Baby Monster Research Articles

The 2‐fusion system of the Monster

AbstractThis paper is part of an effort to determine a certain class of simple 2‐fusion systems, and to use that result to simplify the proof of the classification of the finite simple groups. The main theorem proves that the 2‐fusion system of the Monster is the unique simple system with a fully centralized involution whose centralizer is the fusion system of the universal covering group of the Baby Monster.

M, B and Co1 are recognisable by their prime graphs

Abstract The prime graph, or Gruenberg–Kegel graph, of a finite group 𝐺 is the graph Γ ⁢ ( G ) \Gamma(G) whose vertices are the prime divisors of | G | \lvert G\rvert and whose edges are the pairs { p , q } \{p,q\} for which 𝐺 contains an element of order p ⁢ q pq . A finite group 𝐺 is recognisable by its prime graph if every finite group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) \Gamma(H)=\Gamma(G) is isomorphic to 𝐺. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which 𝐺 is one of the 26 sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, M \mathrm{M} , the Baby Monster, B \mathrm{B} , and the first Conway group, Co 1 \mathrm{Co}_{1} . We prove that these three groups are recognisable by their prime graphs.

A characterization of the Mathieu–Conway–Monster series of locally projective graphs

Let I=(V,L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let Γ be the collinearity graph of I and let G be a group of automorphisms of I. Then Γ is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I; (b) the stabilizer G(x) of a vertex x∈V in G induces on the set of lines containing x, the group Ln(2) for some n≥2 in its natural doubly transitive action of degree 2n−1. Thus a locally projective graph in dimension n has valency 2n−1 or 2(2n−1) depending on whether it is thin or thick. The G(x)-invariant structure of a projective GF(2)-space of linear dimension n on the set of lines containing x, will be denoted by Πx. In the thin case we further assume that (c) the action of G on Γ is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge {x,y} of Γ an element in G which swaps x and y can be chosen to centralize G(x)∩G(y) modulo O2(G(x)∩G(y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) GF(2)-spaces. Less trivial classical examples are the dual polar GF(2)-spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some sporadic simple groups. The fundamental problem is to establish an upper bound on |G(x)| in terms of the dimension n, by finding an absolute bound d such that the vertex-wise stabilizer in G of the ball of radius d in Γ centred at a vertex x is always trivial (independently of the graph): Gd(x)=1. In the classical examples G2(x)=1. The first examples of Petersen and Tilde geometries were constructed in the early 1980s to disprove the belief that the classical bound holds in general. In fact, for the vertex-transitive graph of the Petersen geometry of the Baby Monster group the smallest d with Gd(x)=1 is 5. In 1990s V.I. Trofimov in a brilliant series of papers proved that G6(x)=1 in the thin case without assuming (c) and (d). In 2002 it has been proved by S.V. Shpectorov and the author that in the thin case under consideration all the examples with the critical parameter d above the classical bound indeed come from Petersen geometries (one way or another). An ongoing project is to extend the classification of locally projective amalgams to the thick case, which is considerably more complicated. Until recently only a characterization of an important class of 3-dimensional locally projective graphs [3] was available. In a recent paper [6] the author has shown that if the standard geometrization of a thick locally projective graph contains planes locally isomorphic to the generalized quadrangle of order (2,2), then Γ contains a family of thin locally projective subgraphs which are densely embedded in Γ in the sense that the stabilizer in G(x) of such a subgraph containing x has index 2n. This result places in a general setting the famous embeddingsG(M22)↪G(M24),G(Co2)↪G(Co1),G(BM)↪G(M) among Petersen and Tilde geometries. In the present paper we characterize the graphs of the geometries on the right hand side of these embeddings in the class of thick locally projective graphs with planes of the above specified type and with critical parameter d above the classical bound.

The intersection graph of a finite simple group has diameter at most 5

Let G be a non-abelian finite simple group. In addition, let Delta _G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of Delta _G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

Duality defect of the monster CFT

We show that the fermionization of the Monster CFT with respect to is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the orbifold, i.e. it enjoys the Kramers–Wannier duality. The Kramers–Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay–Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay–Thompson series is invariant under the genus-zero congruence subgroup 16D 0 of .

Classification of some vertex operator algebras of rank 3

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellekens list holomorphic VOA. These include the affine algebra $E_{8,2}$ and H\"ohn's Baby Monster VOA $\mathbf{VB}^\natural_{(0)}$ but the other $13$ seem to be new. The idea in the proof of our Main Theorem is to exploit properties of a family of vector-valued modular forms with rational functions as Fourier coefficients, which solves a family of modular linear differential equations in terms of generalized hypergeometric series.

Verification of the ordinary character table of the Baby Monster

We prove the correctness of the character table of the sporadic simple Baby Monster group that is shown in the ATLAS of Finite Groups.

Monster anatomy

We investigate the two-dimensional conformal field theories (CFTs) of c=frac{47}{2} , c=frac{116}{5} and c = 23 ‘dual’ to the critical Ising model, the three state Potts model and the tensor product of two Ising models, respectively. We argue that these CFTs exhibit moonshines for the double covering of the baby Monster group, 2;cdotp;mathbb{B} , the triple covering of the largest Fischer group, 3 · Fi24′ and multiple-covering of the second largest Conway group, 2 · 21+22 · Co2. Various twined characters are shown to satisfy generalized bilinear relations involving Mckay-Thompson series. We also rediscover that the ‘self-dual’ two-dimensional bosonic conformal field theory of c = 12 has the Conway group Co0 ≃ 2 · Co1 as an automorphism group.

The number of linear factors of supersingular polynomials and sporadic simple groups

The set of prime numbers p such that the supersingular j-invariants in characteristic p are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields.

Supersingular Elliptic Curves and Moonshine

We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.

From Vertex Operator Algebras to Conformal Nets and Back

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A_V. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c=1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and J\"{o}rss gives back the strongly local vertex operator algebra V from the conformal net A_V and give conditions on a conformal net A implying that A= A_V for some strongly local vertex operator algebra V.

(2+)-Replication and the Baby Monster

The definitions of replicable and completely replicable functions are intimately related to the Hecke operators for the modular group. We define the notions of "$(2+)$-replicable" and "completely $(2+)$-replicable" functions by considering the Hecke operators for $\Gamma_0(2)^+$. We prove that the McKay-Thompson series for $2\cdot\mathbb{B}$, as computed by H\"ohn, are completely $(2+)$-replicable.

Universal RCFT correlators from the holomorphic bootstrap

We elaborate and extend the method of Wronskian differential equations for conformal blocks to compute four-point correlation functions on the plane for classes of primary fields in rational (and possibly more general) conformal field theories. This approach leads to universal differential equations for families of CFT’s and provides a very simple re-derivation of the BPZ results for the degenerate fields ϕ1,2 and ϕ2,1 in the c < 1 minimal models. We apply this technique to compute correlators for the WZW models corresponding to the Deligne-Cvitanović exceptional series of Lie algebras. The application turns out to be subtle in certain cases where there are multiple decoupled primaries. The power of this approach is demonstrated by applying it to compute four-point functions for the Baby Monster CFT, which does not belong to any minimal series.

Modular constraints on conformal field theories with currents

We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne’s exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by mathcal{W} -algebras of various type and observe that the bounds on the gap depend on the choice of mathcal{W} -algebra in the small central charge region.

Complete spelling rules for the Monster tower over three-space

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

Two-dimensional RCFT’s without Kac-Moody symmetry

Using the method of modular-invariant differential equations, we classify a family of Rational Conformal Field Theories with two and three characters having no Kac-Moody algebra. In addition to unitary and non-unitary minimal models, we find "dual" theories whose characters obey bilinear relations with those of the minimal models to give the Moonshine Module. In some ways this relation is analogous to cosets of meromorphic CFT's. The theory dual in this sense to the Ising model has central charge 47/2 and is related to the Baby Monster Module.

Finding 47:23 in the Baby Monster

In this paper we describe methods for finding very small maximal subgroups of very large groups, with particular application to the subgroup 47:23 of the Baby Monster. This example is completely intractable by standard or naïve methods. The example of finding 31:15 inside the Thompson group $\text{Th}$ is also discussed as a test case.

A Moonshine Dialogue in Mathematical Physics

Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d’enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell’s theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it.

THE 2-LOCAL RANKS OF LARGE SPORADIC GROUPS

In this paper we mainly study sporadic groups in terms of p-local rank, especially the 2-local ranks of some sporadic groups (Conway simple group Co1, Janko simple group J4, Fisher simple groups, the Baby Monster 𝔹 and the Monster 𝕄) will be given which complete the problem of determining the p-local ranks of all sporadic simple groups for all primes p.

Orbit invariants and an application to the Baby monster

We prove that the minimal base size for the permutation action of the sporadic simple Baby monster group B on the cosets of its 7th and 8th maximal subgroup (in decreasing order of size) is 3 and 2 respectively. Motivated by the large sizes of these permutation actions, we develop new computational methods to prove that an orbit is regular and to show that two orbits are disjoint.