aa r X i v : . [ m a t h - ph ] J u l The Ponzano-Regge asymptoticof the supersymmetric 6jS symbols
Lionel Br´ehametJuly 23, 2018
Abstract
We adapt the Gurau’s proof (2008) about the asymptotic limit of Ponzano-Regge formula tosupersymmetric 6jS symbols according to their intrinsic parities alpha, beta, gamma. Thebehaviour at a large scaling shows significant differences depending on these parities. Thedecay is slowed and the angles of the oscillating parts in cosine are generally shifted or morealtered. Our results should be relevant in 3-D Quantum Supergravity and Spin Foam models.PACS: 03.65.Fd Algebraic methodsPACS: 04.60.-m Quantum gravityPACS: 11.30.Pb SupersymmetryMSC: 81Q60; 83D05.Keywords. 6j symbol; Quantum Gravity; supersymmetric 6jS symbol. Introduction
Since the growing expansion of quantum gravity theory [1] the SU (2) 6 j symbols acquireda considerable importance by becoming the basic building blocks of all spin networks. Theyappear to represent a quantum tetrahedron with quantized edges and even can be viewed aseigenfunctions of a discrete Schr¨odinger equation [2]. As well known the classical Ponzano-Regge partition function Z P R [3] was expressed as a sum of products of 6 j symbols. In [4] theoccurrence of supersymmetric 6 j S symbols [8, 9], related to OSP (1 | Z sugra . A possible different divergencecompared to the classical case was sketched.Guided by the conceptual approach of Gurau [10], our present task is to give also an elementaryproof of the Ponzano-Regge asymptotic of the 6 j S supersymmetric symbols.Let us recall the formulation of Gurau: Under a rescaling of all its arguments by a large k the j symbol associated to an Euclideantetrahedron behaves like (cid:26) kj kj kj kJ kJ kJ (cid:27) = 1 √ πk V cos n π X ι =1 (cid:2)(cid:0) kj ι + 12 (cid:1) θ j ι + (cid:0) kJ ι + 12 (cid:1) θ J ι (cid:3)o , (1) where V is the tetrahedron volume and θ j ι , θ J ι are the exterior dihedral angles of the tetrahedroncorresponding to the edges j ι and J ι respectively. Tetrahedron of volume V J J J j j j j and j S symbols and notations We rewrite our formulas [9] with notations as close as possible to those used by Gurau [10] andreplace j i ↔ J i , p → v , p → v , p → v , p → v , q → p . Notations used in [10] were v = j + j + j , v = J + j + J , v = J + J + j , v = j + J + J , (2) p = j + J + j + J , p = j + J + j + J , p = j + J + j + J . (3) Diagrammatic representation of a j symbol (cid:26) j j j J J J (cid:27) : the four triangles of any j the three columns pairs v v v v p p p v i is the sum of the values of the three circled spins just above v i in the diagrams. In the sameway, p j is the sum of the values of the four circled spins above p j . The variables p , v satisfy theequation X j =1 p j = X i =1 v i . (4)2s a result any spin is determined by two v i and one p j according to2 j = v + v − p , j = v + v − p , j = v + v − p , (5)2 J = v + v − p , J = v + v − p , J = v + v − p , (6)or by two p j and two v i ‘complementary’ according to2 j = p + p − v − v , j = p + p − v − v , j = p + p − v − v , (7)2 J = p + p − v − v , J = p + p − v − v , J = p + p − v − v . (8)Formulas for a 6 j symbol are well known, however here we shall use an expression [11], usedelsewhere long ago. That avoids the usual triangle coefficients △ and contains only terms in v i , p j . (cid:26) j j j J J J (cid:27) = √ R X t ( − t ( t +1)! Q i =4 i =1 ( t − v i )! Q j =3 j =1 ( p j − t )! , (9)where the radical R under square root is the prefactor used by Gurau R = Q j =3 j =1 Q i =4 i =1 ( p j − v i )! Q i =4 i =1 ( v i + 1)! . (10) A short historical review about the supersymmetric j S symbols: The conceptualization of supersymmetric 3 j or 6 j symbols was done in several years spread overtime, with sometimes different notations. It started with Pais and Rittenberg [5] in 1975 with asemisimple graded Lie algebra named “graded su ( )”. Later in 1977, Scheunert et al. [6] havecomputed the super-rotation Clebsch-Gordan coefficients as a product of usual rotation Clebsch-Gordan coefficients with a scalar factor. Finally, in 1981, Berezin and Tolstoy [7] suggestedthat the (iso)scalar factors form a pseudo-orthogonal matrix. Subsequently these works werematerialized in the paper by Daumens et al. [8] dated 1993 with the introduction of properlydefined 6 j S symbols.Our definitions of supersymmetric 6 j Sπ symbols [9] of parity π = α, β, γ were published in2006, more than a decade after the paper of Daumens et al. .These 6 j Sπ symbols were expressed by a single sum formula over an index t [involving monomialsΠ π ( t )] as shown below: (cid:26) j j j J J J (cid:27) Sπ = ( − P j ι J ι p R Sπ X t ( − t t !Π π ( t ) Q i =4 i =1 ( t − [ v i + ] ) ! Q j =3 j =1 ( [ p j + ] − t ) ) ! , (11)where R Sπ is a supersymmetric prefactor given by R Sπ = Q j =3 j =1 Q i =4 i =1 [ p j − v i ] ! Q i =4 i =1 [ v i + ] ! . (12)Detailed expressions for R Sα , R Sγ , R Sβ are given in Appendix. Notation [ ] means ’integer part of’.In contrast to SU (2), OSP (1 |
2) triangles v i can be integer or half-integer, however they stillsatisfy the well known triangular inequalities.Let us recall the definitions of the parities π and monomials Π π ( t ) of degree 0 or 1 in t : π = α if ∀ i ∈ [1 , v i integer ,π = β if ∃ only two distinct v i , v j integer ,π = γ if ∀ i ∈ [1 , v i half-integer . (13) Ref. [8] was explicitly used in 2004 by Livine and Oeckl [4] . On that date our work [9] was not known. β , both integer triangles shall be denoted by v, v ′ , both other half-integer by v, v ′ .The single integer quadrangle is denoted by p , both other half-integer by p, p ′ .In this case eq. (4) transforms into p + p + p ′ = v + v ′ + v + v ′ . (14)Thus a parity π just depends on the quality half-integral or integer of a triangle v i . Let us notethat only the γ parity may contain supersymmetric symbols whose all the spins are half-integers.Our monomials Π π ( t ) were defined in [9] asΠ α ( t ) =1 , (15)Π β ( t ) = − t (2 j ⋆β + 1) + ( p + )( p ′ + ) − vv ′ , (16)Π γ ( t ) = − t +2( J j + J j + J j )+( J + j + J + j + J + j )+ 12 . (17)All constants Π α (0), Π β (0), Π γ (0) are positive integers .The special spin j ⋆β was identified [9] as the vertex common to both half-integer triangles v, v ′ :2 j ⋆β = p + p ′ − v − v ′ = v + v ′ − p. (18)Other shorter formulas are available: J j + J j + J j = X i = ι j ι J ι and J + j + J + j + J + j = 12 X j =1 p j . (19)Oddly enough these ‘supersymmetric’ quantities are reflected in most parameters of the discrim-inant necessary for the saddle points computation [10], while the background is that of standard6 j .An appropriate formulation more adapted to the manipulations of P t in [10] is the followingΠ β ( t ) = − ( t + 1)(2 j ⋆β + 1) + [(2 j ⋆β + 1) + ( p + )( p ′ + ) − vv ′ ] , (20)Π γ ( t ) = − ( t + 1) + [2 X ι =1 j ι J ι + 12 X j =1 p j + 32 ] . (21)The supersymmetric frontal phase ( − P j ι J ι is worthwhile of attention. From eq. (21) in [10] A = 2 X ι =1 j ι J ι = X i 12 )!( k ( p − v ′ ) − 12 )! × ( k ( p − v ))!( k ( p − v ′ ))!( k ( p − v ) − 12 )!( k ( p − v ′ ) − 12 )! × ( k ( p ′ − v ))!( k ( p ′ − v ′ ))!( k ( p ′ − v ) − 12 )!( k ( p ′ − v ′ ) − 12 )! × (cid:0) kv ! kv ′ ! (cid:1) − (cid:0) kv + 12 )!( kv ′ + 12 )! (cid:1) − . (5.1)11rom eqs. (32-33) the numerator of R Sβ ( k ) becomes NumPre β = (2 π ) e [ k ln( k ) − k ](4 P j p j − P i v i )+ k [ P j,i ( p j − v i ) ln( p j − v i )] × e [ln( p − v )+ln( p − v ′ )+ln( p − v )+ln( p − v ′ )+ln( p ′ − v )+ln( p ′ − v ′ )]+3 ln( k ) . (5.2)and the denominator is DenPre β = (2 π ) e [ k ln( k ) − k ]( v + v ′ + v + v ′ ) × e k [ v ln( v )+ v ′ ln( v ′ )+ v ln( v )+ v ′ ln( v ′ )] × e [ln( kv )+ln( kv ′ )]+ln( kv )+ln( kv ′ ) = (2 π ) e [ k ln( k ) − k ] P i v i + k [ P i v i ln( v i )] × e k ) e { ln( v )+ln( v ′ ) } +ln( v )+ln( v ′ ) . (5.3)Note the lack of frontal factor in k m in R Sβ ( k ).Whence R Sβ ( k ) = (2 π ) e k [ P j,i ( p j − v i ) ln( p j − v i ) − P i v i ln( v i )] × e ln (cid:8) ( p − v )( p − v ′ )( p − v )( p − v ′ )( p ′− v )( p ′− v ′ )( vv ′ )( v v ′ (cid:9) . (5.4)Thanks to the correlation table and the definition of the spin J ∗ β it can be proved that e ln (cid:8) ( p − v )( p − v ′ )( p − v )( p − v ′ )( p ′− v )( p ′− v ′ )( vv ′ )( v v ′ (cid:9) = ( p − v )( p − v ′ ) v v ′ e h J ∗ β , (5.5)where the header fraction is dimensionless. Then R Sβ ( k ) =(2 π ) e k [ P j,i ( p j − v i ) ln( p j − v i ) − P i v i ln( v i )] × ( p − v )( p − v ′ ) v v ′ e h J ∗ β . (5.6)Finally q R Sβ ( k ) =(2 π ) r ( p − v )( p − v ′ ) v v ′ e h J ∗ β + k h ( i,J ) . (5.7) Σ β and F β ( x ) From eqs.(11) and (20) we define two sums Σ ′ β , Σ ′′ β to be gathered to form a a Σ β :Σ ′ β = t max X t min ( − t ( t +1)!( t − v )!( t − v ′ )!( t − v − )!( t − v ′ − )!( p − t )!( p + − t )!( p ′ + − t )! , (5.8)Σ ′′ β = t max X t min ( − t t !( t − v )!( t − v ′ )!( t − v − )!( t − v ′ − )!( p − t )!( p + − t )!( p ′ + − t )! , (5.9)so that Σ β = − (2 j ⋆β + 1)Σ ′ β + [(2 j ⋆β + 1) + ( p + 12 )( p ′ + 12 ) − vv ′ ] Σ ′′ β , (5.10)12ith t min = max ( v, v ′ , v + , v ′ + ) and t max = min ( p, p + , p ′ + ) . (5.11)Computation of Σ ′ β is very similar to that of the standard sum Σ analyzed by Gurau whereasΣ ′′ β looks like our Σ α explicited in sect. .With all spins multiplied by the factor k , and changes t → kx, p, v → kp, kv , Σ ′ β ( k ) rewrites asthe sum P x = t max x = t min over the quotient below ( − kx ( kx +1)!( k ( x − v ))!( k ( x − v ′ ))!( k ( x − v ) − )!( k ( x − v ′ ) − )!( k ( p − x ))!( k ( p − x )+ )!( k ( p ′ − x )+ )! . (5.12)By using factorial approximations for large k we write Σ ′ β in detail asΣ ′ β = 1(2 π ) x = t max X x = t min G ′ β ( x ) , (5.13)with G ′ β ( x ) = e [ k ln( k ) − k ] x + k { ıπx + x ln( x ) } + ln( k x ) × e − (cid:8) ln k ( x − v )+[ k ln( k ) − k ]( x − v ′ )+ k ( x − v ′ ) ln( x − v ′ )+ ln k ( x − v ′ ) (cid:9) × e − (cid:8) [ k ln( k ) − k ]( x − v )+ k ( x − v ) ln( x − v ) (cid:9) × e − (cid:8) [ k ln( k ) − k ]( x − v )+ k ( x − v ) ln( x − v )+[ k ln( k ) − k ]( x − v ′ )+ k ( x − v ′ ) ln( x − v ′ ) (cid:9) × e − (cid:8) [ k ln( k ) − k ]( p − x )+ k ( p − x ) ln( p − x )+ ln k ( p − x ) (cid:9) × e − (cid:8) [ k ln( k ) − k ]( p − x )+ k ( p − x ) ln( p − x )+ln k ( p − x ) (cid:9) × e − (cid:8) [ k ln( k ) − k ]( p ′ − x )+ k ( p ′ − x ) ln( p ′ − x )+ln k ( p ′ − x ) (cid:9) . (5.14)All rearrangements done it remains G ′ β ( x ) = 1 k e ln x x − vi )Π( pj − x ) + ln ( x − v )( x − v ′ )( p − x )( p ′− x ) × e k { ıπx + x ln( x ) − P ( x − v i ) ln( x − v i ) − P ( p j − x ) ln( p j − x ) } . (5.15)It results in Σ ′ β ( k ) = 1(2 π ) x = t max X x = t min k e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F β ′ ( x )+ kf β ′ ( x ) , (5.16)with F β ′ ( x ) = 12 ln x Π( x − v i )Π( p j − x ) ≡ F ( x ) , (5.17) f β ′ ( x ) ≡ f ( x ) . (5.18)In the same way we deriveΣ ′′ β ( k ) = 1(2 π ) x = t max X x = t min k e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F β ′′ ( x )+ kf β ′′ ( x ) , (5.19)13ith F β ′′ ( x ) = 12 ln x Π( x − v i )Π( p j − x ) ≡ F ( x ) − ln( x ) , (5.20)so that e F β ′′ ( x ) = 1 x e F ( x ) and f β ′′ ( x ) ≡ f ( x ) . (5.21) β ) Identifying Σ ′ β of eq. (5.16) as a Riemann sum with k → dx leads toΣ ′ β ( k ) = 1(2 π ) k Z t max t min dx e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F ( x )+ k f ( x ) . (5.22)Equation (5.10), with j ⋆β → k j ⋆β , gives the following dominant contribution − (2 k j ⋆β + 1)Σ ′ β ( k ) = − j ⋆β (2 π ) Z t max t min dx e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F ( x )+ k f ( x ) . (5.23)In the same way Σ ′′ β ( k ) = 1(2 π ) k Z t max t min dx x e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F ( x )+ k f ( x ) . (5.24)Its coefficient in (5.10), [(2 j ⋆β + 1) + ( p + )( p ′ + ) − vv ′ ], is rescaled into[(2 k j ⋆β + 1) + ( kp + 12 )( kp ′ + 12 ) − k vv ′ ] = k [ p p ′ − vv ′ ] + k [2 j ⋆β + 12 ( p + p ′ )] + 34 . (5.25)The Σ ′′ β ( k ) contribution is also dominant and can be written down as1 x [ p p ′ − vv ′ ](2 π ) Z t max t min dx e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F ( x )+ k f ( x ) . (5.26)Finally the integral approximation has the formΣ β = (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x (cid:19) π ) Z x = t max x = t min e ln ( x − v )( x − v ′ )( p − x )( p ′− x )] + F ( x )+ k f ( x ) . (5.27)Writing f ( x ) as the equation (27) used by Gurau for the saddle points f ( x ± ) = j f j ( x ± )+ j f j ( x ± )+ j f j ( x ± )+ J f J ( x ± )+ J f J ( x ± )+ J f J ( x ± ) , (5.28)using its result, see eqs. (26-27-44) in [10], ℜ ( f j ∗ β ( x + )) = ℜ ( f j ∗ β ( x − )) = − h j ∗ β (not depending on x + or x − ) , (5.29)our correlation table for j ∗ β shows that the following identification holds ℜ (cid:26) ln ( x ± − v )( x ± − v ′ )( p − x ± )( p ′ − x ± ) (cid:27) ≡ ℜ{ f j ∗ β ( x ± ) } = − h j ∗ β . (5.30)In contrast the imaginary part is ℑ (cid:0) f j ∗ β ( x ± ) (cid:1) = ± θ j ∗ β . (5.31)14t follows that Σ β ( x ± ) = e − h j ∗ β (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x ± (cid:19) π ) e ± ı θ j ∗ β + F ( x ± )+ k f ( x ± ) . (5.32)After the Laplace’s approximation the final header factor is simply 1 √ πk .Ultimately from eq. (5.7) the saddle points contributions for parity β have the form Sdl β ( x ± ) = 1 √ πk r ( p − v )( p − v ′ ) v v ′ e h J ⋆β × e − h j ∗ β F ( x ± ) p − f ′′ ( x ± ) (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x ± (cid:19) × e ± ı θ j ∗ β + k [ h ( i,J )+ f ( x ± )] . (5.33)From eqs. (32-33) in [10] clearly − f ′′ ( x − ) = + ı √△ x − Π(( p j − x − ) . (5.34)From eq. (35) in [10] we obtain Sdl β ( x ± ) = 1 √ πk r ( p − v )( p − v ′ ) v v ′ e h J ⋆β × e k [ j ( h j + f j )+ j ( h j + f j )+ j ( h j + f j )] × e k [ J ( h J + f J )+ J ( h J + f J )+ J ( h J + f J )] × e [ f j + f j + f j + f J + f J + f J ] × e − h j ∗ β p ∓ ı √△ (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x ± (cid:19) e ± ı θ j ∗ β . (5.35)From △ = (24) V and √∓ ı = e ± ı π we derive1 p ∓ ı √△ = 1 √ V e ± ı π . (5.36)The result for Sdl β ( x ± ) becomes the complex expression Sdl β ( x ± ) = 1 √ πkV r ( p − v )( p − v ′ ) v v ′ e h J ⋆β × e k [ j ( h j + f j )+ j ( h j + f j )+ j ( h j + f j )+ J ( h J + f J )+ J ( h J + f J )+ J ( h J + f J )] × e [ f j + f j + f j + f J + f J + f J ] × e − h j ∗ β e ± ı π (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x ± (cid:19) e ± ı θ j ∗ β . (5.37)From eq. (44) in [10] the real part of e k [ j ( h j + f j )+ j ( h j + f j )+ j ( h j + f j )+ J ( h J + f J )+ J ( h J + f J )+ J ( h J + f J )] 15s zero, then its remains only the imaginary part (the h j i ,J i being real) which is e ± ık [ j θ j + j θ j + j θ j + J θ J + J θ J + J θ J ] whence Sdl β ( x ± ) = 1 √ πkV r ( p − v )( p − v ′ ) v v ′ e h J ⋆β × e ± ık [ j θ j + j θ j + j θ j + J θ J + J θ J + J θ J ] × e ± ı [ θ j + θ j + θ j + θ J + θ J + θ J ] × e ℜ [ f j + f j + f j + f J + f J + f J ] × e − h j ∗ β e ± ı π (cid:18) − j ⋆β + [ p p ′ − vv ′ ] x ± (cid:19) e ± ı θ j ∗ β . (5.38)That is Sdl β ( x ± ) = 1 √ πkV r ( p − v )( p − v ′ ) v v ′ × e − [ h j + h j + h j + h J + h J + h J + h j ∗ β − h J ⋆β ] × (cid:18) [ v + v ′ − p − p ′ ] + [ p p ′ − vv ′ ] x ± (cid:19) e ± ı ( π + ϕ θkβ ) , (5.39)where a new angle ϕ θk β depending on θ j ∗ β is defined via ϕ θk β = ι =3 X ι =1 [( kj ι + 12 ) θ j ι + ( kJ ι + 12 ) θ J ι ] + 12 θ j ∗ β . (5.40)We have 1 x + = ( B − ı p △ )2 C and 1 x − = ( B + ı p △ )2 C , (5.41)whence adding the contribution + and − , taken into account C = Π( v i ), √△ = 24 V , we obtain vi ) n(cid:8) v i )[ v + v ′ − p − p ′ ]+ B [ p p ′ − vv ′ ] (cid:9) cos ( π + ϕ θkβ )+24 V [ p p ′ − vv ′ ] sin ( π + ϕ θkβ ) o . A preliminary result is then (cid:26) kj kj kj kJ kJ kJ (cid:27) Sβ ≈ ( − ( v + v ′ − p ) √ πkV Π( v i ) r ( p − v )( p − v ′ ) v v ′ × e − [ h j + h j + h j + h J + h J + h J ] − [ h j ∗ β − h J ⋆β ] × n(cid:8) v i )[ v + v ′ − p − p ′ ]+ B [ p p ′ − vv ′ ] (cid:9) cos ( π + ϕ θkβ )+24 V [ p p ′ − vv ′ ] sin ( π + ϕ θkβ ) o . (5.42)Factor h J ⋆β is compensatory, i.e. h J ⋆β term vanishes whereas h J ⋆β becomes h J ⋆β .Unfortunately this evidence will be more or less lost in the general formulas we try to explicitbelow.Computation of e h J ∗ β − h j ∗ β yields e h J ∗ β − h j ∗ β = e ln (cid:8) ( p − v )( p − v ′ )( p ′− v )( p ′− v ′ )( p − v )( p − v ′ ) v v ′ ( p − v )( p − v ′ )( p − v )( p − v ′ ( p ′− v )( p ′− v ′ ) v v ′ (cid:9) , (5.43)16hence e − [ h J ∗ β − h j ∗ β ] = s ( p − v )( p − v ′ )( p − v )( p − v ′ ( p ′ − v )( p ′ − v ′ ) v v ′ ( p − v )( p − v ′ )( p ′ − v )( p ′ − v ′ )( p − v )( p − v ′ ) v v ′ . (5.44)Besides from eqs. (8-9-11) in [10] we can write e − H ( i,J ) = e − P j =1 ( h jj + h Jj ) = e − ln Π i,j ( pj − vi ) v i = q Π i,j [ v i / ( p j − v i )] (5.45)We conclude that (cid:26) kj kj kj kJ kJ kJ (cid:27) Sβ ≈ ( − ( v + v ′ − p ) √ πkV Π( v i ) r ( p − v )( p − v ′ ) v v ′ × q Π i,j [ v i / ( p j − v i )] s ( p − v )( p − v ′ )( p − v )( p − v ′ ( p ′ − v )( p ′ − v ′ ) v v ′ ( p − v )( p − v ′ )( p ′ − v )( p ′ − v ′ )( p − v )( p − v ′ ) v v ′ × n(cid:8) v i )[ v + v ′ − p − p ′ ]+ B [ p p ′ − vv ′ ] (cid:9) cos ( π + ϕ θkβ )+24 V [ p p ′ − vv ′ ] sin ( π + ϕ θkβ ) o = ( − ( v + v ′ − p ) √ πkV p Π i,j [( p j − v i ) v i ] × q ( p − v )( p − v ′ ) v v ′ r ( p − v )( p − v ′ )( p − v )( p − v ′ ( p ′− v )( p ′− v ′ ) v v ′ ( p − v )( p − v ′ )( p ′− v )( p ′− v ′ )( p − v )( p − v ′ ) v v ′ × n(cid:8) v i )[ v + v ′ − p − p ′ ]+ B [ p p ′ − vv ′ ] (cid:9) cos ( π + ϕ θkβ )+24 V [ p p ′ − vv ′ ] sin ( π + ϕ θkβ ) o . (5.46)Note that denominator p Π i,j [( p j − v i ) v i ] has the dimension of an area like √ Π v i in the formula(3.18) for approximating the supersymmetric symbol 6 j Sα . The following line with √ and √ isdimensionless. Prior to properly present results as similar as possible with standard 6 j symbols we will use aformula to shift the angular arguments, namely: a cos( x ) + b sin( x ) = N cos ( x − ψ ) , (6.1)where N = p a + b and tan( ψ ) = ba . (6.2)Then for each parity α , γ , β , we get N α = p B + (24 V ) , ψ α = arctan (cid:16) VB (cid:17) , (6.3) N γ = N α , ψ γ = − ψ α , (6.4) N β = √ (2Π( v i )[ v + v ′ − p − p ′ ]) +(24 V [ p p ′ − vv ′ ]) , (6.5) ψ β = arctan (cid:16) V [ p p ′− vv ′ ]2Π( vi )[ v + v ′− p − p ′ ] (cid:17) . (6.6)17e recall that V is the tetrahedron volume and B which also has the dimension of a volume isgiven by B = (cid:16) X ι =1 j ι J ι X j =1 p j (cid:17) + 2( j j j + j J J + j J J + j J J ) . (6.7) Parity α , : Under a rescaling of all its arguments by a large k a supersymmetric j Sα symbol behaves like (cid:26) kj kj kj kJ kJ kJ (cid:27) Sα = 1 √ πkV √ Π v i h N α cos (cid:16) π ϕ θk − ψ α (cid:17) i , (6.8)where the angle ϕ θk is defined by ϕ θk = ι =3 X ι =1 ( kj ι + 12 ) θ j ι + ( kJ ι + 12 ) θ J ι . (6.9) θ j ι , θ J ι are the exterior dihedral angles of the tetrahedron corresponding to the edges j ι and J ι respectively. Parity γ : [If k is even, refer to formula (6.8)] Under a rescaling of all its arguments by a large k (odd) a supersymmetric j Sγ behaves like (cid:26) kj kj kj kJ kJ kJ (cid:27) Sγ = ( − (1+ P p j ) √ πkV h N α cos (cid:16) π ϕ θk γ + ψ α (cid:17) i , (6.10)where ϕ θk γ = ι =3 X ι =1 k ( j ι θ j ι + J i θ J ι ) . (6.11) Parity β : [If k is even, refer to formula (6.8)] Under a rescaling of all its arguments by a large k (odd) a supersymmetric j Sβ behaves like (cid:26) kj kj kj kJ kJ kJ (cid:27) Sβ = ( − ( v + v ′ − p ) √ πkV p Π i,j [( p j − v i ) v i ] r ( p − v )( p − v ′ ) v v ′ × r ( p − v )( p − v ′ )( p − v )( p − v ′ ( p ′− v )( p ′− v ′ ) v v ′ ( p − v )( p − v ′ )( p ′− v )( p ′− v ′ )( p − v )( p − v ′ ) v v ′ h N β cos (cid:16) π ϕ θk β − ψ β (cid:17)i , (6.12)where a new angle depending on θ j ∗ β is defined as ϕ θk β = ι =3 X ι =1 [( kj ι + 12 ) θ j ι + ( kJ ι + 12 ) θ J ι ] + 12 θ j ∗ β . (6.13)This means that only one term among the six ( kj ι + ) θ j ι , ( kJ ι + ) θ J ι transforms into ( kj j ∗ β +1) θ j ∗ β .Denominator in √ has the dimension of an area like √ Π v i . Compared to a standard 6 j symbol [ SU (2)] a major difference with the supersymmetric 6 j Sπ [ OSP (1 | k . Indeed it becomes18 k instead of q k . The standard term in cos contains always angular arguments depending onthe six tetrahedral angles θ j ι , θ J ι . However all the angles are shifted from their standard valuesby an angle different according to each parity π .For parity α the usual expressions like ( kj ι + ) θ j ι are unchanged. For parity γ (and k odd) theterms in vanish so that it remains ( kj ι ) θ j ι and so on. For parity β (and k odd) the angulardependence takes a form where a special angle θ j ∗ β modifies the standard formula into( kj j ∗ β + 1) θ j ∗ β + P ι ( kj ι + ) θ j ι + ( kj ι + ) θ J ι | jι,Jι = j ∗ β .If k is even all supersymmetric 6 j Sπ have the same asymptotic behaviour, ie that of parity α .Pertinent interpretations are clearly within the Quantum Supergravity framework. Appendix: Expression of supersymmetric prefactors R Sα = Q j =3 j =1 Q i =4 i =1 ( p j − v i )! Q i =4 i =1 ( v i )! , (A.1) R Sγ = Q j =3 j =1 Q i =4 i =1 ( p j − v i − )! Q i =4 i =1 ( v i + )! , (A.2) R Sβ =( p − v )!( p − v ′ )!( p − v − 12 )!( p − v ′ − 12 )! × ( p − v )!( p − v ′ )!( p − v − 12 )!( p − v ′ − 12 )! × ( p ′ − v )!( p ′ − v ′ )!( p ′ − v − 12 )!( p ′ − v ′ − 12 )! × (cid:0) v ! v ′ ! (cid:1) − (cid:0) ( v + 12 )!( v ′ + 12 )! (cid:1) − (A.3) References [1] C. Rovelli, Quantum Gravity (Cambridge University Press, 2004).[2] V. Aquilanti, D. Marinelli and A Marzuoli, “Hamiltonian dynamics of a quantum of space:hidden symmetries and spectrum of the volume operator, and discrete orthogonal polyno-mials”, Journal of Physics A: Mathematical and Theoretical. (17), 175001-175303 (2013).[3] G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients ,(Spectroscopic and grouptheoretical methods in physics, North-Holland, New York, 1968).[4] E.R. Livine and R. Oeckl, “Three-dimensional Quantum SuperGravity and SupersymmetricSpin Foam Models”, Adv.Theor Math. Phys. (6), 951-1001(2004).[5] A. Pais and V. Rittenberg, “ Semisimple graded Lie algebras”, J. Math. Phys. (10),2062-2073 (1975).[6] M. Scheunert, W. Nahm and V. Rittenberg, “Irreducible representations of the osp ( , )and spl ( , ) graded Lie algebras”, J. Math. Phys. (1), 155-162 (1977).[7] F. A. Berezin, V. N. Tolstoy, “The group with Grassmann structure UOSP(1.2)”, (3),409-428 (1981). 198] M. Daumens, P. Minnaert, M. Mozrzymas and S. Toshev, “The super-rotation Racah-Wigner calculus revisited”, J. Math. Phys. (6), 2475-2507 (1993).[9] L. Br´ehamet, “Analytical complements to the parity-independent Racah-Wigner calculusfor the superalgebra osp ( | ) Part I”, Il Nuovo Cimento (3), 241-274 (2006).[10] R. Gurau, “The Ponzano-Regge Asymptotic of the 6 j Symbol: An Elementary Proof”,Ann. Henri Poincar´e (7), 1413-1420 (2008).[11] L. Br´ehamet, “Regge Symmetry of 6- j or super 6-j S Symbols: a Re-Analysis with Parti-tion Properties”, Pioneer Journal of Mathematical Physics and its Applications6