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Poisson-Lie T-duality and complex geometry in N = 2 superconformal WZNW models
El ELSEVIER
Nuclear Physics B 510 [PM] (1998) 623-639
Poisson-Lie T-duality and complex geometry in N = 2 superconformal WZNW models S.E. P a r k h o m e n k o
1
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
Received 1 July 1997; revised 22 September 1997; accepted 23 October 1997
Abstract Poisson-Lie T-duality in N = 2 superconformal WZNW models on the real Lie groups is considered. It is shown that Poisson-Lie T-duality is governed by the complexifications of the corresponding real groups endowed with Semenov-Tian-Shanskysymplectic forms, i.e. Heisenberg doubles. Complex Heisenberg doubles are used to define on the group manifolds of the N = 2 superconformal WZNW models the natural actions of the isotropic complex subgroups forming the doubles. It is proved that with respect to these actions N = 2 superconformal WZNW models admit Poisson-Lie symmetries. The Poisson-Lie T-duality transformation maps each model onto itself but acts non-trivially on the space of classical solutions. (~ 1998 Published by Elsevier Science B.V. PACS: 11.25.Hf; 11.25.Pm
Keywords: Strings; Duality; Superconformalfield theory
1. I n t r o d u c t i o n N = 2 superconformal field theories ( S C F T ' s ) play the role of building blocks in the superstring vacua construction. The investigations in this direction were initiated in the pioneering works of Refs. [ 1,2], where it was shown that N = 2 SCFT's may describe Calabi-Yau manifolds compactifications of superstrings. Since then the N = 2 SCFT's, and the profound structures associated with them are an area of investigation. On the other hand, it is well known that the string vacua, considered as conformal field theories, in general, have deformations under which the geometry of the target space changes. I E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Publishedby Elsevier Science B.V. All rights reserved. Pll S0550-32 l 3 (97) 00712-8
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These deformations include the discrete duality transformations, so called T-duality, which are symmetries of the underlying conformal field theory [3,4]. A well-known example of T-duality is mirror symmetry in the Calabi-Yau manifold compactifications of the superstring [5]. The Poisson-Lie (PL) T-duality, recently discovered by Klimcik and Severa [6] is a generalization of the standard non-Abelian T-duality [7-11]. The main idea of the approach [6] is to replace the requirement of isometry of a o--model with respect to some group by the weaker condition of the Poisson-Lie symmetry of the theory. This generalized duality is associated with two groups forming a Drinfeld double [ 12] and the duality transformation exchanges their roles. This approach has been developed further in Refs. [ 13-17]. In order to apply PL T-duality in superstring theory one needs to have dual pairs of conformal and superconformal ~r-models. A simple example of a dual pair of conformal o--models associated with the 0 ( 2 , 2) Drinfeld double was presented in Ref. [ 18]. Then, it was shown in Refs. [ 14,15] that WZNW models on compact groups are natural examples of PL dualizable o--models. The supersymmetric generalization of PL T-duality was considered in Refs. [ 1921]. In particular, due to the close relation between N = 2 superconformal WZNW (SWZNW) models and Drinfeld's double (Manin triple) structures on the corresponding group manifolds (Lie algebras) [22,23], it was shown in Ref. [20] that N = 2 SWZNW models possess a very natural PL symmetry, and PL T-dual o--models for N = 2 SWZNW models associated with real Drinfeld's doubles were constructed. Then the first example of PL T-duality in N = 2 SWZNW models on the compact groups was obtained in Ref. [21] for U(2)-SWZNW model. In the present paper we generalize the results of our preceding paper [21] to the case of N = 2 superconformal WZNW models on compact groups of higher dimensions. These models correspond to the complex Manin triples endowed with hermitian conjugation which conjugates isotropic subalgebras forming the Manin triples. After a brief review of the classical N = 1 superconformal WZNW models in Section 1, we describe in Section 2 the complex geometry of N = 2 SWZNW models on compact groups. We show that the Heisenberg double [24,25] of the complexification of the N = 2 SWZNW model group manifold plays a central role in PL T-duality: on the one hand, the Lagrangian of the model can be expressed (locally) in terms of the components of Semenov-Tian-Shansky symplectic form of the Heisenberg double. On the other, there is the natural action of the isotropic subgroups forming the double on the space of fields of the model. In Section 3 we show that each N = 2 SWZNW model admits PL symmetries with respect to these actions which we use to show that the PL T-duality transformation maps the model onto itself but acts non-trivially on the space of solutions. Though our results are concerned with N = 2 SWZNW on compact groups they can be straightforwardly generalized to real non-compact groups.
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2. The elassleal N = 1 supereonformal W Z N W model
In this section we briefly review the N = 1 SWZNW models using the superfield formalism [26]. We parametrize the super world-sheet introducing the light-cone coordinates x±, and Grassmann coordinates O±. The generators of supersymmetry and the covariant derivatives are 3
3
Qm = 30---~ + tOiOm'
D m - O0±
lO±cgm"
( 1)
They satisfy the relations {D+, D:t:} = - { Q + , Q+} = -12o+, {O+, Din} = {Q+, Qm} = {Q, D} = 0,
(2)
where the brackets {, } denote the anticommutator. The superfield of the N = 1 SWZNW model,
G = g + tO_g,+ + tO+~,_ + iO_O+F,
(3)
takes values in a Lie group G. We will assume that its Lie algebra g is endowed with an ad-invariant non-degenerate inner product ( , ) . The inverse group element G - l is defined by the relation
G-IG = 1
(4)
and has the decomposition G - I = g - i _ tO_g-l~+g-1 _ tO+g-l~_g-i
-lO_O+g -1 ( F + ~ _ g - l g % _ ~ + g - l ¢ _ )g-l.
(5)
The action of the N = 1 SWZNW model is given by Sswz = f dZxd20 ((G-1D+G, G - 1 D - G } )
-- [ d 2 x d 2 { ~ d t
dt
(6)
The classical equations of motion can be obtained by making a variation of (6),
D _ ( G - I D + G ) = D + ( D _ G G - l ) = 0.
(7)
The action (6) is invariant under the super-Kac-Moody
6a, O(x+, x_, 0+, O- ) = a+(x_, O+)O(x+, x_, 0+, O_ ), fi,,_ G(x+, x_, 0+, O_ ) = - G ( x + , x_, 0+, O _ ) a _ (x+, O_ ), where a± are g-valued superfields and N = 1 supersymmetry transformations [26]
(8)
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S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
G-13E+ G = ( G - I e + Q + G ) ,
(9)
8,_GG -I = e-Q-GG -l •
In the following we will use the supersymmetric version of the Polyakov-Wiegmann formula [ 27 ] S s w z [ G H ] = Sswz[G] + Sswz[H] +
fd2xd20(G-ID+G,D-HH-1).
(10)
It can be proved as in the non-supersymmetric case.
3. Complex geometry in N = 2 superconformal W Z N W models
In Refs. [ 22,23,28 ] supersymmetric WZNW models which admit extended supersymmetry were studied and the correspondence between extended supersymmetric WZNW models and finite-dimensional Manin triples was established in Refs. [22,23]. By definition [ 12], a Manin triple (g, g+, g_ ) consists of a Lie algebra g, with a non-degenerate invariant inner product ( , ) and isotropic Lie subalgebras g± such that g = g+ ® g_ as a vector space. The corresponding Sugawara construction of N = 2 Virasoro superalgebra generators was given in Refs. [22,23,28,29]. To make a connection between the Manin triple construction of Ref. [ 22,23 ] and the approach of Ref. [28] based on complex structures on the Lie algebras the following comment is relevant. Let g be a real Lie algebra and J be a complex structure on the vector space g. J is referred to as the complex structure on the Lie algebra g if J satisfies the equation [ J x , Jy] - J [ J x , y] - J [ x , Jy] = [ x , y ]
(11)
for any elements x, y from g. It is clear that the corresponding Lie group is a complex manifold with left (or right) invariant complex structure. In the following we will denote the real Lie group and the real Lie algebra with the complex structure satisfying (11 ) as the pairs (G, J) and (g, J) correspondingly. Suppose the existence of the non-degenerate invariant inner product (,) on g so that the complex structure J is skew-symmetric with respect to (,). In this case it is not difficult to establish the correspondence between complex Manin triples and complex structures on the Lie algebras. Namely, for each complex Manin triple (g, g+, g_ ) there exists a canonic complex structure on the Lie algebra g such that the subalgebras g+ are its i t eigenspaces. On the other hand, for each real Lie algebra g with non-degenerate invariant inner product and skew-symmetric complex structure J on this algebra one can consider the complexification gC of g. Let g± be i t eigenspaces of J in the algebra gO. Then (gC, g+, g_) is a complex Manin triple . Moreover, it can be proved that there exists a one-to-one correspondence between the complex Manin triple endowed with an antilinear involution which conjugates isotropic subalgebras ~" : g± ---+g:F and the real
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Lie algebra endowed with an ad-invariant non-degenerate inner product ( , ) and the complex structure J which is skew-symmetric with respect to ( , ) [22]. Therefore we can use this conjugation to extract the real form from the complex Manin triple. If the complex structure J on the Lie algebra is fixed then it defines the second supersymmetry transformation [28]
(G-I&7+ G) " =rI+(JI)~(G-ID+G) b, (67 GG-1)a=rI_(Jr)~,(D_GG-1) b,
(12)
where Jl, Jr are the left invariant and right invariant complex structures on G which correspond to the complex structure J. To specify our presentation we concentrate in this paper on N = 2 SWZNW models on compact groups (the extension to non-compact groups is straightforward), i.e. we shall consider complex Manin triples such that the corresponding antilinear involutions will coincide with the hermitian conjugations. Hence it will be implied in the following that G is a subgroup in the group of unitary matrices and the matrix elements of the superfield G satisfy the relations ~,,,,,= ( g - l ) n m ,
~ n = (///-1)7,
pmn = ( F - l ) n m ,
(13)
where we have used the following notations:
(s~l = _g-l~+g-l,
F -1 = --g-l(F + ~ _ g - l ¢ + -~b+g-l~b_)g -1.
(14)
Now we have to consider some geometric properties of the N = 2 SWZNW models closely related with the existence of the complex structures on the corresponding groups. Let us fix some compact Lie group with the left invariant complex structure (G, J) and consider its Lie algebra with the complex structure (g, J). The complexification gC of g has the Manin triple structure (gC,g+,g_). The Lie group version of this triple is the double Lie group (GC, G + , G _ ) [24,30,25], where the exponential subgroups G± correspond to the Lie algebras g+. The real Lie group G is extracted from its complexification with the help of the hermitian conjugation r, G = {g C GClr(g) = g - l } .
(15)
Each element g C G C from the vicinity Gl of the unit element from G c admits two decompositions,
g = g+gZi = ~_~+l,
(16)
where ~± are dressing transformed elements of g+ [30]: ~,+ = (g~l)uv.
(17)
Taking into account (15) and (16) we conclude that the element g (g ~ G1) belongs to G iff
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
628
r(g+) = ~,~l.
(18)
These equations mean that we can parametrize the elements from C1 -- Gl f3 G
(19)
by the elements from the complex group G+ (or G _ ) , i.e. we can introduce complex coordinates (they are just matrix elements of g+ (or g _ ) ) in the cell C1. To do this one needs to solve, with respect to g_, --1
r(g_) = (g+)g-
(20)
[to introduce G_ coordinates on Gl one needs to solve, with respect to g+, --I
r(g+) = (g_)g+ ].
(21)
Thus the formulas (16), (20), [ (21) ] define the map ~b+ • G+ ~ Cl,
(22)
[~bi- : G_ ~ C~].
(23)
For the N = 2 SWZNW model on the group G we obtain from (16) the decompositions for the superfield (4) (which takes values in CI )
G(x+,x_) = G+(x+,x_)G-l(x+,x_ ) = G_(x+,x_ )G+l(x+,x_).
(24)
Due to (24), (10) and the definition of Manin triple we can rewrite the action (6) for this superfield in the manifestly real form Sswz = - ~
1/
d2xd20 ( ( p + , p Z ) + {/5+,/5+_)),
(25)
where the superfields
p+ = G~_IDG+,
/5+ = G ~ l D G +
(26)
correspond to the left invariant 1-forms on G~:,
r ± = g~Jdg+,
~± = ~T_ld~±.
(27)
To proceed further we need to prove the following:
Lemma 1. (1) The g+-valued 1-form r + considered as a form on (CL, J) is holomorphic. (2) the g_-valued 1-form ? - considered as a form on (Cl, J) is antiholomorphic.
Proof We identify the Lie algebra gC with the space of complex left invariant vector fields on the group G. Let
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629 (28)
{Ri, i = 1 . . . . . d}
be the basis in the Lie subalgebra g+ and { R i , i = l . . . . . d}
(29)
the basis in the Lie subalgebra g_ so that (28), (29) constitute the orthonormal basis in gO: (30)
(R i, R.i ) = 6j.
Denote by ( the canonic gO-valued left invariant 1-form on the group G. It is obvious that ( = ~ J R . i + ~ j R J,
j(k = _t(~,
J(k = t(k.
(31)
We can also write r + = r~ei,
r - = ri Ri,
?+ = ~iRi,
r
= ri Ri,
I + ~ dg+g+ 1 = liRi,
l - =- d g _ g -1 = li Ri,
/+ ~ d,~+~+ t = [iRi,
[- =- d ~ _ ~ -1 = [iR i.
(32)
(1) Using the first decomposition from (16) we get (33)
sc = g _ r + g -1 - I - .
Let us introduce the matrices g _ R i g - j = M i j R J + NJi Rj, g+Ri g+ 1 = piJ Rj + Q j R j, g _ R i g -1 = ( N * ) ~ R j, g+RigT l = (Q*)JRj.
(34)
Using these matrices and (30) we can express the 1-forms r i, ri, I i, li in terms of sci, r.i=
(N-1)j~ i,
~k, li = --S~i -~ M ji ~t N - l " Jd kb ri = ( ( N * ) -1 )f ( _ ¢ j + M n j ( N - I )~¢k), Ii = ( N - I ) ~ ( Q * ) ~ s
c~.
(35)
Taking into account ( 3 l ) , (35) we obtain Jr k = - t r k,
Jl ~ = - t l ~.
(36)
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Thus r k, l t, k = 1 . . . . . d are (1,0)-forms and their complex conjugates ?k, rk, k = 1 . . . . . d are (0,1)-forms on (C1, J ) . Because the forms r k, ~ are linearly independent on ( C l , J ) and linearly independent on G+ we conclude that the map ~bl is holomorphic [31 ], where the complex structure on G+ is given by the multiplication by t. Since the forms r k are holomorphic as the forms on G+ statement (1) of the lemma follows. (2) Using the second decomposition from (16) we can write
= g+r ~-~1 _ [4-.
(37)
Introducing the matrices
~_Ri~, -1 = 2(4ijRj + N~ Rj, a~+Ri~+ 1 = piJ Rj + O_.}RJ, ~_ gi~ -1 = ( g * ) ~ g j, ~+Ri~+ l = ( O*)~ Rj
(38/
and using (30), one can express the 1-forms P, ri, 7i, [i in terms of ~:i,
~,j = ( 0 -1 )~Zi, yi = __(i _.~_p j i ( O--1 ) jk..~:k,
~. = (Q --1 ) jk( N * )i~k" j
(39)
In view of (31), (39) we obtain
JPk = tPk,
J[k = t[~.
(40)
Thus fk, [k, k = 1 . . . . . d are (0,1)-forms and their complex conjugates Fk, [k, k = 1 . . . . . d are (1,0)-forms on (C~, J ) . Because the forms ?k, ~k are linearly independent on (C1, J ) and linearly independent on G _ we conclude that the map ~b~- is holomorphic, where the complex structure on G _ is given by the multiplication by - t . Since the forms ?k are antiholomorphic as the forms on G _ , statement (2) of the lemma follows. Due to L e m m a 1 the forms
{r i,Pi},
i= 1. . . . . d
(41)
constitute the basis of holomorphic and antiholomorphic 1-forms in the open subset C1. Consequently, in the basis of (28), (29) we can write the components of g+-valued 1-forms p - , / 5 + as follows:
Pi = ELTPj + EijP / ,
/5i = /5i = EfiP i "+"ETjPJ,
where we have used the notations
(42)
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639 E'6 =
Eft,
Eij =
631
ETj.
(43)
Thus the Lagrangian of the action (25) has the following form: A = pi+ ( P - ) i - ~ i (P+)i i -J - p~_pJ+) " " = ½((E(i - Eft) -}- (E(/+ Eji))pi+p/_ +Ei3(p+p_
+½( (E(7- E~) + (Eo + E)7)),b~_/3./_.
(44)
Let us show that
Eij + Eji =
EO+E37= 0.
(45)
Using (16) we can represent the kinetic term of the action (6) in the following form: i / + Ei)(pi+[9j_ - pi ~+) -2 = (Eij+ E -ji)P+P_
+(
-j + E~)p+p_.
(46)
This expression means that the bilinear form ( , ) on G written in the basis (41) has the following non-zero components:
--gij = Eij + Eji,
- g i ] = 2Ei3,
-K;3 = E~) + E3;.
(47)
On the other hand, the complex structure J is skew-symmetric with respect to (, >, and therefore (45) should be satisfied. L e m m a 2. The Lagrangian A can be expressed in terms of the Semenov-Tian-Shansky symplectic form ,(2, l , o --c a b A = ~Zcba~p+p_,
(48)
where we have used the common notation pa, a = 1. . . . . 2d for the 1-forms pi, ~i. P r o o f In the open subset CI, where the decomposition (16) holds, the Semenov-TianShansky symplectic form /2 can be represented as follows [25,16]: 12 = r i/~ r i
+ ~i A r i .
(49)
Due to the forms ri, Fi c a n be expressed in terms of r i, ~i in perfect analogy to Pi, ~i the restriction o f / 2 on any world-sheet (which should not be confused with the super world-sheet of the N = 2 SWZNW model under consideration) is given by 12 [z = dx+ A d x _ ( ~1 ( Eij - Eji) (ri+r~ - r i r~ ) + 2E(?(ri+~'~ - r i F'~ ) -
-:
).
(50)
Comparing (44) with (50) and taking into account (45) and Lemma 1 we obtain (48).
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To generalize (16), (18) one has to consider the set W (which we shall assume in the following to be discrete and finite) of classes G + \ G C / G _ and pick up a representative w for each class [w] C W. It gives us the stratification of G ¢ [25]: Gc=
U
G+wG_=
[wICW
U
Gw.
(51)
GW.
(52)
[w]EW
There is a second stratification, GO=
U
G_wG+=
U [w]EW
[wIEW
We shall assume, in the following, that the representatives w satisfy the unitarity condition r ( w ) = w -I
(53)
It allows us to generalize (16), (18) as follows: (54)
g = g+wg - l = ~_w~+ 1.
It will be more convenient to rewrite this decomposition in another form, (55)
g = wg+g - I = wg_g+ 1.
The group elements g+, ~,+ from this formula should not be confused with the group elements from (54) (of course they are mutually related, but do not coincide). To make the decompositions (55) unambiguously determined we should demand that W g+ C G+,
~_ C G~_' ,
(56)
where G~ = G+ N w - l G + w ,
Gw _ = G_ N w-lG_w.
(57)
A helpful example, where the formulas ( 5 1 ) - ( 5 3 ) , (55) hold, is the Bruhat decomposition of the complexification of an even-dimensional semisimple compact Lie group with an appropriate maximal torus decomposition. In order for the element g to belong to the real group G the elements g+, ~± from (55) should satisfy (18). Thus the formulas (55), (56), (20) [(21)] define the map
4'~ : G÷W ---, Cw --- Gw n G,
(58)
[~bw : G w ~ Cw-= Gw N G ] .
(59)
We can obtain the corresponding generalization of Lemma 1 for the g~_ components of the form r +, where g~_ = g+ n w - l g + w [gW components of the form ? - , where gW = g_ n w - l g _ w ] proving by analogy with the proof of the Lemma 1 that the map (58) [ (59) ] is holomorphic.
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Using the appropriate generalization of (24) and taking into account (56), (57) we can conclude that the action for the map into the cell Cw is given by (25). As the map (58) is holomorphic, the same is true for the Lagrangian: it is given by (48). where paRa C g+. It is clear that Eq. (25) is correct inside the super world-sheet domain where the superfields take values in the cell Cw. On the boundaries of these domains, where jumps from one cell to another take place, some additional terms should be added, but as will be explained below, for our purposes it will suffice to ignore these terms. Eqs. (20), (55) and (25) mean that there is a natural action of the complex group G+ on G, and the set W parametrizes the G+ orbits Cw. It is obvious that there is also the action of the complex group G_ on G so that the analogies of Eqs. (44), (45), (48) can be obtained in a similar way.
4. Poisson-Lie s y m m e t r y in N = 2 S W Z N W model
In view of (20), (55), (25) we can consider the (G, J ) - S W Z N W model as a o-model on the orbits of the complex Lie group G+ and find the equations of motion by making a varying the action (25) under the right action of this group on itself. We will consider the variations which are non-zero only inside the super world-sheet domains, where jumps from one cell Cw to another one take place. Thus we can not take into account the corresponding boundary terms omitted in the action (25). The set of holomorphic maps {~bw + , w c W} defines the action of G+ on the group G:
h+ .g ==-wg+h+gS l(g+h+),
h+ E G+,
(60)
where g-J (g_h+) means the solution of (20) with the argument g+h+. The vector fields {Si, i = 1 . . . . . d} generating this action are the holomorphic vector fields on the cells Cw. Note that in the case when G is an even-dimensional semisimple compact Lie group and ( 5 1 ) - ( 5 3 , (55) are given by the Bruhat decomposition, these vector fields will coincide with the classical screening currents in the Wakimoto representations of [32]. This is due to the fact that the classical screening currents in the Wakimoto representations are given by the right action of the maximal nilpotent subgroup N+ on the big cell of the corresponding flag manifold G / B _ , where B_ is the Borel subgroup of the group G (N+ E B+) [33]. Let us consider a variation of (25) under the vector field Z = zisi + Z;Si for the map onto the cell Ci. We obtain on the extremals
D + ( A _ ) i + D_ (A+)i - LsiA=O, D + ( A - ) 7 + D+(A+) 7 - LLA=O, where Ls,, Ls, mean the Lie derivatives along the vector fields St, currents Ai, A; are given by
(61)
Si
and the Noether
634
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639 ( A _ )i = E o p )_ -4- EijP)--, ( A+ ) i =-Ei~pt+ - EjiP)+,
(,4a:)i = (A+)7.
(62)
As the fields Si are holomorphic, we get from (48)
Ls, A~b = t j~Ls~S2cb. 2
(63)
Now one has to find the Lie derivative of the Semenov-Tian-Shansky symplectic form. Because Lsj = isjd + disj and s2 is closed, we have
LsiO = disjS2 = 2drj, LL.O = di~jS2 = 2d?j,
(64)
where we have used (49). In view of (27) drj, d~j can be expressed in terms of r.i, ~i again,
drj=
lf~kriArl~,
d~j=_lfjk~iA?l~,
(65)
where f~k are the structure constants of the Lie algebra g_. Thus the action (60) is the Poisson action [24] with respect to the Poisson structure defined by the symplectic form .(2. Therefore
LsiA= t f~k ( jp+ )i(P- ) k, LL A = tflk( j~+ )i(~_ )k.
(66)
Taking into account the relations (A-)i
= (P-)i,
(67)
(A+)i = t(Jp+)i,
where the second equality is due to (62) and Lemma 1, we get that the following PL symmetry conditions are satisfied on the extremals of N = 2 (G, J)-SWZNW model:
Ls~A= f ] k ( A + ) j ( A - ) ~ , LL A = ~k(A+)3(a_)k"
(68)
By demanding the closure of (68), [Ls,, Lsj] = f~Lsk, we have the consistency condition
f~j fkm = f~" f~n - f~k fm _ fnm f~ n + f~k f,]~,
(69)
which is satisfied due to the Jacobi identity in the Lie algebra gC. As is easy to see from (61), Eqs. (68) are equivalent to the zero-curvature equations for the F+_-component of the super stress tensor FMu, (F+-)i ~ D+(A-)i
+ D-(A+)i
- finm(A+)n(A-)m
(F+_); =- D+(A_)7 + D - ( A + ) 7 -
f'/'(a+)~(a_)~,
= O, = 0.
(70)
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Using the standard arguments of the super Lax construction [34] one can show that from (67) it follows that the connection is flat, FMN = 0 ,
M,N=
(+,--,+,-).
(71)
Eqs. (70) are the supersymmetric generalization of the Poisson-Lie symmetry conditions from Ref. [6]. Indeed, the Noether currents Ai, A 7 are generators of the g+ action, while the structure constants in (70) correspond to the Lie algebra g_, which is Drinfeld's dual to g+ [35]. Now we turn to the maps onto the remainder cells Cw, w E W. For each w ~ W the action and the Lagrangian for the map onto Cw are given by (25) and (48) with the restriction (56). Consequently, the corresponding equations of motion will coincide with (70), where only the g~_ components of the Noether currents (62) are not identically zero. Thus we have shown that the N = 2 (G, J ) - S W Z N W model admits the Poisson-Lie symmetry with respect to the complex group G+.
5. P o i s s o n - L i e T-self-duality of N = 2 S W Z N W models
The PL T-dual to the (G, J ) - S W Z N W o--model should obey conditions as (70) but with the roles of the Lie algebras g+ interchanged [6]. To find the action of this model we start from the maps onto the cell C~. Due to (71) we may associate to each extremal surface G + ( x + , x _ , O + , O_) C G+, a map ("Noether charge") V_ (x+, x _ , O+, O_ ) from the super world-sheet into the group G_ such that ( A+ )i = - ( D ± V - V_-1 )i.
(72)
Now we build the following surface in the double GC: F ( x+, x_, O+, O_ ) = G+ ( x+, x _, O+, O_ ) V_ ( x+, x_, O+, O_ ).
( 73 )
In view of (67) it is natural to represent V_ as the product V_ = GSJH51 ,
(74)
where G_ is determined from (20) and H_ satisfies the equation D _ H _ = 0.
(75)
Therefore the surface (73) can be rewritten in the form F ( x ± , O± ) = G ( x ± , O+ ) H _- l (x+, O_ ),
(76)
where G ( x ± , O ± ) E Cl is the solution of the (G)-SWZNW model restricted to the corresponding domain of the super world-sheet.
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The solution and the "Noether charge" of the dual o--model are given by a "dual" parametrization of the surface (73) [6]
F(x+,x_,O+,O_) = G_(x+,x_,O+,~9_)fZ+(x+,x_,O+,O_),
(77)
where G_(x+,x_,O+,O_) 6 G_ and fZ+(x+,x_,6)+,O_) 6 G+. Thus in the dual o-model Drinfeld's dual group to the group G+should acts, i.e. it should be a o'-model on the orbits of the group G _ and with respect to this action the PL symmetry conditions dual to (68) should be satisfied,
Ls, fl= f l k ( i + ) J ( A - ) k, L~zi = f ] k ( A + ) J ( i - )
where {S i, ~i,
~,
(78)
i = 1 . . . . . d} are the vector fields which generate the G_ action, it,
i~:, i ~ are the Lagrangian and the Noether currents in the dual o--modeh Taking into account the second decomposition from (16) and holomorphicity of ~b~-, it is easy to see that the right action of G_ on itself defines the action on the cell Cj:
h_.g=_~_h_~,+l(~_h_),
h_ c G _ ,
(79)
so that the vector fields generating this action are the holomorphic vector fields. Therefore, due to Eq. (49) the Lagrangian A of the initial N = 2 SWZNW model on the cell Cl written in G_ coordinates satisfies (78). Thus the dual o--model on the cell C1 is governed by the action (25) and we can rewrite the right-hand side of (77) analogously to (76),
F(x±, O:a:) = G(x+, O+ )/V/~l (x+, O-- ),
(80)
where t~(x+, ~9+) E CI and D_/:/+ = 0.
(81)
For the maps onto the remainder cells Cw we also have the surfaces in the double G c written in two ways,
F(x+, x_, 0+, 0_) = wG+(x+,x_, 0+, O_)V_(x+,x_,O+, 0 _ ) =wG_(x+,x_,O+,O_ )~'+(x+,x_,¢9+,O_ ),
(82)
where G+(x~,~9~),(/+(x±,O+) 6 G~_, G _ ( x ± , O ± ) , V _ ( x ± , O ± ) E G w. The arguments we have used for the maps onto C1 can be applied (with the appropriate modifications due to the restrictions (56)) to this case also because for each w 6 W the map ~bw is holomorphic, so we conclude that the dual tr-model on each cell Cw is governed by the action (25). Hence, we can rewrite (82) in two more ways:
F(x±, O± ) = wG(x±, O+ )H -1 (x+, O_ ) = wG(x±, O± )/r./+ 1 (x+, ~9_ ),
(83)
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637
where G , G E Cw, H _ E GW_,/:/+ E G~_. It is clear that the set of equations (76), (80), (83) defines the map from the total super world-sheet onto the double Go: F ( x ± , O± ) = G ( x ± , O ± ) H -1 (x+, O_ )
= G(x4-, (9±)F/+ 1(x+, 69_ ),
(84)
where G ( x + , 04,) is the solution of ( G ) - S W Z N W model and G ( x + , O4.) E G. From this equation we see that G ( x + , (94.) is the classical solution of the (G, J ) - S W Z N W model also because it is obtained from the solution G ( x ± , (94-) by the right multiplication on the G-valued function H - H21[-I+ that satisfies the equation D _ H = 0.
(85)
Thus the dual o--model coincides with the initial one, i.e. the N = 2 SWZNW model on the real Lie group is PL T-self-dual and the PL T-duality transformation is a special type of super Kac-Moody symmetry.
6. Conclusions We have shown that Poisson-Lie T-duality in the classical N = 2 SWZNW models on compact Lie groups is governed by complexifications of these groups endowed with the Semenov-Tian-Shansky symplectic forms, i.e. Heisenberg doubles. Each N = 2 SWZNW model admits very natural PL symmetries with respect to the natural actions of the isotropic subgroups forming the complex Heisenberg double. Under the PL T-duality transformation the N = 2 SWZNW model maps onto itself but this transformation acts as some special super Kac-Moody symmetry on the classical solutions. Thus N = 2 SWZNW models on compact groups are PL T-self-dual. This result bears a resemblance to the T-self-duality in the self-dual torus compactifications [ 3 ]. Note also that in order to apply PL T-duality transformations we used only the left invariant complex structure on the group manifold. Since the (2, 2) Virasoro superalgebra of symmetries also requires the right invariant complex structure, an interesting question which certainly deserves further attention is as to what happens with the right invariant complex structure under this transformation. Another point which is believed to be interesting is the N = 4 SWZNW model's generalization of PL T-duality. The N = 4 superconformal algebra demands the existence of an infinite number of complex structures on the Lie algebra of the model which are parametrized by the points of a two-dimensional sphere [23,28]. It is reasonable to expect that the quaternionic geometry will appear in these models. We expect the PL T-duality to exist also in N = 2 superconformal Kazama-Suzuki models [36] since these models can be represented as the cosets (N = 2 G-SWZNW m o d e l ) / ( N = 2 H-SWZNW model), where H is a subgroup of G [37]. Another interesting question is related to the quantum picture of the PL T-duality in N = 2 SWZNW models. Because the Poisson-Lie groups is nothing but a classical
638
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
limit of the quantum groups [ 12], there appears to be an intriguing possibility of the relevance of quantum groups in the T-duality and other superstring applications for example in D-branes [ 15,38].
Acknowledgements I am very grateful to B. Feigin for discussions. I would like to thank the VolkswagenStiftung for financial support as well as the members of the group of Prof. Dr. R. Shrader for hospitality at Freie Universit~it of Berlin, where a significant part of this work was done. This work was supported in part by grants INTAS-95-IN-RU-690, CRDF RP1-277, RFBR 96-02-16507.
References [ 1 ] D. Gepner, Nucl. Phys. B 296 (1988) 757; Phys. Lett. B 199 (1987) 380. [21 T. Eguchi, H. Ooguri, A. Taormina and S.K. Yang, Nucl. Phys. B 315 (1989) 193; A. Taormina, The N = 2 and N = 4 superconformal algebras and string compactification, CERNTH.5409/89. 131 A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 79. [4] E. Alvarez, L. Alvarez-Gaum6 and Y. Lozano, An introduction to T-duality in string theory, CERNTH.7486/94, hep-th/9410237. 15[ P. Candelas, M Lynker and R. Schimmrigk, Nucl. Phys. B 341 (1990) 383; P. Candelas, X. De la Ossa, P.S. Green and L. Parkes, Nucl. Phys. B 359 (1991) 21; B.R. Greene and R. Plesser, Nucl. Phys. B 338 (1990) 15. [61 C. Klimcik and P. Severa, Phys. Lett. B 351 (1995) 455. [71 X. De la Ossa and F. Quevedo, Nucl. Phys. B 403 (1993) 377. [81 A. Giveon and M. Rocek, Nucl. Phys. B 421 (1994) 173; E. Alvarez, L. Alvarez-Gaume and Y. Lozano, Phys. Lett. B 336 (1994) 183; A. Giveon, E. Rabinovichi and G. Veneziano, Nucl. Phys. B 322 (1989) 167. [9] B.E. Fridling and A. Jevicki, Phys. Lett. B 134 (1984) 70. 110] E.S. Fradkin and A.A. Tseytlin, Ann. Phys. 162 (1985) 31. l I 11 T. Curtright and C. Zachos, Phys. Rev. D 49 (1994) 5408; D 52 (1995) R573. 1121 V.G. Drinfeld, Quantum groups, Proc. Int. Cong. Math., Berkeley, CA (1986) p. 798. [ 131 C. Klimcik and P. Severa, Poisson-Lie T-duality and loop groups of Dfinfeld doubles, CERN-TH/95330, hep-th/9512040. [141 C. Klimcik and P. Severa, Non-abelian momentum-winding exchange, CERN-TH/96-142, hepth / 9605212. 1151 C. Klimcik and P. Severa, Open strings and D-branes in WZNW models, CERN-TH/96-254, hepth/9609112. [ 161 C. Klimcik and P. Severa, T-duality and the moment map, IHES/P/96/70, hep-th/9610198. [ 171 E. Tyurin and R. von Unge, Poisson-Lie T-duality: the path-integral derivation, hep-th/9512025. I 181 A.Yu. Alekseev, C. Klimcik and A.A. Tseytlin, Quantum Poisson-Lie T-duality and WZNW model, CERN-TH/95251, hep-th/9509123. 1191 K. Sfetsos, Poisson-Lie T-duality and supersymmetry, THU-96/38, hep-th/9611199. [201 S.E. Parkhomenko, Pisma Zh. Eksp. Teor. Fiz. 64 (December 1996) 823; Poisson-Lie T-duality and N = 2 superconformal WZNW models, Landau TMP/12/96, hep-th/9612034. 12 l I S.E. Parkhomenko, Poisson-Lie T-duality in N = 2 superconformal WZNW models on compact groups, Landau TMP/05/97, hep-th/9705233. 1221 S.E. Parkhomenko, Zh. Eksp. Teor. Fiz. 102 (1992) 3. 1231 S.E. Parkhomenko, Mod. Phys. Lett. A 11 (1996)445.
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[24] M.A. Semenov-Tian-Shansky, Dressing Transformations and Poisson-Group Actions, RIMS, Kyoto Univ. 21 (1985) 1237. [251 A.Yu. Alekseev and A.Z. Malkhin, Commun. Math. Phys. V162 (1994) 147. [261 P. Di Veccia, V.G. Knizhnik, J.L. Petersen and P. Rossi, Nucl. Phys B 253 (1985) 701. [27] A. Polyakov and P. Wiegmanu, Phys. Lett. B 131 (1983) 121. 128 ] Ph. Spindel, A. Sevrin, W. Troost and A. Van Proeyen, Nucl. Phys. B 308 (1988) 662; B 311 (1988/89) 465. [291 E. Getzler, Manin pairs and topological field theory, M1T-preprint (1994). 130] J.-H. Lu and A. Weinstein, J. Diff. Geom. 31 (1990) 501. [ 31 I S. Kobayasi and K. Nomizu, Foundations of Differential Geometry 11 (lnterscience Publishers, 1969). 132 ] M.Wakimoto, Commun, Math. Phys. 104 (1994) 605. [331 B. Feigin and E. Frenkel, Commun. Math. Phys. 128 (1990) 161; P. Bouwknegt, J. McCarthy and K. Pilch, Prog. Theor. Phys. Suppl. 102 (1990) 67; B. Feigin and S. Parkhomenko, Regular representations of Affine Kac-Moody algebras, RIMS, Kyoto Univ. 943 (1993). 1341 J. Evans and T. Hollowood, Nucl. Phys. B 352 (1991) 723. 1351 V.G. Drinfeld, DAN SSSR V262 (1983) 285. 136J Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989) 232. 1371 CM. Hull and B. Spence, N = 2 current algebra and coset models, QMC/PH/89/24 (1989). 138 ] C. Klimcik and E Severa, Poisson-Lie T-duality: Open strings and D-branes, CERN-TH/95-339, hepth/9512124.
Nuclear Physics B 510 [PM] (1998) 623-639
Poisson-Lie T-duality and complex geometry in N = 2 superconformal WZNW models S.E. P a r k h o m e n k o
1
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
Received 1 July 1997; revised 22 September 1997; accepted 23 October 1997
Abstract Poisson-Lie T-duality in N = 2 superconformal WZNW models on the real Lie groups is considered. It is shown that Poisson-Lie T-duality is governed by the complexifications of the corresponding real groups endowed with Semenov-Tian-Shanskysymplectic forms, i.e. Heisenberg doubles. Complex Heisenberg doubles are used to define on the group manifolds of the N = 2 superconformal WZNW models the natural actions of the isotropic complex subgroups forming the doubles. It is proved that with respect to these actions N = 2 superconformal WZNW models admit Poisson-Lie symmetries. The Poisson-Lie T-duality transformation maps each model onto itself but acts non-trivially on the space of classical solutions. (~ 1998 Published by Elsevier Science B.V. PACS: 11.25.Hf; 11.25.Pm
Keywords: Strings; Duality; Superconformalfield theory
1. I n t r o d u c t i o n N = 2 superconformal field theories ( S C F T ' s ) play the role of building blocks in the superstring vacua construction. The investigations in this direction were initiated in the pioneering works of Refs. [ 1,2], where it was shown that N = 2 SCFT's may describe Calabi-Yau manifolds compactifications of superstrings. Since then the N = 2 SCFT's, and the profound structures associated with them are an area of investigation. On the other hand, it is well known that the string vacua, considered as conformal field theories, in general, have deformations under which the geometry of the target space changes. I E-mail: [email protected] 0550-3213/98/$19.00 (~) 1998 Publishedby Elsevier Science B.V. All rights reserved. Pll S0550-32 l 3 (97) 00712-8
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These deformations include the discrete duality transformations, so called T-duality, which are symmetries of the underlying conformal field theory [3,4]. A well-known example of T-duality is mirror symmetry in the Calabi-Yau manifold compactifications of the superstring [5]. The Poisson-Lie (PL) T-duality, recently discovered by Klimcik and Severa [6] is a generalization of the standard non-Abelian T-duality [7-11]. The main idea of the approach [6] is to replace the requirement of isometry of a o--model with respect to some group by the weaker condition of the Poisson-Lie symmetry of the theory. This generalized duality is associated with two groups forming a Drinfeld double [ 12] and the duality transformation exchanges their roles. This approach has been developed further in Refs. [ 13-17]. In order to apply PL T-duality in superstring theory one needs to have dual pairs of conformal and superconformal ~r-models. A simple example of a dual pair of conformal o--models associated with the 0 ( 2 , 2) Drinfeld double was presented in Ref. [ 18]. Then, it was shown in Refs. [ 14,15] that WZNW models on compact groups are natural examples of PL dualizable o--models. The supersymmetric generalization of PL T-duality was considered in Refs. [ 1921]. In particular, due to the close relation between N = 2 superconformal WZNW (SWZNW) models and Drinfeld's double (Manin triple) structures on the corresponding group manifolds (Lie algebras) [22,23], it was shown in Ref. [20] that N = 2 SWZNW models possess a very natural PL symmetry, and PL T-dual o--models for N = 2 SWZNW models associated with real Drinfeld's doubles were constructed. Then the first example of PL T-duality in N = 2 SWZNW models on the compact groups was obtained in Ref. [21] for U(2)-SWZNW model. In the present paper we generalize the results of our preceding paper [21] to the case of N = 2 superconformal WZNW models on compact groups of higher dimensions. These models correspond to the complex Manin triples endowed with hermitian conjugation which conjugates isotropic subalgebras forming the Manin triples. After a brief review of the classical N = 1 superconformal WZNW models in Section 1, we describe in Section 2 the complex geometry of N = 2 SWZNW models on compact groups. We show that the Heisenberg double [24,25] of the complexification of the N = 2 SWZNW model group manifold plays a central role in PL T-duality: on the one hand, the Lagrangian of the model can be expressed (locally) in terms of the components of Semenov-Tian-Shansky symplectic form of the Heisenberg double. On the other, there is the natural action of the isotropic subgroups forming the double on the space of fields of the model. In Section 3 we show that each N = 2 SWZNW model admits PL symmetries with respect to these actions which we use to show that the PL T-duality transformation maps the model onto itself but acts non-trivially on the space of solutions. Though our results are concerned with N = 2 SWZNW on compact groups they can be straightforwardly generalized to real non-compact groups.
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2. The elassleal N = 1 supereonformal W Z N W model
In this section we briefly review the N = 1 SWZNW models using the superfield formalism [26]. We parametrize the super world-sheet introducing the light-cone coordinates x±, and Grassmann coordinates O±. The generators of supersymmetry and the covariant derivatives are 3
3
Qm = 30---~ + tOiOm'
D m - O0±
lO±cgm"
( 1)
They satisfy the relations {D+, D:t:} = - { Q + , Q+} = -12o+, {O+, Din} = {Q+, Qm} = {Q, D} = 0,
(2)
where the brackets {, } denote the anticommutator. The superfield of the N = 1 SWZNW model,
G = g + tO_g,+ + tO+~,_ + iO_O+F,
(3)
takes values in a Lie group G. We will assume that its Lie algebra g is endowed with an ad-invariant non-degenerate inner product ( , ) . The inverse group element G - l is defined by the relation
G-IG = 1
(4)
and has the decomposition G - I = g - i _ tO_g-l~+g-1 _ tO+g-l~_g-i
-lO_O+g -1 ( F + ~ _ g - l g % _ ~ + g - l ¢ _ )g-l.
(5)
The action of the N = 1 SWZNW model is given by Sswz = f dZxd20 ((G-1D+G, G - 1 D - G } )
-- [ d 2 x d 2 { ~ d t
dt
(6)
The classical equations of motion can be obtained by making a variation of (6),
D _ ( G - I D + G ) = D + ( D _ G G - l ) = 0.
(7)
The action (6) is invariant under the super-Kac-Moody
6a, O(x+, x_, 0+, O- ) = a+(x_, O+)O(x+, x_, 0+, O_ ), fi,,_ G(x+, x_, 0+, O_ ) = - G ( x + , x_, 0+, O _ ) a _ (x+, O_ ), where a± are g-valued superfields and N = 1 supersymmetry transformations [26]
(8)
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S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
G-13E+ G = ( G - I e + Q + G ) ,
(9)
8,_GG -I = e-Q-GG -l •
In the following we will use the supersymmetric version of the Polyakov-Wiegmann formula [ 27 ] S s w z [ G H ] = Sswz[G] + Sswz[H] +
fd2xd20(G-ID+G,D-HH-1).
(10)
It can be proved as in the non-supersymmetric case.
3. Complex geometry in N = 2 superconformal W Z N W models
In Refs. [ 22,23,28 ] supersymmetric WZNW models which admit extended supersymmetry were studied and the correspondence between extended supersymmetric WZNW models and finite-dimensional Manin triples was established in Refs. [22,23]. By definition [ 12], a Manin triple (g, g+, g_ ) consists of a Lie algebra g, with a non-degenerate invariant inner product ( , ) and isotropic Lie subalgebras g± such that g = g+ ® g_ as a vector space. The corresponding Sugawara construction of N = 2 Virasoro superalgebra generators was given in Refs. [22,23,28,29]. To make a connection between the Manin triple construction of Ref. [ 22,23 ] and the approach of Ref. [28] based on complex structures on the Lie algebras the following comment is relevant. Let g be a real Lie algebra and J be a complex structure on the vector space g. J is referred to as the complex structure on the Lie algebra g if J satisfies the equation [ J x , Jy] - J [ J x , y] - J [ x , Jy] = [ x , y ]
(11)
for any elements x, y from g. It is clear that the corresponding Lie group is a complex manifold with left (or right) invariant complex structure. In the following we will denote the real Lie group and the real Lie algebra with the complex structure satisfying (11 ) as the pairs (G, J) and (g, J) correspondingly. Suppose the existence of the non-degenerate invariant inner product (,) on g so that the complex structure J is skew-symmetric with respect to (,). In this case it is not difficult to establish the correspondence between complex Manin triples and complex structures on the Lie algebras. Namely, for each complex Manin triple (g, g+, g_ ) there exists a canonic complex structure on the Lie algebra g such that the subalgebras g+ are its i t eigenspaces. On the other hand, for each real Lie algebra g with non-degenerate invariant inner product and skew-symmetric complex structure J on this algebra one can consider the complexification gC of g. Let g± be i t eigenspaces of J in the algebra gO. Then (gC, g+, g_) is a complex Manin triple . Moreover, it can be proved that there exists a one-to-one correspondence between the complex Manin triple endowed with an antilinear involution which conjugates isotropic subalgebras ~" : g± ---+g:F and the real
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
627
Lie algebra endowed with an ad-invariant non-degenerate inner product ( , ) and the complex structure J which is skew-symmetric with respect to ( , ) [22]. Therefore we can use this conjugation to extract the real form from the complex Manin triple. If the complex structure J on the Lie algebra is fixed then it defines the second supersymmetry transformation [28]
(G-I&7+ G) " =rI+(JI)~(G-ID+G) b, (67 GG-1)a=rI_(Jr)~,(D_GG-1) b,
(12)
where Jl, Jr are the left invariant and right invariant complex structures on G which correspond to the complex structure J. To specify our presentation we concentrate in this paper on N = 2 SWZNW models on compact groups (the extension to non-compact groups is straightforward), i.e. we shall consider complex Manin triples such that the corresponding antilinear involutions will coincide with the hermitian conjugations. Hence it will be implied in the following that G is a subgroup in the group of unitary matrices and the matrix elements of the superfield G satisfy the relations ~,,,,,= ( g - l ) n m ,
~ n = (///-1)7,
pmn = ( F - l ) n m ,
(13)
where we have used the following notations:
(s~l = _g-l~+g-l,
F -1 = --g-l(F + ~ _ g - l ¢ + -~b+g-l~b_)g -1.
(14)
Now we have to consider some geometric properties of the N = 2 SWZNW models closely related with the existence of the complex structures on the corresponding groups. Let us fix some compact Lie group with the left invariant complex structure (G, J) and consider its Lie algebra with the complex structure (g, J). The complexification gC of g has the Manin triple structure (gC,g+,g_). The Lie group version of this triple is the double Lie group (GC, G + , G _ ) [24,30,25], where the exponential subgroups G± correspond to the Lie algebras g+. The real Lie group G is extracted from its complexification with the help of the hermitian conjugation r, G = {g C GClr(g) = g - l } .
(15)
Each element g C G C from the vicinity Gl of the unit element from G c admits two decompositions,
g = g+gZi = ~_~+l,
(16)
where ~± are dressing transformed elements of g+ [30]: ~,+ = (g~l)uv.
(17)
Taking into account (15) and (16) we conclude that the element g (g ~ G1) belongs to G iff
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628
r(g+) = ~,~l.
(18)
These equations mean that we can parametrize the elements from C1 -- Gl f3 G
(19)
by the elements from the complex group G+ (or G _ ) , i.e. we can introduce complex coordinates (they are just matrix elements of g+ (or g _ ) ) in the cell C1. To do this one needs to solve, with respect to g_, --1
r(g_) = (g+)g-
(20)
[to introduce G_ coordinates on Gl one needs to solve, with respect to g+, --I
r(g+) = (g_)g+ ].
(21)
Thus the formulas (16), (20), [ (21) ] define the map ~b+ • G+ ~ Cl,
(22)
[~bi- : G_ ~ C~].
(23)
For the N = 2 SWZNW model on the group G we obtain from (16) the decompositions for the superfield (4) (which takes values in CI )
G(x+,x_) = G+(x+,x_)G-l(x+,x_ ) = G_(x+,x_ )G+l(x+,x_).
(24)
Due to (24), (10) and the definition of Manin triple we can rewrite the action (6) for this superfield in the manifestly real form Sswz = - ~
1/
d2xd20 ( ( p + , p Z ) + {/5+,/5+_)),
(25)
where the superfields
p+ = G~_IDG+,
/5+ = G ~ l D G +
(26)
correspond to the left invariant 1-forms on G~:,
r ± = g~Jdg+,
~± = ~T_ld~±.
(27)
To proceed further we need to prove the following:
Lemma 1. (1) The g+-valued 1-form r + considered as a form on (CL, J) is holomorphic. (2) the g_-valued 1-form ? - considered as a form on (Cl, J) is antiholomorphic.
Proof We identify the Lie algebra gC with the space of complex left invariant vector fields on the group G. Let
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629 (28)
{Ri, i = 1 . . . . . d}
be the basis in the Lie subalgebra g+ and { R i , i = l . . . . . d}
(29)
the basis in the Lie subalgebra g_ so that (28), (29) constitute the orthonormal basis in gO: (30)
(R i, R.i ) = 6j.
Denote by ( the canonic gO-valued left invariant 1-form on the group G. It is obvious that ( = ~ J R . i + ~ j R J,
j(k = _t(~,
J(k = t(k.
(31)
We can also write r + = r~ei,
r - = ri Ri,
?+ = ~iRi,
r
= ri Ri,
I + ~ dg+g+ 1 = liRi,
l - =- d g _ g -1 = li Ri,
/+ ~ d,~+~+ t = [iRi,
[- =- d ~ _ ~ -1 = [iR i.
(32)
(1) Using the first decomposition from (16) we get (33)
sc = g _ r + g -1 - I - .
Let us introduce the matrices g _ R i g - j = M i j R J + NJi Rj, g+Ri g+ 1 = piJ Rj + Q j R j, g _ R i g -1 = ( N * ) ~ R j, g+RigT l = (Q*)JRj.
(34)
Using these matrices and (30) we can express the 1-forms r i, ri, I i, li in terms of sci, r.i=
(N-1)j~ i,
~k, li = --S~i -~ M ji ~t N - l " Jd kb ri = ( ( N * ) -1 )f ( _ ¢ j + M n j ( N - I )~¢k), Ii = ( N - I ) ~ ( Q * ) ~ s
c~.
(35)
Taking into account ( 3 l ) , (35) we obtain Jr k = - t r k,
Jl ~ = - t l ~.
(36)
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S.E. Parkhomenko/Nuclear Physics B 510 [PM] (19981 623-639
Thus r k, l t, k = 1 . . . . . d are (1,0)-forms and their complex conjugates ?k, rk, k = 1 . . . . . d are (0,1)-forms on (C1, J ) . Because the forms r k, ~ are linearly independent on ( C l , J ) and linearly independent on G+ we conclude that the map ~bl is holomorphic [31 ], where the complex structure on G+ is given by the multiplication by t. Since the forms r k are holomorphic as the forms on G+ statement (1) of the lemma follows. (2) Using the second decomposition from (16) we can write
= g+r ~-~1 _ [4-.
(37)
Introducing the matrices
~_Ri~, -1 = 2(4ijRj + N~ Rj, a~+Ri~+ 1 = piJ Rj + O_.}RJ, ~_ gi~ -1 = ( g * ) ~ g j, ~+Ri~+ l = ( O*)~ Rj
(38/
and using (30), one can express the 1-forms P, ri, 7i, [i in terms of ~:i,
~,j = ( 0 -1 )~Zi, yi = __(i _.~_p j i ( O--1 ) jk..~:k,
~. = (Q --1 ) jk( N * )i~k" j
(39)
In view of (31), (39) we obtain
JPk = tPk,
J[k = t[~.
(40)
Thus fk, [k, k = 1 . . . . . d are (0,1)-forms and their complex conjugates Fk, [k, k = 1 . . . . . d are (1,0)-forms on (C~, J ) . Because the forms ?k, ~k are linearly independent on (C1, J ) and linearly independent on G _ we conclude that the map ~b~- is holomorphic, where the complex structure on G _ is given by the multiplication by - t . Since the forms ?k are antiholomorphic as the forms on G _ , statement (2) of the lemma follows. Due to L e m m a 1 the forms
{r i,Pi},
i= 1. . . . . d
(41)
constitute the basis of holomorphic and antiholomorphic 1-forms in the open subset C1. Consequently, in the basis of (28), (29) we can write the components of g+-valued 1-forms p - , / 5 + as follows:
Pi = ELTPj + EijP / ,
/5i = /5i = EfiP i "+"ETjPJ,
where we have used the notations
(42)
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639 E'6 =
Eft,
Eij =
631
ETj.
(43)
Thus the Lagrangian of the action (25) has the following form: A = pi+ ( P - ) i - ~ i (P+)i i -J - p~_pJ+) " " = ½((E(i - Eft) -}- (E(/+ Eji))pi+p/_ +Ei3(p+p_
+½( (E(7- E~) + (Eo + E)7)),b~_/3./_.
(44)
Let us show that
Eij + Eji =
EO+E37= 0.
(45)
Using (16) we can represent the kinetic term of the action (6) in the following form: i / + Ei)(pi+[9j_ - pi ~+) -2
+(
-j + E~)p+p_.
(46)
This expression means that the bilinear form ( , ) on G written in the basis (41) has the following non-zero components:
--gij = Eij + Eji,
- g i ] = 2Ei3,
-K;3 = E~) + E3;.
(47)
On the other hand, the complex structure J is skew-symmetric with respect to (, >, and therefore (45) should be satisfied. L e m m a 2. The Lagrangian A can be expressed in terms of the Semenov-Tian-Shansky symplectic form ,(2, l , o --c a b A = ~Zcba~p+p_,
(48)
where we have used the common notation pa, a = 1. . . . . 2d for the 1-forms pi, ~i. P r o o f In the open subset CI, where the decomposition (16) holds, the Semenov-TianShansky symplectic form /2 can be represented as follows [25,16]: 12 = r i/~ r i
+ ~i A r i .
(49)
Due to the forms ri, Fi c a n be expressed in terms of r i, ~i in perfect analogy to Pi, ~i the restriction o f / 2 on any world-sheet (which should not be confused with the super world-sheet of the N = 2 SWZNW model under consideration) is given by 12 [z = dx+ A d x _ ( ~1 ( Eij - Eji) (ri+r~ - r i r~ ) + 2E(?(ri+~'~ - r i F'~ ) -
-:
).
(50)
Comparing (44) with (50) and taking into account (45) and Lemma 1 we obtain (48).
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
632
To generalize (16), (18) one has to consider the set W (which we shall assume in the following to be discrete and finite) of classes G + \ G C / G _ and pick up a representative w for each class [w] C W. It gives us the stratification of G ¢ [25]: Gc=
U
G+wG_=
[wICW
U
Gw.
(51)
GW.
(52)
[w]EW
There is a second stratification, GO=
U
G_wG+=
U [w]EW
[wIEW
We shall assume, in the following, that the representatives w satisfy the unitarity condition r ( w ) = w -I
(53)
It allows us to generalize (16), (18) as follows: (54)
g = g+wg - l = ~_w~+ 1.
It will be more convenient to rewrite this decomposition in another form, (55)
g = wg+g - I = wg_g+ 1.
The group elements g+, ~,+ from this formula should not be confused with the group elements from (54) (of course they are mutually related, but do not coincide). To make the decompositions (55) unambiguously determined we should demand that W g+ C G+,
~_ C G~_' ,
(56)
where G~ = G+ N w - l G + w ,
Gw _ = G_ N w-lG_w.
(57)
A helpful example, where the formulas ( 5 1 ) - ( 5 3 ) , (55) hold, is the Bruhat decomposition of the complexification of an even-dimensional semisimple compact Lie group with an appropriate maximal torus decomposition. In order for the element g to belong to the real group G the elements g+, ~± from (55) should satisfy (18). Thus the formulas (55), (56), (20) [(21)] define the map
4'~ : G÷W ---, Cw --- Gw n G,
(58)
[~bw : G w ~ Cw-= Gw N G ] .
(59)
We can obtain the corresponding generalization of Lemma 1 for the g~_ components of the form r +, where g~_ = g+ n w - l g + w [gW components of the form ? - , where gW = g_ n w - l g _ w ] proving by analogy with the proof of the Lemma 1 that the map (58) [ (59) ] is holomorphic.
S.E. Parkhomenko/NuclearPhysics B 510 [PM] (1998) 623-639
633
Using the appropriate generalization of (24) and taking into account (56), (57) we can conclude that the action for the map into the cell Cw is given by (25). As the map (58) is holomorphic, the same is true for the Lagrangian: it is given by (48). where paRa C g+. It is clear that Eq. (25) is correct inside the super world-sheet domain where the superfields take values in the cell Cw. On the boundaries of these domains, where jumps from one cell to another take place, some additional terms should be added, but as will be explained below, for our purposes it will suffice to ignore these terms. Eqs. (20), (55) and (25) mean that there is a natural action of the complex group G+ on G, and the set W parametrizes the G+ orbits Cw. It is obvious that there is also the action of the complex group G_ on G so that the analogies of Eqs. (44), (45), (48) can be obtained in a similar way.
4. Poisson-Lie s y m m e t r y in N = 2 S W Z N W model
In view of (20), (55), (25) we can consider the (G, J ) - S W Z N W model as a o-model on the orbits of the complex Lie group G+ and find the equations of motion by making a varying the action (25) under the right action of this group on itself. We will consider the variations which are non-zero only inside the super world-sheet domains, where jumps from one cell Cw to another one take place. Thus we can not take into account the corresponding boundary terms omitted in the action (25). The set of holomorphic maps {~bw + , w c W} defines the action of G+ on the group G:
h+ .g ==-wg+h+gS l(g+h+),
h+ E G+,
(60)
where g-J (g_h+) means the solution of (20) with the argument g+h+. The vector fields {Si, i = 1 . . . . . d} generating this action are the holomorphic vector fields on the cells Cw. Note that in the case when G is an even-dimensional semisimple compact Lie group and ( 5 1 ) - ( 5 3 , (55) are given by the Bruhat decomposition, these vector fields will coincide with the classical screening currents in the Wakimoto representations of [32]. This is due to the fact that the classical screening currents in the Wakimoto representations are given by the right action of the maximal nilpotent subgroup N+ on the big cell of the corresponding flag manifold G / B _ , where B_ is the Borel subgroup of the group G (N+ E B+) [33]. Let us consider a variation of (25) under the vector field Z = zisi + Z;Si for the map onto the cell Ci. We obtain on the extremals
D + ( A _ ) i + D_ (A+)i - LsiA=O, D + ( A - ) 7 + D+(A+) 7 - LLA=O, where Ls,, Ls, mean the Lie derivatives along the vector fields St, currents Ai, A; are given by
(61)
Si
and the Noether
634
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639 ( A _ )i = E o p )_ -4- EijP)--, ( A+ ) i =-Ei~pt+ - EjiP)+,
(,4a:)i = (A+)7.
(62)
As the fields Si are holomorphic, we get from (48)
Ls, A~b = t j~Ls~S2cb. 2
(63)
Now one has to find the Lie derivative of the Semenov-Tian-Shansky symplectic form. Because Lsj = isjd + disj and s2 is closed, we have
LsiO = disjS2 = 2drj, LL.O = di~jS2 = 2d?j,
(64)
where we have used (49). In view of (27) drj, d~j can be expressed in terms of r.i, ~i again,
drj=
lf~kriArl~,
d~j=_lfjk~iA?l~,
(65)
where f~k are the structure constants of the Lie algebra g_. Thus the action (60) is the Poisson action [24] with respect to the Poisson structure defined by the symplectic form .(2. Therefore
LsiA= t f~k ( jp+ )i(P- ) k, LL A = tflk( j~+ )i(~_ )k.
(66)
Taking into account the relations (A-)i
= (P-)i,
(67)
(A+)i = t(Jp+)i,
where the second equality is due to (62) and Lemma 1, we get that the following PL symmetry conditions are satisfied on the extremals of N = 2 (G, J)-SWZNW model:
Ls~A= f ] k ( A + ) j ( A - ) ~ , LL A = ~k(A+)3(a_)k"
(68)
By demanding the closure of (68), [Ls,, Lsj] = f~Lsk, we have the consistency condition
f~j fkm = f~" f~n - f~k fm _ fnm f~ n + f~k f,]~,
(69)
which is satisfied due to the Jacobi identity in the Lie algebra gC. As is easy to see from (61), Eqs. (68) are equivalent to the zero-curvature equations for the F+_-component of the super stress tensor FMu, (F+-)i ~ D+(A-)i
+ D-(A+)i
- finm(A+)n(A-)m
(F+_); =- D+(A_)7 + D - ( A + ) 7 -
f'/'(a+)~(a_)~,
= O, = 0.
(70)
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
635
Using the standard arguments of the super Lax construction [34] one can show that from (67) it follows that the connection is flat, FMN = 0 ,
M,N=
(+,--,+,-).
(71)
Eqs. (70) are the supersymmetric generalization of the Poisson-Lie symmetry conditions from Ref. [6]. Indeed, the Noether currents Ai, A 7 are generators of the g+ action, while the structure constants in (70) correspond to the Lie algebra g_, which is Drinfeld's dual to g+ [35]. Now we turn to the maps onto the remainder cells Cw, w E W. For each w ~ W the action and the Lagrangian for the map onto Cw are given by (25) and (48) with the restriction (56). Consequently, the corresponding equations of motion will coincide with (70), where only the g~_ components of the Noether currents (62) are not identically zero. Thus we have shown that the N = 2 (G, J ) - S W Z N W model admits the Poisson-Lie symmetry with respect to the complex group G+.
5. P o i s s o n - L i e T-self-duality of N = 2 S W Z N W models
The PL T-dual to the (G, J ) - S W Z N W o--model should obey conditions as (70) but with the roles of the Lie algebras g+ interchanged [6]. To find the action of this model we start from the maps onto the cell C~. Due to (71) we may associate to each extremal surface G + ( x + , x _ , O + , O_) C G+, a map ("Noether charge") V_ (x+, x _ , O+, O_ ) from the super world-sheet into the group G_ such that ( A+ )i = - ( D ± V - V_-1 )i.
(72)
Now we build the following surface in the double GC: F ( x+, x_, O+, O_ ) = G+ ( x+, x _, O+, O_ ) V_ ( x+, x_, O+, O_ ).
( 73 )
In view of (67) it is natural to represent V_ as the product V_ = GSJH51 ,
(74)
where G_ is determined from (20) and H_ satisfies the equation D _ H _ = 0.
(75)
Therefore the surface (73) can be rewritten in the form F ( x ± , O± ) = G ( x ± , O+ ) H _- l (x+, O_ ),
(76)
where G ( x ± , O ± ) E Cl is the solution of the (G)-SWZNW model restricted to the corresponding domain of the super world-sheet.
S.E. Parkhomenko/NuclearPhysicsB 510 [PM] (1998)623-639
636
The solution and the "Noether charge" of the dual o--model are given by a "dual" parametrization of the surface (73) [6]
F(x+,x_,O+,O_) = G_(x+,x_,O+,~9_)fZ+(x+,x_,O+,O_),
(77)
where G_(x+,x_,O+,O_) 6 G_ and fZ+(x+,x_,6)+,O_) 6 G+. Thus in the dual o-model Drinfeld's dual group to the group G+should acts, i.e. it should be a o'-model on the orbits of the group G _ and with respect to this action the PL symmetry conditions dual to (68) should be satisfied,
Ls, fl= f l k ( i + ) J ( A - ) k, L~zi = f ] k ( A + ) J ( i - )
where {S i, ~i,
~,
(78)
i = 1 . . . . . d} are the vector fields which generate the G_ action, it,
i~:, i ~ are the Lagrangian and the Noether currents in the dual o--modeh Taking into account the second decomposition from (16) and holomorphicity of ~b~-, it is easy to see that the right action of G_ on itself defines the action on the cell Cj:
h_.g=_~_h_~,+l(~_h_),
h_ c G _ ,
(79)
so that the vector fields generating this action are the holomorphic vector fields. Therefore, due to Eq. (49) the Lagrangian A of the initial N = 2 SWZNW model on the cell Cl written in G_ coordinates satisfies (78). Thus the dual o--model on the cell C1 is governed by the action (25) and we can rewrite the right-hand side of (77) analogously to (76),
F(x±, O:a:) = G(x+, O+ )/V/~l (x+, O-- ),
(80)
where t~(x+, ~9+) E CI and D_/:/+ = 0.
(81)
For the maps onto the remainder cells Cw we also have the surfaces in the double G c written in two ways,
F(x+, x_, 0+, 0_) = wG+(x+,x_, 0+, O_)V_(x+,x_,O+, 0 _ ) =wG_(x+,x_,O+,O_ )~'+(x+,x_,¢9+,O_ ),
(82)
where G+(x~,~9~),(/+(x±,O+) 6 G~_, G _ ( x ± , O ± ) , V _ ( x ± , O ± ) E G w. The arguments we have used for the maps onto C1 can be applied (with the appropriate modifications due to the restrictions (56)) to this case also because for each w 6 W the map ~bw is holomorphic, so we conclude that the dual tr-model on each cell Cw is governed by the action (25). Hence, we can rewrite (82) in two more ways:
F(x±, O± ) = wG(x±, O+ )H -1 (x+, O_ ) = wG(x±, O± )/r./+ 1 (x+, ~9_ ),
(83)
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
637
where G , G E Cw, H _ E GW_,/:/+ E G~_. It is clear that the set of equations (76), (80), (83) defines the map from the total super world-sheet onto the double Go: F ( x ± , O± ) = G ( x ± , O ± ) H -1 (x+, O_ )
= G(x4-, (9±)F/+ 1(x+, 69_ ),
(84)
where G ( x + , 04,) is the solution of ( G ) - S W Z N W model and G ( x + , O4.) E G. From this equation we see that G ( x + , (94.) is the classical solution of the (G, J ) - S W Z N W model also because it is obtained from the solution G ( x ± , (94-) by the right multiplication on the G-valued function H - H21[-I+ that satisfies the equation D _ H = 0.
(85)
Thus the dual o--model coincides with the initial one, i.e. the N = 2 SWZNW model on the real Lie group is PL T-self-dual and the PL T-duality transformation is a special type of super Kac-Moody symmetry.
6. Conclusions We have shown that Poisson-Lie T-duality in the classical N = 2 SWZNW models on compact Lie groups is governed by complexifications of these groups endowed with the Semenov-Tian-Shansky symplectic forms, i.e. Heisenberg doubles. Each N = 2 SWZNW model admits very natural PL symmetries with respect to the natural actions of the isotropic subgroups forming the complex Heisenberg double. Under the PL T-duality transformation the N = 2 SWZNW model maps onto itself but this transformation acts as some special super Kac-Moody symmetry on the classical solutions. Thus N = 2 SWZNW models on compact groups are PL T-self-dual. This result bears a resemblance to the T-self-duality in the self-dual torus compactifications [ 3 ]. Note also that in order to apply PL T-duality transformations we used only the left invariant complex structure on the group manifold. Since the (2, 2) Virasoro superalgebra of symmetries also requires the right invariant complex structure, an interesting question which certainly deserves further attention is as to what happens with the right invariant complex structure under this transformation. Another point which is believed to be interesting is the N = 4 SWZNW model's generalization of PL T-duality. The N = 4 superconformal algebra demands the existence of an infinite number of complex structures on the Lie algebra of the model which are parametrized by the points of a two-dimensional sphere [23,28]. It is reasonable to expect that the quaternionic geometry will appear in these models. We expect the PL T-duality to exist also in N = 2 superconformal Kazama-Suzuki models [36] since these models can be represented as the cosets (N = 2 G-SWZNW m o d e l ) / ( N = 2 H-SWZNW model), where H is a subgroup of G [37]. Another interesting question is related to the quantum picture of the PL T-duality in N = 2 SWZNW models. Because the Poisson-Lie groups is nothing but a classical
638
S.E. Parkhomenko/Nuclear Physics B 510 [PM] (1998) 623-639
limit of the quantum groups [ 12], there appears to be an intriguing possibility of the relevance of quantum groups in the T-duality and other superstring applications for example in D-branes [ 15,38].
Acknowledgements I am very grateful to B. Feigin for discussions. I would like to thank the VolkswagenStiftung for financial support as well as the members of the group of Prof. Dr. R. Shrader for hospitality at Freie Universit~it of Berlin, where a significant part of this work was done. This work was supported in part by grants INTAS-95-IN-RU-690, CRDF RP1-277, RFBR 96-02-16507.
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