We consider a space-time invariant duality symmetric action for a free
Maxwell field and an $SL(2,{\bf R})\times SO(6,22)$ invariant effective action
describing a low-energy bosonic sector of the heterotic string compactified on
a six-dimensional torus. The manifest Lorentz and general coordinate invariant
formulation of the models is achieved by coupling dual gauge fields to an
auxiliary vector field from an axionic sector of the theory.Comment: 9 pages, LaTeX, no figures, references added, misprints eliminate

Pasti, Paolo

Sorokin, Dmitrij

Tonin, Mario

English

arXiv

We consider a space--time invariant duality symmetric action for the free Maxwell field and an SL(2,{\bf R})\times SO(6,22) invariant effective action describing a low--energy bosonic sector of the heterotic string compactified on a six--dimensional torus. The manifest Lorentz and general coordinate invariant formulation of the models is achieved by coupling dual gauge fields to an auxiliary vector field from an axionic sector of the theory

Pasti, P

Sorokin, D P

Tonin, M

CERN Document Server

Po
st
Sc
rip
t〉  p
ro
ce
ss
ed
 b
y 
th
e 
SL
A
C/
D
ES
Y
 L
ib
ra
rie
s o
n 
19
 Ju
n 
19
95
.
H
EP
-T
H
-9
50
61
09
Preprint DFPD 95/TH/37
hep-th/9506xxx
June 1995
Duality symmetric actions with manifest space{time
symmetries

Paolo Pasti, Dmitrij Sorokin
y
and Mario Tonin
z
Universita Degli Studi Di Padova, Dipartimento Di Fisica \Galileo Galilei"
ed INFN, Sezione Di Padova, Via F. Marzolo, 8, 35131 Padova, Italia
Abstract
We consider a space{time invariant duality symmetric action for the free Maxwell eld and
an SL(2;R)SO(6; 22) invariant eective action describing a low{energy bosonic sector
of the heterotic string compactied on a six{dimensional torus. The manifest Lorentz
and general coordinate invariant formulation of the models is achieved by coupling dual
gauge elds to an auxiliary vector eld from an axionic sector of the theory.
PACS numbers: 11.15-q, 11.17+y

This work was carried out as part of the European CommunityProgramme \Gauge Theories, Applied
Supersymmetry and QuantumGravity" under contract SC1{CT92{D789, and supported in part by M.P.I.
y
on leave from Kharkov Institute of Physics and Technology, Kharkov, 310108, Ukraine.
e{mail: sorokin@pd.infn.it
z
e{mail: tonin@pd.infn.it
1
1 Introduction
The understanding of the important role played by duality in the theory of strings, su-
pergravity and (super){Yang{Mills theories has allowed one to make new insight into the
structure of these theories and to nd deep relationship between their dierent (dual)
versions. If target space duality (T{duality) and S{duality (which is the generalization
of the electric{magnetic duality) are the exact symmetries of string theory [1, 2, 3], it
is natural to assume that there should be a version of the theory where these duality
symmetries are manifest.
A T{duality symmetric string action was proposed by Tseitlin [4] and generalized to
the case of the heterotic string by Schwarz and Sen [5] This required modication of space{
time transformations of elds. Note, however, that eective supersymmetric eld actions
which describe the low{energy behavior of superstrings do have a global non{compact
symmetry related to T{duality, while S{duality is only a symmetry of the equations of
motion [6]{[9].
In [9] Schwarz and Sen proposed models for describing antisymmetric gauge elds in
D{dimensional space{time, where S{duality symmetry was lifted to the level of action,
their results being the generalization of earlier work by Floreanini and Jackiw [10] and
Henneaux and Teitelboim [11] who constructed actions for describing self{dual tensor
elds in (4p+2){dimensional space{time (p=0,1,..). The general feature of the models
[4, 5, 9, 10, 11] is that, due to the explicit xing of the time direction, they loose manifest
Lorentz and general coordinate invariance, which, however, are replaced by some modied
transformations. This is an example how duality eects the symmetry structure of the
theory.
Anyway, one may try to look for a formulation of duality symmetric actions in which
conventional space{time symmetries are restored. An attempt to do this for a duality
symmetric version of Maxwell theory [9]
1
was undertaken in [13]. Having been inspired by
this paper we proposed in [14] a manifestly space{time and duality symmetric formulation
of (supersymmetric) Maxwell theory by enlarging the Schwarz{Sen model [9] with an
auxiliary vector eld u
m
(x) and an antisymmetric tensor eld 
mn
(x) (m=0,1,2,3) in
such a way that upon solving for equations of motions and gauge xing additional local
symmetries associated with the auxiliary elds the model [14] could be reduced to that
of Schwarz and Sen. The physical nature of u
m
(x) and 
mn
(x) was supposed to be a relic
of gravitational vielbein and an axion{like potential, respectively.
In the present paper we further develop the duality symmetric model of [14]. In
particular, we will show that the model indeed has a local symmetry (anticipated in
[14]) allowing one to choose u
m
(x) to be a unit{norm time{like vector and, thus, to
demonstrate that the duality symmetric models [9], with their non{conventional space{
1
Earlier an alternative version of duality invariant Maxwell action was considered in [12]
2
time symmetries, correspond to a denite gauge choice for auxiliary elds in corresponding
duality symmetricmodels with ordinary Lorentz and general coordinate invariance. Using
the example of a model for two abelian gauge elds in D=1+3, we shall show that duality
between these two elds arises due to their specic coupling to a pseudoscalar sector of
the theory through the eld u
m
(x), and the latter can be gauge xed to a constant time{
like vector by use of a local counterpart of a Peccei{Quinn symmetry of axion models.
Thus, u
m
(x) can originate, in fact, from the pseudoscalar sector of the theory. In Section
3 we will present a space{time and SL(2;R)  O(6; 22) invariant eective action which
describes a low{energy limit of a toroidally compactied heterotic string.
2 Duality symmetric Maxwell action
Let us start with an action describing a free Maxwell eld A
m
(x) and a pseudoscalar
(`axion') eld a(x) in D=1+3 Minkovski space:
S =
Z
d
4
x

 
1
4
F
mn
F
mn
 
1
2
(@
m
a(x)  u
m
(x))(@
m
a(x)  u
m
(x))  
pqmn
u
p
@
q

mn

; (1)
where the rst term is the ordinary Maxwell Lagrangian with F
mn
= @
m
A
n
(x)  @
n
A
m
(x)
and the last two terms form the Lagrangian which, upon solving the equations of motion
for u
m
(x) and using a local symmetry of (1) under transformations
a(x) = '(x); u
m
(x) = @
m
'(x); (2)
produces (see, for the details [15, 16, 17, 18, 19] and references therein) the free Lagrangian
for the scalar eld a(x)
L =  
1
2
@
m
a(x)@
m
a(x); (3)
or its dual
L =
1
3!
@
[m

np]
@
[m

np]
; (4)
with the duality relation between a(x) and the antisymmetric eld 
np
to be
@
l
a(x) = 
lmnp
@
m

np
: (5)
The role of u
m
(x) in (1) is to be the gauge eld of the symmetry (2) and to ensure (5) on
the mass shell.
Besides (2) the action (1) is invariant under the abelian gauge transformations of
A
m
(x) and 
mn
(x):
A
m
(x) = @
m
b(x); 
mn
= @
[m
b
n]
(x): (6)
Note that without violating local symmetry (2) one can couple elds from the pseu-
doscalar sector of (1) to the gauge eld in an axion{like fashion:
S
int
=  
Z
d
4
x(@
m
a  u
m
)
mnlp
A
n
F
lp
: (7)
3
The sum of (1) and (7) is explicitly invariant under (2). It is also invariant under the gauge
transformations (6) if one requires that the variation of 
mn
acquires the contribution



mn
= (x)F
mn
. Eliminating either u
m
and 
mn
or u
m
and a(x) by use of equations of
motions and local symmetries one gets the dual versions of the axion theory [16, 17, 18, 19].
Now the question arises whether it is possible to replace (1) with an action which
would be duality symmetric in the Maxwell eld sector, still possess ordinary Lorentz
invariance and be equivalent (at least classically) to the action (1). The answer turns out
to be positive [14].
From [9] we learn that for making the electric{magnetic duality manifest at the level
of action one has to double the number of abelian elds (introducing A

m
(x) (=1,2)) and
construct a duality symmetric action in such a way that equations of motion obtained
from this action lead to the vanishing of the self{dual tensor
F

mn
= L

F

mn
  F

mn
=
1
2

mnpq
L

F
pq
; (8)
where L

=  L

(L
12
= 1), F

mn
=
1
2

mnlp
F
lp
.
When
F

mn
= 0 (9)
one of the abelian elds becomes completely determined through another one, and for
the latter we get the free Maxwell equations by dierentiating (9). Then the duality (9)
between the two gauge elds reduces to the duality between the electric and magnetic
strength of the gauge eld which has been chosen to be independent.
The duality symmetric action proposed in [9], which gives (9), has the following form:
S =  
1
2
Z
d
4
x(B
i
L

E

i
+B
i
B

i
); (10)
where
E

i
= F

0i
= @
0
A

i
  @
i
A

0
; B
i
=
1
2
"
ijk
F

jk
= "
ijk
@
j
A

k
; (11)
and i; j; k = 1; 2; 3 are spacial indices. The action (10) is invariant under modied space{
time transformations of A

i
(in the gauge A

0
= 0):
A

i
= x
0
v
k
@
k
A

i
+ v
k
x
k
@
0
A

i
+ v
k
x
k
L

F

0i
; (12)
where the rst two terms describe the ordinary Lorentz boosts along a constant velocity
v
i
and the third term vanishes on the mass shell (9).
The model constructed this way [9] is classically and quantum mechanically [20] equiv-
alent to the free Maxwell theory.
The covariantization of (10) is achieved by coupling the self{dual tensor (8) to the
auxiliary eld u
m
(x) from the pseudoscalar part of the Lagrangian (1) as follows:
S =
R
d
4
x( 
1
8
F

mn
F
mn
+
1
4( u
l
u
l
)
u
m
F

mn
F
np
u
p
 
1
2
(@
m
a  u
m
)(@
m
a  u
m
)  
mnpq
u
m
@
n

pq
): (13)
4
Action (13) diers from that considered in [14] at the following point. In [14] u
m
(x)
was required to have the negative unit norm
u
2
 u
m
u
m
=  u
0
u
0
+ u
i
u
i
=  1 (14)
and played the role of a component of a local Lorentz frame. In (13) we weakened the
normalization condition by introducing the norm of u
m
(x) only into the term containing
F

mn
. We shall demonstrate the relationship between the two versions of the model later
on.
The necessity to introduce the norm of u
m
(x) into the F

mn
{term is dictated by the
requirement to preserve the local symmetry (2). The action (13) is invariant under the
transformations (2) provided A

m
(x) and 
mn
(x) are transformed as follows
A

m
=
'(x)
u
2
L

F

mn
u
n
; 
mn
=
'(x)
(u
2
)
2
F
r
m
u
r
F
s
n
u
s
L

: (15)
Then, since the solution to the equation of motion of 
mn
, obtained from (13), is
u
m
(x) = @
m
'^(x); (16)
where '^(x) is a scalar function, we can use the transformations (2) to put u
m
= 
0
m
.
2
In
this gauge the action (13) reduces to (10), and the transformation of A

m
in (15) (with
' = x
i
v
i
) is combined with the corresponding Lorentz transformation producing (12).
We see that u
m
(x) plays a double role. From the one hand side it is the gauge eld of
local Peccei{Quinn symmetry and from the other hand it corresponds to a component of
a local Lorentz frame.
The action (13) has another local symmetry [14] (which generalizes that of (10) [9])
under the following transformations of A

m
and 
mn
:
A

m
! A

m
+ u
m
'

(x); (17)

mn
! 
mn
 
'

u
2
F

mp
u
p
u
n
+
'

u
2
F

np
u
p
u
m
:
This symmetry allows one to reduce the general solution of the equations of motion
of A

m

lmnp
@
m
(u
n
F

pr
u
r
) = 0 (18)
to eq. (9) (see [9, 14] for the details). In the gauge where F

mn
= 0, the equations of
motion of u
m
lead to the same expressions for a(x) and 
mn
that follow from (1).
To transit from (13) to (1) we must solve eqs. (18) for one of the gauge elds in terms
of another one and substitute the solution back to (13) [9, 14]. If we skip the kinetic term
for a(x) in (1) and (13), 
mn
becomes a pure gauge eld as well, and the three actions (1),
2
To escape singularities we should require the norm of u
m
to be nonzero.
5
(10) and (13) become classically equivalent and describe dynamics of a single Maxwell
eld.
Now we shall demonstrate how the action (13) is related to the version considered in
[14]. There the vector eld u
m
(in (13)) was subjected to the normalization condition
(14) (we shall denote the normalized vector as u^
m
). This caused a problem of establishing
the explicit invariance of the model under the transformations (2), (15). To ensure this
invariance one should couple (13) with the normalized u^
m
[14] to scale invariant gravity
[21]. Then the action takes the form (we skip, for simplicity, the kinetic term of a(x)):
S =
R
d
4
x
p
 g( 
1
8
F

mn
F
mn
+
1
4
u^
m
F

mn
F
np
u^
p
 
1
p
 g

pqmn
u^
p
@
q

mn
+R
2
+ 6@
m
@
m
); (19)
where g
mn
(x) is a metric, g = det g
mn
, R(x) is the scalar curvature and (x) is a conformal
scalar eld. The action (19) is invariant under (2), (15) provided g
mn
(x) and (x) subject
to the following scale transformations:
g
mn
(x) = (u
m
@
m
')
2
g
mn
(x); (x) =  (u
m
@
m
')(x): (20)
Making redenition of the metric and u^
m
as follows
g
mn
(x) ! 
2
g
mn
; u^
m
(x) !
1

u^
m
; (21)
(which preserves the condition (14)
3
) we can rewrite (20) in the form which describes
coupling the model to the Einstein gravity:
S =
Z
d
4
x
p
g( 
1
8
F

mn
F
mn
+
1
4
u^
m
F

mn
F
np
u^
p
 
1
p
g

pqmn
u^
p
@
q

mn
+R): (22)
The only place where (x) is present in (22) is the term 
pqmn
u^
p
@
q

mn
. Then, putting
(x)u^
m
(x) = u
m
(x), making use of (14) and taking the at limit we get the action (13).
This concludes the establishment of the links between dierent versions of the duality
symmetric formulation of free Maxwell theory.
3 Low{energy eective action in string theory with
manifest SL(2;R)O(6; 22) and space{time symme-
try
In this section we present manifest space{time invariant generalization of the SL(2;R)
O(6; 22) invariant low-energy eective action [9] describing heterotic string theory com-
pactied on a six{dimensional torus [8, 22]. To do this we should introduce 28 dual
3
Note that one can introduce (14) into (19) with a Lagrange multiplier which transforms under the
scale transformations in an appropriate way
6
pairs [23] A
;a
m
(a=1,...,28) of abelian gauge elds and couple them to scalar elds in an
SL(2;R)O(6; 22) covariant way. This is achieved by modifying the self{dual tensor (8)
as follows:
F
;a
mn
= L

L
ab
F
;b
mn
  (L
T
ML)

(L
T
ML)
ab
F
;b
mn

p
 g
2
( L
T
M)

(L
T
M)
ab

mnpq
F
;b pq
;
(23)
where 2 2 matrix valued scalar eld
M =
1

2
(x)
 
1 
1
(x)

1
(x) 
2
1
+ 
2
2
!
; (24)
satises the following conditions:
M
T
=M; MLM
T
= L: (25)
M, L and A
;a
m
transform under the global SL(2;R) transformations ! as follows:
M ! !
T
M! !L!
T
= L A
m
= !
T
A
m
: (26)
A 28  28 matrix{valued scalar eld M(x) satises the conditions
M
T
=M; M
T
LM = L; (27)
where
L =
0
B
B
@
0 I
6
0
I
6
0 0
0 0  I
16
1
C
C
A
: (28)
Under a global O(6; 22) rotation M , L and A
;a
m
transform as follows:
M ! 

T
M
; 

T
L
 = L A
m
! 

T
A
m
: (29)
(see ref. [9] for the details). The transformation law of the self{dual tensor (23) under
SL(2;R)O(6; 22) is
F
mn
! !
 1


 1
F
mn
: (30)
Using the properties of the elds described above one may convince oneself that the
following general coordinate invariant action has SL(2;R)O(6; 22) symmetry:
S =
Z
d
4
x
p
 g( 
1
8
F
;a
mn
(L
T
ML)

(L
T
ML)
ab
F
;b mn
+
1
4u
2
u
m
F
;a
mn
M

M
ab
F
;b np
u
p
 
1
4
g
mn
tr(@
m
ML@
n
ML) +
1
8
g
mn
Tr(@
m
ML@
n
ML) 
1
p
g

pqmn
u
p
@
q

mn
+R): (31)
It can be rewritten in a simpler form
S =
Z
d
4
x
p
 g(
1
2u
2
u
m
F
;a
mn
F
;a np
u
p
 
1
p
 g

pqmn
u
p
@
q

mn
+R
7
 1
4
g
mn
tr(@
m
ML@
n
ML) +
1
8
g
mn
Tr(@
m
ML@
n
ML)) (32)
which, upon xing the gauge u
m
=
1
p
 g
00

0
m
, 
mn
= 0, directly reduces to the Schwarz{
Sen action [9].
Note that we did not add the kinetic term of a(x) (1), (13) to the actions (31) and (32).
This is because we would like to identify a(x) with 
1
(x) which has already entered the
actions (31) and (32) as part of the dilaton{axion matrixM (24). The coupling ofM to
the gauge elds ensures the manifest SL(2;R) symmetry but brakes local transformations
of 
1
(x) (or a(x) in (2)) down to the global Peccei{Quinn shifts which form a subgroup
of the global SL(2;R) (26)
4
. Such a coupling violates duality between the pseudoscalar
eld and 
mn
in favor of the former and makes 
mn
an auxiliary eld, which can be
eliminated from (31), (32) by solving for the equations of motion of u
m
and 
mn
, and
substituting u
m
(x) back into (31), (32) in the form u
m
= @
m
'^(x) (16).
It is tempting to look for a version of the low{energy eective string action which
would be manifestly duality symmetric not only in the gauge sector but in the axion
sector (1), (7) as well. Might it imply a localization of the SL(2;R) ?
4 Conclusion and discussion
We have constructed the space{time invariant duality symmetric action for the free
Maxwell theory and the SL(2;R)  O(6; 22) invariant eective action describing low{
energy bosonic sector of the heterotic string compactied on a 6{dimensional torus. This
has been achieved by coupling the self{dual stress tensor, constructed out of the dual
gauge elds, to the auxiliary vector eld from the axionic sector of the theory.
One can add to the bosonic action (13) the kinetic term for neutral fermions:
S
F
=  i
Z
d
4
x
 
m
@
m
 : (33)
Then the full action becomes supersymmetric [9, 14] under the following transformations
with odd constant parameters 

= i
5
L



:
A

m
= i 
m


;
 =
1
8
F
mn

m

n


+
1
4
L

u
p
F
pm
u
n

m

n


; (34)
all other elds being inert under the supersymmetry transformations.
We see that the supersymmetric transformation law for the fermion  (x) (34) is non{
conventional and reduces to the ordinary one only on the mass shell (8). This reminds
the problem with the Lorentz transformations (12) which we have just solved. Using the
4
Note that the action (31), (32) is still invariant under the local transformations of u
m
(2), A
m
and

mn
(15)
8
same reasoning as lead us to introducing u
m
(x) one may try to nd a superpartner of
u
m
(x) whose presence in the theory gives rise to a local fermionic symmetry (being a
counterpart of (2, 15)) which involves  (x) and leads to (34) upon gauge xing the local
fermionic symmetry. This construction may arise from coupling the models discussed
above to supergravity, from which, in fact they originate.
The covariantization procedure for duality symmetric actions [10, 11, 4, 9] proposed in
[14] and developed herein is applicable to abelian tensor elds in space{time dimensions
other than D = 4 and may turn out to be useful for nding new dual versions of D = 10
supergravity [9].
Acknowledgements. The authors are grateful to I. Bandos for constant interest to
this work and fruitful comments. D.S. would also like to thank A. Khoudeir, F. Quevedo
and A. Sen for useful discussion.
References
[1] K. Narain, Phys. Lett. B169, 41 (1986); K. Narain, H. Sarmadi and E. Witten, Nucl.
Phys. B279, 369 (1987).
[2] A. Font, L. Iba~nez, D. Lust and F. Quevedo, Phys. Lett. B249, 35 (1990); S. J. Rey,
Phys. Rev. D43, 526 (1991).
[3] A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A6, 2677 (1991).
[4] A. Tseytlin, Phys. Lett. B242, 163 (1990); Nucl. Phys. B350, 395 (1991).
[5] J. H. Schwarz and A. Sen, Phys. Lett. B312, 105 (1993).
[6] M. de Roo, Nucl. Phys. B255, 515 (1985).
[7] A. Sen, Nucl. Phys. B404, 109 (1993).
[8] J. Maharana and J. H. Schwarz, Nucl. Phys. B390, 3 (1993)
[9] J. H. Schwarz and A. Sen, Nucl. Phys. B411, 35 (1994).
[10] R. Floreanini and R. Jackiw, Phys. Rev. Lett. 59, 1873 (1987).
[11] M. Henneaux and C. Teitelboim, in Proc. Quantum mechanics of of fundamental
systems 2, Santiago, 1987, p. 79; Phys. Lett. B206, 650 (1988).
[12] S. Deser and C. Teitelboim, Phys. Rev. D13, 1592 (1976); S. Deser, J. Phys. A:
Math. Gen. 15, 1053 (1982).
[13] A. Khoudeir and N. Pantoja, hep{th/9411235.
9
[14] P. Pasti, D. Sorokin and M. Tonin, Phys. Lett. B352, 59 (1995); hep-th/9503182.
[15] V. I. Ogievetsky and I. V. Polubarinov, Soviet. J. Nucl. Phys. 4, 156 (1967); M. Kalb
and P. Ramond, Phys. Rev. D9 2273 (1974).
[16] M. J. Bowick, S. B. Giddings, J. A. Harvey, G. T. Horowitz and A. Strominger,
Phys. Rev. Lett. 61, 2823 (1988); S Coleman, J. Preskill and F. Wilczek, Nucl. Phys.
B378, 175 (1992).
[17] H. Nicolai and P. K. Townsend, Phys. Lett. B98, 257 (1981).
[18] R. Kallosh, A. Linde, D. Linde and L. Susskind, Preprint SU-ITP-95-2, hep{
th/9502069.
[19] F. Quevedo, Preprint NEIP95{007, hep{th/9506081.
[20] I. Martin and A. Restuccia, Phys. Lett. B323, 311 (1994).
[21] S. Deser, Ann. Phys. 59, 1689 (1970).
[22] A. Sen, Phys. Lett. B303, 22 (1993).
[23] R. Kallosh and T. Ortin, Phys. Rev. D48, 742 (1993).
10


Duality Symmetric Actions with Manifest Space-Time Symmetries

Abstract

Similar works

Full text

Available Versions

CERN Document Server