non abelian t-duality for open strings

ELSEVIER Nuclear Physics B (Proc. Suppl.) 56B (1997) 36-41

PROCEEDINGS SUPPLEMENTS

Non Abelian T-duality for open strings* Stefan FBrste,at Alexandros A. Kehagiasb t and Stefan SchwagerC 5

aSektion Physik,Universitat Miinchen, TheresienstraBe 37, 80333 Miinchen, Germany

“Physik Department, Technische Universitat Miinchen, James fianck StraBe 85748 Garching, Germany

CSektion Physik,Universitat Miinchen, TheresienstraBe 37, 80333 Miinchen, Germany

In the first part of the talk we discuss T-duality for a free boson on a world sheet with boundary in a setting suitable for the generalization to non-trivial backgrounds. The gauging method as well as the canonical transformation are considered. In both cases Dirichlet strings as T-duals of Neumann strings arise in a generic way. In the second part the gauging method is employed to construct the T-dual of a model with non-Abelian isometries.

1. Introduction

During the last years the appealing picture arose that all string theories are just suitable de- scriptions of one underlying unique theory. This picture is supported by the discovery that many string theories are linked to each other via duality- transformations. Among those dualities T- duality is around already for quite some time [1,2]. It is a weak coupling/weak coupling duality relating small compactification radii on one side to large compactification radii on the dual side. In some cases it maps different looking string theories on each other like type HA on type IIB [3-61 or the heterotic Es x Es on a heterotic SO(32) string (with symmetry breaking Wilson lines)[7]. Already some time ago it has been ob- served that Dirichlet boundary conditions arise in the T-dual description of toroidally compacti- tied open strings with the usual Neumann boundary conditions [3,8,9]. That is the T-dual picture

*Talk given by S. F5rste at the International Symposium on the Theory of Elementary Particles Buckow, August 27 - 31, 1996; Work partly supported by the EC programme SCl-CT92-0789 and the European Commission TMR pro- grammes ERBFMRX-CT96-0045 and ERBFMRX-CTBB 0090. temail: stefan.foersteQphysik.uni-muenchende ternail: kehagiasOphysik.tu-muenchende Bemail: stefan.schwagerOphysik.uni-muenchen.de

includes extended objects (D-branes) in a natural way, (for an excellent review see [lo]). Those D-branes play an important role in establishing string dualities like e.g. type I/heterotic duality [ll) or in identifying heterotic/heterotic duality in six dimensions with T-duality in an open string theory [12]. Therefore it appears to be interest- ing to generalize the T-duality transformation for open strings living in a non trivial background.

That in backgrounds possessing a Poisson-Lie symmetry T-duality maps Neumann on Dirich- let conditions has been shown in [13]. Later on, T-duality was carried out in general backgrounds with Abelian isometries [ 14,151. The generalization for backgrounds with non-Abel&i isometries is worked out in [16], which will be partly re- ported in the present talk. There have been some publications discussing T-duality corresponding to non-Abelian isometries [17], [18-231. Here we will follow the initial work of [17], i.e. we have to restrict ourself on semi-simple isometry groups.

Non-Abelian T-duality for open strings has been discussed in [24] as a canonical transformation. In [24] results differing from ours have been obtained. Below we will argue that a proper in- clusion of boundary effects in the canonical transformation gives Dirichlet conditions as a dual for free varying ends of the string, generically.

0920-5632/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII: SO920-5632(97)00307-l

S. FCrste et al. /Nuclear Physics B (Pnx. Suppl.) 568 (1997) 3641 37

In the next section we will discuss T-duality for the simplest example, viz. a free boson. We will use the gauging method [25] and we will also discuss a canonical transformation approach. Knowing the results for the closed string it will be straightforward to extend those considerations to less trivial backgrounds. In section three we gauge the non-Abelian isometries of a sigma model with boundary and sketch the T-duality transformation. The last section concludes the talk also mentioning some open problems. In a brief appendix we give basic ideas about the canonical transformation.

2. The free boson

Combining the following discussion of the free boson with known results of closed string theory it will be straightforward to modify closed string results to the open string. The action for the free boson is given by

s = s

d2z a,xa”x. (1) c In addition we specify boundary conditions

bV,X1sn = 0, (2)

with b being some two dimensional vector. If b is tangent to the boundary we have, up to a constant, Dirichlet conditions and otherwise Neu- mann or mixed conditions. This model is invariant under global shifts of X. Now we are going to gauge that symmetry and at the same time to suppress excitations of the gauge fields, i.e. to rewrite the theory to an equivalent one. Integrat- ing out the gauge fields will then give the T-dual model. When suppressing gauge field excitations one also has to care about global excitations in order to obtain a full equivalence. That issue will be neglected here since in the non-Abelian case this is an unresolved problem also in the closed string [l&20]. We want to construct an action invariant under

x 4 x + f(z)

in combination with

R, + R, - q-&f.

(3)

(4

Defining covariant derivatives via

D,X = 8,X + R,.

a gauge invariant action is given by

(5)

S gauged = J, dLz (D,XD”X + XF) +jax ds (d” + Ic(s)b”) R,, (f-3

where F is the field strength corresponding to the isometry gauge field R,

F = fnbd&,, (7)

X is a Lagrange multiplier forcing the field strength F to vanish, t is the tangent vector on the boundary dC and c is an arbitrary constant. The second Lagrange multiplier n forces the b- component of the gauge field to vanish at the boundary. Integrating out X will constrain the gauge fields to be pure gauge, integrating out IE will imply boundary conditions on the gauge parameter such that the gauge parameter can be absorbed by a shift in X and the equivalence to the ungauged model is established.

Before integrating out the gauge fields it is useful to employ partial integrations such that the action does not contain derivatives of R. Then the R integral is ultra local and factorizes into an integral where the argument of the gauge field is in the bulk times an integral where the argument of the gauge field is on the boundary. The first one gives the dual action in terms of the dual coordinate X whereas the later integration results in a two dimensional delta function

6(‘) (ct, + nb, + At,) . (8)

In the case that b is not tangent to the boundary (i.e. Neumann or mixed conditions) the second Lagrange multiplier IE has to vanish and the re- maining one dimensional delta function imposes Dirichlet conditions on the dual coordinate

x1en = -c. (9)

On the other hand if we started with Dirichlet conditions (up to a constant), i.e. b is tangent to the boundary, (8) just relates the first to the second Lagrange multiplier on the boundary and no boundary conditions for the dual coordinate are specified, i.e. the ends of the dual string vary

38 S. Fiirste et al. /Nuclear Physics B (Proc. Suppl.) 56B (1997) 36-41

Figure 1. T-duality for open strings interchanges free varying ends with fixed ends.

freely, (the infinite factor 6(O) can be thought of as arising from specifying Dirichlet conditions only up to a constant). Finally we end up with the picture drawn in figure 1.

Before considering non trivial backgrounds we would like to discuss how to obtain the very same result as a canonical transformation. Now, let us choose as a specific world sheet a strip in the upper half plane bounded by the g = 0 and the c = R axis. The time r is the coordinate parallel to the boundary. Furthermore, let us assume that the time asymptotic of the fields is such that the boundary can be closed at infinite times. The action for a free boson X is given by

s = J cl%+ [(&x)2 - (a,X)“] +$ (S,=() - S,=,) (+c&X,

(10)

with c being some arbitrary constant. The boundary term is a total derivative integrated over a closed path and hence vanishes. It will prove useful to keep it, nevertheless. The canonical momentum is defined by

P=& T

(11)

leading to

P = 8,X for u E (O,n), (12)

p= f {

for f-7 = 0 -f for u=?r (13)

for free varying ends, i.e. the net momentum flow through the boundary is zero. The generating functional for the canonical transformation5 is given by the expression

F [x,X] = ; lr dir (X&X - X&X) (14)

from which one obtains the canonical transformation via

SF = P6X - i%ff.

with

(15)

bF = S; du (-6X&$ + CL&X)

+a sxx - xsx ( ) lT=lr (16) 0=0

and (12) we get

a,2 = -4,x F = -0,x

for (T E (0,7r). (17)

Eq. (13) together with (16) gives

X = -c for d = 0,7r (18)

which in turn freezes the X variation to vanish at the boundary and leaves the boundary value of the dual momentum open, i.e. we see already at this stage that a Dirichlet string arises in the dual picture, which will turn out to be a generic

5Some basics about canonical transformations are given in the appendix.

S. Fijrste et al. /Nuclear Physics B (Proc. Suppl.) 568 (1997) 3641 39

result as long as the original string has free varying ends. (Note however that adding additional boundary contributions to the generating functional will change that result.)

3. The gauged model and non-Abelian T- duality

In this section we are going to gauge a model with non trivial background following [26,27] and to construct the T-dual model in analogy to the previous discussion. The sigma model under con- sideration is given by

The non trivial fields are the target space metric G the dilaton 9 (coupling to the scalar curvature R12) in the bulk and to the geodesic curvature k on the boundary), and the U(1) gauge field A coupling to the boundary. (The antisymmetric tensor field is taken to be trivial for simplicity.) Further let us assume that there are isometries forming a semi-simple Lie group,

[t,<.J] = fIJK‘tK.

Then the coordinates Xm transform as

(20)

6X” = EIJJ? (21)

The conditions that the action (20) is invariant under these (global) transformations is that the Lie derivative of the dilaton vanishes and that the Lie derivative of the U(1) gauge field is a pure gauge

LcIArn = am41 . (22)

Note that the right hand side of (22) is invariant under constant shifts of 41. F’rom now on we will neglect the dilaton and comment on it in the conclusions. By evaluating L[c,,~,lA, we find that 41 must satisfy the consistency conditions

‘+$J - &,+I = fIJK4K - krJ, (23)

where kIJ are constants transforming under the above mentioned constant shifts of $1. The gauged model is obtained by introducing isometry gauge fields R and demanding invariance of the action under

6Xrn = &Y)

f5O;i = d,f'+f~~'RfE~. (24)

Since we have no antisymmetric tensor field in the bulk the gauge invariant bulk part of the action is obtained by replacing partial derivatives with covariant ones

D,Xm = 8,X” - [,mfl;f . (25)

To construct an action with an invariant boundary term we employ Noether’s method, i.e. we add

$1) = I dsCrf-2;

and determine the fields CI such that the full action is gauge invariant. That leads to two equations (for details see [16])

Cr = -Am<l;n +$I + XI (27) .&,CJ = -~JI~CK, PQ

with XI being some constant. These equations are compatible provided that

k1.J - fIJKAK = 0, (29)

implying an integrability condition on the constants klJ

fIJKkKr, + QCl.pWL = 0. (30)

The gauged model is finally given by

S =

-4 SC d”oGm,D,XmDaXn - Jds (Am&Xm - C,@) . (31)

Now, one can perform a T-duality transformation for the considered model in the same way as described in the beginning for the free boson. The details are given in [16]. In addition to the coordinates Xm there might be additional coordinates X* transforming as singlets under the isometry. So, we add a Lagrange multiplier constraining the isometry gauge field strength to vanish. This La- grange multiplier transforms in the adjoint of the

40 S. FZirste et al. /Nuclear Physics B (Proc. Suppl.) S6B (1997) 36-41

isometry group. A second Lagrange multiplier is added to give the correct boundary conditions on the gauge parameters, and as we have seen in the beginning it finally drops out, (here we start with the Neumann string). Then rewriting the action such that there are no derivatives of the isometry gauge fields and factorizing the R integral into a bulk part and a boundary part will give the dual bulk action and Dirichlet conditions on the dual coordinates AI

AI -I- CI = 0 ondC, (32)

being covariant with respect to the isometry group. The gauge fixing is done by fixing the X”‘, and in general one has to fix some AI as well in order to remove a residual symmetry.

of the gauge

4. Conclusions

We have seen that T-duality in open string models interchanges Neumann with Dirichlet boundary conditions. Specifically, Neumann boundary conditions correspond to free varying ends of the string and are not imposed as an exter- nal constraint whereas Dirichlet conditions arise as constraints imposed by a delta function. We argued that a canonical transformation gives the same result as long as the generating functional has no additional boundary contributions. Since this is in agreement with the RoEek-Verlinde [25] method it seems to be natural not to add boundary contributions to the generating functional.

We finish this talk by mentioning some open problems. One problem is, as in the closed string case, the treatment of global issues. Anot her problem is the dilaton shift. In a general sigma model with boundary there could be two dilatons, one coupling to R@) in the bulk and another one coupling to the geodesic curvature on the boundary. String theory tells us that there is only one dilaton coupling to the Gauss-Bonnet density. So, a natural guess would be that the dual dilaton couples also to the Gauss-Bonnet density and re- ceives the same shift as in the closed string case [17]. A proof for this guess is missing, so far. It might be as well possible that the dilaton re- ceives (infinite) contributions at the position of

the D-brane [ll) which would result in different boundary and bulk dilatons.

Acknowledgments

S. F. would like to thank Javier Borlaf, Harald Dorn and Konstadinos Sfetsos for fruitful discus- sions. The work of S. S. is supported by DFG - Deutsche Forschungs Gemeinschaft. A. K. is supported by the Alexander von Humboldt Foun- dation and S. F. is supported by GIF- German Israeli Foundation for Scientific Research.

A. Canonical transformation for the open string - preliminaries

In the Hamiltonian picture the action for the open string is given by

s [X, P] = Jc d2a (7-l + Pk) +JBcd7 (h(X,P) +pR). (33)

The equation of motion arising from g reads

* c%i 631 ’ -- P=ax dx’ ( > in the bulk and

qac= (-$+g)6x,sc (35)

on the boundary. Consider now an equivalent action

s [_%,“] ,

This action is equivalent to the first one provided that

6 (s - 3) = / dr $dF [X, a%] defining the generating functional F. Taking into account the equations of motion, (the ones from GS/bP are not needed explicitly), (36) leads to

P6X - i%k = 6F (37)

also in the case of open strings.

S. Fd;rste et al. /Nuclear Physics B (Proc. Suppl.) 56B (1997) 36-41 41

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non abelian t-duality for open strings

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