14 research outputs found

    Lattices and automorphisms of compact complex manifolds

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    This work makes use of well-known integral lattices to construct complex algebraic varieties reflecting properties of the lattices. In particular the automorphism groups of the lattices are closely related to the symmetries of varieties. The constructions are to two types: generalised Kummer manifolds and toric varieties. In both cases the examples are of the most interest. A generalised Kummer manifold is the resolution of the quotient of a complex torus by some finite group G. A description of the construction for certain cyclic groups G by given in terms of holomorphic surgery of disc bundles. The action of the automorphism groups is given explicitly. The most important example is a compact complex 12-dimensinoal manifold associated to the Leech lattice admitting an action of the finite simple Suzuki group. All these generalised Kummer manifolds are shown to be simply connected. Toric varieties are associated to certain decompositions of Rn into convex cones. The automorphism groups of those associated to Weyl group decompositions of Rn are calculated. These are used to construct 24-dimensional singular varieties from some Neimeier lattices. Their symmetries are extensions of Mathieu groups and their singularities closely related to the Golay codes

    Beauty And The Beast Part 2: Apprehending The Missing Supercurrent

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    The Moonshine module is a c=24c=24 conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in \cite{Dixon:1988qd} that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference \cite{Dixon:1988qd} did not provide an explicit construction of a superconformal current. The present paper fills that gap. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in \cite{Harvey:2020jvu}. In particular, our construction relies on error correcting codes.Comment: 66 page

    Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

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    Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces (Ω,σ)(\Omega,\sigma) and appropriate spaces of functions inside L2(Ω,σ)L^2(\Omega,\sigma). The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups

    Automorphisms on algebraic varieties: K3 surfaces, hyperkähler manifolds, and applications on Ulrich bundles

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    One of the main tools to study the geometry of complex algebraic varieties is the group of automorphisms. The first part of this thesis concerns the study of symplectic automorphisms of finite order on K3 surfaces, and birational symplectic maps of finite order on projective hyperkähler manifolds which are deformation equivalent to the Hilbert scheme of K3 surfaces. In the second part of this thesis, the automorphism groups of rational homogeneous spaces are used to study Ulrich bundles in smooth projective varieties

    Mock Modular Mathieu Moonshine Modules

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    We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4 superconformal algebra. Similarly, any subgroup of Co_0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2 superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N=4 (resp. N=2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman--Sims, are also discussed.Comment: 94 pages, including 56 pages of tables; v2: updated references and minor revisions to abstract, introduction and sections 8 and
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