Umbral Moonshine Research Articles

Sl(2)ˆ Decomposition of denominator formulae of some BKM Lie superalgebras - II

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did not appear in that case. We restrict our analysis to the first N terms in the expansion as a first attempt at deconstructing the Siegel modular forms and unravelling the structure of potentially new Lie algebras that occur for N=5,6.

Formula omitted] decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2)ˆ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2)ˆ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

Optimal mock Jacobi theta functions

We classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that many of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients.

Properties of extremal CFTs with small central charge

We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $c\leq 24$. These are theories whose only operators with dimension smaller or equal to $c/24$ are the vacuum and its (super)Virasoro descendents. The prototypical example is the monster CFT, whose famous genus zero property is intimately tied to the Rademacher summability of its twined partition functions, a property which also distinguishes the functions of Mathieu and umbral moonshine. However, there are now several additional known examples of extremal CFTs, all of which have at least $\mathcal N=1$ supersymmetry and global symmetry groups connected to sporadic simple groups. We investigate the extent to which such a property, which distinguishes the monster moonshine module from other $c=24$ chiral CFTs, holds for the other known extremal theories. We find that in most cases, the special Rademacher summability property present for monstrous and umbral moonshine does not hold for the other extremal CFTs, with the exception of the Conway module and two $c=12, ~\mathcal N=4$ superconformal theories with $M_{11}$ and $M_{22}$ symmetry. This suggests that the connection between extremal CFT, sporadic groups, and mock modular forms transcends strict Rademacher summability criteria.

BKM Lie superalgebras from counting twisted CHL dyons – II

We revisit earlier work by one of us which lead to a periodic table of Borcherds-Kac-Moody algebras that appeared in the context of the refined generating function of quarter-BPS states (dyons) in N=4 supersymmetric four-dimensional string theory. We make new additions to the periodic table by making use of connections with generalized Mathieu moonshine as well as umbral moonshine. We show the modularity of some Siegel modular forms that appear in umbral moonshine associated with Niemeier lattices constructed from A-type root systems and further show that the same Siegel modular forms appear for generalized Mathieu moonshine in some cases. We argue for the existence of a new kind of BKM Lie superalgebras that arise from the dyon generating functions for the Z5 and Z6 CHL orbifolds.

Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules

In this work we consider an association of meromorphic Jacobi forms of half-integral index to the pure D-type cases of umbral moonshine, and solve the module problem for four of these cases by constructing vertex operator superalgebras that realise the corresponding meromorphic Jacobi forms as graded traces. We also present a general discussion of meromorphic Jacobi forms with half-integral index and their relationship to mock modular forms.

Moonshine modules and a question of Griess

We consider the situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded trace functions are weakly holomorphic modular forms. Under a mild hypothesis we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. In particular, we show that our hypothesis holds for umbral moonshine and for automorphism groups of certain vertex operator algebras.

Generalised Umbral Moonshine

Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon which connects finite groups and distinguished modular objects. In this paper we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data for it. A central role is played by the deformed Drinfel'd (or quantum) double of each umbral finite group $G$, specified by a cohomology class in $H^3(G,U(1))$. We conjecture that in each of the 23 cases there exists a rule to assign an infinite-dimensional module for the deformed Drinfel'd double of the umbral finite group underlying the mock modular forms of umbral moonshine and generalised umbral moonshine. We also discuss the possible origin of the generalised umbral moonshine.

K3 Elliptic Genus and an Umbral Moonshine Module

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 D4 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}}.

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.

Super vertex algebras, meromorphic Jacobi forms and umbral moonshine

The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups.

K3 string theory, lattices and moonshine

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 K3 times {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.

Weight one Jacobi forms and umbral moonshine

We analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that nonzero holomorphic Jacobi forms of weight one do not exist for many combinations of index and level, and use this to establish a characterization of the McKay–Thompson series of umbral moonshine in terms of Rademacher sums.

Heterotic sigma models on T8 and the Borcherds automorphic form \u03a612

We consider the spectrum of BPS states of the heterotic sigma model with (0, 8) supersymmetry and T8 target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds’ automorphic form Φ12, a modular form which exhibits automorphy for O(2, 26; ℤ). We comment on possible implications for Umbral moonshine and theories of AdS3 gravity.

3D string theory and Umbral moonshine

The simplest string theory compactifications to 3D with 16 supercharges—the heterotic string on T 7, and type II strings on —are related by U-duality, and share a moduli space of vacua parametrized by . One can think of this as the moduli space of even, self-dual 32-dimensional lattices with signature (8,24). At 24 special points in moduli space, the lattice splits as . can be the Leech lattice or any of 23 Niemeier lattices, while is the E8 root lattice. We show that starting from this observation, one can find a precise connection between the Umbral groups and type IIA string theory on K3. This may provide a natural physical starting point for understanding Mathieu and Umbral moonshine. The maximal unbroken subgroups of Umbral groups in 6D (or any other limit) are those obtained by starting at the associated Niemeier point and moving in moduli space while preserving the largest possible subgroup of the Umbral group. To illustrate the action of these symmetries on BPS states, we discuss the computation of certain protected four-derivative terms in the effective field theory, and recover facts about the spectrum and symmetry representations of 1/2-BPS states.

The umbral moonshine module for the unique unimodular Niemeier root system

We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.

Landau-Ginzburg orbifolds and symmetries of K3 CFTs

Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3ellipticgeneratwinedbydiscretesymmetriesthataremanifestintheUVdescription, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M11 ⊂ 2.M12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.

Derived equivalences of K3 surfaces and twined elliptic genera

We use the unique canonically twisted module over a certain distinguished super vertex operator algebra—the moonshine module for Conway’s group—to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective complex K3 surface that fixes a stability condition in the distinguished space identified by Bridgeland. According to work of Huybrechts, following Gaberdiel–Hohenegger–Volpato, any such derived equivalence determines a conjugacy class in Conway’s group, the automorphism group of the Leech lattice. Conway’s group acts naturally on the module we consider. In physics, the data of a projective complex K3 surface together with a suitable stability condition determines a supersymmetric non-linear sigma model, and supersymmetry-preserving automorphisms of such an object may be used to define twinings of the K3 elliptic genus. Our construction recovers the K3 sigma model twining genera precisely in all available examples. In particular, the identity symmetry recovers the usual K3 elliptic genus, and this signals a connection to Mathieu moonshine. A generalization of our construction recovers a number of Jacobi forms arising in umbral moonshine. We demonstrate a concrete connection to supersymmetric non-linear K3 sigma models by establishing an isomorphism between the twisted module we consider and the vector space underlying a particular sigma model attached to a certain distinguished K3 surface.

Proof of the umbral moonshine conjecture

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay–Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here, we establish the existence of the umbral moonshine modules in the remaining 22 cases.

Umbral Moonshine and K3 Surfaces

Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. In this paper we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices.