Simone Speziale Oxford, 3 Jan. 2017 Twistors in Spin Networks and Loop

ξl ˜ Nl Nl

Twistors in spin networks and loop quantum gravity

τs τt

˜ Nl Nl s(l) j l t(l) Simone Speziale Oxford, 3 Jan. 2017

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X34 Intro and overview Loop quantum gravity: a background independent approach to quantum gravity based on the use of real Ashtekar variables and spin networks

• Main results: space is quantised, geometric operators have discrete spectra and non-commuting properties; applications to cosmology and black holes

• Main open questions: recovering smooth geometries following Einstein’s equations in the semiclassical limit; systematically compute transition amplitudes

Each `Feynman diagram’ of the theory corresponds to a truncation to a ﬁnite number of degrees of freedom: this truncation describes the evolution of a fuzzy quantum geometry.

In recent years many properties of this fuzzy quantum geometry have been clariﬁed, and twistors have emerged as natural tools to provide a covariant description of the geometry, to answer open questions, and to lead to new research directions.

It is a rich research area, and I will only present some isolated results, more closely related to twistors, without attempting due to the limited time a comprehensive and coherent overview.

1. SU(2) spin networks represent a collection of fuzzy polyhedra glued together along faces 2. Their spacetime embedding, based on SU(2) SL(2,C), is naturally described by twistors ! 3. Using twistors to construct SU(2,2) spin networks I. Loop Quantum Gravity Spin networksQuanta and quantumof space in geometry loop quantum gravity QuantumQFT ﬁeld theory Loop quantumLQG gravity

= n = F n H H H

n, pi,hi quanta of ﬁelds ,je,iv quanta of space | i! | i! • number of quanta • number of quanta and their relations Spin• momenta networks are eigenstates of geometric• volumes operators of regions such as surface areas • helicites • areas of interconnecting surfaces spins quanta of area • 7! Fuzzy spinning particles Fuzzy linked polyhedra intertwiners quanta of volumes • 7! dynamics described by dynamics described by Feynman diagrams spin foams Key result

geometric operators turn out to have discrete spectra with minimal excitations proportional to the Planck length

diagrams can be organised diagram organisation not yet established! in PT or EFT hands-on approach for the moment:

Speziale — LQG and polyhedracompute Motivations andin overviewa given truncation, then change truncation 8/38 Classical limit of a spin network

L N ,jl,in = L2[SU(2) /SU(2) ,dµHaar]= j n Inv l n Vj | i2 H l ⌦ ⌦ 2 l ⇣ ⇥ ⇤⌘

On a single link: L [SU(2),dµ ] T ⇤SU(2) the classical phase space of a spinning top 2 Haar !

On a single node: Inv V a certain phase space corresponding to ⌦i ji ! SKM h i shapes of polyhedra (a classical counterpart to Penrose’s spin-geometry theorem)

L On the full graph: T ⇤SU(2) // C~ turns out to describe the H ! n intrinsic and extrinsic geometry of a discretised space-like hypersurface II. SU(2) singlets and polyhedra (Freidel-Krasnov-Livine ’09, Bianchi-Dona-S ’10) SU(2) singlets and polyhedra

n 2 j i=1Sji iVji 3 ⇥ ! ⌦ j2 ~ j123 C =0 j12 # # j1 ...

2 := S // C~ KM Inv i Vji j1,j2,...; j12,j123,... SKM ⇥i ji S ! ⌦ | i h i dim = 2(n 3) virtual spins = n 3 SKM

It follows from three theorems: A4n4

A3n3 • Kapovich-Millson:

reduction by the closure constraint gives a symplectic manifold A2n2 with action angle variables the diagonals and dihedral angles of the polygon A1n1 O • Guillermin-Stenberg: (for compact orbits) quantisation commutes with reduction

• Minkowski: a closed sum of vectors in R3 defines a unique bounded, convex polyhedron Polyhedra reconstruction Bianchi, Doná, and Speziale Bianchi and HMH Bohr-Sommerfeld Minkowski theorem: a convex flat polyhedron at fixed areas has 2(n-3) degrees of freedom, and can be numerical reconstruction ’11 tetrahedral volume spectrum ’11 uniquely reconstructed from the normalsGeometry of polyhedra

14 Explicit reconstruction procedure: Xl edge lengths, volume, adjacency matrix N~i jiN~i =0 ﬁÑ { } 12 i For F 4 there are many X di↵erent combinatorial structures, or classes ° 10

F 6 Dominant: Codimension 2: Notice that a polyhedron has in general many possible adjacency classes: the adjacency matrix and valence of “ 8 each face is uniquely determined by the normals 6 Codimension 1: Codimension 3:

@ 4 The classes are all connected by 2-2 Pachner moves @ ‚ Ø @ (they are all tessellations of the 2-sphere) @ 2 It is the conﬁguration of normals to determine the class 0 0 2 4 6 8 10 The phase space F can be mapped in regions corresponding to di↵erent classes. ‚ S Dominant classes have all 3-valent vertices. ´ Adapting an algorithm by Lasserre, numerical explicit reconstruction [maximal n. of vertices, V 2 F 2 ,andedges,E 3 F 2 ] HMH, Han,“ p ´ Kamiq Òski,“ Riellop ´ q curved HMH described full phase diagram (Bianchi-DonàSubdominant and S ’10) classes are special conﬁgurations ´ with lessertetrahedron edges and vertices, reconstruction and span ’16 of pentahedral adjacencies ’13 measure zero subspaces. 5 [lowest-dimensionalO O classO forO maximal=1l number,O of triangulari SO faces](3) 3d slice of 6,cuboidsblue 4 3 2 1 œ ≈∆ S 4

Speziale — Twistors and LQG From loop quantum gravity to twisted geometries 25/33

1 1 2 3 4 5

2 4. Spinors 63

the Regge action. This is an encouraging result for the model, however the limiting procedure is delicate to handle on a full triangulation, and it has been argued that only flat solutions are compatible with the saddle point equations of the large spin limit [93]. This has started a debate in the literature on whether non-trivial curvature is properly accounted for, and if not, what is the problem with the model or the limiting procedure. See [94, 29, 44] for some references. Furthermore, the result relies on the special geometric properties of the 4- simplex, and it is not clear how to interpret the most general spin foams that are not dual to a triangulations [84, 95]. These partial results show the importance of improving our understanding of the dynamics already 4. Spinors 63 at the classical level. Actions for twisted geometries and their relations to the Regge action on-shell of 4. Spinors the shape matching conditions have been studied in [51, 88, 96, 75, 97,63 98], but there is as of yet no clear the Regge action. This is an encouragingconsensus on result the meaning for the ofmodel, curvature however and the dynamics limiting away procedure from the is delicate shape matching to subspace, nor on the the Reggehandle action. on a This full triangulation, is an encouragingo and↵-shell it result has role been for of the the argued torsion model, that a however priori only flat kinematically the solutions limiting are present procedure compatible in the is delicate theory with the [99]. to saddle An interesting development in handlepoint on a fullequations triangulation, of the large and spin itthis has limit sense been [93]. is argued the This alternative that has startedonly interpretation flat a solutions debate in of are the the compatible literature same phase on with space whether the in termssaddle non-trivial of discrete geometries with torsion point equationscurvature of is the properly large accounted spin limitalong [93]. for, the and This edges if has not, proposed started what is a the in debate [100]. problem in Studying the with literature the the model dynamics on whether or the of limiting the non-trivial non-shape-matched procedure. configurations is also important to understand whether they admit a continuum limit reproducing general relativity, or whether curvatureSee is [94, properly 29, 44] accounted for some references. for, and if Furthermore, not, what is the the problem result relies with on the the model special or thegeometric limiting properties procedure. of the 4- the latter property can be satisfied only by the Regge configurations. Lastly, spinors have also found many See [94,simplex, 29, 44] for and some it is references.not clear how Furthermore, to interpret the the result most relies general on spin the special foams that geometric are not properties dual to a of triangulations the 4- applications in the study of the quantum dynamics of spin foam models, e.g. [101, 102, 103, 104, 105, 106]. simplex,[84, and 95]. it These is not clear partial how results to interpret show the the importance most general of improving spin foams our that understanding are not dual to of a the triangulations dynamics already [84, 95].at These the classical partial resultslevel. Actions show the for importance twisted geometries of improving and our their understanding relations to of the the Regge dynamics action already on-shell of at thethe classical shape level. matching Actions conditions for twisted4.1 havePolyhedra been geometries studied and as in [51, their a Grassmannian 88, relations 96, 75, 97,to the 98], Regge but there action is as on-shell of yet of no clear consensus on the meaning of curvature and dynamics away from the shape matching subspace, nor on the the shape matching conditions have2( beenn 3) studied variables in [51,parametrizing 88, 96, 75, 97,can 98], be but given there by is KM as of variables, yet no clearor as we have seen by angles up to o↵-shell role of the torsion a priori kinematically present in theKM theory [99]. An interesting development in ij consensus on the meaning of curvaturetheir and linear dynamics dependencies, away from or by the theS shape scale-less matching triple products subspace,w nor, upon tothe their linear dependencies encoded this sense is the alternative interpretation of the same phase space in terms of discrete geometriesijk with torsion o↵-shell role of the torsion a priori kinematicallyin Plücker relations. present in the theory [99]. An interesting development in this sensealong is the the alternative edges proposed interpretation in [100].Alternative of Studying the same to the phase the dynamics KM space parametrization, in of terms the non-shape-matched of discrete another geometries simple configurations one with that torsion already is also solves the linear dependencies important to understand whether they admit a continuum limit reproducing general relativity, or whether along the edges proposed in [100]. Studyingis the one the in terms dynamics of cross of the ratios. non-shape-matched This was pointed configurations out in (Krasnov-Livine-Freidel is also 09), and it is easy to the latter property can be satisfied only by the Regge configurations. Lastly, spinors have also found many important to understand whether theyprove: admit first, a we continuum use GS theorem: limit reproducing general relativity, or whether applications in the studyPolyhedra and projective geometry of the quantum dynamics of spin foam models, e.g. [101, 102, 103, 104, 105, 106]. the latter property can be satisfied only by the Regge configurations. Lastly, spinors1 n have~ also1 foundn many KM (CP ) // C (CP ) /SL(2, C) (3.111) applications in the study of the quantum dynamics of spin foam models,S e.g.' [101, 102,' 103, 104, 105, 106]. 4.1 Polyhedra as a GrassmannianA n Then, we recall4 that4 there is a bijection (⇣1,...⇣n)=(g, Z1,...Zn 3)whereg SL(2, C) and 2 Many alternative parametrization possible: KM := i S // C~ A3n3 2 4.1 2(Polyhedran 3) variablesS as parametrizing a⇥ Grassmannianji KM can be given by KM variables, or as we have seen by angles ij up to S KM variables,(⇣ 1or normals,⇣2)(⇣3 ⇣ i) their linear dependencies, or by the scale-less tripleA2n2 products wijk, upZ toi = their linear dependencies encoded (3.112) Zdef dim KM = 2(n 3) or dihedral(⇣ angles,⇣ )(⇣ ⇣ ) { } 2(n 3)in variables Plücker relations. parametrizing KM can be given by KM variables, or as we have seen1 by3 angles2 i ij up to S S A1n1 their linear dependencies, or by the scale-less tripleO products w ,or up triple to their products, linear or dependencies cross-ratios encoded Alternative to the KM parametrization,are the cross ratios another (or thesimple alternativeijk one that (⇣ already⇣ )( solves⇣ ⇣ the)/(⇣ linear⇣ )( dependencies⇣ ⇣ )). (Notice that while the cross in Plücker relations. 1 3 2 i 2 3 1 i is the one in terms of cross ratios.ratios are This perfectly was pointed rotational out in invariant (Krasnov-Livine-Freidel quantities, the reconstruction 09), and it is of easy the to polyhedron from them isn’t; Alternative to the KM parametrization, another simple one that already solves the linear dependencies • Guillermin-Stenberg: prove: first, we use GS theorem:one fixes three normals ⇣ arbitrarily, typically 1, 0, ; then there is a unique boost that closes the oriented is the one in terms of cross ratios. This was pointed out ini (Krasnov-Livine-Freidel 09), and it is easy to (for compact orbits) quantisation commutes with reduction, tetrahedron (1, 0, ,Z ); or alternatively in real terms,1 one fixes one normal parallel toz ˆ, one to lies in the 1 n 1~ 1 n prove:and symplectic reduction is obtained dividing by the orbits of the complexified group first, we use GS theorem: KM (CP 1) // C (CP ) /SL(2, C) (3.111) (xy) plane,S ' then its longitude' as well as the two components of the third normal are derived from the closure relation. All1 n this~ just to stress1 n that I do not see my projective geometry in the use of these cross ratios as Then, we recall that there is aKM bijection(CP ()⇣1//,...C ⇣n)=((CP g,) Z/1SL(2,...Z, Cn)3)whereg SL(2, C) and (3.111) Svariables).' ' 2 The same cross ratios (3.112) can also be understood as solutions to a Plucker embedding of as a Then, we recall that there is a bijection (⇣1,...⇣n)=((g,⇣1 Z1,...Z⇣2)(⇣3n 3)where⇣i) g SL(2, C) and KM • Parametrization in terms of cross-ratios: projective variety.Zi = Indeed, (Etera-Laurent 09)2 show that (3.112) Zdef S (⇣1 ⇣3)(⇣2 ⇣i) { } (⇣1 ⇣2)(⇣3 ⇣i) • Zi = 2n (3.112) Zdef Phase space of shapes as a Grassmannian are the cross ratios (or the alternative(Freidel-Krasnov-Livine ’09) (⇣ ⇣ )(⇣ ⇣ )/(⇣ ⇣ )(Gr⇣ C(2⇣,n)).) (NoticeP C /SL(2 that, C while) the cross (3.113) EL (1⇣1 3⇣3)(2⇣2 i⇣i) 2 3 1 i ' { } { } ratios are perfectly rotational invariant quantities, the reconstruction of the polyhedron from them isn’t; which I will prove below. For the time being(and identification of the let us observe that on the LHS we have a Grassmannian, which are theone cross fixes ratios three (or normals the alternative⇣ arbitrarily, (⇣1 typically⇣3)(⇣2 1,⇣0i),/(⇣;2 then⇣3)( there⇣1 is⇣i a)). unique (Notice boost that that while closes the the cross oriented i is known to be a projective variety in P ( n Plucker embedding n), defined by the quadratic Plücker relations. Let me call ratios are perfectly rotational invariant quantities, the reconstruction1 of theC polyhedronC from them isn’t; tetrahedron (1, 0, ,Z1); or alternatively0 1 in0 real1 terms, one fixes onenof S normalKM^n in a projective space) parallel toz ˆ, one to lies in the 1 Fij = zi zj zj zi the coordinates for C C , and their set of Plückers FijFkl FikFjl + FilFjk = 0. one fixes(xy three) plane, normals then its⇣i longitudearbitrarily, as typically well as the 1, 0 two, ; components then there is of a the unique third^ boost normal that areP closes derived the from oriented the closure Then 1 tetrahedronrelation. (1, 0 All, this,Z1); just or alternatively to stress that in I realdo not terms, see my one projective fixes one normal geometry parallel in the toF usez12 ˆ,F one of3i these to lies cross in the ratios as 1 Zi = , (3.114) ZF (xy) plane,variables). then its longitudeAnalytical adjacency matrix of pentahedron from Desargues’ cross ratios? as well as the two components of the third normal are derivedF13F from2i the closure { } relation. AllThe this same just cross to stress ratios(wip Haggard and S’) that (3.112)as I do I will can not also also see prove be my understood projective below. Then, geometry as solutions on the in RHS the to a of use Plucker (3.113) of these embedding we cross can use ratios of asKM as a variables).projective variety. Indeed, (Etera-Laurent 09) show that S 2n n The same cross ratios (3.112) can also be understood as solutions toP aC Plucker/SL(2 embedding, C) P C of KMKM as a (3.115) totti 2n ' ⇥ SS EL { } projective variety. Indeed, (Etera-Laurent 09)Gr showC(2 that,n) P C /SL(2, C) (3.113) ' 2n n 1 n n { } which trivially follows from P C P C (CP ) . This shows that P C KM is a projective variety which I will prove below. For(which the time I will being later let review us2n observe how it that gets' on identified the LHS⇥ with we have the space a Grassmannian, of ‘framed’ polyhedra which⇥ S at varying shapes, areas GrC(2,n) P C /SL(2, C) (3.113) EL is known to be a projective variety in P (Cn' Cn), defined by the quadratic Plücker relations. Let me{ call} 0 1 0 1 n ^n whichF Iij will= provez z below.z z the For coordinates the time being for letC us observeC , and thattheir on the set LHS of Plückers we haveF aij Grassmannian,Fkl FikFjl + whichFilFjk = 0. i j j i ^ P is knownThen to be a projective variety in P (Cn Cn), defined by the quadratic Plücker relations. Let me call 0 1 0 1 n ^n F12F3i Fij = zi zj zj zi the coordinates for C C , and Zi their= set of, Plückers FijFkl FikFjl + FilFjk = 0.(3.114) ZF Then ^ P F13F2i { } as I will also prove below. Then, on the RHS ofF12 (3.113)F3i we can use Zi = , (3.114) ZF F13F2i { } 2n n P C /SL(2, C) P C KM (3.115) totti as I will also prove below. Then, on the RHS of (3.113) we can' use ⇥ S { } 2n n 1 n n which trivially follows from P C 2nP C (CP ) .n This shows that P C KM is a projective variety P C' /SL(2⇥, C) P C KM ⇥ S (3.115) totti (which I will later review how it gets identified with' the⇥ spaceS of ‘framed’ polyhedra at varying shapes, areas{ } 2n n 1 n n which trivially follows from P C P C (CP ) . This shows that P C KM is a projective variety (which I will later review how it gets' identified⇥ with the space of ‘framed’ polyhedra⇥ S at varying shapes, areas III. Twisted geometries and twistors (Freidel and S ’10, S and Wieland ’12) Twisted geometries

Geometry on the graph A spin network is then a collection of polyhedra glued together

The associated intrinsic geometry is a generalisation of Regge’s: Each classical holonomy-flux configuration a face seen from adjacent polyhedra has a unique area, on a fixed graph describes but can have different shapesacollectionofpolyhedraandtheirembedding each polyhedron is locally flat: curvature emerges at the faces, as in Regge calculus ‚ extrinsic geometry encoded in the parallel transport g between adjacent polyhedra Where is the information on the extrinsic geometry? ‚ i 0i 1 i jk the structure defines a piecewise discrete metric, in generalA = !discontinuous✏ jk! :(E) 2 1. Embed the SU(2) spin networks in SL(2,C) two neighbouring polyhedra share a face with same area but di↵erent shape (this γ-dependent embedding is the discrete equivalent of the use of real Ashtekar-Barbero connection) shape-matching conditions can be written explicitly ‚ k 2. Define the extrinsic curvature àDittrich la Regge via the boost `dihedral angle’ between 4d normals to the and S ’08 l This subcase describes ordinary Regge calculus polyhedra i j τs τt shape-matchings ˜ Twistors have proved crucial to address these two steps kinematical LQG Nl ReggeNl geometries ›Ñ (twisted geometries) The result is a symplectic reduction from flat twistors s(l) j l t(l) to twisted geometries

Speziale — Twistors and LQG From loop quantum gravity to twisted geometries 26/33 LQG phase space: smearing Di↵eomorphism covariance requires a distributional smearing of the Poisson algebra ‚ A on 1d paths E on 2d surfaces g exp AXETwistors to twisted geometries “ l “ l˚

˜ ≥ Take a collection of flat twistors, ≥ Zl each associated to a half-link of the oriented graph Poisson structures: Zl ‚ continuous on each link : ùñ i j Imposing the following set of constraints gives a symplectic reduction to A C A x ,A y 0 the phase space of loop gravity: g B ,g D 0 t ap q bp qu “ t u“ i b i b 3 • on each link, helicity and dilatation matching : defines the phase space T*SL(2,C) Aa x ,Ej y j ap q x, y Xi,g ⌧ig t p q p qu “ p • qton each half-link, simplicity constraints selecting a space-like plane : reduces to Ashtekar-Barbero T*SU(2) u“´ a b E x ,E y 0 • on each node, a closure condition for the SU(2) generatorsXi,Xj ✏ijkXk t i p q j p qu “ t u“ 2L ~ L ~ T // Cl,Sl, Cn T ⇤SU(2) // Cn Non-commutativity introduced by the discretization (same{ happens} ' in LGT) LQG phase space on a fixed graph: Conversely, a simple Schroedinger-like quantisation of twistors gives SU(2) spin networks ‚ (seen as embedded in unitary SL(2,C) irreps of the principal series, )⇢ = k, k = j L P T ˚SU 2 “ p q Quantization gives the holonomy-flux algebra on ‚ L L2 SU 2 H “ r p q s In the same sense as L , q,ˆ pˆ i is a quantization of T R, q, p 1 p 2rRs r s“ ~q p t u“ q

Speziale — Twistors and LQG Motivations and goals 9/30 In terms of this decomposition, the Poisson brackets (A.1) read

Li,Lj = ✏ijkLk, Li,Kj = ✏ijkKk, Ki,Kj = ✏ijkLk, { } { } { } P I ,JJK = ⌘IJP K + ⌘IKP J , CI ,JJK = ⌘IJCK + ⌘IKCJ , (A.4) { } { } CI ,PJ = ⌘IJD J IJ, D, P I = P I , D, CI = CI . { } { } { } In the main text we refer to the three Casimir invariants of su(2, 2). These are given by

(2) = M M ab =2C 2D2 4P C, (A.5) C ab 1 · 1 (3) = ✏ M abM cdM ef =3DC 6W C, (A.6) C 8 abcdef 2 · (4) = M M bcM M da C ab cd =2C 2 + C 2 +2D4 +8J J JKP I C +8DJ P I CJ +8D2P C +4P 2C2 + 4(P C)2, (A.7) 1 2 IJ K IJ · · where takes the form of a Lie derivative of the group character, and when interpreted in terms of spinors, it acts as 1 IJ 2 2 2 2 1 IJ 2 2 C1 = JIJJa boost= L inK mixing= 2(⇧ the+ ⇧¯ source),C and2 = target(?J)IJ spinors.J =2K InL spite=2i( of⇧ its⇧¯ simplicity,) (A.8) its action on holonomies and fluxes 2 is completely non-linear, and has no2 resemblance with· usual dilatations. This is unavoidable, as the discrete are the Lorentz Casimirs, and phase space has compacttakes the form directions, of a Lie so derivative there is noof the usual group meaning character, of linear and when dilatations. interpreted On in the terms other of spinors, hand, the it acts as 1 JK L generator has thea boost propertyWI = in✏IJKL mixing ofM preserving theP source the and symplectic target spinors. structure, In spite and of(A.9) its all simplicity, transformed its action quantities on holonomies recover andthe fluxes 2 expected behaviouris completely in the continuum non-linear, limit. and has no resemblance with usual dilatations. This is unavoidable, as the discrete (4) is the Pauli-Lubanski vector,All which our results, satisfiesphase computingWI space= sP hasI for the compact a action massless directions, of particle. SU(2, so2) In there dilatations, deriving is noC usual,weusedthe the meaning relation of between linear dilatations. the di↵erent On the boost other dihe- hand, the identities dral angles in thegenerator literature has and the propertyproving theof preserving equivalence the ofsymplectic di↵erent structure, secondary and simplicity all transformed constraints, quantities as recover well the 1 expected1 behaviour in the continuum limit. J J JKP I C =as computing✏ J IJW theK CL finiteJ actionJ IJP ofC, the J discreteJ JKJ holonomy-fluxJ LI =2C 2 + C dilatation,2. (A.10) rely heavily on the use of twistorial IJ K 2 IJKL All2 ourIJ results,· computingIJ theKL action of SU(21 , 2)2 dilatations, the relation between the di↵erent boost dihe-

formalism, and are a demonstration of its usefulness to loop quantum gravity. 27 dral angles in the literature and proving the equivalence of di↵erent secondary simplicity constraints, as well Casimir evaluations: with one twistor as computing the finite action of the discrete holonomy-flux dilatation, rely heavily on the use of twistorial (2) =6formalism,s2, (3) and= 6 ares3, a demonstration(4) =6s4 of its usefulness to loop(A.11) quantum gravity. 2 TwistorsC andC twisted C geometries With two twistors: ↵ A 2 ¯ 2 ¯ A˙ A twistor can be2 described Twistors as a pair andof spinors, twistedZ =(! geometries,i⇡¯A˙ ) C C ⇤ =: T. It has a dual, Z↵ =( i⇡A, !¯ ), (2) = 4(s2 +˜s2 + ⇢ 2) + 2(s s˜)2, 2 (A.12) which defines a pseudo-Hermitian[e.g. 1005.2090] norm of signature (+ + ), C | | ↵ A 2 ¯ 2 ¯ A˙ A twistor can be described as a pair of spinors, Z=(! ,i⇡¯ ˙ ) C C ⇤ =: T. It has a dual, Z↵ =( i⇡A, !¯ ), (3) = 6(s +˜s)[(s s˜)2 + ⇢ 2], A(A.13)2 which defines a pseudo-Hermitian↵ norm of signatureA (+ + ), C | | Z Z¯↵ = 2Im(⇡A! ), (1) (4) = 2(s +˜s)2 + 4[(s s˜)2 + ⇢ 2]2 (A.14) C | | Z↵Z¯ = 2Im(⇡ !A), (1) preserved byStep 1: T*SL(2,C) from twistors SU(2,2) transformations. It is well-known [8] that↵ these transformationsA can be realized by Hamil-

B SU(2) casetonian revisited vector ﬁelds,preserved if we by equip SU(2,2) the transformations.space with canonical It is well-known Poisson brackets, [8]’s, or the that real one these transformations can be realized by Hamil-

tonian vector fields, if we equip the space with canonicalF Poisson brackets, ↵ ¯ ↵ Z , Z = i . (2) C Formularium: to be removed ↵ ¯ ↵ Twistor space carries a representation of SL(2,C), which is Hamiltonian wrt Z , Z = i . (2) Twistor’s definitions: In Penrose’s abstract index convention, variety:A projective embedded in via spinor bilinears relations and Plücker the generators of SU(2,2) can be written as In Penrose’s abstract index convention, the generators of SU(2,2) can be written as A ↵ ! ↵ IJ¯ (A↵ B) A˙B˙ B B I AA BA˙ AB I A A˙ A Z = , ZJ, Z==!i⇡, ✏ IJ⇡A+cc, ! (,PA=B)A ,A˙B˙ = i!⇡ , ⇡¯ ,C=I ✏ A =A˙ i! (C.15)!¯ I,DA A=Re(˙ ⇡A! ), A (3) i⇡¯ ˙ { } J{ = !} ⇡ ✏ +cc{ ,P} = i⇡ ⇡¯ ,C= i! !¯ ,D=Re(⇡ ! ), (3) ✓ A ◆ A and and their finite actionand their is realized finite action via isthe realized exponential via the map, exponential map, 0 2 ¯ ¯˙ A¯ To represent the covariant holonomy-flux algebra of LQG, ⌘↵˙ = , Z↵ = ⌘↵˙ Z =( i⇡AIJ, ! )IJ IJ (C.16) IJ 2 0 exp(! M )exp(. f !:=IJM exp()!. f :=M exp(, !IJ)f.M , )f. (4) (4) we seek a representation of the symplectic manifold T*SL(2,C) ✓ ◆ IJ (S and Wieland ’12)IJ{ ·} { ·} The i factor allows to see the twistor as a Dirac bi-spinor in the chiral basis,A and it is used in the LQG 1 Half the twistor norm,A s := Im(⇡AA! ), isA called helicity of the1 twistor, since literature, as well as theHalf bar over the⇡ twistorto keep norm, track ofs its:= anti-selfdual Im(⇡A! nature:), is called⇡LQG helicity= i⇡¯P enrose of the. As twistor, a consequence,since dilatation generators and helicity are inverted: I I 2 I IW = sP , (5) T?//C T ⇤SL(2, C) W = sP , (5) ' 1 ⇡! := ⇡ !A = DI + is,IJKL s = Z¯ Za. (C.17) A whereA W := 1/2✏ P J a is the Pauli-Lubanski vector.’s? The complex scalar ⇡ ! = D + is, for which we I IJKL 2J KL A A where W := 1/2also✏ useP theJ JKL index-freeis the Pauli-Lubanskinotation [⇡ ! , is vector. invariant The under complex the Lorentz scalar subalgebra.⇡A! = D + is, for which we 2 ↵ ˜↵ A | i ijk T? : 2 twistors, each with also(Z use, Z the) index-freeThe notation representation⇡A! =0 [⇡ ! , of is the invariant conformal under algebra the constructed Lorentz subalgebra. in this way is not the most general one, since the

6 | i W The representationalgebra of has the 152 conformal dimensions, algebra whereas constructed the carrying space in this wayhas only is not 8 dimensions. the most general In fact, one,the generators since the (3) are Spinor’s norms: areas Spinor’s extrinsic to geometry (related of each face phases: a framing ) Spinor’s A A I I C = ⇡A! ⇡˜algebraA!˜ has: matches helicity and dilatation of the two twistors 15 dimensions,not independent: whereasP theand carryingC are null space and relatedhas only to D 8and dimensions. the Lorentz In generators fact, the by generators (3) are I I not independent: P and C are null and related[I J to] D and the LorentzIJ generators2 by 2

2P C =(DWhat is the+ preciseprojectivity relation ands btwn the? the real) complex projectivity Grassmannian JCan of we the analytically adjacencies? ,P reconstructgeometry, the either adjacency the using complex projective oneof of the the C = (D + s ). (6) Breaks SU(2,2) to SL(2,C) · [I J] IJ 2 2 Accordingly, the2P threeC Casimir=(D + operatorss?) J ,P of su(2, 2)C (see= the(D Appendix+ s ). for deﬁnitions) are not independent:(6)

Fixed total area total Fixed phase total Fixed · Connectionsthey with projective geometrynice were papers already are highlighted by Freidel, in Krasnov, & all Livine; Using proportional spinors they showed explicitly the isomorphism: to the only conformallyQuestions: invariant quantity in ,thehelicity: Accordingly, the three Casimir operators of su(2, 2) (see the Appendix for definitions) are not independent: they are all proportional to the only conformallyC(2) invariant=6s2,C quantity(3) = in6 s3,C,thehelicity:(4) =6s4. (7) Hence, the representationC(2) =6 ofs2 the,C algebra(3) on= twistor6 s3,C space is(4) special,=6s4 and. furthermore reducible, with irreducibles(7) labelled by s. In this sense, Penrose refers to twistors as the spinors of the conformal group. Hence, the representationIrreducible of the representations algebra on twistor with independent space is special, real values and furthermore of the Casimirs reducible, can be with obtained irreducibles working with 2 ˜ labelled by s. Inthe this larger sense, carrying Penrose space refers tomade twistors of pairs as of the twistors spinors (Z, ofZ the), with conformal generators group. constructed by linearity. This space has now 4 conformally invariant quantities, given by the two helicities plus the pseudo-hermitian product Irreducible representations↵ with independent real values of the Casimirs can be2 obtained working with Z¯↵Z˜ , and the Casimirs are independent functions of these four variables. has been considered in the twistor the larger carrying space 2 made of pairs of twistors (Z, Z˜), with generators constructed by linearity. This 1 space has now 4 conformallyIn the twistor invariant literature, quantities, the helicity is given usually by given the by two the real helicities part. The plus di↵erence the pseudo-hermitian comes from the extra producti used here in the definition of the twistor. In this way we can match Dirac’s conventions for a bi-spinor, and bridge more easily with the notation ↵ 2 Z¯↵Z˜ , and the Casimirsused in loop are quantum independent gravity. Notice functions also that of wethese use metric four variables. conventions withhas mostly been plus, considered thus we map in vectors the to twistor anti-hermitian matrices, see Appendix A. 1In the twistor literature, the helicity is usually given by the real part. The di↵erence comes from the extra i used here in the definition of the twistor. In this way we can match Dirac’s conventions for a bi-spinor, and bridge more easily with the notation used in loop quantum gravity. Notice also that we use metric conventions with3 mostly plus, thus we map vectors to anti-hermitian matrices, see Appendix A.

3 The representation of the conformal algebra constructed in this way is not the most general one, since the algebra has 15 dimensions, whereas the carrying space has only 8 dimensions. In fact, the generators (3) are not independent: P I and CI are null and related to D and the Lorentz generators by

2P [I CJ] =(D + s?) J IJ,PI C = (D2 + s2), (6) I where ? is the Hodge-star. Accordingly, the three Casimir operators of su(2, 2) (see the Appendix for deﬁnitions) are not independent: they are all proportional to the only conformally invariant quantity in ,thehelicity:

(2) =6s2, (3) = 6 s3, (4) =6s4. (7) C C C Hence, the representation of the algebra on twistor space is special, and furthermore reducible, with irreducibles labelled by s. In this sense, Penrose refers to twistorsThe gritty details as the spinors of the conformal group. Irreducible representations with independent real values of the Casimirs can be obtained working with the larger carrying space 2 made of pairs of twistors (Z, Z˜), with generators constructed by linearity.4 This space has now 4 conformally invariant quantities, given1 by the two helicities plus the pseudo-hermitian product The¯ functions˜↵ T ⇤G⇧, ⇧gandGh are( related⇧,h) by ⇥ =Tr(⇧dhh ) 2 Z↵Z , and the' Casimirs⇥ 3 are independent functions of these four variables. has been considered in the twistor literature [11,12], among other things in relation⇡! to ambitwistor1 theory and to build representations of massive ⇧ = h⇧h , (11) 2 ⇡! ˜ particles.generators as left-invariant vector fields, their adjoint right-invariant Bute as shown in [13], has another interesting property⇧ := thatAdh( links⇧) it to loop quantum gravity: it contains T SL(2, ) as a symplectic submanifold, obtained by Hamiltonian reduction with respect to the first 4 and span 14 out⇤ of 16C dimensions of T2. They obey the Poisson brackets class constraint e ee ↵ ˜↵ A To construct T*SL(2,C), The functions⇧i,h ⇧=Take 2 twistors, each with , ⇧hand⌧i, h areC⇧:=i related,h [⇡(=!Z⌧ byi,h,Z[˜⇡ )!˜ =0, ⇡A! =0 and (12a) (8) | i | i 6 A C 2C AC AC referred to as (complex) area-matchingh B,h D = condition. Explicitly,✏ ⇧BD + ✏ weBD⇧ take⇡!. 1 (12b) (⇡!)(⇡!) ⇧ = h⇧h , (11) e ⇡! e A A On the constraint hypersurfaceIJ (A BC) =0A˙B˙ ,twokeypropertieshold:Firstly,theadjointrepresentationrelatestheA !˜ ⇡B ⇡˜ !B IJ (A B) A˙B˙ J 1 = ! ⇡ ✏ + cc, h = 2 , J˜ = !˜ ⇡˜ ✏ + cc, (9) fluxes, ⇧ = h⇧h .Secondly,thecomponentsoftheholonomycommute.Hence,werecoverthePoissonand span 14 out of 16 dimensionsB of T . They obeye the Poisson brackets ee [⇡ ! [˜⇡e!˜ algebra of T ⇤SL(2, C) with ⇧ and ⇧ as the (chiral) left- and right-invariant| i | i Hamiltonianee vector fields on the ⇧i,h = h⌧i, ⇧i,h = ⌧ih, (12a) group manifold: p p 2 and following the conventions of [13], we take opposite signs for the2C twistor brackets : 4 e k A C A ACC AC ⇧i, ⇧j = ✏ij ⇧k, ⇧i,h = h⌧i, h B⇧,hi,hD ==⌧ih, h B,h✏ D⇧BD +=0✏BD. ⇧ . (13) (12b) e B B (⇡!)(B⇡!A) CA=0 • these quantities are invariant under the ⇡A, ! C orbits = A = C⇡Ã=, !˜⇡Ae! ⇡Ã!˜ (10) The functions ⇧In, ⇧ fact,and weh are can related easily see by that this procedure { amounts } exactly to{ a symplectic} reduction T ⇤SL(2,C) On2 the constraint hypersurface C =0,twokeypropertieshold:Firstly,theadjointrepresentationrelatesthe ' SL(2, C) sl(2, C) T //C.OntheC =01constraint hypersurface, the Hamiltonian vector field XC = C, It is easy⇥ to check' fluxes, that (9)⇧ = are hC⇧-invariant.h⇡!.Secondly,thecomponentsoftheholonomycommute.Hence,werecoverthePoisson1 Whene the constraint (8) is satisfied, thee generators{ ·} are related generates• thethey further satisfy orbits ⇧ = h⇧h , ee 2 (11) by the adjoint actionalgebra of h ofandT ⇤SL(2 thus, C⇡ (9)!) with span⇧ aand 12-d⇧ as submanifold the (chiral) of left- and, for right-invariant which one recovers Hamiltonian the Poisson vector fields on the e z z z z brackets of T ⇤SL(2groupexp, )with (zX manifold:C +¯JzXandC¯ ):(J˜!acting, ⇡, !, ⇡) respectively(e !, e ⇡ as, e left-invariant!, e ⇡),z andC. right-invariant(14) vectors fields on C 2 7! 2 and span 14 outthe of group 16 dimensions manifold, of parametrizedT . Theye obey by theh as Poisson a left-handedk brackets group element. This symplectic reductionA holdsC provided The functions ⇧, ⇧ and h are invariante⇧i, under⇧j = such✏ij gauge⇧k, transformations,⇧i,h = h and⌧i, thus span⇧i,h the= space⌧ih, obtainedh B by,h D =0. (13) [⇡ ! = 0, which we exclude from ouree analysis frome now on, and is 2-to-1: there is a 2 symmetry associatedC=0 symplectic reduction.⇧i,h = h⌧i, ⇧i,h = ⌧ih, Z(12a) | i6 e e e e withLet us the add simultaneous two more remarksIn sign fact, flip to we complete of can the easily spinors. the analysis. see thatSee [13–15] First, this procedure the for map more between amounts details. twistors exactly and to holonomy-flux a symplectic reduction T ⇤SL(2, C) A C 2C 2 AC AC ' variables is 2–to–1,eh sinceSL(2,h exchanging, C) =sl(2 spinors, C) asT //C✏ .Onthe⇧ + ✏C =0⇧ constraint. hypersurface, the Hamiltonian vector field XC = C, The symplecticB manifoldD ⇥ T ⇤SL(2,'C) is the buildingBD BD block of projectede spin networks(12b) [16], which appear as { ·} generates the( orbits⇡!)(⇡!) boundary states of covariant spin foame models [17, 18]. There, a copy of T ⇤SL(2, C) decorates each link of an and an additional symmetry: Z2 (!, ⇡, !, ⇡) (⇡, !, ⇡, !) (15) oriented graph,C and=0 the orientation is used to7! associate unambiguously zZ toz thez sourcez node, and Z˜ to the On the constraint hypersurface ,twokeypropertieshold:Firstly,theadjointrepresentationrelatestheexp (zXC +¯zXC¯ ):(!, ⇡, !, ⇡) (e !, e ⇡, e !, e ⇡),zC. (14) leaves both1 holonomy (10) and flux (2) unchanged. Hence, to arrive at the reduced phase space, we also fluxes, ⇧ = htarget.⇧h .Secondly,thecomponentsoftheholonomycommute.Hence,werecoverthePoisson Each twistor is then subject to the primary simplicitye constraints,7! which reduce the covariant2 boundary have to divide out this residual Z2 symmetry⇧, ⇧ ee (15).h Second, because of the restriction ⇡! =0,thespinorial algebra of T ⇤SL(2states, C) towith those⇧ and relevantThe⇧ as functions the for (chiral)the SU(2) left-and spin andearee networks right-invariant invariant ofe undere loop Hamiltonian such quantum gauge transformations, gravity. vector fields6 The on constraints and the thus span are the deduced space obtained by parametrization cannot cover the submanifold of (h, ⇧):⇧AB⇧ = Tr ⇧2 =0, and what we truly find group manifold:from a discretisationsymplectic of the gravitational reduction. action in AshtekarAB variables,e e and relate thee tetrade to the bivector B is T ⇤SL(2, C) removed fromLet allus add its null two configurations. more remarks toThe complete complete the isomorphism analysis. First, could the be mapdefined between through twistors and holonomy-flux e asuitabletreatmentofthedegenerateconfigurations,seee.g.theanaloguesituationinthewhich is thek field canonically conjugated to the connection. ToA representC the constraintsSU(2) case at the [30]. discrete level, ⇧i, ⇧j = ✏ij ⇧k, variables⇧i,h is= 2–to–1,h⌧ei, since exchanging⇧i,h = ⌧ih, spinors ash B,h D =0. (13) However,we introduce below we a time wille identify direction⇧ withNI ( then) on Plebanski each node field smearedn of the over graph. 2-dimensional In theirC=0 linear surfaces form in a introducedt = const. in [19] (see In fact, we slice canalso of easily [8, initial 20, see 21])data. that the Be thisprimary this hypersurface procedure simplicity amountsspace-like, constraints exactly the restriction read to a symplectic is(! automatically, ⇡, !, ⇡) reduction ( fulfilled,⇡, !, ⇡,T! and⇤)SL(2 has, C no) physical (15) consequence2 for the following. 7! ' SL(2, C) sl(2, C) T //C.OntheC =0constraint hypersurface, the Hamiltonian vector field XC = C, ⇥ ' leaves both holonomy (10)e and flux (2)IJ unchanged. Hence, to arrive at{ the·} reduced phase space, we also generates the orbits NI B =0, (11) have to divide out this residual Z2 symmetry (15). Second, because of the restriction ⇡! =0,thespinorial e e ABe e 2 6 parametrizationB. cannot Twistors coverz from thez the submanifoldz LQGz action of (h, ⇧):⇧ ⇧AB = Tr ⇧ =0, andIJ whatKL we truly find for all bivectorsexp (zXCB+¯associatedzXC¯ ):(!, with⇡, !, ⇡ links) sharing(e !, e the⇡, e node!, en,⇡ and),z implyC. simplicity of B, namely (14) ✏IJKLB B = is T ⇤SL(2, C) removed7! from all its null configurations.2 The complete isomorphism could be defined through 0. The bivector B is related to the Lorentz generators via B =( ?)J,where R is the Immirzi parameter. The functionsThe⇧, ⇧ interestand h inare the invariant phaseasuitabletreatmentofthedegenerateconfigurations,seee.g.theanaloguesituationinthe space under of suchSL(2, gaugeC) holonomies transformations, and fluxes and comes thus from span loop the quantum space obtained2 gravity. We by work SU(2) case [30]. inSee the [8, first-order 22, 23] for tetrad details,However, formalism, and below [24] and wefor start will extensions from identify the often-called to⇧ with the case the Holst Plebanski of a action null hypersurface. field for general smeared relativity. over 2-dimensional In terms surfaces in a t = const. symplectic reduction. e e I e e of chiralIt variables,is customary it isslice the of fix initial the data. time Be gauge this hypersurfaceN =(1, 0 space-like,, 0, 0), but the the restriction construction is automatically extends to fulfilled, an arbitrary and has no physical Let us add two more remarks to complete the analysis. First, the map between twistors and holonomy-fluxi variables is 2–to–1,gauge since [22]. exchanging In thisconsequence gauge, spinors we as identify for the following. the left-handed generators with 3-vectors ⇧ , whose real and imaginary e i i ~ +ii Ai i B I parts are rotations L andSHolst boosts[A, e]=K ,via⇧ =(⌃L B+(e)iKF)/2.A(A The)+cc normal., N is conserved(16) by the canonical ` 2 i ^ (!, ⇡, !, ⇡) P (⇡, !,Z⇡M, !) (15) 2This is convenient for the geometric interpretation7! of the theory. Switching to the alternative conventions with equal-sign ` = 8⇡ G /c3 > 0 B. Twistors from the LQG actioncc. leaves both holonomywherebracketsP (10)also used and~ inN flux the literature (2)is the unchanged. Planck is straightforward, length, Hence, toviais arrive the the map Barbero–Immirzi at!A the⇡ reducedA, ⇡A parameter, phase!A. space, and we “ also”denotes complex conjugation of everything preceding. See e.g. [31–34] for the case7! with boundary7! terms. The action (16) have to divide out this residualp Z2 symmetry (15). Second, because of the restriction ⇡! =0,thespinorial is a non-analytic functionalThe ofinterest the left-handed ine thee phasesl(2, spaceCAB) econnectione of SL(2, CA),andthefoursolderingformsholonomies2 and fluxes6 comese transforming from loop quantum gravity. We work parametrization cannot cover( the1 , 1 ) submanifold of (h,SL(2⇧):, )⇧ ⇧AB =F ATr ⇧ =0, and what⌃ weA truly(e) find in the irreducible 2 2inrepresentation the first-order of tetradC formalism,. Curvature and4 startB and from Plebanski the often-called 2-form HolstB are action uniquely for general relativity. In terms is T ⇤SL(2, C)determinedremoved from by the all equations itsof null chiral configurations. variables, it is The complete isomorphism could be defined through asuitabletreatmentofthedegenerateconfigurations,seee.g.theanaloguesituationintheSU(2) case [30]. ¯ i i However, below we willA identify A⇧ withA the PlebanskiC fieldA smearedAC over 2-dimensional~ A +i i surfaceslA m in a0Bt =i const. F B(A)=dA B + A C A B, ⌃ B(e)=SeHolst[A,eB eC¯]== ⌧ Bi ✏ lm⌃e Be(e) + eF Ae(A)+cc, .,(17) (16) slice of initial data. Be this hypersurface space-like,^ the restriction is^ automatically2` 2 i fulfilled,2 and^ has^ no^ physical P ✓ ZM ◆ consequence for the following. 3 where `P = 8⇡~GN/c is the Planck length, > 0 is the Barbero–Immirzi parameter, and “cc.”denotes complex conjugation of everything preceding. See e.g. [31–34] for the case with boundary terms. The action (16) p is a non-analytic functional of the left-handed sl(2, C) connection A,andthefoursolderingformse transforming B. Twistors from1 1 the LQG action A A in the irreducible ( 2 , 2 ) representation of SL(2, C). Curvature F B and Plebanski 2-form ⌃ B(e) are uniquely determined by the equations The interest in the phase space of SL(2, C) holonomies and fluxes comes from loop quantum gravity. We work in the first-order tetrad formalism, and startA from theA often-calledA HolstC actionA for generalAC¯ relativity.i InA termsi i l m 0 i F (A)=dA + A A , ⌃ (e)=e e ¯ = ⌧ ✏ e e + e e , (17) of chiral variables, it is B B C ^ B B ^ BC 2 Bi 2 lm ^ ^ ✓ ◆ ~ +i A B SHolst[A, e]= ⌃ B(e) F A(A)+cc., (16) ` 2 i ^ P ZM 3 where `P = 8⇡~GN/c is the Planck length, > 0 is the Barbero–Immirzi parameter, and “cc.”denotes complex conjugation of everything preceding. See e.g. [31–34] for the case with boundary terms. The action (16) p is a non-analytic functional of the left-handed sl(2, C) connection A,andthefoursolderingformse transforming 1 1 A A in the irreducible ( 2 , 2 ) representation of SL(2, C). Curvature F B and Plebanski 2-form ⌃ B(e) are uniquely determined by the equations

A A A C A AC¯ i A i i l m 0 i F (A)=dA + A A , ⌃ (e)=e e ¯ = ⌧ ✏ e e + e e , (17) B B C ^ B B ^ BC 2 Bi 2 lm ^ ^ ✓ ◆ takes the form of a Lie derivative of the group character, and when interpreted in terms of spinors, it acts as a boost in mixing the source and target spinors. In spite of its simplicity, its action on holonomies and fluxes is completely non-linear, and has no resemblance with usual dilatations. This is unavoidable, as the discrete phase space has compact directions, so there is no usual meaning of linear dilatations. On the other hand, the generator has the property of preserving the symplectic structure, and all transformed quantities recover the expected behaviour in the continuum limit. All our results, computing the action of SU(2, 2) dilatations, the relation between the di↵erent boost dihedral angles in the literature and proving the equivalence of di↵erent secondary simplicity constraints, as well as computing the finite action of the discrete holonomy-flux dilatation, rely heavily on the use of twistorial formalism, and are a demonstration of its usefulness to loop quantum gravity.

2 Twistors and twisted geometries

↵ A 2 ¯ 2 ¯ A˙ A twistor can be described as a pair of spinors, Z =(! ,i⇡¯A˙ ) C C ⇤ =: T. It has a dual, Z↵ =( i⇡A, !¯ ), which defines a pseudo-Hermitian norm of signature (+ + ),2 SU(2)Step 2: Simplicity constraints and constrained incidence relation subgroup of the Lorentz group, and allows us to define the Hermitian scalar product between spinors, AA˙ ↵ ¯ A 2 ! ⇡ := ⇡A!¯A˙ . We willZ alsoZ↵ use= 2Im( the short-hand⇡A! ), notation ! := ! ! for the norm. (1) h | i i i k k h | i Consider the following constraintThe constraints (11) read K + L = 0, and are equivalent to imposing the matching of left-handed and preserved by SU(2,2)right-handed transformations. sectors Itup is to well-known a phase, [8] that these transformations can be realized by Hamil- tonian vector fields, if we equip the spaceIJ with canonical Poissoni brackets,i✓ i ✓ SU(2) subgroup of the LorentzNI ( group,?)J and=0 allows us⇧ to= definee ⇧¯ the, Hermitian = cot scalar. product between spinors, (12) AA˙ , 2 2 ! ⇡ := ⇡A!¯A˙ . We will also use the short-hand↵ ¯ notation↵ ! := ! ! for the norm. h | i Using (3), the constraintsi i haveZ simple, Z spinorial= i . equivalentsk k h [13,| 14],i (2) Thenamely the matching up to a phase constraints (11) read K + of left and right su(2) algebrasL = 0, and are equivalent to imposing the matching of left-handed and Inright-handed Penrose’s abstract sectors index up to convention, a phase, the generatorsF of= SU(2,2)D s can=0,F be written= as! ⇡ =0. (13) 1 ✓ 2 h | i i i✓ ¯ i 2 IJ (A B) A˙B˙ ⇧I = Ae A˙⇧ , I A= cotA !A˙ . AA˙ A (12) Using J = ! ⇡ ✏ +cc,Pthe constraints read= i⇡ ⇡¯ ,C!= i=! !¯|| ||,D ⇡¯ =Re(˙ ⇡ ! ), (3) The constraint F1, real and Lorentz-invariant, is solved( posing2 i)j A A Using (3), the constraints have simple spinorial equivalents [13, 14], and their finite action is realized via the exponential map, [⇡ ! =( + i)j, j R (14) | i 2 • One of these 3 is a Casimir equation: F1 = DIJ s =0,F2 =IJ ! ⇡ =0. (13) and fixes the relativeexp(! globalIJM phase). f := between exp(!IJ theM spinors.h, |)f.i F2, which explicitly depends on(4) the chosen time { ·} A˙ direction, implies that one spinor is proportional to the parity transform of the other, ⇡A ˙ !¯ .Putting The constraint• The rest aligns the null poles of the spinors to add up to F , real and Lorentz-invariant,A is solved posingN Selects a 2d space-like plane 1 / AA Half the twistor norm,the1 twos := conditions Im(4 ⇡A! together,), is called the helicity constraints of the can twistor, be) solvedsince expressing oneResearch spinor Report in terms of the other and the (which will act as the polyhedron’s face) norm j: I I Results. Using twistors,[ the⇡ ! classicalW =(= formalism sP+ i), canj, be extended( j + iR in)j a natural way˙ to null hypersurfaces, (5) (14) | The areas come from the values of the helicities: i ⇡ = 2 !¯A. (15) with the Euclidean polyhedra replacedimportance of working with full twistor space not simply the projective one by null polyhedraA with space-like2 AA faces,˙ and SU(2) by the little I IJKL group ISO(2). The main di↵erence is that a larger symmetry! group appears, due toA the fact that on a whereandW fixes:= the 1/2 relative✏ PJ globalJKLnull hypersurface,is thephase Pauli-Lubanski the between isometry group the is vector. larger: spinors. the The primaryF complex2 simplicity,k whichk constraints scalar explicitly⇡ presentA! in= depends theD formalism+ is, on for the which chosen we time also use the index-freeThere notationbecome is an [⇡ all alternative! first, class, is invariant and removing parametrization under the gauge the orbits Lorentz of during the the solution symplectic subalgebra. space,reduction selectsmotivated only the helicity by the canonicalA˙ structure of the direction, implies that one spinor| i is proportional to the parity transform of the other, ⇡A AA˙ !¯ .Putting The representationconstraints: ofγ- thenullsubgroup conformal twistors with incidence constrained to a time-like direction while of theF2 littleis algebra secondgroup. This constructed class, fact hasF an1 appealingis infirst this counterpart class. way A is in good particle not the coordinate theory: most as well-known, general among the one, its/ orbits since the is the SU(2) norm the two conditions! together,2, andrepresentations the theF -gauge-invariant constraints of massless particles can spinor only be depend solved is on expressing the spin quantum one number, spinor the translations in terms being of the other and the algebra has 15 dimensions,k k whereasredundant1 gauges. the carrying In our setting, space the gauge orbitshas have only the 8 geometric dimensions. interpretation In of fact, shifts along the the generators null (3) are norm j: I Idirection of the hypersurface. not independent: P and C are null and related to D and the LorentzA generators by As a consequence, information on the(A shapes + i of)j the polyhedra! A˙ is lost, ⇡ = !¯ . NR (15) and the result is a much simpler,A abelianz geometric:= 2 2j picture.AA˙ i It+1 can, be z = 2j. (16) [I J] IJ ! ! 2 2k k NL Remark: can be easily extended to time-like and null hypersurfaces described2P C by an=( EuclideanD + singulars?) J structure,P on the 2-dimensionalCk=k (D space-+ s ). (6) k k NR like surface defined by a foliation of space-time byp null· hypersurfaces. This p NL Null twisted geometries characterised by the 2d space-like data only 2 There is an alternativeHere j = 0geometric by parametrization assumption, structure is naturally and of we decomposed the have solution restricted into a conformal space, to metricj> motivated0 and using the byZ thesymmetry canonicalS0 of thestructure symplectic of the reduction Accordingly,constraints: the while threeF Casimiris6 secondscale factors, operators class, formingF locally of issu conjugate(2 first, 2) class. pairs. (see Proper the A action-anglegood Appendix coordinate variables for definitions) among its are orbits not independent: is the SU(2) norm by C2. Seeon [13] the gauge-invariant for details. phase1 space are described by the eigenvectors of the they are2 all proportional to the only conformally invariant quantity in ,thehelicity:0 Figure 2: Spacetime foliation by ! , and the F1-gauge-invariantThe solutionLaplacian spaceof the spinor dual is graph. thus is Finally, a five-dimensional we quantised the phase manifold space and its ,null dependent hypersurfaces on the Immirzi parameter (via ✓) and k k algebra using Dirac’s algorithm, obtaining a notion of spin networks for null the choice of time(2) gauge,2 that can(3) be parametrized3 (4) by one of4 the two Lorentzian spinors, plus the norm j; or hypersurfaces.C =6 Suchs spin,C networks are labelled= 6 bys SO(2),C quantum numbers,=6s and. are embedded non-trivially (7) A A alternatively,in the by unitary, the infinite-dimensional reduced spinor irreduciblez !, plus representations the norm of the! Lorentz: group. zA := 2j , z =k k 2j. (16) Hence, the representation of theThe algebra results, obtained on twistor in with PhD space! studenti is+1 special, Mingyi Zhang, andk k have furthermore been published in reducible, Phys. Rev. D with [5], irreducibles and have been presented at various internationalk k 0 conferences.= !A,j The result= z thatA, the! kinematical. quanta are (17) labelled by s. In this sense,carried Penrose by a refers co-dimension to twistors twop surface could as the point spinors towards{ a of holographic} thep{ conformal structurek k} of group.the theory, a fact HereIrreduciblej = 0 by representations assumption,which and with would we haveindependent have far reaching restricted consequences. real tovaluesj> However, of0 using the many Casimirs non-trivial the 2 stepssymmetry can are be needed obtained beforeof the these symplectic working with reduction The two normsresults canj beand extended! toplay include complementary the complete dynamics roles of the in system. theZ First geometric of all, our interpretation analysis needs to of loop quantum gravity : 6 2 ˜ theby largerC. See carrying [13] forj spaceis details. the areabemade complemented of the of face pairs withk dualk a of continuum to twistors the canonical link, (Z, while analysisZ), with of! theis generators Plebanski related action to constructed the for general extrinsic relativity by curvature, on linearity. a as weThis will recall below. null hypersurface. This motivated my next piece of research.||0 || spaceThe has now solution 4 conformallyWe space refer is to thus invariant twistors a five-dimensional quantities,satisfying the given simplicity manifold by the constraints two , helicities dependent (13), plus oron the equivalently the pseudo-hermitian Immirzi (15), parameter as simple product (viatwistors.✓) and3 ¯ ˜↵ 2 Z↵theZ choice, and the of Casimirs time gauge,On are a link,1.6 independent that Spin we can impose foam be dynamicsfunctions parametrized the same of set these of by constraints four one variables. of the on two both Lorentzianhas source been and considered spinors, target twistors, in plus the the twistor norm j; or A alternatively,1 by the reducedDescription spinor of thez problem., plusThe the current norm state of the! art: for the dynamics of LQG uses the Engle-Pereira- In the twistor literature, the helicity is usually given by the realA part.k k TheA di↵erenceA comes fromA the extra i used here in the definition of the twistor. In thisRovelli-Livine way we can spin match foam model Dirac’s to implement conventionsz the Hamiltonian for! a bi-spinor, constraintz˜ and and reduce bridge the!˜ more kinematical easily scalar with the notation product among spin network states to a physical= one. The model, is supported= by a number, of results. (18) used in loop quantum gravity. Notice also that we use metric0 = conventions!Az,j =! withziA+1, mostly! . plus,z˜ thus we!˜ i map+1 vectors to anti-hermitian (17) Explicit evaluation of transition amplitudesk k is on thek otherk hand hinderedk k by thek sheerk complexity of the matrices, see Appendix A. model. This is especially true for Lorentzian{ signature} { (the modelk k exists} for both Lorentzian and Euclidean ¯ ˙ ¯ The two norms jwithand induced!signatures),play Poisson complementary where thebrackets definition ofzA theroles, z¯ vertexB in= amplitude theiA geometricB includes= unboundedzÃ interpretation, z˜¯B integrals,, and due reduced to of the loop non- area quantum matching gravity condition : C = 2 compact2 nature of the Lorentz group.{ The field}2 has focused2 for a number{ of years} on technical and4 formal z k z˜kdevelopments,= 0. The and partial latter studies implies of the semiclassicalz = limit.z˜ It=2 is myj, opinion and reducesthat the most this urgentC questionsymplectic space to T ⇤SU(2), as j is the area of thek k facek dualk to the link, while ! 3kisk relatedk k to the extrinsic curvature, as we will recall below. shown in [25],in this parametrizedapproach is to show by how to|| do calculations,|| and compute. I have recently obtained two important 3 We refer to twistors satisfyingresults in the this direction. simplicity First, I constraints have shown how the (13), model or can equivalently be recasted in a way (15), in which as thesimple vertex twistors. amplitudes are purely SU(2). All the non-compacteness goes into integrals over boosts associated with edges. On a link, we impose the same set of constraints1 on both sourcez˜ z + andz˜][z target~ twistors,1 In this way, the model resemblesX~ = a latticez ~ z gauge,g theory= model,| ih with| the| plaquette|, weightX˜ decomposed= z˜ into~ z˜ , (19) edge weights. Secondly, the edge integrals can be related in a precise way to Clebsch-Gordan coecients for A 2h A| | i A z zÃ 2h | | i SL(2, C). Using andz extending results! in the relevant literature,z˜ Ik havekk!˜ beenk able to derive a number of finite and asymptotic properties,= and more importantly,, set up the= ground for systematic, numerical evaluations. (18) where ~ is the vector ofz Pauli matrices! i+1 and X2z˜= X˜ 2 =!˜ ji2.+1 Again, X and X˜ are related on-shell of C by the k k k k k k k k adjoint action2 General of g, and relativity act as right-invariant on null hypersurfaces and left-invariant vector fields. with induced PoissonThat brackets is, twistorszA constrained, z¯B¯ = i byAB˙ the= areazÃ matching, z˜¯B¯ , and and reduced primary simplicity area matching constraints condition describeC the= phase A central feature of general relativity is the presence of non-linear constraints among the initial data on a 2 2 space T SU(2) of loop{ quantum}2 gravity.2 When{ SU(2)} gauge-invariance4 holds, by means of the closure conditions z z˜ = 0. The⇤ latterCauchy implies surface, whichz prevent= an explicitz˜ =2 descriptionj, and of the reduces physical degrees this of freedom,C symplectic and complicate the space to T ⇤SU(2), as k k k k dynamics. Taming themk k is a crucialk partk of the theory, be it for formal studies and quantisation attempts, or 3 shown in [25], parametrizedThe termfor numericalsimple by twistors relativity is and also applications used in to the astrophysics twistor andliterature, gravitational but waves. in reference An interesting to a bi-twistor approach being simple in the same sense of a simple bivector as defined above. 1 z˜ z + z˜][z ~ 1 X~ = z ~ z ,g= | ih | | |, X˜ = z˜ ~ z˜ , (19) 2h | | i z z˜ 2h | | i k kk k 5 where ~ is the vector of Pauli matrices and X2 = X˜ 2 = j2. Again, X and X˜ are related on-shell of C by the adjoint action of g, and act as right-invariant and left-invariant vector fields. That is, twistors constrained by the area matching and primary simplicity constraints describe the phase space T ⇤SU(2) of loop quantum gravity. When SU(2) gauge-invariance holds, by means of the closure conditions

3The term simple twistors is also used in the twistor literature, but in reference to a bi-twistor being simple in the same sense of a simple bivector as deﬁned above.

5 10 Chapter 2. General relativity in connection variables and loop quantum gravity

1.2 Plebanski action. 10 Chapter 2. General relativity inAnticipating connection on variables the canonical and loop analysis quantum to gravity appear below, the structure of the action (2.1) shows that the IJ pull-back !a of the connection on a spatial slice will be conjugated to a wedge product of two tetrad fields, more precisely to the bi-vector 1.2 Plebanski action. aIJ 1 abc I J B := ✏ eb ec . (2.30) Anticipating on the canonical analysis to appear below, the structure of the action (2.1) shows that the 2 IJ 10 Chapter 2. General relativity in connectionIJ variables and loop quantum gravity pull-back !a of the connection on a spatial slice will be conjugatedRecall that to a a bivector wedge productB which of two is tetrad the exterior fields, product of two vectors is called simple, and satisfies the IJ KL more precisely to the bi-vector simplicity constraint ✏IJKLB B = 0. Notice that for each component a, the bi-vector in the internal 1.2 Plebanski action.aIJ 1 abcindicesI J is of a special form, called simple Hence one way to do the canonical analysis is to start from BF B := ✏ eb ec . (2.30) 2 and add so called simplicity constraints reducing the B field to the exterior product of two tetrads. This Anticipating on the canonical analysis to appear below, the structure of the action (2.1) shows that the IJ equivalent reformulation of the theory is also interesting because it gives a direct link to TQFTs. Recall that a bivectorpull-backB which!IJ isof the the exterior connection product on a spatial of two slice vectors will be is conjugated called simple to a, andwedge satisfies product the of two tetrad fields, IJ aKL simplicity constraint ✏moreIJKL preciselyB B to= the 0. bi-vector Notice that for each component a, the bi-vector in the internal indices is of a special form, called simple Hence one way toConsider do the canonical the1 following analysis action, is to start from BF BaIJ := ✏abceI eJ . (2.30) and add so called simplicity constraints reducing the B field to the exterior2 productb c of two tetrads. This IJ 1 IJ 1 ⇤ IJ KL equivalent reformulationRecall of the that theory a bivector is alsoB interestingwhichS is because the exterior it gives product aS direct(B, of! two, link)= vectors to TQFTs.( is+ called?)BsimpleF,IJ and(!) satisfies theIJKL + ✏IJKL B B , (2.31) IJ { KL} ^ 2 6 ^ simplicity constraint ✏IJKLB B = 0. Notice that for each componentZ a, the bi-vector in the internal✓ ◆ Consider the followingindices action, is of a special form, called simple Hence one way to do the canonical analysis is to start from BFIJ and add so called simplicity constraintsSimplicity constraints from the Plebanski actionwhere reducingF (!) the is theB field curvature to the exterior of an so product(4) connection of two tetrads.!, and ThisB a 2-form with values in the algebra. This equivalent reformulation1 IJ of the theory1action is also can interesting be⇤ called because non-chiral,IJ it givesKL to a direct distinguish link to TQFTs.it from the original Plebanski action, which is identical but S S(B,!, )= ( + ?)B F (!) + ✏ B B , (2.31) IJKL { } ^ IJ 2usesIJKLg = su2.6 IJKL⇤ is the cosmological^ constant, and the field a Lagrange multiplier, symmetric under The simplicity constraints come from writing GR as a BF theory (Plebanski action)✓ ◆ ConsiderZ the following action, exchange of the first and the second pair, and antysimmetric within each pair. In addition, we impose on IJ IJKL where F (!) is the curvature of an so(4) connection !, and theB constrainta 2-form with✏IJKL values in the=0 algebra.. It has thereforeThis the same symmetries of the Riemann tensor, and 20 action can be called non-chiral, to distinguish it from the1components. originalIJ Plebanski Its variation1 action, gives which⇤ the is identical followingIJ but equations,KL known as simplicity constraints, S S(B,!, )= ( + ?)B FIJ(!) IJKL + ✏IJKL B B , (2.31) { } IJKL^ 2 6 ^ uses g = su2. ⇤ is the cosmological constant, andZ the field a Lagrange✓ multiplier, symmetric◆ under IJ KL 1 IJKL exchange of the first and the second pair, and antysimmetricsimpl within each pair. InIJ addition, weB imposeB on = ✏ B ?B . (2.32) whereIJKLF (!) is the curvature{ of an}so(4) connection !, and B a 2-form with values^ in the algebra.12 Thish ^ i the constraint ✏IJKL =0. It has therefore the same symmetriesIJKL of the Riemann tensor, and 20 action can beLagrange multiplier satisfies and leads to 20 equations imposing the simplicity of called non-chiral, to distinguish" it fromIJKL the=0 original Plebanski action, which is identical but B : components. Its variationuses givesg = su the2. ⇤ followingis the cosmological equations, knownconstant,Solutions as and simplicity satisfying the field constraints, theIJKL non-degeneracya Lagrange multiplier, condition symmetricB ?B under= 0 can be divided in two sectors [12], h ^ i6 exchange of the first and the second1 pair, and antysimmetric within each pair. In addition, we impose on simpl the constraintB✏IJ BKLIJKLBsols= =0✏.IJKLIt hasB therefore?B . the same symmetriesBIJ = of the(1/ Riemann2)(2.32)✏IJ eK tensor,eL,B and 20IJ = eI eJ . (2.33) { } IJKL { } KL components. Its variation^ gives12 the followingh ^ equations,i known as simplicity constraints,± ^ ± ^ Solutions satisfying the non-degeneracy condition B ?BWe= will 0 can review be divided the derivation in two sectors below. [12], In both cases, the 20 constraints (4.12) reduce the initial 36 independent h ^ i6IJ KL 1 IJKLIJ I simpl Remarks: BcomponentsB = of B✏ µ⌫ downB to?B 16,. parametrized by a tetrad e(2.32)µ, and the action reduces to (2.1). gets further { } IJ IJ K L correctlyÎJ identified12I J providedh ^ i we choose the first of the four solutions in (2.33). All other 3 correspond to Bsols B •= : self-dual formulation =(1/2)i ✏KLe e ,B= e e . (2.33) { } ± ± ^ simple redefinitions± ^ of . Solutions satisfying• For real, two Urbantke metrics; the constraints make them coincide the non-degeneracy condition B ?B = 0 can be divided in two sectors [12], We will review the derivation below. In both cases, the 20 constraintsIt is also (4.12)h useful^ reducei6 to review the initial the 36 on-shell independent meaning of the Lagrange multiplier. To that end, let us look at IJ • In the Hamiltonian formulation, the constraints lead to secondary constraints which impose part of the IJ I IJ K L IJ I J components ofBsolsB down to 16, parametrized by aB tetrad= e(1,/2) and✏KL thee actione ,B reduces= to (2.1).e e .gets further (2.33) { µ⌫} ±theµ other field^ equations. These± ^ read correctly identified provided we choosetorsion-less condition the first of the four solutions in (2.33). All other 3 correspond to We will review the derivation below. In both cases, the 20 constraints (4.12) reduce the initial 36 independentτs τt simple redefinitions of . IJ Self-duality and metricityI components of Bµ⌫ down to 16, parametrized by a tetrad eµ, and the action reduces to (2.1). gets further It is also useful to review the on-shell meaning of the Lagrange multiplier. To that end, let us look at ˜ Discretisation: one normal per node, one generator per link Nl Nl correctly identified provided we chooseIt the is intriguing first of the to four unravel solutions the inmechanism (2.33). All behaind other 3 this correspond mechanism. to Indeed, the key to the formalism is the the other field equations.simple These redefinitions read of . fact that the 2-forms B can always be used to introduce a metric, regardless of any constraints. The role It is also useful to review the on-shellIJ meaning of the Lagrange multiplier. To that end, let uss(l) look at jl constraints: NnI B =0 t(l) Self-duality and metricitythe other field equations. These readl of the constraints is rather to single out this metric, namely to freeze the remaining components of the 2-forms which do not enter the definition of the metric. To be more specific, given an su2-valued 2-form It is intriguing to unravel the mechanism behaind this mechanism.i Indeed, the key to the formalismBIJ is=( the ?)J IJ Self-dualitycanonical conjugate variable to the connection: and metricity Bµ⌫ , i =1, 2, 3, a metric canJ such that be defined through the well-knownl Urbantke formula [13, 14], fact that the 2-forms B can always be used to introduce a metric, regardless of any constraints. The role IJ of the constraints is ratherIt is intriguing to single to out unravel this the metric, mechanism namely behaind to freeze this the mechanism. remaining Indeed, components the keyU of to the the1 formalism↵ is thei j k gU NnI ( ?)Jl =0 gU g = ✏ ✏ B B B . (2.34) 2-forms which do notfact enter that the the definition 2-forms B ofcan the always metric.{ } be To used be to more introduce specific, a metric, given an regardlesssu2-valued of any 2-formµ⌫ constraints.12 ijk The roleµ↵ ⌫ i of the constraints is rather to single out this metric, namely to freeze the remaining components of the Bµ⌫ , i =1, 2, 3, a metric can be defined through the well-known Urbantke formula [13, 14], p 2-forms which do not enter the definitionNotice of in the this metric. formula To be the more completely specific, antisymmetric given an su2-valued tensor 2-form✏ijk, the unique singlet in the tensor product of i Bµ⌫ , i =1, 2, 3, a metric can1 be definedthree through adjoint the representations well-known Urbantke of SU(2). formula The [13,B 14],field needs to be complex for this metric to have Lorentzian gU gU gU = ✏ ✏↵Bi Bj Bk . (2.34) { } µ⌫ 12 ijk signature,µ↵ while⌫ a real field yields Euclidean signature. The same mechanism can be applied to both the 1 originalU U Plebanski↵ actionsi j fork general relativity and the modified theories. In the original case, the action gU p g gµ⌫ = ✏ijk ✏ Bµ↵BB⌫. (2.34) Notice in this formula{ } the completely antisymmetric tensorgives✏ijk, quadratic the12 unique field singlet equations, in the tensor some product of which of are the “metricity constraints” which freeze the remaining three adjoint representations of SU(2). The B field needsp to be complex fori this metric to have Lorentzian Notice in this formula the completelycomponents antisymmetric of tensorB not✏ijk captured, the unique by singlet (2.34), in and the tensor the rest product reduce of to the Einstein equations for (2.34). If we signature, while a realthree field adjoint yields representations Euclidean signature. of SU(2). Theconsider The sameB field a mechanism modification needs to can be where complex be applied the for constraints this to both metric the are to removed, have Lorentzian not only we get new field equations, but the extra original Plebanski actionssignature, for general while a relativity real field and yields the Euclidean modified signature. theories. In The the same original mechanism case, the can action be applied to both the gives quadratic field equations,original Plebanski some of actions which for are general the “metricity relativity constraints” and the modified which theories. freeze the In the remaining original case, the action components of Bi notgives captured quadratic by (2.34), field equations, and the rest some reduce of which to the are the Einstein “metricity equations constraints” for (2.34). which If freeze we the remaining consider a modificationcomponents where the constraints of Bi not captured are removed, by (2.34), not only and we the get rest new reduce field to equations, the Einstein but equations the extra for (2.34). If we consider a modification where the constraints are removed, not only we get new field equations, but the extra Summary

Twistors provide powerful tools to study the semiclassical geometry of loop quantum gravity, and permit an elegant covariant description of the classical limit of spin networks

Spin networks (Flat) Twistors

Twisted geometries

Regge geometries IV. SU(2,2) spin networks and a self-dual octahedron (wip Maciej Dunajski, Miklos Langvik and S) Some motivations…

As mentioned in the Road to Reality, Penrose originally intended to extend SU(2) spin networks to SU(2,2), as a way of introducing a notion of scale and distances via the Poincare generators, while keeping semi- simplicity of the group

This line of thought then blossomed in twistor theory… And conversely, LQG solved the issue of distances via the Ashtekar variables, which endow the SU(2) spins with quanta of areas and volumes

But how would do SU(2,2) spin network relate to the SU(2) ones of LQG? Are the associated notions of scales and distances comparable?

Can we use SU(2,2) spin nets as tools to investigate questions as breaking of conformal symmetry in LQG?

Motivated by these questions, one is led to look for a twistorial embedding of T*SU(2,2) ab InIn the the following, following, we we will will denote denote the the abstract set set of of 15 15 susu(2(2,,2)2) generators generators by by antisymmetric antisymmetric matrices matricesMMab, , ((a,a, b b)=0)=0,...,...5.5. The The action action is is Hamiltonian Hamiltonian with respect respect to to the the canonical canonical SU(2 SU(2,,2)-invariant2)-invariant Poisson Poisson structure structure ↵ ↵ Z↵ , Z¯¯ = i↵ . (4) ZZ {Z , Z }= i. (4) ZZ{ } { } ab { } Our goal isIn to the extend following, this we representation will denote the to abstract the larger set of Poisson 15 su(2, algebra2) generators carried by by antisymmetric the symplectic matrices manifoldM , Our goal( isa, b to)=0 extend,...5. this The representation action is Hamiltonian to the with larger respect Poisson to the algebra canonical carried SU(2, 2)-invariant by the symplectic Poisson structure manifold TT⇤⇤SU(2SU(2,,2).2). Let Let us us fix fix some some notations notations and provide explicit explicit expressions, expressions, which which will will be be useful useful references references for for later later ↵ ¯ ↵ results. Using the isomorphism between a Lie algebraZ and, Z the= i (co-)tangent . vector space at the origin of(4) its ZZ results. Using the isomorphism between a Lie algebra{ and the} (co-)tangent vector space at the origin of its { } ↵↵ LieLie group group manifold, manifold,Our goalTT⇤⇤SU(2SU(2 is to,, extend2)2) SU(2SU(2 this representation, 2) susu(2(2,,2)2) to can can the be be larger parametrized parametrized Poisson algebra by by a a carried unitary unitary by 4x4 the 4x4 symplectic matrix matrixGG manifold,, and and ↵↵ '' ⇥ algebraalgebra generators generatorsT ⇤SU(2MM, 2). Letinin usthe the fix fundamental fundamental some notations irrep. and provide Usingand Using (2) explicit (2) and and expressions, its its inverse inverse which with with upstairwill upstair be useful indices, indices, references we we write write for later the the conditioncondition for forresults. unitarity unitarity Using as as follows, follows, the isomorphism between a Lie algebra and the (co-)tangent vector space0 at2 the origin¯ of its ¯˙ A¯ ⌘↵˙ = , Z↵ = ⌘↵˙ Z =( i⇡A, ! ) (C.16) 1↵ ↵↵ ↵↵↵˙↵˙ ˙˙ 2 0 ↵ Lie group manifold, T ⇤SU(2, 2)GSU(2=, 2)G† su(2, 2)⌘ canG¯¯ be⌘ parametrized by a unitary 4x4 matrix G , and(5) Gunitary ↵ G' = G⇥† ⌘ G ↵˙↵˙⌘˙˙ ✓ ◆ (5) Gunitary algebra generators M in the fundamental irrep.The⌘⌘ Usingi factor (2) and allows its toinverse see the with twistor upstair as indices, a Dirac we write bi-spinor the in{ { the chiral} } basis, and it is used in the LQG abab↵↵ baba↵↵ WeWe denote denote by bycondition == for unitarity ,(,(a,a, as b b follows,)=0,...5,5, a a basis basis of of susu(2(2,,2)2) in in the the fundamental fundamental 4x4 4x4 representation representation (which (which A A literature, as well as˙ the bar over ⇡ to keep track of its anti-selfdual nature: ⇡LQG = i⇡¯P enrose. As a consequence, 1↵ ↵ ↵↵˙ ¯ Gunitary cancan be be written written from from tensor tensor products products of identitiesG and anddilatation Pauli Pauli= G† matrices, matrices, generators⌘ G see see↵˙ ⌘ and (C.45)˙ (C.45) helicity for for are explicit explicit inverted: formulas); formulas); so so that that(5) ↵↵ abab abab↵↵ ⌘ { } MM == MM ,, and andab↵ the the representation representationba↵ (3) (3) can can be be written written as as a a bilinear bilinear in in the the twistors, twistors, a

= arg Z arg Z˜ = 0 (16) 2This is di↵erent from the previous cases, where the equivalent of the helicity constraints played the role of both relating M and M, but also closing theimposes Jacobidet identityG =1 for, the makes reducedU algebra.(and thus TheU origin˜) pure of this gauges, di↵erence and is describesin the di↵erent a symplectic structure of space the of 30 real dimensions group element. There di↵isomorphicerent dyads did to notT ⇤SU(2 commute, 2), a parametrized priori. Here they by do. (14) above. See Appendix C for detailed demonstration, I still need to type it here in a short and clear form. 3 3 Self-dual octahedron

The various constraints that we have considered have interesting geometric meanings. Let us begin with the incidences.

2This is di↵erent from the previous cases, where the equivalent of the helicity constraints played the role of both relating M and M, but also closing the Jacobi identity for the reduced algebra. The origin of this di↵erence is in the di↵erent structure of the group element. There di↵erent dyads did not commute a priori. Here they do.

3 One constraint surface and the self-dual octahedron

4 (non-null) twistors mutually incident: ⇢ = Z¯ Z =0 ij i · j

4 16 6 complex equations in T C '

Constraint surface is ten-dimensional Let us look at the projective twistor space and the associated six-dimensional constraint surface TWELVECOMPLEXNULLRAYS 3 in complexiﬁed Minkowski space X12 23 4 α and 4 β planes mutually intersecting 14 24 along null complex lines, X24 X32 21 with vertices coinciding with 34 13 31 43 the intersections of pairs of α planes X13 12 X14 1 42 Self-dual in the sense that X = " X¯ ij 2 ijkl kl 41 32

X34 Recall that generate a U(4) algebrasi, ⇢ij mixing the twistors Fig. 4. The octahedral conﬁguration of twelve complex null rays in MC.

The conﬁgurations of octahedra should be spanned by the orbits of the generatorsMaciej Dunajski, Cambridge, February⇢ij 27th, 2015. A fun generalisation of the problem solved by Tod a long time ago of the symplectic reduction to Souriau phase space

6 Conclusions

Twistors provide powerful tools to study the semiclassical geometry of loop quantum gravity, and permit an elegant covariant description of the classical limit of spin networks

The relation is for the moment restricted to ﬂat twistors, and a ﬁnite number of them!

Still much more to explore, conformal spin networks, curved twistors, hyperbolic twisted geometries, continuum limit… and if the extension to curved twistors works, could the γ-embedding provide new cues to the googly problem?