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Mathieu Moonshine, the observation that the Fourier coefficients of the elliptic genus on K3 can be interpreted as dimensions of representations of the Mathieu group M24, has been proven abstractly, but a conceptual understanding in terms of a representation of the Mathieu group on the BPS states, is missing. Some time ago, Taormina and Wendland showed that such an action can be naturally defined on the lowest non-trivial BPS states, using the idea of `symmetry surfing, i.e., by combining the symmetries of different K3 sigma models. In this paper we find non-trivial evidence that this construction can be generalized to all BPS states.
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The elliptic genus of K3 is an index for the 1/4-BPS states of its sigma model. At the torus orbifold point there is an accidental degeneracy of such states. We blow up the orbifold fixed points using conformal perturbation theory, and find that this fully lifts the accidental degeneracy of the 1/4-BPS states with h=1. At a generic point near the Kummer surface the elliptic genus thus measures not just their index, but counts the actual number of these BPS states. We comment on the implication of this for symmetry surfing and Mathieu moonshine.
In this paper we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on $K3times mathbb R^6$, where Niemeier lattices play an important role. This extends earlier results by including points in the moduli space with enhanced gauge symmetries in spacetime, or, equivalently, where the world-sheet CFT becomes singular. After classifying the symmetries as abstract groups, we study how they act on the BPS states of the theory. In particular, we classify the conjugacy classes in the T-duality group $O^+(Gamma^{4,20})$ which represent physically distinct symmetries. Subsequently, we make two conjectures regarding the connection between the corresponding twining genera of $K3$ CFTs and Conway and umbral moonshine, building upon earlier work on the relation between moonshine and the $K3$ elliptic genus.
textit{Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=bigoplus_{ngg-infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width $sinmathbb{Z}^+$. For each $1leq rleq s$ and each irreducible character $chi_i$, we employ Frobenius $r$-character extension $chi_i^{(r)} colon G^{(r)}rightarrowmathbb{C}$ to define textit{width $r$ McKay-Thompson series} for $V_G^{(r)}:=V_Gtimescdotstimes V_G$ ($r$ copies) for each $r$-tuple in $G^{(r)}:=Gtimescdotstimes G$ ($r$ copies). These series are modular functions which then reflect differences between $r$-character values. Furthermore, we establish orthogonality relations for the Frobenius $r$-characters, which dictate the compatibility of the extension of weak moonshine for $V_G$ to width $s$ weak moonshine.
The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs and gauge particles -- more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (K{a}hler potential and the metric) of the moduli space of the Calabi-Yau manifold, thus providing a way to calculate the metric of Calabi-Yau manifold itself directly from the experimental data.
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
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