ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaThe Geometry of WZW BranesG. FelderJ. Fr�ohlichJ. FuchsC. Schweigert

Vienna, Preprint ESI 751 (1999) September 10, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

hep-th/9909030ETH-TH/99-24September 1999THE GEOMETRY OF WZW BRANESGiovanni Felder , J�urg Fr�ohlich , J�urgen Fuchs and Christoph SchweigertETH Z�urichCH { 8093 Z�urichAbstractThe structures in target space geometry that correspond to conformally invariantboundary conditions in WZW theories are determined both by studying the scat-tering of closed string states and by investigating the algebra of open string vertexoperators. In the limit of large level, we �nd branes whose world volume is a regularconjugacy class or, in the case of symmetry breaking boundary conditions, a `twined'version thereof. In particular, in this limit one recovers the commutative algebraof functions over the brane world volume, and open strings connecting di�erentbranes disappear. At �nite level, the branes get smeared out, yet their approximatelocalization at (twined) conjugacy classes can be detected unambiguously.It is also established that in any rational conformal �eld theory the structure con-stants of the algebra of boundary operators coincide with speci�c entries of fusingmatrices.1

1 IntroductionConformally invariant boundary conditions in two-dimensional conformal �eld theories haverecently attracted renewed attention. By now, quite a lot of information on such boundaryconditions is available in the algebraic approach, including boundary conditions that do notpreserve all bulk symmetries. In many cases, the conformal �eld theory of interest has alsoa description as a sigma model with target space M . It is then tempting to ask what thegeometrical interpretation of these boundary conditions might be in terms of submanifolds(and vector bundles on them or, more generally, sheaves) of M . Actually, this question makesan implicit assumption that is not really justi�ed: It is not the classical (commutative) geometryof the target M that matters, but rather a non-commutative version [1] of it.In the present note, we investigate the special case of WZW conformal �eld theories. Formost of the time we restrict our attention to the case where the torus partition function is givenby charge conjugation. Then the classical target space is a real simple compact connected andsimply connected Lie group manifoldG. In particular, the underlying manifold is parallelizable,i.e. its tangent bundle is a trivial bundle.The latter property of Lie groups will allow us to apply methods that were developped in [2],by which geometric features of the D-brane solutions of supergravity in at 10-dimensionalspace-time were recovered from the boundary state for a free conformal �eld theory. The basicidea of that approach was to compute the vacuum expectation value of the bulk �eld thatcorresponds to the closed string state ���1 ~���1jqij~qi (1.1)on a disk with a boundary condition � of interest. Here our convention is that quantitieswithout a tilde correspond to left-movers, while those with a tilde correspond to right-movers.The operator ��n is the nth mode of the u(1) current in the �-direction of the free conformal�eld theory. The symmetric traceless part of the state (1.1) corresponds to the graviton, theantisymmetric part of the state to the Kalb--Ramond �eld, and the trace to the dilaton, all ofmomentum q.Let us explain the rationale behind this prescription. At �rst sight it might seem morenatural to employ graviton scattering in the background of a brane for exploring the geometry.This would correspond, in leading order of string perturbation theory, to the calculation of atwo-point correlation function for two bulk �elds on the disk. However, by factorization of bulk�elds such an amplitude is related to (a sum of) products of three-point functions on the spherewith one-point functions on the disk. Since the former amplitude is completely independent ofthe boundary conditions, all information on a boundary condition that can be obtained by useof bulk �elds will therefore be obtainable from correlators involving a single bulk �eld. Similarfactorization arguments also encourage us to concentrate on world sheets with the topology ofa disk.The idea of testing boundary conditions with vacuum expectation values of bulk �elds�nds an additional justi�cation in the following reasoning. In terms of classical geometry,boundary conditions are related to vector bundles over submanifolds of the target manifoldM , the Chan--Paton bundles. Such bundles, in turn, should be regarded as modules over thering F(M) of functions on M . Heuristically, we may interpret the algebra of (certain) bulk2

�elds as a quantized version of F(M). The expectation values of the bulk �elds on a diskthen describe how the algebra of bulk �elds is represented on the boundary operators or, moreprecisely, on the subspace of boundary operators that are descendants of the vacuum �eld. (Asa side remark we mention that boundary conditions are indeed most conveniently described interms of suitable classifying algebras. These encode aspects of the action of the algebra of bulkoperators on boundary operators.)The relevant information for computing the one-point functions on a disk with boundarycondition � is encoded in a boundary state B�, which is a linear functionalB� : Mq;~q HqH~q ! C (1.2)on the space of closed string states. (For an uncompacti�ed free boson, the left- and right-moving labels of the primary �elds are related as ~q= q.) We are thus led to compute thefunction G��� (q) := B�(���1~���1jqij~qi) : (1.3)Using the explicit form of the boundary state, this quantity has been determined in [2]. UponFourier transformation, it gives rise to a function ~G��� (x) on position space. It has been shownthat the symmetric traceless part of the function G� reproduces the vacuum expectation value ofthe graviton in the background of a brane, while the antisymmetric part gives the Kalb--Ramond�eld, and the trace the dilaton.In order to see how these �ndings generalize to the case of (non-abelian) WZW theories, letus discuss the structural ingredients that enter in these calculations. Boundary states can beconstructed for arbitrary conformal �eld theories, in particular for WZW models. Moreover,since group manifolds are parallelizable, it is also straightforward to generalize the oscillatormodes ���n: they are to be replaced by the corresponding modes Jan of the non-abelian currentsJa(z). Here the upper index a ranges over a basis of the Lie algebra of G, a=1; 2; ::: ;dimG,and n2Z. Together with a central elementK, these modes span an untwisted a�ne Lie algebrag, according to [Jan; J bm] =Xc fabc J cn+m +K �ab�n+m;0 : (1.4)Here fabc and �ab are the structure constants and Killing form, respectively, of the �nite-dimensional simple Lie algebra �g whose compact real form is the Lie algebra of the Lie groupmanifold G. Notice that the generators of the form Ja0 form a �nite-dimensional subalgebra,called the horizontal subalgebra, which can (and will) be identi�ed with �g.We �nally need to �nd the correct generalization of the state jqi. To this end we note thatjqi is the vector in the Fock space of charge q with lowest conformal weight. For WZW theories,instead of this Fock space, we have to consider the following space. First, we must choose anon-negative integer value k for the level, i.e., the eigenvalue of the central element K. Thespace of physical states of the WZW theory with charge conjugation modular invariant is thenthe direct sum M�2PkH�H�+ ; (1.5)3

where H� is the irreducible integrable highest weight module of g at level k with highest weight�, and Pk is the (�nite) set of integrable weights � of g at level k. Every such g-weight �corresponds to a unique weight of the horizontal subalgebra �g (which we denote again by �),which is the highest weight of a �nite-dimensional �g-representation. However, for �nite levelk, not all such highest weights of �g appear; this truncation will have important consequenceslater on.Unlike in the case of Fock modules, the subspace of states of lowest conformal weight inthe module H� is not one-dimensional any longer. Rather, it constitutes the irreducible �nite-dimensional module �H� of the horizontal subalgebra �g. Therefore in place of the function (1.3)we now consider Gab� (v~v) := B�(Ja�1 ~J b�1 jvij~vi) (1.6)for v ~v 2 M�2Pk �H� �H�+ : (1.7)As a matter of fact, one may also look at analogous quantities involving other modes Jan, orcombinations of modes, or even without any mode present at all. It turns out that qualitativelytheir behavior is very similar to the functions (1.6); they all signal the presence of a defect atthe same position in target space. Our results are therefore largely independent on the choiceof the bulk �eld we use to test the geometry of the target.The function Gab� can be determined from known results about boundary conditions in WZWmodels. This allows us to analyze WZW brane geometries via expectation values of bulk �elds.Another approach to these geometries is via the algebra of boundary �elds. While the secondsetup focuses on intrinsic properties of the brane world volume, the �rst perspective o�ers anatural way to study the embedding of the brane geometry into the target. Both approacheswill be studied in this paper.We organize our discussion as follows. In section 2 we compute the function Gab� for thoseboundary conditions which preserve all bulk symmetries. To relate this function to classicalgeometry of the group manifold G, we perform a Fourier transformation. We then �nd thatthe end points of open strings are naturally localized at certain conjugacy classes of the groupG. At �nite level k, the locus of the end points of the open string is, however, smeared out,though it is still well peaked at a de�nite regular rational conjugacy class. The absence ofsharp localization at �nite level k shows that, even after having made the relation to classicalgeometry, the brane exhibits some intrinsic `fuzziness'. It should, however, be emphasized thatat �nite level the very concept of both the target space and the world volume of a brane asclassical manifolds are not really appropriate.The algebra of boundary �elds for symmetry preserving boundary conditions is analyzedin section 3. We show that for any arbitrary rational conformal �eld theory the boundarystructure constants are equal to world sheet duality matrices, the fusing matrices, according toC�A� B C��L � = (FL�C;A�B[ � �� + ])� : (1.8)Furthermore, we are able to show that in the limit of large k the space of boundary operatorsapproaches the space of functions on the brane world volume. In the same limit open strings4

connecting di�erent conjugacy classes disappear, while such con�gurations are present at every�nite value of the level.In section 4 we discuss symmetry breaking boundary conditions in WZW theories for whichthe symmetry breaking is characterized through an automorphism ! of the horizontal subalge-bra �g. 1 It turns out that the end points of open strings are then localized at twined conjugacyclasses, that is, at sets of the formC!G(h) := fgh!(g)�1 j g 2Gg (1.9)for some h2G. The derivation of our results on symmetry breaking boundary conditionsrequires generalizations of Weyl's classical results on conjugacy classes. (The necessary tools,including a twined version of Weyl's integration formula, are collected in an appendix.) Insection 5, we extend our analysis to non-simply connected Lie groups. We �nd features thatare familiar from the discussion of D-branes on orbifold spaces, such as fractional branes, andpoint out additional subtleties in cases where the action of the orbifold group is only projective.2 Probing target geometry with bulk �eldsWe start our discussion with the example of boundary conditions that preserve all bulk sym-metries. In this situation the correlators on a surface with boundaries are speci�c linear com-binations of the chiral blocks on the Schottky double of the surface [3]. The boundary statedescribes the one-point correlators for bulk �elds on the disk and, accordingly, it is a linearcombination of two-point blocks on the Schottky cover of the disk, i.e., on the sphere. The lat-ter { which in the present context of correlators on the disk also go under the name of Ishibashistates { are linear functionals B� : H�H�+ ! C (2.1)that are characterized by the Ward identitiesB� � �Jan 1+ 1 Ja�n� = 0 : (2.2)Choosing an element v~v 2 �H� �H�+, we can use the invariance property (2.2) and the commu-tation relations (1.4) to arrive atB�(Ja�1v J b�1~v) = �B�((1 Ja1J b�1) (v~v)) = �B�((1 [Ja1 ; J b�1]) (v~v))= �Xc fabcB�(v J c0~v)� �abk B�(v~v) : (2.3)There is one symmetry preserving boundary condition for each primary �eld � in the theory[4]. The coe�cients in the expansion of the boundary states with respect to the boundary blocksare given by the so-called (generalized) quantum dimensions:B� = X�2Pk S�;�S;� B� : (2.4)1 Not all symmetry breaking boundary conditions of WZW theories are of this form.5

Here S is the modular S-matrix of the theory and refers to the vacuum primary �eld. Towrite the state (2.4) in a more convenient form, we use the fact that the generalized quantumdimensions are given by the values of the �g-character of �H� at speci�c elements y� of the Cartansubalgebra of the horizontal subalgebra �g or, equivalently, by the values of the G-character of�H� at speci�c elements h� of the maximal torus of the group G. Concretely, we haveS�;�=S;� = ��(h�) (2.5)with ��(h) := tr �H�R�(h) (2.6)and h� := exp(2�iy�) with y� := �+�k+g_ (2.7)for any level k and �g-weight �. In formula (2.7), � denotes the Weyl vector (i.e., half the sumof all positive roots) of �g and g_ is the dual Coxeter number. The boundary state thus readsB� = X�2Pk ��(h�)B� : (2.8)The function Gab� de�ned in formula (1.7) is then found to beGab� (v~v) = ���(h�) �B�(v [Ja0 ; J b0]~v) + �abk B�(v~v)� (2.9)for v~v 2 �H� �H�+.We recall that the group character � is a class function, i.e. a function that is constant onthe conjugacy classes CG(h) := fghg�1 j g 2Gg (2.10)of G. It is therefore quite natural to associate to a symmetry preserving boundary conditionthe conjugacy class CG(h�) of the Lie group G that contains the element h�. We would like toemphasize that CG(h�) is always a regular conjugacy class, that is, the stabilizer of h� underconjugation is just the unique maximal torus containing this element.Our next task is to perform the analogue of the Fourier transformation between G�� and~G�� in [2]. To this end we employ the fact that left and right translation on the group manifoldG give two commuting actions of G on the space F(G) of functions on G and thereby turn thisspace into a G-bimodule. By the Peter--Weyl theorem, F(G) is isomorphic, as a G-bimodule,to an in�nite direct sum of tensor products of irreducible modules, namelyF(G) �= M�2P �H� �H�+ (2.11)Here P �Pk=1 is the set of all highest weights of �nite-dimensional irreducible �g-modules. Wemay identify the conjugate module �H�+ with the dual of �H�. Then the isomorphism (2.11)sends v~v 2 �H� �H�+ to the function f on G given byf(g) = ~v(R�(g)v) � h~vjR�(g)jvi (2.12)6

for all g 2G. Using the scalar product on F(G), we can therefore associate to every linearfunctional �: F ! C a function (respectively, in general, a distribution) ~� on the group manifoldby the requirement that �(v~v) = ZGdg ~�(g)� h~vjR�(g)jvi (2.13)for v2 �H�. After introducing dual bases fvig of �H� and f~vjg of �H�+, the orthonormalityrelations for representation functions then allow us to write~�(g) =X�2PXi;j �(vi~vj)� h~vjjR�(g)jvii : (2.14)According to (1.7), at �nite level we have to deal with the �nite-dimensional truncationsFk(G) := M�2Pk �H� �H�+ (2.15)of the space (2.11) of functions on G. For every k, the space Fk(G) can be regarded as asubspace of F(G). We will do so from now on; thereby we arrive at a picture that is closeto classical intuition. The level-dependent truncation (2.15) constitutes, in fact, one of thebasic features of a WZW conformal �eld theory. (This is a typical e�ect in interacting rationalconformal �eld theories, which does not have an analogue for at backgrounds.)Next we relate the linear function Gab on Fk to a function ~Gab on the group manifold G bythe prescription ~Gab(g) := X�2PkXi;j Gab(vi~vj)� h~vj jR�(g)jvii (2.16)for g 2G. By direct calculation we �ndGab(g) = �X�2Pk (�abk S��;�S;� tr �H�R�(g) +Xc fabc S��;�S;� tr �H�J cR�(g))= ��abkX�2Pk ��(h�)� ��(g)�X�2Pk ��(h�)� tr �H�[Ja; J b]R�(g) : (2.17)In analogy with the situation for at backgrounds [2] we are led to the following interpretationof this result. The �rst term in the expression (2.17) is symmetric and hence describes thevacuum expectation value of dilaton and metric that is induced by the presence of the brane,while the second term, which is antisymmetric, corresponds to the vacuum expectation valuefor the Kalb--Ramond �eld.To proceed, we introduce, for every k 2Z>0, a function 'k on G�G by'k(g; h) := X�2Pk ��(g) ��(h)� : (2.18)In the limit k!1 the integral operator associated to 'k reduces to the �-distribution on thespace of conjugacy classes. Indeed, because of limk!1 Pk =P , for every class function f on Gwe have ZGdh limk!1'k(g; h) f(h) = f(g) ; (2.19)7

which is a consequence of the general relationZGdg �V (g)� �W (g) = dim(HomG(V;W )) (2.20)valid for any two G-modules V; W . Comparison with the result (2.17) thus shows that, in thelimit of in�nite level, the brane is localized at the conjugacy class CG(h�) of G. It is worthemphasizing, however, that this holds true only in that limit. In contrast, at �nite level, thebrane world volume is not sharply localized on the relevant conjugacy class CG(h�). Rather, itgets smeared out or, in more fancy terms, its localization is on a `fuzzy' version of a conjugacyclass. Nevertheless, already at very small level the localization is su�ciently sharp to indicateunambiguously what remains in the limit.For concreteness, we display a few examples for boundary conditions with g= sl(2) in �gure1. The functions of interest arefk;�(h) := N J(h) jX�2Pk ��(h�)��(h)j2 ; (2.21)where J is the weight factor in the Weyl integration formula (see (A.3) and (A.8)) and N is anormalization constant which is determined by the requirement that RT dh f(h)= 1. For sl(2),we have T = [0; 2] and J(z)= sin2(�z). The functions (2.21) are then given byfk;x(z) = Nk;x ( kX�=0 sin((�+1)�x) sin(�z))2 (2.22)for z 2 [0; 1), where k 2Z>0 and x=(�+1)=(k+2) with �2 f0; 1; ::: ; kg. The examples plottedin �gure 1 are for conjugacy classes x=1=6 and 1=2 and for levels k=4; 10 and 28.Closer inspection of the sl(2) data also shows that the sharpness of the localization scaleswith k+g_. More speci�cally, for any given conjugacy class x and any fraction p of �= (k+2)�1,the integrated density Ix;p :=R x+�=px��=p dz f(z) depends only very weakly on the level. In fact, wehave collected extensive numerical evidence that even after taking this rescaling into account,the localization improves when the level gets larger, i.e., that Ix;p(k) is monotonically increasingwith k. (The improvement is not spectacular, though. For instance, Ix=�;1 rises from :9829 atk=3 to :9889 at k=20, and Ix=�;3 rises from :6063 to :6074.)Note in particular that all brane world volumes are concentrated on regular conjugacyclasses, and that already at small level the overlap with the exceptional conjugacy classes (i.e.,x=0 and x=1 for g= sl(2)) is negligible. Indeed, as is clearly exhibited by the last mentioneddata, even the level-dependent allowed conjugacy classes that, at �xed level, are closest to anexceptional class (i.e., x=� for g= sl(2)) are not driven into the exceptional one in the in�nitelevel limit. (Thus, in this respect, our �ndings do not agree with the prediction of the semi-classical analysis in [5] and in [6]. The origin of this discrepancy appears to be the absence ofthe shift k 7! k+g_ in the classical setup. This shift occurs also naturally in other quantitieslike e.g. in character formul� and conformal weights.) This result will be con�rmed by theinvestigation of the algebra of boundary operators, to which we now turn our attention.8

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Figure 1: The function (2.22) for branes centered at the conjugacy classes x=1=6 and x=1=2for levels 4; 10; 28. 9

3 The algebra of boundary �eldsIn this section we focus our attention on the operator product algebra of (WZW-primary)boundary �elds. As a �rst basic ingredient, we need to determine by which quantum numberssuch a �eld is characterized. Boundary points are precisely those points of a surface thathave a unique pre-image on its Schottky cover. Accordingly, on the level of chiral conformal�eld theory, boundary operators are characterized by a single primary label �. 2 When all bulksymmetries are preserved, this label takes its values in the set of chiral bulk labels. (In thepresence of symmetry breaking boundary conditions, the analysis has to be re�ned, see [7,8].)Moreover, a boundary operator typically changes the boundary condition; therefore it carriestwo additional labels �; � which indicate the two conformally invariant boundary conditions atthe two segments adjacent to the boundary insertion.Boundary operators are therefore often written as � �� (x). But, in fact, this is still notsu�cient, in general. The reason is that �eld-state correspondence requires to associate toevery state that contributes to the partition functionA��(t) =X� A��� ��( it2 ) (3.1)for an annulus with boundary conditions � and � a separate boundary �eld. For symmetrypreserving boundary conditions, the annulus coe�cients A��� are known [4] 3 to coincide withfusion rule coe�cients: A��� = N ��� : (3.2)The fusion rules N ��� are not, in general, zero or one; as a consequence one must introduceanother degeneracy label A, taking values in f1; 2; ::: ;N�+��g [7, 6, 9]. The complete labellingof boundary operators therefore looks like� A�� (x) : (3.3)We remark in passing that in more complicated situations, like e.g. symmetry breaking bound-ary conditions [7] or non-trivial modular invariants [10], the degeneracy spaces relevant for theboundary operators still admit a representation theoretic interpretation that involves appropri-ate (sub-)bundles of chiral blocks.The boundary �elds satisfy an operator product expansion which schematically reads�A�� (x)� B � (y) � X� N ���XL=1 N ��XC=1 C�A� B C��L � [�C � (y) + ::: ] ; (3.4)where L2 f1; 2; ::: ;N ��� g labels a basis of the space of chiral couplings from � and � to �.We claim that, for every (rational) conformal �eld theory, the structure constants C�A� B C��L �2 For WZW models, we are actually also interested in the horizontal descendants, which have the sameconformal weight as the primary �eld. Then � should be regarded as a pair consisting of both the highestweight and the relevant actual �g-weight. Correspondingly, in the operator product (3.4) below one must thenin addition include the appropriate Clebsch--Gordan coe�cients of �g.3 Compare also [6] for arguments in a lagrangian setting.10

appearing here are nothing but suitable entries of fusing matrices F:C�A� B C��L � = (FL�C;A�B[ � �� + ])� : (3.5)Recall that the fusing matrices describe the transition between the s- and the t-channel offour-point blocks; pictorially, in our conventions, this relation looks like���@@@ @@@��� ?R� I�� ���KL = X� N ���XM=1 N �+��XN=1 FK�L;M�N [��� � ] � @@@��� ���@@@-R� I�� ���M N (3.6)To establish this identity, we observe that the operator product coe�cientsC�A� B C��L � furnisha solution of the sewing constraint [11,12] that arises from the two di�erent factorizations of acorrelation of four boundary �elds. Including all degeneracy labels, this sewing relation readsN ��XE=1 C�A� B E��K � C C �D �E�� L �+ C�E E �� �+ =X� N ���XM=1 N �+��XN=1 N ���XF=1C� B C � F��M � C�A� F �D�� N �+ C�D �D ��+ � � FM�N;K�L[��� � ] ; (3.7)where F[��� � ] is the fusing matrix which relates the two di�erent factorizations. For WZWmodels, the fusing matrices coincide with the 6j-symbols of the corresponding quantum groupwith deformation parameter at a k+g_th root of unity. The constraint (3.7) can be solvedexplicitly in full generality, without reference to the particular conformal �eld theory underinvestigation. First note that the structure constants C�A� B C��L � depend on six chiral and fourdegeneracy labels, so that their label structure is precisely the same as the one of the fusingmatrices. The key observation is then to realize the similarity between the constraint (3.7) andthe pentagon identityN ��XE=1 FK�+A;B E[ ��+� �+ ] � FL�+E;C�D[ � � �+ ]=X� N ���XM=1 N �+��XN=1 N ���XF=1 FM�F ;B C[ � �� �+ ] � FN�+A;F�D[ � �� �+ ] � FL�+K;M�N [� ��� ] (3.8)for the fusing matrices. 4 Indeed, it is not too di�cult to show that under the identi�cation (3.5)the constraint (3.7) becomes { after exploiting the tetrahedral (S4-) symmetry [13] of the fusing4 The structural similarity between the factorization constraint (3.7) and the pentagon identity was alsoobserved in [9]. In this context, we would like to stress that the fusing matrices are entirely de�ned in terms ofchiral conformal �eld theory. 11

matrices in order to replace 5 some of the fusing matrix entries by di�erent ones { nothing butthe (complex conjugated) pentagon identity (3.8).A more direct way to understand the result (3.5) is by interpreting the boundary �elds�A�� as (ordinary) chiral vertex operators, which pictorially amounts to the prescription�A�� = � �?�� �A (3.9)The operator product (3.4) then describes the transition� � �? ?� �� � A B �! @@@ ���R ?� �� ��LC� (3.10)from which one can read o� the desired identi�cation (3.5) between boundary structure con-stants and fusing matrices.Thus we conclude that the boundary structure constants are indeed nothing but suitableentries of fusing matrices. (For the case of Virasoro minimal models, this has already beenobserved in [14] using special properties of those models.) It should be noted, however, that thefusing matrices are not completely determined by the pentagon identity and their tetrahedralsymmetry. Rather, there is a gauge freedom related to the possibility of performing a change ofbasis in the spaces of chiral three-point couplings. In the present setting, the gauge invariancecorresponds to the freedom in choosing a basis in the space of all boundary �elds �A�� with�xed � and �; �. Once the gauge freedom has been �xed at the level of chiral three-pointcouplings, it is natural to make the same gauge choice also for the boundary �elds.We now show that, upon appropriately taking the limit of in�nite level, the algebra ofthose boundary operators that do not change the boundary condition approaches the algebraF(G=T ) of functions on the homogeneous space G=T , where T is a maximal torus of the Liegroup G. This space G=T is of interest to us because every regular conjugacy class of G is, asa di�erentiable manifold, isomorphic to G=T . Our result therefore perfectly matches the factthat the fusing matrices of WZW models can be expressed with the help of k+g_th roots ofunity, i.e. again the level k gets shifted by the dual Coxeter number g_. Correspondingly, theweights are shifted by the Weyl vector so that, again, we are naturally led to regular conjugacyclasses.We start our argument by regarding the algebra F(G=T ) as a left G-module only, ratherthan as an algebra. The module F(G=T ) is fully reducible and can be decomposed as follows.5 Note that the tetrahedral transformations that do not preserve the orientation of the tetrahedron involvecomplex conjugation of F. Also, by the tetrahedral symmetry, the structure constants involving the vacuum label are just combinations of quantum dimensions, and these precisely cancel against the quantum dimensionscoming from the other tetrahedral transformations that have to be performed. More details will be givenelsewhere. 12

According to (2.11), the space F(G) of functions on G is a G-bimodule under left and righttranslation. Since the right action of T on G is free, we can then identify F(G=T ) with thesubspace of T -invariant functions on G,F(G=T ) = F(G)T �= M�2P �H� ( �H�+)T : (3.11)Furthermore, invariance under the maximal torus T just picks the weight space for weight zero.Thus we �nd F(G=T ) �= M�2P mult(�)0 �H� (3.12)as an isomorphism of �g-modules, where mult(�)0 is the multiplicity of the weight �=0 in theirreducible module �H� with highest weight �. Thus, in particular, only modules belonging tothe trivial conjugacy class of �g-modules appear in the decomposition (3.12). Recall that theboundary operators are organised in terms of modules H� of the a�ne Lie algebra g. Ouraim is to show that the algebra of boundary �elds that correspond to states of lowest grade inthe modules H�, i.e. which are either primary �elds or horizontal descendants, 6 carries a leftG-module structure that in the limit of in�nite level coincides with the decomposition (3.12).When restricting to this �nite subspace Fk of boundary operators, via �eld-state correspon-dence the annulus amplitudes tell us that Fk carries the structure of a G-module, and as aG-module it is isomorphic to the direct sumFk �= M�;�2Pk A��� �H� (3.13)of irreducible G-modules. Thus to be able to perform a more quantitative analysis of the alge-bra of boundary operators, we need to control the values of the annulus coe�cients A��� . Sinceaccording to the identity (3.2), as long as all bulk symmetries are preserved, these numbersjust coincide with the fusion rules coe�cients, A��� =N�+��, we are interested in concrete ex-pressions for the fusion rules. It turns out that they can be expressed through suitable weightmultiplicities in the following convenient form:N��+� = 1jW j Xw1;w22W�(w1) �(w2) X�2L_mult(�)�w1(�+�)+w2(�+�)��(k+g_) : (3.14)Here W is the Weyl group of �g and � its sign function, and the sum over � extends over thecoroot lattice L_ of �g. The relation (3.14) can be derived by combining the Kac--Walton formulafor WZW fusion coe�cients with Weyl's character formula and Weyl's integration formula. Fordetails, see appendix A.We are interested in the behavior of the numbers (3.14) in the limit of large k. As forboundary conditions, this limit must be taken with care. As we have seen, they can be labelled6 In general conformal �eld theories, there is no underlying `horizontal' structure, unlike in WZW models.It is not clear whether, in general, it is the set of states of lowest conformal weight, or e.g. the quotient (called`special subspace') of H� that was introduced in [15], that is relevant in this argument. But already for WZWmodels this truncation is not su�ciently well understood.13

either by conjugacy classes of group elements h� or by the corresponding �g-weights �. But therelation (2.7) between these two types of data involves explicitly the level k, so that we haveto decide which of the two is to be kept �xed in the limit. In the present context, we keepthe conjugacy classes �xed. Accordingly we consider two sequences, denoted by �k and �k, ofweights such that y� := �k+�k+g_ and y� := �k+�k+g_ (3.15)do not depend on k (and exp(2�iy�) and exp(2�iy�) are regular elements of the maximal torusT of G).In terms of these quantities, in formula (3.14) the multiplicity of the weight�k := (k+g_) (�w1(y�) + w2(y�)� �) (3.16)appears, with �xed y� and y� (and �xed w1; w2 and �). At large k this weight becomes largerthan any non-zero weight of the module �H�, except when the relation�w1(y�) + w2(y�)� � = 0 (3.17)is satis�ed. As a consequence, at large level, the action of element w2 of the Weyl group Wof �g on the regular element y� must coincide with the action of the element (w1; �) of thecorresponding a�ne Weyl group W on y�. This, however, is only possible when y�= y� , andthen it follows that �=0 as well as w1=w2. Thus the requirement (3.17) has jW j manysolutions. We thus obtain in the limit of in�nite levellimk!1N�+k �k� = �y�;y� mult(�)0 : (3.18)In view of the relation between fusion rules and annulus coe�cients we thus learn that, inthe limit of large level, only those pairs of boundary conditions contribute which correspond toidentical conjugacy classes, or in other words, only those open strings survive which start andend at the same conjugacy class. (For every �nite value of k, however, such open strings arestill present.) Moreover, in this limit the non-vanishing annulus partition function becomelimk!1Ay� y�(t) =X�2P mult(�)0 ��(it=2) ; (3.19)so that (3.13) simpli�es to limk!1Fk �= M�2P mult(�)0 �H� : (3.20)This space is indeed nothing but F(G=T ) as appearing in (3.12). Our result indicates inparticular that the algebra of boundary operators that do not change the boundary conditionis related to the space of functions on the brane world volume. This should be regarded asempirical evidence for a statement that is not obvious in itself, since in general non-trivialvector bundles over the brane can appear as Chan--Paton bundles, so that boundary operatorsmight as well be related to sections of non-trivial bundles rather than to functions.In summary, in the limit of in�nite level those boundary operators that belong to statesin the �nite-dimensional subspace �H��H� of lowest conformal weight furnish a G-module14

that is isomorphic to the algebra F(G=T ) of functions on a regular conjugacy class, seen as aG-module.So far we have considered the spaces of our interest only as G-modules. But we wouldlike to equip both F(G) and the space of boundary operators also with an algebra structure.The operator product algebra of boundary operators, whose structure constants are, as wehave seen, fusing matrices, obeys certain associativity properties. These properties are notimmediately related to ordinary associativity, because the de�nition of the operator productinvolves a limiting procedure.Several proposals have been made recently for the relation between the operator productalgebra of boundary operators and the algebra F(G=T ). An approach based on deformationquantization was proposed in [16]. The de�nition of the product then involves �xing theinsertion points of the two boundary �elds at prescribed positions in parameter space. As thetheory in question is not topological, one is thus forced to introduce arbitrary and non-intrinsicdata { in contrast to the situation with topological theories studied in [17]. Another proposal[18] starts from a restriction of the operator product algebra to �elds that correspond to thestates of lowest conformal weight in the a�ne irreducible modules. This destroys associativity.The prescription in [18] also allows only for open strings that have both end points on one andthe same brane. This is di�cult to reconcile with the fact that (compare formula (3.19) above)open strings connecting di�erent branes can only be ignored in the limit of in�nite level.4 Symmetry breaking boundary conditionsWe now turn to boundary conditions of WZW models that break part of the bulk symmetries.One important class of consistent boundary conditions can be constructed by prescribing anautomorphism ! of the chiral algebra that connects left movers and right movers in the presenceof a boundary. In this case the boundary condition is said to have automorphism type !. Wepoint out, however, that also boundary conditions are known for which no such automorphismexists. A WZW example is provided by so(5) at level 1; in this example, there is a conformalembedding with a subalgebra isomorphic to sl(2) at level 10, and one can classify boundaryconditions (see [9]) that preserve only the sl(2) symmetries. However, no general theory forsuch boundary conditions without automorphism type has been developped so far, and we willnot consider them in the present paper.Every boundary condition preserves some subalgebra �A of the full chiral algebra A; becauseof conformal invariance, �A contains the Virasoro subalgebra of A. For boundary conditionsthat do possess an automorphism type !, the preserved subalgebra �A�A can be characterizedas an orbifold subalgebra, namely as the algebra �A=A<!> consisting of elements that are �xedunder !. A theory treating such boundary conditions for arbitrary conformal �eld theories hasbeen developped in [7,8]. In the case of interest to us, the relevant automorphisms of the chiralalgebra are induced by automorphisms ! of the horizontal subalgebra �g of the untwisted a�neLie algebra g that preserve the compact real form of �g. Via the construction (1.4) of a�ne Liealgebras as centrally extended loop algebras, every such automorphism ! extends uniquely toan automorphism of g. By a slight abuse of notation we denote this automorphism by !, too.We brie y summarize some of the results that we will derive in this section. In the same15

way that symmetry preserving boundary conditions are localized at regular conjugacy classes,the boundary conditions of automorphism type ! are localized at the submanifoldsC!G(h) := fgh!(g)�1 j g 2Gg ; (4.1)to which we refer as twined conjugacy classes. In the case of sl(2) or, more generally, whenever !is an inner automorphism of g, the twined conjugacy classes are just tilted versions of ordinaryconjugacy classes. More precisely, they can be obtained from ordinary conjugacy classes byright translation, CAdsG (h)= CG(hs) s�1.In the case of outer automorphisms, the dimension of twined conjugacy classes di�ers fromthe dimension of ordinary ones. While ordinary regular conjugacy classes are isomorphic tothe homogeneous space G=T , twined conjugacy classes for outer automorphisms turn out tobe isomorphic to G=T !0 , where T !0 is a subtorus of the maximal torus T . For instance, forg= sl(3) the dimension of regular conjugacy classes is dim(G=T )= 8�2= 6, while for outerautomorphisms twined conjugacy classes have dimension dim(G=T !0 )= 8�1= 7. The increase inthe dimension actually generalizes a well-known e�ect in free conformal �eld theories, where allautomorphisms are outer, to the non-abelian case. Namely, in a at d-dimensional backgroundthe relevant automorphism, which is an element of O(d), determines the dimension of the braneand a constant �eld strength on it. In particular, non-trivial automorphisms can change thedimension of the brane.The boundary states B!� for symmetry breaking boundary conditions of automorphism type! are built from twisted boundary blocks B!� [8]. For the latter, the Ward identities (2.2) getgeneralized to B!� � �Jan 1+ 1!(Ja�n)� = 0 : (4.2)To proceed, we need some further information on automorphisms of �g that preserve the compactreal form. Such automorphisms are in one-to-one correspondence to automorphisms of theconnected and simply connected compact real Lie group G whose Lie algebra is the compactreal form of �g. For each such automorphism ! there is a maximal torus T of G that is invariantunder !. The complexi�cation t of the Lie algebra of T is a Cartan subalgebra of �g. The torusT is not necessarily pointwise �xed under !. The subgroupT ! := ft2 T j!(t)= tg (4.3)of T that is left pointwise �xed under ! can have several connected components [19, 20]. Theconnected component of the identity will be denoted by T !0 .The automorphism ! of �g can be written as the composition of an inner automorphism,given by the adjoint action Ads of some element s2 T , with a diagram automorphism !�:!(g) = !�(sgs�1) ; (4.4)without loss of generality, s can be chosen to be invariant under !, !(s)= s. Let us recall thede�nition of a diagram automorphism. Any symmetry _!� of the Dynkin diagram of �g inducesa permutation of the root generators Ei� that correspond to the simple roots of �g with respectto the Cartan subalgebra t, according toEi� 7! !�(Ei�) := E _!�i� : (4.5)16

This extends uniquely to an automorphism !� of g that preserves the compact real form andis called a diagram automorphism of �g. When ! is an inner automorphism then the diagramautomorphism in the decomposition (4.4) is the identity; in general, !� accounts for the outerpart of !. Also note that, for inner automorphisms, T ! is the full maximal torus T .As ! leaves a Cartan subalgebra t invariant, there is an associated dual map !? on theweight space t? of �g. Applying the condition (4.2) for the zero modes, i.e. n=0, one sees thatnon-zero twisted boundary blocks only exist for symmetric weights, i.e. weights � satisfying!?(�)= �. Note that relation (4.4) implies that !?(�)=!�?(�), so that in the case of innerautomorphisms all integrable highest weights � contribute.Next, we explain what the coe�cients in the expansion of the symmetry breaking boundarystates with respect to the twisted boundary blocks are, i.e., what the correct generalization ofthe numbers S�;�=S;� appearing in formula (2.4) is. We have seen in (2.5) that for != id,these coe�cients are given by the characters �� of G, evaluated at speci�c elements (2.7) of themaximal torus T . For general !, the analogous numbers have been determined in [21]. For thepresent purposes it is most convenient to express them as so-called twining characters [22,23],evaluated at speci�c elements h� of T .Let us explain what a twining characters is. To any automorphism ! of �g we can associatetwisted intertwiners �!, that is, linear maps�! : �H� ! �H!?� (4.6)between �g-modules that obey the twisted intertwining property�! �R�(x) = R!?�(!(x)) ��! (4.7)for all x2 �g. By Schur's lemma, the twisted intertwiners are unique up to a scalar. Forsymmetric weights, !?(�)= �, the twisted intertwiner �! is an endomorphism. In this case we�x the normalization of �! by requiring that �! acts as the identity on the highest weightvector. For symmetric weights, the twining character �!� is now de�ned as the generalizedcharacter-valued index �!�(h) := tr �H��!R�(h) : (4.8)Character formul� for twining characters of arbitrary (generalized) Kac--Moody algebras havebeen established in [22,23].Finally we describe at which group elements g 2G the twining character must be evaluatedin order to yield the coe�cients of the boundary state. The integral �g-weights form a latticeLw consisting of all elements of t? of the form �= Prank�gi=1 �i�(i) that obey �i 2Z for all i.Both the Weyl group W and the automorphism !? act on this lattice. We will also need alatticeMw! that contains the lattice Lw! of integral symmetric �g-weights, i.e., of integral weightssatisfying !?(�)= � or, equivalently, � _!�i=�i for all i=1; 2; ::: ; rank �g. The latticeMw! consistsof symmetric �g-weights as well, but we weaken the integrality requirement by imposing only thatNi�i 2Zfor all i. Here Ni denotes the length of the corresponding orbit of the Dynkin diagramsymmetry _!�. For brevity we call this lattice Mw! the lattice of fractional symmetric weights.By construction, the lattice Mw! is already determined uniquely by the outer automorphismclass of !. In particular, when ! is inner, then both Mw! and the symmetric weight lattice Lw!just coincide with the ordinary weight lattice Lw.17

The lattice Lw of integral weights of �g has as a sublattice the lattice L_ of integral linearcombinations � = rank�gXi=1 �i �(i)_ (4.9)of simple coroots �(i)_. In analogy to what we did before for weights, we also introduce anotherlatticeM_!, the lattice of fractional symmetric coroots, by requiring that !?(�)=� and Ni�i 2Zfor all i. We have the inclusions L_! �M_! and Lw!�Mw!.On both the lattice Mw! of fractional symmetric weights and the lattice M_! of fractionalsymmetric coroots, we have an action of a natural subgroup W ! of the Weyl group W , namelyof the commutant W ! := fw2W jw!?=!?wg : (4.10)The group W ! depends only on the diagram part !� of !; in particular, for inner automor-phisms, !? is the identity and henceW ! =W . For outer automorphisms,W ! can be describedexplicitly [23] as follows. For the outer automorphisms of A2n, W ! is isomorphic to the Weylgroup of Cn; for A2n+1 to the Weyl group of Bn+1; for Dn to the one of Cn�1; and for E6 to theWeyl group of F4. Finally, for the diagram automorphism of order three of D4 one obtains theWeyl group of G2. (This whole structure allows for a generalization to arbitrary Kac--Moodyalgebras, and the commutant of the Weyl group can be shown to be the Weyl group of someother Kac--Moody algebra, the so-called orbit Lie algebra [22].) The group W ! also acts onthe �xed subgroup T ! of the maximal torus T . One can show that the twining characters (4.8)are invariant under the action of W !, which generalizes the invariance of ordinary charactersunder the full Weyl group W .To characterize the symmetry breaking boundary conditions, we now choose some fractionalsymmetric weight �2Mw!. It is not hard to see that the group elementh� := exp(2�iy�) ; (4.11)where y� is the corresponding dual element in the Cartan subalgebra, i.e. y� := �+�k+g_ , dependson � only modulo fractional symmetric coroots. Moreover, the subgroup W ! of the Weyl groupW acts freely on the set of all h�; there are as many di�erent orbits as there are symmetricintegrable weights. Accordingly, we should actually regard the label � of a boundary conditionof automorphism type ! as an element� 2Mw!=(W !n(k+g_)M_!) : (4.12)A boundary condition is then uniquely characterized by an element of this �nite set. Letting� run over this set, we obtain all conformally invariant boundary conditions of automorphismtype !.Let us list a few other properties of the group element h�. It is an element of the �xedsubgroup T ! of the maximal torus, or more precisely, of the connected component T !0 of theidentity of T !. Moreover, it is a regular element of G.Furthermore, it should be mentioned that in the special case of outer automorphisms of�g=A2n, there is an additional subtlety in the description of the twined conjugacy classes. Itarises from the fact [21] that in this case the extension of the diagram automorphism of �g to the18

a�ne Lie algebra g does not exactly give the diagram automorphism of g. The additional innerautomorphism of g is taken into account by the adjoint action of an appropriate element s� ofthe maximal torus. Namely, denote by x� the dual of the weight 14(�(n)+�(n+1)), i.e. the Cartansubalgebra element such that (x�; x)= 14(�(n)+�(n+1))(x) for all x in the Cartan subalgebra of�g. Then, for outer automorphisms of A2n, formula (4.11) must be generalized toh� := exp(2�iy�) exp(2�ix�) : (4.13)We are now �nally in a position to write down the boundary states explicitly; we haveB!� = X�2P!k �!�(h�)B!� (4.14)with P !k the set of symmetric weights in Pk. For trivial automorphism type, != id, we recoverformula (2.4).Fortunately, all the group theoretical tools that we used in the previous sections have gen-eralizations to the case of twining characters (for details see appendix B). Therefore, oncewe have expressed the boundary states in the form (4.14), we are also able to generalize thestatements of sections 2 and 3 to the case of symmetry breaking boundary conditions. Forinstance, recall that ordinary characters are class functions,��(ghg�1) = ��(h) ; (4.15)i.e. they are constant on conjugacy classes CG (2.10). Combining the cyclic invariance of thetrace and the twisted intertwining property (4.7) of the maps �!, one learns that twiningcharacters are twined class functions in the sense that�!�(gh!(g)�1) = �!�(h) : (4.16)As a consequence, the twined conjugacy classesC!G(h) := fgh!(g)�1 j g 2Gg (4.17)and the twined adjoint action Ad!g : h 7! g h!(g)�1 (4.18)(i.e. the twined version of the adjoint action Adg of g 2G) will play exactly the roles forsymmetry breaking boundary conditions that ordinary conjugacy classes CG(h) and ordinaryadjoint action Adg play in the case of symmetry preserving boundary conditions. We refrainfrom presenting details of the calculations; for some hints and for the necessary group theoreticaltools, such as a twined version of Weyl's integration formula, we refer to appendix B.We summarize a few properties of twined conjugacy classes (for details see appendix B).Every group element g 2G can be mapped by a suitable twined adjoint map to T !0 . For regularelements h2G, the twined conjugacy class is isomorphic, as a manifold with G-action, to thehomogeneous space C!G(h) �= G=T !0 : (4.19)19

For outer automorphisms, the following intuition appears to be accurate. The twined con-jugacy classes are submanifolds of G of higher dimension. To characterize them by the intersec-tion 7 with elements of the maximal torus, it is therefore su�cient to restrict to the symmetricpart T ! of the maximal torus (and even to the connected component T !0 of it). In contrast, foran inner automorphism !=Ads with s2G, the twined conjugacy classes have the same shapeas ordinary conjugacy classes; indeed, they are just obtained by right-translation of ordinaryconjugacy classes: CAdsG (h) = CG(hs) s�1 : (4.20)The twined analogue of the formula (3.17) requires only the symmetric part of the weightto vanish (because in the twined analogue of (A.6) only equality of the symmetric parts of theweights is enforced by the integration). As a consequence, at �xed automorphism type ! thelarge level limit (3.20) of the boundary operators gets replaced bylimk!1F !k �= M�2P mult(�)0;! �H� ; (4.21)where mult(�)0;! stands for the sum of the dimensions of all weight spaces of �H� for weights whosesymmetric part vanishes. The limit limk!1 F !k again yields the algebra of functions on thebrane world volume which in this case is isomorphic, as a manifold, to the homogeneous spaceG=T !0 .5 Non-simply connected group manifoldsIn this section we extend the results of the previous two sections to Lie group manifolds Gthat are not simply connected. Before we present our results in more detail, we brie y outlinethem for the group G=SO(3). As is well-known, SO(3) is obtained as the quotient of thesimply connected group SU(2) by its centerZ2. We will see that to every symmetry preservingboundary condition for SO(3) we can again associate a conjugacy class of SO(3). The latter areprojections of orbits of conjugacy classes of the covering group SU(2) under the action of thecenter Z2. Thinking of the group manifold SU(2) as the three-sphere S3 with the north polebeing the identity element +11 and the south pole the non-trivial element �11 of the center,the action of the center is the antipodal map on S3. The conjugacy classes that are relatedby the center are then those having the same `latitude' on S3. Those conjugacy classes whichdescribe boundary conditions must obey the same integrality constraints as in the SU(2) theory.Explicitly, at level k the two SU(2) conjugacy classes (�+�)=(k+g_) and (k��+�)=(k+g_)give rise to a single boundary condition for SO(3). An additional complication arises for the`equatorial' conjugacy class �= k=2, which is invariant under the action of the center; it givesrise to two distinct boundary conditions. Also note that all automorphisms of SO(3) are inner,and thus in one-to-one correspondence with automorphisms of SU(2). Symmetry breakingboundary conditions of SO(3) therefore correspond to tilted SO(3) conjugacy classes.7 One word of warning is, however, in order. The orbits of twined conjugation intersect T!0 in several points,but, in contrast to the standard group theoretical situation, the intersections are not necessarily related by theaction of W!. Rather, a certain extension W (T!0 ) of W!, to be described in appendix B, is needed [19,20].20

This picture is reminiscent of the phenomena one encounters in orbifold theories, and indeedthe WZW theory based on the group SO(3) can be understood [24,25] as an orbifold of the SU(2)WZW theory. Branes of the orbifold theory correspond to symmetric brane con�gurations inthe covering space; branes at �xed point sets give rise to several distinct boundary conditions,known as `fractional branes' [26]. We point out, however, the following additional feature thatis revealed by our analysis. Namely, in case the orbifold group admits non-trivial two-cocycles,branes at �xed points sets do not necessarily split. To what extent a splitting occurs is controlledby the cohomology class of the relevant two-cocycles.Let us now describe our results more explicitly. For the time being, we restrict our attentionto boundary conditions that preserve all bulk symmetries. The compact connected simple Liegroup G can be written as the quotient of a simply connected, compact and connected universalcovering group ~G by an appropriate subgroup � of the center of ~G. There is a natural projection� : ~G! G (5.1)whose kernel is the �nite group �. As a consequence, the WZW theory based on G can be seenas an `orbifold' of the theory based on ~G. (It should be pointed out, however, that the term`orbifold' is used in this context in a broader sense than is commonly done in the representationtheoretic formulation of orbifolds in conformal �eld theory, compare e.g. to [27].)It is known [24, 25] that the WZW theory on a non-simply connected group manifold isdescribed by a non-diagonal modular invariant that can be constructed with the help of simplecurrents. The relevant simple currents are in one-to-one correspondence with the elements ofthe subgroup � of the center of ~G. In the most general situation, the non-diagonal modularinvariant in question is obtained by applying a so-called simple current automorphism to achiral conformal �eld theory that is itself constructed from the original diagonal theory by asimple current extension [28]. For the sake of simplicity, in the sequel we will discuss only suchconformally invariant boundary conditions for which only one of the two mechanisms, i.e., eithera simple current automorphism or a simple current extension, is present. For ~G=SU(2), bothcases correspond to the non-simply connected quotient SO(3)= SU(2)=Z2; the former arises forlevels of the form k=2 mod 4Z, where one deals with a modular invariant of Dodd-type, whilethe latter appears for levels k=0 mod 4Zand corresponds a modular invariant of Deven-type.We �rst consider simple current extensions. We can then invoke the general result thatboundary conditions preserving all bulk symmetries are labelled by the primary �elds of therelevant conformal �eld theory, which is now not the WZW theory corresponding to ~G, but theconformal �eld theory that is obtained from it by the simple current extension. This extendedtheory can be described as follows [29]. Its primary �elds correspond to certain orbits of theaction of � on the primary �elds of the unextended theory. But only a certain subset of orbits isallowed, e.g. for G=SO(3) only those that correspond to integer spin highest weights. We willsee later, however, that the other orbits describe conformally invariant boundary conditions aswell. Those boundary conditions do not preserve all symmetries of the extended chiral algebra,but they still preserve all symmetries of the chiral algebra for the ~G-theory.We also must account for the fact that the action of � on the set of orbits is not necessarilyfree. 8 When it is not free, then there are several distinct primary �elds associated to the same8 While the (left or right) action of � on individual group elements is obviously free, the action on conjugacyclasses can be non-free, since h and �h with �2� can belong to the same conjugacy class.21

orbit. For determining the number of primaries coming from such an orbit , one must take intoaccount the fact that the action of the simple current group is in general only projective; analgorithm for solving this problem has been developped in [29]. We summarize these �ndingsin the statement that the boundary conditions of the WZW theory based on G correspond toorbits of conjugacy classes of ~G under the action of �, with multiplicities when this action isnot free.Next, we study the case of automorphism modular invariants. For this situation the bound-ary conditions that preserve all bulk symmetries have been found in [12] for ~G=SU(2) andin [10] for the general case. They are labelled by orbits of the action of � on primary �elds,or, equivalently, on conjugacy classes. Again, when this action is not free, then there are sev-eral inequivalent boundary conditions associated to the same orbit. On disks with boundaryconditions that come from the same orbit, bulk �elds in the untwisted sector possess identicalone-point functions, but the one-point functions of bulk �elds in the twisted sector are di�erentfor di�erent boundary conditions of this type. They di�er in sign, and the absolute values arecontrolled by the matrices SJ that describe the modular S-transformation of one-point chiralblocks on the torus with insertion of the relevant simple currents J [10].To provide a geometrical interpretation of these results, we �rst relate conjugacy classes ofthe group G to conjugacy classes of its covering group ~G. The conjugacy class CG(h) of anelement h2G in the non-simply connected group G can be written as the image under the map� (5.1) of several conjugacy classes C ~G of the universal covering group ~G. We claim that��1(CG(h)) =[�2� C ~G(�~h) ; (5.2)where ~h2 ~G is any a lift of h, i.e. �(~h)= h. To see that the set on the right hand side of (5.2)is contained in the set on the left hand side, we note that its elements are of the form ~g �~h ~g�1for some ~g 2 ~G and some �2�. Further, we have�(~g �~h ~g�1) = �(~g)�(~h)�(~g�1) = ghg�1 (5.3)where g is the projection �(~g); since ghg�1 lies in CG(h), indeed ~g �~h ~g�1 is contained in theleft hand side of (5.2). Conversely, assume that h0 2G is conjugate to h2G, which meansthat ghg�1= h0 for some g 2G. There exists a ~g 2 ~G such that �(~g)= g, and every element of��1(ghg�1) is of the form (�1~g) (�2~h) (�3~g�1) for suitable elements �1; �2; �32�. Using that the�i are central in ~G, this means that ��1(h0) lies in the set on the right hand side of (5.2).Let us now consider those conjugacy classes which are left invariant by some subgroup �0 of�. (For example, the group manifold ~G=SU(2) is a three-sphere S3, and the regular conjugacyclasses are isomorphic to spheres S2 of �xed latitude; thus there is a single conjugacy classthat is �xed by the action of the center Z2 of ~G, namely the equatorial conjugacy class. Atlevel k, it corresponds to the weight �= k=2 that is a �xed point with respect to fusion withthe non-trivial simple current of the sl(2) WZW theory.) The �nite subgroup �0 acts freelyon such an invariant conjugacy class C�. Therefore the space F(C�) of functions on C� can bedecomposed into eigenspaces under the action of �0. In the simplest case, the subspaces justconsist of odd and even functions, respectively. In general, the decomposition readsF(C�) = M 2�0� F (C�) ; (5.4)22

where the eigenvalues are given by characters of �0.It follows that the boundary conditions for non-simply connected groups G can be describedby conjugacy classes of G itself, with the important subtlety that those conjugacy classes whichare invariant under the action of the group � give rise to several distinct boundary conditions.Our analysis reproduces, in particular, the following familiar features of D-branes on orbifoldspaces. Brane con�gurations on the original space ~G that are symmetric under the action of �give rise to boundary conditions in the quotient G. Individual branes that are invariant undera subgroup �0 of the orbifold group � yield several boundary conditions which di�er in thecontribution from the twisted sector; the coe�cients in their boundary states are reduced by acommon factor, which is precisely the e�ect of fractional branes [26].We can also describe the analogue of the decomposition (5.4) of functions on invariantbranes for boundary operators. Again, we discuss simple current extensions and automorphismsseparately. In the case of automorphisms, it was shown in [10] that the annulus multiplicities aregiven by the rank of the sub-vector bundle of chiral blocks with de�nite parity under the simplecurrent automorphism. In the case of simple current extensions, the annulus multiplicities are,according to [4], fusion rules of the ~G-theory. Moreover, general results [29] on the fusion rulesof a simple current extension show that the fusion rules of the extended theory { that is, in ourcase, of the G-theory { are given by sub-bundles of de�nite parity as well. Just like for simplyconnected groups, our analysis therefore con�rms the general idea that the algebra of boundaryoperators should be a quantization of the algebra of functions on the brane world volume.We also would like to point out one important subtlety in the analysis of invariant orbits.The exact analysis [7] reveals that not all invariant orbits necessarily split o� and give riseto several boundary conditions. Rather, it can happen that the action of the stabilizer of theorbifold group in the underlying orbifold construction is only projective, and in this case evenan invariant conjugacy class can give rise to only a single boundary condition. An example isgiven by ~G=Spin(8)=Z2�Z2; at level 2, there is a single conjugacy class that is �xed under �,and yet, due to the appearance of a genuine untwisted stabilizer [29], it gives rise to a singleconformally invariant boundary condition. For more details, we refer to [8].We proceed to brie y discussing some aspects of symmetry breaking boundary conditionsfor WZW theories on non-simply connected group manifolds. We �rst discuss which automor-phisms can be used. While every automorphism of �g that preserves the compact real formgives rise to an automorphism of the universal covering group ~G, such automorphisms do notnecessarily descend to the quotient group G. Rather, every automorphism of ~G restricts toan automorphism of the center Z( ~G) of ~G; for an inner automorphism this restriction is theidentity. The automorphisms of ~G that descend to automorphisms of G= ~G=� are preciselythose that map � to itself. Notice that the group of inner automorphisms of ~G and G coin-cide; in both cases this group is equal to the adjoint group ~G=Z( ~G)�=G=Z(G). The symmetrybreaking boundary conditions for non-simply connected group manifolds that come from auto-morphisms are therefore related to twined conjugacy classes of G in much the same way as inthe simply connected case, with the same subtleties arising for twined conjugacy classes thatare left invariant by some element of the center.We �nally remark that in the case of extensions, such as those for A1 at level k=0 mod4Z, another type of symmetry breaking boundary condition exists for the G-theory, namelyboundary conditions which only preserve the symmetries of the unextended theory, i.e. of the23

~G-theory. These come from automorphisms of the extended chiral algebra that act as theidentity on the unextended one. It has been demonstrated [7] that such boundary conditionsare labelled by ~G-primaries as well. As already mentioned, they correspond to those �-orbitsof conjugacy classes of ~G that are projected out in the ~G theory. For G=SO(3), for instance,they are obtained by projection from conjugacy classes of SU(2) that are related to half-integerspin highest weights. We can therefore describe also this type of boundary conditions by orbitsof ~G-conjugacy classes which by (5.2) project, in turn, to G-conjugacy classes.AcknowledgementWe would like to thank A. Lerda, K. Gaw�edzki and O. Grandjean for stimulating discussions.

24

A Fusion rulesIn this appendix derive the relation (3.14) between fusion rule coe�cients and weight multi-plicities. We start with the observation that a character can, on one hand, be written in termsof weight multiplicities ��(h) =X� m(�)� e�(h) ; (A.1)and on the other hand can be expressed in terms of Weyl's character formula as��(h) = X�1(h)Xw2W �(w) ew(�+�)(h) : (A.2)Here the sum is over the Weyl group W of �g, � is the sign function on W , andX(h) := e�(h)Y�>0(1 � e��(h)) (A.3)is the well-known expression for the denominator. (Up to an exponential e�+�, X�1 is just thecharacter of the corresponding Verma module of highest weight �.)Next we recall the Kac--Walton formula [30, 31] for WZW fusion rules. It expresses thefusion coe�cients N��� as an alternating sum over a certain subset W� of the a�ne Weyl groupW . W� consists by de�nition of those elements of W that map the fundamental Weyl alcoveto some alcove in the fundamental Weyl chamber. The set W� furnishes a distinguished setof representatives for the coset W=W , but W� is not a group. The representatives can becharacterized by the fact that they have minimal length. The Kac--Walton rule yieldsN��+� = Xw�2W� �(w�)Lw�(�+�)��;�+;� (A.4)where Lw�(�)�+� is the dimension of the space of singlets in the tensor product �Hw�(�) �H�+ �H�of the three �g-modules �Hw�(�), �H�+ and �H�. This dimension, in turn, can be expressed in termsof an integral over the corresponding characters asLw�(�+�)��;�+;� = ZGdg �w�(�+�)��(g)��+(g)��(g)= 1jW j ZTdhJ(h)�w�(�+�)��(h)��+(h)��(h) ; (A.5)where in the second line we have used Weyl's integral formula to reduce the integral to anintegral over a maximal torus T of G.The next step is to insert the formula (A.5) into the Kac-Walton rule (A.4) and to recombinethe summations overW andW� . At the same time, we use the Weyl character formula to rewritethe characters �w�(�+�)�� and ��+(g), while the third character is expressed in terms of weight25

space multiplicities. We then arrive atN��+� = 1jW j ZTdh Xw�2W� �(w�)J(h)�w�(�+�)��(h)��+(h)��(h)= 1jW j Xw1;w22W �(w1)�(w2) X�2L_ ZTdh ew1(�+�)�w2(�+�)+(k+g_)�(h)��(h)= 1jW j Xw1;w22W �(w1)�(w2)X� X�2L_ ZTdh ew1(�+�)�w2(�+�)+(k+g_)�(h)mult(�)� e�(h)= 1jW j Xw1;w22W �(w1)�(w2) X�2L_mult(�)�w1(�+�)+w2(�+�)�(k+g_)� ; (A.6)so that we have �nally arrived at the relation (3.14). Here in the second line we have also usedthe following two simple relations. First, the characters of two conjugate modules are relatedas ��+(h) = ��(h�1) : (A.7)Second, the Jacobian factor J in Weyl's integration formula can be expressed in terms of X asJ(h) = X(h)X(h�1) : (A.8)Together they allow us to cancel the two Weyl denominators against the volume factor J . The�-summation in the third line of (A.6) is over the weight system of �H�, and in the last line theintegral over the maximal torus T was evaluated explicitly.B Twined conjugationTo investigate the properties of the twined conjugation (4.18), it turns out to be helpful torelate it to the theory of non-connected Lie groups. The non-connected Lie groups for whichthe connected component of the identity is isomorphic to a given real, compact, connected andsimply connected Lie group G can be related to subgroups of the group of automorphisms of theDynkin diagram of the Lie algebra �g whose compact real form is the Lie algebra of the groupG. (This should not be confused with the relation between non-simply connected groups andautomorphisms of the extended Dynkin diagram.) Namely, for every subgroup �� of diagramautomorphisms of �g, one can construct a Lie group �G with the group of connected componentsgiven by �0( �G)= �� as the semi-direct product of the Lie group G and the �nite group ��.Conversely, if g is any element of a Lie group �G that is not in the connected component of theidentity, then the adjoint action of g on the Lie algebra �g is given by an outer automorphism!g and therefore corresponds to a symmetry of the Dynkin diagram of �g.The non-trivial connected components of �G are, as di�erentiable manifolds with metric,isomorphic to G. We �x a connected component G _! that corresponds to the element _! ofthe group of Dynkin diagramsymmetries of �g. The adjoint action of any element g 2G of theconnected component of the identity maps G _! to itself. Taking any arbitrary element g _! 2G _!,26

we can write every element in G _! as hg _! with h2G, and we have !(g)= g _!gg�1_! . For theadjoint action of g 2G we then �ndAdg(hg _!) = g h!(g)�1 g _! = Ad!g (h) g _! (B.1)with !�!g _! . We see that, after choosing an origin g _! for G _!, ordinary conjugation by g 2Gacts on h like twined conjugation. Changing the origin g _! changes the relevant automorphismby an inner automorphism.Now denote by N!(T !0 ) := fg 2G j gt!(g)�1 2 T !0 for all t 2 T !0 g (B.2)the twined normalizer of the connected component T !0 in the �xed subgroup of the maximaltorus T . The quotient W (T !0 ) := N(T !0 )=T !0 (B.3)is called the Weyl group of T !0 . It can be shown [19, 20] that W (T !0 ) is the product of thesubgroup W ! of the Weyl group that was de�ned in (4.10) and a �nite abelian group �(G;!).Moreover, the mapping degree of the mappingq! : G=T !0 � T !0 ! G(gT !0 ; t) 7! gt!(g)�1 (B.4)is [19, 20] deg q!= jW (T !0 )j. In particular, the mapping degree is positive, so q! is surjective.This, in turn, implies that any group element of G can be mapped a suitable by twined conju-gation (4.18) into T !0 , which generalizes the well-known conjugation theorems for the maximaltorus.The determinant of q! can be computed. One �nds at the point (1; h) with h2 T !0det q! = j�(G;!)j jY��>0 (1 � e2�i��(h))j2 =: j�(G;!)j � J!(h) ; (B.5)where the product is over a set of weights that are constructed from !?-orbits of positive �g-rootsand which can be shown [22] to be isomorphic to the set of positive roots of the so-called orbitLie algebra that is associated to �g and !. (Recall that W ! is isomorphic to the Weyl group ofthe orbit Lie algebra.) Application of Fubini's theorem then yields the twined generalizationZGdg f(g) = 1jW!j ZT!0 dhJ!(h)(ZG=T!0 d(gT !0 ) f(gt!(g)�1)) (B.6)of Weyl's integration formula. Here dg, dh and d(gT !0 ) are the Haar measures on the Lie groupsG and T !0 and on the homogeneous spaceG=T !0 , respectively. Obviously, the integration formulais particularly useful for twined class functions �! (see (4.16)), for which it reduces toZGdg �!(g) = 1jW!j ZT!0 dhJ!(h)�!(h) : (B.7)27

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