Abstract
We investigate the two-dimensional conformal field theories (CFTs) of c=frac{47}{2} , c=frac{116}{5} and c = 23 ‘dual’ to the critical Ising model, the three state Potts model and the tensor product of two Ising models, respectively. We argue that these CFTs exhibit moonshines for the double covering of the baby Monster group, 2;cdotp;mathbb{B} , the triple covering of the largest Fischer group, 3 · Fi24′ and multiple-covering of the second largest Conway group, 2 · 21+22 · Co2. Various twined characters are shown to satisfy generalized bilinear relations involving Mckay-Thompson series. We also rediscover that the ‘self-dual’ two-dimensional bosonic conformal field theory of c = 12 has the Conway group Co0 ≃ 2 · Co1 as an automorphism group.
Highlights
Potts model and the tensor product of two Ising models, respectively. We argue that these conformal field theories (CFTs) exhibit moonshines for the double covering of the baby Monster group, 2 · B, the triple covering of the largest Fischer group, 3 · Fi24 and multiple-covering of the second largest Conway group, 2 · 21+22 · Co2
47 2 satisfies the bilinear relation. Another example is a pair of rational CFTs of c = 8 and c = 16 having no Kac-Moody symmetry but finite group symmetry [13]
We propose that the characters of the above RCFT (4.10) obey intriguing bilinear relations with those of the critical Ising model (2.2) to give the baby Monster modules (2.7), f0(τ ) = f0(τ )g0(τ ) + f (τ )g1(τ ) + fσ(τ )g2(τ ), f (τ ) = f0(τ )g1(τ ) + f (τ )g3(τ ) + fσ(τ )g4(τ ), fσ(τ ) = f0(τ )g2(τ ) + f (τ )g4(τ ) + fσ(τ )g5(τ )
Summary
The simplest unitary minimal model M(4, 3) describes the critical point of the secondorder phase transition of the Ising model. We in turn consider a rational CFT dual to the tensor product of the critical Ising model, the simplest example of the c = 1 CFTs studied in [22,23,24] This theory has nine primaries of conformal weights h1. We conjecture that the nine-character rational CFT of c = 23, dual to the product of two critical Ising model, has 2 · 21+22 · Co2 as an automorphism group. We propose that the characters of the above RCFT (4.10) obey intriguing bilinear relations with those of the critical Ising model (2.2) to give the baby Monster modules (2.7), f0(τ ) = f0(τ )g0(τ ) + f (τ )g1(τ ) + fσ(τ )g2(τ ), f (τ ) = f0(τ )g1(τ ) + f (τ )g3(τ ) + fσ(τ )g4(τ ), fσ(τ ) = f0(τ )g2(τ ) + f (τ )g4(τ ) + fσ(τ )g5(τ ).
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