The explanation of the deformed Schild string

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by  Rie Kuriki Maegawa

The explanation of the deformed Schild string
Rie Kuriki Maegawa ∗
6 Department of Applied Physics
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Fukui University, Fukui 910-8507, Japan
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n February 7, 2008
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Abstract
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5 The author comments on [1]. One of the deformed actions can express
5 the Neveu-Schwarz-Ramond superstring under three gauge conditions. One
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of these depends on a matrix induced by the string coordinate.
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PACS code : 11.25.-w
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0 Key words : Schild string, Neveu-Schwarz-Ramond superstring
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h
t
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p
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h 1 The review of the Schild string and the defor-
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i mation
X
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We summarize the properties of the Schild string and the results in [1]. In next
section we will find the relation between the deformed Schild string in [1] and the
Neveu-Schwarz-Ramond superstring.
The Schild string is given by the following action
Γ
S = d2x {XA,XB}2 +∆e , (1)
Schild
Z (cid:20)e (cid:21)
where A = 0,1,···,D − 1 and XA are coordinates for a string propagating in D
space-time dimensions [2][3]. e, which is a scaler density, is an auxiliary field. Γ and
∆ are arbitrary constant. The bracket of {XA,XB} is defined by
{XA,XB} = ǫmn∂ XA∂ XB, (2)
m n
∗
Internet address: [email protected]
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where ǫmn is ǫmn = −ǫnm and ǫ01 = 1.
“What are the impediments in constructing new Schild type superstring action
from the Neveu-Schwarz-Ramond superstring ?” is the honest motivation of [1]. Su-
persymmetry can be made obvious by formulating the theory in the super-space.
There we deformed the actionof the Schild string. It is based on thesuper-space for-
mulation of the Neveu-Schwarz-Ramond superstring. The Neveu-Schwarz-Ramond
superstring is described in N=1 superconformal flat superspace as
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S = − d2xd2θED YADαY , (3)
NSR α A
4π Z
where D is the covariant derivative for the local supersymmetry and the index α
α
denote the chiralities of two dimensional spinor [5]. YA is the superfield
i
YA = XA +iθ¯ψA + θ¯θBA, (4)
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where BA is an auxiliary field which is introduced to close the gauge algebra of the
supersymmetry on the field ψA. E is the superdeterminant of the supervierbein.
The supervierbein has been found in [4],
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E a = e a +iθ¯γaχ + θ¯θe aA
m m m m
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1 1 1
E α = χ α + θµ(γ ) αω − θµ(γ ) αA
m m 5 µ m m µ
2 2 4
3i 1
− θ¯θχ αA− θ¯θ(γ )αβφ , (5)
m m β
16 4
E α = iθλ(γα) ,
µ λµ
i
E α = δ α − θ¯θδ αA,
µ µ µ
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where e a is the vierbein and χ α is the Rarita-Schwinger field. The supergauge
m m
transformation of the χ α contains an auxiliary field A. By integrating the super-
m
coordinates, we have the action of the Neveu-Schwarz-Ramond superstring,
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S = − d2xe(gmn∂ XA∂ X +iψ¯Aγm∂ ψ −BAB
NSR m n A m A A
2π Z
1
−iχ¯ γmγnψA∂ X + ψ¯ψχ¯ γnγmχ ), (6)
n m A m n
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where e = dete a. We proposed two actions by using the supervierbein. One of
m
these is expressed as
Γ
S = d2xd2θ deth +∆E , (7)
2 αβ
Z (cid:26)E (cid:27)
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where
C
h = D Y D Y . (8)
αβ α β C
The determinant of h is represented as a bracket
αβ
1 A B 2
deth = − Y ,Y , (9)
αβ
2 h i
where
YA,YB = ǫαβD YAD YB. (10)
α β
h i
Integrating θ we obtained
Γ 3i
S = d2x [ B BBψ¯ ψA −B2ψ¯ψ + BAψ¯ψχ¯ γmψ −2BAψ¯ γnψB∂ χ
2 A B m A A n B
Z e 2
+ ψ¯ψ 2 3iA+ 1gmnχ¯ χm + 1 ǫnmχ¯ γ5χ
n m n
(cid:16) (cid:17) (cid:18) 8 2 16e (cid:19)
5i
+ψ¯ψ( iψ¯Bγm∂ ψ +gnm∂ X ∂ XA − gnmχ¯ ψA∂ X
m B n A m n m A
2
i
+ ǫml∂ XAψ¯Aγ5χ )−ψ¯AψBgmn∂ X ∂ X
m l m A n B
2e
1 ∆ 1
{X ,XB}ψ¯ γ5ψB ]− d2x ǫmnχ¯ γ5χ +ieA . (11)
A A m n
e Z 2 (cid:18)2 (cid:19)
This is the second deformation in [1]. It is invariable under the volume-preserving
diffeomorphism in the superspace, in which the infinitesimal parameter ξM of the
super gauge transformation satisfies
∂ ξM = 0. (12)
M
By expanding (12) in terms of θ we found three differential equations,
i
∂ fm + χ¯ γmζ = 0, (13)
m m
2
i 1
i∂ (γmζ) + (γ γmζ) ω + χ ζ¯γnγmχ = 0, (14)
m α 5 α m mα n
2 4
∂ (ζ¯γnγmχ ) = 0, (15)
m n
where fm and ζµ are the parameters of the two dimensional general coordinate
transformation and the local supersymmetry, respectively. The solution of (15) is
ζ¯γnγmχ = C′m, (16)
n
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where C′m is an arbitrary constant. Notice that in the virtue of the two dimansional
gamma-matrix, γnγ γ = 0, (16) is invariable under the local fermionic transfor-
m n
mation which is defined by the shift of the Rarita-Shwinger field as
χ → γ η , (17)
m m W
where η , which is Majorana spinor, is the infinitesimal parameter. We easily found
W
δS 3iΓ ∆i
2 = (ψ¯ψ)2 − e = 0, (18)
δA 8e 2
in (11). In the next section we rearrange the terms in (11) by using (18).
2 The relation between the deformed Schild string
and the Neveu-Schwarz-Ramond superstring
The deformed action (11) is expressed as the e = 1 gauge fixed Neveu-Schwarz-
Ramond superstring action and additional terms as follows. The result is
S = ±kΓS +K +U, (19)
2 NSR(e=1) ±
where k is expressed by Γ and ∆ as
4∆
k2 ≡ . (20)
3Γ
We have introduced k as
4∆
(ψ¯ψ)2 = e2. (21)
3Γ
As the result of (18) (19) is explicitly written by
S = d2x[ gnm∂ XA∂ XA +iψ¯Bγm∂ ψ −B2
NSR(e=1) n m m B
Z
1
−iχ¯ γmγlψA∂ X + χ¯ γnγmχ ψ¯ψ ]. (22)
l m A m n
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∆ 3
K = d2x eχ¯ γnγmχ ∓ ikΓ∂ XAψ¯Aγmγnχ (23)
± n m m n
Z (cid:20) 2 2 (cid:21)
Γ 3i
U = d2x {B BBψ¯ ψA ± BAeχ¯ γmψ −2BAψ¯ γnψB∂ X
A B m A A n B
Z e 2
1
−ψ¯AψBgmn∂ X ∂ X + {X ,X }ψ¯ γ ψ }. (24)
m A n B e A B A 5 B
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In the calculation we have used
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γmγn = gmn − ǫmnγ5, (25)
e
where γm = γae m. If we define “a matrix like gamma-matrix” as follows,
a
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Γ Γ ≡ G − Υ γ5,
A B AB AB
e
G ≡ gmn∂ X ∂ X +B B , (26)
AB m A n B A B
Υ ≡ {X ,X },
AB A B
then we can express (24) as
Γ 3i
U = d2x {−ψ¯AΓ Γ ψB ± BAeχ¯ γmψA −2BAγnψB∂ X }. (27)
A B m n B
Z e 2
The construction in (26) is inspired by (25). We pay attention that the index n,m of
the gamma-matrix represents the two-dimensional surface of string world-sheet and
theindexA,B ofΓ denotesthespace-timedimensionwhereastringispropagating.
A
That we can neatly reexpress (11) by introducing a matrix defined in the space-time
is interesting point here.
Finally we find that if we impose gauge conditions
γmχ = 0, (28)
m
A
Γ ψ = 0, (29)
A
and
A
B = 0, (30)
then the deformed action (11) is equal to the e = 1 gauge fixed Neveu-Schwarz-
Ramond superstring action. Notice that the third gauge condition is not imposed
as the equation of motion BA derived from (11).
In the last section we sum up [6] in where the author treats similar gauge con-
ditions. Then we discuss our gauge conditions.
3 Discussion and summary
In [6] the author considers the super-Weyl invariant regularization of the two-
dimmensional supergravity. He uses the superfield formulation and he finally con-
structs the super-Liouville action which possesses the super-Weyl invariance (the
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Weyl invariance and the local fermionic symmetry) and the super-area preserving
diffeomorphism. To define which part of the two-dimensional superdiffeomorphism
group is compatible with the super-Weyl symmetry, he searchs the two dimensional
diffeomorphism and the local supersymmetry which preserve the gauge conditions
e = 1, and γmχ = 0.
m
Finally he found the constraints which the parameters of the supersummetry and
the two dimensional diffeomorphism should satisfy. In our case (19) becomes the
e = 1 gauge fixed Neveu-Schwarz-Ramond superstring action, which is invariable
under the area-preserving diffeomorphism, the local fermionic transformation and
the restricted supersymmetry, by virtue of (28), (29) and (30). Moreover we should
notice that (13) represents the area-preserving diffeomorphism
∂ fm = 0, (31)
m
under γmχ = 0. (31) is one of the constraints which he found in [6]. Another he
m
found is similar to (14).
In conclusion, the superfield formulation of superstring and supergravity is quite
interesting and usuful. In the superspace we can manifestly keep the supersymmme-
try. However it is not trivial to define the supersymmetry which is compatible with
other symmetires. We expect that the direction of our Schild string deformation
which preserves the global structure of the bracket (2) would possess the dynamics
of full superstring.
The author is grateful to the applied physics department of Fukui university for
its gracious hospitality.
References
[1] R.Kuriki, S.Ogushi and A. Sugamoto, Modern Physics Letters A, Vol.14,
No.17(1999)1123-1131, Deformation of Schild String
[2] A.Schild, Phys, Rev.D16,1722(1997)
[3] T.Yoneya, Prog.Theor.Phys.97,949(1997)
[4] P.S.Howe, J.Phys.A12,393(1978)
[5] P.S.Howe, Phys.Lett.B70, 453(1977)
[6] D.R.Karakhanyan, Phy.Lett.B365(1996)56
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