Abstract
The monster sporadic group is the automorphism group of a central charge \(c=24\) vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its \(c=24\) stress tensor T(z), this theory contains many other conformal vectors of smaller central charge; for example, it admits 48 commuting \(c=\frac{1}{2}\) conformal vectors whose sum is T(z). Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups \({}^2{\textit{E}}_6(2)\) and \({\textit{F}}_4(2)\) of Lie type. Many of these examples are naturally associated to McKay’s \(\widehat{E_8}\) correspondence, and we use the structure of Norton’s monstralizer pairs more generally to organize our presentation.
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Notes
A subquotient is a quotient of a subgroup.
A group is said to be perfect if it admits no non-trivial abelian quotients. In particular, any non-abelian simple group is a perfect group.
In this paper, \({\textit{V}}{\mathbb {B}}^\natural \) will always denote a \({\mathbb {Z}}\)-graded VOA. This differs slightly from Höhn’s \(\frac{1}{2}{\mathbb {Z}}\)-graded vertex operator super algebra, which is constructed by taking a direct sum of \({\textit{V}}{\mathbb {B}}^\natural \) and its unique irreducible module of highest weight \(\frac{3}{2}\).
The notation \({\textit{VF}}^\natural \) was used in [28], though we use \({\textit{VF}}_{24}^\natural \) in anticipation of our construction of similar VOAs \({\textit{VF}}_{23}^\natural \) and \({\textit{VF}}_{22}^\natural \) associated with the other Fischer groups \({\textit{Fi}}_{23}\) and \({\textit{Fi}}_{22}\).
In particular, our examples are suggestive of the existence of a (unique) functor from the category of monstralizer pairs to the category of monstralizing commutant pairs.
We use the convention that \(D_n\) is the symmetry group of a regular n-gon, i.e. \(|D_n|=2n\).
A few of these algebras inherit an extra order 2 outer automorphism from the monster.
Although the 4A and 2B cases involve the McLaughlin group and Conway’s second group, both of which belong to the happy family, we hesitate to give them the names \({\textit{VMcL}}^\natural \) and \({\textit{VCo}}_2^\natural \) in case it is possible in the future to define chiral algebras with \({\textit{McL}}\) and \({\textit{Co}}_2\) symmetry on the nose, as opposed to our current constructions which realize extensions of these groups.
A subVOA \(\mathcal {W}\) need not have the same stress tensor as \(\mathcal {V}\). In the case that it does, we say that \(\mathcal {W}\) is a full subVOA.
Technically there is a vector space of highest weight states, so e.g. \(\varphi ^{(\alpha )}\) should be thought of as carrying an extra index which we are suppressing.
In fact, n is the order of X if X is non-anomalous; in general, n is some multiple of the order of X.
We are not aware of a general condition which determines when such a lifting goes through, but we are confident that it does in all the cases that we invoke this structure.
A simple-current is an operator J such that the OPE of J with any primary contains only a single term, which is itself a primary.
Their fusion categories are “braid-reversed equivalent”
At least for the Hecke operators considered in this paper, though they may generalize.
Convergence of this sum has not been proven in general, however we conjecture that in all the cases we consider in this paper, the sum dooes in fact converge.
Similar formulae appear in [91] for weight \(\frac{1}{2}\) vector-valued modular forms transforming in the Weil representation.
We are not aware of any theorem that requires the highest weight subspaces of irreducible modules of a (suitably nice) VOA to transform as irreducible representations of the inner automorphism group, and we suspect that this niceness is related to the fact that \((G,{\widetilde{G}})\) form a monstralizer pair.
In e.g. [27], the notation \({\textit{V}}{\mathbb {B}}^\natural \) is actually used to denote the vertex operator superalgebra obtained by taking a direct sum of \({\widetilde{\mathcal {W}}}_{{\mathbb {Z}}_{\mathrm {2A}}}\) with its irreducible module of highest weight \(\frac{3}{2}\). This VOSA has automorphism group \({\mathbb {Z}}_2 \times {\mathbb {B}}\), where the extra \({\mathbb {Z}}_2\) is generated by \((-1)^F\).
Unlike in the case of the \(\widehat{E_8}\) correspondence, these VOAs are not always generated by their two central charge \(\frac{7}{10}\) conformal vectors.
We use the notation \(N_G(S) = \{ g\in G \mid gS=Sg\} \) to denote the normalizer of the set S in G.
This also justifies the consideration of the group \(3.{\textit{Fi}}'_{24}\) in [99].
The integers \(\ell = 7, 11, 13, 17\) also satisfy \(23 + \ell ^2 = 0 \ \text {mod} \ 24\). However, \(G_7,\) \(G_{11},\) \(G_{13}\), and \(G_{17}\) have negative entries and so we do not consider them.
Whenever a group has multiple irreducible representations of dimension d, we use the notation \({\mathbf {d}}_{(i)}\) to denote the ith irrep of dimension d, according to how they are ordered in Gap.
The naive prediction from the monstralizer \([D_{\mathrm {2A}}\circ 2^2.{^2}{\textit{E}}_6(2)].S_3\) is that the full automorphism group should be \({^2}{\textit{E}}_6(2).S_3\) as opposed to \({^2}{\textit{E}}_6(2).2\). However, if one decomposes the Griess algebra \(V^\natural _2\) into representations of \({^2}{\textit{E}}_6(2).S_3\) one does not find any singlets besides the usual stress tensor, which indicates that a deconstruction is not possible. This example is the reason why we say in general that \(\mathrm {Aut}(\mathcal {W}_{{\widetilde{G}}}) = ({\widetilde{G}}/Z({\widetilde{G}})).H'\) for \(H'\) a subgroup of the group H which appears in the monstralizer \([G\circ {\widetilde{G}}].H\). It is the only example we consider for which \(H'\ne H\).
Irreps of \(2.{^2}{\textit{E}}_6(2).2\) are the same as irreps of \(2^2.{^2}{\textit{E}}_6(2).2\) in which the central \({\mathbb {Z}}_2\) which is generated by the 2A conjugacy class acts trivially. In the Gap ordering, the first 320 irreps of \(2^2.{^2}{\textit{E}}_6(2).2\) map onto the irreps of \(2.{^2}{\textit{E}}_6(2).2\).
In most examples, the bilinear which pairs \(\chi _\alpha \) with its Hecke images to produce the J-function can be obtained as a linear combination of the matrices \(G_\ell \) described in Sect. 2.3.2. This is one of the few instances for which this is not true, i.e. the matrix which relates \({\mathsf {T}}_{11}\chi \) to \({\widetilde{\chi }}\) cannot be realized as a linear combination of the matrices \((G_\ell )^T\). This suggests that there are modular invariant ways to combine characters with their Hecke images beyond the bilinears studied in [42], however we leave their study to future work.
See Appendix C for an alternative derivation of these characters.
We use the fact that \(2^{1+22}.{\textit{Co}}_2\) has a \({\mathbb {Z}}_2\) normal subgroup which leaves \(2^{22}.{\textit{Co}}_2\) after taking the quotient [114].
The Niemeier lattices are the even, positive-definite, unimodular lattices of rank 24.
The algebra e.g. \({\widetilde{\mathcal {W}}}_{B(\mathrm {1A})}\) is defined to be the commutant of \(\mathcal {W}_{B(\mathrm {1A})}\) in \({\textit{V}}{\mathbb {B}}^\natural \), while the algebra \({\widetilde{\mathcal {W}}}_{D_{\mathrm {2A}}}\) is defined to be the commutant of \(\mathcal {W}_{D_{\mathrm {2A}}}\) in \(V^\natural \); we hope that this notation will not cause confusion.
The appearance of the module \(\mathcal {L}(\tfrac{6}{7},5)\) is related to the fact that the conformal vector is “derived.”
Such a subalgebra exists, as proven in [66].
We use the term fictitious because these characters e.g. do not lead to consistent fusion rules, and the bilinear they participate in, equation (C.5), has fractional coefficients \(\alpha _i\).
C.f. the previous footnote.
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Acknowledgements
We thank G. Mason for very helpful correspondence and understand that he and C. Franc have also been looking at commutants of subVOAs of \(V^\natural \). We also thank L. F. Alday, H. Choi, S. Harrison, T. Johnson-Freyd, B. Julia, Y. Lin, G. Moore, J. Sempliner, S. Shao, and Y. Wu for discussions. JH would like to thank L. Dixon for collaboration on deconstructing CFTs roughly thirty years ago. JH and SL gratefully acknowledge the hospitality of the Aspen Center for Physics (under NSF Grant No. PHY-1066293) for providing an excellent atmosphere for collaboration. We have made use of Gap [108], Sage [117], and Magma [103] for various computational aspects of our analysis. JH acknowledges support from the NSF(Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.) under grant PHY 1520748. BR gratefully acknowledges support from the NSF under grant PHY 1720397. KL and SL are supported in part by KIAS Individual Grants PG006904 and PG056502, and by the National Research Foundation of Korea Grants NRF2017R1D1A1B06034369 and NRF2017R1C1B1011440. The work of JB is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant No. 787185).
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Appendices
Fusion Rules for the \({\widetilde{\mathcal {W}}}_{D_{n\mathrm {X}}}\) Algebras
In this appendix, we present the fusion rules for some of the theories discussed in the main text. The structure constants \(\mathcal {N}_{\alpha \beta }^{\gamma }\) are computed using the S-matrix of each theory and the Verlinde formula, equation (2.17). It turns out that each theory has \(\mathcal {N}_{\alpha \beta }^{\gamma } = 0 \ \text {or} \ 1\) for all \(\alpha ,\beta ,\gamma \). In the cases that we used the block-diagonalization method to determine the characters of our models (c.f. Sect. 2.1.2), the consistency of the fusion rules provides a non-trivial check on our results. Because fusion algebra is associative, \(\mathcal {N}_{\alpha \beta }^{\gamma } = 1\) imposes that \(\mathcal {N}_{\beta \alpha }^{\gamma } = 1\). Below, we present all the non-vanishing fusion algebra coefficients of the five theories in Sects. 3.2.2, 3.2.3, 3.2.9, 3.2.5 and 3.2.6.
1.1 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \({\widetilde{\mathcal {W}}}_{D_{\mathrm {2A}}}\):
1.2 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \( {{\textit{VT}}}^\natural \):
1.3 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \( {{\textit{VHN}}}^\natural \):
1.4 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \( {{\textit{VF}}}_{23}^\natural \):
1.5 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \( {{\textit{VF}}}^\natural _{22}\):
1.6 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \({\widetilde{\mathcal {W}}}_{D_{\mathrm {4A}}}\):
1.7 List of non-vanishing \(\mathcal {N}_{\alpha \beta }^{\gamma }\) for \({\widetilde{\mathcal {W}}}_{D_{\mathrm {4B}}}\):
Group Theory Data
In this appendix we provide, for a few examples, the group theoretic data necessary for analyzing twined bilinears.
Let us start with a somewhat general discussion of how characters of \(\mathcal {W}\) and \({\widetilde{\mathcal {W}}}\), twined by inner automorphisms, can be bilinearly combined to produce twined characters of the VOA \(\mathcal {V}\) in which they sit as commutant pairs. We assume for simplicity that \(\mathcal {V}\) is a meromorphic CFT with partition function \(\mathcal {Z}\), that \(\mathrm {Inn}(\mathcal {W})\times \mathrm {Inn}({\widetilde{\mathcal {W}}})\subset \mathrm {Aut}(\mathcal {V})\), and that the inner automorphism groups are realized honestly on the modules of \(\mathcal {W}\) and \({\widetilde{\mathcal {W}}}\) which appear in the decomposition of \(\mathcal {V}\) (as opposed to projectively). Under these assumptions, there will be generalized bilinear relations of the form
which arise by taking the graded trace of both sides of the decomposition
Because the graded characters are class functions of the associated groups, one only needs to know how the conjugacy classes of \(\mathrm {Inn}(\mathcal {W})\times \mathrm {Inn}({\widetilde{\mathcal {W}}})\) fuse into the conjugacy classes of \(\mathrm {Aut}(\mathcal {V})\). For illustrative purposes, we take \(\mathcal {V}=V^\natural \) and provide the necessary data for the cases \((\mathcal {W},{\widetilde{\mathcal {W}}}) = (\mathcal {W}_{5A},{\textit{VHN}}^\natural )\) and \((\mathcal {W}_{D_{\mathrm {3C}}},{\textit{VT}}^\natural )\) in Tables 11 and 12: namely, information about how conjugacy classes of \({\textit{HN}}\cong \mathrm {Inn}({\textit{VHN}}^\natural )\) and \({\mathbb {Z}}_3\times {\textit{Th}}\subset \mathrm {Inn}(\mathcal {W}_{D_{\mathrm {3C}}})\times \mathrm {Inn}({\textit{VT}}^\natural )\) fuse into conjugacy classes of \({\mathbb {M}}\). One can use this data to conduct checks on our proposals regarding the implementation of the symmetry groups in these two examples. For example, a prediction of Table 12 is that
where \(\chi _{\omega ,\alpha }\) are the characters of \(\mathcal {W}_{D_{\mathrm {3C}}}\) twined by the generator of its \({\mathbb {Z}}_3\) automorphism, \(\chi _{\mathrm {2A},{\textit{VT}}^\natural (\alpha )}\) are the characters of \({\textit{VT}}^\natural \) twined by an element of the 2A conjugacy class of \({\textit{Th}}\), and \(J_{\mathrm {6F}}\) is the McKay-Thompson series of the 6A conjugacy class in \({\mathbb {M}}\) (c.f. Sect. 3.2.9 for more details). To compute \(\chi _{\mathrm {2A},{\textit{VT}}^\natural (\alpha )}\) to low order in its q-expansion, one can use the character table of \({\textit{Th}}\), Tables 7, 8, 9 and 10, as well as the decompositions of the graded-components \({\textit{VT}}^\natural (\alpha )_h\) into \({\textit{Th}}\) representations, Table 6.
Alternative Derivation of the Characters of \( {{\textit{VF}}}^\natural _{22}\)
In this appendix, we give an alternative derivation of the characters of \({\textit{VF}}^\natural _{22}\). The basic idea is that, although the Hecke method does not work out of the box, one can perform intermediate deconstructions for which the Hecke method is effective. Although we work purely at the level of modular forms, our steps are motivated by the following algebraic manipulations.
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1.
We first decompose the moonshine module into (an extension of) one of its \({\mathcal {L}}(\tfrac{1}{2},0)\otimes {\mathcal {L}}(\tfrac{4}{5},0)\otimes {\mathcal {L}}(\tfrac{7}{10},0)\) subalgebrasFootnote 33 and its commutant. Here, the Hecke method is effective in producing the dual characters.
-
2.
It is straightforward to infer from the previous step how the moonshine module decomposes into just \({\mathcal {L}}(\tfrac{1}{2},0)\otimes {\mathcal {L}}(\tfrac{4}{5},0)\) and its commutant. From the fact that \({\mathcal {P}}(2)\cong {\mathcal {L}}(\tfrac{1}{2},0)\) and \({\mathcal {P}}(3) \cong {\mathcal {L}}(\tfrac{4}{5},0)\oplus {\mathcal {L}}(\tfrac{4}{5},3)\), we will be able to re-interpret this as a decomposition of \(V^\natural \) into a \({\mathcal {P}}(2)\otimes {\mathcal {P}}(3)\) subalgebra and its commutant.
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3.
The previous step will give us a bilinear of the form \(J(\tau ) = \sum _i g_i(\tau ) {\widetilde{g}}_{i}(\tau )\) which we can set equal to the bilinear in equation (3.122) to extract expressions for the dual characters \(\chi _{{\textit{VF}}^\natural _{22}(\alpha )}(\tau )\).
We start by constructing the characters of an extension of \({\mathcal {P}}(2)\otimes {\mathcal {P}}(3)\otimes {\mathcal {P}}(6)\), using the block-diagonalization method outlined in Sect. 2.1.2. We label the S-matrices of the VOAs \({\mathcal {P}}(2)\), \({\mathcal {P}}(3)\), and \({\mathcal {P}}(6)\) as \(\mathcal {S}^{(2)}, \mathcal {S}^{(3)}\), and \(\mathcal {S}^{(6)}\). One can show that the matrix of the tensor product theory, \(\mathcal {S}^{(2)} \otimes \mathcal {S}^{(3)} \otimes \mathcal {S}^{(6)}\), can be block-diagonalized into a \(14 \times 14\) block and its complement. This suggests the existence of a unitary RCFT described by 14 characters which can be expressed in terms of parafermion characters as
The modular properties of the characters in (C.1) are governed by the S-matrix \(\mathcal {S}\)
where \(\alpha = \sqrt{1-\frac{1}{\sqrt{5}}}, \ \beta = \sqrt{1+\frac{1}{\sqrt{5}}}\), and \(s_1 = \text {sin}\left( \frac{\pi }{8}\right) \). The T-matrix \(\mathcal {T}\) reads
One can check that these matrices furnish a representation of \({\textit{PSL}}_2({\mathbb {Z}})\), namely \(\mathcal {S}^2 = 1\) and \((\mathcal {S} \cdot \mathcal {T})^3 = 1\).
Our goal is to find the characters dual to those in equation (C.1). The solution we present, although inspired by the idea of performing a series of intermediate deconstructions, unfortunately features steps which do not quite have consistent VOA interpretations. The contents of the remainder of this appendix should therefore be thought of strictly as manipulations at the level of modular forms which produce the right answer for the characters \(\chi _{{\textit{VF}}_{22}^\natural }(\tau )\), although we do believe it should be possible to improve upon our results.
Step 1 We start off by constructing the fictitious charactersFootnote 34 of an extension \(\mathcal {U}\) of \({\mathcal {L}}(\tfrac{1}{2},0)\otimes {\mathcal {L}}(\tfrac{4}{5},0)\otimes {\mathcal {L}}(\tfrac{7}{10},0)\). This extension has 16 states of conformal weight
whose characters \(f_j\) with conductor \(N=60\) can be written in terms of minimal model characters as
where \(\chi ^{(5)}_{1,+} \equiv \chi _{1,1}^{(5)} + \chi _{1,5}^{(5)}\) and \(\chi ^{(5)}_{2,+} \equiv \chi _{2,1}^{(5)} + \chi _{2,5}^{(5)}\).
The central charge of \({\mathcal {L}}(\tfrac{1}{2},0)\otimes {\mathcal {L}}(\tfrac{7}{10},0)\otimes {\mathcal {L}}(\tfrac{4}{5},0)\) is 2, and its commutant in \(V^\natural \) has central charge \(22=2\cdot 11\), so we have a chance at finding the dual characters as the Hecke image of f under \({\mathsf {T}}_{11}\). We provide the q-expansions of the components of \({{\mathsf {T}}}_{11}f\) below,
One can check that the Hecke images (C.4) satisfy the following bilinear,
where the \(\alpha _{i}\) take the values
for \(i=0,1, \dots , 15\).
Step 2 Next, let us construct the fictitious charactersFootnote 35 of a particular extension of \({\mathcal {L}}(\tfrac{1}{2},0)\otimes {\mathcal {L}}(\tfrac{4}{5},0)\); its irreducible modules have highest weights
and its characters can be written as
We would like to find the dual characters, which by assumption should satisfy a bilinear of the form
By comparing (C.5) and (C.9), one can express \({\widetilde{g}}_{i}(\tau )\) in terms of (C.4) and the characters of \({\mathcal {L}}(\tfrac{7}{10},0)\). We find
Step 3 Note that the parafermion theories \({\mathcal {P}}(2)\) and \({\mathcal {P}}(3)\) are the same as \({\mathcal {L}}(\tfrac{1}{2},0)\) and \({\mathcal {L}}(\tfrac{4}{5},0)\oplus {\mathcal {L}}(\tfrac{4}{5},3)\), respectively. Thus, we can replace the characters of the \({\mathbb {Z}}_2\) parafermion theory \(\psi ^{(2)}_{\ell ,m}\) with the characters \(\chi ^{(3)}_{r,s}\) in equation (C.1). The relation between Ising and \({\mathcal {P}}(2)\) characters is
Similarly, we can substitute \(\chi ^{(5)}_{r,s}\) for the characters of the \({\mathbb {Z}}_3\) parafermion theory \(\psi ^{(3)}_{\ell ,m}\) as
The next step is to find the relations among the characters \({\widetilde{g}}_{i}(\tau )\) and \(\chi _{{\textit{VF}}^\natural _{22}(\alpha )}\) by comparing equations (C.9) and (3.122). Setting them to be equal, we get a relation of the form
Here, \({\widetilde{\chi }}_{\alpha }(\tau )\) is short notation for \(\chi _{{\textit{VF}}^\natural _{22}(\alpha )}(\tau )\). Using the expressions for the \({\widetilde{g}}_{i}(\tau )\) in equation (C.10), one can find the q-expansions of \(\chi _{{\textit{VF}}^\natural _{22}(\alpha )}(\tau )\). Here, we assumed \({\widetilde{\chi }}_{8}(\tau )={\widetilde{\chi }}_{9}(\tau )\) and \({\widetilde{\chi }}_{12}(\tau )={\widetilde{\chi }}_{13}(\tau )\), because \({\chi }_{8}(\tau )={\chi }_{9}(\tau )\) and \({\chi }_{12}(\tau )={\chi }_{13}(\tau )\). With these extra conditions, (C.13) recovers the q-expansions (3.123).
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Bae, JB., Harvey, J.A., Lee, K. et al. Conformal Field Theories with Sporadic Group Symmetry. Commun. Math. Phys. 388, 1–105 (2021). https://doi.org/10.1007/s00220-021-04207-7
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DOI: https://doi.org/10.1007/s00220-021-04207-7