Mathieu Group M24 Research Articles

Moonshine, superconformal symmetry, and quantum error correction

Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a c = 24 two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a c = 12 CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group M24 has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving (4, 4) superconformal symmetry was studied in beautiful work by Gaberdiel, Taormina, Volpato, and Wendland [41]. The present paper shows that in both the GTVW and c = 12 theories the construction of superconformal generators can be understood via the theory of quantum error correcting codes. The automorphism groups of these codes lift to symmetry groups in the CFT preserving the superconformal generators. In the case of the N = 1 supercurrent of the GTVW model our result, combined with a result of T. Johnson-Freyd implies the symmetry group is the maximal subgroup of M24 known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on [41] the Ramond-Ramond sector of the GTVW model is related to the Miracle Octad Generator which in turn leads to a role for the Golay code as a group of symmetries of RR states. Moreover, (4, 1) superconformal symmetry suffices to define and decompose the elliptic genus of a K3 sigma model into characters of the N = 4 superconformal algebra. The symmetry group preserving (4, 1) is larger than that preserving (4, 4).

Hodge-Elliptic Genera and How They Govern K3 Theories

The (complex) Hodge-elliptic genus and its conformal field theoretic counterpart were recently introduced by Kachru and Tripathy, refining the traditional complex elliptic genus. We construct a different, so-called chiral Hodge-elliptic genus, which is expected to agree with the generic conformal field theoretic Hodge-elliptic genus, in contrast to the complex Hodge-elliptic genus as originally defined. For K3 surfaces X, the chiral Hodge-elliptic genus is shown to be independent of all moduli. Moreover, employing Kapustin's results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic Hodge-elliptic genus of K3 theories, while the complex Hodge-elliptic genus does not. This new invariant governs part of the field content of K3 theories, supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic Hodge-elliptic genus, and thereby the complex elliptic genus. Mathematically, this space is modelled by the sheaf cohomology of the chiral de Rham complex of X. It decomposes into irreducible representations of the N=4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the Mathieu group M24 that is predicted by Mathieu Moonshine. This is interpreted as evidence in favour of the ideas of symmetry surfing, which have been proposed by Taormina and the author, along with the claim that the sheaf cohomology of the chiral de Rham complex is a natural home for Mathieu Moonshine. These investigations also imply that the generic chiral algebra of K3 theories is precisely the N=4 superconformal algebra at central charge c=6, if the usual predictions on infinite volume limits from string theory hold true.

On base sizes for almost simple primitive groups

Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G0=An and Ω is an orbit of subsets or partitions of {1,…,n}, or if G0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

Gravitational couplings in mathcal{N}=2 string compactifications and Mathieu Moonshine

We evaluate the low energy gravitational couplings, Fg in the heterotic E8 ×E8 string theory compactified on orbifolds of K3 × T2 by g′ which acts as a ℤN automorphism on K3 together with a 1/N shift along T2. The orbifold g′ corresponds to the conjugacy classes of the Mathieu group M24. The holomorphic piece of Fg is given in terms of a polylogarithm with index 3−2g and predicts the Gopakumar-Vafa invariants in the corresponding dual type II Calabi-Yau compactifications. We show that low lying Gopakumar-Vafa invariants for each of these compactifications including the twisted sectors are integers. We observe that the conifold singularity for all these compactifications occurs only when states in the twisted sectors become massless and the strength of the singularity is determined by the genus zero Gopakumar-Vafa invariant at this point in the moduli space.

Calabi-Yau manifolds and sporadic groups

A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M24 was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau d-folds for d > 3 have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.

A combinatorial construction of an M12-invariant code

A ternary [66, 10, 36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see [N. Pace: New ternary linear codes from projectivity groups, Discrete Mathematics 331 (2014), 22-26]. We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5, 6, 12). We also present a proof that the Pace code does indeed have minimum distance 36.

The Pace code, the Mathieu group [formula omitted] and the small Witt design [formula omitted

A ternary [66,10,36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see Pace (2014). We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5,6,12). We also present a proof that the Pace code does indeed have minimum distance 36.

Mathcal{N} =2 heterotic string compactifications on orbifolds of K3 \xd7 T2

We study mathcal{N} = 2 compactifications of E8 × E8 heterotic string theory on orbifolds of K3 × T2 by g′ which acts as an {mathbb{Z}}_N automorphism of K3 together with a 1/N shift on a circle of T2. The orbifold action g′ corresponds to the 26 conjugacy classes of the Mathieu group M24. We show that for the standard embedding the new supersymmetric index for these compactifications can always be decomposed into the elliptic genus of K3 twisted by g′. The difference in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of K3 twisted by g′. We work out in detail the case for which g′ belongs to the equivalence class 2B. We then investigate all the non-standard embeddings for K3 realized as a {T}^4/{mathbb{Z}}_{nu } orbifold with ν = 2,4 and g′ the 2A involution. We show that for non-standard embeddings the new supersymmetric index as well as the difference in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the difference in number of hypermultiplets and vector multiplets in the spectrum.

A twist in the M 24 moonshine story

Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z2-orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z2-orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3. The 45-dimensional irreducible representation of M24 exhibits a twist, which we prove can be undone in the case of Z2-orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group Z2^4 : A8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.

Eta products, BPS states and K3 surfaces

Inspired by the multiplicative nature of the Ramanujan modular discriminant, Delta, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 surfaces admitting Nikulin automorphisms. In due course, we will present some identities arising from q-expansions as well as relations to the sporadic Mathieu group M24.

Enriques moonshine

We propose a new moonshine phenomenon associated with the elliptic genus of the Enriques surface ( of the elliptic genus of K3) with the symmetry group given by the Mathieu group M12.

Mathieu Moonshine and symmetries of K3 σ‐models

AbstractA recent observation by Eguchi, Ooguri and Tachikawa (EOT) suggests a relationship between the largest Mathieu group 𝕄24 and the elliptic genus of K3. This correspondence would be naturally explained by the existence of a non‐linear σ‐model on K3 with the Mathieu group as its group of symmetries. However, all possible symmetry groups of K3 models have been recently classified and none of them contains 𝕄24. We review the evidence in favour of the EOT conjecture and discuss the open problems in its physical interpretation.

Moonshine for $M_{24}$ and Donaldson invariants of $\mathbb{C} \mathrm{P}^2$

Eguchi, Ooguri, and Tachikawa recently conjectured a new moonshine phenomenon. They conjecture that the coefficients of a certain mock modular form H(tau), which arises from the K3 surface elliptic genus, are sums of dimensions of irreducible representations of the Mathieu group M24. We prove that H(tau) surprisingly also plays a significant role in the theory of Donaldson invariants. We prove that the Moore-Witten u-plane integrals for H(tau) are the SO(3)-Donaldson invariants of CP^2. This result then implies a moonshine phenomenon where these invariants conjecturally are expressions in the dimensions of the irreducible representations of M24. Indeed, we obtain an explicit expression for the Donaldson invariant generating function Z(p,S) in terms of the derivatives of H(tau).

Depth and Symmetry in Conway's M13 Puzzle

SummaryWe analyze a sliding tile puzzle (due to J. H. Conway) which gives a presentation of the Mathieu group M12, one of the sporadic simple groups. We use the symmetries of a finite projective plane to classify automorphisms of M12 which preserve the depth of puzzle positions, and give an algorithm for solving the puzzle by hand.

Note on twisted elliptic genus of K3 surface

We discuss the possibility of Mathieu group M24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M24. In this Letter we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.

A special class of simple 24-vertex polyhedra and tetrahedrally coordinated structures of gas hydrates

It is established that the eight-dimensional lattice E(8) and the Mathieu group M(12) determine a unique sequence of algebraic geometry constructions which define a special class of simple 24-vertex, 14-face polyhedra with four-, five- and six-edge faces. As an example, the graphs of the ten stereohedra that generate most known tetrahedrally coordinated water cages of gas hydrates have been derived a priori. A structural model is proposed for the phase transition between gas hydrate I and ice.

Base sizes for sporadic simple groups

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) ⩽ 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.

K3 surfaces, $\mathcal{N}=4$ dyons and the Mathieu group $M_{24}$

A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural interpretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new for the largest Mathieu group is connected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3xT^2-compactified type II string theory. This insight on the symmetry of the theory sheds new light on the generalised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spectrum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d=4, N=4 string theory, twisted by elements of the group of stringy K3 isometries.

Symmetrical features and local phase transitions of ordered solid structures: Tetravalent structures of gas hydrates

The interrelation between the topological invariants of the surface of polyhedra and the algebraic invariants has been analyzed. It has been demonstrated that this interrelation allows one to consider the processes of the removal of the geometric degeneracy characterized by the presence of high-symmetry critical points in local regions of the solid system. An analysis has been made of algebraic automorphisms that permit one to replace the stringent crystallographic requirement of the translation invariance by topological constraints which make it possible to determine ordered solid structures and a special type of local phase transitions. It has been established that the eight-dimensional lattice E8, the Galois field GF(11), and the Mathieu group M12 determine the only sequence of constructions of the algebraic geometry that identifies the unique class of 24-vertex tetradecahedral simple polyhedra with tetragonal, pentagonal, and hexagonal faces. The graphs of ten stereohedra of this class, including the Fedorov-Kelvin parallelohedron, have been a priori derived as an example. A model is proposed for local phase transformations in structures of gas hydrates in which the majority of the known tetravalent water frameworks are formed by polyhedra belonging to the class under consideration.

The Leech lattice Λ and the Conway group ⋅𝑂 revisited

We give a new, concise definition of the Conway group ⋅ O \cdot \mathrm {O} as follows. The Mathieu group M 24 \mathrm {M}_{24} acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of ( 24 4 ) \left ({24}\atop {4} \right ) tetrads. We use this action to define a progenitor P P of shape 2 ⋆ ( 24 4 ) : M 24 2^{\star \left ( 24 \atop 4 \right )}:\mathrm {M}_{24} ; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P P by this relator results in ⋅ O \cdot \mathrm {O} . Consideration of the lowest dimension in which ⋅ O \cdot \mathrm {O} can act faithfully produces Conway’s elements ξ T \xi _T and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ⋅ O \cdot \mathrm {O} of the integral vectors in R 24 {\mathbb R}_{24} .