1 Introduction

A lot of information about the dynamics of D-branes is encoded in their charges. In particular, the D-brane charges constrain possible decay processes, and thus play an important role in stability considerations. There is evidence that these charges take values in (twisted) K-theory [1, 2, 3]. For D-branes on a simply connected group manifold \(G\), the charge group is conjectured to be the twisted K-theory \(^{k+h^\vee}K (G)\) [4, 5], where the twist involves an element of the third cohomology group \(H^{3} (G,\mathbb{Z})\), the Wess-Zumino form of the underlying Wess-Zumino-Witten (WZW) model at level \(k\).

For all simple, simply connected Lie groups \(G\), the twisted K-theory has been computed in [6] (see also [7, 8]) to be \[\tag{1.1} ^{k+h^\vee}K (G)\ =\ \underbrace{\mathbb{Z}_{M(G,k)}\oplus \dotsb \oplus \mathbb{Z}_{M(G,k)}}_{2^{{\rm rk} (G)-1}}\ ,\] where \(M(G,k)\) is the integer \[M(G,k)\ =\ \frac{k+h^\vee}{{\rm gcd}(k+h^\vee,L)} \ .\] Here \(h^\vee\) is the dual Coxeter number of the finite dimensional Lie algebra \(\bar{\mathfrak{g}}\), and \(L\) only depends on \(G\) (but not on \(k\)). In fact, except for the case of \(C_n\) that will not concern us in this paper, \(L\) is \[L \ =\ {\rm lcm} \{1,2,\ldots,h-1\} \ ,\] where \(h\) is the Coxeter number of \(\bar{\mathfrak{g}}\). For \(\bar{\mathfrak{g}}=A_n\) this formula was derived in [9] (see also [10]), while the formulae in the other cases were checked numerically up to very high levels in [11]. For the classical Lie algebras and \(G_2\) an alternative expression for \(M\) was also derived in [8].

These results should be compared with the charges that can be determined directly in terms of the underlying conformal field theory. The idea behind this approach is that brane configurations that are connected by RG flows should carry the same charge. These constraints were used in [12] to determine the charge group of su\((2)\). The constraint equations were generalised in [9] to the branes \(a\in \mathcal{B}_{k}^{\omega}\) of an arbitrary WZW model that preserve the full affine symmetry algebra \(\mathfrak{g}\) up to some automorphism \(\omega\). There it was argued that the charges \(q_a\) satisfy \[\tag{1.2} \dim (\lambda)\ q_{a}\ =\ \sum_{b\in \mathcal{B}_{k}^{\omega }} \mathcal{N}_{\lambda a}{}^{b}\ q_{b} \ ,\] where \(\lambda\in \mathcal{P}_{k}^{+} (\bar{\mathfrak{g}})\) is a dominant highest-weight representation of the affine Lie algebra \(\mathfrak{g}\) at level \(k\), \(\dim(\lambda)\) is the Weyl-dimension of the corresponding representation of the horizontal subalgebra \(\bar{\mathfrak{g}}\), and \(\mathcal{N}_{\lambda a}{}^{b}\) are the NIM-rep coefficients appearing in the Cardy analysis. In this paper we shall ignore the low level (\(k=1,2\)) subtleties discussed in [11] and assume that \(k\) is sufficiently big (\(k\geq 3\)).

For the trivial automorphism (\(\omega =\text{id}\)), the branes can be labelled by dominant highest weights of \(\mathfrak{g}\), \(\mathcal{B}_{k}^{\text{id}}\cong \mathcal{P}_{k}^{+} (\bar{\mathfrak{g}})\). In this case, the constraints (1.2) were evaluated in [9, 11]. The charges are given (up to rescalings) by the Weyl-dimensions of the corresponding representations, \(q_{\lambda}=\dim (\lambda)\), and the charge is conserved only modulo \(M(G,k)\). Thus, the untwisted branes account for one summand \(\mathbb{Z}_{M(G,k)}\) of the K-group (1.1).

For nontrivial outer automorphisms, a similar analysis was carried through in [13]. Here, the D-branes are parametrised by \(\omega\)-twisted highest weight representations \(a\) of \(\mathfrak{g}_{k}\) [14, 15, 16], and the NIM-rep coefficients are given by twisted fusion rules [16]. The twisted representations can be identified with representations of the invariant subalgebra \(\bar{\mathfrak{g}}^{\omega}\) consisting of \(\omega\)-invariant elements of \(\bar{\mathfrak{g}}\), and we can view \(\mathcal{B}_{k}^{\omega}\) as a subset of \(\mathcal{P}_{k'}^{+} (\bar{\mathfrak{g}}^{\omega })\), where \(k'=k+ h^\vee(\bar{\mathfrak{g}}) -h^\vee(\bar{\mathfrak{g}}^{\omega})\). It was found that the charge \(q_a\) of \(a\in \mathcal{B}_{k}^{\omega}\) is again (up to rescalings) given by the Weyl dimension¹ of the representation of \(\bar{\mathfrak{g}}^{\omega}\), \(q_{a}=\dim (a)\), and that the charge identities are only satisfied modulo \(M(G,k)\). Thus each such class of twisted D-branes accounts for another summand \(\mathbb{Z}_{M(G,k)}\) of the charge group. Since the number of automorphisms does not grow with the level, these constructions do not in general account for all the charges of (1.1); for the case of the \(A_n\) series, a proposal for the D-branes that may carry the remaining charges was made in [17, 18] (see also [19]).

The analysis of [13] was only done for all order-2 automorphisms of the classical Lie groups. There exist two ‘exceptional’ automorphisms, namely the order-3 automorphism of \(D_4\) (triality), and the order-2 automorphism of \(E_6\) (charge conjugation). These two cases are the subject of this paper. We will find again that with the charge assignment \(q_{a}=\dim (a)\), the charge identities are satisfied modulo \(M(G,k)\). Thus each such class of D-branes accounts for another summand of the charge group.²

The plan of the paper is as follows. In the remainder of this section we shall explain the main steps in proving these results that are common to both cases. The details of the analysis for the case of the triality automorphism of \(D_{4}\) (whose invariant subalgebra is \(G_{2}\)) is given in section 2. The corresponding analysis for the charge conjugation automorphism of \(E_6\) (whose invariant subalgebra is \(F_4\)) is given in section 3.

1.1 Some notation and a sketch of the proof

We begin by briefly introducing some notation. The \(\omega\)-twisted D-branes are characterised by the gluing conditions \[\tag{1.3} \left( J^a_n + \omega(\bar{J}^a_{-n}) \right)\, |\!| a\rangle\!\rangle \ =\ 0 \ ,\] where \(J^a_n\) are the generators of \(\mathfrak{g}\). Every boundary state can be written in terms of the \(\omega\)-twisted Ishibashi states \[\tag{1.4} |\!| a\rangle\!\rangle \ =\ \sum_{\mu \in {\mathcal E}_k^\omega} \psi_{a\mu}\, |\mu\rangle\!\rangle^\omega \ ,\] where \(|\mu\rangle\!\rangle^\omega\) is the (up to normalisation) unique state satisfying (1.3) in the sector \({\mathcal H}_\mu \otimes \bar{\mathcal H}_{\mu^\ast}\). The sum in (1.4) runs over the so-called exponents that consist of the weights \(\mu\in {\mathcal P}^{+}_k(\bar{\mathfrak{g}})\) that are invariant under \(\omega\). The NIM-rep coefficients are determined by the Verlinde-like formula \[\tag{1.5} {\mathcal N}_{\lambda a}{}^{b} \ =\ \sum_{\mu \in {\mathcal E}_k^\omega} \frac{ \psi_{b\mu}^\ast\, S_{\lambda \mu}\, \psi_{a\mu}}{S_{0\mu}} \ ;\] they define a non-negative integer matrix representation (NIM-rep) of the fusion rule algebra. (For a brief review of these matters see for example [13] and [20].)

It is clear on general grounds (see [13]) that for any charge assignment \(q_a\) for \(a\in \mathcal{B}_{k}^{\omega}\), the charge identity (1.2) can at most be satisfied modulo \(M(G,k)\). Our strategy will therefore be to construct a solution that solves (1.2) modulo \(M(G,k)\). This solution is again given by \(q_a = \dim(a)\). Furthermore, we can show that this solution of the charge equation is unique (up to trivial rescalings).

Our arguments will depend on the particularities of the two cases, but the general strategy is the same. The key observation of our analysis in both cases is a relation of the form \[\tag{1.6} \mathcal{N}_{\lambda a}{}^{b}\ =\ \sum_{\gamma,i} \varphi_{\lambda}{}^{\gamma}\,\varepsilon_{i}\, N_{\gamma a}{}^{\rho_{i}(b)}\] that expresses the NIM-rep coefficients \(\mathcal{N}_{\lambda a}{}^{b}\) in terms of the fusion rules \(N_{\gamma a}{}^{\rho_{i} (b)}\) of the affine algebra corresponding to \(\bar{\mathfrak{g}}^\omega\). Here \(\varphi_{\lambda}{}^{\gamma}\) is the branching coefficient which denotes how often the representation \(\gamma\) of \(\bar{\mathfrak{g}}^{\omega}\) appears in the restriction of the representation \(\lambda\) to \(\bar{\mathfrak{g}}\). The \(\rho_{i}\) are maps \(\rho_{i}:\mathcal{B}_k^{\omega}\to \mathcal{P}_{k'}^{+}(\bar{\mathfrak{g}}^{\omega})\) and \(\varepsilon_{i}\) is a sign attributed to the map \(\rho_{i}\). Furthermore \(k'\) is defined as before, \(k'=k+h^\vee(\bar{\mathfrak{g}}) - h^\vee(\bar{\mathfrak{g}}^\omega)\). In the cases studied in [13] analogous formulae for the NIM-rep coefficients were used for which the \(\rho_i\) could be expressed in terms of simple currents. In the current context where the invariant algebras are \(G_2\) and \(F_4\), such simple currents do not exist. Nevertheless it is possible to find such maps \(\rho_{i}\) (see (2.11) and (3.5) below for the specific formulae) that ’mimic’ the action of the simple currents.

The different maps \(\rho_i\) have disjoint images, and we can write \[{\mathcal P}_{k'}^{+}(\bar{\mathfrak{g}}^{\omega}) \ =\ \bigcup_i \rho_i\left(\mathcal{B}_k^{\omega}\right) \ \ \cup \ {\mathcal R}_k \ ,\] where \({\mathcal R}_k\) denotes the remainder. The second key ingredient in our proof are the relations \[\tag{1.7} \begin{aligned} \dim (\rho_{i} (a)) \ & = \ \varepsilon_{i}\dim (a), \quad a\in \mathcal{B}_k^{\omega}\\ \dim (b)\ & = \ 0, \quad b\in {\mathcal R}_k \ . \end{aligned}\] Both of these identities hold modulo \(M(G^\omega,k')\). Finally we observe by explicit inspection of the above formulae for \(M(G,k)\) that in the two cases of interest \[M(G,k) \ =\ M(G^\omega,k') \ .\] This then allows us to reduce the proof of the charge identities for the twisted D-branes of \(G\) to that of the untwisted D-branes of \(G^\omega\). In fact, the argument is simply \[\begin{aligned} \sum_{b \in \mathcal{B}_{k}^\omega} \mathcal{N}_{\lambda a}^{\phantom{\lambda a}b}\, \dim_{G^\omega}(b)\ & =\ \sum_{b \in \mathcal{B}_{k}^{\omega}}\sum_{i}\sum_{\gamma} \varepsilon_{i}\,\varphi^{\phantom{\lambda}\gamma}_{\lambda} N_{\gamma a}^{\phantom{\lambda a}\rho_{i}(b)}\, \dim_{G^\omega}(b) & \nonumber\\ & =\ \sum_{b \in \mathcal{B}_{k}^{\omega}} \sum_{i}\sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda} N_{\gamma a}^{\phantom{\lambda a}\rho_{i}(b)}\, \dim_{G^\omega}(\rho_{i}(b)) & {\rm mod}\ M(G,k) \nonumber\\ & =\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda} \sum_{b \in \mathcal{P}_{k'}^{+}(\bar{\mathfrak{g}}^\omega)} N_{\gamma a}^{\phantom{\lambda a}b}\, \dim_{G^\omega}(b) & {\rm mod}\ M(G,k) \nonumber\\ &=\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda}\, \dim_{G^\omega}(\gamma)\,\dim_{G^\omega}(a) & {\rm mod}\ M(G,k) \nonumber \\ & =\ \dim_{G}(\lambda)\,\dim_{G^\omega}(a) \ . & \end{aligned}\] In the following two sections we shall give the details for how to define the maps \(\rho_i\), and prove the various statements above. We shall also be able to show that our charge solution is unique up to trivial rescalings.