3 The \(E_{6}\) case with charge conjugation

The analysis for the case of \(E_6\) is fairly similar to the \(D_4\) case discussed in the previous section, and we shall therefore be somewhat briefer. For \(E_{6}\) the invariant subalgebra under charge conjugation is \(\bar{\mathfrak{g}}^{\omega}=F_{4}\). Again we have the identity \[M(E_{6},k)\ =\ M(F_{4},k+3)\ =\ \frac{ k+ 12}{{\rm gcd}(k+12,2^3 \cdot 3^2\cdot 5 \cdot 7 \cdot 11)} \ .\] As before we therefore expect that \[\tag{3.1} \dim_{E_{6}}(\lambda)\,q_a \ =\ \sum_{b\in \mathcal{B}_{k}^\omega} \mathcal{N}_{\lambda a}^{\phantom{\lambda a}b} \, q_b \qquad \text{mod} \;M(F_{4},k+3) \ ,\] where \[\tag{3.2} q_b \ =\ \dim_{F_4}(b) \ .\] The order 2 automorphism \(\omega\) of \(E_{6}\) maps the Dynkin labels \[(\mu_{0};\mu_{1},\mu_{2},\mu_{3},\mu_{4},\mu_{5},\mu_{6})\] to \((\mu_{0};\mu_{5},\mu_{4},\mu_{3},\mu_{2},\mu_{1},\mu_{6})\), and thus the set of exponents is \[\mathcal{E}^{\omega}_{k} \ =\ \{(\mu_{0};\mu_{1},\mu_{2},\mu_{3},\mu_{2},\mu_{1},\mu_{6}) \in \mathbb{N}_{0}^{7} \, \vert \, \mu_{0}+2\mu_{1}+4\mu_{2}+3\mu_{3}+2\mu_{6}=k\}\ .\] The twisted algebra here is \(E_{6}^{(2)}\). The set of boundary states at level \(k\) is explicitly given by \(\mathcal{B}_{k}^\omega=\{(b_{0};b_{1},b_{2},b_{3},b_{4})\in \mathbb{N}_{0}^{5}\,\vert\, b_{0}+2b_{1}+3b_{2}+4b_{3}+2b_{4}=k\}.\) As in the last section, it is possible to identify the \(\psi\)-matrix of \(E_6\) at level \(k\) with the \(S\)-matrix of \(F_4\) at level \(k+3\) (see also [16]), \[\psi_{b\mu}\ =\ 2\,S^{\prime}_{b\tilde{\mu}}\ ,\] where \(\tilde{\mu}\) is now defined by \[\tag{3.3} \mu \ \mapsto\ \tilde{\mu}= (\mu_{0};2\mu_{1}+1,2\mu_{2}+1,\mu_{3},\mu_{6})\ .\] As before we observe that if \(\mu\in{\mathcal E}^\omega_k\), then \(\tilde\mu\in \mathcal{P}^+_{k+3}(F_4)\equiv \mathcal{P}_{k+3}\), where the latter is explicitly defined as \(\mathcal{P}_{k+3}= \{(\tilde\mu_{1},\tilde\mu_{2},\tilde\mu_{3},\tilde\mu_{4})\in \mathbb{N}^{4}_{0}\,\vert\, \tilde\mu_{1}+2\tilde\mu_{2} +3\tilde\mu_{3}+2\tilde\mu_{4}\leq k+3\}\). Furthermore, we can express ratios of \(S\)-matrices of \(E_{6}\) by those of \(F_{4}\), \[\frac{S_{\lambda\mu}}{S_{0\mu}} \ =\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda}\ \frac{S^{\prime}_{\gamma\tilde{\mu}}}{S^{\prime}_{0\tilde{\mu}}} \ .\] Here \(S\) denotes the \(E_{6}\) \(S\)-matrix at level \(k\), \(S^{\prime}\) is the \(F_{4}\) \(S\)-matrix at level \(k+3\), and \(\varphi^{\phantom{\lambda}\gamma}_{\lambda}\) describes the branching \(E_6\supset F_4\); for the six fundamental representations of \(E_6\) the branching rules are \[\begin{aligned} (1,0,0,0,0,0) & \simeq & (0,0,0,0,1,0) \, \to\, (1,0,0,0) \oplus (0,0,0,0) \\ (0,1,0,0,0,0) & \simeq & (0,0,0,1,0,0) \, \to\, (0,1,0,0) \oplus (0,0,0,1)\oplus (1,0,0,0) \\ (0,0,1,0,0,0) & \to & (0,0,1,0) \oplus (1,0,0,1) \oplus 2\cdot (0,1,0,0) \oplus (0,0,0,1) \end{aligned}\] and \[\tag{3.4} (0,0,0,0,0,1) \to (0,0,0,1) \oplus (1,0,0,0) \ .\] The relevant affine mappings are in this case \[\tag{3.5} \begin{array}{l} \rho_0(b)\ =\ (b_{1},b_{2},b_{3},b_{4})\\ \rho_1(b)\ =\ (k-2b_{1}-3b_{2}-4b_{3}-2b_{4},1+b_{1}+b_{2},b_{3},b_{4})\\ \rho_2(b)\ =\ (k+1-b_{1}-3b_{2}-4b_{3}-2b_{4},b_{2},b_{3},b_{4})\\ \rho_3(b)\ =\ (k-2b_{1}-3b_{2}-4b_{3}-2b_{4},b_{1},b_{2}+b_{3}+1,b_{4}) \ , \end{array}\] which map boundary states at level \(k\) to dominant weights of \(F_{4}\) at level \(k+3\), i.e. to elements of \(\mathcal{P}_{k+3}\). There is a similar identity to (2.12) for the \(S\)-matrices \[\tag{3.6} S^{\prime}_{\rho_{0}(b)\, \nu} +S^{\prime}_{\rho_{1}(b)\, \nu} -S^{\prime}_{\rho_{2}(b)\, \nu} -S^{\prime}_{\rho_{3}(b)\, \nu} \ = \ \left\{ \begin{array}{ll} 4 \, S^{\prime}_{b\, \nu} \qquad & \text{if}\;\nu_{1}=\nu_{2}=1\;\text{mod}\; 2 \\ 0 \qquad & \text{otherwise}, \end{array} \right.\] where \(b\in \mathcal{B}_{k}^\omega\) and \(\nu \in \mathcal{P}_{k+3}\). Again, the elements which satisfy \(\nu_{1}=\nu_{2}=1\) mod \(2\) are precisely the images \(\nu=\tilde{\mu}\) of an element \(\mu\) of \(\mathcal{E}_k^{\omega}\) under the mapping (3.3). By \(\rho(\mathcal{B}_{k}^\omega)\) we denote the union of the images of \(\mathcal{B}_{k}^\omega\) under the maps \(\rho_{i}\), \(\rho (\mathcal{B}_{k}^\omega)= \bigcup_{i=0}^{3}\rho_{i} (\mathcal{B}_{k}^\omega)\). The elements of \(\mathcal{P}_{k+3}\) which are not reached by the maps form the set \(\mathcal{R}_{k}=\mathcal{P}_{k+3}\setminus \rho(\mathcal{B}_{k}^\omega)\).

Using essentially the same arguments as for the case of \(D_4\) discussed in the last section, we can then show that the \(E_{6}\) NIM-rep can be expressed in terms of \(F_{4}\) fusion matrices as \[\tag{3.7} \begin{aligned} \mathcal{N}_{\lambda a}^{\phantom{\lambda a} b} \ &=\ \sum_{\gamma}\varphi^{\phantom{\lambda}\gamma}_{\lambda}\, \Big(N_{\gamma a}^{\phantom{\gamma a}\rho_{0}(b)} +N_{\gamma a}^{\phantom{\gamma a}\rho_1(b)}-N_{\gamma a}^{\phantom{\gamma a}\rho_2(b)}-N_{\gamma a}^{\phantom{\gamma a}\rho_3(b)}\Big)\\ &=\ \sum_{i=0}^{3}\sum_{\gamma}\varepsilon_{i}\, \varphi^{\phantom{\lambda}\gamma}_{\lambda}N_{\gamma a}^{\phantom{\gamma a}\rho_{i}(b)} \ , \end{aligned}\] where the \(\varepsilon_{i}\) account for the signs. [Explicitly, \(\varepsilon_1=\varepsilon_2=+1\) and \(\varepsilon_3=\varepsilon_4=-1\).] Following the argument of section 1.1, it thus only remains to show that \[\tag{3.8} \dim_{F_{4}}(\rho_{i}(b))=\varepsilon_{i}\,\dim_{F_{4}}(b) \qquad \text{mod } M(F_{4},k+3)\] and for all \(r \in \mathcal{R}_{k}\) \[\tag{3.9} \dim_{F_4}(r)=\ 0 \qquad \text{mod } M(F_4,k+3).\] To prove equation (3.8) we note that \[\dim_{F_{4}}(\rho_{i}(b))\ =\ \varepsilon_{i}\,\dim_{F_{4}}(b) +\frac{M(F_4,k+3)}{F}\, p^{23}(b)\] where \[F\ =\ \frac{2^{15}\cdot 3^7\cdot 5^4\cdot 7^2\cdot 11}{\textrm{gcd}(k+12,2^3\cdot 3^2\cdot 5\cdot 7\cdot 11)}\] and \(p^{23}\) is a \(k\) and \(\rho_{i}\)-dependent polynomial (with integer coefficients) of degree \(23\) in the labels \(b_i\). Now \(M(F_4,k+3)\) and \(F\) are coprime whenever \(2^4\), \(3^3\), \(5^2\) and \(7^2\) do not divide \(k+12\); in this case (3.8) is proven as before. Otherwise the analysis is more involved and many cases would have to be distinguished. We have not attempted to analyse all of them in detail, but we have performed a numerical check up to fairly high levels. This seems satisfactory, given that the identities for \(M(E_6,k)\) and \(M(F_4,k)\) have also only be determined numerically.

Finally, we need to show the identity (3.9). This requires a good description of the set \(\mathcal{R}_{k}\). Here it is convenient to write it as the union of four disjoint subsets which are defined by \[\begin{aligned} \mathcal{R}_{k}^1\ &=\ \left\{ \begin{array}{p{11.2cm}l} \{b \in \mathcal{P}_{k+3}\,\vert\, b=(1+2j_1,j_2,j_3,(k+2)/2-j_1-j_2-2j_3) \} \\ \{b \in \mathcal{P}_{k+3}\,\vert\, b=(2j_1,j_2,j_3,(k+3)/2-j_1-j_2-2j_3)\} \\ \end{array} \right.\\ \mathcal{R}_{k}^2\ &=\ \left\{ \begin{array}{p{11.2cm}l} \{b \in \mathcal{P}_{k+3}\,\vert\, b=(2j_1,2j_2,j_3, (k+2)/2-j_1-3j_2-2j_3) \} \\ \{b \in \mathcal{P}_{k+3}\,\vert\, b=(1+2j_1,2j_2,j_3,(k+1)/2-j_1-3j_2-2j_3)\} \\ \end{array} \right.\\ \mathcal{R}_{k}^3\ &=\ \left\{ \begin{array}{p{11.2cm}l} \{b \in \mathcal{P}_{k+3}\,\vert\, b=(1+2j_1,1+2j_2,j_3, (k-2)/2-j_1-3j_2-2j_3) \} \\ \{b \in \mathcal{P}_{k+3}\,\vert\, b=(2j_1,1+2j_2,j_3,(k-1)/2-j_1-3j_2-2j_3)\} \\ \end{array} \right.\\ \mathcal{R}_{k}^4\ &=\ \left\{ \begin{array}{p{11.2cm}l} \{b \in \mathcal{P}_{k+3}\,\vert\, b=(j_1,1+2j_2,j_3, (k-2)/2-j_1-3j_2-2j_3) \} \\ \{b \in \mathcal{P}_{k+3}\,\vert\, b=(j_1,2j_2,j_3,(k+1)/2-j_1-3j_2-2j_3)\} \\ \end{array} \right.\end{aligned}\] where the top line corresponds to the even case and the bottom line to the odd case and where \((j_1,j_2,j_3) \in \mathbb{N}_{0}^3\). The same arguments as before show that the elements \(r\) in these sets satisfy \(\dim_{F_4}(r)=0\) \(\text{mod}\; M(F_4,k+3)\). Again, this is proven only if \(k+12\) is not divisible by \(2^4,3^3,5^2\) or \(7^2\); for the other levels we have only performed numerical checks.

Finally, by counting the elements of the different sets we can confirm (as before) that we have correctly identified the set \(\mathcal{R}_k\). This completes the proof for the case of \(E_6\).

3.1 Uniqueness

The proof of uniqueness is analogous to the \(D_{4}\) case. It only remains to show that all fundamental representations of \(F_{4}\) can be obtained as restrictions of \(E_{6}\) representations. From the branching rules (3.4) we see immediately that this is true for \((1,0,0,0)\), \((0,0,0,1)\) and \((0,1,0,0)\). The remaining fundamental representation \((0,0,1,0)\) appears in the decomposition of \((0,0,1,0,0,0)\), but it comes together with \((1,0,0,1)\). The latter representation can be obtained from the other fundamentals by the \(F_{4}\) tensor product \[(1,0,0,0)\otimes (0,0,0,1)\ \to \ (1,0,0,1)\oplus (1,0,0,0)\oplus (0,1,0,0)\ .\] Hence, also \((0,0,1,0)\) can be written in terms of the restriction of \(D_{4}\)-representations.

While we were in the process of writing up this paper we became aware of [23] which contains closely related work.

Acknowledgements: This research has been partially supported by the Swiss National Science Foundation and the Marie Curie network ‘Constituents, Fundamental Forces and Symmetries of the Universe’ (MRTN-CT-2004-005104). The work of S.F. was supported by the Max Planck Society and the Max Planck Institute for Gravitational Physics in Golm. We thank Terry Gannon for useful communications. This paper is largely based on the Diploma thesis of T.M. [24].