Dirac Matrices for Chern-Simons Gravity

Dirac Matrices for Chern–Sim ons Gra vit y F ernando Izaurieta, ∗ Ricardo Ram ´ ırez, † and Eduardo Rodr ´ ıguez ‡ Dep artamento de Matem´ atic a y F ´ ısic a Apli c adas, Universidad Cat´ olic a de la Sant ´ ısima Conc ep ci´ on, Alonso de R ib er a 2850, 4090541 Conc ep ci´ on, Chi le (Dated: No vember 11, 2018) A genuine gauge theory for the P oincar´ e, d e Sitter or anti-de Sitter algebras can b e constructed in (2 n − 1)-dimensional spacetime by means of the Chern–Simons form, yielding a gra vitational theory that differs from General Relativity but shares man y of its prop erties, such as second order field equations for the metric. The particular form of the Lagrangian is determined by a rank n , symmetric tensor inv arian t under the relev ant algebra. In practice, the calculatio n of this inv arian t tensor can b e redu ced to th e computation of t he trace of th e symmetrized p rodu ct of n Dirac Gamma matrices Γ ab in 2 n -dimensional spacetime. While straigh t forward in principle, this calculation can b ecome extremely cumbersome in practice. F or large enough n , existing comput er algebra pack ages take an inordinate long time to pro d uce the answer or plainly fail having u sed up all av ailable memory . In this talk w e show that the general form ula for the trace of th e symmetrized prod uct of 2 n Gamma matrices Γ ab can b e written as a certain sum ov er the integer partitions s of n , with every term b eing multiplied by a numerical co efficient α s . W e then give a general algorithm t h at computes t he α -coefficients as t he solution of a linear system of equations generated by ev aluating the general formula for different sets of tensors B ab with random numerical entries. A recurrence relation b etw een different coefficients is sho wn to hold and is used in a second, “minimal” algorithm to greatly sp eed up the compu t ations. Run t ime of th e minimal algorithm stays below 1 min on a typical desktop computer for up to n = 25, which easily cov ers all foreseeable applications of the trace formula. I. INTRO D UCTION There’s more to higher-dimensio nal g ravit y tha n Ein- stein and Hilb ert [1 – 7]. Chern–Simons (CS) gravit y in d = 2 n − 1 dimensions is a gauge theor y for the Poincar´ e, de Sitter or anti-de Sitter (AdS) a lgebras, dep ending on the v alue of the cosmo log- ical constant [8 , 9]. Let us fo cus o n the AdS algebr a, so ( d − 1 , 2) . A conv enient matrix r epresentation is pr ovided by Γ AB = Γ [ A Γ B ] , where Γ A are Dirac matrices in D = d + 1 = 2 n dimensions: 1 J ab = 1 2 Γ ab , (1) P a = 1 2 Γ a,d . (2) The Lagra ng ian for CS gravity is sha p ed to a great extent b y a rank - n , AdS-inv a riant s ymmetric p olyno mial h· · · i . This p oly no mial can b e identified with any of the fol- lowing tra ces: • T r { Γ A 1 B 1 · · · Γ A n B n } (Lore ntz sca lar) • T r (Γ ∗ { Γ A 1 B 1 · · · Γ A n B n } ) (Lorentz pseudo scalar), ∗ fizaurie@ucsc.cl † ricramirez@ucsc.cl ‡ edurodri guez@ucsc.cl 1 The indices r un as f ollows: A, B = 0 , 1 , . . . , D − 1, a, b = 0 , 1 , . . . , d − 1. where {· · · } denotes sy mmetrized matrix pro duct. The pseudoscalar trace re a ds T r (Γ ∗ { Γ A 1 B 1 · · · Γ A n B n } ) = γ ǫ A 1 B 1 ··· A n B n , (3) where γ is a numerical co efficient. Use o f this inv a riant po lynomial brings in the La nczos–Lovelock [2, 9] fa mily of Lagrang ians into CS gravit y . The scalar trace, on the other hand, is more inv olved. In this work we pr ovide tw o algor ithms that can be used to efficiently compute the sca lar trace for a n y n and for an y spacetime dimension d (without an y implied relation betw een n and d ). II. FORMULA TION OF T HE PROBLEM AND RESUL TS Let us consider Dirac ma trices Γ a , a = 0 , . . . , d − 1, in d -dimensional Minko wsk i spacetime. By definition, they satisfy the Clifford algebr a [10] Γ a Γ b + Γ b Γ a = 2 η ab 1 , (4) where η ab = ( − + · · · +) is the usual Minko wski metric and 1 stands for the m × m unit matrix, with m = 2 ⌊ d/ 2 ⌋ . The Γ-matrices which a re the sub ject of this w o rk are defined as Γ ab = Γ [ a Γ b ] = 1 2 (Γ a Γ b − Γ b Γ a ) . (5) F or completeness, let us define the sy mmetr ized pro d- uct of n matrices M i , i = 1 , . . . , n , as { M 1 · · · M n } = 1 n ! X π ∈ S n M π (1) · · · M π ( n ) , (6) 2 where the sum extends over all p er m utations π in the symmetric group S n . Exp erience shows that the trace is most efficiently writ- ten with all matrices multiplied by arbitra r y an tisym- metric tensor s . T ake, for insta nce, the tr ace of the sym- metrized pro duct o f t wo Gamma matrices, and compare the following equations: T r { Γ ab Γ cd } = m ( η ad η bc − η ac η bd ) , (7) A ab B cd T r { Γ ab Γ cd } = 2 mA a b B b a . (8) The tw o terms on the r ight-hand side of eq. (7) hav e collapsed into one in eq. (8). Gr eater simplifications are achiev ed for mor e complicated ca ses. If desire d, eq . (7) can b e recovered from eq. (8) by means of the forma l replacement A ab → δ ab cd , B ab → δ ab cd , where δ ab cd is the generalized Kronecker delta. Let B ab i , i = 1 , 2 , 3 , . . . , be arbitra ry a ntisymmetric tensors, and let us define β i = B ab i Γ ab . (9) The symmetrized pro duct of n β -ma trices can b e written as a linear combination of matrices Γ a 1 ··· a p = Γ [ a 1 · · · Γ a p ] , with p = 0 , 4 , 8 , . . . , 2 n (for n even) or p = 2 , 6 , 10 , . . . , 2 n (for n odd). The only term that con- tributes to the tra c e is that prop or tional to the identit y matrix ( p = 0). F o r o dd d , how ever, the Γ a 1 ··· a d matrix is also prop ortiona l to the identit y and must b e gener i- cally taken in to acco unt when computing the trace. The expansion of the symmetrized pro duct of the β -matric e s includes only Γ-matrices with an even num b er of indices, so that the Γ a 1 ··· a d -term never actually shows up in our case. In particular , this means that the tra ce of the sym- metrized pro duct of an odd num b er of β -matr ic e s v an- ishes identically . The trace of the symmetrized pro duct of 2 n β - matrices, on the other hand, can be written as T r { β 1 · · · β 2 n } = m X s ⊢ n α s B ( s ) , (1 0) where the notation s ⊢ n [11] indicates that the sum must be p er formed over all integer par titions s of n , and B ( s ) stands for the following sum of c o nt ractions of B -tenso r s: B ( s ) = X h i 1 ··· i 2 n i r Y j =1 D B i 2 s 1 + ··· +2 s j − 1 +1 · · · B i 2 s 1 + ··· +2 s j E . (11) In eq . (11), the notation h i 1 · · · i 2 n i is used to indicate that the sum m ust b e per formed ov er all i 1 , . . . , i 2 n ∈ { 1 , . . . , 2 n } , with the restriction that they be all differ- ent . This implements the p ermutation of all β - matrices. Every term in the sum contains the pro duct of r factors of the form h B 1 · · · B q i , where r is the length of the par- tition s = ( s 1 , . . . , s r ), with n = s 1 + · · · + s r . The j -th factor in the pro duct repres ent s the trace of the pro duct of 2 s j B -tens ors, i.e., h B 1 · · · B q i = ( B 1 ) c 1 c 2 ( B 2 ) c 2 c 3 · · · ( B q ) c q c 1 , (12) with q = 2 s j . T o every term in eq. (10), i.e ., to e very partition s of n , there co rresp onds an α s co efficient. Numerical v alues for the α -co efficients co rresp onding to the par titio ns of n = 1 , . . . , 7 are given in T able I. The following examples for n = 1 , . . . , 4 sho uld help clarify the meaning of eqs. (10) and (11): T r { β 1 β 2 } = m X h ij i α 1 h B i B j i , (13) T r { β 1 · · · β 4 } = m X h ij kl i [ α 2 h B i B j B k B l i + + α 11 h B i B j i h B k B l i ] , (14) T r { β 1 · · · β 6 } = m X h i 1 ··· i 6 i [ α 3 h B i 1 · · · B i 6 i + + α 21 h B i 1 · · · B i 4 i h B i 5 B i 6 i + + α 111 h B i 1 B i 2 i h B i 3 B i 4 i h B i 5 B i 6 i ] , (15) T r { β 1 · · · β 8 } = m X h i 1 ··· i 8 i [ α 4 h B i 1 · · · B i 8 i + + α 31 h B i 1 · · · B i 6 i h B i 7 B i 8 i + + α 22 h B i 1 · · · B i 4 i h B i 5 · · · B i 8 i + + α 211 h B i 1 · · · B i 4 i h B i 5 B i 6 i h B i 7 B i 8 i + + α 1111 h B i 1 B i 2 i h B i 3 B i 4 i h B i 5 B i 6 i h B i 7 B i 8 i ] . (16) The pro of o f eq. (10) is by exhaustion; the right-hand side includes all pos sible terms that ma y c o nt ribute to the trace of the symmetrized pro duct of 2 n β - matrices. 2 Our a pproach to the co mputatio n o f the α -co efficients is the sub ject of se ction II I. II I. METHOD A. General A lgorithm The central observ atio n b ehind the algor ithm used in the co mputation o f the α -co efficients shown in T able I is the fact that eq. (10) is v alid for arbitr ary tensors B ab i . F or illustration purp oses , let us focus first on the n = 3 case. E q. (15 ) s implifies grea tly if w e choose a ll B -tensor s to b e e q ual, since in this case the s um ov er all different 2 The formu la f or T r (Γ ∗ { β 1 · · · β n } ) includes pseudoscalar terms that appear in certain dimensions d (e.g., ǫ abcd B ab i B cd j for d = 4) but are absent f rom T r { β 1 · · · β 2 n } , where only Lorentz scalars are allo wed. Here Γ ∗ = Γ 0 · · · Γ d − 1 is the d - dimensional gener- alization of γ 5 in d = 4. 3 T ABLE I. α -co efficien ts corresp onding to the partitions of n = 1 , . . . , 7. n s α s 1 1 1 2 1 + 1 1 / 2 2 − 2 / 3 3 1 + 1 + 1 1 / 6 2 + 1 − 2 / 3 3 32 / 45 4 1 + 1 + 1 + 1 1 / 24 2 + 1 + 1 − 1 / 3 2 + 2 2 / 9 3 + 1 32 / 45 4 − 272 / 315 5 1 + 1 + 1 + 1 + 1 1 / 120 2 + 1 + 1 + 1 − 1 / 9 2 + 2 + 1 2 / 9 3 + 1 + 1 16 / 45 3 + 2 − 64 / 135 4 + 1 − 272 / 315 5 1 5872 / 14175 6 1 + 1 + 1 + 1 + 1 + 1 1 / 720 2 + 1 + 1 + 1 + 1 − 1 / 36 2 + 2 + 1 + 1 1 / 9 2 + 2 + 2 − 4 / 81 3 + 1 + 1 + 1 16 / 135 3 + 2 + 1 − 64 / 135 3 + 3 512 / 2025 4 + 1 + 1 − 136 / 315 4 + 2 544 / 945 5 + 1 15872 / 14 175 6 − 707584 / 4 67775 7 1 + 1 + 1 + 1 + 1 + 1 + 1 1 / 5040 2 + 1 + 1 + 1 + 1 + 1 − 1 / 180 2 + 2 + 1 + 1 + 1 1 / 27 2 + 2 + 2 + 1 − 4 / 81 3 + 1 + 1 + 1 + 1 4 / 135 3 + 2 + 1 + 1 − 32 / 135 3 + 2 + 2 64 / 405 3 + 3 + 1 512 / 2025 4 + 1 + 1 + 1 − 136 / 945 4 + 2 + 1 544 / 945 4 + 3 − 8704 / 141 75 5 + 1 + 1 7936 / 14 175 5 + 2 − 31744 / 42 525 6 + 1 − 707584 / 4 67775 7 8947302 4 / 42567525 per mut ations o f i 1 , . . . , i 6 ∈ { 1 , . . . , 6 } is trivially p er- formed. The r esult reads 1 6! m T r  β 6  = α 3  B 6  + α 21  B 4   B 2  + α 111  B 2  3 . (17) W e wish to cast eq. (17) as a linear equation with three unknowns, namely , α 3 , α 21 and α 111 . T o do this we need to b e able to a ssign numerical v alues to the left-hand side and to the v arious h B q i -terms that appea r on the right-hand side. W e a c complish this b y (i) picking some antisymmetric tensor B ab with ra ndo m numerical en tr ies and (ii) choos ing an explicit r epresentation for the Γ- matrices. 3 W e emphasize that the p ossibility o f cho osing the B -tensor s at will relies up on the fact that eq. (10) is v alid for arbitrary B i ’s. T o b e able to s olve for the α -co efficients we need tw o more equations. These are readily obtained by randomly selecting tw o further B -tensors . D enoting the three dif- ferent choices for the B -tenso rs by B k , with k = 1 , 2 , 3 , we obtain the following 3 × 3 linea r system: Z (111) 1 α 111 + Z (21) 1 α 21 + Z (3) 1 α 3 = T 1 , (18) Z (111) 2 α 111 + Z (21) 2 α 21 + Z (3) 2 α 3 = T 2 , (19) Z (111) 3 α 111 + Z (21) 3 α 21 + Z (3) 3 α 3 = T 3 , (20) where T k = 1 6! m T r  β 6 k  , (21) Z (111) k =  B 2 k  3 , (22) Z (21) k =  B 4 k   B 2 k  , (23) Z (3) k =  B 6 k  . (24) The metho d to co mpute the α -co efficients for any v alue of n is now clear and c an be s ummarized in the following sequence: 1. Let p = p ( n ) b e the nu m ber of partitions of n . 4 2. Cho ose an explicit represen tation for the Γ- matrices (see, e.g ., Ref. [12]). 3. F or k = 1 , . . . , p , do : (a) Pick an a n tis y mmetric tens o r B ab k with ra n- dom n umer ical entries. (b) Compute T k = 1 (2 n )! m T r  β 2 n k  , (25) where β k = B ab k Γ ab . 3 See section IV for a discussion of the c hoice of spacetime dimen- sion d i n which to carry out the computation. 4 The function p ( n ) is called the “partition f unction” in the math- ematical literature [11 ]. 4 (c) F or every partitio n s ⊢ n , with n = s 1 + · · · + s r , compute Z ( s ) k = r Y j =1 D B 2 s j k E . (26) The notation h B q k i stands for [see eq . (12)] h B q k i = ( B k ) c 1 c 2 ( B k ) c 2 c 3 · · · ( B k ) c q c 1 . (27) 4. The α -co efficient s ar e the so lution to the p × p linear system of equations X s ⊢ n Z ( s ) k α s = T k ( k = 1 , . . . , p ) . (28) B. Minimal A lgorithm Careful insp ection of the α -co efficients shown in T a- ble I shows that there exis ts a recur rence relatio n among different co efficient s. Let s be a pa rtition of n . The frequency representa- tion [1 1] of s is the notatio n s = (1 µ 1 2 µ 2 · · · ), where µ j represents the m ultiplicity o f j , i.e., the num b er of times that a given in teger j app ear s in s . W e find that the c o efficient α s corres p o nding to the partition s = (1 µ 1 2 µ 2 · · · ) can b e written a s α s = n Y j =1 α µ j j µ j ! , (29) where α j are the co efficient s as so ciated with the “elemen- tary” partitions 1 = 1, 2 = 2, 3 = 3, etc. F or example, all co efficients a sso ciated with the non- elementary partitions of n = 1 , 2 , 3 can b e computed from α 1 , α 2 and α 3 by means of the equations α 11 = α 2 1 2! α 0 2 0! = 1 2 , (30) α 111 = α 3 1 3! α 0 2 0! α 0 3 0! = 1 6 , (31) α 21 = α 1 1 1! α 1 2 1! α 0 3 0! = − 2 3 . (32) Of course, this r ecurrence relation also holds for mor e complicated cases, such as α 3211 = α 2 1 2! α 1 2 1! α 1 3 1! α 0 4 0! α 0 5 0! α 0 6 0! α 0 7 0! = − 32 135 . (33) When a pplied to an elemen ta ry co efficient, eq. (29) yields an ident ity . The recurr ence relation in eq. (29) can b e used to com- pute the v alues for the α -co efficie nts as so ciated with all the non-elementary partitions of n . Its use, how ever, re- quires knowledge of the element ary co efficient s, for whic h no clo sed formula is av ailable. This situation sugg ests a “minimal” a lgorithm that (i) ca lculates elementary co ef- ficient s in a manner analogous to tha t of the “ general” algorithm and (ii) computes non- element ary co efficients from eq. (29). The following seq uence describ es such an algorithm: 1. Let N b e the maximum in teg er for which we wish to calculate the α - c o efficients. 2. Cho ose an explicit represen tation for the Γ- matrices. 3. Pick an antisymmetric tensor B ab with ra ndo m nu- merical entries. 5 4. F or n = 1 , . . . , N , do: (a) Compute T = 1 (2 n )! m T r  β 2 n  , (34) where β = B ab Γ ab . (b) F or every pa rtition s ⊢ n , with n = s 1 + · · · + s r , compute Z ( s ) = r Y j =1  B 2 s j  . (35) (c) Use the recurre nce rela tio n (29) to calculate all non-elementary co efficients asso cia ted with the par titions of n (this step is empt y for n = 1). (d) Solve X s ⊢ n Z ( s ) α s = T (36) for α n (this is a linear equation with one un- known). IV. DISCUSSION AND CONCLUSIONS The algorithms describ ed in section I I I turn aro und the problem of finding formulas for the trace o f a pro d- uct o f Gamma matrices. The usual textb o ok approa ch starts with eq. (4) and de duc es the required for mulas from there. Our approach he r e w ork s the other w ay around. W e star t by identifying the general form o f the equation for the trace of the symmetriz e d pro duct of 2 n β - ma trices. E q. (10 ) amo un ts to such an identification, 5 W e took d = 2 and B 01 = +1, since a t wo-index antisymmet- ric tensor has only one degree of freedom in tw o spacetime di- mensions, and ov erall numerical f actors are not si gnifican t f or the calculation. See section IV for a discussion of the ch oice of spacetime dimension d in which to carry out the computation. 5 since it contains a ll p ossible sums of B -contractions that may contribute to the trace. The α -co efficients app ear as undetermined parameter s, which are computed by de- manding v alidit y of eq. (10) in s everal nontrivial cases. As stress e d in section II I , our metho d w o r ks b ecaus e eq. (10) holds for arbitr ary antisymmetric tensors B ab i . W e have used B -tensor s with random numerical entries to g enerate the linea r sys tem of equations whose solu- tion provides the α -coefficients. In this sense our ap- proach b ear s some resemblance to Monte Carlo metho ds, where random num b ers play a crucial role. The use o f random matrices, 6 how ever, is not essential to our cal- culation. All that is r equired for the gener a l algo rithm to succeed is a set of B -tenso rs such that ev er y iteratio n pro duces an equation for the α -co e fficien ts that is lin- early indep e ndent from the rest, yielding a full-rank Z matrix [cf. eq. (28)]. The solution we find is, o f course, indep endent of the choice of B -tenso rs; this is conceptually clear , but can also be verified b y r unning the algorithm several times with different sets of (ra ndomly generated) B -tenso rs. The fac t that the same solutio n is obtained e very time confirms b oth this indep endence and the correctness of eq. (10), i.e., that no other terms ca n b e added to the trace. The α -co efficients ar e a lso independent of the space- time dimension d , which means tha t the alg orithm should in principle work for any d we choo se. There is, how ever, an imp o rtant caveat. T o pr o duce a solv able sys tem o ne needs the B -tenso rs to hav e a sufficient num b er o f inde- pendent comp onents, so tha t the successive iteratio ns of the a lgorithm y ield linearly indep endent equatio ns. W e find that there is a minimum spacetime dimensio n d = 2 n that allows the Z matrix to achiev e full ra nk. This means that the ge ne r al a lg orithm must b e run with d ≥ 2 n in order for a solution to b e pr o duced. The minimal algor ithm, with only o ne linear equation to b e solved, works even with a minimum spa cetime di- mension of d = 2. Is our a pproach any better than the textbo o k metho d? One w ay to pro be into this questio n is to co mpare the runtime of b oth. The textb o ok metho d can be imple- men ted in, e.g., Kas p er Peeters’ e x cellent computer alge- bra system “ Cadabra” [13, 14]. W e w ere able to deduce, starting o nly from the definitio n of Dirac ma trices, the α -co efficients for n = 1 , 2 , 3. The n = 3 case to o k some 30 min to b e solved on a typical desktop computer, 7 while the n = 4 c ase ca us ed the progr am to c rash. T his ap- proach, of c ourse, requires har dly any input a nd pro duces the full soug ht-after formula. Starting fro m eq. (10), we progra mmed o ur gener al alg orithm in the co mputer al- gebra system “Maxima ” [1 5] and were able to run it suc- cessfully for n = 1 , . . . , 7. The n = 8 case caused Maxima to r un out of memory . Runtime for n = 1 , . . . , 4 was neg- ligible, while the n = 7 case to ok under ha lf a n hour. The minimal algo rithm, which we als o progr ammed in Max - ima, had neglig ible runtime even for N = 25 . T able I I summarizes runtime fo r these different scena r ios. Complexity for the genera l alg orithm grows ex po nen- tially with n . Complexity fo r the minimal alg orithm, on the o ther hand, gr ows linearly with p . 8 All foreseeable applications of the f ormula for the trace o f a pr o duct of 2 n Gamma matrices ar e well cov ered b y the minimal algorithm with negligible runtime. T ABLE I I. Approximate runtime for the text b ook metho d (as implemented in Cadabra) and the general and minimal algorithms (as implemented in Maxima) on a typical desk- top computer. F or the minimal algorithm, the first column is understo o d to mean N , the maxim u m integer for whic h the α -co efficients are computed. The second column lists the partition function of n , whic h corresp onds to the number of α -coefficients to b e determined . n p T extb o ok General Minimal Method Algorithm Algorithm 1 1 negligible negligible negligible 2 2 negligible negligible negligible 3 3 ∼ 30 min n egligible negligible 4 5 crashed negli gible negligi ble 5 7 negligible negligible 6 1 1 ∼ 1 min negligible 7 1 5 ∼ 24 min negligible 8 2 2 crashed negligible 9 3 0 negligible . . . . . . . . . 26 2436 ∼ 1 min 28 3718 ∼ 2 min 30 5604 ∼ 5 min ACKNO WLEDGMENTS The authors w is h to thank T o m´ as Bar rios and Ruth Sandov al for many friendly , helpful and enlightening co n- versations on the sub ject of this w o rk. F. I. and E. 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