Supergravity from Bosonic String

0.13 MB·

by  B. B. Deo

Supergravity from Bosonic String
B. B. Deo
Department of Physics, Utkal University, Bhubaneswar, India
(Dated: February 1, 2008)
Deser-Zumino[1]N=1supergravityaction forthegraviton and gravitinopairin fourdimensions
is deduced from the Nambu-Goto action [2, 3] of the 26 dimensional bosonic string theory using
Principle of Equivalence.
PACSnumbers: 04.65.+e,12.60.J
Keywords: supergravity,supersymmetry,superstring
Notlongago,Casher,Englert,NicolaiandTaormina [4,5]showedthatconsistentsuperstringscanbethe solutions
ofD=26bosonicstringtheoryandthelaterappearsasthefundamentalstringtheory. Thishasalsobeendemonstrated
5 bymeandincollaborationwithL.Maharanainrecentreferences[6]and [7,8,9]. Detailsofconstructonofconsistent
0 superstring are given there. None of the existing four dimensional string theories contain construction of gravitino
0
and graviton ground states from string theory excitation quanta leading to supergravity. As far as I know, this is
2
the first attempt in this direction. I repeat some of them so that the preprints can be skipped without missing the

n deductionsofthe maintheme whichisofimmediate informationtostringtheorists. The physicalopenbosonicstring
a theory with the coordinates Xµ in the world sheet (σ,τ) is given by the action
J

6
S = 1 d2σ ∂αXµ(σ,τ) ∂ X (σ,τ), µ=0,1,2..........,25. (1)
5 −2π Z α µ
v
0 It is well known that we can introduce 44 Majorana fermions φ as proven basically by Mandelstam [10] for (1+1)
λ
3 quantum field theory, which are scalars in the world sheet and four bosonic coordinates and the action
1
8 44
1
0 S = d2σ∂αXµ(σ,τ) ∂αXµ(σ,τ) i φ¯jρα∂αφj, µ=0,1,2,3. (2)
4 −2π Z −
Xj=1
0  
/
h where we have taken
t
- 0 i 0 i
p ∂α =(∂σ,∂τ), ρ0 =(cid:18) i −0 (cid:19), ρ1 =(cid:18) i 0(cid:19) and φ¯=φ†ρ0 (3)
e
h
In the Chapter-6.1.2 in [11], it has also been proved and stated that two Majorana fermions in 1+1 dimensions are
:
v equivalent to one real boson in the infinite volume limit. It is also true in anomaly free string theory on a finite
Xi interval or a circle even one like in Veneziano model. Observingthat there are also MajoranaLorentz fermions ψµ in
the bosonic representation 0f SO(3,1), we can further rewrite equation (2) as [6]
r
a
11
1
S = d2σ∂αXµ(σ,τ) ∂αXµ(σ,τ) i ψ¯µ,jρα∂αψµ,j (4)
−2π Z −
Xj=1
 
presuming this to be anomalyfree. This has been provedin reference[6]. The upper index refer to rowandthe lower
index to a column. Searching for a Ψµ, to copartner to Xµ, one has to further divide the elevens into two groups.
There is a group of six, j=1,2,.,6, where the negative and positive frequency parts are ψµ,j = ψ(+)µ,j +ψ(−)µ,j and
another group of five, k=7,8,..11 with φµ,k = φ+µ,k φ−µ,k. The negative sign is absorbed in creation operators as
allowedby the freedom of phase of Majoranafermion−s. I introduce matrices (ej,ek) which are rowsof ten zeroesand
only one 1 in the jth or kth place. eje =6 and eke =5. With these the world sheet action,
j k
1
S = d2σ ∂αXµ(σ,τ) ∂ X (σ,τ) iψ¯µ,jρα∂ ψ +iφ¯µ,kρα∂ φ (5)
α µ α µ,j α µ,k
−2π Z −
(cid:2) (cid:3)
is invariant under supersymmetric transformations,
δXµ = ǫ¯ ejψµ ekφµ , (6)
j − k
δψµ,j = (cid:0)iǫ ejρα∂ Xµ,(cid:1) (7)
α
−
and δφµ,k = iǫ ekρα∂ Xµ. (8)
α
−
2
ǫ is a constant anticommuting spinor.
In action(5), there are 4 bosons but elevenfermions which are Lorentzvectors ofSO(3,1) whichis possible only in
1+1 dimensions [11]. This will show up in two supersymmetricsuccessive transformations. In our case,in particular,
the transformation leads to spatial translation with the coefficient aα =2iǫ¯1ραǫ but constrained by
2
ψµ =e Ψµ, φµ =e Ψµ (9)
j j k k
which defines the nature of e and e and also provides the much needed linear combination for the fermion SUSY
j k
partner of Xµ
Ψµ =ejψµ ekφµ. (10)
j − k
There is no need for additional auxilliary field. The action (5) reduces to
1
S = d2σ ∂αXµ(σ,τ) ∂ X (σ,τ) iΨ¯µρα∂ Ψ (11)
α µ α µ
−2π Z −
(cid:2) (cid:3)
Using equations (6) to (8) and equation (11) itself, it is easy to verify
δXµ = ǫ¯Ψµ, δΨµ = ieρα∂ Xµ (12)
α
−
[δ ,δ ]Xµ = aα∂ Xµ, [δ ,δ ]Ψµ,j =aα∂ Ψµ,j. (13)
1 2 α 1 2 α
The actions given in equations(5) and (13) are equivalent. Ψµ, while in jth or kthsite emits and absorbs quanta by
quantising ψµ,j or φµ,k according to the equation (5) which should be unitary, free of ghosts and nomalies. It should
be free of tachyonsandmodular invariant. These aspects have been studied in detailin references [9] and [7], where,
also currents and energy momentum tensors have been given in light cone basis as deduced from local 2d-global
symmetric extension of the above action. We shall not need these details here.
The action in (11) yields the only nonvanishing equal time commutator and anticommutators [11] as
π
[∂ Xµ(σ,τ),∂ Xν(σ′,τ)] = ηµνδ′(σ σ′) (14)
± ±
±2 −
ψµ(σ,τ),ψν(σ′,τ) = πηµνδ(σ σ′)δ (15)
{ A B } − AB
φµ(σ,τ),φν (σ′,τ) = πηµνδ(σ σ′)δ (16)
{ A B } − − AB
Ψµ(σ,τ),Ψν (σ′,τ) = πηµνδ(σ σ′)δ (17)
{ A B } − AB
On a light cone basis, the Noether supercurrents and energy momentum tensors are
J = ∂ X Ψµ (18)
+ + µ +
J = ∂ X Ψµ (19)
− − µ −
i i
T = ∂ Xµ∂ X + ψ¯µ,j∂ ψ φ¯µ,k∂ φ (20)
++ + + µ 2 + + +µ,j − 2 + + +µ,k
i i
T = ∂ Xµ∂ X + ψ¯µ,j∂ ψ φ¯µ,k∂ φ (21)
−− − − µ 2 − − −µ,j − 2 − − −µ,k
Itfollowsfromthevariationofthelocal2-dsupersymmetricactionthattheaboveequations(18)to(21)vanish. This
enables one to construct physical states without ghosts in the Fock space. One quantises the action (5) as [6]
1
Xµ(σ,τ) = xµ+pµτ +i αµ e−inτ Cos(nσ) (22)
n n
nX6=0
+∞
1
or ∂ Xµ = αµ e−in(τ±σ) (23)
± 2 n
X−∞
and [ αµ, αν ] = m δ ηµν (24)
m n m+n
With Neveu-Schwartz(NS) and Ramond (R) [13, 14] boundary conditions
Ψµ = 1 Bµ e−ir(τ±σ), with bµ,j =ejBµ and b′µ,k =ekBµ NS (25)
± √2 r r r r
r∈XZ+21
= 1 Dµ e−im(τ±σ), with dµ,j =ejBµ, and d′µ,k =ekBµ R (26)
√2 m m m m m
Xm
3
and the anticommutation relations are
Bµ, Bν =ηµνδ , Dµ, Dν =ηµνδ . (27)
{ r s} r+s { m n} m+n
To construct admissible states, it is essential to give the Super Virasoro generators and their algebra in notations of
reference [13, 14]
1 π
L = dσeimσT (28)
m π Z ++
−π
∞
1 1 1
= :α α :+ (r+ m):(b b b′ b′ ): NS (29)
2 −n· m+n 2 2 −r· m+r− −r· m+r
X−∞ r∈Xz+12
∞ ∞
1 1 1
= :α α :+ (n+ m):(d d d′ d′ ),: R (30)
2 −n· m+n 2 2 −n· m+n− −n· m+n
X−∞ n=X−∞
√2 π ∞
G = dσeirσJ = α ejb ekb′ , NS (31)
r π Z + −n· n+r,j − n+r,k
−π n=X−∞ (cid:0) (cid:1)
∞
F = α ejb ekb′ . R (32)
m −n· n+r,j − n+r,k
X−∞ (cid:0) (cid:1)
The Super Virasoro algebra is
C
[L ,L ] = (m n)L + (m3 m)δ , (33)
m n m+n m,−n
− 12 −
1
[L ,G ] = ( m r)G , NS (34)
m r m+r
2 −
C 1
G ,G = 2L + (r2 )δ , (35)
r s s+r r,−s
{ } 3 − 4
1
[L ,F ] = ( m n)F , R (36)
m n m+n
2 −
C
F ,F = 2L + (m2 1)δ , m=0. (37)
m n m+n m,−n
{ } 3 − 6
The central charge C of the action (5) is 26.
Inthissuperstringtheory,therearefourbosons(C=4),twentytwotransversefermions(C=11)(ψ1,2,φ1,2)belonging
j k
to two transverse superfermionic modes Ψ1,2 (equation 9), the usual conformal ghosts [11, 12] (C=-26) and lastly
the superconformal ghosts (β,γ) contributing a central charge 11 in a supplemented Hilbert space. The last action
is identical to the action of the 22 longitudinal ghost modes (ψ0,3,φ0,k) or two superfermionic modes Ψ0,3. This
j k
identity has been proved in the reference [9]. Thus the central charge is zero and there is Lorentz invariance and the
superstring theory is anomaly free.
The physical states are defined as
NS: (L 1)φ> = 0, L φ>=0, G φ>=0. r,m > 0. (38)
o m r
− | | |
R: (L 1)ψ > = (F2 1)ψ > =0, L ψ > =0, F ψ >=0. m > 0. (39)
o− | α o − | α m| α r|
α is the spinor component index. Admissible Fock space states are
NS Eigenstates : αµ Bν 0>, (40)
−n −m |
Yn,µmY,ν(cid:8) (cid:9)(cid:8) (cid:9)
R Eigenstates : αµ Dν 0>u. (41)
−n −m |
Yn,µmY,ν(cid:8) (cid:9)(cid:8) (cid:9)
G.S.O. projectionoperators(1+( 1)F) [11] is implicit for the NS eigenstates. Supersymmetry forbids the existence
−
of tachyonic vacuum. Here, there are two vacua which are tachyonic in the scalar NS sector and the other in the
fermionic Ramond sector. Their self energies cancel as in the supersymmetry, not showing up in overall Fock space.
4
In the Ramond or Fermionic sector, I aim at the 3-spin gravitino. In four flat dimensions, the Rarita-Schwinger
2
equation is
ǫµνλσγ γ ∂ ψ =(gµνγσ gµσγν +gνσγµ γµγνγσ)∂ ψ =0 (42)
5 λ ν σ ν σ
− −
As shown by GSO in reference [15], this is equivalent to
pµψ =γµψ =γ ψ =0 (43)
µ µ µ
·
in off-shell momentum space. Condition (43) ensures that there is no admixture of spin 1 fermions.
2
Let the Fock ground state above the spurious tachyonic state of the R-sector be
φ >=αµ 0,p>u orDµ 0,p>u ; (44)
| o −1| µ −1| 1µ
u , u are spinors. This is possible because there is Ramond tachyon along with Neveu-Schwartz. In either case
µ 1µ
L φ >=0, so that this state is massless. Constraints L φ >=0 and F φ >=0 imply that
0 0 1 0 1 o
| | |
pµ u =0, γµ u =0 (45)
µ µ
It is known that Dµ γµ, the Dirac gamma matrices and αµ =pµ. So pµu =0 and γµu =0 as well. There is the
freedom to choose ao f∼our component spinor ψµ such that u 0=γ5u and 1µ 1µ
1µ µ
ψ =(1+γ )u (46)
µ 5 µ
By virtue of the above equations
γµψ =pµψ =0 (47)
µ µ
Further, F is essentially γ p. The wave equation is
0
·
F ψ γ pψ =0 (48)
0 µ µ
∼ ·
So ψ is the Rarita-Schwingerspin 3 vectorialspinor without spin 1 component. The generalrelativity equation can
µ 2 2
befoundbyinvokingtheprincipleofequivalence. Theeverycurvedspacetimepointonthemanifold,atangentspace
is constructed. The vector indices are a,b,.. The vierbein matrices like ea takes the x space objects to the tangent
µ
space and vice versa
eaeb =gµν, eaebµ =δab (49)
µ ν µ
For completely specifying the spin connections ωab, the covariant derivative D is such that
µ µ
D ea =0 (50)
µ λ
With defination of γ matrices
γaeµ =γµ, (51)
a
the covariant derivative of a spinor ψ is
D ψ =(∂ +ωabσ )ψ (52)
µ µ µ ab
σab is the antisymmetric product of two gamma matrices. Basically, the pricniple of equivalence states that in the
curved space time, the Rarita-Schwinger equation is
γ γ5D ψ ǫµνσρ =0 (53)
ν σ ρ
With e=√g, the invariant action in the curved space time is
i
S = d4x e ψ¯ γ γ5D ψ ǫµνσρ (54)
gravitino µ ν σ ρ
−2Z
5
For the graviton, the problem is more involved but interesting. Let Cij be the coefficients of the probabilities of
emission and absorption of the superpartners Ψµ in the ith and jth sites at the same instance. Then consider the
string state(dropping the creation operator suffix ‘-1/2’) above the spurious NS tachyonic state,
h†µν(p)= Cij bµbν +bνbµ 2ηµνbλbλ 0,p>. (55)
i j i j − i j |
Xi,j (cid:0) (cid:1)
The presence of NS tachyonhelp build the massless gravitonstate. This is symmetric and traceless. Further, L will
0
betakenasthefreeHamiltonianH intheinteractionrepresentation. Operatingonthisstate,onegetsL h†µν(p)=0.
o 0
So this is massless. Also p h†µν(p)=p h†µν(p)=0 if Cij =Cji. If Cij is symmetric, a little algebra shows that
µ ν
G12h†µν(p)=hG12, h†µν(p)i=0. (56)
Thus h† (p) satisfy all physical state conditions due to Virasoro algebraic relations. This should be graviton. The
µν
commutator is
[h (p), h (q)]=2π C 2δ4(p q). (57)
µν µν
| | −
To switch over quantum field theory, we define the gravitationalfield by space time Fourier transform of h (p)
µν
1 d3p
h (x)= h (p) eipx+h† (p) e−ipx , (58)
µν (2π)3 Z √2p µν µν
0 (cid:2) (cid:3)
with the commutator
1 d3p
[h (x), h (y)]= eip.(x−y) e−ip.(x−y) fµν,λσ, (59)
µν λσ (2π)3 Z 2p0 h − i
where
fµν,λσ =gµλgνσ+gνλgµσ gµνgσλ. (60)
−
The Feynman propagator is
1
µν,λσ(x y)=<0T h (x)h (y)0>= d4p µν,λσ(p) eip.(x−y), (61)
△ − | µν λσ | (2π)4 Z △
where
1 1
µν,λσ(p)= fµν,λσ . (62)
△ 2 p2 iǫ
−
Thisis thepropagatorofthe gravitoninthe interactionrepresentation. Butwedonotyethavethe generalrelativity.
I take the clue from the case of gravitino and follows it up. In four dimensions, the fourier transform is
d4p
h (x)= h (p) eipx (63)
µν Z (2π)4 µν
and the hermitian
d4p
h† (x)= h† (p) e−ipx (64)
µν Z (2π)4 µν
They are related by
h†µhλ =δ (65)
λ ν µν
by choosing the normalisation
<0,p0,q>=(2π)4δ4(p q) (66)
| −
6
and absorbing δ(4)(0) in C 2. In tangent space,
| |
h (x)=ea(x)eb(x)T (67)
µν µ ν ab
with h (x)=T =0. After some calculation, one gets
µµ ab
h (x)=eaeb(x)T , with h (x)=T =0. (68)
µν µ ν ab µµ bb
After some calculation, one gets by equation(68)
[D ,D ]hλ =eλ e RacTcb =0 (69)
µ λ ν a νb µλ
where the Riemannian Curvature tensor is
Rac =∂ ωac+∂ ωdc (µ λ) (70)
µλ µ λ µ λ − ↔
Inverting equation(69)
[D ,D ]hλ =R hλ =0 (71)
µ λ ν µλ ν
as the parallel transport relation. Using the equation (65), one gets
R =h†λR hσ =h†λ[D ,D ]hσ (72)
µν ν µσ λ ν µ σ λ
At this point, it is appropriate to quote the words of Misner, Thorne and Wheeler [16] ’The Laws of Physics written
in abstract geometric form, differ in no way whatsoever between curved space time and flat space time; this is
guaranteed by and in fact is a mere rewording of the equivalence principle. In equation (72), the right hand side
[D ,D ]=[∂ ,∂ ]=0 in flat space time. So anywhere in the universe with curved or flat space time
µ σ µ ν
R =0 (73)
µν
This is the Einstein equation of General Relativity deduced from string theory. The graviton action is, with R =
gµνR ,
µν
1
S = d4x e R (74)
gravitation
−2κZ
κisthegravitationalcouplingconstant,whosesquareisproportionaltotheNewton’sconstant. TheN=1supergravity
action in four dimensions is the sum of equations (43) and (74),
1 1
S = d4x e R+ψ¯ γ γ5D ψ ǫµνσρ (75)
−2Z (cid:18)κ2 µ ν σ ρ (cid:19)
The local supergravity transformations that leave it invariant are
1 i
δψ = D ǫ(x) and δ em = κ ǫ¯(x)γmψ (76)
µ κ µ µ −2 µ
I havesuccededin derivingthe 4d, N=1supergravitytheory asit developsfromarenormalisableopen26dbosonic
string. This has unlimited useful consequences in physics.
I thank Prof L. Maharana but for whose discussions and suggestions, this article could not have been completed.
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7
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