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Register a new userK3 Elliptic Genus and an Umbral Moonshine Module
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Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of $K3$ string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the $K3$ elliptic genus. Inspired by the above two relations between moonshine and $K3$ string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of $D_4$ root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group $G^{D_4^{oplus 6}}$ whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems $A_5^{oplus 4}D_4$, $A_7^{oplus 2}D_5^{oplus 2}$ , $A_{11}D_7 E_6$, $A_{17}E_7$, and $D_{10}E_7^{oplus 2}$.
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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.
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.
We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi--Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.
The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of some of these invariants. The theory of mock modular forms makes a surprise appearance, shedding light on the integrality properties of some well-known examples.
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.
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