of think to way Another hsteeaefieprmtr o h oa symmetries: local the for parameters five are there Thus etratn nsm v-iesoa metric, five-dimensional some on acting vector V ie httefilsaeidpneto h e fhdi- fifth new the of independent are fields the that vided S action, Einstein–Hilbert five-dimensional de- the be by should scribed it that implies invariance diffeomorphism five-dimensional have to theory five-dimensional the For ercta et hsciei,adtease a be can answer the and be: criteria, to found this meets that metric ecie bv.(etk atdojcst efive- be to objects hatted take and dimensional (We above. transformations local described four-dimensional the reproduce ere,cnoecmiete ofr v vector five a form to sym- them local combine the one for can parameters five metries, are there given lows: ain of mations n h igescalar single the and inrpoue h ordmninlato ( action four-dimensional the reproduces tion metric the field thus The field. is scalar content a to the- coupled dimensions Einstein–Maxwell four exceptional is in ory point or starting double The produce theory. field so and supergravity in netn h naz( ansatz the Inserting δ of transformations gauge the and L diffeomor- by derivative, Lie given the by are described as theory phisms, the of symmetries local ∂ transformations: gauged and fields all for i.e. mension omtosol eeddo ordmnin n oto so and dimensions four on depended only formations S by given is fields h orvco htgnrtstediffeomorphisms, the generates that vector four the tential 2 g ˆ ˆ x 5 µ V A ˆ = µ ˆ 5 ν ˆ = U µ h hleg hni ofidtefive-dimensional the find to is then challenge The h auaKenie a hnb xrse sfol- as expressed be then may idea Kaluza–Klein The Z = = uhta ifoopim eeae yti five this by generated diffeomorphisms that such = Z µ " d 0. = ∂ F d g 4 A µ µν 5 V x µν χ µ x p ρ = . q + n h clrfield scalar the and A ∂ φ − ∂ ρ µ φ − 2 e g U µ . A 2 R g ˆ A A µ µ φ ν µ + µ ¡ ˆ ( − R A g ˆ ∂ = ν ∂ .(5) ). ( ρ g χ ν ( V φ µ ) A 4 htgnrtsteguetransfor- gauge the generates that φ − 2 µ ,5).) noti v-iesoa ac- five-dimensional this into ) µ U A 2 4 1 stefil teghfor strength field the is ν ρ F # g (2) , µν µν (4) . F h n-omvco po- vector one-form the , µν φ h cinfrthese for action The . A − µ 2 1 , ∂ µ φ∂ µ φ ¢ (1) , g A ˆ 1 µ µ ˆ pro- ) ν ˆ The . will V (6) (3) .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. µ , write: uses then One dimension. fifth the in derivative the with i shr oiaiea roiagmn htti a to had this that argument priori a an imagine to hard is (it ( hsojcswt oetmi h fhdrcinwill direction fifth the in momentum with objects Thus hn ic h ordmninlo-hl eainis relation on-shell four-dimensional the since Then, tti on n a s bu lcrccag quanti- charge electric about ask may one point this At ie n dnie h ordmninleeti charge electric four-dimensional the identifies one vided i to derivative a as operator wave momentum particle the probe and function the between relationship usual the e ofu iesosb ( by dimensions four to ted transformations. local the generating parameters l oecetv ic ocmiedfemrhs with diffeomorphism combine lit- to a be since to creative have more will tle we follow that sections the In Kaluza–Klein theory). about knowing already without case the be diffeomorphisms five-dimensional into combine transformations nicely one-form and diffeomorphisms of tion identity. connected the not to and or finite are diffeomorphisms that transformations i.e. gauge so transformations examined ‘large’ not in- called have this and seen transformations only finitesimal have we speaking Strictly formations. trans- gauge one-form and diffeomorphisms dimensional h oet oc a o lcrccagsi eoee pro- recovered is charges electric for law force Lorentz the ( ansatz metric the oemr sti ilpoieagiighn later. approach hand guiding Kaluza–Klein a provide the will this examine as more us some let this, doing Before theory. higher-dimensional a in diffeomorphisms condition: rclycagdsae nteter.S a h action the far So theory. the in states charged trically qain o rb atcei h five-dimensional the ( in ansatz the particle with theory probe a for equations higher- new the from sources perspective? electric dimensional add we can How in( tion P obey: will dimensions five in object light-like A spective. per- four-dimensional the from charges electric as appear erdc hsw edtecntan ( constraint the need we this reproduce P ain foerequires one If sation. le htteetaKlz–li ieto scompact, is direction Kaluza–Klein extra the that plies p M 1 ∂ fr ag rnfrain ilntpoueusual produce not will transformations gauge -form µ µ ecie isenMxelter ihn currents. no with theory Einstein–Maxwell describes ) ˆ 5 = P P hs h v-iesoa ifoopim restric- diffeomorphisms five-dimensional the Thus, nsm es ewr uk nta h combina- the that in lucky were we sense some In o htw aetehge-iesoa hoy( theory higher-dimensional the have we that Now h nwri ie ucl ycluaigtegeodesic the calculating by quickly given is answer The = µ µ ˆ Q 6 P = − = e h etse st okfrteoii fteelec- the of origin the for look to is step next the ) 5 . 0 = M Q 2 e hsipisthat implies this , . P 5 P 5 + 4 P n h osritfrtereduc- the for constraint the and ) µ Q P 4 .Digti,oedsoesthat discovers one this, Doing ). e µ ob unie hnti im- this then quantised be to = .(8) 0. 6 r qiaett four- to equivalent are ) P 5 = M n hsteBPS the thus and 6 oapyt the to apply to ) a il Theory Field nal (7) (9) 5 ), Proceedings 3 d B (1) C (14) (13) d (2). reduc- . Topo- = g SU (2) G . hidden dimen- (1) gauge theo- ¢ − . (11) ¢ B d U 2 2 ) ∧ y d (4) ∂φ the field strength of ( H G 1 2 B + which is the group man- . (12) ∧ d − 2 3 ) r g 2 copies of (4) i S = , respectively. y H + G d H 1 d 2 1 1 (3) 12 i with field strengths + = A ]: H − + ∧ H (4) 22 R with µνρ 2 ¡ G C z (1) φ , , µν 1 2 48 C µ (d B − 1 H , C − − − k φ ge , ∂ (3) 2 (2) H − k which controls the twist is related to the har- C (2). One then carries out a so called G µν + i p . ij d 4 1 g 2 ǫ x A ¡ ħ t 1 2 = SU , this is an object with magnetic charge n 10 d Z µ = d π (4) ] A − 2 their number. Once one has this thenthere is a ques- using twist matrices often called a Scherk–Schwarz j G Z =− There are then a set of obvious extensions to this idea. The action is given by: = A d 2 = i s [ S ries. One can do something more interestingnon-Abelian and gauge recover fields and Yang–Mills theorydimensional if space our is a grouptake the manifold. hidden So space for to example be From the perspective of fourfield dimensions and the gauge monic function as follows: ∂ ifold of logically the magnetic chargethe is circle the fibration first and Chern isuse class thus the of quantised. One relationship can betweenand then the radius electric charge of to produce the the Diraccondition: Hopf quantisation circle eg and Gross and Perry [ requires the fibre to be a circlemust and be integer. its Thus twist demanding over a the magnetic base solutionquires re- the Kaluza–Klein isometric direction to beRemarkably a even circle. though there is noon metric this direction dependence the topology of howto it the is magnetic fibred gives charge. rise One can have yet more hidden dimensions,by let us denote tion about their geometry. Havingsions the be a torus just gives 1.3 M-theory pergravity whose bosonic fieldsNS-NS split sector into which is the the sopotential: metric, called dilaton and two-form d tion reduction. These twist matrices arethe left-invariant essentially Maurer–Cartan given one-forms on by and the RR sector whoseform fields potentials are one-form and three- and The field The quantisation now though is topological in origin, it The type IIA string low energy effective action is IIA su- ] 2 S 21 over then , one 1 3 5 S S P . (10) × 2 4 in the har- is the time − 2 M d t p such that it is y R 4 will have a dis- d 5 + M P is an integer. This 2 is the radial direc- ¢ t n + , it is independent of 5 R , where x 2 1)d is a circle fibred over an S 3 − × S 1 where . Such solutions are known − + 0 n R R P H ( × = − = ħ t 5 5 5 R , and the parameter P x r P d = / ¡ 4 p H ). This is in contrast to the usual quan- + M is a harmonic function of the transverse 6 + one needs a Killing symmetry in that di- 1 2 H 5 t locally. In fact, to construct a monopole one ] sense. To write a solution whose ADM mo- = P and allow a fibration of the resulting for the Hopf fibration as coming from the an- d 1 ) 1 20 S 2 3 r − ( S S × H 4 H . Then the momentum operator coordinate and obeys the Kaluza–Klein constraint 5 M R x =− For the monopole solution in gravity one then takes base, . This solution was found independently by Sorkin [ 2 + s 2 declares that the circle is fibred over the tum mechanical intuition whereby fieldseigenstates in depend momentum exponentially on the associated coor- dinate. At this point, without quantum mechanics charge quantisation is mysterious from thepoint purely gravitational of view. Thesestruct null for waves us are electrically charged solutionsity objects that in from con- five pure grav- dimensions. Theis next how natural question one toing may ask Dirac’s construction construct of magnetic magnetic monopolestromagnetism in charges? indicates elec- that Follow- insteadKaluza–Klein total of space as thinking a product of space, the mentum is rection. To require the mass to be also given by typically one takesdius this direction to be a circle of ra- uses the Hopf fibration where only using only a gravitationalseeks degrees of a freedom gravitational oneKaluza–Klein solution then direction. Momentum with is then momentum thought ofthe in in ADM [ the in that there is noalert mentioning of reader magnetic may charges. The feelthat although slightly the dissatisfied electromagnetic field withcharges is the are geometric the not. fact Thisclue may is be to take remedied seriously as the follows. identification of The momentum is the usual way Kaluza–Klein quantises electric charge, also needs a Killing direction ina time null and wave the such solution be that d and are called pp-waves. The solution is as follows: crete spectrum with S gular polar coordinates on the space transverse toIn a point. other words, the base space, direction which plays no role and monic function will be thethis momentum. has a Note Killing that symmetry since along on the metric ( the to make tion. One now fibres the Kaluza–Klein circle over the R The function with charge and mass. Now to realise charge and mass which differs slightly from the usual Dirac quantisation Proceedings hnteds ete n slf ihtefloigeleven- following the with left is one settles dust the When ihteeietfiain n fe elsaigo the of scaling Weyl a after and identifications these With aei h lvnhdrcin)Fnly foetit the twists one if Finally, direction.) eleventh the in wave where C C hr ehv dnie h Rone-form RR the identified have we where hstesrn opiglmti h ii nwhich in limit the is limit coupling strong the Thus h xetto au of value expectation The lvndmninllfig n te hn ont is note to the by thing restructured other are One M-theory fields lifting. of IIA eleven-dimensional structure the how rich from the comes of higher-dimen- Much the in theory. monopoles sional and waves become to brane. D6 the ob- tains one then direction eleventh mag- the null using Kaluza–Klein monopole a netic a make take to should circle one eleven-dimensional implies thus (The which direction. 11th BPS the is in D0 momentum the is brane D0 iesoa I Rthree-form RR IIA dimensional eae sflos he-omptnilo lvndi- eleven of potential three-form mensions, A be follows. then must as fields related other The field. vector Kaluza–Klein B d with: lift Kaluza–Klein tional rgam n ittemti n n-ompotential one-form and metric the lift Kaluza–Klein the and out programme carry to now is M-theory of essence inlthree-form sional tigculn nIAte sgvnb the by given is then IIA in coupling string S action: dimensional h e iesoa erci h I cina follows: as action IIA the in metric dimensional of ten terms the in metric eleven-dimensional identifi- the this relates allow cation super- to scaling eleven-dimensional Weyl of necessary The sector gravity. bosonic the the- as eleven-dimensional the ory identify can we then metric httecagdojc soitdto associated object charged sees one the intuition that Kaluza–Klein the following symme- Again eleven-dimensional try. full the recovers theory the 4 g g µν s s (11) 2 11 µ (10) (11) = 11 2 = µν ec,Mter emtie h 0adD branes D6 and D0 the geometrises M-theory Hence, oa lvndmninlmti.Ti sjs tradi- a just is This metric. eleven-dimensional an to Z µ = = wihi h SN w-omo I)adteten the and IIA) of two-form NS-NS the is (which R = d l G g R 11 p 11 µν B (11) 1 11 = x µν (10) ¶ p 2 3 g d d C µν C (10) . − x 3 , (11) µ gR stefil teghfrteeleven-dimen- the for strength field the is d . x oe rmcmiigtetwo-form, the combining from comes − ν C C + Z µνρ (11) h e seto h iti htthe that is lift the of aspect key The . R 48 1 11 2 = G (d e C 2 φ x µνρ (10) − stesrn coupling string the is 11 Z − (16) . C 6 1 C G 3 (10) µ ∧ d x htis: That . G µ ∧ ) 2 C C µ R (18) , 11 hc sthe is which C sfollow: as µ ihthe with g s The . (15) (19) (17) .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. ehv obndtemti ihteoefr Rfield RR one-form the with metric the combined have We h prahi a rmbiguiesl hsi what in Thus universal. being from far is approach The where ewn ocmieteprmtr ftelclsymme- local the of parameters the combine to want we etrt e2 be to vector S olw ewl att obn l the all combine to want will we follows fields? RR other the and two-form NS job. the about the what of but fraction small a done it only retrospect have in we but like results looks non-trivial of number huge a will reductions theories. Different string way. perturbative one different different give theory a eleven-dimensional in the reduce can has one once that nldstedltnfield dilaton the includes h ercadso and metric the ewe I n I n osnteiti h Heterotic the in in action The exist theories. not I type does or and IIB differ and fields IIA RR between the of spectrum the whereas pergravities metric, erc e smv otenx ipetcs ftyn to field, trying gauge of two-form case NS-NS simplest the next combine the the to move with us fields M- let gauge metric, (and one-form theory combine Kaluza–Klein to how in theory) seen have we that Now field double supergravity, NS-NS Lifting 2 object. higher-dimensional single a of part h i eiaiea eoecmie ihthe with combined before field as derivative vector Lie the a by generated feomorphisms omguetasomto hc sgnrtdb one- a parameter by gauge generated form is which transformation gauge form δ re noasingle a into tries case Kaluza–Klein the from intuition our following Now, meitl n esta hsrqie the requires this that sees one Immediately hr sa nsa rpryta h first the that property unusual an is there et ftevco r otaain ihrsetto respect with contravariant are vector the of nents ino h agn udeof bundle tangent exten- the This of bundle. sion cotangent and bundle tangent the of for Now bundle. tangent the the of sections as fields vector of thinks one Mathematically covariantly. transform ponents second the while diffeomorphisms dimensional B = µν theory hseee-iesoa itn a biul e to led obviously has lifting eleven-dimensional This h oa ymtisaenwthe now are symmetries local The generalised Z = d H d ∂ g [ x = µν µ p χ d hsN-Ssco scmo oaltesu- the all to common is sector NS-NS This . ν − B ] ge . d stefil teghof strength field the is etrfil ti eto ftedrc sum direct the of section a is it field vector dmninl othat so -dimensional, − 2 geometrise φ generalised ¡ R − χ µ 12 1 φ sfollows: as H sgvnby: given is l h ed n aethem make and fields the all 2 M − etrfield vector d ytectnetbundle cotangent the by 2 1 ( iesoswihalso which dimensions ∂φ B ) 2 d . p I ¢ dmninldif- -dimensional fr ed with fields -form (20) , = 1,...,2 V B a il Theory Field nal µν v I generalised µ d = ihthe with , U through ( compo- 1 two- (1) d d v Also, . µ com- , (21) χ ν d ). - Proceedings - ). 5 d ts d 24 (25) where -dimen- ν d χ . The next ]) we find , µ is the field g ¢ ∂ 24 p α d that does the = φ µ is not gauge in- N 2 B I J ∂ − -field, the super- ν µν µν e -field. (This is just M M ∂ B B F indices contracted. ∂ B = − δ − d d α 2 dimensions acting on MN ν MK − − e d B (1) one-form fields.) One M M µ 2 L KL ∂ by ∂ + dU M φ = KL N MN ∂ α µν M F M N KL ∂ N -dimensional two-form gauge trans- M this field strength ∂ d d M MN µν ). Remarkably, this can be done. (Given ∂ M where B M ∂ one-form fields. Under the gauge transfor- 24 2 1 2 α MN d − − µν diffeomorphisms do not contain the gauge ) for M transformations that are generated by a gener- F 8 1 d d 21 ¡ α generalised metric subject to the constraint ( d 2 µν − d F e First let us find the action by brute force. We will take = 1/4 S the 2 transformations of some additional surface terms as given in [ formations. This statement canamining be the seen Lie directly derivative by in ex- 2 under this symmetry. This isthere disappointing, is as it there means notRiemannian geometry a in lift higher dimensions. of Or in a other two-form gauge theory to manding that the actionter reduces imposing to ( the correct one af- the usual transformation of could just give up now and itexplains is why the perhaps Kaluza–Klein this lift result of which theoriesform with gauge higher fields has onlystead recently we been will developed. persevere In- with the knowledge that theit lift will be our goal to uncover its structure. a completely general two derivativealised action metric of and the dilaton gener- with all 2 gravity action has over 10that this terms, a priori thus this is it possible.) Fixing is these very coefficien unclear sional Einstein–Hilbert− action along with the term strength for related to the usual dilaton mation ( dimensional Einstein–Hilbert action gives the there is no antisymmetrisationnot the on gauge transformation indices for the and is thus that there are just sixcontractions, different and thus terms ignoring for an theaction overall alternative scaling this of means the justthe five bosonic coefficients NS-NS supergravity are action. fixed In termsderivative to of give a action two for the metric and for each new dimension. Under this reduction, the 2 step is to determine the local symmetries. Thethe idea full is 2 that the action for the generalised metric job is: variant and thus the action is manifestly not symmetric The coefficient for each term will then be fixed by de- The usual Lie derivative generates: (and allowing some integration by parts which implies words, 2 where we have introduced a new rescaled dilaton field will not be usual geometry but some generalisation and . - ) d d µ d µν di- ˜ x g (22) , (24) d µ x ( reduction = . I d dimensions. ). Reduction into a single from the nor- X developed by d ν 20 µ µν B B vector fields, one d etc. runfrom1,...,2 and J -dimensional Einstein– , denotes the original d I µν g M dimensional extended space . (23) d ). An impetuous reader might ]. etc. run from 1,..., ! 6 β ν 23 , αβ µ µ g B generalised geometry αν . forthis2 B d 2 α ν µ Ê α B B − coordinates. That is (for now) we will demand, are the new novel coordinates of the doubled M µν d ν ∗ g ˜ extra dimensions. x à T 0 is often denoted by: d = ⊕ = Inspired by Kaluza and Klein we then seek a so called The next step is to construct an action functional for M I J ν To be able to do soisomorphic we to will assume that the space is locally We take the doubled coordinates to be ˜ x M generalised metric that will combine the fields and the Greek indices generalised geometric object. This generalisedgiven metric by: is cussed in more detail later;it for as now it a is natural sufficient generalisation to of see the Kaluza–Klein metric additional dimensions but withnew the novel peculiarity dimensions that have the been assigned a metric One sees it follows the Kaluza–Klein metric ansatz for space. The capital Latin indices the generalised metric (and dilaton) that under mal Kaluza–Klein perspective are and is the basis for 1 of TM Hilbert action with generalised metric, dimensional manifold. Following our Kaluza–Klein intu- ition indicates that wegent should space not but extend only the extend space the itself tan- to 2 quickly do the reduction of the 2 mensions with the generalisedton). metric This (and does usual not dila- give the right answer. To see why Hitchin and Gualtieri [ be tempted to try the Einstein–Hilbert action in 2 here means that wethe remove new functional dependence on ∂ that: Kaluza–Klein constraint ( The origin of this metric and its properties will be dis- 1 with where we can use our previous Kaluza–Klein calculation to when acting on the fields. This is the equivalent of the will be equal to the supergravity action ( We then introduce coordinates on this doubled space. Proceedings ( where where hc en htoecnrieo oe nie ihei- with indices lower or raise can one that means which O sagoal defined globally a is hssadtofr rnfrain ( transformations two-form and phisms { deriv Lie usual the for tive: that recall Lie us generalised let First, the algebra derivative. by the generated study transformations and the itself of symmetry local the of tency Y hand at be: to case determined the is Y-tensor For the requirements. above the on determined be based to is that tensor invariant defined globally η 6 new the that be guid- The will deformed. principle be ing will derivative Lie the so and ture atarpeettv facoset: a of representative a fact L follows: as deformation a deriva- and Lie tive usual the of terms in derivative Lie generalised field vector alised lsdLedrvtv culygnrtslclcontinuous local generates actually derivative Lie alised iewe eue sn ( using reduced when tive toestecondition: the obeys It hrtegnrlsdmti or metric generalised the ther o swa losoet pi h oriae nthe on coordinates the split to one into space allows what is sor rnfrain ( transformations under invariant from indeed is far action is the Fortunately, obvious. transformations local under symmetry and the inverse so its and metric the in derivatives, written partial scalar of Ricci terms Basi- the of geometry. equivalent Riemannian the have in we cally, would covari- we using as constructed objects been ant not has check action non-trivial the a under since is invariant This derivative. is Lie metric generalised generalised the the con- whether check for the must action examine we the First, to above. is the of step all next of sistency The space. the on tion hogotteter.Tegnrlsdmti ( metric generalised The theory. the throughout M O 26 L ˆ J I ( J I ( V J I d O d U sta tlae the leaves it that is ) U = L K , ) ( , = d d L × I à rnfrain.This transformations. ) η = , 0 1 1 0 L = O V d IK η } L V ) ( d ! J I = M U V x ) η L U L K I µ . L K , steuulLedrvtv and derivative Lie usual the is [ x U + η ˜ ν (27a) , LJ , Y V hti talw st aeapolarisa- a have to us allows it is That . 26 V ] J I (29) , (30) . I .Te ems xmn h consis- the examine must we Then ). L K O ilpeev oeadtoa struc- additional some preserve will ( U d η , K J I d ∂ esr rca rpryof property crucial A tensor. ) 24 nain n hstegener- the thus and invariant J V hudgive should ) L O η (26) , J I ( d setal this Essentially . generalised , d tutr appears structure ) 21 .W rt the write We ). d Y diffeomor- i deriva- Lie J I 23 L K sin is ) η (27b) (28) ten- sa is .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. a- hr h atdtrsaetegnrlsdLederivates Lie generalised the are terms hatted the where ( ekrcntan ssilakycalnefrDul Field Double for challenge key a still is constraint weaker with hsi biul ovdb h auaKenlk con- like Kaluza–Klein the by solved obviously is This h eodtr nteagba( algebra the in term second The hc sdfie as: defined is which ie y( by given { that: discover we Now ∂ alternative the with theory larger the choice: in solved constraint be above the can that see one however away Straight ue hoy tl by qiesrnet constraint, stringent) (quite ‘unre- a the obeys Here still constraints. theory’ no duced has theory unreduced bv agba fgnrlsdLedrvtvs( derivatives Lie generalised the of this ‘algebra’ to addition above In worry. should one point this At ing. η constraint’: ( ‘strong constraint following weak the called products so on the constraint then fields the of implement how to about expects here said one be if to more much is There theorists. the obeys only that theory full the with Working theory. ( straint η on constraint covariant by a given is- impose fields these all must one to that resolution is The identity. sues Jacobi the obey not auaKenprdg,terdcino h hoyby ( the theory from the obeys departure of definition reduction clear the first paradigm, the Kaluza–Klein see we though Here ∂ ∂ theory: the construct to using been have we that straint coordi- the of terms in ap- nates constraint constraint the this out many with Writing later disappear plied. see also will will we issues As other closes. algebra the then plied [ U 33 L x x x J I J I ˜ ˆ µ µ µ , U .Nt httecntan ( constraint the that Note ). ∂ ∂ φ φ ∂ V I I , x ˜ φ L = = ] φ∂ ∂ µ x C M ˆ φ J n edi h hoy hnti osriti ap- is constraint this When theory. the in field any 0. 0, µ V φ 35 J = , = } ψ x = ˜ = U n oi scranysilardcino the of reduction a still certainly is it so and ) 26 ν ∂ = 0 , µ L P n h [.,.] the and ) ∂ ˆ 0. ∂ [ µ ˜ U P φ V , V = M ] C 35 .(34) 0. + − u o auaKenter the theory Kaluza–Klein for but ) a 2 1 η η J I MN ∂ C I η ∂ stes ald‘C-bracket called so the is PQ J 33 (31) , U 31 swae hntecon- the than weaker is ) P rvnsi rmclos- from it prevents ) ∂ N V Q − ( 33 a il Theory Field nal U becomes ) ↔ 31 V does ) ). (36) (37) (33) (35) (32) Proceedings g 7 (40) (38) (39) -field in B . Relating the two g p -field transform when there is B , ˜ φ isometries, the transformations form d . ]). Then as expected the ADM momentum φ . 2 ) of generalised Lie derivatives that captures 27 g − . ˜ – d φ and its dual ) group. This is immediately realised in double , 2 ge H -dimensional section being 25 − d d φ ambiguity. The different choices will then imply ∗ , ( e p d will be associated with the electric charge of the 2 -dimensional space-time then we need a measure d O ν Z = = ( ˜ x Z d φ d = O p 2 2 e − − the local symmetries of theton theory. under The T-duality is shift captured in byton the relating to dila- the the DFT usual dilaton dila- andfor requiring the the usual measure field theory as a linear transformation onmetric. the It generalised is in this sense that double field theory makes local an an isometry are knownis as manifest the in Buscher the rules. doubleof All field of the theory this as ambiguity a inan consequence isometry. section For choice in the presence of higher dimensions). These differentthrough choices what are is related called T-duality anding the how rules the determin- metric and three-form field strength: different choices of metric for space-time (and in time. However when we do thistify that different means components we of will the iden- generalisedthe metric space-time metric. with Thus with ais single a isometry there dual theories then produces the usualthat dilaton the shift. Note generalised metric hasdoes unit not determinant contribute and to so the measure;is in the DFT the DFT measure dilaton.our Once we pick a section and choose e on this space which we doDFT with dilaton a as field follows, redefinition of the e For different choice of section this willrise be different to givin the induced dilationdilaton transformation between the 2.2 Charged states the charged states ofprobes the of theory. the Rather backgroundthe than and approach we will examine their take will geodesic be to equation have look for ADM solutions that type of charges. First, we should developpeople an [ Again we follow the Kaluza–Klein intuition and look at ADM formalism for DFT (this has been done by a set of without isometries there is no duality but there is still the Q ‘duality manifest’. It is important to realise though that . - - ˜ ˜ z x z d d in , co- x ˜ z ˜ x -field in and B z only. There is ν ˜ x coordinates or all coordinates is like hav- -dimensional NS-NS su- space is ambiguous. One x d d nor on its canonical pair d z or the space with coordinates z coordinates. Reducing the theory to , ν x ˜ x ) as a choice for solving the strong con- coordinates. To be concrete let us consider 37 d . This background obviously solves the strong -dimensional supergravity. The independence ˜ x -dimensional subspace within the doubled 2 d d tensor gives a pairing between every coordinate isometries (although written in non-covariant lan- η d dimensional space in whichconstructed. When the we solve double the strong field constraint andtermine de- theory which coordinates the is fields depend on thengives this us a canonical choice foris the choice subspace, of section, for that whichif those one are uses the ( coordinates. Now, constraint and even does so degenerately for and every ordinate which we will label ductions of the doublepergravity. field Above we theory chose (DFT) all that give su- 2.1 T-duality in double field theory pergravity. could take the choice of section to be the space equipped straint then the canonical choicespanned of by section the is the space so that now the fields will depend on these choice of section then reproduces somethingpected. unex- One again reduces to Since neither coordinate appearsbe in included the trivially solution in our it choice can of what we call space- dimensional subspace of the 2 no Kaluza–Klein equivalent statement.terpret How this can choice we of in- First solution let of us the discuss strongtime how constraint? within we the identify doubled our space.straint We physical solve to space- the discover strong that con- ourton generalised depend metric on and dila- those a coordinates as set corresponding to those of of ourtime. coordinates. space- We We then then make identify tion’ what (the is strong constraint known is as alsodition called a the by ‘choice section some). of con- The sec- choicetion of of a section is the identifica- that the fields dothe not section depend condition on prescription to either determine of the them. Now ordinates but obviously one can make a mixedlands set on pro- a particular background given by a metric and guage). Now let us considerditional the isometries case and where the we generalisedon have less metric ad- than depends of the generalised metric on ing vided the strong constraint is obeyed. In each case one (the which there was no dependence on some particular co- with coordinates We see in the above that there are thus a variety of re- Proceedings hntesolution the Then h betta are uha lcrccag sthe is charge electric an such carries that object The ( hsDTwv ouinmynwb xmndwt an with examined be now may solution wave DFT This with n hsteTda bet fwv n tigaeasin- a are string and wave of objects T-dual the thus And foeo h ol fsrn hoy ormv singulari- remove to theory, solutions. string supergravity from of ties goals realisation the a have of thus one We of is. solution string the though given papers. prescription these the in using are we DFT in solution the of 8 inta has that usual tion in string the of orientation space-time.) the to corresponds rne ae efis nrdc h coordinates: plane the null introduce a first of We structure wave. the fronted have will it approach Klein 2 tig Tedrcino h oetmi the in momentum the of direction (The string. lentv hieo eto reuvlnl utrotate to just propagation of equivalently direction or the section of choice alternative e rl.Teeuvln oteehv encntutdin constructed [ been in have DFT inte- these like to Komar equivalent the The and mass grals. ADM the allows of This definition asymptotically). means the least turn (at in symmetry which Killing global charge a a conserved be a to for fit- needs As symmetry there exist. not intuition Noether’s do they with solution ting generic a construct for to fact care in as and some such require currents momentum conserved and of energy gen- notion in the that aware relativity be eral will string reader versed a well The as wave. space-time or in DFT what interpretation the is the momentum charge determines the single of a orientation with The DFT momentum. in solution wave gle supergravity. of solution wave usual the recovers one so t d t , 2 s , z ieeeeti rbeai ic ti not is it since a problematic of is concept element the line though conveniently even to metric as generalised so the elements encode line as solutions DFT give We d 2 z , oe h aeslto satal o iglreven singular not actually is solution wave the Note, enwse ouint h F qain fmo- of equations DFT the to solution a seek now We = , t = = ˜ y , osat (41b) constant. z m ( ˜ M H + + ietoswt the with directions , J I 25 t ˜ d 2( − , d z y – ˜ 2)(d H X , 2 27 y ˜ + I − p m ota hnw ee otemomentum the to refer we when that so ] d d x t ˜ 1)(d otewv ilb retdi the in oriented be will wave the so ) X ν 2 y ˜ − J oetmadfloigteKaluza– the following and momentum 2 2 d ab rte sflos[ follows as written maybe t d z z 2 ˜ ) + − d H t ˜ y d (d z and t ) ˜ 2 + x − y ˜ pc.We n does one When space. d ietostransverse. directions z ˜ 2 O ) + ( d , d ) -invariant. 28 ]: x ˜ X space (41a) M .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. = lclsn’tebaeo h icei oehn htleads that something is circle the on brane the ‘localising’ ewl o aetefir ob nthe in be to fibre the take now will We h F iao snow: is dilaton DFT The hsi ngnrlntpsil.Ti osrcindoes construction This possible. not general in is This hr ifrn hie fscingv T-duals. give section of choices different where Q hc steitga fthe of integral the is which h aecodnt ytma before: as system coordinate same the oooe o oedtisseteppr[ paper the see details more For monopole. and e rs–er–ooi nazadfir iceaon a around the circle follow transverse a will We fibre charges. and magnetic ansatz the Gross–Perry–Sorokin find to is step lorqiiga diinlcrl ota n a use: can one that so circle by additional so an does requiring but also cohomology in flux three-form a produce gerbe. non-trivial for space geometric total of a sense producing the In gerbe. the geometrising to close perilously r -uls gi eaepouigasnl solution single a producing are we again so T-dual five-brane are the and produces monopole solu- section The The of Kaluza–Klein-monopole. choice string. alternative the the of the with as dual tion surprise magnetic no the is is This five-brane metric). the in appearing tion ( cycle rn nacrl.Mkn oegnrlflxsadalso and fluxes general more pro- Making not five- circle. NS a is smeared on a one but brane fact five-brane NS in general Thus the ducing fibration. Hopf a through oy avyadMoe[ Moore and Gre- Harvey by gory, solution. 1998 in the back predicted of first corrections’ were corrections mode Such ‘winding called so to xmn hsslto ihteuulcoc fscini.e. section of choice usual the with solution this examine ei charge: netic n h sa oooecntuto oform to construction monopole usual the and x H d − s M 3 pc hni steN v-rn ouinwt mag- with solution five-brane NS the is it then space 2 2 ( d o htw aedn h lcrccags h next the charges, electric the done have we that Now o h tetv edrmywryw aecome have we worry may reader attentive the Now S = = = A = 1 i Z M H H × He and + + + (1 MN S H hudntb ofsdwt h amncfunc- harmonic the with confused be not should 2 η H 2 − + H ab ) S ( 2 H d = δ φ H − 2 d X j i 0 1 r eae swt h Gross–Perry–Sorokin the with as related are H sbfr ofr h offirto of fibration Hopf the form to before as x − (42b) , A M + 2 a 2 i d A d ( (d H S x X 2 2 )d − y b N ) 2 i + H z d A 2 η z 1 ˜ i + ( ab A − S 30 H j 1 δ d )d H ,(44) ), j i x − n hnraie nDTin DFT in realised then and ] ˜ a flxoe rnvrethree transverse a over -flux 1 y d d d i y d ˜ z x ˜ ˜ j y 2 b d + j . z + ) + H − x 1 ˜ δ pc n use and space j i 29 a il Theory Field nal d .We we When ]. y ˜ i d y ˜ H j (42a) + 2 (43) ( S S 2 3 ) . Proceedings 9 ) coset. ) coset. d d ( ( O SO × -form poten- )/ ) p ]. These gener- d d ( ( 42 O ) curved space in- – GL d )/ ( 39 d . , d GL d E ( ) cosets but with different O d , ] and investigate different ex- -field into a generalised met- d ( B ]. One unfortunate property of 43 O ] for an excellent review of non- 44 38 supergravity, exceptional ) tangent space index. Generalised ge- d ( d SO 11 In the case of usual Kaluza–Klein theory, one could ] has explored having field theory -form potentials such as the RR forms of type theories 43 p terises this coset. What[ then for other cosets? Recently ometry is concerned with the dex and one possibly asymmetric tangent space groups. ThisDouble leads field to theory describing a non-RiemannianNow, can space. we ask about cosetsRemarkably, of exceptional the cosets exceptional groups? will provideright us geometry with the for eleven-dimensionalfact one supergravity. can In then follow [ tials of eleven-dimensional supergravity thenall we can of do string theory.from We an can even also more comebe formal though at of direction. as this the Usual description question gravity of the can ism where the vierbein has one or the three-form and six form ofpergravity? eleven-dimensional The su- RR sector ofdimensional string supergravity theory as described lifts in our to description eleven- of M-theory above so if we can lift the ric of aspace with twice the dimension. What about other ceptional cosets leading to a host ofometries non-Riemannian in ge- M-theory [ the exceptional groups is that typically one hasto to make a deal general statement for geometric fluxes. consider more sophisticatedScherk–Schwarz-type where reductions one introduces such twistces matri- to as allow non-Abelian gauge the fieldsory. in One the can reduced do the- this alsoScherk–Schwarz with reductions. DFT This and leads have generalised togravity gauged where the super- gauging is related to theis twist field. described This in aalised Scherk–Schwarz series reductions of are one worksplications of [ the of key double ap- field theory.pergravities They that allow otherwise gauged would su- from have a no higher-dimensional theory. lift or origin 3 Lifting Kaluza–Klein theory incorporates vectormetric fields in into one dimension the higher.corporates Double the field two-form theory in- in the solution. See [ The generalised metric of the previous section parame- This is most conveniently seen in the vierbein formal- with them on a case by case basis rather than being able ] 2 2 ˜ )- 1 R di- d 35 , ˜ x -flux d f ( O -flux and direction H ˜ x dependence . Remarkably ˜ 2 x − ˜ R -flux was shown ] known as the 5 Q = then the string ten- ′ 1 36 α ]. This is one of the ap- →∞ with a scale. One might 34 ˜ , R ˜ x , (45) flux seemed something new 33 , Q -branes. With the abc 29 p R , 32 → , c ]. As such, exotic branes although so- 31 ab are the usual three-form and geomet- 37 Q f and the momentumquantised in units of → ˜ R bc and a f flux even more exotic and contentious since it H -flux and the so called twisted torus. They are toy → H R 0. In this case the string tension would go to zero and But what more? Now we have an extended theory Finally, in order for this to work we require the flux was sourced by an NS5-brane and the → abc ˜ H H brane which although it locally solvestions supergravity equa- of motion it requires a patching with an a series of works [ so called geometric flux (the measuretwist). of This the was geometrical done with a toy model of the three-torus this then gives the string tension as all be it with anew constraint. novel We can directions non-trivially while fibreto still the make obeying non-trivial the constraint fluxes. Sometime ago people [ sion goes to infinity and we only have supergravity. models because they are not solutions of supergravitystring (or theory low energy effective) equations of motion. lutions of string theory are not globallyof defined solutions supergravity. They areof of DFT. course Branes source bona-fide R-flux solutions canare also even be found more but exotic they choice of as section condition so they that there require is a non-canonical looked at what happens when one T-dualises in the right variableseled (those with of a Poisson DFT) equation. they may be mod- plications of DFT, one simplysource of allows the the harmonic delta function to function ing space be and localised one in recovers a wind- harmonic function that was ric fluxes in supergravity. by the Kaluza–Klein monopole.to The be sourced by an exotic brane [ then the string tension is spontaneouslycompactification generated i.e. from having one is tempted to think we area describing string tensionless theory phase. in Equally if ask what happens in the decompactificationR limit where and seemed to require a T-dualisation inrection. a Looking non-isometric at di- actual solutions in supergravity, the transformation [ rection to be a circle andrection then must the be momentum quantised in and thisDirac so di- we quantisation then for get the usual This gave rise to the following sequence of fluxes: with where winding mode dependencies. Those are the stringworld-sheet theory instanton corrections. Thiswindings works of string because are dual to momentum modes and so with radius Proceedings G where h eeaie ifoopimwl egnrtdb a by generated be will diffeomorphism generalised The eddi obefil hoyaditouecoordinates introduce and theory field double in did we hr h motn oeo hsfco nUdaiyi dis- is U-duality in factor this of role important the where esatdwt n-omo auaKenadte the then and Kaluza–Klein of one-form a with started we h edrwridaotteoealfco fdet( of factor overall the about worried reader The ,.,.Teguetasomto of transformation gauge The 1,...,4. form inlsaei em fteuulmetric usual the of terms in space sional oriae ilb e ftasaingnrtr rgen- or generators momenta: translation eralised of set a be will coordinates oewt h indices the with note 10 ie yteetro eiaieof derivative exterior the by given ersnain h sa coordinates usual The representation. eeaie vector generalised with diffeomorphism’. symmetry ‘generalised local one this onto combine diffeomorphism to want we before as ic hr r orcmoetof component of nents four are there since SL gener- this coset: the fact of representation In a two-forms. is metric on alised metric induced the is { oeby note eti h metric the is tent later). discussed be will as natural M-theory in bedding o nAeinp-form, Abelian an for M T but geometry Riemannian on: with as bundle tangent usual cussed. h biu etse,tersrcint 4 to restriction the step, next obvious the is theory three-form the supergravity, NS-NS of two-form four in that (Given potential space. involved three-form higher-dimensional a issues a of to the dimensions lifting of the most study illustrate and will that ample h eeaie erci norgdt ed[ read to encouraged is metric generalised the g y x [ µν J I αβ µ (5)/ , n hnwie onamti nti e dimen- ten this on metric a down writes then One ecmieteecodntst rdc an produce to coordinates these combine We esflo h aepoesa eoe h edcon- field The before. as process same the follow Lets oe hta ihDTti snwamti o nthe on not metric a now is this DFT with as that Note, omk rges ewl o osdrasml ex- simple a consider now will we progress, make To , = σδ ] y ⊕ C [ oriae eoethe become coordinates (det( µν Λ SO = µνρ 2 ] C htflostesm structure. same the follows that } χ T 1 2 (5). µνρ ( [ ∗ sfollows: as g µν g M )) µσ ] n o ae h oefloigwhat following move the makes now One . n ic eaei ordimensions four in are we since and − . g 1/2 νδ à g − P g p V µν 1 g µν I 2 I . µδ C I δ n h he-omwihw de- we which three-form the and + = , ν C g J γδ 1 2 (3) ( νσ = C v µ ,(46b) ), µ ,.,0 soitdt these to Associated 1,...,10. = , ǫτ χ d C 10 [ χ µν νǫτ (2) ] of C .Where ). n t edstrength field its and , g p scalled is SL 1 v γδ C 2 µ x C , 5 hc ede- we which (5) σρ µ steuulone usual the is d µ u i compo- six but σρ n e novel new and g ae h em- the makes µν I ! 45 H = (46a) , n three- and n [ and ] = 1,...,10, d µ C SL g , (48) (47) So . in ) ν 46 (5) .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. = ] uldsuso fteEcpinlFedTer (EFT) Theory Field Exceptional the of discussion full A eke eedneon dependence keep we hr ermv h eedneo h oe coordi- novel the on dependence the remove we where hti when is That ihthe with ie en nthe in being dices h -esrbiggvnby: given being Y-tensor with the achieved is This transformations. gauge and three-form diffeomorphisms become must transformations cal a banete 11 we either space-time) obtain choosing being can as mean identify we we that this coordinates (by the section of choices different make we su- when now IIB that says and This respectively. pergravity supergravity eleven-dimensional to related oa ler sgvni [ in given is algebra local ic ehv nymngdt e hoyi or(or four in theory are a get dimensions) to ten managed indeed only have (or we eleven since the where out group. U-duality the realise to sufficient U- isometries of have T- the number must to to one lead led again section transformations; isometries duality of of choices presence different the Now duality. in of DFT choices in Different section dimensions). mem- spatial the two has because brane (essentially dimension preserve not does M-theory in moment and winding exchanging since Y alge- the case, DFT unless: the close in not seek will as bra now Just can algebra. the one of derivative closure Lie generalised the Given ibr iha appropriate an with Hilbert osti s[ is this does ae evn swt h original the with us leaving nates oewr htteeaetoidpnetsltosto solutions [ independent condition two this are there that work some constraint the under and tensors ant from constructed be must Y-tensor the Now ie ytegnrlsdLedrvtv si qain( equation in as derivative Lie are generalised symmetries the by local given The symmetries. is local step the next at the look action, to the and space the on metric alised n o rvt n he-omoc n eoe the removes one coordinates once novel the three-form of dependence a and gravity for one space. novel new the with space usual the ing S J I = h suerae ilb tti on rigt work to trying point this at be will reader astute The gi n a e auaKensyeo structure of style Kaluza–Klein a see can one Again ohvn osrce h oriae,tegener- the coordinates, the constructed having So h etse st n nato htrdcst the to reduces that action an find to is step next The PQ 12 1 + ∂ M 84 I 1 ∂ C MN M J fil en h f ignlcmoet mix- components diagonal off the being -field = MN ∂ 17 0. M ∂ [ ]: ( µν M 48 M ] PQ = · pt rva rnfrain.Oeis One transformations. trivial to up ] L K 10 d ∂ ∂ h cinwl eoe Einstein– becomes will action the 0 N M rIBi 10 in IIB or of M M SL PQ y L K 47 14 5 and (5) − Y )( .Oecnte hwwith show then can One ]. , y H J I H 2 1 24 M PQ 2 PQ , d y MN em h cinthat action The term. x hsi o surprise a not is This . ∂ 34 = µ i N h te swhere is other The . hs hie are choices These . ǫ y ∂ H en a being Ji I [ M µν PQ ∂ M ǫ [ ] µν PQi rmtefields. the from ). PQ ] a il Theory Field nal = · SL ∂ with 5 P 5 invari- (5) h lo- the 0 M of NQ I SL , J (50) (49) 26 + (5). in- ). Proceedings ] 11 µν [ y , µ 1 x 16 10 27 56 R 248 . There is ] direction or ) . In fact since ] ] C µνρστ (8) µν [ ∗ (16) [ (4,4) ∗ y (2,3) ∗ y (5, µνρστ [ SU SO y SO SO USp (5) , internal space. SO (8) d (8) (5) (16) × and the dimension of the coordinate SO SU SO (5) USp H / SO G direction or as a monopole with Hopf ] 5. We will now reverse the logic from the µν [ (5) (5,5) 7(7) 8(8) 6(6) y = E E E SL d SO The cosets 5 7 8 4 6 dG H H Once we have sufficient number of dimensions to in- Before moving on lets consider some of the cases describe this. But when wethen we have must include 5 coordinates dimensions for wound orthus five-branes augmenting more the previous set with then one more subtlety, which is that weD6-branes can have which wound from the eleven-dimensionaltive perspec- is a Kaluza–Klein monopole withrections 6 and world-volume di- one Hopf fibrehave direction 6 and antisymmetrised thus coordinates it and would nate. a When one Hopf takes coordi- this into accountdimension then of the one coordinate gets representations the in the table. clude the five-brane winding coordinates something new happens when we considernates for solutions. both We the have wrapped membranefive-brane coordi- and the but wrapped these areother. electromagnetic Thus duals following the of previous each two logic ways there to would get be the the five-brane same as solution. a One null wave could in describe the above. In the previouslift instances the local we symmetries, lookedsolutions form at that an how correspond action to to andspace. then strings The novel find and coordinates branes wereing then in related modes the to of wind- these branes.edge Now of we branes will in useand M-theory our then to knowl- one establish can whatM-theory follow to has the membranes expect and same five-branes. When procedure we con- sider as the before. d dimensional internal spacebranes we must can ask winding what in thisis space. Up only to the 5 membrane dimensions it and so the coordinates in other dimensions and gainble some of intuition cosets. for the Letstance ta- us of see what happens in the next in- as a monopole with Hopf fibre given by representation for dimension electromagnetic duality isgroup contained these must be in the same the solution infield the theory. U-duality exceptional In fact one wouldment then for make the the same membrane argu- and describe it as either a null Table 1 wave in the ] 1 I d a of R A 52 M – d 49 will be en- ] puts this 7 52 M – . We have also along with G 1 49 . The generalised Lie ). The section condi- d 7 E E directions. Thus we asso- for a given dimension directions give a membrane µν H ]. . At thispointthe / µν 7 y 56 G M – × 53 4 ). and then add novel coordinates to 7 d M E − reproduces a wrapped five brane. This is = 11 ] (5) generalised geometry and 7 dimensions 11 M µν [ SL M × y for when the internal space is Lorentzian. d ∗ space. What is now called exceptional field theory M H d = − The relevant coset, The next steps then are to look at the states of the the- Finally, the reader will have noticed we have rather The next step is to consider the new coordinates fi- 11 11 M M the ‘internal space’ and describes howthe one other. is The fibred over local symmetriesconsistent between need both spaces. to All be these considerations madelead to to a be highly complex theoryries that of is works described in initially a by se- Hohm and Samtleben [ three) dimensions. The way one thinks about thislows. is Take as fol- the internal space are listed in the table tirely trivial e.g. a seven torus which nothing depends on. additional 6 novel dimensions to givescribed 10 by dimensions de- described by usual Riemannian geometry.have Therefore 17 we dimensions in total before thenchoice making to a go section down to elevenortendimensions.general Thisisthe story. We split the eleven-dimensional space as and then by others [ back in and allows forover the arbitrary other. The fibrations space we of have been oneby dealing the space with name goes ‘internal space’ andmannian the space is here called neglected the Rie- ‘external space’. Thethat are have field indices both spaces most importantly a field to make them a representation of listed tion coming from the closurethe of same the once written algebra in will termsto also of and be the including Y-tensor (again up ory just as we didmomenta in in Kaluza–Klein the theory novel and DFT. Null derivative will be the same, only the Y-tensor will change as developed by Hohm and Samtleben [ brutally truncated the theory by completely ignoring the the state with momentum givesstate. the electrically charged exactly keeping with the intuition from previous cases bred non-trivially over theKlein space-time type monopole base. solution A whoseordinates Kaluza– Hopf fibre has co- solution wrapped on the ciate the new novel directionsmodes. with membrane winding Then we augment the four-dimensional space with the which is a one-formin the ‘externalspace’ and a vector in which is the coordinate representation of (this is true up to and including where the fibred solution gives the magnetic state and Proceedings ihtemmnu nteeeet ieso.This dimension. eleventh the in momentum the with hr efda tigslto ie iet h ttsin states the to rise gives solution string self-dual a where Tewr nti eto sbsdo olbrto with collaboration a on based is section this in work (The iesosta rnfr ne h 4d the under transform that dimensions 4 12 hoyi h atta h yeI ueagba ntndi- superalgebra. eleven-dimensional ten unique in the superalgebras to lifted II be type can mensions the that fact the is theory dnicto a ese ietyb oprn h BPS the comparing by directly seen be can identification ten D0-brane the the identify to must associated we charge algebra central IIA dimensional type the lifting In faShr–cwr-yeast o euigET[ EFT reducing for ansatz Scherk–Schwarz-type a of theory. higher-dimensional the sin- in a to solution from gle lifted come states be these can that and theory dimensions the higher that suggests some under it transforming group states duality with theory true. a always have is we this If that U- suspect the might under One transform group. that duality states brane the to way rise different in gives reduction Its string. self-dual the of sion dimensions 6 in theory (2,0) the states. with bound familiar their those and For M-theory in branes the describe to shown and constructed were EFT to solutions self-dual by given fibre in htata hi ore hsi eotdi depth in reported is This source. [ solu- their in brane as exotic act appropriate that the tions and DFT de- fluxes for exotic scribed the of generalisations M-theory to leads ieto.Ti srpre ndti n[ in detail in dual reported electromagnetic is the This one in direction. in monopole wave a a be and will direction it that meaning ‘self-dual’ be must acl er [ Perry Malcolm Superalgebras 4 EFT. using constructed full the use and the by given ansatz. Scherk–Schwarz is the turn of matrix in twist which tensor embedding so the called gauged by determined produces is is gauging result the where The supergravities consistent. is it show can 60 esoa euto natrs hsi nlgu.The analogous. [ is in this described torus, solution a on reduction mensional { Q .Ti rasteuulscincniinadytone yet and condition section usual the breaks This ]. α 61 olwn h auaKenitiinlast h idea the to leads intuition Kaluza–Klein the Following ial,w a osdrmr opiae fibrations complicated more consider can we Finally, = , Q – P β 63 µ } ( = hr uesetu fsltoshv been have solutions of spectrum huge a where ] C Γ µ ) αβ y [ 64 µνρστ + E d Z ) ehp h trigpitfrM- for point starting the Perhaps ]). smer opthsltos This solutions. patch to -symmetry µν ] h nwri httesolution the that is answer The . ( 57 C Γ sasr fgaiainlver- gravitational of sort a is ] µν ) αβ + Z µ 1 ··· µ 57 5 ( C hr these where ] SL Γ µ 2 fe di- after (2) 1 ··· µ 5 ) αβ (51) 58 .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. s – ewl hneaieterpeetto hoyfrthis for theory representation the examine then will We w ilol consider only will (we hsi o e da ubro uhr aepur- have authors of number A idea. new a not is This h 0baeietfiainwt h ulwv nM- in wave null the with identification brane D0 The 4 Y theory, con- the quadratic in momenta obey the to on straints shown be then will resentations ueagba The superalgebra. { schematically: is that charges, central without gebra u f ytefc httecnrlcags( charges central the that fact the by off put theory. dimensional e iesos ewl hseaiesm pcfi ex- dimension, specific with some amples examine thus will We dimensions. these new in momenta as superalgebra charges central eleven-dimensional the interpreting the and lifting of idea the P wave: null a for equation the to dawsntcda h eynisneo xeddsu- the extended of charges of this central naissance the of where very seed in the the persymmetry at fact In noticed forms. was various idea in idea this sued dimensions. extra the in momenta their from would dimensions arise eleven only in tensions tensionless effective be their would and in branes null-waves the from all result i.e. would dimensions extra branes central dimensions? the the extra all then in all so momenta If reinterpret from arising could as one charges Can again. trick rlsdgoer ecie bv sstu od where do to up the set is above described geometry eralised ieso.Loiga h lvndmninlsuperal- ( eleven-dimensional gebra the di- at eleventh ten the Looking in in dimension. momentum its mass of result effective a just Its is mensions massless. actually is brane rlcharge, cen the tral with similarity formal this realises exactly theory P equation: state saLrnztofr n v om loteobjects i.e. the charges Also central form. the five with and contracted two-form Lorentz a as xlrdfrtes called central so the the for interpreting explored were of interpretations alternative idea immedi- and momenta the This as charges end algebra! Clifford to a seems obey ately to appear not do X Q d 0 0 J I I α 2 2 = h edrtmtdb hsie ilimdaeybe immediately will idea this by tempted reader The ie h icsinaoei em aua orevisit to natural seems it above discussion the Given oee,ti seatywa h oto xeddgen- extended of sort the what exactly is this However, hoywr ecie scmn rmsm higher- some from coming as described were theory PQ generalised , | = | = Q ( x β P P Z 51 µ } I | | , 2 = P 2 ti epigt e foecudtytesame the try could one if see to tempting is it ) y . J µν ( C = Z , Γ 0, y I µ oriae nteetne emtyare geometry extended the in coordinates en dnie with identified being ) 1 αβ ··· P µ massless P 5 I I .(54) ). P (55) . d J δ < J I d -hoyalgebra M-theory 6): = = i h eeaie es)rep- sense) generalised the (in 01 ihasml superal- simple a with 10,16 .(56) 0. P 11 hsteD0- the Thus . C 51 [ Γ 12 a il Theory Field nal µν transform ) , 65 , C ]. N Γ µ 1 (53) (52) = ··· µ 2, 5 - Proceedings , H 13 (5). and (60) a SO 0 and its g αβ (5)/ ) ). C SL 60 ]. Thus instead (2,3). Along with 67 ]. To see why this is SO 68 , (59a) ´ 1,...,4. Consequently we kl = kl a , ) (59b) V α 2 is 14 with signature given by mn nk 1 (5) theory that is suited to what p g g 3 SL ml to be given by bef g kl C − a . (59c) ef V as follows: nl 1,...,3) which gives a total of six nega- is a metric on the coset of with which we can lower and raise spinor a 1,...,5 (61) g C = I J klpq = 2 1 αβ abc mn mk ) η i G a C b 1 g + ( − V is Lorentzian. The reason for this, aside from pq (2,3). 2 1 2 ab a C 1 p g C = , ab SO ³ (with g β = kl , = α In order to make the theory supersymmetric we will We will now introduce spinors of the local group (2,3), with spinor index ) (5)/ kl ab i Not to be confused with the dimension of the space itself on which this metric acts which is of course 10. I J 0 a mn Γ g any desire to constructtion, a simply theory comes with from the ain usual temporal various restriction dimensions direc- of and spinors signatures [ SO follows. require that the spacethus have a Lorentzian signature and ous description of the this we have the charge conjugation matrix ( then have the associated gamma matrices: ( is the induced metric on antisymmetric bi-vectors and coset structures are discussed in [ 3 of the usual Euclidean 14-dimensionalsignature coset we will with need (0,14) to work with the coset: SL the appropriate choice offields coset in the we coset simply and count examine the numberwithrections. the These negativedi- negative directions are given by: tive directions leaving 8 positive directions whichmatches indeed the counting of the coset given in ( inverse ( indices respectively through left multiplication. three-form This is just a more convenient rewriting from the previ- V The metric This coset’s dimension which form the Clifford algebra for we have defined (10,14)-(4,6)=(6,8). The various choices of Lorentzian where C G , of 66 and (57) (58) 10 1 and ab = and the g (5) as fol- a where the 1,...,4),six x 1,...,10. As- SL 1234 physical sec- = η (5) = of a J (5) U-duality in- , are in the ,( SL I 10 a ] SL x ij [ light cone z become the ab . (5) but the spinors of the theory y are introduced to make up a ten di- cd ] SO y ab [ y (5)/ abcd that was discovered previously from de- η SL a is the alternating symbol, i.e. 1 2 x = = . 5 abcd I a ab η z z P 1,...,5 and the coordinates ] and the second condition is just the statement of = = = geometry, the simple example of = ] ] This shows the remarkable interplay between the We now recombine the coordinates to produce an One then writesdown a generalised metricon this ten 47 (5) representation. The usual coordinates (5) which we denote with the index , j ij ij [ [ , z lows: realisation of membranes windings i 4.1 U-duality and its realisation in an extended global duality symmetry, the local symmetries and thepersymmetry su- in the extended space. Thisstraint quadratic on momenta con- restricts the effective degreesdom of and free- avoids the various no-go theoreming for construct- supersymmetric theories in dimensionseleven. higher than then lie in the interior of the cone. new coordinates mensional space. (We move here tocate Latin we indices must to consider indi- different possible signatures).key The to making supersymmetrylive work in is the local that group the H. Thus spinors for the being massless (in the generalisedsentations theory). will Other not repre- obeybe this contained constraint in and the so usualextended will formulation geometry where not of the generalised section or conditionimposed. must be In terms of themassless usual theory states these will generalised becondition the determines a 1/2 generalised BPS states. The section SL tion condition manding closure of the algebra a local symmetries [ SL dimensional space in terms of the usual metric P sociated with these coordinatestion will generators be or generalised a momenta: set of transla- variant action the bosonic sector is given by a nonlinear Along side the usual four coordinates, 1/2 BPS lie, the other states with less supersymmetry 18 The first quadratic constraint is exactly the will be a representation of Spin(5). Where Proceedings h ssplmne yteLrnzagbafor algebra Lorentz the by supplemented is The where hc ilatnto h etrrpeetto u on but representation vector the on not act will which ihteovosietfiainof identification obvious the with aihs n hwta hyfr‘hr multiplets’. form‘short they that show and vanishes, e { the set and space-time four-dimensional usual the in menta r ler sn hssto eeaie am matri- gamma generalised supersymme- a of { form set ces, to this able using be algebra to try wish we Now tions. P as: decomposes momentum generalised the Similarly, menta the ain ..weeteqartcCsmro momentum of Casimir quadratic the where represen- i.e. massless the tations examine to wishes one when case Casimirs. its through theory generalised state the the the of classify to is relevant be it will case that algebra Poincare this la- In to algebra representation. the the of Casimirs bel the use may we and algebra Poincar the of of representations algebra irreducible elemen- are the states Wigner, tary Following use states. will classify we to theory space-time field quantum as Just any space-time. in generalised the for group motion the ( nite: Γ 14 ymtyagba hycnb eopsdinto decomposed be can they algebra, symmetry the in are { given by: is superalgebra complete the generalised Thus the group. Poincare of generators the only has sector bosonic supercharges, fatsmersdpout fthe of products antisymmetrised of G, group global the of representation priate y( by Γ ( Q C Γ I I arcsjs sw i ihtecodntsdescribed coordinates the with did we as just matrices [ − α j i = = ecluaetesur f( of square the calculate We ewl rce xcl si h sa superalgebra usual the in as exactly proceed will We rmthese From 1 57 , 10 ] Q Z Γ Γ P ) Γ α I ): [ β cd [ C j i j i ) I β of P } αβ ,testo eeaie momenta generalised of set the }, ] aesmmnai h oe xedddirec- extended novel the in momenta labels } ] scag ojgto arxfor matrix conjugation charge is = = I = = SO 10 ( ( ( ehv h eeaie onaegopi.e. group Poincare generalised the have we C C ( Γ ( Γ P Γ I 23 ota hncmie ihtemo- the with combined when that so (2,3) Γ of [ ) a a J I α Q SO 5 5] ) ) β , βγ SL αβ α , 1 2 , Γ ocnrlcagsaerqie;the required; are charges central No . (2,3) P η P 5.T opr ihteuulsuper- usual the with compare To (5). [ cd abcd I I P [ , ] J .(63) ). I Γ ≥ Z = arcsw a omteappro- the form can we matrices .(66) 0. cd ,.,0(62) 1,...,10 Q (64) ) α , P 55 I ] hc spstv defi- positive is which ) = Γ P ,[ 0, i a matrices: 5 SO = SL P 23 spinors. (2,3) P P a I 5.Teset The (5). , I P ihmo- with n the and J SO SO ] = (2,3) (1,3) (65) 0, .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. e s h rttr stesadr P odto ( condition BPS standard the is term first The erpoueteqartccntanso h gener- the on constraints quadratic the reproduce we lsdmmnab usiuigtedcmoiin( decomposition the substituting by momenta alised P are: charges central the and of momenta terms four-dimensional in constraints the means This separately. ish aeaqartccntan ntegnrlsdmomenta, P generalised the on constraint quadratic a have h asesmliltof space multiplet because massless extended works the Supersymmetry the usual massless. of are the states geometry view BPS extended of in Thus point algebra. the Poincare from in less [ in calculated calculation. as same BPS the tially 1/2 be to state sec- the the for required and constraints charge quadratic the central are the equations two to ond equal be mass the ing no( into setal led perdi h ieauei h con- the in literature the in has appeared calculation already this essentially of foundation The momenta. alised fdgeso reo stemsiemliltin multiplet massive the as freedom of degrees of hs qain become: equations these ftesprymtyagbastrtstebud( bound the saturates algebra supersymmetry the of the closes. i.e. derivatives, Lie geometry generalised extended of algebra the of algebra symmetry cal condition section ( P C a ǫ I [ klm j i . P − j i ewl o eemn hscntan ntegener- the on constraint this determine now will We eadn htti szr,w edec iet van- to line each need we zero, is this that Demanding hnb eadn htti on sstrtd we saturated, is bound this that demanding by Then hspoue sprsigsio indices): spinor (suppressing produces This ntrso the of terms In h eodeuto ( equation second The h rteuto ( equation first The hsw e htfo eadn h representation the demanding from that see we Thus 1 = a ] Γ P 66 = 2 I [ ¡ + + j i )( P n eadn h on ssaturated. is bound the demanding and ) Z η ] C 4 2 [ ab ab j i = P Z Γ ] P Z ab a P P J ) Z a ab [ I P kl Z P P ab I cd ] , b I P Γ = = − Γ J htw edt moes httelo- the that so impose to need we that b .(69b) 0. abcd = + ,(69a) 0, Z ab P SL 69b a Z . Z 5 eeaie momenta, generalised (5) cd ab SO mle httesaei mass- is state the that implies ) 69b ( η = ac 23 a h aenumber same the has (2,3) 0, speieythe precisely is ) η bd − η Z ad ab η Z bc cd ) 69 a il Theory Field nal ǫ ¢ abcd Á yessen- by ] 52 + physical SO requir- ) = (1,4). 0. P (68) (67) 66 63 [ j i ] ) ) Proceedings = 15 (5) d (75) (2,3) SO is now × SO represen- there is a H (5) (5)/ d . In terms of I SO SL z massless . and manifest I is different from the . Thus we have com- Γ . ) I H H P C 1,...,5 (74) ) and again reproduce the ). The algebra = C (5,5) acts is 16-dimensional C to give (5, ν . , i i (5, µ Γ Γ (5, SO 5 SO and the set of generalised gamma . SO SO I I = ), ] = P P (5) and the corresponding complex d H (5,5) invariant action are described in ) is saturated for the (5,5)/ SO µνρστ . (5,5)/ [ I SO 66 y SO Γ , ] SO µν ; one then does the following: [ (5,5) and y H 5 of , ) rather than the more customary / . µ SO C = G x β ( = ated Clifford algebra, metrised products of by the closure of the algebra of generalised Lietives, deriva- also known as the section condition. tation to give constraints on matrices alised momenta a representation of G which we thinkalised of momenta as the gener- α ) In case the reader is worried that the This is the structure that we will replicate for the dif- Step 1, we construct spinor of d ]. (5, U-duality in = µ G I Γ 71 z in 5, coset i) Construct spinors of H and write down the associ- ii) Form a representation of G using a sum of antisym- case was somehow degenerate andnow we carry got out this lucky procedure we explicitly for will the case of vi) This constraint should be the same as that required geometry is then just from consideringresentations the of massless rep- this algebration — being the 1/2 BPS massless as representa- usual. ferent cosets. In summary, for dimension SO physical section condition for the theory. 5 Case, plex spinors of Clifford algebra: ( v) Demand( iv) Rewrite the superalgebra in terms of only the gener- iii) Combine the momenta and central charges to form because again we willsignature choose and a so metric the local with group, Lorentzian Euclidean case. One discovers thatsion the 25 coset with has signature dimen- (15,10) asspace it itself should. on The which extended the five-dimensional representations: [ that is, usual coordinates augmentedordinates with and two-form five-form co- coordinates. Thespace metric and on the this with coordinates that we will denoted by We will now follow the above instruction set for the case cd ) in and (71) (72) (70) en- Z on 4 55 ] αβ ¢ 69 ) to give: . ab 70 Γ cd 5 Z Γ , supersymmetry αβ . (73) (4) ¢ cd d C cd Z ab Γ Γ αβ (1,3) language so that abcd C ) η abcd 1 2 cd (2,3) charge conjugation SO η ¡ Γ abcd 1 2 (2,3). Thus the four dimen- + SO η (4) 1 2 a C SO ¡ ( P =− ]. What this calculation shows + + αβ ab 70 a ) a Γ 5 I P P 5 I Γ P Γ a αβ P αβ ) ) Γ αβ (2,3), the local Lorentz group of the ex- 5 ) a 5 αβ Γ I ) Γ Γ matrices: I I a (5)-symmetry with no central charges. All SO 1. We can then insert this into ( Γ Γ Γ (4) (4) Γ SL C C C C C algebra in [ ( ( ( ( ( (5) the global symmetry of the extended space. =− 0 and 5 = = = = = 2 11 ) SL Γ } (1,3) charge conjugation matrix which we denote } = will be related to the 5 E } β β (1,3). Obviously, the Spin(2,3) spinors also can be Γ (4) a Q Q SO (4) C , , Γ Thus, we have seen how the usual 4 Finally, we now rewrite the supersymmetry algebra , C SO α α 5 = tion to the similarity between theand physical the section 1/2 conditi BPS constraints. We thank Boris Pioline in particular for first drawing our att Q Q Γ { { matrix by: 4 text of U-duality multiplets for 1/2 BPS states [ sion by tended space and their Cliffordtions algebra, of and representa- { dimensional is the connection between:sions, these 1/2 states in BPS the states 10-dimensional extendedthe in space, spinors 4 of dimen- of the generalised space with no central charges ( more recently the condition has beenof rewritten the in terms terms of four-dimensional quantitiesand central i.e. charges as 4d follows: momenta thought of as Spin(1,3) Dirac. Crucially, thegation charge matrix conju- will be different because oftwo the time presence like of directions in of the central charges come from momentatended in directions. the The novel section ex- condition of the extended algebra with central charges may be lifted to an algebra Now we wish to think of this in with ( C where we have used the elementary properties of four- we can reinterpret the spinors as being Dirac spinors with global Proceedings ehv enta nDTw obetecodntsbut coordinates the double we DFT in that seen have We h osrit r thus: are constraints The h ypetcfr sue odtrietepolarisa- the determine to used is form symplectic The h hs pc a aoia oriae { coordinates canonical has space phase The h uesio odto ntegnrlsdmomenta, generalised the P on condition spinor pure the [ [ ously. er w aefudGMA[ GAMMA found have (we gebra ntrso ersnain of representations of terms In seupe ihasmlci form symplectic a with equipped is se.Te n ead httebs fteln bundle line the of base the that demands one Then ishes. n o ems osm okwt h 5 the with work some do must we now and clqatmbnl hc en n eemnsa determines one means submanifold, which Lagrangian bundle phys- quantum the to ical bundle pre-quantum the take to space phase 16 tp2 ems o omarpeetto of representation a form now must we 2, Step r-unu udeaewv ucin npaespace phase φ on functions wave are bundle pre-quantum Γ matrices: gamma of set lxln udeoe hs pc ih2 with space phase over bundle line plex analogy the and theory field Double 6 tp3 ewietegnrlsdmmnai em fa of terms in momenta momentum generalised 5d the write we 3, Step P intruhdmnigta h ulbc of back pull the that demanding through tion submanifold). coisotropic and isotropic d-dimensional incniin ota h omls losu ohave to of us functions allows be formalism to fields the that ‘sec- so or condition’ constraint al- tion strong do the we of Crucially, solutions alternative constraint. low strong the solve to in fields order on dependence coordinate the restrict later then P tp4 edmn ( demand we 4, Step { 73 56 C I ( µ I I x Γ = Z ihgoercquantisation geometric with = ,weeoehsi h r-unu udeacom- a bundle pre-quantum the in has one where ], ]. n ace rcsl h hsclscincondition section physical the precisely matches and hsi ihyaaoost emti quantisation geometric to analogous highly is This ems hnips ocle plrsto’on ‘polarisation’ called so a impose then must We , = I µν p ( , ( Γ C P 2( ). µ = + µ Γ η , , J Γ and 0 4 µν Z } P [ P µν [ µν P µ I ] P µ Z , ] , P Γ J µν Z P [ ν = µνρστ [ Γ µ µνρστ + ν n eta hre sfollows: as charges central and Z + µν 66 2( ] .(76) ). ] Z Z .(77) ). odtriecntanson constraints determine to ) Z [ µν µν [ x µν Z Σ or + Z ρσ ρσ Z fpaesae(hti a is (that space phase of x ˜ ] µ ] + 1 utntbt simultane- both not just G 72 + ··· P µ hsi qiaetto equivalent is this , P ,vr sflfrthis): for useful very ], τ 5 τ Z Z Z τµνρσ µ ω τµνρσ 1 etoso the of Sections . ··· µ 5 d ) = d ) Á dimensions. Γ G + .(79) 0. ω lfodal- Clifford µνρσ x rmthis from to , p Σ . and } van- (78) P .S emn auaKenApoc oDul n Exceptio and Double to Approach Kaluza–Klein A Berman: S. D. I , ain ne aoia rnfrain,wihaethe are which transformations, canonical under variant sidpnetof independent is de- coordinates submanifold with Lagrangian scribed the to natural restricted be osrc oaiainusing polarisation a construct r.Tepaesaeo h tigcnan h coordi- the contains string the { of nates space phase The ory. their and discussed. much possible is also There equivalence schemes quantisation. quantisation other under are Moyal symmetry as symplectic known man- ifest maintaining is of advantage This the has product. and quantisation star the called often iae npaesaesc httesmlci omis form symplectic by the given that such space phase on dinates coor- submanifold). the Lagrangian the on of depends dinates only lan- function wave mundane the more that (In guage, submanifold. Lagrangian the is cee hr n ep h r-unu udeand bundle have pre-quantum thus the keeps one where scheme, eetto uhta h arninsbaiodi de- is by submanifold scribed Lagrangian the that such rep- momentum resentation the have and choice alternative an make lci ru u t obecvrwihi nw sthe as known [ is group which metaplectic cover double sym- its the but of group representation plectic a not is func- fact function in wave wave the mean the that polarisation in of choices phases different of under changes tion pick to These function phase. wave a the allow up can one fact in Further this, transform. than to needs form volume the polari- then of sation choices the different on between function moves wave one the As of space). norm the define can (this one submanifold so Lagrangian is quantisation) the form’ on ‘half form volume as a known pick is what (in must polarisation one of choices different makes one When above. exposes. Hamiltonian canonical the the that hides co- symmetry it space but footing) and same the time be- on treating ordinates of (in advantage relativistic the naturally has is ing symmetry Lagrangian this The Lagrangian, manifest. a not using system descrip- the usual of the tion in that Note, structure. symplectic preserve the that space phase of transformations coordinate in- is Physics system. the of symmetry symplectic hidden l fti shgl nlgu oDTadsrn the- string and DFT to analogous highly is this of All omk l hscnrt,ltu dp abu coor- Darboux adopt us let concrete, this all make To oe hr sa nieyatraiequantisation alternative entirely an is there Note, o nfc iei o ut ssml sdescribed as simple as quite not is life fact in Now x , x ˜ ω .Tesrn osritipista n must one that implies constraint strong The }. φ p = ( and x d , p p u o eomtepouto fields, of product the deform now but ) φ ∧ p ( d p 73 and q .Teedfeetcocsrflc the reflect choices different These ). ]. aefunctions Wave . φ q ( q uhta h aefunction wave the that such .Nt,hwvr n could one however, Note, ). η ..a i.e. , d dmninlsub- -dimensional φ ( x , a il Theory Field nal p hncan then ) Proceedings ]. = ]. . ) 17 . 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