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Moonshine

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[1204.2779] Umbral Moonshine. Umbral moonshine. In mathematics, umbral moonshine is the name for a mysterious connection between the Mathieu group M24 and K3 surfaces, observed by Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa (2011).

Umbral moonshine

Cheng, Duncan & Harvey (2012) observed that some of the functions appearing in umbral moonshine are Ramanujan's Mock theta functions. Cheng, Miranda C. N.; Duncan, John F. R.; Harvey, Jeffrey A. (2012), Umbral Moonshine Eguchi, Tohru; Hikami, Kazuhiro (2009), "Superconformal algebras and mock theta functions", Journal of Physics. Mathieu group M24. In mathematics, the Mathieu group M24, introduced by Mathieu (1861, 1873), is a 5-transitive permutation group on 24 objects, of order Constructions of the Mathieu group[edit] The Mathieu group can be constructed in various ways.

Mathieu group M24

Permutation groups[edit] M24 has a simple maximal subgroup of order 6072 and this can be represented as a linear fractional group on the field F23. One generator adds 1 to each element (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). These constructions were cited by Carmichael (1956, pp.151, 164, 263). Modular curve. Analytic definition[edit] The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.

Modular curve

The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N Γ(N), for some positive integer N, where The minimal such N is called the level of Γ. A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface commonly denoted Y(Γ). Compactified modular curves[edit] Supersingular prime (moonshine theory)

This definition is related to the notion of supersingular elliptic curves as follows.

Supersingular prime (moonshine theory)

For a prime number p, the following are equivalent: The modular curve X0+(p) = X0(p) / wp, where wp is the Fricke involution of X0(p), has genus zero.Every supersingular elliptic curve in characteristic p can be defined over the prime subfield Fp.The order of the Monster group is divisible by p. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then conjectural) existence of a sporadic simple group having exactly these primes as prime divisors. Monstrous moonshine. In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Conway and Simon P.

Monstrous moonshine

Norton in 1979, used to describe the unexpected connection between the monster group M and modular functions, in particular, the j function. It is now known that lying behind monstrous moonshine is a certain conformal field theory having the Monster group as symmetries. The conjectures made by Conway and Norton were proved by Richard Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. ATLAS: Monster group M. Order = 808017424794512875886459904961710757005754368000000000 = 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71.

ATLAS: Monster group M

Monster group. It is a simple group, meaning it does not have any proper non-trivial normal subgroups (that is, the only non-trivial normal subgroup is M itself).

Monster group

Existence and uniqueness[edit] Griess's construction showed that the monster existed. Monster Lie algebra. E8 lattice. The norm[1] of the E8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8.

E8 lattice

The existence of such a form was first shown by H. J. S. Smith in 1867,[2] and the first explicit construction of this quadratic form was given by A. Korkine and G. Lattice points[edit] The E8 lattice is a discrete subgroup of R8 of full rank (i.e. it spans all of R8). All the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), andthe sum of the eight coordinates is an even integer. In symbols, It is not hard to check that the sum of two lattice points is another lattice point, so that Γ8 is indeed a subgroup.

An alternative description of the E8 lattice which is sometimes convenient is the set of all points in Γ′8 ⊂ R8 such that Properties[edit] Hermitian symmetric space. In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure.

Hermitian symmetric space

First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). Hermitian symmetric spaces of compact type[edit] Definition[edit] admits a decomposition where.

Conway group. In mathematics, the Conway groups Co1, Co2 and Co3 are the three sporadic groups discovered by John Horton Conway.

Conway group

Ramanujan–Petersson conjecture. In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 satisfies when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms. Ramanujan L-function[edit] The Riemann zeta function and the Dirichlet L-function satisfy the Euler product, and due to their completely multiplicative property Are there L-functions except for the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Tau-function. The Ramanujan tau function, studied by Ramanujan (1916), is the function defined by the following identity: where with and. II25,1. In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector.

In particular it is closely related to the Leech lattice, and has the Conway group Co1 at the top of its automorphism group. Construction[edit] Write Rm,n for the m+n dimensional vector space Rm+n with the inner product of (a1,...