Duality Defect of the Monster CFT

TL;DR

This work analyzes the Monster CFT through fermionization with respect to a non-anomalous $\\mathbb{Z}_{2A}$ symmetry, revealing a tensor product structure with the Baby Monster CFT and a Majorana-Weyl fermion. It identifies a non-invertible duality defect $\\mathcal N$ that extends the symmetry to an Ising-type defect category, and defines the defect McKay-Thompson series $Z^{\\mathcal N}$ whose Fourier coefficients decompose into dimensions of $2.\\mathbb B$ irreps. The authors establish modular properties of $Z^{\\mathcal N}$, showing invariance under the genus-zero subgroup $16D^0$ of $PSL(2,\\mathbb{Z})$, and provide detailed decompositions of defect-related partition functions into Ising and Baby Monster characters. These results illuminate deep connections between Moonshine, topological defects, and the Baby Monster symmetry, with implications for generalized moonshine and higher-categorical structures in holomorphic CFTs.

Abstract

We show that the fermionization of the Monster CFT with respect to $\mathbb{Z}_{2A}$ is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the $\mathbb{Z}_{2A}$ orbifold, i.e. it enjoys the Kramers-Wannier duality. The Kramers-Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay-Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay-Thompson series is invariant under the genus-zero congruence subgroup $16D^0$ of $PSL(2,\mathbb{Z})$.

Figures (7)

• Figure 1: The bosonization/fermionization of the Monster CFT with respect to the $\mathbb{Z}_{2A}$ and the $\mathbb{Z}_{2B}$ symmetries. Here "B$\&$B" stands for the "Beauty and the Beast" ${\cal N}=1$ SCFT of Dixon:1988qd, while "Baby" and "MW" stand for the Baby Monster CFT and the Majorana-Weyl fermion, respectively. The Baby $\otimes$ MW fermionic CFT is invariant under stacking of the (1+1)$d$ invertible spin TQFT $(-1)^{\text{Arf}}$, since all its partition functions with odd spin structure vanish. Correspondingly, the Monster CFT is self-dual under the $\mathbb{Z}_{2A}$ orbifold.
• Figure 2: Topological defects are generalizations of global symmetries.
• Figure 3: The duality defect line $\cal N$ defines a (non-invertible) map $\widehat{\cal N}:{\cal H}\rightarrow {\cal H}$ on the Hilbert space of local operators by encircling a local operator $\phi(x)$ and shrinking the circle.
• Figure 4: Four resolutions of $Z_{\cal N}^{\cal N}$. The boundary square represents a torus with opposite sides identified. The solid line (in the interior of the square) stands for the duality defect $\cal N$, while the dashed line stands for the $\mathbb{Z}_{2A}$ defect line $\eta$.
• Figure 5: $F$-moves relating the different resolutions for $Z_{\cal N}^{\cal N}$, $Z_{\cal N}^\eta$, and $Z_\eta^{\cal N}$. There are two independent resolutions for $Z_{\cal N}^{\cal N}$, and we choose to use the two without an intermediate $\eta$ line. The replacement rule for the other two (the ones with an intermediate $\eta$ line) can be obtained by solving the first two equations. There is one independent resolution for $Z_{\cal N}^\eta$ and $Z_\eta^{\cal N}$, and we choose to use the ones on the left.