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Topological Modularity of Monstrous Moonshine

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Abstract

We explore connections among Monstrous Moonshine, orbifolds, the Kitaev chain and topological modular forms. Symmetric orbifolds of the Monster CFT, together with further orbifolds by subgroups of Monster, are studied and found to satisfy the divisibility property, which was recently used to rule out extremal holomorphic conformal field theories. For orbifolds by cyclic subgroups of Monster, we arrive at divisibility properties involving the full McKay–Thompson series. Orbifolds by non-abelian subgroups of Monster are further considered by utilizing the data of Generalized Moonshine.

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Notes

  1. Holomorphic CFTs are tautologically (0,1) SQFTs with a trivial supersymmetric sector.

  2. In the absence of a global gravitational anomaly, the \(\text {S}_n\) permutation symmetry is non-anomalous [23].

  3. An invertible field theory as introduced by Freed [38] is a topological quantum field theory with a unique vacuum. The characterization of the Arf invariant as a mod 2 index was given by Atiyah [39].

  4. The twisted denominator formula [30, 31] without inversion also figured in the work by Carnahan [40].

  5. In terms of topological defect lines, the h defect runs upwards in the time direction, the g defect runs toward the right in the space direction, and the group multiplication at a vertex reads

  6. While many conventions exist in the literature, e.g., Dijkgraaf, Vafa, Verlinde, Verlinde [44, (4.18)] or Dong and Mason [45, (1.4)], we choose to follow Tuite [43, (14)] closely since his paper contains many useful formulae for our computations.

  7. In terms of topological defect lines, we nucleate a loop of \(x \in G\) on the torus and perform the obvious moves to turn \(Z_h^g\) into \(Z_{xhx^{-1}}^{xgx^{-1}}\).

  8. Since group elements are labeled by integers in this subsection, to avoid confusion with the rth power of Z, we write \(Z^r_0\) instead of \(Z^r\) to denote the partition function with a temporal twist and no spatial twist.

  9. A sufficient condition for this property is if the outer automorphism group \(\text {Aut}(\mathbb {Z}_N\)) is a symmetry. Note that the CPT symmetry implies \(Z[\mathcal {T}]^r_0 = Z[\mathcal {T}]^{N-r}_0\). The author thanks Yifan Wang for a discussion.

  10. Conjugacy classes related by inversion must share the same McKay–Thompson series, as decreed by the CPT symmetry. All coincidences of the McKay–Thompson series can be explained this way, except for 27AB. The author thanks Yifan Wang for a discussion.

  11. See [50, 51] for a physics derivation of Monstrous Moonshine. The most nontrivial aspect of the Monstrous Moonshine [32] is that the McKay–Thompson series are Hauptmoduls of genus-zero subgroups of \(\text {PSL}(2,\mathbb {R})\). Since this property plays little role in the present pursuit, we make no further mention.

  12. Johnson-Freyd and Treumann [52] further showed that the \(\mathbb {Z}_{24}\) generated by \({V^\natural }\) is a direct summand of \(\text {H}^3(\mathbb {M},\text {U}(1))\), and \(\text {H}^3(\mathbb {M},\text {U}(1)) \ominus \mathbb {Z}_{24}\) has order dividing 4. The author thanks Theo Johnson-Freyd for explaining these results.

  13. While it is not true that the conjugacy classes \(\text {Cl}(g^r)\) only depend on \(\gcd (N,r)\) for every Monster group element g, it is true up to inversion. For instance, \(g\in \text {23A}\) and \(g^2\in \text {23B}\), despite \(\gcd (23,1) = \gcd (23,2) = 1\); however, because \(g^{-1}\in \text {23B}\), the conjugacy classes 23AB are related by inversion, and must share the same McKay–Thompson series by the CPT symmetry. The author thanks Yifan Wang for a discussion.

  14. It is important to series expand first in p and then in q.

  15. For a single copy \(n=1\), it is well-known that \({V^\natural }/{\langle {p\text {B}} \rangle }\) with \(p=2,3,5,7,13\) gives the Leech lattice CFT, with 24 spin-one conserved currents coming from the twisted sector; by contrast, (3.14) gives none.

  16. The (possible) class fusions for these subgroups were given in [54]. \(L_2(71)\) was proven to be a subgroup of \(\mathbb {M}\) by [55].

  17. For \(G = L_2(29), ~ \textrm{A}_5, ~ L_2(71)\), when computing the twisted sector of an order-2 element h, Gaiotto and Yin [33] only performed the partial projection by the cyclic subgroups \({\langle {h} \rangle } \cong \mathbb {Z}_{14}, ~ \mathbb {Z}_2, ~ \mathbb {Z}_{36}\) of the full centralizers \(\text {D}_{28}, ~ \mathbb {Z}_2 \times \mathbb {Z}_2, ~ \text {D}_{72}\), and obtained \(Z[{V^\natural }^{\otimes 2}/\text {S}_2 \times G]_{q^0} = 222, 3816, 112\), respectively. Using Norton’s Generalized Moonshine [34] data, we are able to uniquely determine the correct answers for \(L_2(29)\) and \(L_2(71)\), but not for \(\textrm{A}_5\) due to the author’s lack of knowledge of the class fusion(s) for \(\textrm{A}_5 < \text {C}_\mathbb {M}(h)\).

  18. The author thanks Scott Carnahan for correspondence and for sharing Norton’s data.

  19. See [7, 13, 59] for studies of the Conway CFT [58] in the context of TMF.

  20. The author thanks Theo Johnson-Freyd for a discussion.

  21. The series divisibility of pJ with base 24 instead of 12 can be explained by considering symmetric orbifolds of the TMF class \(j \Delta ^{-1}\), instead of \(j \Delta ^{-2}\) that is realized by Monster. This comment is due to Theo Johnson-Freyd.

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Acknowledgements

The author thanks Jan Albert, Scott Carnahan, Chi-Ming Chang, Theo Johnson-Freyd, Justin Kaidi and Yifan Wang for discussions and comments on the draft.

Funding

This work was supported by the Simons Collaboration Grant on the Non-Perturbative Bootstrap.

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  1. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Ying-Hsuan Lin

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Lin, YH. Topological Modularity of Monstrous Moonshine. Ann. Henri Poincaré 25, 2427–2452 (2024). https://doi.org/10.1007/s00023-023-01352-8

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