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Register a new userThe Monster, the Baby Monster and Traces of Singular Moduli
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Authors
Victor Godet
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In this note, we provide evidence for new (super) moonshines relating the Monster and the Baby monster to some weakly holomorphic weight 1/2 modular forms defined by Zagier in his work on traces of singular moduli. They are similar in spirit to the recently discovered Thompson moonshine.
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These notes stem from lectures given by the first author (JM) at the 2008 Moonshine Conference in Kashiwa and contain a number of new perspectives and observations on Monstrous Moonshine. Because many new points have not appeared anywhere in print, it is thought expedient to update, annotate and clarify them (as footnotes), an editorial task which the second author (YHH) is more than delighted to undertake. We hope the various puzzles and correspondences, delivered in a personal and casual manner, will serve as
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, $2cdot mathbb{B}$, the triple covering of the largest Fischer group, $3cdot text{Fi}_{24}$ and multiple-covering of the second largest Conway group, $2cdot 2^{1+22} cdot text{Co}_2$. 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 $text{Co}_{0}simeq2cdottext{Co}_1$ as an automorphism group.
Let $f$ and $g$ be weakly holomorphic modular functions on $Gamma_0(N)$ with the trivial character. For an integer $d$, let $Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $iinfty$ under the action of $Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{hat{H}(f)}(r)$ of the sum $H(f)(z) = sum_{d>0}Tr_d(f)e^{2pi idz}$ is a special value of a regularized twisted $L$-function defined by $Tr_d(f)$ for $dleq0$. It is proved that the regularized $L$-function is meromorphic on $mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{hat{H}(f)}(r)=Q_{hat{H}(g)}(r)$ for all $r$ equivalent to $i infty$ under the action of $Gamma_0(4N)$, then $Tr_d(f)=Tr_d(g)$ for all integers $d$.
The most distant quasar yet discovered sets constraints on the formation mechanism of black holes. Its light spectrum has tantalizing features that are expected to be observed before the reionization epoch ended.
We present the analysis of the extraordinarily bright Gamma-Ray Burst (GRB) 130427A under the hypothesis that the GRB central engine is an accretion-powered magnetar. In this framework, initially proposed to explain GRBs with precursor activity, the prompt emission is produced by accretion of matter onto a newly-born magnetar, and the observed power is related to the accretion rate. The emission is eventually halted if the centrifugal forces are able to pause accretion. We show that the X-ray and optical afterglow is well explained as the forward shock emission with a jet break plus a contribution from the spin-down of the magnetar. Our modelling does not require any contribution from the reverse shock, that may still influence the afterglow light curve at radio and mm frequencies, or in the optical at early times. We derive the magnetic field ($Bsim 10^{16}$ G) and the spin period ($Psim 20$ ms) of the magnetar and obtain an independent estimate of the minimum luminosity for accretion. This minimum luminosity results well below the prompt emission luminosity of GRB 130427A, providing a strong consistency check for the scenario where the entire prompt emission is the result of continuous accretion onto the magnetar. This is in agreement with the relatively long spin period of the magnetar. GRB 130427A was a well monitored GRB showing a very standard behavior and, thus, is a well-suited benchmark to show that an accretion-powered magnetar gives a unique view of the properties of long GRBs.
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