Eleven-Dimensional Gauge Theory for the M Algebra as an Abelian Semigroup Expansion of osp(32|1)

0.29 MB·English

by  Fernando Izaurieta

Eleven-Dimensional Gauge Theory for the M Algebra
as an Abelian Semigroup Expansion of osp(32|1)
Fernando Izaurieta∗ and Eduardo Rodr´ıguez†
Departamento de Matem´aticas y F´ısica Aplicadas,
Universidad Cat´olica de la Sant´ısima Concepci´on, Concepci´on, Chile
Patricio Salgado‡
Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile
(Dated: December 17, 2006)
A new Lagrangian realizing the symmetry of the M Algebra in eleven-dimensional space-time is
presented. By means of the novel technique of Abelian Semigroup Expansion, a link between the
M Algebra and the orthosymplectic algebra osp(32|1) is established, and an M Algebra-invariant
symmetrictensorofranksixiscomputed. Thissymmetricinvarianttensorisakeyingredientinthe
8
construction of the new Lagrangian. The gauge-invariant Lagrangian is displayed in an explicitly
0
Lorentz-invariant way by means of a subspace separation method based on the extended Cartan
0
homotopy formula.
2

n
a I. INTRODUCTION lation of a CS/Transgression form theory for the M-
J
Algebra is clear. A priori, the construction of a CS

3 Supergravity for the M Algebra would seem something
String Theory and eleven-dimensional Supergravity
2
straightforwardto do, especially since CS Supergravities
became inextricably linked after the arrival of the M-

3 Theory Paradigm. All efforts notwithstanding, the low- for osp(32|1) are already well-known [2, 3, 5]. This is
however not the case, and the construction is actually
v energy regime of M Theory remains better known than
5 its non-perturbative description. However, the possi- highly nontrivial. The reason is that in both cases, for
2 bility has been pointed out that M Theory may be CS and Transgression forms, the key ingredient in the
2 constructionisthe invarianttensor. Andpreciselyinthe
non-perturbatively related to, or even formulated as, an
6 case of the non-semisimple M Algebra, the direct option
eleven-dimensional Chern–Simons theory [1, 2, 3] (see
0
ofusingthesupertraceasinvarianttensorisnotafruitful
also [4]).
6
one.
0 Chern–Simons (CS) Theory has quite compelling fea-
/ tures. On one hand, it belongs to the restricted class of This problem has been dealt with in Refs. [13, 14] us-
h
gaugefieldtheories,withaone-formgaugeconnectionas ing a physicist’s approach: the Noether method. Start-
t
- the sole dynamical field. On the other hand, and in con- ingfromthe Poincar´eCSLagrangian,aCSFormforthe
p
trast with usual Yang–Mills theory, there’s no a priori M Algebra is recursively constructed, adding new terms
e
h metric needed to define the CS Lagrangian, so that the to finally reach an invariant Lagrangian. After the La-
: theory turns out to be background-free. CS Supergrav- grangianisconstructed,itispossibletoreadbackthein-
v
ities (see, e.g., [5] and references therein) exist in every variant tensor. This approach has proved succesful, but
i
X odddimension;three-dimensionalGeneralRelativitywas it has some drawbacks: (i) it requires a lot of physicist’s
r famously quantized by making the connection to CS [6]. insight and cleverness and (ii) as the authors of [13, 14]
a
Inrecenttimes,anevenmoreappealinggeneralization make clear, the method does not rule out the possiblity
of this idea has been presented, the so-called Transgres- of extra terms in the Lagrangian.
sion form Lagrangians. Transgression forms [7, 8, 9, 10,
Ontheotherhand,amorematemathicalpointofview
11, 12] are the matrix where CS forms stem from. The
has been developed in Ref. [15], where the M Algebra
maindifferencebetweenCSandTransgressionformscon-
has been shown to correspond to an expansion [24] of
cerns a new, regularizing boundary term which renders
osp(32|1). Expansions stand out among other algebra
the Transgression form fully gauge invariant. As a con-
manipulation methods (such as contractions, deforma-
sequence, the boundary conditions and Noether charges
tions and extensions) as the only one which is able of
computedfromatransgressionactionhavethechanceto
changingthedimensionofthealgebra;ingeneral,itleads
be physically meaningful.
toalgebraswithadimensionalityhigherthantheoriginal
Since a gaugefield theory for the M Algebramay take one.
usonestepclosertounderstandingthe non-perturbative
Inanutshell,theexpansionmethodconsiderstheorigi-
description of M Theory, the importance of the formu-
nalalgebraasdescribedbyitsassociatedMaurer–Cartan
(MC) forms on the group manifold. Some of the group
parametersarerescaledbyafactorλ,andthe MC forms
are expanded as a power series in λ. This series is fi-
∗Electronicaddress: fi[email protected]
†Electronicaddress: [email protected] nally truncated in a way that assures the closure of the
‡Electronicaddress: [email protected] expanded algebra. The subject is thoroughly treated in
2
Refs. [15, 16, 17, 18]. WhenthesemigroupS canbedecomposedinsubsetsS ,
p
In the expansions approach, the algebra is formulated S = p∈ISp, such that they satisfy the condition [26]
intermsoftheMCforms,andtherefore,theCSformfor
S
theMAlgebramustbewrittenthroughafreedifferential S ·S ⊂ S , (3)
p q r
algebraseriesfromthefullosp(32|1)-CSform. Again,to
extractfromthere aninvarianttensor forthe MAlgebra r∈\i(p,q)
proves to be nontrivial. then we have that
BothapproachesfocusonconstructingdirectlytheCS
form. In this article, a third alternative is considered: G = S ×V (4)
R p p
the Lie Algebras S-expansion method, which focuses on p∈I
M
theconstructionoftheinvariant tensor. Thisprocedure,
developedingeneralinRef.[19],isformulatedintermsof isa‘resonantsubalgebra’ofS×g(seeTheorem4.2from
the original Lie algebra generators and an abelian semi- Ref. [19]).
group S. Given this originalLie algebra and the abelian Anevensmalleralgebracanbeobtainedwhenthereis
semigroup as inputs, the S-expansion method gives as a zero element in the semigroup, i.e., an element 0S ∈S
output a new Lie algebra, and besides it, general ex- such that, for all λα ∈ S, 0Sλα = 0S. When this is
pressions for the invariant tensor for it in terms of the the case, the whole 0S ×g sector can be removed from
semigroup structure. the resonant subalgebra by imposing 0S ×g = 0. The
Thepaperisorganizedasfollows. Insec.IIthederiva- remaining piece, to which we refer to as 0S-reduced al-
tion of the M Algebra as an abelian semigroup expan- gebra, continues to be a Lie algebra (for a proof of this
sion of osp(32|1) is performed, and a way to construct fact and some more general cases, see 0S-reduction and
an M algebra-invarianttensor is found. Some aspects of Theorem 6.1 from Ref. [19]).
the transgressionLagrangianarereviewedinSectionIII, In the next section these mathematical tools will be
where useof the subspaceseparationmethodproduces a used in order to show how the M algebra can be con-
new explicit action for an eleven-dimensional transgres- structed from osp(32|1).
sion gauge field theory. In sec. IV we comment on the
dynamicsproducedbythetransgressionLagrangian. We
close with conclusions and some final remarks in sec. V. B. M Algebra as an S-expansion
Inthissectionweroughlysketchthestepstobeunder-
II. THE M ALGEBRA AS AN S-EXPANSION takeninordertoobtaintheMalgebraasanS-Expansion
OF osp(32|1) of osp(32|1).
As with any expansion, the first step consists in split-
Inthis sectionwe briefly reviewthe generalmethodof ting the osp(32|1) algebra in distinct subspaces. This is
abelian semigroup expansion and its application in ob- accomplished by defining
taining the M Algebra as an S-Expansion of osp(32|1).
We refer the reader to [19] for the details. V = J(osp) , (5)
0 ab
n o
V = Q(osp) , (6)
1
A. The S-Expansion Procedure
V =nP(osp)o,Z(osp) . (7)
2 a a1···a5
ConsideraLiealgebragandafiniteabeliansemigroup n o
S ={λα}. According to Theorem 3.1 from Ref. [19], the Here V0 corresponds to the Lorentz algebra, V1 to the
direct product S×g is also a Lie algebra. Interestingly, fermionsandV2 totheremainingbosonicgenerators,na-
there are cases when it is possible to systematically ex- mely AdS boosts and the M5-branepiece. The algebraic
tractsubalgebrasfromS×g. Start by decomposing g in structuresatisfiedbythesesubspacesiscommontoevery
a direct sum of subspaces, as in superalgebra,as can be seen from the equations
[V ,V ]⊂V , (8)
g= V , (1) 0 0 0
p
[V ,V ]⊂V , (9)
p∈I 0 1 1
M
[V ,V ]⊂V , (10)
0 2 2
where I is a set of indices. The internal subspace struc-
[V ,V ]⊂V ⊕V , (11)
ture of g can be codified through [25] the mapping 1 1 0 2
i : I ×I → 2I, where the subsets i(p,q) ⊂ I are such [V1,V2]⊂V1, (12)
that [V ,V ]⊂V ⊕V . (13)
2 2 0 2
[V ,V ]⊂ V . (2) The second step is particular to the method of S-
p q r
expansions,and deals with finding an abelian semigroup
r∈Mi(p,q)
3
TABLE I: The M algebra can be regarded as an S(2)- TABLE II: (Anti)commutation relations for the M algebra.
Expansionofosp(32|1). ThetableshowstherelationbetwEeen HereΓa are Dirac matrices in d=11.
generators from both algebras. The three levels correspond
to the three columns in Fig. 1 or, alternatively, to the three
subsets into which S(2) has been partitioned. Jab,Jcd =δeacbdfJef, (25)
E h i
GR Subspaces Generators hJab,Pci=δeacbPe, (26)
S0×V0 ZJaabb == λλ20JJaa((oobbsspp)) hJab,Zcdi=δ1eacbdfZef, (27)
S1×V1 Q00 === λλλ331JQQa(((oboosssppp))) hJab,Z[Jc1a·b·,·cQ5i]==−4!21δdaΓcba1eb·1·Q···c·5e,4Zde1···e4, ((2289))
Pa = λ2Pa(osp) [Pa,Pb]=0, (30)
S2×V2 Zabcde = λ2Za(obscdpe) [Pa,Zbc]=0, (31)
0 = λ3Pa(osp) [Pa,Zb1···b5]=0, (32)
0 = λ3Za(obscdpe) [Zab,Zcd]=0, (33)
[Zab,Zc1···c5]=0, (34)
[Za1···a5,Zb1···b5]=0, (35)
S which can be partitioned in a ‘resonant’ way with re-
[Pa,Q]=0, (36)
spect to (8)–(13). This semigroup exists and is given by
[Z ,Q]=0, (37)
S(2) ={λ ,λ ,λ ,λ }, with the defining product ab
E 0 1 2 3 [Z ,Q]=0, (38)
abcde
λ , when α+β ≤2,
λαλβ = λ3α,+β otherwise. (14) Q,Q¯ = 81„ΓaPa− 12ΓabZab+ 51!ΓabcdeZabcde«. (39)
(cid:26) ˘ ¯
A straightforward but important observation is that,
for each λ ∈S(2), λ λ = λ , so that λ plays the rˆole
α E 3 α 3 3
of the zero element inside S(2). osp(32|1)arerepresentedonthehorizontalaxis,andthe
E semigroup elements on the vertical one. The shaded re-
Consider now the partition S(2) =S ∪S ∪S , with
E 0 1 2 gion on the left corresponds to the resonant subalgebra,
including the λ ×osp(32|1) sector, which is mapped to
S ={λ ,λ ,λ }, (15) 3
0 0 2 3 zero via the 0 -reduction. The gray sector on the right
S
S1 ={λ1,λ3}, (16) corresponds to the M algebra itself. The diagram allows
S ={λ ,λ }. (17) us to graphically encode the subset partition (15)–(17)
2 2 3
on each column, and makes checking the closure of the
This partition is said to be resonant, since it satisfies algebra a straightforwardmatter.
[compare eqs. (8)–(13) with eqs. (18)–(23)] Large sectors of the resonant subalgebra are abelian-
ized after imposing the condition λ × osp(32|1) = 0.
3
S ·S ⊂S , (18)
0 0 0 Thisconditionalsoplaysafundamentalrˆoleintheshap-
S ·S ⊂S , (19) ing of the invariant tensor for the M algebra as an S-
0 1 1
S ·S ⊂S , (20) expansionofosp(32|1). Inthisway,itseffects arefeltall
0 2 2
thewaydowntothetheory’sspecificdynamicproperties.
S ·S ⊂S ∩S , (21)
1 1 0 2
S ·S ⊂S , (22)
1 2 1
S ·S ⊂S ∩S . (23)
2 2 0 2 C. M-Algebra Invariant Tensor
Theorem 4.2 from Ref. [19] now assures us that
Finding all possible invariant tensors for an arbitrary
G =(S ×V )⊕(S ×V )⊕(S ×V ) (24) algebra remains, to the best of our knowledge, as an im-
R 0 0 1 1 2 2
portant open problem. Nevertheless, once a matrix rep-
is a resonant subalgebra of S(2)×g. resentation for a Lie algebra is known, the (super)trace
E
As a last step, impose the condition λ × g = 0 on alwaysprovideswithaninvarianttensor. Butpreciselyin
3
G and relabel its generators as in Table I. This proce- ourcase,thisisnotawisechoice: ingeneral,itispossible
R
dure gives us the M algebra, whose (anti)commutation to prove that when the condition 0 ×g=0 is imposed,
S
relations are recalled in Table II. thesupertracefortheS-expandedalgebrageneratorswill
A clearer picture of the algebra’s structure can be ob- correspond to just a very small piece of the whole (su-
tained from the diagram in Fig. 1. The subspaces of per)trace for the g-generators. For the particular case
4
n-selector K ρ, which is defined as
α1···αn
1, when ρ=γ(α ,...,α ),
K ρ = 1 n (41)
α1···αn (0, otherwise.
Theorem 7.1 from Ref. [19] states that
T ···T =α K γhT ···T i
(A1,α1) (An,αn) γ α1···αn A1 An
(42)
(cid:10) (cid:11)
correspondstoaninvarianttensorfortheS-expandedal-
gebra without 0 -reduction,where α are arbitrarycon-
S γ
stants.
When the semigroup contains a zero element 0 ∈ S,
S
a smaller algebra can be obtained by ‘0 -reducing’ the
S
S-expanded algebra,i.e., by mapping all elements of the
form0 ×g to zero. Writing λ for the nonzeroelements
S i
of S, Theorem 7.2 from Ref. [19] assures that
T ···T =α K jhT ···T i (43)
(A1,i1) (An,in) j i1···in A1 An
isa(cid:10)ninvarianttensorf(cid:11)orthe0 -reducedalgebra,withα
S j
being arbitrary constants. As can be seen by comparing
eq. (42) with eq. (43), this invariant tensor corresponds
to a ‘pruning’ of (42).
In the M-algebra case, one must compute the com-
ponents of K j for S(2). Using the multiplication
i1···i6 E
law (14), these are easily seen to be
K j =δj , (44)
i1···i6 i1+···+i6
where δ is the Kronecker delta.
Using eqs. (43) and (44), we have that the only non-
FIG.1: (a) Theshaded region denotestheresonant subalge- vanishing components of the M algebra-invariant tensor
braG . (b)Shadedareascorrespond totheMalgebra itself, are given by
R
which is obtained from G by mapping the λ ×osp(32|1)
R 3
sector to zero. hJa1b1···Ja6b6iM =α0hJa1b1···Ja6b6iosp, (45)
hJ ···J P i =α hJ ···J P i ,
a1b1 a5b5 c M 2 a1b1 a5b5 c osp
(46)
of the M algebra, the only non-vanishing component of
the supertrace is Tr(Ja1b1···Janbn). A CS Lagrangian hJa1b1···Ja5b5Za6b6iM =α2hJa1b1···Ja6b6iosp, (47)
constructed with this invariant tensor would lead to an hJ ···J Z i =α hJ ···J Z i ,
a1b1 a5b5 c1···c5 M 2 a1b1 a5b5 c1···c5 osp
‘exotic gravity’, where the fermions, the central charges
(48)
and even the vielbein would be absent from the invari-
ant tensor. For this reason, it becomes a necessity to QJa1b1···Ja4b4Q¯ M =α2 QJa1b1···Ja4b4Q¯ osp,
work out other kinds of invariant tensors; very interest- (49)
(cid:10) (cid:11) (cid:10) (cid:11)
ing work on precisely this point has been developed in
Refs. [13, 14], where an invariant tensor for the M alge- where α0 and α2 are arbitrary constants.
braisobtainedfromthe Noether method, finallyleading ItisnoteworthythatthisinvarianttensorfortheMal-
to a CS M-algebra Supergravity in eleven dimensions. gebra,evenifitpossessesmanymorenonzerotermsthan
the supertrace [which would consist of (45) alone], still
InthecontextofanS-expansion,Theorems7.1and7.2
misses a lot of other terms present in that for osp(32|1).
from Ref. [19] provide with non-trivial invariant tensors
This is a common feature of 0 -reduced algebras. In
different from the supertrace. S
stark contrast, S-expanded algebras which do not arise
Letλ ,...,λ ∈S bearbitraryelementsofthesemi-
α1 αn from a 0 -reduction process do have invariant tensors
group S. Their product can be written as S
larger than the one for the original algebra. This fact
shapes the dynamics of the theory to a great extent, as
λα1···λαn =λγ(α1,...,αn). (40) we shall see in section IV.
The supersymmetrized supertrace will be used to pro-
This product law can be conveniently encoded by the vide an invariant tensor for osp(32|1), with the 32×32
5
Dirac matrices in eleven dimensions as a matrix repre- with the Lagrangian
sentation for the bosonic subalgebra, sp(32). The rep-
resentation with Γ1···Γ11 = +11 was chosen. In order LT(2n+1) A,A¯ =kQ(A2←n+A¯1)
to write the Lagrangian, field equations and boundary 1
conditions, it is very useful to have the components of (cid:0) (cid:1)=(n+1)k dthθFtni. (55)
theosp(32|1)-invarianttensorwithitsindicescontracted Z0
with arbitrary tensors. An explicit calculation gives us Here A denotes an M algebra-valued, one-form gauge
connection
La1b1···La5b5BchJ ···J P i =
1 5 1 a1b1 a5b5 c osp A=ω+e+b +b +ψ¯, (56)
2 5
1
2εa1···a11L1a1a2···L5a9a10B1a11, (50) and similarly for A¯. In Eq. (56) each term takes values
on a different subspace of the M algebra, namely
La11b1···La66b6hJa1b1···Ja6b6iosp = 13 ω = 21ωabJab, (57)
σX∈S6 e=eaPa, (58)
1
(cid:20)4Tr Lσ(1)Lσ(2) Tr Lσ(3)Lσ(4) Tr Lσ(5)Lσ(6) + b2 = 12ba2bZab, (59)
−Tr (cid:0)L L L(cid:1) (cid:0)L Tr L(cid:1) (cid:0)L + (cid:1) 1
σ(1) σ(2) σ(3) σ(4) σ(5) σ(6) b = babcdeZ , (60)
16 5 5! 5 abcde
+15(cid:0)Tr Lσ(1)Lσ(2)Lσ(3)Lσ(cid:1)(4)L(cid:0)σ(5)Lσ(6) (cid:1), (51) ψ¯=ψ¯αQα. (61)
(cid:21)
(cid:0) (cid:1) In Eq. (54), k is an arbitrary constant, θ = A−A¯,
A =A¯+tθ, and F =dA +A2. The Lagrangian(55)
La1b1···La5b5Bc1···c5hJ ···J Z i = t t t t
1 5 5 a1b1 a5b5 c1···c5 osp corresponds to a transgression form [7, 8, 9, 10, 11, 12].
1 5 Transgression forms are intimately related to CS forms,
ε − La1a2···La7a8 L Bbca9a10a11+
3 a1···a11 4 σ(1) σ(4) σ(5) bc 5 since they can be written as the difference of two CS
σX∈S5(cid:20) (cid:2) (cid:3) formsplusaboundaryterm. Thepresenceofthiscrucial
+10Lσa1(1a)2Lσa3(2a)4Lσa5(3a)6 Lσ(4) a7b Lσ(5) a8cB5bca9a10a11+ boundary term cures some pathologies present in stan-
dard CS Theory, such as ill-defined conserved charges
1
+ 4Lσa1(1a)2Lσa3(2a)4Lσa5(3a)6B(cid:2)5a7···a1(cid:3)1Tr(cid:2) Lσ(4(cid:3))Lσ(5) + (see Ref. [11]).
The general form of the Lagrangian given in Eq. (55)
−Lσa1(1a)2Lσa3(2a)4 Lσ(3)Lσ(4)Lσ(5) a5(cid:0)a6B5a7···a11(cid:1), (52) suffices in order to derive field equations, boundary con-
(cid:2) (cid:3) i ditions and Noether charges. Nevertheless, an explicit
version is highly desirable because it clearly shows the
La1b1···La4b4χ¯ ζβ QαJ ···J Q¯ = physical content of the theory; in particular, a separa-
1 4 α a1b1 a4b4 β osp
tioninbulkandboundarycontributionsisessential. This
1
− ε La1a(cid:10)2···La7a8χ¯Γabcζ+ (cid:11) important task can be painstakingly long if approached
240 a1···a8abc 1 4
na¨ıvely,i.e. throughthesoleuseofLeibniz’srule. Away
1 3
+ Tr L L La1a2La3a4χ¯Γ ζ+ out of the bog is provided by the subspace separation
60 4 σ(1) σ(2) σ(3) σ(4) a1···a4 method presented in Refs. [9, 12]. This method serves
σX∈S4(cid:20) (cid:0) (cid:1) a double purpose; on one hand, it splits the Lagrangian
−2La1a2 L L L a3a4χ¯Γ ζ+
σ(1) σ(2) σ(3) σ(4) a1···a4 in bulk and boundary terms and, on the other, it allows
3 the separationofthe bulkLagrangianinreflectionofthe
(cid:2) (cid:3)
+ Tr L L Tr L L χ¯ζ+
4 σ(1) σ(2) σ(3) σ(4) algebra’s subspace structure. The method is based on
−Tr L(cid:0) L L(cid:1) L(cid:0) χ¯ζ , (cid:1) (53) the iterative use of the ‘Triangle Equation’
σ(1) σ(2) σ(3) σ(4)
where T(cid:0)r stands for the trace(cid:1)in t(cid:3)he Lorentz indices, i.e. QA(2←n+A¯1) =Q(A2←n+A˜1)+QA(˜2←n+A¯1)+dQ(A2←n)A˜←A¯, (62)
Tr(LiLj)=(Li)ab(Lj)ba. Eq. (62) expresses a transgression form Q(2n+1) as the
A←A¯
sumoftwotransgressionformsdependingonanarbitrary
one-form A˜ plus a total derivative. This last term has
III. THE M-ALGEBRA LAGRANGIAN
the form
We consider a gaugetheory on anorientable (2n+1)- QA(2←n)A˜←A¯ ≡
dimensional manifold M defined by the action 1 t
n(n+1) dt ds A−A˜ A˜−A¯ Fn−1 ,
st
S(2n+1) A,A¯ = L(2n+1) A,A¯ , (54) Z0 Z0 D(cid:16) (cid:17)(cid:16) (cid:17) E(63)
T T
ZM
(cid:2) (cid:3) (cid:0) (cid:1)
6
where All three boundary terms that should in principle ap-
pear in (78) cancel due to the very particular properties
Ast =A¯+s A−A˜ +t A˜−A¯ , (64) of the invariant tensor chosen [cf. eqs. (45)–(49)].
F =dA +(cid:16)A2 . (cid:17) (cid:16) (cid:17) (65) The tensors Ha, Hab, Habcde and R are defined as
st st st
A first splitting of the Lagrangian (55) is achieved by Ha ≡ R5Pa M, (79)
introducing the intermediate connection A˜ =ω¯, Hab ≡(cid:10)R5Za(cid:11)b M, (80)
L A,A¯ =Q(A1←1)ω¯ +Q(ω¯1←1)A¯+dQ(A1←0)ω¯←A¯, (66) Habcde ≡(cid:10)R5Zabc(cid:11)de M, (81)
and a s(cid:0)econd(cid:1)one by separating Q(11) through ω: Rαβ ≡(cid:10)QαR4Q¯β(cid:11)M. (82)
A←ω¯
(cid:10) (cid:11)
Explicitly using the invariant tensor (50)–(53) one finds
Q(A1←1)ω¯ =Q(A1←1)ω+Q(ω1←1)ω¯ +dQ(A1←0)ω←ω¯. (67)
α
After these two splittings, the Lagrangian(55) reads H = 2R(5), (83)
a 64 a
L A,A¯ =Q(A1←1)ω−Q(A¯1←1)ω¯ +Q(ω1←1)ω¯ +dB(10), (68) Hab =α2 25 R4− 34R2R2 Rab+5R2Ra3b−8Ra5b ,
with(cid:0) (cid:1) (cid:20) (cid:18) (cid:19) (cid:21)
(84)
B(10) =Q(A1←0)ω←ω¯ +Q(A1←0)ω¯←A¯. (69) Habcde =−156α2 5R[abRc(4d)e]+40Rf[aRgbRc(3d)e]fg+
The first two terms in (68) are identical (with the ob- −R2R(3) h+4R(2) R3 fg , (85)
vious replacements), and we shall mainly concentrate abcde abcdefg
obne uannraellyaztiendgttohetmhe.twTohefortmhierrd; tinermparwtiiclullbare,sithocwann btoe R=−α2 R4− 3R2R2(cid:0) 11(cid:1)+ i1 R(4)Γabc+
40 4 96 abc
madetovanishwithoutaffectingtherest. Theboundary (cid:26)(cid:18) (cid:19)
term(69)canbe writtenina moreexplicitwayby going −3 R2Rab− 8 R3 ab RcdΓ . (86)
abcd
back to eq. (63) and replacing the relevant connections 4 3
(cid:20) (cid:21) (cid:27)
and curvatures. The result is however not particularly (cid:0) (cid:1)
illuminating and,as its explicit formis notneeded in or- Here we have used the shortcuts [27]
der to write boundary conditions, we shall not elaborate
any longer on it. Rn =Ra1a2···Rana1, (87)
Let us examine the transgression form Q(A1←1)ω. The Ranb =Rac1Rc1c2···Rcn−1b, (88)
subspace separation method can be used again in order
R(n) =ε Rb1b2···Rb2n−1b2n. (89)
to write down a closed expression for it. To this end we a1···ad−2n a1···ad−2nb1···b2n
introduce the following set of intermediate connections:
OnsectionIV weshallcommentonthe dynamicspro-
A0 =ω, (70) ducedbythisLagrangian;herewemayalreadynotethat
A1 =ω+e, (71) no derivatives of ea, ba2b or ba5bcde appear. This can be
traced back to the particular form of the invariant ten-
A =ω+e+b , (72)
2 2 sor (45)–(49), which contains no nonzero components of
A3 =ω+e+b2+b5, (73) the form J3PZ , etc.
2
A =ω+e+b +b +ψ¯. (74) The last contribution to the Lagrangian (68) comes
4 2 5 (cid:10) (cid:11)
fromthe Qω←ω¯ term. Takingintoaccountthe definition
The Triangle Equation (62) allows us to split the trans-
of a transgression form and the form of the invariant
gression Q(A141←) A0 following the pattern tensor,itisstraightforwardtowritedowntheexpression
Q(11) =Q(11) +Q(11) +dQ(10) , (75) 1
A4←A0 A4←A3 A3←A0 A4←A3←A0 Q(ω1←1)ω¯ =3 dtθabLab(t), (90)
Q(A131←) A0 =Q(A131←) A2 +Q(A121←) A0 +dQ(A130←) A2←A0, (76) Z0
Q(11) =Q(11) +Q(11) +dQ(10) . (77) where
A2←A0 A2←A1 A1←A0 A2←A1←A0
ProceedingalongtheselinesonearrivesattheLagran- L (t)= R5J (91)
ab t ab M
gian
(cid:10) (cid:11)
and
1
Q(11) =6 H ea+ H bab+
A4←A0 a 2 ab 2 1
(cid:20) R = [R ]abJ , (92)
t t ab
1 5 2
+ H babcde− ψ¯RD ψ . (78)
5! abcde 5 2 ω (cid:21) [Rt]ab =R¯ab+tDω¯θab+t2θacθcb. (93)
7
An explicit version for L (t) reads off one finds that there are severaldistinct sectors which
ab
arebythemselvesinvariant,sothatitisperfectlysensible
5 3
L (t)=α R4− R2R2 [R ] + to associate them with different couplings.
ab 0 2 t 4 t t t ab TheneteffectontheLagrangian(78)concernsonlythe
(cid:20) (cid:18) (cid:19)
+5R2[R ]3 −8[R ]5 . (94) explicit expressions for the tensors defined in (79)–(82);
t t ab t ab the new versions read
i
Afewcommentsareinorder. Asseenin(94),Q(ω1←1)ω¯ is H = α2R(5), (97)
proportional to α , as opposed to all other terms, which a 64 a
0
are proportional to α . This is a direct consequence of 5 3
2 H =α κ R4− γ R2R2 R +
the choice of invariant tensor. Being the only piece in ab 2 2 15 4 5 ab
(cid:20) (cid:18) (cid:19)
the Lagrangian unrelated to α , it can be removed by
2 +5κ R2R3 −8R5 , (98)
simply picking α = 0. This independence also means 15 ab ab
0
tThhaits Qis(ω1r←1e)lω¯atiesdbtyoittsheelffaincvtatrhiaantttuhnisdetrertmhecMorraeslgpeobnrdas. Habcde =−156α2 5R[abRc(4d)e(cid:3)]+40Rf[aRgbRc(3d)e]fg+
to the only surviving component when the supertrace is −κ R2R(h3) +4R(2) R3 fg , (99)
15 abcde abcdefg
used to construct the invariant tensor.
Becauseofits form, Q(ω1←1)ω¯ apparentlycontains abulk R=−α2 κ3R4− 3(5γ9−(cid:0)4)R(cid:1)2Ri2 11+
interaction of the ω and ω¯ fields. This is no more than 40 4
(cid:26)(cid:20) (cid:21)
an illusion; in order to realize this, it suffices to use the 1
+ R(4)Γabc+
‘Triangle Equation’ with the middle connection set to 96 abc
zero, −3 κ R2Rab− 8 R3 ab RcdΓ . (100)
9 abcd
Q(ω1←1)ω¯ =Q(ω1←1)0−Q(ω¯1←1)0+dQ(ω1←0)0←ω¯. (95) 4(cid:20) 3(cid:0) (cid:1) (cid:21) (cid:27)
Here Q(11) and Q(11) correspond to two independient The constants κn and γn are not, as it may seem, an
ω←0 ω¯←0 infinite tower of arbitrary coupling constants, but are
CSexotic-gravityLagrangiansandQ(10) corresponds
ω←0←ω¯ rather tightly constrained by the relations
to the boundary piece relating them.
n
κ =1+ (κ −1), (101)
m n
m
A. Relaxing Coupling Constants n
γ =γ + −1 (κ −1). (102)
m n n
m
Allresultssofarhavebeenobtainedfromtheinvariant (cid:16) (cid:17)
These two sets of constants replace the above β and
tensor given in eqs. (50)–(53). This in turn was derived 4+2
β ;oncearepresentativefromeveryoneofthemhas
from the supersymmetrized supertrace of the product of 2+2+2
been chosen, the rest is univocally determined by (101)–
six supermatrices representing as many osp(32|1) gener-
(102). In other words, fixing one particular κ sets the
ators. Inparticular,wehaveused32×32DiracMatrices n
values of all others. Once all κ are fixed, choosing one
in d = 11 to represent the bosonic sector, so that the n
γ ties together all the γ’s.
bosonic components of the invariant tensor correspond n
The original coupling constants β and β can
to their symmetrized trace [20, 21]. 4+2 2+2+2
be expressed in terms of the new κ and γ as [28]
Differentinvarianttensorsmaybe obtainedbyconsid- n n
eringsymmetrizedproductsoftraces,asinhFpihFn−pi.
1
To exhaust all possibilities one must consider the parti- β = n(κ −1), (103)
4+2 n
tions of six (which is the order of the desired invariant Tr(11)
tensor). Amoment’sthoughtshowsthat,apartfromthe 15
β = (γ −κ ). (104)
alreadyconsidered6=6partition,onlythe6=4+2and 2+2+2 [Tr(11)]2 n n
6 = 2+2+2 cases contribute, as all others identically
vanish. We are thus led to consider the following linear It is also worth to notice that
combination:
β =0 ⇔ κ =1, (105)
4+2 n
h···i =h···i +β h···i +β h···i .
M 6 4+2 4+2 2+2+2 2+2+2 β =0 ⇔ γ =κ . (106)
(96) 2+2+2 n n
(The coefficient in front of h···i can be normalized to
6
unity without any loss of generality).
The amazing result of performing this exercise is that B. Comparison between the S-Expansion
Lagrangian and the HTZ Lagrangian
nonewtermsappearintheinvarianttensor(96);rather,
the original rigid structure found in (50)–(53) is relaxed
into one which takes into account the new coupling con- InRef. [14], anactionfor aneleven-dimensionalgauge
stants β andβ . Turning these constantson and theoryfor the Malgebrawasfound throughthe Noether
4+2 2+2+2
8
procedure. ThecorrespondingLagrangiancanbe castin IV. DYNAMICS
the form
A. Field Equations and Four-Dimensional
1 1 5
L =G ea+ G bab+ G babcde− ψ¯QD ψ, Dynamics
α a 2 ab 2 5! abcde 5 2 ω
(107)
where ThefieldequationsforAandA¯ arecompletely analo-
gous,andtherefore in this sectionthey will be presented
G =R(5), (108) onlyforA. Thegeneralexpressionforthefieldequations
a a
reads
G =−32(1−α) R4−2R2R2 R +
ab ab
+5R2Ra3b−4R(cid:2)a5(cid:0)b , (cid:1) (109) F5TA M =0, (115)
G =− 5 (64α)R (cid:3)R(4) , (110) where {TA,A=1,...(cid:10),dim(g(cid:11))} is a basis for the algebra
abcde 16 [ab cde] and F is the curvature.
64 1 Thefieldequationsobtainedbyvaryingea,bab,ba1···a5
Q= R(4)Γabc+ 2 5
5 96 abc and ψ are given by
(cid:20)
−1(1−α) R2R −R3 R Γabcd . (111) Ha =0, (116)
2 ab ab cd
(cid:21) Hab =0, (117)
(cid:0) (cid:1)
H =0, (118)
Here α is an arbitrary constant. abcde
In our work we have obtained the Lagrangian(78), RDωψ =0, (119)
1 1 5 whereexplicitexpressionsforHa,Hab,Habcde andRcan
L=Haea+2Habba2b+5!Habcdeba5bcde−2ψ¯RDωψ, (112) befoundinEqs.(97)–(100). The fieldequationobtained
from varying ωab reads
where H , H , H and R are given in Eqs. (97)–
a ab abcde L −10 D ψ¯ Z (D ψ)+
(100). ab ω ab ω
TheadvantageofwritingbothLagrangiansinthisway +5H (cid:0) Tc+(cid:1) 1 ψ¯Γcψ +
isthatitmakeseasiertocompareeq.(107)witheq.(112) abc 16
(cid:18) (cid:19)
just by matching the coefficients H , H , H and R
a ab abcde 5 1
with Ga, Gab, Gabcde and Q. +2Habcd Dωbcd− 16ψ¯Γcdψ +
Besidesanoverallmultiplicative constant[29], the La- (cid:18) (cid:19)
1 1
grangian (107) possesses two tunable independent con- + H D bc1···c5 + ψ¯Γc1···c5ψ =0, (120)
stants, κ and γ , and the Lagrangian (112) posseses 24 abc1···c5 ω 16
n n (cid:18) (cid:19)
just one, α. An interesting question is if there is some
where we have defined
particular choice of κ’s and γ’s which allows us to reob-
taintheHTZLagrangian. Interestingly,theanswerisno. L ≡ R5J , (121)
ab ab M
As a matter of fact, it can be seen by simple inspection
of the expressions for Habcde and Gabcde that in the S- (Zab)αβ ≡(cid:10)QαR3J(cid:11)abQ¯β M, (122)
expansion Lagrangian new terms arise which cannot be Habc ≡(cid:10)R4JabPc M,(cid:11) (123)
wiped out by a simple choice of the κ and γ constants.
Nevertheless, there are some choices which bring both Habcd ≡(cid:10)R4JabZc(cid:11)d M, (124)
Lagrangians closer. For example, the identification Habcdefg ≡(cid:10)R4JabZcde(cid:11)fg M. (125)
α−1 Explicit versions for th(cid:10)ese quantities(cid:11)are found using
κ15 = (113) the invariant tensor (50)–(53):
5
8
γ5 = 15(α−1) (114) Lab =α0 25 R4− 43R2R2 Rab+5R2Ra3b−8Ra5b ,
(cid:20) (cid:18) (cid:19) (cid:21)
(126)
allows us to identify some terms of H with the ones
ab
in G . In the same way, the attempt to match (100)
ab α 3 1
and (111) leads to a system of equations which has a Zab = 402 2 Ra3b− 4R2Rab 11− 48Ra(3b)cdeΓcde+
solution under some conditions. (cid:26) (cid:18) (cid:19)
3 1
Thus the comparison between the Lagrangians (78) − R Rcd− R2δcd RefΓ +
and (107) shows the independence between them. The 4 ab 2 ab cdef
(cid:18) (cid:19)
LagrangianwhicharisesfromtheS-expansionprocedure − δcgR RhdRef −Rc Rd Ref+
ab gh a b
containsallthetermsoftheHTZLagrangian,alongwith
new terms which cannot be made to vanish by a simple +(cid:2)1δef R3 cd Γ , (127)
choice of constants. 2 ab cdef
(cid:21) (cid:27)
(cid:0) (cid:1)
9
α
H = 2R(4), (128) two sets of mathematical tools. The first of these sets
abc 32 abc
was providedin Ref. [19], where the procedure of expan-
sion is analyzed using abelian semigroups and non-trace
3 invariant tensors for this kind of algebras are written.
Habcd =α2δaefbδcgdh 4R2RefRgh−Re3fRgh−RefRg3h+ The problem of the invariant tensor is far from being a
(cid:20) trivial one; as discussed in Ref. [19], the 0 -reduction
4 S
− R R3 +R3 R −R2 R2 + procedure which was necessary in order to construct the
5 eh fg eh fg eh fg
M algebra from osp(32|1) also renders the supertrace,
1(cid:0) 1 (cid:1)3
+ R2R R + η R4− R2R2 + usually used as invariant tensor, as almost useless. The
2 eh fg 8 [ef][gh] 4 other set of tools is related with properties of transgres-
(cid:18) (cid:19)
8 sion forms, and especially with the subspace separation
−η R2R2 − R4 , (129)
fg eh 5 eh method [9, 12], used in order to write down the Lagran-
(cid:18) (cid:19)(cid:21) gian in an explicit way.
From a physical point of view, it is very compelling
α 5 1
H = 2δcdefg − R(3) R − R R(3) + that, using the methods of ‘dynamical dimensional re-
abc1···c5 80 c1···c5 3 abcde fg 6 ab cdefg duction’introducedin[13,14], somethingthat lookslike
(cid:20)
+10R(2) Rp Rq − 2R(1) R3 pq+ a ‘frozen’ version of four-dimensional Einstein–Hilbert
abcdepq f g 3 abcdefgpq gravitywithpositivecosmologicalconstantisobtainedby
+ 1Rp Rq R(2) − 1Rq R(2) (cid:0)Rp(cid:1)+ simplyabandoningtheprejudicethatthevacuumshould
3 a b cdefgpq 3 a bcdefgp q satisfy F =0. This dynamics ‘freezing’ is a consequence
1 1 oftheconstrainedformoftheinvarianttensor: theMal-
+ 4R2Ra(2b)cdefg+ 3RqbRa(2c)defgpRpq+ gebrahas more generatorsthan osp(32|1), but less non-
vanishing components on the invariant tensor. For this
10 10
− 3 ηgaRb(3c)depRpf + 3 ηgbRa(3c)depRpf+ reason, the equations of motion associated to the vari-
ations of ea, bab and ba1···a5 become simply constraints
5 2 5
− η R(4) . (130) on the gravitationalsector. But the poor form of the in-
24 [ab][cd] efg
(cid:21) varianttensorisadirectconsequenceofthe0S-reduction
procedure. As shown in Theorem 7.1 from Ref. [19], an
They satisfy the relationships
invarianttensorforagenericS-expandedalgebrawithout
H = 1RabH , (131) 0S-reduction has more non-vanishing components than
c abc its 0 -reduced counterpart and, in general, even more
2 S
1 components than the invariant tensor of the original al-
Hcd = RabHabcd, (132) gebra.
2
1 Theaboveconsiderationsmakeitevidentthatitwould
H = RabH , (133)
cdefg 2 abcdefg be advisable to avoid the 0S-reduction. The M alge-
R= 1RabZ . (134) braarisesasthe 0S-reductionofthe resonantsubalgebra
ab given by eq. (24). This resonant subalgebra itself looks
2
verymuchliketheMalgebra,inthesensethatithasthe
The problem of finding a ‘true vacuum’ can be ana- anticommutator
lyzed in a similar way to the Refs. [13, 14], leading to
some results of the above-mentionedreferences: it is not
1 1 1
possible to reproduce four-dimensional General Relativ- Q,Q¯ = ΓaP − ΓabZ + Γa1···a5Z ,
8 a 2 ab 5! a1···a5
ity because there are too many constraints on the four- (cid:18) (cid:19)
(cid:8) (cid:9) (135)
dimensional geometry. [30]
but it also has an osp(32|1) subalgebra (spanned by
There are several ways in which one could deal with
λ J , λ P , λ Z and λ Q; let us remember
this problem; as we will discuss in the conclusions, the 3 ab 3 a 3 a1···a5 3
that λ λ = λ ). The ‘central charges’ are no longer
excessof constraintsis stronglyrelatedto the semigroup 3 3 3
abelian; rather, their commutators take values on the
choicemadeinordertoconstructtheMalgebraandalso
λ ×osp(32|1) sector. This algebra has a much bigger
to the 0 -reduction. When other semigroups are cho- 3
S
tensor than the ‘normal’ M algebra (see Theorem 7.1
sen, different algebras can arise which reproduce several
from Ref. [19]), and therefore, an ‘unfrozen’ dynamics
features of the M algebra without having its ‘dynamical
which has good chances of reproducing four-dimensional
rigidity’ [19].
Einstein–Hilbert Gravity.
AmoreelegantalgebrachoiceisalsoshowninRef.[19].
V. SUMMARY AND CONCLUSIONS Replacing the M algebra’s semigroup S(2) for the cyclic
E
groupZ , a resonantsubalgebra of Z ×osp(32|1) is ob-
4 4
The construction of a transgressiongauge field theory tained. It has very interesting features, like having two
for the M algebrahas been developedthroughthe use of fermionic charges, Q and Q′ with an M algebra-like an-
10
ticommutator settled. As already discussed, in order to solve these
problems it might be wise to take into account the fact
Q′,Q¯′ = Q,Q¯ that the M algebra is but one possible choice within a
1 1 1 family of superalgebras. Other members of this family
(cid:8) (cid:9)=(cid:8)8 ΓaP(cid:9)a− 2ΓabZab+ 5!Γa1···a5Za1···a5 . [obtained from osp(32|1) using different abelian semi-
(cid:18) (cid:19) groups, for instance] might also play a rˆole in finding
(136)
a truly fundamental symmetry.
Two sets of AdS boost generators, P and P′, and two
a a
(non-abelian) ‘M5’ generators, Z and Z′ , are
alsopresent. Thisdoublinginsevear1a··l·ag5eneratoras1·m··aa5kesit Acknowledgments
speciallysuitabletoconstructatransgressiongaugefield
theory. On the other hand, since Z is a discrete group, F. I. and E. R. wish to thank P. Minning for hav-
4
it does not have a zero element; therefore, it has from ing introduced them to so many beautiful topics, es-
the outset very good chances of having unfrozen four- pecially that of semigroups. They are also grateful to
dimensionaldynamics. Work regardingthis issue will be D. Lu¨st for his kind hospitality at the Arnold Sommer-
presented elsewhere. feld Center for Theoretical Physics in Munich, where
At this point, it is natural to ask ourselves what the part of this work was done. F. I. and E. R. were
relationship between this M algebra or M algebra-like supported by grants from the German Academic Ex-
transgression theories and M Theory could be. It has change Service (DAAD) and from the Universidad de
been proposed that some CS supergravity theories (see Concepci´on (Chile). P. S. was supported by FONDE-
Refs. [1, 2, 3, 23]) in eleven dimensions could actually CYT Grant 1040624 and by Universidad de Concepci´on
correspond to M Theory, but the potential relations to throughSemilla Grants 205.011.036-1Sand 205.011.037-
standardCJSsupergravityandStringtheoryremainun- 1S.
[1] R.Troncoso,J.Zanelli,New gaugeSupergravity inSeven (2007) 127. arXiv: hep-th/0603061.
andElevenDimensions.Phys.Rev.D58(1998)101703. [13] M. Hassa¨ıne, R. Troncoso, J. Zanelli, Poincar´e Invari-
arXiv:hep-th/9710180. ant Gravity with Local Supersymmetry as a Gauge The-
[2] P. Hoˇrava, M Theory as a Holographic Field Theory. ory for the M-Algebra. Phys. Lett. B 596 (2004) 132.
Phys.Rev.D 59 (1999) 046004. arXiv: hep-th/9712130. arXiv: hep-th/0306258.
[3] H. Nastase, Towards a Chern–Simons M Theory of [14] M.Hassa¨ıne,R.Troncoso,J.Zanelli,11DSupergravityas
OSp(1|32)×OSp(1|32). arXiv: hep-th/0306269. a Gauge Theory for the M-Algebra. Proc. Sci. WC 2004,
[4] M. Bagnoud, L. Carlevaro, A.Bilal, Supermatrix models 006 (2005). arXiv: hep-th/0503220.
for M-theory based on osp(1|32,R). Nucl. Phys. B 641 [15] J. A. de Azc´arraga, J. M. Izquierdo, M. Pic´on, O. Va-
(2002) 61. arXiv: hep-th/0201183. rela, Generating Lie and Gauge Free Differential (Su-
[5] J. D. Edelstein, J. Zanelli, (Super-)Gravities of a per)Algebras by Expanding Maurer–Cartan Forms and
different sort. J. Phys. Conf. Ser. 33 (2006) 83. Chern–Simons Supergravity. Nucl. Phys. B 662 (2003)
arXiv:hep-th/0605186. 185. arXiv:hep-th/0212347.
[6] E. Witten, (2+1)-Dimensional Gravity as an Exactly [16] J.A.deAzc´arraga,J.M.Izquierdo,M.Pic´on,O.Varela,
Soluble System. Nucl.Phys. B 311 (1988) 46. Extensions, Expansions, Lie Algebra Cohomology and
[7] A. Borowiec, M. Ferraris, M. Francaviglia, A Covari- Enlarged Superspaces. Class. Quant. Grav. 21 (2004)
antFormalismforChern–SimonsGravity.J.Phys.A36 S1375. arXiv:hep-th/0401033.
(2003) 2589. arXiv: hep-th/0301146. [17] J. A. de Azc´arraga, J. M. Izquierdo, M. Pic´on, O. Va-
[8] A. Borowiec, L. Fatibene, M. Ferraris, M. Francaviglia, rela, Expansions of Algebras and Superalgebras and
CovariantLagrangianFormulationofChern–Simonsand some Applications. Invited lecture delivered at Work-
BF Theories. Int. J. Geom. Meth. Mod. Phys. 3 (2006) shop on Deformations and Contractions in Mathemat-
755. arXiv: hep-th/0511060. icsandPhysics,Oberwolfach,Germany,15–21Jan2006.
[9] F. Izaurieta, E. Rodr´ıguez, P. Salgado, On Transgres- arXiv: hep-th/0703017.
sion Forms and Chern–Simons (Super)Gravity. Preprint [18] J. A. de Azc´arraga, J. M. Izquierdo, Lie Groups, Lie
LMU-ASC-77-05. arXiv:hep-th/0512014. Algebras, CohomologyandSomeApplicationsinPhysics.
[10] P. Mora, Transgression Forms as Unifying Principle in Camb. Univ.Press, 1995.
FieldTheory.Ph.D.Thesis,UniversidaddelaRepu´blica, [19] F. Izaurieta, E. Rodr´ıguez, P. Salgado, Expanding Lie
Uruguay(2003). arXiv:hep-th/0512255. (super)algebras through Abelian Semigroups. J. Math.
[11] P.Mora, R. Olea, R.Troncoso, J. Zanelli, Transgression Phys. 47 (2006) 123512. arXiv: hep-th/0606215.
FormsandExtensionsofChern–SimonsGaugeTheories. [20] A. Van Proeyen, Tools for Supersymmetry. Lectures
JHEP 0602 (2006) 067. arXiv: hep-th/0601081. given at Spring School on Quantum Field Theory: Su-
[12] F. Izaurieta, E. Rodr´ıguez, P. Salgado, The Extended persymmetry and Superstrings, Calimanesti, Romania,
Cartan Homotopy Formula and a Subspace Separation 24–30 Apr1998. arXiv: hep-th/9910030.
Method for Chern–Simons Theory.Lett.Math.Phys.80 [21] U.Gran,GAMMA:AMathematicaPackageforperform-