United Nations Educational Scientific and Cultural Organization
and
International Atomic Energy Agency


THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS





Supersymmetric Quantum Corrections
and
Poisson-Lie T-Duality





F. Assaoui* and T. Lhallabi*
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.





Abstract
The quantum actions of the (4,4) supersymmetric non-linear sigma model and its dual in
the Abelian case are constructed by using the background superfield method. The propagators
of the quantum superfield and its dual and the gauge fixing actions of the original and dual
(4,4) supersymmetric sigma models are determined. On the other hand, the BRST
transformations are used to obtain the quantum dual action of the (4,4) supersymmetric non-
arXiv: 28 Jul 2000 linear sigma model in the sense of Poisson-Lie T-duality.
*






* Permanent address: Section of High Energy Physics, H. E. P. L, University MohammedV, Scientific Fac,
Rabat, Morocco.
E-mail: Lhallabi@fsr.ac.ma


1 - Introduction


Various T-duality transformations [1] connecting two seemingly different sigma models

or strings backgrounds, have aroused a considerable amount of interest. The non-Abelian T-

duality transformation of the isometric sigma model on a group manifold G gives non-

isometric sigma model on its lie algebra [2,3]. As a result, it was not known how to perform

the inverse duality transformation to get back to the original model. In order to solve this

problem C. Klimcik and P. Severa [4] proposed a generalization of the Abelian and traditional

non-Abelian dualities called Poisson-Lie T-duality. The main idea of this approach is to

replace the requirement of isometry of sigma model with respect to some group by a weaker

condition, which is the Poisson-Lie symmetry of the theory. This generalized duality is

associated with two groups forming a Drinfeld double [5] and the duality transformation

exchanges their roles. This approach has received further developments in a serie of works

[6]. Furthermore, the Abelian and non-Abelian T-duality of the two-dimensional (4,4)

supersymmetric sigma model is treated classically [7] and its Poisson-Lie T-duality is

discussed [8].

On the other hand, the quantum equivalences of Poisson Lie T-duality related sigma

models were studied perturbatively in [9] and [10]. It was shown that Poisson-Lie

dualizability is compatible with renormalization at 1-loop. In the present work we start by

studying the quantization of the dually related (4,4) supersymmetric sigma models in the

Abelian case, by using the covariant background superfield formalism [11,12]. Thereafter, we

discuss the quantum equivalence of (4,4) supersymmetric sigma models related by Poisson-

Lie T-duality.

The organization of this paper is as follows. In section 2, we construct the quantum

actions of the (4,4) supersymmetric non-linear sigma model and its dual in the Abelian case

by using the background superfield method. The propagators of the quantum superfield and

its dual are obtained. Furthermore, the BRST transformations of the quantum and background

superfields and their duals are given. This leads to the gauge fixing actions of the original and

dual (4,4) supersymmetric sigma models. In section 3, we study the quantum Poisson-Lie T-

duality of the (4,4) supersymmetric non-linear sigma model by using the BRST

transformations associated with the transformation of the group elements. The generalized

Cartan-Maurer equation is written in terms of the quantum supercurrents and the Poisson-Lie



1


symmetry conditions are given. Finally, in section 4 we make concluding remarks and discuss

our results.


2 - Quantization of the Dually Related (4,4) Supersymmetric Sigma Models in the
Abelian Case


We consider a supermanifold M with metric G a = 1, . . ., d and antisymmetric tensor
ab


B which determines a supersymmetric generalized Wess-Zumino term [13]. The action of
ab


the two- dimensional (4,4) supersymmetric sigma model [7] is given by



S (2.1)
( 4,4) = d (
{G B D D
ab + ab ) ++ a --b}

where d d
= 2 d
y 2 +
d2 -
du
+ - is the measure of the twodimensional (4,4) analytic subspace [14]


and is a scalar superfield satisfying the analycity conditions [7]

+ a - a
D- = 0 = D+ (2.2)

with


+ -
D = , D = 
- D
- + + , and  are the harmonic derivatives namely

+ -


++ ++ + +
D = -
2
+ + --
(2.3)
-- -- - -
D = -
2
- - ++


with




= U
U

In order to set up a manifestly supersymmetric covariant background field formalism for the


(4,4) supersymmetric non-linear sigma model based on the parallel transport equation [11,12],


we introduce the unconstrained prepotential a
(t) (0 t )
1 defined along the non-


geodesic curves a
(y, +
, -
,U )
+ - [15] as:


2 2
a + - a
= D D
- + (2.4)





2


which connects the background superfield with the total superfield


a a a
= + (2.5)
cl



with


+ 2 - 2 a a
D D
- + ( )
0 =
cl (2.6)
+ 2 - 2 a a a
D D
- + )
1
( = +
cl




The non analytic superfield a
satisfies the equation of parallel transport namely [16]:


2
d a
d b

a
d c
+ ( (2.7)
bc ) = 0
2
dt dt dt

where the solution

a a a 1 2 2
= D D (2.8)
cl + - a (
bc ) b + - c
cl - + +...
2

is obtained by using the initial conditions

a a
( )
0 = .
cl

a
d (2.9)
a
=
dt t=0

a
is the background prepotential and a
is the quantum superfield which is an unconstrained

superfield. Consequently

a a 1 2 2
= - a ( D D (2.10.a)
bc ) b + - c
cl - + +...
2

with

2 2
a + - a
= D D .
- + (2.10.b)



In order to generate a covariant set of background superfield vertices in powers of the

quantum superfield a
we insert the expansion (2.6) and (2.10) in the action (2.1). By using

the developments of G and B [11] which are available in any coordinate system, the
ab ab


background superfield expansion of the two dimensional (4,4) supersymmetric sigma model

is given by




3


S 2 2
S () - S ( ) (4,4) D D
cl - ( ) + - c
c cl - + =
( 4,4) (4,4)

d F D D D D
T { 2 2
(
ab ) ++ a -- + -
b
cl - +

1 2 2 2 2 2 2
- + - d + - e ++ + - a -- b
R D D D D D D D D
abde - + - + - +
3
1 2 2
- d + - e ++ a b
R D D D D
abde - + --
cl
3 cl

1 2 2 2 2
- + - d + - e ++ a -- b
R D D D D D D
abde - + - + cl +.. }.
3

(2.11)

with 2 + 2 - G B
d = d d d , F
- + = G B
+ and R = R + R (2.12)
T ab ab ab abde abde abde

which is the curvature tensor of the supermanifold M. - b
D
- is the supercovariant derivative

defined by:

-- b -- b b n -- m
D = D + D (2.13)
nm cl


From the action (2.11) we see that the coefficients at all orders in the quantum non analytic

superfield a
are constructed from geometrical tensors that are functions of the background

analytic superfield . However, the formulation of manifestly supersymmetric Feynman
cl


rules in terms of the analytic superfield a
is not quite suitable in diagrammatic calculations.

For this reason we introduce n-bein i
e ( ) where n is the dimension of the supermanifold:
a cl


i
e ( ) e ( ) = F ( ) (2.14)
a cl ib cl ab cl


by defining

i i a
= e ( ) (2.15)
a cl


the kinetic term in (2.11) becomes

++ i 2
-- + - i
d 2
D D D
D - + (2.16)
T


with

-- i -- i --i j
D = D + V
j (2.17)
--i i -- m
V ( ) V D
j cl mj cl


where - i
V - is the superconnection on the supermanifold. Consequently the propagator of the
j


quantum superfield i
is as follows:


4


G(z , u , z , u ) i
= (z ,u ) j
(z ,u )
1 1 2 2 1 1 2 2 (2.18)
2 2
ij
= - (D++ D--D+ D- ) 1-
- + (z - z ) (u ,u )
1 1 1 1 1 2 1 2


with (z - z ) is the full (4,4) harmonic superspace function and (u ,u ) is the harmonic
1 2 1 2


function [17]. It is clear that the Feynman rules constructed from the expanded action (2.11)

lead to manifestly covariant quantum corrections written as integrals on the full harmonic


(4,4) superspace. The divergences can be removed by counterterms namely:


S
(4,4)
= d L

 (2.19)
T



where the lagrangian counterterm L
is a scalar function of the background superfield of

dimension (-2) and zero Lorentz and Cartan-Weyl charges.

The above procedure is adequate to establish a covariant background superfield


expansion but the use of the unconstrained superfield a
means that the action (2.11) has a

quantum gauge invariance which must be gauge fixed. In fact, the gauge transformations of


quantum and background superfields are given by


a
= a

(2.20)
cl = 0

where a
is a superparameter. The covariant gauge fixing term is obtained by introducing a

BRST operator [14] namely


a
s = a
C , scl = 0 (2.21)
a
sC = 0


with a
C an anticommuting ghost superfield associated to the superparameter a
.

Furthermore, we introduce an antighost a
C' and a commuting a
b superfields such that:



sC'a = a
b
(2.22)
a
sb = 0 ,




5


which allow to obtain a (4,4) supersymmetric gauge fixing term with ghost number zero. By


using dimensional arguments and postulating BRST invariance of the action
S , the s-
(4,4)



invariant gauge fixing action is given by

~
S
( 4,4) = d s F ( ) C
( ' D++ D-- +C'
-- ++ )
GF T { a b a b
ab cl cl (2.23)
++ a b ++ -- e -- d
+ R D C' D D D
abed cl cl }

where and are coupling constants. The use of (2.21), (2.22) and (2.14) in the expansion of

(2.23) leads to

S 
( 4,4) = d b D++ D-- + C' D++D--C + b
++ --
GF T { i i i
i i cl i

++ a i ++ -- j -- d
+ R D b D D D (2.24)
aijd cl cl

++ a i ++ -- j -- d
+ R D C' D D C D
aijd cl cl }

Eliminating the auxiliary superfield b by using its equation of motion we obtain the following


constraint

++ --
D D R D D D D (2.25)
i + a j d
++ --cli + ++
aijd ++ -- --
cl cl = 0

Consequently the gauge fixing action (2.24) becomes:

S  (2.26)
( 4,4) = d C' D ++ D --C + R D ++ C D ++ D --C D --
'
GF T { i a i j d
i aijd cl cl }

Let us now give the quantization of the dual (4,4) supersymmetric non linear sigma model


which is given by [7]:

~ ~ ~
S  (2.27)
( 4,4) = d { ++ a --
G
( B ) D Y D Y
ab + b
ab }
with

a
Y = {~ i
, } and
~ - ~ - ~
G = G 1 G = G 1B G = G 1-B
00 00 0i 00 0i i0 00 0i
~ ~ - ~
B = 0 B = G G
1 B = - G 1-G (2.28)
00 0i 00 0i i0 00 0i
~
G = G - G 1- G G + B B
ij ij 00 [ 0i j 0 0i j 0 ]
~
B = B + G 1- G B + G B
ij ij 00 [ 0i j 0 0 j 0i ]





6


As previously, the background superfield expansion of the dual (4,4) supersymmetric sigma

~
model action is given by using the developments of the dual metric tensor G and the dual
ab

~
antisymmetric tensor B which are available in any coordinate system [11]
ab




~
~ ~ S 2 2
( 4,4) + - ~
S (Y ) - S (Y ) Y D D
cl - ( ) b
b cl - + =
( 4,4) (4,4) Y
d F Y D D D D
T {~ ++ ~ 2 2
a -- + - ~
( ) b
ab cl - +

1 ~ 2 2
+ - ~ 2 2
d + - ~ 2 2
e ++ + - ~ a -- ~
- b
R D D D D D D D D
abde - + - + - +
3
1 ~ ~ 2 2
d + - ~
- e ++ a -- b
R D D D Y D Y
abde - +
3 cl cl

1 ~ 2 2
+ - ~ 2 2
d + - ~e ++ ~
- a -- b
R D D D D D D Y
abde - + - + cl + .. }
.
3

(2.29)

with

~ ~ ~
F = G + B (2.30)
ab ab ab

~ ~
~
G B
R = R + R (2.31)
abde abde abde


and

~ a a
= Y = ~ ,
cl cl { i
cl }
~ a a
= Y = {~ ~0 i i
+ + = Y + ~
, cl } a a
cl


where

~ a + 2 - 2 ~ a 1 ~ a ~ ~b +2 -2 ~c
= D
- D+ - ( D D
bc )
cl - + +.. .
2
~ ~
a
= { 0 i
, }


By insering the Buscher's formulas (2.28) in the definition of the curvature tensor (2.31) we

obtain the following relations





7


~
G -2 G
R = - G
( ) R
00de oo 00de
~
G
R = - G -2
( ) B R - G R
0ide oo [ G B
0i 00de oo 0ide ]
~ (2.32)
G G
R = R - G 1
( )- G R + G R - B R - B R
i jde i jde oo [ G G B B
0 j 0ide oi 0 jde 0 j 0ide oi 0 jde ]
+ G -2
( ) G G - B B R
oo [ 0i 0j oi oj ] G00de
and

~ ~
B B
R = 0 , R = - G 2
( )- G R - G R
00de 0ide oo [ G G
0i 00de oo 0ide ]
~
B B
R = R - G 1
( )- B R + G R - B R - G R (2.33)
i jde i jde oo [ G B G B
0 j 0ide oi 0 jde 0i 0 jde oj 0ide ]
+ G -2
( ) G B - B G R
oo [ 0i 0j oi oj ] G00de
We note that these relations are similar to that of the ordinary case given in Ref [18] by


imposing a quantum duality condition. It is clear that the Feynman rules constructed from the


expanded dual action (2.29) will yield manifestly covariant quantum corrections written as


integrals over the full harmonic (4,4) superspace. The divergences can be removed by


conterterms which are integrals in the (4,4) harmonic superspace of globally defined scalar


functions of the dual background superfield

~ ~
S  (2.34)
(4,4) = d T L

On the other hand, the propagators of the dual quantum superfield a
~ are not standard as in

the original theory. This can be surmoved by introducing the n-dual bein ~ k
e (Y ) namely
a cl




k ~
~ ~
e (Y ) e (Y ) = F (Y ) (2.35)
a cl b
k cl ab cl


by defining

~ k ~ k a
~
= e Y
( ) (2.36)
a cl



The insertion of (2.35) and (2.36) in the action (2.29) leads to the following kinetic term




++ ~ k 2 2
-- + - k
d ~
D D D
D - + (2.37)
T





8


with

-- ~ ~ ~ ~
k -- k --k l
D = D + V
l
~ (2.38)
--k ~k -- n
V Y
( ) V D Y
l cl nl cl

~
where - k
V - is the dual superconnection of the supermanifold. Therefore, the propagator of
l


the dual quantum superfield k
~ is given by
~ ~k ~
G(z ,u , z , u ) = (z ,u ) l
(z ,u )
1 1 2 2 1 1 2 2 (2.39)
2 2
kl
= - (D++D--D+ D- ) 1-
- + (z - z ) (u ,u )
1 1 1 1 1 2 1 2


Furthermore, the dual action (2.29) has a quantum gauge invariance, which must be gauge

fixed leading to Faddeev-Popov ghosts in the usual way. In fact, the gauge transformations of

quantum and background dual superfields are given by

~ a ~
= a
(2.40)
( a
Y )
cl = 0

with a
~ a superparameter, and their corresponding BRST transformations are
~ a ~
s = a
C , a
Y
s cl = 0
~ (2.41)
a
sC = 0

~
C is a dual ghost superfield associated with the superparameter a
~ . However the s-invariant

gauge fixing dual action is given by


~ ~ ~ ~ ~
S
( 4,4) = d s F Y
( ) C
( ' D++ D-- + C
' ' Y
)
GF T { a b a b
ab cl -- ++ cl (2.42)
~ ++ ~
a b ++ -- e -- d
+ 'R D Y C' D D D Y
abed cl cl }

where '
and '
are coupling constants and


~ a ~
sC' = a
b
~ (2.43)
a
sb = 0

By using (2.35), (2.41) and (2.43) in the dual gauge fixing action (2.42) we obtain





9


~ ~ ~ ~ ~ ~
S 
( 4,4) = d b D++ D-- + C' D++D--C + b' Y

GF T { k k li
k k ++ -- cl k
~ ++ ~ ~
a k ++ -- l -- d
+ ' R D Y b D D D Y (2.44)
akld cl cl
~ ++ ~ ~
a k ++ -- l -- d
+ 'R D Y C' D D C D Y
akld cl cl }
~
Eliminating the auxiliary dual superfield b by using its equation of motion we obtain the

following constraint



++ -- ~ ' ' ~ ++ a ++ -- ~
D D Y R D Y D D D Y (2.45)
k + l d
++ -- cl k + --
akld cl cl = 0


We note that for ' '
= =0 we obtain the equation of motion of the dual action
~
S which is equivalent to the equation of motion of the original action S .
( 4,4) Y cl =
cl =0 ( 4,4) 0


Moreover, the constraints of the (4,4) supersymmetric sigma model and its dual are equivalent

by using the following equalities



~ = ~0 ~
, =
a { i i }
a
Y = ~ , (2.46)
cl { a
cl }
' ~R = R
akld akld





3 - Quantum Equivalence of (4,4) Supersymmetric Sigma Models Related by Poisson-lie

T-Duality

Let us consider the two-dimensional (4,4) supersymmetric sigma model [7,8] which is

described on the target supermanifold M by a metric G , a = 1, . . . , d and antisymmetric
ab


tensor Bab

S  (3.1)
( 4,4) = d {F ( ) D D
ab ++ a --b}

where F = G + B . The structure group G of the target space defines a left (right) group
ab ab ab


action namely

a i a
= (3.2)
i


with i= 1, . . . , dimG, i
are the world-sheet dependent superparameters and a
are the
i


correspondingly right (left) invariant frames in the lie superalgebra G of the group G which

satisfy the relation



10


[ , = f (3.3)
i j ]a k a
ij k


where k
f are the structure constants of the lie group G. Furthermore, the variation of the
ij


group G element is as follows


g =
g (3.4)

or equivalently

-
g
1 g = (3.5)



On the other hand, the BRST transformations associated with (3.2) are given by



s a = i a
C i (3.6)
a
s C = 0

In order to define a (4,4) supersymmetric quantum action with ghost number zero we

a
introduce a superfield C ' and an auxiliary superfield a
b such that



s C'a = a
b
(3.7)
a
sb = 0

Since the classical lagrangian namely

++ a -- b
L = F () D D
cl ab


is invariant under the BRST transformations (3.6) thenL can be replaced by
cl


L = L sL^
+ (3.8)
Q cl


Consequently, the (4,4) supersymmetric quantum sigma model action is given by



S ^
 (3.9)
( 4,4)Q = d [Lcl + sL]
with

L^ = F ( D++C'
) D-- + D++ D--C '
ab [ a b a b ]

This quantum action is expanded as:




11


S  ( ) ( ) ( )
( 4,4 = d F D++ D-- + C D-- F D++C '
) - D++ F D--C
'
Q { a b j
ab [ a b
a a
ba j ab j ] (3.10)
i
C [ b a
d -- ' ++ a d ++ ' -- b
F D C D + F D C D
d ab j d ab i ]
where we have used the constraint

++
D ( --
F D D F D (3.11)
ab b) + -- ( ++
ba b) = 0

which is obtained by eliminating the auxiliary superfield a
b . Now let us give the variation of

the action (3.10) under the BRST transformations (3.6) and (3.7) namely



j
sS 
(4,4) = d C D-- J ++ - D++ J -- + F D ++ D -- + sF D --C ' D ++

Q { b
d a b d a
Qj Qj d ab j d ab j

a b
d ++ ' -- b d -- ' ++ i a
+ sF D C D + F D C D C
d ab j d ab j i

a
d b ++ ' -- i
+ F D C D C
d ab j i }

(3.12)

where the following constraint

--
D [ d ++
F D D C F D F D D C F (3.13)
d ab j a - ++ j a
ab j ]+ ++ [ d --
b ba j a + -- j a
ba j ]= 0


which is deduced by eliminating the auxiliary superfield, is used and the quantum

supercurrents 
J are given by
Q


++ a ++ b a ++ b
J = F D + (sF ) D C'
Qj ba j ba j (3.14)
-- a -- b a -- b
J = F D + (sF ) D C'
Qj ab j ab j


Consequently, the BRST invariance of the quantum action (3.10) leads to the following


equation

-- ++
D J D J F D D D C D C D C D C
Qj - ++ --
Qj + d [ ++ a -- b -- 'b ++ i a b ++ '
d ab j + i + a -- i ]
i (3.15)
+ d
sF [ -- 'b ++ a ++ '
D C D D C D
d ab j + a --b]=0

which can be rewritten as

(d + s)J l L
Qj + ( )
(3.16)
Q = 0
j




where l denotes the lie derivative namely
j





12


d
l F = F

ab d ab j
j (3.17)
d
l sF = sF

ab d ab j
j



d is the exterior derivative on the analytic subspace [8] and

-- ++ ++ --
J (3.18)
Qj = J d
Qj + JQj
d


are the Noetherian 1-forms on the world-sheet where the harmonic differentials are given by

  

d = U dU

On the other hand the action (3.10) possesses non-commutative conservation laws if the


generalized Cartan-Maurer equation holds on shell


1 ~
(d + s)J f J J (3.19)
Qi + jk
i Q j Q k = 0
2
~ ~
where jk
f are the structure constants of the dual target space with lie superalgebra G .
i


Thereafter, from (3.16) and (3.19) we deduce that

1 jk
l (L )
= ~
f J J (3.20)
Q i Q j Q k
i 2

Furthermore, the expansion of (3.19) in terms of ghost number leads to

1 ~
dJ f J J
Qi + jk
i Q j Q k = 0
2 (3.21)
sJQi = 0

which are equivalent to


 JQi = 0
- 
-
D J D J
Qi = 0 = + 
+ Qi (3.22)

sJQi = 0
-- ++ ++ -- ++ ~
D J D J J f J
Qi - Qi + jk --
Q j i Q k = 0



By using the expressions of the supercurrents (3.14), the component equations (3.22) give the


following conditions for F and its BRST transformations namely
ab





13


~
c jk d
l (F ) = F
f F (3.23.a)
ab ca j i db k
i

~
c jk d
l (sF ) = F
f sF
ab ca j i db k
i
~ (3.23.b)
c jk d
l (sF ) = sF
f F
ab ca j i db k
i


a ~ jk d ++ 'e -- 'b
sF f sF D C D C (3.24)
ea j i db k = 0


Thereafter, the equation (3.24) implies that

++ ~
jk --
J f J
i =
' ' 0
C j C k



or equivalently

J J = (3.25)
' ' 0
C C
j k





The equations (3.23.a) and (3.23.b) are the conditions of the Poisson-Lie symmetry

formulated at the level of the (4,4) supersymmetric sigma model quantum lagrangian, as for

the ordinary situation [4,6,19,8], by adding the condition (3.23.b) given by the lie derivative

of the BRST transformation of the tensor F ( . However, we conclude that the (4,4)
ab )


supersymmetric quantum sigma model with the action of the group G on its target space

~
admits a Poisson-Lie dual model for some dual group G [6,20]. Therewith, the dual quantum

action of (3.1) is given by


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
S d F ( ) D D s F ( )(D C ' D D D C ' (3.26)
( 4,4)Q = { ++ a --
ab b + [ ++ a -- b ++ a --
ab + b ]}

which satisfies the following conditions

~ ~ ~ ~
ab ca j i k ~ db
l~ (F ) = F f F
c j k d
i
~ ~ ~ ~ ~ ~ ~ ~ (3.27)
ab ca j i k db ca j i k ~ db
l~ ( ) = =
i sF F f sF sF f F
c j k d c j k j



and where the backgrounds are related by



[ -
F ( F
ab = )
0 ] 1 ~ab ~
= ( = )
0
[ (3.28)
-
-
sF ( s F
ab = )
0 ] 1 = [ ~ab ~
( ( = ))
0 - ] 1
1





14


Finally, we note that our results on Poisson-Lie T-duality and especially our general

formulation of T-duality in non isometric backgrounds may applied to a wide class of non

linear sigma models [21]. On the other hand, supersymmetric quantum cosmologies may be

derived from the non-linear sigma model with appropriate linearity conditions [22].

Therefore, it would be interesting to investigate N = 4 supersymmetric quantum cosmologies

from the (4,4) supersymmetric non-linear sigma models associated with duality [23].



4 - Conclusion

In this paper, we have constructed the quantum actions of the (4,4) supersymmetric non-

linear sigma model and its dual in the Abelian case by using the background superfield

method, which is based on the parallel transport equation. Furthermore, we have deduced the

relations between the curvature tensor of the supermanifold M and its dual by using the

Buscher's formulas. The propagators of the quantum superfields and its dual are determined

by introducing the n-bein a
e ( ) and its dual ~a
e (Y ) .
i cl i cl


On the other hand, by using the BRST transformations associated with the left (right)

group action on the superfields of the supermanifold M, we have constructed the quantum

(4,4) supersymmetric non-Abelian dual sigma model action. This is obtained in the sense of

Poisson-Lie T-duality which generalizes the Abelian and non-Abelian dualities. The quantum

action and its dual obey the same conditions of the Poisson-Lie symmetry but with the tilted

and untilted variables interchanged. Thus, the non-commutative conservation laws for the

quantum (4,4) supersymmetric sigma model are given in terms of the quantum

supercurrents 
J . However, the investigation of the N = 4 supersymmetric quantum
Q


cosmologies from the (4,4) supersymmetric non-linear sigma models associated with duality

is under study [23].



Acknowledgments

The authors would like to thank Professor S. Randjbar-Deami for reading the manuscript
and Professor M. Virasoro, the International Atomic Energy Agency and UNESCO for
hospitality at the Abdus Salam International Centre for Theoretical Physics, Trieste. This
work is supported by the program PARS n0 phys. 27.372/98 CNR and by the frame work of
the Associate and Federation Schemes of the Abdus Salam International Centre for
Theoretical Physics, Trieste.




15


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