Charges of twisted branes: the exceptional cases

2 The \(D_4\) case with triality

In the \(D_{4}\) case the relevant automorphism \(\omega\) is triality which maps the Dynkin labels \(\mu=(\mu_{0};\mu_{1},\mu_{2},\mu_{3},\mu_{4})\) to \((\mu_{0};\mu_{4},\mu_{2},\mu_{1},\mu_{3})\). The set of exponents labelling the \(\omega\)-twisted Ishibashi states is therefore \[\mathcal{E}^{\omega}_{k}=\{ (\mu_{0};\mu_{1},\mu_{2},\mu_{1},\mu_{1})\in \mathbb{N}_{0}^{5}\,\vert\,\mu_{0}+3\mu_{1}+2\mu_{2}=k\}\ .\] The \(\omega\)-twisted boundary states are labelled by the level \(k\) integrable highest weights of the twisted Lie algebra \(\mathfrak{g}^{\omega}=D_{4}^{(3)}\), which are \(\mathcal{B}_{k}^\omega=\{(b_{0};b_{1},b_{2}) \in \mathbb{N}^{3}_{0} \,\vert\, b_{0}+2b_{1}+3b_{2}=k\}\). The states of lowest conformal weight of these representations form irreducible representations of the invariant subalgebra \(\bar{\mathfrak{g}}^{\omega}=G_2\) with highest weights \((b_{1},b_{2})\). For this reason we propose that the corresponding D-brane charge is the Weyl dimension of these irreducible representations, i.e. \[\tag{2.1} q_{b}\ =\ \dim_{G_2}(b_{1},b_2)\ =\ \dim_{G_2}(b)\ .\] In this section we shall prove that (2.1) solves the charge constraint \[\tag{2.2} \dim_{D_4}(\lambda)\,q_a\ =\ \sum_{b\in \mathcal{B}_{k}^{\omega}}\mathcal{N}_{\lambda a}^{\phantom{\lambda a}b} \, q_b\] modulo \(M(G,k)\) and that this solution is unique (up to rescalings).

2.1 The solution

To show that (2.1) indeed solves the charge constraint (2.2) we trace the problem back to the case of untwisted branes in \(G_{2}\). So we need to express ’twisted \(D_{4}\) data’ by ’untwisted \(G_{2}\) data’. We first note, as already mentioned in section 1.1, that the integer \(M\) for \(G_{2}\) at level \(k+2\) equals the integer for \(D_{4}\) at level \(k\), \[M(D_{4},k)\ =\ M(G_{2},k+2)\ =\ \frac{k+6}{{\rm gcd}(k+6,60)}\ .\] As is also explained there, the key result (1.6) that we need to prove expresses the NIM-rep \(\mathcal{N}\) of \(D_4\) in terms of the fusion rules of \(G_{2}\). The first step in providing such a relation is the identification of the \(D_{4}\) \(\psi\)-matrix at level \(k\) with the (rescaled) \(S\)-matrix of \(G_{2}\) at level \(k+2\) (in the following we shall denote the \(S\)-matrix of \(G_2\) by \(S'\) in order to distinguish it from the \(S\)-matrix of \(D_4\))3 \[\tag{2.3} \psi_{b\mu}\ =\ \sqrt{3}\,S^{\prime}_{b\tilde{\mu}}\ ,\] where \(\tilde{\mu}\) is defined by \[\tag{2.4} \mu \ \mapsto\ \tilde{\mu} = (\mu_{0};3\mu_{1}+2,\mu_{2}) \ .\] Note that if \(\mu \in {\mathcal E}^\omega_k\), then \(\tilde{\mu}\in\mathcal{P}^{+}_{k+2}(G_{2})\equiv \mathcal{P}_{k+2}=\{(\tilde\mu_{1},\tilde\mu_{2})\in \mathbb{N}_{0}^{2}\,\vert\,\tilde\mu_{1}+2\tilde\mu_{2}\leq k+2\}\). The identity (2.3) can be proven as follows. Define \(\kappa =k+6\) and \(c(x)=\cos\big(\frac{2\pi x}{3\kappa} \big)\). The \(\psi\)-matrix is given by (see [16]), \[\tag{2.5} \begin{aligned} \psi_{b\mu} = \frac{2}{\kappa} \big( c (pp'+2pq'+2qp'+qq') + c (2pp'+pq'+qp'-qq')\\ +c (-pp'+pq'+qp'+2qq') - c (2pp'+pq'+qp'+2qq')\\ - c (pp'+2pq'-qp'+qq') -c (pp'-pq'+2qp'+qq') \big) , \end{aligned}\] where \(p=b_{1}+b_{2}+2\), \(q=b_{2}+1\) and \(p'=3\mu_{1}+\mu_{2}+4\), \(q'=\mu_{2}+1\). On the other hand, if we define \(m=\lambda_{1}+\lambda_{2}+2\), \(n=\lambda_{2}+1\), \(m^{\prime}=\nu_{1}+\nu_{2}+2\) and \(n^{\prime}=\nu_{2}+1\), then the \(S\)-matrix of \(G_{2}\) at level \(k+2\) is [22] \[\tag{2.6} \begin{aligned} S'_{\lambda\nu} =\frac{-2}{\sqrt{3}\kappa} \big( c(2mm^{\prime}+mn^{\prime}+nm^{\prime}+2nn^{\prime}) +c(-mm^{\prime}-2mn^{\prime}-nn^{\prime}+nm^{\prime})\\ +c(-mm^{\prime}+mn^{\prime}-2nm^{\prime}-nn^{\prime}) -c(-mm^{\prime}-2mn^{\prime}-2nm^{\prime}-nn^{\prime})\\ -c(2mm^{\prime}+mn^{\prime}+nm^{\prime}-nn^{\prime}) -c(-mm^{\prime}+mn^{\prime}+nm^{\prime}+2nn^{\prime})\big)\ . \end{aligned}\] For (2.6) we also use the abbreviated notation \[\tag{2.7} S'_{\lambda\nu}\ =\ \frac{-2}{\sqrt{3}\kappa}\{c(u_{1}) +c(u_{2})+c(u_{3})-c(u_{4})-c(u_{5})-c(u_{6})\} \ .\] By comparing (2.5) and (2.6) one then easily proves (2.3).

Next we observe from (1.5) that in order to obtain fusion matrices of \(G_{2}\) we also need to express the quotient \(\frac{S_{\lambda\mu}}{S_{0 \mu}}\) in terms of \(G_{2}\) \(S\)-matrices. The relevant relation is \[\tag{2.8} \frac{S_{\lambda\mu}}{S_{0\mu}} \ =\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda}\ \frac{S^{\prime}_{\gamma\tilde{\mu}}}{S^{\prime}_{0\tilde{\mu}}} \ .\] Here, \(\varphi^{\phantom{\lambda}\gamma}_{\lambda}\) denotes the \(D_{4}\supset G_{2}\) branching rules; the most important ones are \[\tag{2.9} (1,0,0,0)\ \to\ (1,0)\oplus (0,0)\quad \text{and}\quad (0,1,0,0)\ \to\ (0,1)\oplus (1,0)\oplus (1,0) \ .\] The easiest way to prove (2.8) is to consider the explicit expressions for the fundamental representations of \(D_{4}\).

Taking all of this together we can now write the \(D_{4}\) NIM-rep as \[\tag{2.10} \mathcal{N}_{\lambda a}^{\phantom{\lambda a}b}\ =\ \sum_{\mu \in \mathcal{E}_{k}^{\omega}} \psi_{a\mu}\,\psi_{b\mu}^{*}\, \frac{S_{\lambda\mu}}{S_{0\mu}} \ =\ 3\sum_{\gamma}\varphi_{\lambda}{}^{\gamma} \, \sum_{\mu \in \mathcal{E}_{k}^{\omega}} S_{a\tilde{\mu}}^{\prime}\, S_{b\tilde{\mu}}^{\prime*}\, \frac{S'_{\gamma\tilde{\mu}}}{S'_{0\tilde{\mu}}} \ .\] Although this formula reminds one of the Verlinde formula, the last sum still does not give the \(G_{2}\)-fusion rules as the range of summation for \(\tilde{\mu}\) is only a subset of \(\mathcal{P}_{k+2}\). To resolve this problem we introduce the affine mappings \[\tag{2.11} \begin{array}{l} \rho_{0}({b})\ =\ (b_{1},b_{2})\\ \rho_{1}({b})\ =\ (k-2b_{1}-3b_{2},1+b_{1}+b_{2})\\ \rho_{2}({b})\ =\ (k+1-b_{1}-3b_{2},b_{2})\\ \end{array}\] which map the set \(\mathcal{B}_{k}^\omega\) of boundary states to disjoint subsets of \(\mathcal{P}_{k+2}=\mathcal{P}_{k+2}^+(G_2)\). They have the crucial property \[\tag{2.12} S^{\prime}_{\rho_{0} (b)\, \nu} +S^{\prime}_{\rho_{1} (b)\, \nu} -S^{\prime}_{\rho_{2}(b)\, \nu} \ =\ \left\{ \begin{array}{ll} 3 \, S^{\prime}_{b\, \nu} \qquad & \text{if}\;\nu_{1}=2\; \text{mod}\;3\\ 0 \qquad & \text{otherwise,} \end{array} \right.\] where \(b \in \mathcal{B}_{k}^\omega\) and \(\nu \in \mathcal{P}_{k+2}\). This follows from the fact that the left hand side can be written as \[\begin{gathered} \frac{-\sqrt{3} \kappa}{2}\big(S^{\prime}_{b\, \nu} +S^{\prime}_{\rho_1(b)\, \nu} -S^{\prime}_{\rho_2(b)\, \nu}\big) \ =\big(1+\cos(v_{1})+\cos(v_{2})\big)\big(c(u_{1})-c(u_{4})\big) \\ +\big(1+\cos(v_{1})+\cos(v_{3})\big)\big(c(u_{2})-c(u_{6})\big) +\big(1+\cos(v_{2})+\cos(v_{3})\big)\big(c(u_{3})-c(u_{5})\big) \\ +\big(\sin(v_{1})+\sin(v_{2})\big)\big(s(u_{1})+s(u_{4})\big) -\big(\sin(v_{1})-\sin(v_{3})\big)\big(s(u_{2})+s(u_{6})\big)\\ -\big(\sin(v_{2})+\sin(v_{3})\big)\big(s(u_{3})+s(u_{5})\big) ,\end{gathered}\] where \(v_{1}=\frac{2}{3}\pi(\nu_{1}+3\nu_{2}+4)\), \(v_{2}=\frac{2}{3}\pi(2\nu_{1}+3\nu_{2}+5)\), \(v_{3}=\frac{2}{3}\pi(\nu_{1}+1)\) and \(s(x)=\sin\big(\frac{2\pi x}{3\kappa}\big)\). (The \(u_i\) are defined as in (2.7).) This is easily seen to agree with the right hand side of (2.12).

Let \(\rho(\mathcal{B}_{k}^\omega) =\rho_{0}(\mathcal{B}_{k}^\omega)\,\cup\, \rho_{1}(\mathcal{B}_k^\omega)\, \cup \, \rho_{2}(\mathcal{B}_{k}^\omega)\) and \(\mathcal{R}_{k}=\mathcal{P}_{k+2}\setminus \rho(\mathcal{B}_{k}^\omega)\). The special elements \(\nu \in \mathcal{P}_{k+2}\) in (2.12) which satisfy \(\nu_{1}=2\) mod \(3\) are precisely the images \(\nu=\tilde{\mu}\) under (2.4) of a suitable element \(\mu\) of \(\mathcal{E}^{\omega}_{k}\). The key relation \(\eqref{eqn:killer1}\), together with (2.10), therefore implies that the \(D_{4}\) NIM-rep can be written as a sum of \(G_{2}\) fusion matrices, \[\tag{2.13} \begin{aligned} \mathcal{N}_{\lambda a}^{\phantom{\lambda a}b} \ & =\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda} \sum_{\mu \in \mathcal{P}_{k+2}} S^{\prime}_{a\mu}\, \frac{S^{\prime}_{\gamma\mu}}{S^{\prime}_{0\mu}}\, \Big(S^{\prime*}_{\rho_{0}(b)\, \mu}+S^{\prime*}_{\rho_{1} (b)\, \mu} -S^{\prime*}_{\rho_{2} (b)\, \mu}\Big)\nonumber\\ &=\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda}\, \Big(N_{\gamma a}^{\phantom{\gamma a} \rho_{0}(b)} +N_{\gamma a}^{\phantom{\gamma a}\rho_{1} (b)}-N_{\gamma a}^{\phantom{\gamma a}\rho_{2} (b)}\Big)\nonumber \\ &=\ \sum_{i=0}^{2}\sum_{\gamma}\varepsilon_{i}\, \varphi^{\phantom{\lambda}\gamma}_{\lambda}N_{\gamma a}^{\phantom{\gamma a}\rho_{i}(b)} \ , \end{aligned}\] where \(N_{\gamma}\) denote \(G_{2}\) fusion matrices at level \(k+2\) and \(\varepsilon_{i}\) accounts for the signs. [Explicitly \(\varepsilon_0=\varepsilon_1=+1\) and \(\varepsilon_2=-1\).] This is the relation (1.6) we proposed in section 1.1. Note that (2.13) is valid for all highest weights \(\lambda\) of \(D_{4}\), not only for the ones appearing in \(\mathcal{P}_{k}^{+} (D_{4})\). In fact we can continue \(\mathcal{N}_{\lambda}\) and \(N_{\gamma}\) outside of the usual domain by rewriting the ratios of \(S\)-matrices appearing in (2.10) as characters of the finite Lie algebras.

According to the argument given in section 1.1, there is only one further ingredient we need to show. This concerns the behaviour of the \(G_{2}\)-Weyl dimensions under the maps \(\rho_{i}\), and is summarised in the relations (1.7). Thus we need to prove that \[\tag{2.14} \dim_{G_{2}}(\rho_{i}(b))=\ \varepsilon_{i}\,\dim_{G_{2}}(b) \qquad \text{mod } M(G_{2},k+2)\] and that any element \(r \in \mathcal{R}_{k}\) satisfies \[\tag{2.15} \dim_{G_{2}}(r)=0, \qquad \text{mod}\; M(G_{2},k+2)\ .\] The dimension of a \(G_{2}\) representation \((b_{1},b_{2})\) is given by \[\begin{gathered} \dim_{G_{2}}(b_{1},b_{2})\ =\ \frac{1}{120} (b_{1}+1)(b_{2}+1)(b_{1}+b_{2}+2)\\ \cdot(b_{1}+2b_{2}+3)(b_{1}+3b_{2}+4)(2b_{1}+3b_{2}+5)\ .\end{gathered}\] To prove (2.14) we find by explicit computation that \[\tag{2.16} \dim_{G_{2}}(\rho_i(b)) \ =\ \varepsilon_i\, \dim_{G_2}(b) +\frac{M(G_2,k+2)}{F}\;p^{5}(b) \ ,\] where \(p^{5}(b)\) denotes a \(k\) and \(\rho_{i}\)-dependent polynomial of order \(5\) in the variables \(b_{1},b_{2}\) with integer coefficients, and \(F=\frac{120}{\textrm{gcd}(k+6,60)}\). Thus it remains to show that \(\frac{p^{5}}{F}\) is an integer. If \(8\) does not divide \(k+6\), then \(M(G_2,k+2)\) and \(F\) are coprime. Since \(\frac{M(G_2,k+2)}{F}p^{5}\) is an integer, \(\frac{p^{5}}{F}\) has to be an integer as well and we are done. If \(8\) is a divisor of \(k+6\), then \(F\) and \(M(G_2,k+2)\) have greatest common divisor \(2\). The result then follows provided that \(p^{5}\) is even, which is easily verified.

To show (2.15) we first have to identify the elements of \(\mathcal{R}_{k}\). It is convenient to write this set as the (disjoint) union of the two subsets \(\mathcal{R}^1_{k}\) and \(\mathcal{R}^2_{k}\). The first of them is defined by \[\mathcal{R}^1_{k} = \{(b_{1},b_{2}) \in \mathcal{P}_{k+2}\, \vert \, (b_{1},b_{2})=(k+2-3j,j), \; j \in \mathbb{N}_{0}\}\ .\] The set \(\mathcal{R}^2_{k}\equiv \mathcal{R}_{k} \setminus \mathcal{R}^1_{k}\) depends in a more complicated manner on \(k\). To describe it explicitly we therefore distinguish the three cases:

For any \(r \in \mathcal{R}_{k}\) one then easily checks that \[\dim_{G_2}(r)\ =\ \frac{M}{F}p^5(j)\ \] with some polynomial \(p^{5}\) in \(j\) of order 5. One finds that the polynomials \(p^5(j)\) are even whenever 8 divides \(k+6\). Using the same arguments as above, this then finishes the proof of (2.15). It remains to check that we have identified the complete set \(\mathcal{R}_{k}\) correctly. Because \(\rho_{i}(\mathcal{B}_{k}^\omega)\,\cap\, \rho_{j}(\mathcal{B}_{k}^\omega)=\emptyset\) for \(i\neq j\) we have \(|\rho(\mathcal{B}_{k}^\omega)| =3 \,|\mathcal{B}_{k}^\omega|\). In order to see that \(\mathcal{P}_{k+2}=\rho(\mathcal{B}_{k}^\omega)\cup \mathcal{R}_{k}\), it is therefore sufficient to count the number of elements of the different sets. One easily finds

\[|\mathcal{P}_{k+2} | = \left\{ \begin{array}{ll} \frac{1}{4}(k+4)^2 & \quad k \; \text{even}\\ \frac{1}{4}(k+3)(k+5) & \quad k \; \text{odd}\\ \end{array}\right.\] as well as \[|\mathcal{R}_{k} | = \left\{ \begin{array}{llr} \frac{1}{2}(k+2) & \quad k=0 & \,\text{mod}\ 6 \\ \frac{1}{2}(k+4) & \quad k=2,4 & \text{mod}\ 6\\ \frac{1}{2}(k+3) & \quad k=3 & \text{mod} \ 6 \\ \frac{1}{2}(k+5) & \quad k=1,5 & \text{mod}\ 6\\ \end{array}\right.\] and \[|\mathcal{B}_{k}^{\omega } | = \left\{ \begin{array}{llr} \frac{1}{12}k^{2}+\frac{1}{2}k+1 & \quad k=0 &\,\text{mod}\ 6 \\ \frac{1}{12}(k+2)(k+4) & \quad k=2,4 & \text{mod}\ 6\\ \frac{1}{12}(k+3)^2 & \quad k=3 & \text{mod} \ 6 \\ \frac{1}{12}(k+1)(k+5)& \quad k=1,5 & \text{mod}\ 6 \\ \end{array}\right.\]

Using these formulae it is then easy to show that \(|\mathcal{P}_{k+2}|=|\rho(\mathcal{B}_{k}^\omega)|\,+\, |\mathcal{R}_{k}|\). This completes the proof.

2.2 Uniqueness

It remains to prove that the solution we found is unique up to an overall rescaling of the charge. To this end we show that any solution of the charge constraint modulo some integer \(M'\) satisfies the relation \[\tag{2.17} q_a\ =\ \dim (a)\, q_0 \quad \mod M' \ ,\] and thus is obtained from our solution (2.1) by scaling with the factor \(q_{0}\).

To prove (2.17) we first want to show that any \(G_{2}\) representation \(a\) can be obtained as restriction of a linear combination of \(D_4\) representations \(\lambda_j\) with integer coefficients \(z_j\). We explicitly allow negative multiplicities and write formally \[a \ =\ \bigoplus_j z_j \lambda_j\big|_{G_2} \ .\] Obviously it is sufficient to prove this for the fundamental representations. Looking at the branching rules (2.9) we see that the representation \((1,0)\) appears in the decomposition of \((1,0,0,0)\), so we can write \[(1,0)\ =\ \big( (1,0,0,0) - (0,0,0,0) \big) \big|_{G_2} \ .\] Similarly, we can express \((0,1)\) as a restriction because it appears exactly once in the decomposition of \((0,1,0,0)\) together only with \((1,0)\) (see (2.9)).

Now consider a boundary state labelled by \(a\). We can use (2.13) to write4 \[\begin{aligned} \nonumber \dim_{G_2}(a)\, q_0\ &=\ \sum_j z_j\, \dim_{D_4} (\lambda_j)\, q_0\\ \nonumber &=\ \sum_{j,b} z_j \, \mathcal{N}_{\lambda_j 0}{}^b q_b \qquad \mod M'\\ \nonumber &=\ \sum_{i,j,\gamma,b} z_j \, \varepsilon_i \, \varphi_{\lambda_j}{}^{\gamma} \, N_{\gamma 0}{}^{\rho_i(b)}\, q_b\\ \nonumber &=\ \sum_{i,b} \varepsilon_i\, N_{a0}{}^{\rho_i(b)} \, q_b\\ &=\ q_a \ . \end{aligned}\] In the last step we used the fact that \(\rho_{i} (\mathcal{B}_{k}^\omega)\) and \(\mathcal{B}_{k}^\omega\) are disjoint for \(i\not= 0\), so that only \(i=0\) contributes. This concludes the proof of (2.17).