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PoS(QGQGS 2011)022 http://pos.sissa.it/ ]), Superstring 1 8 [ = N 4, = , d andreas.thurn@.fau.de , Supergravities as their low energy limits and thus provide a PACS: 04.65.+e, 04.60.Pp, 04.20.Fy, 04.50.-h d [email protected] theories are argued to reproduce 10 UV-completion of those theories. Loop (LQG)background on independent the other and hand non-perturbative is a approach manifestly ity, to however the presently quantisation limited of to General fourthese Relativ- two dimensions. approaches to In order quantumsions to gravity, and make we new contact extend matter between the fieldsour appearing techniques framework in of for Supergravity further LQG theories. research to aretwo higher The comparisons theories, between dimen- immediate namely symmetry applications reduced and of sectors black holes. of those Supergravity was originally introduced inshortcoming the of hope perturbative non-renormalisability. ofbeen finding Although reached a today this theory (with goal of the does possible gravity not exception without of seem the to have Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

3rd Quantum Gravity and Quantum GeometryFebruary School, 28 - March 13, 2011 Zakopane, Poland [email protected] Norbert Bodendorfer, Thomas Thiemann andInstitute Andreas for Thurn Theoretical III, FAU ErlangenGermany - Nürnberg, Staudtstr. 7, 91058E-mail: Erlangen, New Variables for Classical and Quantum (Super)-Gravity in all Dimensions PoS(QGQGS 2011)022 ], it 13 appears ]. ) 8 3 ( 1 dimensions + ] and references 1 and supersym- 6 , + 5 , 4 , 3 and its conjugate momentum , the momentum conjugate to to form the Ashtekar-Barbero i j ab i a δ P j in a vierbein formulation of Gen- e b e ) i a 3 e , 1 ( = are spatial indices on the leaves of the ab D q ,.., 1-dimensional case, an extended phase space 1 ], in which a globally hyperbolic spacetime of + = 17 2 ,.. b , indices in the fundamental representation. Finally, one a ) 3 ( annihilating the co-dreibein -dimensional constant time hypersurfaces with Euclidean sig- ]. In order to understand the problem, it is more instructive to , where ]. The reason for this can be traced back the very foundation of D ai j 3 are so ab 16 Γ 15 q ], which by symmetry arguments are expected to be the low energy which in turn enters the definition of ,.., 14 1 ab = gauge symmetry in the 3 K gauge theory. From a Lagrangian point of view, the group SO ) ,.. ) 3 j ( 3 , ( i 1 is foliated into ] and also a microscopic derivation of the entropy is available [ + 1 spacetime dimensions by dimensionally reducing . The problem how- 12 , where ] in 1986. During the next 10 years, the technical foundations of the programme were D + 2 bi e to the spin connection ab i a ]. Since the cosmological sector of String Theory has been addressed in the context of String K K The programme of was initiated by Ashtekar’s discovery of his new 1 dimensional reduction of String Theory and it is not clear which one should be preferred. In 10 . To obtain an SO = , ], and the derivation of the Bekenstein-Hawking entropy from a microscopic analysis [ i + a ab 9 7 adds q is introduced, coordinatised by a co-dreibein defined by spacetime foliation. The remainingbedding information information of of the the spacetime constant metric,in time the more hypersurfaces extrinsic into curvature precisely the the spacetime em- , are encoded K the LQG programme, namely theclassically possibility as of an SO rewriting Generalafter Relativity gauge in fixing the 3 time-like part oferal the Relativity. Lorentz It group thus SO sounds plausible thatwhich the derivation however can is simply not be repeated possible in [ any dimension, dimension nature endowed with a metric look at a Hamiltonian derivationformulation of of the Hamiltonian connection General formulation. Relativity [ Here, one starts from the ADM therein. More importantly, for the firstfamous time Wheeler-DeWitt-equation it and was construct possible a to1996 well make until defined mathematical today, the sense Hilbert focus out space. of of researchtably the In has Loop shifted the Quantum towards period Cosmology, applications the from of quantisation the[ of framework, most cosmological no- models using LQGWhile techniques todays measurements indicate the dimensionality of spacetime to be 3 cosmology [ sounds natural to try tosults compare in these these two symmetry-reduced approaches situations. toresults in quantum Presently, 3 gravity, the starting best with one their could re- do is to compare these 10/11 dimensions, the fundamental dimensions ofto Superstring/ be M-Theory, made much in less order choices todimensional have describe Supergravities [ the theories. Therefore,effective applying field the techniques theory of limits LQG of tobridge Superstring/M-Theory, between higher these seems approaches more to favourable QuantumDespite in Gravity its and order we obvious to will applications, build follow thisSupergravities the a route is literature in rather on this scarce letter. generalisations [ of LQG to higher dimensional ever, also known as the3 landscape, is that many choices have to be made in order to arrive at a variables [ New Variables for Classical and Quantum (Super)-Gravity in all Dimensions developed and it transpired thatcrete LQG eigenvalues naturally of provides geometric a operators quantum theory such of as geometry area with or dis- volume, see [ metry has not been found yet,a the lot ideas of of interest, mostquantum and notably gravity which in are aims the still at attracting context providing[ the of framework String for Theory, a a unified quantum mainly theory perturbative of approach all to forces PoS(QGQGS 2011)022 ) ], = D (1) , a 19 1 ( . ) ] that the second D ]. By performing , 19 1 18 ( is singled out. ) 0, 3 ( , to the spin connection, trans- = ) . 3 ) } ( y ) , y x ( ( b k , δ E k k j a , , ) δ K x a b ( aIJ i jk j a K γδ Lie algebra indices, as E γε ) + − { + D 3 , = = 1 ai j HYB aIJ ( } } Γ Γ ) ) y y ( ( = = k b k b A A ai j aIJ , , (which will be explained below) and a second class partner, ) ) A A x x S being SO ( ( j j a a 1-dimensional Palatini formulation of [ D A E { . + { ,.. 0 G D ] and properties II. and III. listed above are therefore not satisfied. = Yang-Mills theory phase space, however, it is subject to second class and its canonically conjugate momentum are real valued, where is compact. dim 20 ,.. ) j a J G D A , ,.., , I 1 1 ( = j ], called the hybrid spin connection, is a generalisation of the usual spin connec- , with 21 is a free parameter called the . Since we are adding the extrinsic and aIJ [ A R D ] of the formulation obtained by gauge unfixing. The idea is to decompose the SO to the case where the internal space is one dimension bigger than the spatial hypersurface. ∈ HYB aIJ 21 ,.., γ Γ 1 ai j Γ I. The connection II. The Poisson algebra is of the usual, simple form , the only restrictions being that: III. The gauge group constraints, the simplicity constraint which in particular impliessponding that the Dirac connection bracket is [ not self-commuting with respect to the corre- the canonical analysis ofon the arrives at an SO connection tion Instead of using the Dirac bracket or choosing time gauge, it was shown in [ class partner of theof simplicity the constraint theory can by be theonly removed machinery first of without gauge class changing unfixing. constraints the One andproperty physical is III. a content is left Poisson-commuting still with connection. not a satisfied connection The because formulation of drawback with the however non-compact is gauge that group SO where New Variables for Classical and Quantum (Super)-Gravity in all Dimensions connection forming in the adjoint representation, it becomesspecial clear groups that and this a construction scan can of only the work for classical very Lie groups reveals that SO The Lagrangian derivation ofHamiltonian this derivation, i.e. connection an formulation extension of now theof suggests which ADM can phase the be space deduced existence of by General of performing Relativity,straint the the a [ symplectic details reduction dual with respect to the simplicity con- It follows that if onewith is the above able properties, to one rewrite basicallyThe gets higher the intuition dimensional quantisation on Supergravities thereof as for how free. higher Yang-Mills to theories dimensions construct can a be connection obtained formulation from with the the procedure above of listed gauge properties unfixing in [ where curvature, transforming in the fundamental representation of SO At the quantum level however, thesary. restriction In fact, to the a quantisation special methods dimensionG are or formulated for gauge any group dimension is and not compact neces- gauge group PoS(QGQGS 2011)022 . S is aIJ nor aJ π ] and e 26 LQG inspired A aIJ π ]. , where ] J | 23 a ] for first steps in e while the remain- I . Neither Yang-Mills theory, have nothing to do [ j a 27 n ab ) e , β 1 j K a ∝ K + is the (up to sign) unique and β aIJ 0, I D ( π n γ ≡ ] = and its conjugate momentum ]. The simplicity constraint ] 0, e π [ [ 22 ab = 3: In the usual formulation, the j is a free parameter. Therefore the j q S 0 0 SPIN a HYB a = R Γ Γ ∈ D − = as a derivative operator [ β can thus be written as a function of j j . On the other hand, in the “time gauge” 0 0 aIJ γ ab LQG a NEW a and q π A A π 3 which nevertheless have the same symplectic ; , l a 4 = K D jkl ]. gauge γε connection and its conjugate momentum 23 − appearing in the quantum theory to be spherical [ ) S ) ] = D 1 , e ≈ [ is the spin connection of the co-dreibein 1 ] ] + ( e π [ [ D is canonically conjugate to the spatial metric whose quantisation SPIN a jk ( Γ 0. The spatial metric SPIN a jk HYB a jk aIJ − Γ Γ A = . It transpires that the Immirzi parameter based standard LQG, the Hamiltonian constraint is modified by a new a I − β e ) LQG a jk ], can be quantised and has to be treated at the quantum level. It restricts I 3 A n ( 25 can be shown to be related to the extrinsic curvature NEW a jk . While this torsion symmetry and the switch of internal signature can be , A aIJ 24 aIJ , i.e. and where K A is an SO(3) connection and is related to the extrinsic curvature by j b aI is the hybrid connection built from e e j a ] and depends on the Immirzi parameter as functions of an SO LQG K π [ ) A = cd LQG K HYB ab A , Γ I K 0 have a “boost” part, the information about the extrinsic curvature is encoded in the rotational cd ) and show that the ADM Poisson brackets are reproduced [ δ q 1 ( = ab SPIN I the pullback of the spacetime vielbein to the spatial hypersurfaces and furthermore imposes constraints on the intertwiningthe maps canonical between theory. them, see [ The following remark is due at this point for the case of irreducible representations of SO unit normal on connection where n As compared to the SO term, which can be interpreted asspect to a the gauge simplicity invariant extension constraint of whichgauge the is transformations original generated an by Hamiltonian integral the part with simplicity of re- of constraint the can the new be connection torsion shown formulation. to of change The understood a from certain a part Hamiltonian point ofcovariant view, it Lagrangian is framework. unclear if The theyspin can foam simplicity be models constraint, translated [ to which a is manifestly already well known from New Variables for Classical and Quantum (Super)-Gravity in all Dimensions The boost part of by ( enters the construction as a bridgeand between the the vielbein proposed formulation. Yang-Mills version In of fact, General Relativity on the constraint surface In this sense, the connection As a next step, one realises thatspace nothing is in the of calculations Lorentzian actually depends orthat on Euclidean the whether Hamiltonian signature, the internal formulations the of reason Lorentzianphase and of Euclidean space, which General can while Relativity lead be the to traced signaturestraint. the back of same Lorentzian to spacetime General the appears Relativity fact as canand a a thus loop relative be quantisation sign expressed can in as be the an performed [ Hamiltonian SO con- Γ part of where information about the extrinsic curvaturedepends is on encoded the in parameter thewith boost each part other and of that the the new newas formulation connection is compared a and to rather the different extension LQG of formulation the even ADM in phase space can be obtained, in simplified terms, by representing ing part turns outthe to process be and pure rewrite gauge.P the Starting ADM from variables, the the ADM spatial phase metric space, one can now reverse PoS(QGQGS 2011)022 on in- . 0 ) ) γ 1 2 ]. On ]. The ( in + ]. Dirac 28 i 30 D 29 are along a ( ) 1 . It is possible ) + 1 D + ( D ( 3 one can generalise the 11, there exist real repre- , = 10 D , 4 = invariant version derived from the 1 ) D + ( D and to rewrite the Hamiltonian constraint ) D . The idea is now to act with SO , ) 1 D ( ( ] we show that for 5 -matrices, which differ only by a factor of 19 γ and thus real fermions can be used, however, due to the ) on spinor space is necessarily complex in those dimensions. D , ) , Dirac’s 1 1 ) ( 1 + + D ( D ( and SO 3, both, the quantisation using Ashtekar Barbero variables and the one em- ) = D , invariant way such that it reduces to the SO , the action SO 1 D i ) ( , transforming in the fundamental (real) representation of SO 1 I N . The essential features remain the same, the connection remains Poisson commuting. + Gauß constraint in the quantum theory built from the new variables and the SU β ) D , . During the canonical analysis of the corresponding action, one uses the time gauge and ( 4 γ ) ( D , 1 ( to write a compound object of this normal and the Rarita-Schwinger field on which the action of real normal The problem can be resolved by observing that the complex “directions” of SO the (complex) spinorial representation space ofin SO an SO canonical analysis upon choosing the time gauge and solving the simplicity constraint [ above factor of the one hand, we have shownthe that simplicity the constraint second can class partners be can classicallyprice be treated to removed as pay by a was gauge first unfixing a classcanonical and new constraint, analysis term on of in the the the other action Hamiltonian hand, ondirection constraint the which thus which the has does spin to foam not be models naturallymake conducted, are appear progress however based. in with the Further these the new research important variables in questions. seemIn this to order be to a be good able startingmatter to fields. point treat to Scalar Supergravities, fields the as new well connection as gauge formulation bosons has can to be be treated extended in to the usual manner [ The situation is however more complicatedobey as a soon Majorana as condition. one turns Insentations to the of Rarita-Schwinger interesting fields, the dimensions which Lorentz group SO variant kinematical whengives using a tentative Ashtekar-Barbero guideline variables for are the implementation unitarilyilar of equivalent requirement the is quantum usually simplicity posed constraints, in andIt spin a should sim- foams. be noted thatfor the comparing new the variables canonical presented approach informulation. of this LQG letter More to also precisely, the provide the spin a quantummodels, foam new simplicity however models starting it constraint which point is is use solved a atwo classically cornerstone approaches path in in could integral the be the usual established spin LQG onlyrelation, foam formulation on e.g. a and kinematical by a level. showing relation In that betweennian order the the constraint, to spin establish it foam a seems path dynamical favourable integral tosince projects start it onto from is the the closer to kernel canonical the of formulationthe starting the presented appearance point in Hamilto- of of this second the letter spin classthe foam constraints canonical quantisation, (second a analysis class constrained has partners BF-theory. been Also, to argued the to simplicity be constraints) in a problem for the quantisation [ Fermions however are a bit moreSO involved, since they transform under thethus spinor reduces representation the of effective gauge symmetry to SO possibility of doing so heavily reliesalgebras of on SO the strong similarity of the representations of the Clifford New Variables for Classical and Quantum (Super)-Gravity in all Dimensions reduction, namely the ADM phaseexposition space. given so In far [ by consideringon a both 2 parameter family of connection formulations depending ploying the new variablesAlthough proposed not here, strictly forced should on atand us, SO the least stronger yield requirement the that the same joint semiclassical kernel physics. of the simplicity Ultimately, for PoS(QGQGS 2011)022 ]. 25 has to be I N 8 Supergravity = Supergravity action N d ]. Interestingly, due to the 33 ] in the light of the proposed 36 Supergravity and the 2-form Kalb- d 6 keeps track of the deviation of the components of I N ] already familiar from the new spin foam models [ 31 ] and its integrability structure [ 35 Supergravities. As an example, we studied the 3-index photon which d ]. Other Supergravity theories only have an on shell formulation, that is, 34 can be defined in such a way that it respects the reality condition of the Majorana spinor ) 1 + NB and AT thank the German National Merit Foundation for financial support and Matthias D ]. More precisely, the gauge constraint is altered by a term which transforms the momentum ]. In a precise sense, the normal field ( 32 31 Blau for correspondence aboutNicolai Hamiltonian for Supergravity. motivational discussions TT about thankscation higher to Dieter dimensional Supergravity Ashtekar Lüst and variables and String and Theory. Hermann their We want appli- to thank the referee for valuable comments. Acknowledgments which can be obtained fromity the theories 11d in theory 10d by and dimensionalcontain reduction 4d. non as Abelian This well p-form by as fields far some forhigher does Supergrav- whose gauge not quantisation theory exhaust most [ all probably requires of techniques them. from Some Supergravity theories quantisation of Supergravity. On top of fields already discussed,sional Abelian Supergravities, p-form most fields notably are the aRamond 3-index natural field photon ingredient of from to 11 the higher 10 dimen- is subject to a complication[ coming from the Chern-Simons term in the 11 non-trivial gauge invariant functions, the Weylthe algebra level is of twisted the by Chern-Simons a theory. contributionThus, proportional we to believe to have providedSupergravity the theories tools in for higher an dimensions, LQG type notably of 11d quantisation Supergravity of and at 4d least a subset of the algebra of local supersymmetry constraints closesthem only second modulo class the equations off of shell. motionkinds making We of do Supergravity not Theories, possibly have ideas yet fromsuch any gauge a concrete unfixing system proposal may first be for class. applied howMany in to questions order deal to are with turn not these answered byThe our precise work, relation for instance: with Stringmost Theory/M certainly Theory not is the not most provided. elegantisfactory. one. Our A The formulation formulation treatment works in of but termsimprove the is on of simplicity this constraint superfields in remains is future unsat- maybe publicationsthe and more AdS/CFT have desirable. correspondence many other [ We applications hope in to mind be such as able revisiting to conjugate to the 3-index photonspace in quantisation a by using non-trivial only way. gaugeof One invariant the quantities can Weyl and elements however find by perform a using a Hilbert a reduced space state phase representation of the Narnhofer-Thirring type [ New Variables for Classical and Quantum (Super)-Gravity in all Dimensions SO [ the Rarita-Schwinger field from thefor real the axis complex and compensates (gauge) in rotations the away compound from (physical) the object initial, real values. 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