# Derived equivalences of K3 surfaces and twined elliptic genera

- John F. R. Duncan
^{1}Email author and - Sander Mack-Crane
^{2}

**3**:1

**DOI: **10.1186/s40687-015-0050-9

© The Author(s) 2016

**Received: **25 November 2015

**Accepted: **7 December 2015

**Published: **25 February 2016

## Abstract

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.

### Mathematics Subject Classification

11F50 14F05 14J28 17B69 20C34 20C35 58J26## 1 Background

The main result of this paper is a construction which attaches weak Jacobi forms to suitably defined autoequivalences of the bounded derived category of coherent sheaves on a complex projective K3 surface.

The origins of our method extend back to the monstrous moonshine phenomenon, initiated by the observations of McKay and Thompson [101, 102], Conway–Norton [29], and Queen [95]. The more recent Mathieu moonshine observation of Eguchi–Ooguri–Tachikawa [51], and its extension [18, 19, 25] to Niemeier lattice root systems is also closely related.

Our results also have physical significance. As we explain presently, they suggest that a certain distinguished super vertex operator algebra is a universal object for supersymmetric non-linear K3 sigma models. This represents a new role for vertex algebra in physics: rather than serving as the “chiral half” of a particular, holomorphically factorizable super conformal field theory, the super vertex operator algebra in question is, evidently, simultaneously related to a diverse family of super conformal field theories.

### 1.1 Monstrous moonshine

*g*] in the monster group \(\mathbb {M}\) by the work [29] of Conway–Norton. Here \(\widehat{\mathbb {C}}:=\mathbb {C}\cup \{\infty \}\simeq \mathbb {P}^1\) denotes the Riemann sphere, we set

*j*-invariant,

We refer to [40] for a recent review of moonshine, including a much fuller description of the above developments, and many more references.

The most obvious connection between moonshine and this article starts with the *multiplicative moonshine* observation of Conway–Norton (cf. §9 of [29]), considered in detail by Queen [95], which attaches analogues of the \(T_g\) of (1.1) to elements of the Conway group \({ Co}_0\), a twofold cover of the sporadic simple group \({ Co}_1\).

More than this, and in direct analogy with \(V^{\natural }\), the \({ Co}_0\)-module \(V^{s\natural }\) comes equipped with a distinguished super vertex operator algebra structure (cf. Sect. 5 for a recap on vertex algebra, and Sect. 8 for the construction and characterization of \(V^{s\natural }\)).

The reader familiar with vertex algebra will no doubt also be aware of the modularity results on trace functions attached to vertex operator algebras (cf. [35, 90, 111]) and super vertex operator algebras (cf. [38, 103, 104]). The results of [35], for example, go a long way to explaining why the right-hand side of (1.6) should define a holomorphic function on \(\mathbb {H}\) that is invariant for some congruence subgroup of \({\text {SL}}_2(\mathbb {Z})\). Interestingly, there is as yet no conceptual understanding of why (1.6) should actually satisfy the much stronger condition of defining an isomorphism as in (1.1), but see [47], or §6 of the review [40], for a conjectural proposal to establish a theory that would achieve this.

### 1.2 K3 surfaces and Jacobi forms

In this article, we use the unique (up to equivalence) canonically twisted \(V^{s\natural }\)-module to attach a Jacobi form \(\phi _g\) [cf. (9.10)] to a suitable derived autoequivalence *g* of a complex projective K3 surface *X*. More precisely, we prove the following result in Sect. 9.

###
**Theorem**

(9.5) Let *X* be a projective complex K3 surface and let \(\sigma \) be a stability condition in Bridgeland’s space. If *g* is a symplectic autoequivalence of the derived category of coherent sheaves on *X* that preserves \(\sigma \), then \(\phi _g\) is a weak Jacobi form of weight 0, index 1, and some level.

Jacobi forms (cf. [52] or Sect. 3) are 2-variable analogues of modular forms, admitting transformation formulas under a group of the form \({\text {SL}}_2(\mathbb {Z})\ltimes \mathbb {Z}^2\) (or a finite index subgroup thereof), that are modeled on those of the classical Jacobi theta functions \({\vartheta }_i(\tau ,z)\) [cf. (3.14)]. Jacobi forms also appear as Fourier coefficients of Siegel modular forms (cf. [53]). Roughly, a Jacobi form has level if it is required to transform only under some congruence subgroup \(\Gamma <{\text {SL}}_2(\mathbb {Z})\), and the term weak refers to certain growth conditions at the cusps of \(\Gamma \).

See Sect. 4 for a brief review of K3 surfaces, their symplectic derived autoequivalences, and stability conditions, and see Sect. 5 for the notion of canonically twisted module over a super vertex operator algebra.

Note that the appearance of Jacobi forms in vertex algebra goes back to the work of Kac–Peterson [79] (cf. also [78]) on basic representations of affine Lie algebras. (In particular, it actually predates Borcherds’ introduction of the notion of vertex algebra in [9]). More general results were established recently in [82], by applying earlier work [89] of Miyamoto. Cf. also [36]. Vertex algebraic constructions were used to attach Jacobi forms to conjugacy classes in the sporadic simple group of Rudvalis in [45, 46].

What is the meaning of the functions \(\phi _g\) of Theorem 9.5? One answer to this question is furnished by physics. More specifically, Theorem 9.5 can be interpreted as a statement about supersymmetric non-linear sigma models on K3 surfaces. For as explained by Huybrechts in [75], the analyses of [2, 4, 91] (cf. [68] for a concise account) suggest the conjecture that the pairs \((X,\sigma )\) with *X* and \(\sigma \) as in Theorem 9.5 are in natural correspondence with the supersymmetric non-linear sigma models on complex projective K3 surfaces.

### 1.3 Sigma models

Witten introduced [83, 108] a construction which attaches a weak Jacobi form for \({\text {SL}}_2(\mathbb {Z})\) to any supersymmetric non-linear sigma model, called the *elliptic genus* of the model in question. It turns out [cf. (9.28)] that \(\phi _g\) is exactly the K3 elliptic genus when \(g=e\) is the identity autoequivalence (in particular, both \(\phi _e\) and the K3 elliptic genus are independent of the choices of *X* and \(\sigma \)).

Generalizing this, the analysis of [68] suggests that we can expect to obtain a Jacobi form with level, called a *twined elliptic genus*, from any supersymmetry-preserving automorphism of a supersymmetric non-linear sigma model. In terms of the pairs \((X,\sigma )\), such automorphisms should correspond to symplectic autoequivalences that preserve \(\sigma \). Thus, it is natural to compare the \(\phi _g\) to twined elliptic genera of supersymmetric non-linear K3 sigma models.

Unfortunately, it is generally a difficult matter to compute twined K3 elliptic genera, for the Hilbert spaces attached to supersymmetric non-linear K3 sigma models are only known in a few special cases. However, it has been shown recently by Gaberdiel–Hohenegger–Volpato [68] (cf. also [75]) that any group of supersymmetry-preserving automorphisms of such a model can be embedded in the Conway group \({ Co}_0\) (actually, \({ Co}_0\) here can be replaced by \({ Co}_1\), but it seems to be more natural to regard \({ Co}_0\) as the operative group). More specifically (and subject to some assumptions about the moduli space of K3 sigma models), the groups of supersymmetry-preserving automorphisms of K3 sigma models are exactly the subgroups of \({ Co}_0\) that pointwise fix a 4-dimensional subspace of \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\), according to [68].

Thus, there is hope that a suitably defined \({ Co}_0\)-module may be used to recover all the twined K3 elliptic genera, bypassing the explicit construction of super conformal field theories attached to K3 sigma models. The present work furnishes strong evidence that this is indeed the case, and that \(V^{s\natural }_\mathrm {tw}\) is precisely the \({ Co}_0\)-module to consider. Indeed, about half of the conjugacy classes in \({ Co}_0\) that fix a 4-space in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\) appear in the explicit computations of [68, 69, 106], and we find precise agreement with the \(\phi _g\), defined via \(V^{s\natural }_\mathrm {tw}\), in every case.

This leads us to the following conjecture, indicating one precise sense in which \(V^{s\natural }\) may serve as a universal object for K3 sigma models.

###
**Conjecture**

(9.6) The twined elliptic genus attached to any supersymmetry-preserving automorphism of a supersymmetric non-linear K3 sigma model coincides with \(\phi _g\) for some \(g\in { Co}_0\) fixing a 4-space in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\).

It will be interesting to see if \(V^{s\natural }\) cannot ultimately shed light on more subtle aspects of K3 sigma models, beyond their twined elliptic genera.

It is at first surprising that the central charge of \(V^{s\natural }\) is twice that of the super conformal field theories attached to K3 sigma models, i.e., 12 rather than 6. In Sect. 11, we give an explanation for this discrepancy by demonstrating an isomorphism of Virasoro modules between \(V^{s\natural }\) and the Neveu–Schwarz sector of the super conformal field theory attached to a particular, distinguished K3 sigma model, which has been considered earlier in [65, 107]. See Proposition 11.1. Note that we naturally obtain a Virasoro module structure of central charge 12 on the sigma model by taking the diagonal copy of the Virasoro algebra, within the two commuting copies that act on left- and right-movers, respectively.

For concreteness, we have chosen to formulate our main results in terms of derived categories of coherent sheaves, and their stability conditions, rather than sigma models. We refer the reader to [3, 12, 39] for introductory expositions of the deep connection between these notions.

### 1.4 Mathieu moonshine

*q*with integer coefficients,

*g*.

The group \(M_{24}\) appears naturally as a subgroup of \({ Co}_0\), in such a way that the definition of \(\chi _g\) just given coincides with (1.9) for \(g\in M_{24}<{ Co}_0\), so we may compare the \(\phi _g\) of Theorem 9.5 to the weak Jacobi forms \(Z^{(2)}_g\) of Mathieu moonshine. Interestingly, \(\phi _g=Z^{(2)}_g\) for *g* in all but 7 of the 26 conjugacy classes of \(M_{24}\). Cf. Table 8. The conjugacy classes of \(M_{24}\) for which \(\phi _g\ne Z^{(2)}_g\) are those named 3*B*, 6*B*, 12*B*, 21*A*, 21*B*, 23*A*, and 23*B* in [32]. Note that \(\phi _g\) is not even defined for *g* in any of the last 5 of these, since such elements of \({ Co}_0\) do not pointwise fix a 4-space in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\).

Regarding the \(M_{24}\)-module \(\check{K}^{(2)}\) of (1.16), Gannon has proven [70] that the candidate \(H^{(2)}_g\) determined in [22, 50, 66, 67] are indeed the graded trace functions attached to a graded \(M_{24}\)-module, but there is, as yet, no analogue for \(\check{K}^{(2)}\) of the vertex algebraic constructions of \(V^{\natural }\) or \(V^{s\natural }\). The fact that \(\phi _g\) recovers \(Z^{(2)}_g\) for so many \(g\in M_{24}\) suggests that \(V^{s\natural }\) may play an important role in determining such a concrete construction.

See [23] for a detailed review of Mathieu moonshine, including explicit descriptions of the \(H^{(2)}_g\). The \(H^{(2)}_g\) are examples of mock modular forms of weight 1/2, a notion which has arisen fairly recently, thanks to the foundational work of Zwegers [112] on Ramanujan’s mock theta functions [96, 97], contemporaneous work [15] of Bruinier–Funke on harmonic Maass forms, and subsequent contributions by Bringmann–Ono [14] and Zagier [110]. We refer to [94, 110] for introductory accounts of mock modular forms. The \(H^{(2)}_g\) for \(g\in M_{24}\) have been constructed uniformly in [24], and related results appear in [13].

### 1.5 Umbral moonshine

The superscripts in \(\check{K}^{(2)}\), \(H^{(2)}_g\), and \(Z^{(2)}_g\) indicate that Mathieu moonshine is but one case of a more generally defined theory. Indeed, the observations of [51] were extended in [18, 19] (cf. also [25]), to an association of (vector-valued) mock modular forms \(H^{(\ell )}_g=(H^{(\ell )}_{g,r})\) to conjugacy classes [*g*] in finite groups \(G^{(\ell )}\) (with \(G^{(\ell )}=M_{24}\) for \(\ell =2\)), for certain symbols \(\ell \), called *lambencies*. The resulting collection of relationships between finite groups and mock modular forms is now known as umbral moonshine.

The lambencies of [19] are in correspondence with the 23 (non-empty) simply laced root systems that arise in even self-dual positive-definite lattices of rank 24. These are the so-called Niemeier root systems (cf. Sect. 2). For example, if *n* is a divisor of 24 and \(k=24/n\), then \(\ell =n+1\) corresponds to the union of *k* copies of the \(A_n\) root system, denoted \(A_n^k\). In particular, \(\ell =2\) corresponds to \(A_1^{24}\).

The group \(G^{(\ell )}\) is, by definition, the outer automorphism group of the self-dual lattice \(N^{(\ell )}\) whose Niemeier root system corresponds to \(\ell \). That is, \(G^{(\ell )}:={\text {Aut}}(N^{(\ell )})/W^{(\ell )}\) where \(W^{(\ell )}\) is the normal subgroup of \({\text {Aut}}(N^{(\ell )})\) generated by reflections in root vectors. Note that all of these groups \(G^{(\ell )}\) embed in \({ Co}_0\).

According to the McKay correspondence [56, 88], the irreducible simply laced root systems are in correspondence with certain surface singularities called du Val singularities (cf. e.g., [49]). Thus, the governing role of simply laced root systems in umbral moonshine suggests a geometric interpretation involving non-smooth K3 surfaces equipped with configurations of du Val singularities. Evidence in support of this idea is developed in [20]. A number of the weak Jacobi forms \(\phi _g\) constructed here appear also in [20].

*m*is a certain positive integer depending on \(\ell \) and \(I^{(\ell )}\) is a certain subset of \(\{1,\ldots ,m-1\}\) (we refer the reader to [19, 25] or §9 of [40] for a fuller discussion of \(\check{K}^{(\ell )}\) and its relation to \(G^{(\ell )}\) and the \(H^{(\ell )}_g\)).

The existence of \(G^{(\ell )}\)-modules \(\check{K}^{(\ell )}\) satisfying (1.20) is one of the main conjectures of umbral moonshine, and has now been proven [41] for all Niemeier root systems. A concrete, vertex algebraic construction of \(\check{K}^{(\ell )}\) has been established recently [42] for the special case that \(\ell =30+6,10,15\), which is the lambency corresponding to the root system \(E_8^3\). Vertex algebraic constructions of \(G^{(\ell )}\)-modules closely related to the \(\check{K}^{(\ell )}\) appear in [48] for the lambencies corresponding to \(A_3^8\), \(A_4^6\), \(A_6^4\), and \(A_{12}^2\), and in [17] for \(D_6^4\), \(D_8^3\), \(D_{12}^2\), and \(D_{24}\).

In the case of \(A_{\ell -1}^k\), where \(k(\ell -1)=24\), the mock modular forms \(H^{(\ell )}_g\), together with certain characters \(\chi _g\) and \(\bar{\chi }_g\) of \(G^{(\ell )}\), can be used to define weak Jacobi forms \(Z^{(\ell )}_g\) of weight 0 and index \(\ell -1\) (and some level depending on *g*), in a natural way (see §4 of [18] for the details of this construction).

In Sect. 10, we present a natural generalization of the construction of \(\phi _g\) in Sect. 9, and in so doing attach a weak Jacobi form \(\phi ^{(\ell )}_g\), of weight 0 and index \(\ell -1\), to any element \(g\in { Co}_0\) that fixes a 2*d*-dimensional subspace of \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\), where \(d=2(\ell -1)\). Interestingly, many of the \(Z^{(\ell )}_g\) of umbral moonshine are realized as (scalar multiples of) \(\phi ^{(\ell )}_g\) for suitable \(g\in { Co}_0\). Thus, we have evidence that \(V^{s\natural }\) may be an important device for realizing a number of the \(\check{K}^{(\ell )}\) explicitly. The particular coincidences between \(Z^{(\ell )}_g\) and \(\phi ^{(\ell )}_g\) are recorded in Sect. B, Tables 8, 9, 10, 11 and 12.

Surprisingly, \(V^{s\natural }\) can be used to attach mock modular forms to conjugacy classes in finite groups beyond those arising as \(G^{(\ell )}\) for some lambency \(\ell \). Indeed, in [16] the canonically twisted \(V^{s\natural }\)-module \(V^{s\natural }_\mathrm {tw}\) is used to attach 2-vector-valued mock modular forms of weight 1 / 2 to conjugacy classes in any subgroup of \({ Co}_0\) fixing a 2-space in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\). In this way, mock modular forms are attached to the conjugacy classes of the sporadic Mathieu groups \(M_{23}\) and \(M_{12}\), McLaughlin’s sporadic group *McL*, and the sporadic group *HS* of Higman and Higman–Sims (cf. [71, 72, 100]). An association of mock modular forms to conjugacy classes in subgroups of \({ Co}_0\) fixing 3-spaces in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\) is also considered in [16]. Consequently, mock modular forms (of a different kind) are attached to \(M_{22}\) and \(M_{11}\). See [21] (and its prequel [8]) for an extension of this method to subgroups of \({ Co}_0\) that fix a line in \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\). This analysis associates mock modular forms (of yet another variety) to the sporadic groups \({ Co}_2\), \({ Co}_3\), and \(M_{24}\).

It is evident from the above-mentioned results that \(V^{s\natural }\) should play an important role in umbral moonshine. Thus, the close relationship between Conway moonshine and monstrous moonshine serves to motivate the possibility that monstrous and umbral moonshine are related in a deep and direct way, potentially sharing a common origin. The results of [93] also motivate this point of view. See the introduction to [18] for related discussion.

### 1.6 Organization

We now describe the structure of the paper. Since the main result involves a number of different topics, not typically seen together in a single work, we begin with a number of brief preliminary sections. We recall some basic facts about even self-dual lattices, and also discuss the Conway group in Sect. 2. We then recall modular forms, Jacobi forms, and certain special examples of such functions in Sect. 3. We review the main results from [75] on derived equivalences of K3 surfaces in Sect. 4. In Sect. 5, we recall basic definitions in vertex algebra theory, and give a brief description of the Clifford module (a.k.a. free fermion) super vertex algebra construction in Sect. 6.

The results of our earlier work [43] play an important role here, and we review these next, recalling some useful formulas relating to spin modules in Sect. 7, and the construction of the distinguished super vertex operator algebra \(V^{s\natural }\) in Sect. 8.

In Sect. 9, we establish our main result: a mechanism which attaches a weak Jacobi form to any symplectic derived equivalence of a K3 surface that fixes a suitable stability condition. See Theorem 9.5.

We also formulate a conjecture relating the Jacobi forms so arising to twined elliptic genera of K3 sigma models. In brief, all the examples of twined K3 elliptic genera available in the literature are recovered from our construction. This suggests that the super vertex operator algebra \(V^{s\natural }\) serves as a universal object for K3 sigma models.

The construction of Sect. 9 easily generalizes so as to recover a number of weak Jacobi forms of umbral moonshine. We discuss this in detail in Sect. 10.

We give some deeper evidence for the conjectural relationship between \(V^{s\natural }\) and K3 sigma models in Sect. 11, by exhibiting an isomorphism of graded vector spaces between \(V^{s\natural }\) and the super conformal field theory arising from a certain distinguished K3 sigma model.

We present data necessary for the computation of all the Jacobi forms appearing in this work in Sect. A. We record coincidences between these Jacobi forms and other functions appearing in the context of K3 sigma models, and umbral moonshine, in Sect. B.

As mentioned earlier, we choose a square root of \(-1\) in \(\mathbb {C}\) and denote it by \({\mathbf {i}}\). We also set \(e(x):=\mathrm{e}^{2\pi {\mathbf {i}}x}\).

## 2 Lattices

An *integral lattice* is a free \(\mathbb {Z}\)-module of finite rank, \(L\simeq \mathbb {Z}^n\), equipped with a symmetric bilinear form \(\langle \cdot ,\cdot \rangle :L\otimes _\mathbb {Z}L:\rightarrow \mathbb {Z}\). An excellent general reference for lattices is [31]. Given a field *k* of characteristic zero, the bilinear form \(\langle \cdot ,\cdot \rangle \) extends naturally to the *n*-dimensional vector space \(L\otimes _\mathbb {Z}k\) over *k*. The *signature* of *L* is the pair (*r*, *s*) where *r* is the maximal dimension of a positive-definite subspace of \(L\otimes _\mathbb {Z}\mathbb {R}\) and *s* is the maximal dimension of a negative-definite subspace of \(L\otimes _\mathbb {Z}\mathbb {R}\). Call *n* the *rank* of *L*, and say *L* is *non-degenerate* if \(n=r+s\). Say *L* is *positive-definite* if \(s=0\), and *negative-definite* if \(r=0\). Say *L* is *indefinite* if \(rs\ne 0\).

*dual*of

*L*by setting

*L*. Say that

*L*is

*self-dual*if \(L^*=L\). Observe that a self-dual lattice is necessarily non-degenerate.

Given \(\lambda \in L\) call \(\langle \lambda ,\lambda \rangle \) the *square-length* of \(\lambda \). A lattice *L* is called *even* if all of its square-lengths are even integers. The set of vectors of square-length \(\pm 2\) in an even lattice is called its *root system*.

*root lattice*, commonly denoted \(E_8\). It is the unique (up to isomorphism) even self-dual lattice of signature (8, 0). The lattice \(I\!I_{1,1}\) is often denoted

*U*, and sometimes called the

*hyperbolic plane*.

We refer to Chapter V of [98] for a proof of the following fundamental result.

###
**Theorem 2.1**

Suppose that *L* is a non-even self-dual integral lattice with signature (*r*, *s*). If \(rs\ne 0\) then \(L\simeq I_{r,s}\). Suppose that *L* is an even self-dual lattice with signature (*r*, *s*). Then \(r=s\mod 8\). If \(rs\ne 0\) then \(L\simeq I\!I_{r,s}\).

*spin lattice*of rank

*n*, and we denote it \(D_n^+\). The \(D_n\)

*root lattice*is the intersection \(I_{n,0}\cap I\!I_{n,0}\) (for any positive

*n*), and is an even lattice of index 2 in \(D_n^+\)

*L*(up to isomorphism) that solve (2.5), according to Niemeier’s classification [92] of even self-dual definite lattices of rank 24 (cf. also [105] and Chapter 16 of [31]). Distinguished amongst these is the

*Leech lattice*, named for its discoverer (cf. [84, 85]) and denoted here by \(\Lambda \), which is the unique even self-dual lattice of signature (24, 0) with an empty root system (i.e., no vectors of square-length 2).

*Conway group*by setting

*L*with signature (

*r*,

*s*), write \(L(-1)\) for the lattice of signature (

*s*,

*r*) obtained by multiplying the bilinear form on

*L*by \(-1\). Then, for example, if

*L*is even self-dual with signature \((k,16+k)\) for some positive integer

*k*, we have

We call \(\Lambda (-1)\) the *negative-definite Leech lattice*.

An *embedding* of lattices \(K\rightarrow L\) is an embedding of abelian groups \(\iota :K\rightarrow L\) such that \(\langle \lambda ,\mu \rangle _K=\langle \iota (\lambda ),\iota (\mu )\rangle _L\) for \(\lambda ,\mu \in K\). A *primitive embedding* is an embedding \(\iota :K\rightarrow L\) such that the quotient group \(L/\iota (K)\) is torsion free.

## 3 Modular forms

Here we recall some basic facts about modular forms and Jacobi forms. For \(\tau \in \mathbb {H}\) and \(z\in \mathbb {C}\), we use the notation \(q:=e(\tau )\) and \(y:=e(z)\), where \(e(x):=\mathrm{e}^{2\pi {\mathbf {i}}x}\).

*unrestricted modular form*of weight

*k*for a group \(\Gamma <{\text {SL}}_2(\mathbb {R})\) if

*cusps*. The orbit containing \(\infty \) is called the

*infinite cusp*of \(\Gamma \), and the

*modular group*\({\text {SL}}_2(\mathbb {Z})\) has only the infinite cusp.

Say that *f* as in (3.1) is a *weakly holomorphic modular form* if it has at most exponential growth at cusps. This amounts to the condition that if \(\sigma \in {\text {SL}}_2(\mathbb {Z})\) then \(f(\sigma \tau )\) admits a Laurent expansion in \(q^{1/w}\), for some positive integer *w*. If the Laurent expansions of the \(f(\sigma \tau )\) are actually Taylor series in \(q^{1/w}\), so that \(f(\sigma \tau )=O(1)\) as \(\mathfrak {I}(\tau )\rightarrow \infty \), then we say that *f* is a *modular form*. A *cusp form* satisfies \(f(\sigma \tau )\rightarrow 0\) as \(\mathfrak {I}(\tau )\rightarrow \infty \), for every \(\sigma \in {\text {SL}}_2(\mathbb {Z})\).

Write \(M_{k}(\Gamma )\) for the space of modular forms of weight *k* for \(\Gamma \). Write \(S_k(\Gamma )\) for the subspace of cusp forms.

*Eisenstein series*\(E_k\) are a family of modular forms for the full modular group \(SL_2(\mathbb {Z})\). For

*k*even and greater than 2, the Eisenstein series \(E_k\) is defined by

*quasi-modular form*, satisfying

*Dedekind eta function*, a modular form of weight \(\frac{1}{2}\) (with non-trivial multiplier), defined by

*Delta function*, a cusp form of weight 12 for \({\text {SL}}_2(\mathbb {Z})\), defined by

*level*is a modular form for some \(\Gamma _0(N)\),

*N*(i.e., a modular form for \(\Gamma _0(N)\)), then \(f(h\tau )\) is a modular form with level

*hN*.

*N*, let

*h*denote the largest divisor of 24 such that \(h^2\) divides

*N*. Set \(n=N/h\). Then the normalizer of \(\Gamma _0(N)\) in \({\text {SL}}_2(\mathbb {R})\) is composed of the matrices

*e*is an

*exact*divisor of

*n*/

*h*(i.e.,

*e*|(

*n*/

*h*) and \((e,n/eh)=1\)), and \(a,b,c,d\in \mathbb {Z}\) are chosen so that \(ade^2-bcn/h=e\).

We write \(\Gamma _0(n|h)\) for the set of matrices (3.10) with \(e=1\). It is a subgroup of \({\text {SL}}_2(\mathbb {R})\) that is conjugate to \(\Gamma _0(n/h)\). For a fixed non-trivial exact divisor *e*|(*n* / *h*), the matrices (3.10) comprise an *Atkin–Lehner involution* of \(\Gamma _0(n|h)\) (really, an Atkin–Lehner involution is a coset of \(\Gamma _0(n|h)\) in its normalizer, an involution in the sense that it defines an order 2 element of the quotient group \(N(\Gamma _0(n|h))/\Gamma _0(n|h)\)).

*unrestricted Jacobi form*of weight

*k*and index

*m*for \(\Gamma \) if it satisfies

*weak Jacobi form*if \(c_\sigma (n/w,r)=0\) whenever \(n/w<0\), for all \(\sigma \in {\text {SL}}_2(\mathbb {Z})\). Note that \(c_\sigma (n/w,r)\) differs from \(c_{\sigma '}(n/w,r)\) only by a root of unity when \(\sigma '\sigma ^{-1}\in \Gamma \). So it suffices to check the \(c_\sigma (n/w,r)\) for just one representative \(\sigma \) of each right coset of \(\Gamma \) in \({\text {SL}}_2(\mathbb {Z})\).

Good references for Jacobi forms include [34] and [52]. Jacobi forms occur naturally as Fourier coefficients of Siegel Modular forms (cf. [52, 53]), but that manifestation will not play an explicit role here.

In this work, a weak Jacobi form with *level*
*N* is a weak Jacobi form for \(\Gamma _0(N)\).

*Jacobi theta functions*, defined as

Proposition 6.1 of [1] states that any weak Jacobi form of even weight can be written as a polynomial in \(\phi _{0,1}\) and \(\phi _{-2,1}\) with modular form coefficients. We record the following special cases of this for use later on.

###
**Proposition 3.1**

*m*for \(\Gamma \) if and only if there exists \(C\in \mathbb {C}\) and modular forms \(F_{2j}\in M_{2j}(\Gamma )\), for \(1\le j\le m\), such that

We conclude this section with formulas that illustrate (3.18) explicitly for the particular combinations of Jacobi theta functions appearing in (3.16) and (3.17).

###
**Lemma 3.2**

Note that the first identity of Lemma 3.2 is just the definition of \(\phi _{-2,1}\) [cf. (3.17)].

###
*Proof*

*principal congruence group of level*

*N*, being the kernel of the natural map \({\text {SL}}_2(\mathbb {Z})\rightarrow {\text {SL}}_2(\mathbb {Z}/N\mathbb {Z})\). First, we will show that

*S*and

*T*are the standard generators for the modular group, then \(\Gamma (2)\) is generated by \(T^2:\tau \mapsto \tau +2\) and \(ST^2S:\tau \mapsto \frac{-\tau }{2\tau -1}\) (see §6 of [58]), so the required transformations under \(\Gamma (2)\) are

*T*and

*S*, using Jacobi’s imaginary transformations

*O*(

*q*). \(\square \)

## 4 Derived equivalences

Let *X* be a complex K3 surface; i.e., a compact connected complex manifold of dimension 2 with \(\Omega _X^2\simeq \mathcal {O}_X\) and \(H^1(X,\mathcal {O}_X)=0\) (good references for K3 surfaces include [5, 7]). Then the intersection form \((\;.\;)\) equips the integral singular cohomology group \(H^2(X,\mathbb {Z})\) with the structure of an even self-dual lattice of signature (3, 19), so we have \(H^2(X,\mathbb {Z})\simeq E_8(-1)^{\oplus 2}\oplus U^{\oplus 3}\) according to (2.8).

*Mukai lattice*of

*X*, being the lattice obtained from

*Hodge structure*of weight 2 on a lattice

*L*is a direct sum decomposition

*L*into complex subspaces \(L^{p,q}<L\otimes _\mathbb {Z}{\mathbb {C}}\) such that the \(\mathbb {R}\)-linear complex conjugation \(v\mapsto \bar{v}\) on \(L\otimes _\mathbb {Z}{\mathbb {C}}\) that fixes the subset \(L\otimes _\mathbb {Z}{\mathbb {R}}\) induces \(\mathbb {R}\)-linear isomorphisms \(L^{p,q}\simeq L^{q,p}\).

*X*is a complex K3 surface, then we naturally obtain a weight 2 Hodge structure

*X*, by setting

*X*. Say that an automorphism

*g*of the lattice \(\widetilde{H}(X,\mathbb {Z})\) is a

*symplectic Hodge isometry*of \(\widetilde{H}(X,\mathbb {Z})\) if the \(\mathbb {C}\)-linear extension of

*g*to \(\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}\mathbb {C}\) fixes \(\widetilde{H}^{2,0}(X)\) (and hence also \(\widetilde{H}^{0,2}(X)\)) pointwise. Note that \(\widetilde{H}^{2,0}(X)\) and \(\widetilde{H}^{0,2}(X)\) are isotropic with respect to the bilinear form on \(\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}{\mathbb {C}}\) induced from \(\widetilde{H}(X,\mathbb {Z})\). The intersection

Following [75], we write \({\text {Aut}}_s(\widetilde{H}(X,\mathbb {Z}))\) for the group of symplectic Hodge isometries of the Mukai lattice \(\widetilde{H}(X,\mathbb {Z})\) of a complex K3 surface *X*. Note that any symplectic automorphism of *X* of finite order naturally induces a symplectic Hodge isometry of \(\widetilde{H}(X,\mathbb {Z})\), via the induced action of *g* on \(H^*(X,\mathbb {Z})\) (cf. §1.2 of [75]), but for general *X* not all symplectic Hodge isometries arise in this way (cf. §1.4 of [75]).

*X*is projective, admitting an embedding in some complex projective space \(\mathbb {P}^n\). Write \({\text {D}}^\mathrm{b}(X)\) for the bounded derived category of coherent sheaves on

*X*and let \({\text {Aut}}({\text {D}}^\mathrm{b}(X))\) denote the group of isomorphism classes of exact \(\mathbb {C}\)-linear autoequivalences of \({\text {D}}^\mathrm{b}(X)\) (see [6, 73] for detailed expositions of this theory). The induced action of an exact autoequivalence of \({\text {D}}^\mathrm{b}(X)\) on \(\widetilde{H}(X,\mathbb {Z})\)—cf. the discussion in §1.2 in [11]—defines a morphism of groups from \({\text {Aut}}({\text {D}}^\mathrm{b}(X))\) to the automorphism group (i.e., orthogonal group) of \(\widetilde{H}(X,\mathbb {Z})\), and we write \({\text {Aut}}_s({\text {D}}^\mathrm{b}(X))\) for the subgroup of

*symplectic*autoequivalences, being those elements of \({\text {Aut}}({\text {D}}^\mathrm{b}(X))\) that map to symplectic Hodge isometries of \(\widetilde{H}(X,\mathbb {Z})\).

*positive*if its induced action on \({\text {Stab}}(X)\) fixes \(\sigma \), and write \({\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) for the group of all (isomorphism classes of) \(\sigma \)-positive exact \(\mathbb {C}\)-linear autoequivalences of \({\text {D}}^\mathrm{b}(X)\).

*central charge*

*Z*, which may be regarded as a morphism of groups \(\widetilde{H}^{1,1}(X,\mathbb {Z})\rightarrow \mathbb {C}\), or equivalently, via Poincaré duality, as an element of \(\widetilde{H}^{1,1}(X,\mathbb {Z})\otimes _{\mathbb {Z}}{\mathbb {C}}\). According to §1.3 of [75], the real subspace of \(\widetilde{H}^{1,1}(X,\mathbb {Z})\otimes _{\mathbb {Z}}{\mathbb {R}}\) spanned by the real and imaginary parts of

*Z*,

*Z*is the central charge of a stability condition in \({\text {Stab}}^{\circ }(X)\). For such a \(Z\in \widetilde{H}^{1,1}(X,\mathbb {Z})\otimes _{\mathbb {Z}}{\mathbb {C}}\), define \({\text {Aut}}_s(\widetilde{H}(X,\mathbb {Z}),Z)\) to be the subgroup of symplectic Hodge isometries of \(\widetilde{H}(X,\mathbb {Z})\) whose \(\mathbb {R}\)-linear extensions to \(\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}\mathbb {R}\) fix the subspace \(P_Z\) pointwise; such an isometry is called \(P_Z\)-

*positive*.

Recall the natural map (4.8). Huybrechts has shown that this map induces an isomorphism between the group of \(\sigma \)-positive symplectic autoequivalences of \({\text {D}}^\mathrm{b}(X)\) and the group of \(P_Z\)-positive symplectic Hodge isometries of \(\widetilde{H}(X,\mathbb {Z})\) when *Z* is the central charge of a stability condition \(\sigma \) in \({\text {Stab}}^{\circ }(X)\).

###
**Proposition 4.1**

*X*be a projective complex K3 surface, let \(\sigma \in {\text {Stab}}^{\circ }(X)\) and let

*Z*be the central charge of \(\sigma \). Then the natural map \({\text {Aut}}_s({\text {D}}^\mathrm{b}(X))\rightarrow {\text {Aut}}_s(\widetilde{H}(X,\mathbb {Z}))\) induces an isomorphism of groups

Note that any positive-definite 4-dimensional subspace of \(\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}\mathbb {R}\), and in particular \(\Pi =P_X\oplus P_Z\), is naturally oriented. For if \(\varsigma \) is a non-zero element of \(H^{2,0}(X)\), then the 4-tuple \((\mathfrak {R}(\varsigma ),\mathfrak {I}(\varsigma ),\mathfrak {R}(Z),\mathfrak {I}(Z))\) defines an oriented basis, and the resulting orientation depends neither on \(\varsigma \) nor on *Z* (cf. §4.5 of [76]).

*X*and \(\sigma \in {\text {Stab}}^{\circ }(X)\) as above, define

Call a primitive embedding as in (4.14) a *Leech marking* of the data \((X,\sigma )\). We may summarize the previous paragraph by saying that the group \(G_\Pi ={\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) is isomorphic to a subgroup of \({ Co}_0\) that fixes a rank 4 sublattice of \(\Lambda \), and the choice of Leech marking \(\iota \) determines this subgroup completely. The main result of [75] states that the converse is also true.

###
**Theorem 4.2**

([75]) For *X* a projective complex K3 surface and \(\sigma \in {\text {Stab}}^{\circ }(X)\), the group \(G_\Pi ={\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) is isomorphic to a subgroup of \({ Co}_0\) whose action on the Leech lattice fixes a sublattice of rank at least 4. Conversely, if \(G_*\) is a subgroup of \({ Co}_0\) that fixes a rank 4 sublattice of the Leech lattice, then there exists a projective complex K3 surface *X*, a stability condition \(\sigma \in {\text {Stab}}^{\circ }(X)\), and a Leech marking \(\iota \) for \((X,\sigma )\) such that \(G_*\) is a subgroup of \(\iota _*G_\Pi \).

Recall from Sect. 2 that the center of \({ Co}_0\) is the group of order 2 generated by \(-{\text {Id}}\), and \({ Co}_1\) denotes the sporadic simple quotient group \({ Co}_1={ Co}_0/\{\pm {\text {Id}}\}\). Observe that if \(G_*\) is a subgroup of \({ Co}_0\) that has a fixed point in its action on \(\Lambda \), then the natural map \({ Co}_0\rightarrow { Co}_1\) induces an isomorphism between \(G_*\) and its image in \({ Co}_1\). Thus, one may replace \({ Co}_0\) with \({ Co}_1\) in the statement of Theorem 4.2.

## 5 Vertex algebra

In this section, we briefly recall super vertex operator algebras and their canonically twisted modules. We refer to the texts [59, 77, 86] for more background on vertex algebra.

A *super vector space* is a simply a vector space with a \(\mathbb {Z}/2\)-grading, \(V=V_{\bar{0}}\oplus V_{\bar{1}}\). A linear operator \(T:V\rightarrow V\) is called *even* if \(T(V_{\bar{j}})\subset V_{\bar{j}}\), and *odd* if \(T(V_{\bar{j}})\subset V_{\overline{j+1}}\).

*V*a super vector space and

*z*a formal variable, write \(V((z)):=V[[z]][z^{-1}]\) for the space of Laurent series in

*z*with coefficients in

*V*. Taking

*z*to be even, we naturally obtain a super structure \(V_{\bar{0}}((z))\oplus V_{\bar{1}}((z))\) on

*V*((

*z*)). Observe that the rational function \(f(z,w)=(z-w)^{-1}\) naturally defines elements of \(\mathbb {C}((z))((w))\) and \(\mathbb {C}((w))((z))\) via formal power series expansions, for we have \(f(z,w)=\sum _{n\ge 0} z^{-n-1}w^n\) in \(\mathbb {C}((z))((w))\) and \(f(z,w)=-\sum _{n\ge 0}w^{-n-1}z^n\) in \(\mathbb {C}((w))((z))\). These rules extend naturally so as to define

*formal expansion*maps

*super vertex algebra*is a super vector space \(V=V_{\bar{0}}\oplus V_{\bar{1}}\) equipped with a

*vacuum vector*\(\mathbf {1} \in V_{\bar{0}}\), an even linear operator \(T:V\rightarrow V\), and a linear map

*vertex operator*

*Y*(

*a*,

*z*). These data should satisfy the following axioms for any \(a,b, c\in V\).

- 1.
\(Y(a,z)b\in V((z))\) and if \(a\in V_{\bar{0}}\) (resp. \(a\in V_{\bar{1}}\)) then \(a_{(n)}\) is an even (resp. odd) operator for all

*n*; - 2.
\(Y(\mathbf {1},z)={\text {Id}}_V\) and \(Y(a,z)\mathbf {1}\in a+zV[[z]]\);

- 3.
\([T,Y(a,z)]=\partial _z Y(a,z)\) and \(T\mathbf {1}=0\);

- 4.If \(a\in V_{p(a)}\) and \(b\in V_{p(b)}\) are \(\mathbb {Z}/2\) homogenous, there exists an elementdepending on$$\begin{aligned} f\in V[[z,w]]\left[ z^{-1},w^{-1},(z-w)^{-1}\right] \end{aligned}$$
*a*,*b*, and*c*, such thatare the formal expansions of$$\begin{aligned} Y(a,z)Y(b,w)c,\quad (-1)^{p(a)p(b)}Y(b,w)Y(a,z)c,\quad \text {and}\quad Y(Y(a,z-w)b,w)c \end{aligned}$$*f*in*V*((*z*))((*w*)),*V*((*w*))((*z*)), and \(V((w))((z-w))\), respectively [cf. (5.1)].

*parity involution*, acting as \((-1)^j\) on \(V_{\bar{j}}\). A

*canonically twisted module*for

*V*is a super vector space \(M=M_{\bar{0}}\oplus M_{\bar{1}}\) equipped with a linear map

*canonically twisted vertex operator*\(Y_\mathrm {tw}(a,z^{1/2})\), which satisfies the following axioms for any \(a,b\in V, u\in M\):

- 1.
\(Y_\mathrm {tw}(a,z^{1/2})u\in M((z^{1/2}))\) and if \(a\in V_{\bar{0}}\) (resp. \(a\in V_{\bar{1}}\)) then \(a_{(n),\mathrm {tw}}\) is an even (resp. odd) operator for all

*n*; - 2.
\(Y_\mathrm {tw}(\mathbf {1},z^{1/2})={\text {Id}}_M\);

- 3.If \(a\in V_{p(a)}\) and \(b\in V_{p(b)}\), there exists an elementdepending on$$\begin{aligned} f\in M\left[ \left[ z^{1/2}, w^{1/2}\right] \right] \left[ z^{-{1/2}},w^{-{1/2}},(z-w)^{-1}\right] \end{aligned}$$
*a*,*b*, and*u*, such thatare the expansions of$$\begin{aligned}&Y_\mathrm {tw}\left( a,z^{1/2}\right) Y_\mathrm {tw}\left( b,w^{1/2}\right) u, \quad (-1)^{p(a)p(b)}Y_\mathrm {tw}\left( b,w^{1/2}\right) Y_\mathrm {tw}\left( a,z^{1/2}\right) u,\\&\quad \text {and }\; Y_\mathrm {tw}\left( Y(a,z-w)b,w^{1/2}\right) u \end{aligned}$$*f*in the spaces \(M((z^{1/2}))((w^{1/2}))\), \(M((w^{1/2}))((z^{1/2}))\), and \(M((w^{1/2}))((z-w))\), respectively; and - 4.
If \(\theta (a)=(-1)^ma\), then \(a_{(n),\mathrm {tw}}=0\) for \(n\notin \mathbb {Z}+\frac{m}{2}\).

*Virasoro*algebra \(\mathcal {V}\) is the Lie algebra spanned by

*L*(

*m*), for \(m\in \mathbb {Z}\), and a central element \(\mathbf{c }\), with Lie bracket

*central charge*

*c*if the central element \(\mathbf{c }\) acts as multiplication by

*c*on

*V*.

*super vertex operator algebra*is a super vertex algebra \(V=V_{\bar{0}}\oplus V_{\bar{1}}\) containing a

*Virasoro element*\(\omega \in V_{\bar{0}}\) such that if \(L(n):=\omega _{(n+1)}\) for \(n\in \mathbb {Z}\) then

- 5.
\(L({-1})=T\);

- 6.
\([L(m), L(n)]=(m-n)L(m+n)+\frac{m^3-m}{12}\delta _{m+n,0}c{\text {Id}}_V\) for some \(c\in \mathbb {C}\);

- 7.
\(L(0)\) is a diagonalizable operator on

*V*, with rational eigenvalues bounded below, and finite-dimensional eigenspaces.

*V*with central charge

*c*.

*V*such a super vertex operator algebra, let us write \(V=\bigoplus _{n\in \mathbb {Q}}V_n\) for the decomposition of

*V*into eigenspaces

^{1}for \(L(0)-\frac{\mathbf{c }}{24}\).

*V*, we also write

*L*(

*n*) for \(\omega _{(n+1),\mathrm {tw}}\) [cf. (5.3)], a linear operator on \(V_\mathrm {tw}\), and we write \((V_\mathrm {tw})_n\) for the eigenspace with eigenvalue

*n*for \(L(0)-\frac{\mathbf{c }}{24}\).

*V*a super vertex operator algebra, suppose to be given an element \(\jmath \in V\) with \(L(0)\jmath =\jmath \) such that if \(J(n):=\jmath _{(n)}\) then

*element*for

*V*, and we call

*k*the

*level*of \(\jmath \). The action of the operator \(J(0)=\jmath _{(0)}\) preserves the eigenspaces for \(L(0)-\frac{\mathbf{c }}{24}\) by hypothesis, and may in addition be diagonalizable. In such a situation, we write \(V=\bigoplus _{n,r}V_{n,r}\) for the corresponding decomposition into bi-graded subspaces for

*V*.

*V*-module, we abuse notation slightly by writing

*J*(0) also for \(\jmath _{(0),\mathrm {tw}}\), an operator on \(V_\mathrm {tw}\), and define

## 6 The Clifford module construction

We now briefly review the standard construction that attaches a super vertex operator algebra, and a canonically twisted module for it, to a vector space equipped with a non-degenerate symmetric bilinear form. We refer to [54] for a very thorough treatment, and to [43] for a fuller description using the same notation that is employed here.

A similar construction produces a canonically twisted module for \(A(\mathfrak {a})\) [cf. (5.3)], which we call \(A(\mathfrak {a})_\mathrm {tw}\). We recall this now, assuming for the sake of simplicity that \(\dim \mathfrak {a}\) is even.

*polarization*of \(\mathfrak {a}\). Now set

*u*(

*m*) as an operator on \(A(\mathfrak {a})_\mathrm {tw}\) by letting

*u*(

*m*) act by left multiplication in case \(m<0\). For \(m\ge 0\), the action of

*u*(

*m*) is determined by the rules

*d*. Moreover, the action of \(J(0):=\jmath _{(0)}\) on \(A(\mathfrak {a})\) is diagonal, with integer eigenvalues, and similarly for \(J(0):=\jmath _{(0),\mathrm {tw}}\) as an operator on \(A(\mathfrak {a})_\mathrm {tw}\). Since it will be useful later in the article, we record a more detailed statement as follows for future use.

###
**Lemma 6.1**

Let \(\jmath \) as in (6.12). Then \(\jmath \) is a \({\text {U}}(1)\) element for \(A(\mathfrak {a})^0\) with level *d*. We have \(J(0)\mathbf {v}=0\) and \(J(0)\mathbf {v}_\mathrm {tw}=\frac{d}{4}\mathbf {v}_\mathrm {tw}\). Also, \([J(0),a_i^{\pm }(r)]=\pm a_i^{\pm }(r)\) for all \(1\le i\le d\) and \(r\in \frac{1}{2}\mathbb {Z}\), and if \(u\in \mathfrak {a}\) is orthogonal to \({\text {Span}}\{a_1^\pm ,\ldots ,a_d^\pm \}\), then *J*(0) commutes with *u*(*r*) for all \(r\in \frac{1}{2}\mathbb {Z}\).

###
*Proof*

*fermonic normal ordering*, defined so that

*Clifford algebra*associated to \(\mathfrak {a}\) and \(\langle \cdot ,\cdot \rangle \) by setting

*I*is the ideal generated by the \(u\otimes u+\langle u,u\rangle \mathbf{1 }\) for \(u\in \mathfrak {a}\). Given \(u_i\in \mathfrak {a}\), write \(u_1\ldots u_k\) for the image of \(u_1\otimes \cdots \otimes u_k\in T(\mathfrak {a})\) in \({\text {Cliff}}(\mathfrak {a})\). Then the relations (6.8) ensure that \({\text {Cliff}}(\mathfrak {a})\) acts naturally on \(A(\mathfrak {a})_\mathrm {tw}\), via \(u_1\ldots u_k\mapsto u_1(0)\ldots u_k(0)\), for \(u_i\in \mathfrak {a}\).

## 7 Lifting to the spin group

Let \(\mathfrak {a}\) be a complex vector space equipped with a non-degenerate symmetric bilinear form \(\langle \cdot ,\cdot \rangle \), as in Sect. 5. To recall the definition of the *spin group* of \(\mathfrak {a}\), denoted \({\text {Spin}}(\mathfrak {a})\), we remind that the *main anti-automorphism*
\(\alpha \) of \({\text {Cliff}}(\mathfrak {a})\) is defined by setting \(\alpha (u_1\ldots u_k):=u_k\ldots u_1\) for \(u_i\in \mathfrak {a}\). The group \({\text {Spin}}(\mathfrak {a})\) is composed of the even, invertible elements \(x\in {\text {Cliff}}(\mathfrak {a})\) with \(\alpha (x)x=\mathbf{1 }\) such that \(xux^{-1}\in \mathfrak {a}\) whenever \(u\in \mathfrak {a}\).

Say that \(\widehat{g}\in {\text {Spin}}(\mathfrak {a})\) is a *lift* of an element \(g\in {\text {SO}}(\mathfrak {a})\) if \(\widehat{g}\) has the same order as *g*, and \(\widehat{g}(\cdot )=g\) [cf. (7.1)]. More generally, say that \(\widehat{G}<{\text {Spin}}(\mathfrak {a})\) is a *lift* of a subgroup \(G<{\text {SO}}(\mathfrak {a})\) if the natural map (7.2) induces an isomorphism \(\widehat{G}\xrightarrow \sim G\).

Suppose we are given an identification \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\) where \(\Lambda \) is the Leech lattice (cf. Sect. 2). In this situation, we set \(G:={\text {Aut}}(\Lambda )\), a copy of the Conway group \({ Co}_0\) which we may naturally regard as a subgroup of \({\text {SO}}(\mathfrak {a})\). Proposition 3.1 in [43] demonstrates that there is a unique lift of \(G<{\text {SO}}(\mathfrak {a})\) to \({\text {Spin}}(\mathfrak {a})\).

###
**Proposition 7.1**

([43]) If \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\) and \(G={\text {Aut}}(\Lambda )<{\text {SO}}(\mathfrak {a})\), then there is a unique subgroup \(\widehat{G}<{\text {Spin}}(\mathfrak {a})\) such that the natural map \({\text {Spin}}(\mathfrak {a})\rightarrow {\text {SO}}(\mathfrak {a})\) induces an isomorphism \(\widehat{G}\xrightarrow {\sim } G\).

*g*, such that the \(\mathfrak {a}^{\pm }:={\text {Span}}_\mathbb {C}\{a_i^\pm \}\) are isotropic subspaces of \(\mathfrak {a}\), and \(\langle a_i^\pm ,a_j^\mp \rangle =\delta _{i,j}\). Write \(\lambda _i\) for the eigenvalue of

*g*attached to \(a_i^+\).

*g*-invariant polarization of \(\mathfrak {a}\), and we may assume that

*i*. We call \(\mathfrak {z}\) as in (7.8) the lift of \(-{\text {Id}}_\mathfrak {a}\)

*associated*to the polarization \(\mathfrak {a}=\mathfrak {a}^-\oplus \mathfrak {a}^+\).

*g*to \(\widehat{G}<{\text {Spin}}(\mathfrak {a})\) is given explicitly by

*V*is a real vector space contained in \(\mathfrak {a}\), such that \(\mathfrak {a}=V\otimes _{\mathbb {R}}\mathbb {C}\), and such that \(\langle \cdot ,\cdot \rangle \) restricts to an \(\mathbb {R}\)-valued bilinear form on

*V*(e.g., \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\) and \(V=\Lambda \otimes _\mathbb {Z}\mathbb {R}\)). Then a choice of orientation \(\mathbb {R}^+\omega \subset \bigwedge ^{24}(V)\) on

*V*also determines a lift of \(-{\text {Id}}_\mathfrak {a}\) to \({\text {Spin}}(\mathfrak {a})\), for given an ordered basis \(\{e_i\}\) of

*V*satisfying \(\langle e_i,e_j\rangle =\pm \delta _{i,j}\) and

*associated*to the orientation \(\mathbb {R}^+\omega \). Evidently, a change in orientation replaces \(\mathfrak {z}'\) with \(-\mathfrak {z}'\).

We see now from Proposition 7.1 that \(\Lambda \) is naturally oriented. For setting \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\) and \(V=\Lambda \otimes _\mathbb {Z}\mathbb {R}\subset \mathfrak {a}\), and taking \(G={\text {Aut}}(\Lambda )<{\text {SO}}(\mathfrak {a})\) and \(\widehat{G}<{\text {Spin}}(\mathfrak {a})\) as in Proposition 7.1, we may choose the preferred orientation on *V* to be the one for which the associated lift \(\mathfrak {z}'\) of \(-{\text {Id}}_\mathfrak {a}\) [cf. (7.14)] belongs to \(\widehat{G}\). By the same token, there is a preferred \({\text {SO}}(\mathfrak {a})\)-orbit of polarizations \(\mathfrak {a}=\mathfrak {a}^-\oplus \mathfrak {a}^+\) of \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\), being the one for which an associated lift \(\mathfrak {z}\) [cf. (7.8)] belongs to \(\widehat{G}\).

## 8 The Conway moonshine module

We now recall the main construction from [43].

###
**Proposition 8.1**

([43]) The \(A(\mathfrak {a})^0\)-module structure on \(V^{s\natural }\) extends uniquely to a super vertex operator algebra structure on \(V^{s\natural }\), and the \(A(\mathfrak {a})^0\)-module structure on \(V^{s\natural }_\mathrm {tw}\) extends uniquely to a canonically twisted \(V^{s\natural }\)-module structure.

The super vertex operator algebra \(V^{s\natural }\) is distinguished. The following abstract characterization of \(V^{s\natural }\) has been established in [43] (cf. Theorem 5.15 of [44]).

###
**Theorem 8.2**

([43]) The super vertex operator algebra \({V^{s\natural }}\) is the unique self-dual \(C_2\)-cofinite rational super vertex operator algebra of CFT type with central charge 12 such that \(L(0)u=\tfrac{1}{2}u\) for \(u\in {V^{s\natural }}\) implies \(u=0\).

We refer to [43] for the precise meanings of the terms self-dual, \(C_2\)-cofinite, rational, and CFT type. Briefly, a super vertex operator algebra *V* is rational if any *V*-module can be written as a direct sum of irreducible *V*-modules. We say that *V* is self-dual if it is irreducible as a module over itself, and if *V* is the only irreducible *V*-module up to isomorphism. As explained in [43], Theorem 8.2 identifies \({V^{s\natural }}\) as an analogue for super vertex operator algebras of the extended binary Golay code, of the Leech lattice \(\Lambda \) (cf. Sect. 2), and (conjecturally) of the moonshine module vertex operator algebra \(V^{\natural }\) (cf. Sect. 1.1).

As explained in Sect. 7, the spin group \({\text {Spin}}(\mathfrak {a})\) acts naturally on the \(A(\mathfrak {a})^j\) and \(A(\mathfrak {a})_\mathrm {tw}^j\), so it acts naturally on \(V^{s\natural }\) and \(V^{s\natural }_\mathrm {tw}\). In particular, given an identification \(\mathfrak {a}=\Lambda \otimes _\mathbb {Z}\mathbb {C}\), we naturally obtain actions of the Conway group \({ Co}_0\) on \(V^{s\natural }\) and \(V^{s\natural }_\mathrm {tw}\), because \(G={\text {Aut}}(\Lambda )<{\text {SO}}(\mathfrak {a})\) admits a unique lift \(\widehat{G}<{\text {Spin}}(\mathfrak {a})\), according to Proposition 7.1.

*g*acting on \(\mathfrak {a}\), as in (7.7), and define \(C_g\) by setting

^{2}

*g*, and \(C_g=0\) exactly when

*g*has a non-zero fixed point in \(\mathfrak {a}\).

###
**Lemma 8.3**

The main result of [43] is that \(T^s_g\) is the normalized principal modulus for a genus zero group \(\Gamma _g<{\text {SL}}_2(\mathbb {R})\), and \(T^s_{g,\mathrm {tw}}\) is also a principal modulus, so long as \(C_g\ne 0\). From the explicit descriptions of the \(\Gamma _g\) in Table 1 of [43], we see that \(T^s_g(2\tau )\) is invariant for some \(\Gamma _0(N)\), with *N* depending on *g*, for every \(g\in { Co}_0\).

###
**Theorem 8.4**

([43]) Let \(g\in { Co}_0\). Then \(T^s_g(2\tau )\) is the normalized principal modulus for a genus zero group \(\Gamma _g<{\text {SL}}_2(\mathbb {R})\) that contains some \(\Gamma _0(N)\). If *g* has a fixed point in its action on the Leech lattice, then the function \(T^s_{g,\mathrm {tw}}(\tau )\) is constant, with constant value \(-\chi _g\). If *g* has no fixed points, then \(T^s_{g,\mathrm {tw}}(\tau )\) is a principal modulus for a genus zero group \(\Gamma _{g,\mathrm {tw}}<{\text {SL}}_2(\mathbb {R})\).

The groups \(\Gamma _g\) and \(\Gamma _{g,\mathrm {tw}}\) are described explicitly in [43].

*Frame shape*of

*g*.

## 9 Twining genera

In this section, we establish the main results of the paper.

Let *X* be a projective complex K3 surface and let \(\sigma =(\mathcal {P},Z)\) be a stability condition in Bridegland’s space \({\text {Stab}}^{\circ }(X)\) (cf. Sect. 4). Presently, we will attach a formal series \(\phi _g\in \mathbb {C}[y^{\pm 1}][[q]]\) to any \(g\in G_{\Pi }={\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) by computing the graded trace of a suitable automorphism of the canonically twisted module \(V^{s\natural }_\mathrm {tw}\) for the distinguished super vertex algebra \(V^{s\natural }\) that was reviewed in Sect. 8 (and studied in detail in [43]). It will develop (see Theorem 9.5) that \(\phi _g\) is a weak Jacobi form of weight zero and index one, with some level (cf. Sect. 3).

As in (6.10) we write \(Y_{\mathrm {tw}}(\omega ,z^{1/2})=\sum _{n\in \mathbb {Z}} L(n)z^{-n-2}\) for the twisted module vertex operator \({V^{s\natural }_\mathrm {tw}}\rightarrow {V^{s\natural }_\mathrm {tw}}((z))\) attached to the Virasoro element \(\omega \in V^{s\natural }\) [cf. (6.5)]. Then *L*(0) acts diagonalizably on \({V^{s\natural }_\mathrm {tw}}\) with eigenvalues in \(\mathbb {Z}+\tfrac{1}{2}\), thus \(L(0)-\frac{\mathbf{c }}{24}\) defines an integer grading on \(V^{s\natural }_\mathrm {tw}\), since the central charge of \(V^{s\natural }\) is \(c=\frac{1}{2}\dim (\mathfrak {a})=12\).

*X*and

*Z*enable us to define a \({\text {U}}(1)\) element (cf. Sect. 5), and hence a second integer grading on \(V^{s\natural }_\mathrm {tw}\). To see this, first recall the spaces \(P_X\) [cf. (4.7)] and \(P_Z\) [cf. (4.9)] from Sect. 4. Let \(\varsigma \) be a non-zero element of \(H^{2,0}(X)\), and choose vectors

*G*to \({\text {Spin}}(\mathfrak {a})\) whose existence and uniqueness is guaranteed by Proposition 7.1. Recall that we write \(g\mapsto \widehat{g}\) for the isomorphism \(G\rightarrow \widehat{G}\). We may assume that \(\mathfrak {z}'\in \widehat{G}\) [cf. (9.1)], for if this is not true for our first choice of \(\Lambda (-1)\), then it becomes true once we replace \(\Lambda (-1)\) with its image under the reflection in the hyperplane defined by a non-zero vector in \(\Pi \).

*G*, we suppress it from notation. Thus, to each \(g\in G_{\Pi }\subset G\) is associated a corresponding element \(\widehat{g}\in \widehat{G}\). We now define \(\phi _g\in \mathbb {C}[y^{\pm 1}][[q]]\) by setting

Our notation \(\phi _g\) obscures the choice of Leech marking for \((X,\sigma )\). We now show that this convention entails no ambiguity.

###
**Proposition 9.1**

The series \(\phi _g\) is independent of the choice of Leech marking \(\iota \).

###
*Proof*

Suppose that \(\Lambda (-1)\subset \mathfrak {a}\) is chosen as above, having full rank \(\mathfrak {a}=\Lambda (-1)\otimes _\mathbb {Z}\mathbb {C}\) in \(\mathfrak {a}\), containing \(\Gamma _\Pi \) as a primitive sublattice, and such that \(\mathfrak {z}'\in \widehat{G}\), for \(\widehat{G}\) the unique lift of \(G:={\text {Aut}}(\Lambda (-1))\simeq { Co}_0\) to \({\text {Spin}}(\mathfrak {a})\), and \(\mathfrak {z}'\) the lift of \(-{\text {Id}}_\mathfrak {a}\) associated to the chosen orientation (9.1) on \(\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}\mathbb {R}\). A second choice of Leech marking leads to a second copy of the negative-definite Leech lattice, \(\Lambda '(-1)\subset \mathfrak {a}\), with \(\mathfrak {a}=\Lambda '(-1)\otimes _\mathbb {Z}\mathbb {C}\) and \(\Gamma _\Pi <\Lambda (-1)\cap \Lambda '(-1)\). Set \(G':={\text {Aut}}(\Lambda '(-1))\) and write \(\widehat{G}'\) for the unique lift of \(G'\simeq { Co}_0\) to \({\text {Spin}}(\mathfrak {a})\), and assume, as we may, that \(\mathfrak {z}'\in \widehat{G}'\).

We have \(g\in G\cap G'\). Write \(\widehat{g}\) and \(\widehat{g}'\) for the respective lifts to \({\text {Spin}}(\mathfrak {a})\), determined by \(\widehat{G}\) and \(\widehat{G}'\). We have \(\widehat{g}=\pm \widehat{g}'\), and we require to show that, in fact, \(\widehat{g}=\widehat{g}'\).

Let *h* be an orthogonal transformation of \(\mathfrak {a}\) that restricts to an isomorphism \(h:\Lambda (-1)\xrightarrow {\sim }\Lambda '(-1)\). By our hypothesis that \(\mathfrak {z}'\in \widehat{G}\cap \widehat{G}'\), we have \(h\in {\text {SO}}(\mathfrak {a})\). Since \(\Gamma _\Pi \) is a primitive sublattice of \(\Lambda (-1)\cap \Lambda '(-1)\), we may choose *h* so that it restricts to the identity on \(\Gamma _\Pi \). Then *h* commutes with *g*, because *g* acts trivially on \(\Gamma _\Pi ^\perp \). More than this, any lift \(\widehat{h}\) of *h* to \({\text {Spin}}(\mathfrak {a})\) commutes with \(\widehat{g}\), because we have \(\widehat{g}=\prod _{i=1}^{12}\mathrm{e}^{\alpha _iX_i}\) [cf. (7.10)], for some basis \(\{a_i^\pm \}\) of eigenvectors for *g*, as in (7.7), with \(X_i\) as in (7.9), and we may assume that \(\alpha _i\ne 0\) only when \(a_i^\pm \in \Gamma _\Pi \otimes _\mathbb {Z}\mathbb {C}\). Then \(\widehat{h}X_i=X_i\widehat{h}\) whenever \(\alpha _i\ne 0\), and so \(\widehat{h}\widehat{g}=\widehat{g}\widehat{h}\). Now \(\widehat{h}\widehat{G}\widehat{h}^{-1}\) is a lift of \(G'\) to \({\text {Spin}}(\mathfrak {a})\), so it must be \(\widehat{G}'\) by Proposition 7.1. So \(\widehat{h}\widehat{g}\widehat{h}^{-1}\) is the lift \(\widehat{g}'\) of \(hgh^{-1}=g\) to \(\widehat{G}'\), so \(\widehat{g}'=\widehat{h}\widehat{g}\widehat{h}^{-1}=\widehat{g}\), as we required to show. \(\square \)

The coefficients of the \(\phi _g\) may be computed explicitly, in direct analogy with (8.9). With this purpose in mind, we define constants \(D_{{g}}\) as follows.

*g*, constituting a pair of dual bases in the sense that \(\langle a^-_i,a^+_j\rangle =\delta _{i,j}\) (cf. the discussion in Sect. 8). We may assume that

*g*, by hypothesis. We may also assume that the lift of \(-{\text {Id}}_\mathfrak {a}\) associated to the polarization \(\mathfrak {a}=\mathfrak {a}^-\oplus \mathfrak {a}^+\)coincides with \(\mathfrak {z}'\) [cf. (9.1)], for if not, then replace \(a_i^\pm \) with \(a_i^\mp \), for some \(i\in \{1,\ldots ,10\}\).

*g*attached to \(a^{\pm }_i\). Set \(X_i:=\frac{{\mathbf {i}}}{2}(a_i^-a_i^+-a_i^+a_i^-)\) as in (7.9). Then, according to the discussion in Sect. 7, we have

*g*has a fixed point in its action on \(\Gamma _\Pi \). In particular, \(D_{g}\) vanishes whenever the sublattice of \(\Lambda \) fixed by

*g*has rank larger than 4.

We are now prepared to present an explicit expression for \(\phi _g\).

###
**Proposition 9.2**

*X*be a projective complex K3 surface and let \(\sigma \in {\text {Stab}}^{\circ }(X)\). Then for \(g\in G_\Pi ={\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) and \(\phi _g\) defined by (9.10), we have

###
*Proof*

The required identity (9.14) may be obtained via direct calculation. We use the decomposition \(V^{s\natural }=A(\mathfrak {a})^1\oplus A(\mathfrak {a})^0_\mathrm {tw}\) along with the formulas (8.2) and (8.3). We also use Lemma 6.1, and the product formulas (3.15) for the Jacobi theta functions \(\vartheta _i\).

\(\square \)

*g*as in Proposition 9.2, and recall the definition (8.10) of \(\chi _g\).

###
**Proposition 9.3**

*X*be a projective complex K3 surface and let \(\sigma \in {\text {Stab}}^{\circ }(X)\). Then for \(g\in {\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\), we have

###
*Proof*

Applying Proposition 3.1 with \(m=1\) to (9.20), we see that \(\phi _g\) is a weak Jacobi form of level *N* so long as \(F_g\) is a modular form of weight 2 for \(\Gamma _0(N)\).

###
**Proposition 9.4**

Let *X* be a projective complex K3 surface and let \(\sigma \in {\text {Stab}}^{\circ }(X)\). Then for \(g\in {\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) the function \(F_g\) is a modular form of weight 2 for \(\Gamma _0(N_g)\), for some positive integer \(N_g\).

###
*Proof*

As in the definition (9.10) of \(\phi _g\), and the proof of Proposition 9.1, we use Proposition 4.1 to identify \(G_\Pi ={\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) with a subgroup of \({\text {SO}}(\mathfrak {a})\) (recall that \(\mathfrak {a}=\widetilde{H}(X,\mathbb {Z})\otimes _{\mathbb {Z}}{\mathbb {C}}\)), and we choose a Leech marking \(\iota \) for \((X,\sigma )\), in order to identify *g* as an element of \(G={\text {Aut}}(\Lambda (-1))\), for a suitable copy of \(\Lambda (-1)\) in \(\mathfrak {a}\). Then \(\eta _{\pm g}(\tau )\) and \(C_{-g}\) depend only on the conjugacy class of *g* in \(G\simeq { Co}_0\), and \(D_g\) is determined by the conjugacy class \([g]\subset G\) up to sign [cf. (9.13)]. The values \(C_{-g}\) and \(D_{g}\), and the Frame shapes \(\pi _{\pm g}\) that determine the \(\eta _{\pm g}(\tau )\) [cf. (8.14)] may be read off from Table 3.

Now the proof is essentially a case-by-case check of the relevant classes of \({ Co}_0\) (rather than, say, all the groups \(G_{\Pi }\) in \({\text {Aut}}(\widetilde{H}(X,\mathbb {Z}))=O(I\!I_{4,20})\)), but we can use the results of [43] to simplify this further, replacing the explicit calculation of modular forms with simple checks on properties of the invariance groups \(\Gamma _g\) of the functions \(T^s_g(2\tau )\) [cf. (8.6), Theorem 8.4].

As a first step toward this goal, observe that if \(D_{g}\ne 0\) then \(\eta _g(\tau )\) is an eta product of weight 2, meaning that \(\sum _{m>0} k_m=4\) for \(\pi _g=\prod _{m>0}m^{k_m}\). Indeed, \(\sum _{m>0}k_m\) is exactly the rank of \(\Lambda ^g\), and it was pointed out in the sentence following (9.13) that \(D_g\) vanishes when \(\Lambda ^g\) has rank larger than 4.

It follows that the third summand in the definition (9.19) of \(F_g(\tau )\) is a modular form of weight 2, and some level, for each *g*. So we may consider \(F'_g(\tau ):=F_g(\tau )-\frac{1}{2}D_{g}\eta _g(\tau )\) (the prime here does not denote differentiation).

*g*has fixed points in \(\mathfrak {a}\) by hypothesis, \(C_g=0\) [cf. (8.5)]. Thus we obtain

*M*is even, so we may focus on \(F_g''\).

Set \(f_g(\tau ):=\frac{1}{2} t_g(\tau )\Lambda _4(\tau )\). Then \(f_g(\tau )\) is a weakly holomorphic modular form of weight 2 and some level, since \(t_g(\tau )\) is a principal modulus for a genus zero group containing some \(\Gamma _0(N)\), according to (8.11) and Theorem 8.4. Precisely, for \(\Gamma _g\) the invariance group of \(t_g(\tau )\) (i.e., as in Theorem 8.4), the function \(f_g\) is a weakly holomorphic modular form of weight 2 for \(\Gamma _g\cap \Gamma _0(4)\).

*k*for \(\Gamma _0(M)\), for any even

*M*,

Modular forms of level four at cusps

\(\Lambda _4(\tau )\) | \(\Lambda _4(\tau +1/2)\) | |
---|---|---|

1 | \(1/2+O(q)\) | \(1/2+O(q)\) |

1 / 2 |
| \(-1/2+O(q)\) |

1 / 4 | \(-1/8+O\left( q^{1/4}\right) \) | \(O\left( q^{1/4}\right) \) |

- 1.
If \(\alpha \in \Gamma _g\infty \) and \(\Gamma _{-g}\infty \) then \(\alpha =1/4\mod \Gamma _0(4)\),

- 2.
If \(\alpha \in \Gamma _g\infty \) and \(\alpha \notin \Gamma _{-g}\infty \) then \(\alpha =1/2\mod \Gamma _0(4)\), and

- 3.
If \(\alpha \notin \Gamma _{g}\infty \) and \(\alpha \in \Gamma _{-g}\infty \) then \(\alpha =1\mod \Gamma _0(4)\).

Observe that the verification of the statements 1, 2, and 3 is generally quite easy. For example, if \(C_{-g}=0\) (which is the case for most conjugacy classes) then we necessarily have \(\Gamma _g=\Gamma _{-g}\), since (9.24) implies that \(t_g\) and \(t_{-g}\) coincide, up to an additive constant. Then the conditions 2 and 3 become vacuous, and we require only to check that if \(\gamma \in \Gamma _g\) and \(\gamma \infty =\frac{a}{c}\) for \(a,c\in \mathbb {Z}\) with \((a,c)=1\), then \(c=0\mod 4\). The remaining cases are handled similarly. \(\square \)

Together, Propositions 9.3 and 9.4 prove our main theorem.

###
**Theorem 9.5**

Let *X* be a projective complex K3 surface and let \(\sigma \in {\text {Stab}}^{\circ }(X)\). Then for \(g\in {\text {Aut}}_s({\text {D}}^\mathrm{b}(X),\sigma )\) the function \(\phi _g\) is a Jacobi form of weight 0, index 1, and some level.

*g*to be the identity in Theorem 9.5 produces a Jacobi form \(\phi _e\) of weight 0, index 1, and level 1; it must be the K3 elliptic genus, up to a constant. The constant can be determined by setting \(z=0\). Taking \(z=0\) in (3.16) and (3.17), and applying (9.20), we see that in fact it is exactly the K3 elliptic genus, but expressed in a rather non-standard way:

###
**Conjecture 9.6**

The twined elliptic genus attached to a supersymmetry-preserving automorphism \(g\in G_\Pi \) of the supersymmetric non-linear K3 sigma model determined by \(\Pi =P_X\oplus P_Z\) coincides with \(\phi _g\).

## 10 Umbral moonshine

In addition to twined K3 elliptic genera, a number of which coincide with weak Jacobi forms of Mathieu moonshine (cf. Table 8), graded traces on \({V^{s\natural }_\mathrm {tw}}\) defined by \({\text {U}}(1)\) elements corresponding to higher dimensional subspaces of \(\mathfrak {a}\) recover functions arising from umbral moonshine, as we will now explain.

*X*a lambency \(\ell \) and a meromorphic Jacobi form \(\psi ^X\) of weight 1, with index given by the Coxeter number of

*X*. For lambencies \(\ell \) occurring in [18], we recover \(Z^{(\ell )}\) from \(\psi ^X\) according to the rule

*d*-dimensional real vector space \(\Pi <\Lambda \otimes _\mathbb {Z}\mathbb {R}\subset \mathfrak {a}\) and let \(\{a^\pm _i\}\) be bases for isotropic subspaces \(\mathfrak {a}^\pm <\mathfrak {a}\) constituting a polarization \(\mathfrak {a}=\mathfrak {a}^-\oplus \mathfrak {a}^+\). Assume that \(\langle a_i^-,a_j^+\rangle =\delta _{i,j}\) and

*d*according to Lemma 6.1.

*g*restricts to the identity on \(\Pi \), then the action of \(\widehat{g}\) on \(V^{s\natural }_\mathrm {tw}\) commutes with that of \(J(0):=\jmath _{(0),\mathrm {tw}}\), and we may define

*g*in

*G*with \(g|_{\Pi }={\text {Id}}\). Assume that the \(a_i^\pm \) are eigenvectors for

*g*and write

*g*on \(a_i^\pm \). We may assume that \(\alpha _i=0\) for \(1\le i\le d\) since

*g*fixes the corresponding \(a^\pm _i\) by hypothesis. Set \(\nu _i:=\mathrm{e}^{\alpha _i {\mathbf {i}}}\) and define

*g*has a fixed point in its action on the orthogonal complement of \(\Pi \) in \(\Lambda \). In particular, \(D^{(\ell )}_{g}\) vanishes whenever the sublattice of \(\Lambda \) fixed by

*g*has rank larger than 2

*d*.

By a method directly similar to the proof of Proposition 9.2, we obtain the following explicit expression for \(\phi ^{(\ell )}_g\).

###
**Proposition 10.1**

*d*-dimensional subspace of \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\). If \(g\in G\) and \(g|_\Pi ={\text {Id}}\), then

*g*. With this in mind, define

*g*as in Proposition 10.1.

###
**Proposition 10.2**

Let \(\ell \) and *g* be as in Proposition 10.1. Then \(F_{0,g}\) is constant, and there exists a positive integer *N* such that \(F_{2j,g}\in M_{2j}(\Gamma _0(N))\) for \(0<j<\ell \).

###
*Proof*

The proof is very similar to that of Proposition 9.4. That is to say, it is ultimately a case-by-case check, but we use the results of [43] to replace the explicit calculation of modular forms with simple checks on properties of the invariance groups \(\Gamma _g\) of the functions \(T^s_g(2\tau )\) [cf. (8.6), Theorem 8.4].

*g*has fixed points in its action on \(\mathfrak {a}\), by hypothesis, so \(C_g=0\). So (9.24) holds, according to Lemma 8.3. Comparing (9.24) to (10.10), we see that \(F_{0,g}(\tau )=2\chi _g\). In particular, \(F_{0,g}\) is constant, as required.

*j*for some \(\Gamma _0(N)\), since the invariance group of \(t_g\) contains some \(\Gamma _0(N)\). Also, \(F_{2j,g}'=T_{2j}(2)f_{2j,g}\), where \(T_k(2)\) denotes the second-order Hecke operator on modular forms of weight

*k*for \(\Gamma _0(M)\), for any even

*M*. [cf. (9.26)]. So \(F'_{2j,g}\) is a weakly holomorphic modular form of weight 2

*j*for some \(\Gamma _0(N)\), and it remains to verify that \(F'_{2j,g}\) has no poles at cusps.

*j*for \(\Gamma _0(4)\). Table 2 is the appropriate analogue of Table 1, presenting the asymptotic behavior of \(G_{2j}(\tau )\) and \(G_{2j}(\tau +1/2)\) at the three cusps of \(\Gamma _0(4)\).

Modular forms of level four at cusps

\(G_{2j}(\tau )\) | \(G_{2j}(\tau +1/2)\) | |
---|---|---|

1 | \(\left( (-2)^j-1\right) {12^{-j}}+O(q)\) | \({\left( (-2)^j-1\right) }{12^{-j}}+O(q)\) |

1 / 2 |
| \(\left( 1-(-2)^j\right) 12^{-j}+O(q)\) |

1 / 4 | \(\left( 1-(-2)^j\right) 48^{-j}+O\left( q^{1/4}\right) \) | \(O\left( q^{1/4}\right) \) |

- 1.
If \(\alpha \in \Gamma _g\infty \) and \(\Gamma _{-g}\infty \) then \(\alpha =1/4\mod \Gamma _0(4)\),

- 2.
If \(\alpha \in \Gamma _g\infty \) and \(\alpha \notin \Gamma _{-g}\infty \) then \(\alpha =1/2\mod \Gamma _0(4)\), and

- 3.
If \(\alpha \notin \Gamma _{g}\infty \) and \(\alpha \in \Gamma _{-g}\infty \) then \(\alpha =1\mod \Gamma _0(4)\).

###
**Proposition 10.3**

*g*be as in Proposition 10.1. Then we have

###
*Proof*

###
**Theorem 10.4**

Let \(d\in \{2,4,6,8,10,12\}\) and \(\ell =\frac{d}{2}+1\). Let \(\Pi \) be a 2*d*-dimensional subspace of \(\Lambda \otimes _\mathbb {Z}\mathbb {R}\). If \(g\in G\) and \(g|_\Pi ={\text {Id}}\), then \(\phi ^{(\ell )}_g\) is a weak Jacobi form of weight 0 and index \(\ell -1\), with some level depending on *g*.

###
*Proof*

Proposition 10.3 shows that \(\phi ^{(\ell )}_g\) is a homogeneous polynomial in \(\phi _{0,1}\) and \(\phi _{-2,1}\) of the form required by Proposition 3.1. Proposition 10.2 verifies that all the coefficients in this expression have the correct modular properties, except for \(-\frac{1}{2}D^{(\ell )}_g\eta _g(\tau )\), which appears as the coefficient of \(\phi _{-2,1}^{\ell -1}\). So the required result follows from Proposition 3.1, as soon as we verify that \(D^{(\ell )}_g\eta _g(\tau )\) is a modular form of weight \(d=2(\ell -1)\) for some \(\Gamma _0(N)\), but this follows from the definition of \(D^{(\ell )}_g\). For it is apparent from (10.6) that \(D^{(\ell )}_g\) can only be non-zero when the rank of \(\Lambda ^g\) is precisely \(2d=4(\ell -1)\). On the other hand, if the Frame shape of *g* takes the form \(\pi _g=\prod _{m>0}m^{k_m}\), then \({\text {rank}}(\Lambda ^g)=\sum _{m>0}k_m\) which is exactly \(\frac{1}{2}\) times the weight of \(\eta _g(\tau )=\prod _{m>0}\eta (m\tau )^{k_m}\). So \(\eta _g(\tau )\) has weight *d*, or \(D^{(\ell )}_g=0\). In either case, \(D^{(\ell )}_g\eta _g(\tau )\in M_{2(\ell -1)}(\Gamma _0(N))\) for some *N*. This completes the proof. \(\square \)

Taking *g* to be the identity in Theorem 9.5 produces a Jacobi form \(\phi ^{(\ell )}_e\) of weight 0, index \(\ell -1\), and level 1. This construction recovers the extremal weak Jacobi forms \(Z^{(\ell )}\) of [18] for \(\ell \in \{2,3,4,5,7\}\).

###
**Proposition 10.5**

If \(d\in \{2,4,6,8,12\}\) then \(\phi _\mathrm{e}^{(\ell )}=\frac{d}{2}Z^{(\ell )}\).

Explicit expressions for the \(\phi ^{(\ell )}_g\) are recorded in Tables 3, 4, 5, 6 and 7. Coincidences with the weight zero weak Jacobi forms of umbral moonshine are recorded in Tables 8, 9, 10, 11 and 12.

Note that \(\phi ^{(6)}_e\), corresponding to \(d=10\), does not correspond to a weak Jacobi form arising in [18]. However, \(\phi ^{(6)}_e\) maps naturally to the meromorphic Jacobi form \(\psi ^X\) for \(X=A_5^4D_4\) via the construction in §4.3 of [19]. Note that the \(\ell \) for which \(\phi ^{(\ell )}_e\) recovers the weight 0 Jacobi form of umbral moonshine correspond to pure *A*-type root systems *X*. It is natural to ask if some modification of our methods can recover the \(Z^{(\ell )}\) corresponding to the remaining pure *A*-type root systems (at \(\ell =9, 13, 25\)).

## 11 Sigma models

In this section, we describe an isomorphism of graded vector spaces relating \(V^{s\natural }\) and its canonically twisted module \(V^{s\natural }_\mathrm {tw}\) to the vector spaces underlying the NS-NS and R-R sectors of an explicitly constructed super conformal field theory arising from a particular, distinguished supersymmetric non-linear K3 sigma model. This model was constructed by Wendland in [107]. Its automorphism group is exceptionally large, as is demonstrated in [65].

In preparation for a description of the relevant sigma model, let \(\Gamma <\mathbb {C}^2\) be a lattice of rank 4 that spans \(\mathbb {C}^2\) (i.e. \(\Gamma \simeq \mathbb {Z}^4\) and \(\Gamma \otimes _\mathbb {Z}{\mathbb {R}}=\mathbb {C}^2\)). Then the quotient \(T=\mathbb {C}^2{/}\Gamma \) is a complex 2-torus. The *Kummer involution* of *T* is the automorphism induced by the map \(\kappa :x\mapsto -x\) on \(\mathbb {C}^2\). A minimal resolution \(X\rightarrow T/\langle \kappa \rangle \) of the quotient (there are 16 points of *T* fixed by \(\kappa \)) is a complex K3 surface and is projective exactly when *T* is.

*V*for \(\mathbb {C}^2\) regarded as a real vector space of dimension 4. Equip

*V*with the symmetric \(\mathbb {R}\)-bilinear form \(\langle \;,\;\rangle \), such that \(e_1=(1,0)\), \(e_2=(i,0)\), \(e_3=(0,1)\), and \(e_4=(0,i)\) form an orthonormal basis, and set

*V*.

*left-movers*and

*right-movers*, respectively. The complex structure on

*V*arising from the identification \(V=\mathbb {C}^2\) reflects the choice of B-field made in [65]. In the Ramond–Ramond sector, we have

At first glance, it now appears that a detailed investigation of the structure of \(\mathcal {H}_X\) will require a review of the construction of lattice vertex algebras and their twisted modules, but we will refrain from doing that here in favor of using an equivalent description in terms of Clifford modules.

*T*. Now the \(U_0\otimes U_0\)-module structure on \(\bigoplus _iU_i\otimes U_i\) naturally extends to a vertex operator algebra structure as is explained in detail in [54]. In fact, this vertex operator algebra is isomorphic to the lattice vertex algebra \(V_{L}\) for

*L*a copy of the \(E_8\) lattice (cf. Sect. 2), and the vertex operator algebra isomorphism \(V_L\simeq \bigoplus _i U_i\otimes U_i\) reflects the coincidence

*diagonal*Virasoro element

Observe that \(A(\mathfrak {e})\otimes V_L\) is precisely the super vertex operator algebra denoted \(_{\mathbb {C}}V_L^f\) in [44] (the symbols \(V_L^f\) denote a real form of \(_\mathbb {C}V_L^f\)) and used there to construct an \(N=1\) super vertex operator algebra whose automorphism group is the largest simple Conway group, \({ Co}_1={ Co}_0/\{ \pm {\text {Id}}\}\) [cf. (2.7)].

*L*to \(V_{L}\), but according to [54] we may realize such an automorphism explicitly in the \(U_0\otimes U_0\)-module description as \(1\otimes \theta \), where \(\theta \) denotes the parity involution on \(A(\mathfrak {e})\oplus A(\mathfrak {e})_\mathrm {tw}\), fixing \(U_0\) and \(U_{\omega }\), and negating \(U_1\) and \(U_{\bar{\omega }}\). Now we may replace \(V_{\Gamma +\gamma _i,\mathrm {tw}}\otimes V_{\Gamma +\gamma _i,\mathrm {tw}}\) with \(U_{i}\otimes U_{i+\omega }\) in the description of \(\mathcal {H}_{X}\), where \(\{0,1,\omega ,\bar{\omega }\}\) is equipped with the obvious 4-group structure. Comparing with [65], we see that the orbifolding symmetry \(\hat{\kappa }\), lifting the Kummer involution on

*T*, should act as \(\theta \otimes 1\otimes \theta \) on \(\mathcal {H}_{T}\) and as \(\theta \otimes \theta \otimes 1\) on its \(\hat{\kappa }\)-twisted module, and in this way we arrive at the isomorphisms

Observe that the right-hand side of (11.15) is precisely the \(U_0\otimes U_0\otimes U_0\)-module description given in (6.4.7) of [44] for the super vertex operator algebra denoted there by \(_\mathbb {C}V^{f\natural }\). A suitably chosen vector \(\tau \in U_{111}\) equips \(_\mathbb {C}V^{f\natural }\) with a representation of the Neveu–Schwarz super Lie algebra with central charge 12, and a main result of [44] is that the subgroup of \({\text {Aut}}(_\mathbb {C}V^{f\natural })\) composed of elements that fix \(\tau \) is exactly \({ Co}_1\).

*triality*, which, at the level of lattices, is the fact that \(\Gamma ^*\) [cf. (11.3)] admits an automorphism of order 3 that stabilizes the type \(D_4\) sublattice \(\Gamma \) [cf. (11.1)], and cyclically permutes its three non-trivial cosets \(\Gamma +\gamma _i\). At the level of vertex operator algebras and their modules, this translates to the existence of an automorphism \(\sigma \) of \(U_0\), and invertible maps \(\sigma :U_i\mapsto U_{\omega i}\) for \(i\in \{1,\omega ,\bar{\omega }\}\), such that

Data for the computation of \(\phi _g=\phi _g^{(2)}\)

\({ Co}_0\) | \({ Co}_1\) | \(\pi _g\) | \(\pi _{-g}\) | \(C_{-g}\) | \(D_{g}\) | \(\Gamma _{g}\) | \(\Gamma _{-g}\) |
---|---|---|---|---|---|---|---|

1A | 1A | \(1^{24}\) | \(\frac{2^{24}}{1^{24}}\) | 4096 | 0 | \(2-\) | \(4+\) |

2B | 2A | \(1^8 2^8\) | \(\frac{2^{16}}{1^8}\) | 0 | 0 | \(4-\) | \(4-\) |

2C | 2A | \(\frac{2^{16}}{1^8}\) | \(1^8 2^8\) | 0 | 0 | \(4-\) | \(4-\) |

2D | 2C | \(2^{12}\) | \(2^{12}\) | 0 | 0 | \(4|2-\) | \(4|2-\) |

3B | 3B | \(1^6 3^6\) | \(\frac{2^6 6^6}{1^6 3^6}\) | 64 | 0 | \(6+3\) | \(12+\) |

3C | 3C | \(\frac{3^9}{1^3}\) | \(\frac{1^3 6^9}{2^3 3^9}\) | \(-8\) | 0 | \(6-\) | \(12+4\) |

3D | 3D | \(3^8\) | \(\frac{6^8}{3^8}\) | 16 | 0 | 6|3 | \(12|3+\) |

4B | 4A | \(\frac{1^8 4^8}{2^8}\) | \(\frac{4^8}{1^8}\) | 256 | 0 | \((8+)^{\bigtriangleup \tfrac{1}{2}}\) | \(8+\) |

4D | 4B | \(\frac{4^8}{2^4}\) | \(\frac{4^8}{2^4}\) | 0 | \(\pm 64\) | \(8-\) | \(8-\) |

4E | 4C | \(1^4 2^2 4^4\) | \(\frac{2^6 4^4}{1^4}\) | 0 | 0 | \(8-\) | \(8-\) |

4F | 4C | \(\frac{2^6 4^4}{1^4}\) | \(1^4 2^2 4^4\) | 0 | 0 | \(8-\) | \(8-\) |

4G | 4D | \(2^4 4^4\) | \(2^4 4^4\) | 0 | 0 | \(8|2-\) | \(8|2-\) |

4H | 4F | \(4^6\) | \(4^6\) | 0 | 0 | \(8|4-\) | \(8|4-\) |

5B | 5B | \(1^4 5^4\) | \(\frac{2^4 10^4}{1^4 5^4}\) | 16 | 0 | \(10+5\) | \(20+\) |

5C | 5C | \(\frac{5^5}{1^1}\) | \(\frac{1^1 10^5}{2^1 5^5}\) | \(-4\) | \(\pm 25\sqrt{5}\) | \(10-\) | \(20+4\) |

6G | 6C | \(\frac{2^5 3^4 6^1 }{1^4}\) | \(\frac{1^4 2^1 6^5}{3^4}\) | 0 | 0 | \(12+3\bigtriangleup \tfrac{1}{2}\) | \(12+3\bigtriangleup \tfrac{1}{2}\) |

6H | 6C | \(\frac{1^4 2^1 6^5}{3^4}\) | \(\frac{2^5 3^4 6^1}{1^4}\) | 0 | 0 | \(12+3\bigtriangleup \tfrac{1}{2}\) | \(12+3\bigtriangleup \tfrac{1}{2}\) |

6I | 6D | \(\frac{1^5 3^1 6^4}{2^4}\) | \(\frac{2^1 6^5}{1^5 3^1}\) | 72 | 0 | \((12+12)^{\bigtriangleup \tfrac{1}{2}}\) | \(12+12\) |

6K | 6E | \(1^2 2^2 3^2 6^2\) | \(\frac{2^4 6^4}{1^2 3^2}\) | 0 | 0 | \(12+3\) | \(12+3\) |

6L | 6E | \(\frac{2^4 6^4}{1^2 3^2}\) | \(1^2 2^2 3^2 6^2\) | 0 | \(\pm 48\) | \(12+3\) | \(12+3\) |

6M | 6F | \(\frac{3^3 6^3}{1^1 2^1}\) | \(\frac{1^1 6^6}{2^2 3^3}\) | 0 | \(\pm 54\) | \(12-\) | \(12-\) |

6O | 6G | \(2^3 6^3\) | \(2^3 6^3\) | 0 | 0 | \(12\vert 2+3\bigtriangleup \tfrac{1}{2}\) | \(12\vert 2+3\bigtriangleup \tfrac{1}{2}\) |

6P | 6I | \(6^4\) | \(6^4\) | 0 | \(\pm 36\) | \(12|6-\) | \(12|6-\) |

7B | 7B | \(1^3 7^3\) | \(\frac{2^3 14^3}{1^3 7^3}\) | 8 | 0 | \(14+7\) | \(28+\) |

8C | 8B | \(\frac{2^4 8^4}{4^4}\) | \(\frac{2^4 8^4}{4^4}\) | 0 | \(\pm 16\) | \((16\vert 2+)^{\bigtriangleup \tfrac{1}{4}}\) | \((16\vert 2+)^{\bigtriangleup \tfrac{1}{4}}\) |

8D | 8C | \(\frac{1^4 8^4}{2^2 4^2}\) | \(\frac{2^2 8^4}{1^4 4^2}\) | 32 | \(\pm 8\) | \((16+)^{\bigtriangleup \tfrac{1}{2}}\) | \(16+\) |

8G | 8E | \(1^2 2^1 4^1 8^2\) | \(\frac{2^3 4^1 8^2}{1^2}\) | 0 | 0 | \(16-\) | \(16-\) |

8H | 8E | \(\frac{2^34^1 8^2}{1^2}\) | \(1^2 2^1 4^1 8^2\) | 0 | \(\pm 32\sqrt{2}\) | \(16-\) | \(16-\) |

8I | 8F | \(4^2 8^2\) | \(4^2 8^2\) | 0 | \(\pm 16\) | \(16\vert 4-\) | \(16\vert 4-\) |

9C | 9C | \(\frac{1^3 9^3}{3^2}\) | \(\frac{2^3 3^2 18^3}{1^3 6^2 9^3}\) | 4 | \(\pm 9\) | \(18+9\) | \(36+\) |

10F | 10D | \(\frac{2^3 5^2 10^1 }{1^2}\) | \(\frac{1^2 2^1 10^3}{5^2}\) | 0 | \(\pm 20\sqrt{5}\) | \(20+5\bigtriangleup \tfrac{1}{2}\) | \(20+5\bigtriangleup \tfrac{1}{2}\) |

10G | 10D | \(\frac{1^2 2^1 10^3}{5^2}\) | \(\frac{2^3 5^2 10^1 }{1^2}\) | 0 | \(\pm 4\sqrt{5}\) | \(20+5\bigtriangleup \tfrac{1}{2}\) | \(20+5\bigtriangleup \tfrac{1}{2}\) |

10H | 10E | \(\frac{1^3 5^1 10^2}{2^2}\) | \(\frac{2^1 10^3}{1^3 5^1}\) | 20 | \(\pm 5\sqrt{5}\) | \((20+20)^{\bigtriangleup \tfrac{1}{2}}\) | \(20+20\) |

10J | 10F | \(2^2 10^2\) | \(2^2 10^2\) | 0 | \(\pm 20\) | \(20\vert 2+5\) | \(20\vert 2+5\) |

11A | 11A | \(1^2 11^2\) | \(\frac{2^2 22^2}{1^2 11^2}\) | 4 | \(\pm 11\) | \(22+11\) | \(44+\) |

12I | 12E | \(\frac{1^2 3^2 4^2 12^2}{2^2 6^2}\) | \(\frac{4^2 12^2}{1^2 3^2}\) | 16 | \(\pm 12\) | \((24+)^{\bigtriangleup \tfrac{1}{2}}\) | \(24+\) |

12L | 12H | \(\frac{1^1 2^2 3^1 12^2}{4^2}\) | \(\frac{2^3 6^1 12^2}{1^1 3^1 4^2}\) | 0 | \(\pm 6\sqrt{3}\) | \((24\vert 2+12)^{\bigtriangleup \tfrac{1}{4}}\) | \((24\vert 2+12)^{\bigtriangleup \tfrac{1}{4}}\) |

12N | 12I | \(\frac{2^2 3^2 4^1 12^1}{1^2}\) | \(\frac{1^2 4^1 6^2 12^1}{3^2}\) | 0 | \(\pm 24\sqrt{3}\) | \(24+3\bigtriangleup \tfrac{1}{2}\) | \(24+3\bigtriangleup \tfrac{1}{2}\) |

12O | 12I | \(\frac{1^2 4^1 6^2 12^1}{3^2}\) | \(\frac{2^2 3^2 4^1 12^1}{1^2}\) | 0 | \(\pm 8\sqrt{3}\) | \(24+3\bigtriangleup \tfrac{1}{2}\) | \(24+3\bigtriangleup \tfrac{1}{2}\) |

12P | 12J | \(2^1 4^1 6^1 12^1\) | \(2^1 4^1 6^1 12^1\) | 0 | \(\pm 24\) | \(24\vert 2+3\) | \(24\vert 2+3\) |

14C | 14B | \(1^1 2^1 7^1 14^1\) | \(\frac{2^2 14^2}{1^1 7^1}\) | 0 | \(\pm 14\) | \(28+7\) | \(28+7\) |

15D | 15D | \(1^1 3^1 5^1 15^1\) | \(\frac{2^1 6^1 10^1 30^1}{1^1 3^1 5^1 15^1}\) | 4 | \(\pm 15\) | \(30+3,5,15\) | \(60+\) |

###
**Proposition 11.1**

The distinguished super vertex operator algebra \(V^{s\natural }\) is isomorphic to \(\mathcal {H}_{X,\text {NS-NS}}\) as a Virasoro module, when the latter is equipped with the diagonal Virasoro element, \(\omega ^{\mathcal {D}}\). Similarly, \(V^{s\natural }_\mathrm {tw}\) is isomorphic as a Virasoro module to \(\mathcal {H}_{X,\text {R-R}}\).

## Notes

## Declarations

### Acknowledgments

We are grateful to Miranda Cheng, Thomas Creutzig, Xi Dong, Tohru Eguchi, Igor Frenkel, Matthias Gaberdiel, Terry Gannon, Sarah Harrison, Jeff Harvey, Yi-Zhi Huang, Gerald Hoehn, Shamit Kachru, Ching Hung Lam, Atsushi Matsuo, Nils Scheithauer, Yuji Tachikawa, Roberto Volpato, Katrin Wendland, and Timm Wrase for helpful discussions on related topics. We are particularly grateful to Miranda Cheng, Jeff Harvey, Shamit Kachru, Katrin Wendland and the anonymous referees, for comments on an earlier draft. The first author gratefully acknowledges the support from the US National Science Foundation (DMS 1203162) and from the Simons Foundation (#316779). The first author thanks the University of Tokyo for hospitality during the completion of this project.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Aoki, H., Ibukiyama, T.: Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Int. J. Math. 16(3), 249–279 (2005)Google Scholar
- Aspinwall, P.S.: $$K3$$ K 3 surfaces and string duality. Fields. strings and duality (Boulder, CO, 1996), pp. 421–540. World Sci. Publ, River Edge (1997)Google Scholar
- Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. In: Progress in string theory, pp. 1–152. World Sci. Publ., Hackensack (2005)Google Scholar
- Aspinwall, P.S., Morrison, D.R.: String theory on $$K3$$ K 3 surfaces. In: Mirror symmetry, II, vol. 1 of AMS/IP Stud. Adv. Math., pp. 703–716. American Mathematical Society, Providence (1997)Google Scholar
- Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2nd edn. Springer, Berlin (2004)Google Scholar
- Bartocci, C., Bruzzo, U., Ruipérez, D.H.: Fourier-Mukai and Nahm transforms in geometry and mathematical physics. Progress in Mathematics, vol. 276. Birkhäuser Boston Inc, Boston (2009)Google Scholar
- Beauville, A., et al.: Géométrie des surfaces $$K3$$ K 3 : modules et périodes. Société Mathématique de France, Paris. Papers from the seminar held in Palaiseau. Astérisque No. 126(1985)Google Scholar
- Benjamin, N., Harrison, S.M., Kachru, S., Paquette, N.M., Whalen, D.: On the elliptic genera of manifolds of Spin(7) holonomy. ArXiv e-prints (2014)Google Scholar
- Borcherds, R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA, 83(10), 3068–3071 (1986)Google Scholar
- Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992)Google Scholar
- Bridgeland, T.: Stability conditions on $$K3$$ K 3 surfaces. Duke Math. J. 141(2), 241–291 (2008)Google Scholar
- Bridgeland, T.: Spaces of stability conditions. In: Algebraic geometry—Seattle 2005. Part 1, vol. 80 of Proc. Sympos. Pure Math., pp. 1–21. American Mathematical Society, Providence (2009)Google Scholar
- Bringmann, K., Duncan, J., Rolen, L.: Maass-Jacobi Poincaré series and Mathieu Moonshine. ArXiv e-prints (2014)Google Scholar
- Bringmann, K., Ono, K.: The $$f(q)$$ f ( q ) mock theta function conjecture and partition ranks. Invent. Math. 165(2), 243–266 (2006)Google Scholar
- Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)Google Scholar
- Cheng, M.C.N., Dong, X., Duncan, J.F.R., Harrison, S., Kachru, S., Wrase, T.: Mock Modular Mathieu Moonshine Modules. Math. Sci.
**2**, 13 (2015)View ArticleMathSciNetGoogle Scholar - Cheng, M.C.N., Duncan, J.F.R.: Meromorphic Jacobi forms of half-integral index and Umbral moonshine modules (2014)Google Scholar
- Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral moonshine. Commun. Number Theory Phys. 8(2) (2014)Google Scholar
- Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral moonshine and the Niemeier lattices. Res. Math. Sci.
**1**(3) (2014). arXiv:1307.5793v1 [math.RT] - Cheng, M.C.N., Harrison, S.: Umbral moonshine and K3 surfaces. ArXiv e-prints (2014)Google Scholar
- Cheng, M.C.N., Harrison, S.M., Kachru, S., Whalen, D.: Exceptional algebra and Sporadic groups at c = 12. ArXiv e-prints (2015)Google Scholar
- Cheng, M.C.N.: $$K3$$ K 3 . Commun. Number Theory Phys. 4(4), 623–657 (2010)Google Scholar
- Cheng, M.C.N., Duncan, J.F.R:.The largest Mathieu group and (mock) automorphic forms. In: String-Math 2011, vol. 85 of Proc. Symp. Pure Math., pp. 53–82. American Mathematical Society, Providence (2012)Google Scholar
- Cheng, M.C.N., Duncan, J.F.R.: On Rademacher sums, the largest Mathieu group, and the holographic modularity of moonshine. Commun. Number Theory Phys. 6(3) (2012)Google Scholar
- Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Weight one Jacobi forms and Umbral moonshine (2015) (in preparation)Google Scholar
- Conway, J.H.: A perfect group of order $$8,315,553,613,086,720,000$$ 8 , 315 , 553 , 613 , 086 , 720 , 000 and the sporadic simple groups. Proc. Nat. Acad. Sci. USA 61, 398–400 (1968)Google Scholar
- Conway, J.H.: A characterisation of Leech’s lattice. Invent. Math.
**7**, 137–142 (1969)View ArticleMathSciNetMATHGoogle Scholar - Conway, J.H.: A group of order \(8,315,553,613,086,720,000\). Bull. Lond. Math. Soc.
**1**, 79–88 (1969)View ArticleMathSciNetMATHGoogle Scholar - Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc.
**11**(3), 308–339 (1979)View ArticleMathSciNetMATHGoogle Scholar - Conway, J.H., Sloane, N.J.A.: Lorentzian forms for the Leech lattice. Bull. Am. Math. Soc. (N.S.), 6(2), 215–217 (1982)Google Scholar
- Conway, J.H., Sloane, N.J.A: Sphere packings, lattices and groups, vol. 290 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, New York, third edition (1999) (wth additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov)Google Scholar
- Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray. Clarendon Press, Oxford (1985)Google Scholar
- Cummins, C.J., Pauli, S.: Congruence subgroups of PSL $$(2,{\mathbb{Z}})$$ ( 2 , Z ) of genus less than or equal to 24. Exp. Math. 12(2), 243–255 (2003)Google Scholar
- Dabholkar, A., Murthy, S., Zagier, D.: Wall crossing, and mock modular forms, Quantum Black Holes (2012)Google Scholar
- Dong, C., Li, H., Mason, G.: Modular invariance of trace functions in orbifold theory and generalized Moonshine. Commun. Math. Phy. 214, 1–56 (2000)Google Scholar
- Dong, C., Liu, K., Ma. X.: Elliptic genus and vertex operator algebras. Pure Appl. Math. Q., 1(4, part 3), 791–815 (2005)Google Scholar
- Dong, C., Mason, G.: Nonabelian orbifolds and the boson-fermion correspondence. Comm. Math. Phys. 163(3), 523–559 (1994)Google Scholar
- Dong, C., Zhao, Z.: Modularity in orbifold theory for vertex operator superalgebras. Comm. Math. Phys. 260(1), 227–256 (2005)Google Scholar
- Douglas, M.R.: Dirichlet branes, homological mirror symmetry, and stability. In: Proceedings of the International Congress of Mathematicians, vol. III (Beijing, 2002), pp. 395–408. Higher Ed. Press, Beijing (2002)Google Scholar
- Duncan, J.F.R., Griffin, M.J., Ono, K.: Moonshine. Res. Math. Sci.
**2**, 11 (2015)View ArticleMathSciNetGoogle Scholar - Duncan, J.F.R., Griffin, M.J., Ono, K.: A proof of the umbral moonshine conjecture. Res. Math. Sci.
**2**, 26 (2015)View ArticleGoogle Scholar - Duncan, J.F.R, Harvey, J.A.: The Umbral moonshine module for the unique unimodular Niemeier root system. ArXiv e-prints (2014)Google Scholar
- Duncan, J.F.R., Mack-Crane, S.: The Moonshine module for Conway’s group. ArXiv e-prints, to appear in Forum of Math., Sigma, 3 (2014)Google Scholar
- Duncan, J.F.: Super-moonshine for Conway’s largest sporadic group. Duke Math. J. 139(2), 255–315 (2007)Google Scholar
- Duncan, J.F.: Moonshine for Rudvalis’s sporadic group I (2008a)Google Scholar
- Duncan, J.F.: Moonshine for Rudvalis’s sporadic group II (2008b)Google Scholar
- Duncan, J.F.R., Frenkel, I.B.: Rademacher sums, moonshine and gravity. Commun. Number Theory Phys. 5(4), 1–128 (2011)Google Scholar
- Duncan, J.F.R., O’Desky, A.: Meromorphic Jacobi forms, Umbral moonshine, and super vertex algebras (2014)Google Scholar
- Durfee, A.H.: Fifteen characterizations of rational double points and simple critical points. Enseign. Math. 25(1–2), 131–163 (1979)Google Scholar
- Eguchi, T., Hikami, K.: Note on twisted elliptic genus of K3 surface. Phys. Lett. B 694, 446–455 (2011)Google Scholar
- Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 surface and the Mathieu group $$M_{24}$$ M 24 . Exper. Math. 20, 91–96 (2011)Google Scholar
- Eichler, M., Zagier, D.: The theory of Jacobi forms. Birkhäuser (1985)Google Scholar
- Feingold, A.J., Frenkel, I.B.: A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2. J. Math. Ann. 263, 87–144 (1983)Google Scholar
- Feingold, A.J., Frenkel, I.B., Ries, J.F.X.: Spinor construction of vertex operator algebras, triality, and $$E^{(1)}_8$$ E 8 ( 1 ) , vol. 121 of contemporary mathematics. American Mathematical Society, Providence (1991)Google Scholar
- Ferenbaugh, C.R.: The genus-zero problem for $$n\vert h$$ n | h -type groups. Duke Math. J. 72(1), 31–63 (1993)Google Scholar
- Ford, D., McKay, J.: Representations and Coxeter graphs. In: The geometric vein, pp. 549–554. Springer, New York (1981)Google Scholar
- Ford, D., McKay, J., Norton, S.: More on replicable functions. Comm. Algebra 22(13), 5175–5193 (1994)Google Scholar
- Frasch, H.: Die Erzeugenden der Hauptkongruenzgruppen für Primzahlstufen. Math. Ann. 108(1), 229–252 (1933)Google Scholar
- Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, vol. 88 of Mathematical surveys and monographs, 2nd edn. American Mathematical Society, Providence (2004)Google Scholar
- Frenkel, E., Szczesny, M.: Twisted modules over vertex algebras on algebraic curves. Adv. Math. 187(1), 195–227 (2004)Google Scholar
- Frenkel, I.B.: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Anal.
**44**(3), 259–327 (1981)View ArticleMathSciNetMATHGoogle Scholar - Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function $$J$$ J as character. Proc. Nat. Acad. Sci. USA, 81(10), 3256–3260 (1984)Google Scholar
- Frenkel, I.B., Lepowsky, J., Meurman, A.: A moonshine module for the Monster. In: Vertex operators in mathematics and physics (Berkeley, Calif., 1983), vol. 3 of Math. Sci. Res. Inst. Publ., pp. 231–273. Springer, New York (1985)Google Scholar
- Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic Press Inc., Boston (1988)Google Scholar
- Gaberdiel, M.R., Taormina, A., Volpato, R., Wendland, K.: A K3 sigma model with \({\rm Z}_2^8\): \({\rm M}_20\) symmetry. JHEP
**1402**, 022 (2014)View ArticleGoogle Scholar - Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu moonshine in the elliptic genus of K3. JHEP 1010, 062 (2010)Google Scholar
- Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu twining characters for K3. JHEP 1009, 058 (2010)Google Scholar
- Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Symmetries of K3 sigma models. Commun. Number Theory Phys. 6(1), 1–50 (2012)Google Scholar
- Gaberdiel, M.R., Volpato, R.: Mathieu Moonshine and Orbifold K3s (2012)Google Scholar
- Gannon, T.: Much ado about Mathieu (2012)Google Scholar
- Higman, D.G., Sims, C.C.: A simple group of order $$44,352,000$$ 44 , 352 , 000 . Math. Z. 105, 110–113 (1968)Google Scholar
- Higman, G.: On the simple group of D. G. Higman and C. C. Sims. Illinois J. Math. 13, 74–80 (1969)Google Scholar
- Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)Google Scholar
- Huybrechts, D.: Introduction to stability conditions. ArXiv e-prints (2011)Google Scholar
- Huybrechts, D.: On derived categories of K3 surfaces, symplectic automorphisms and the Conway group. ArXiv e-prints (2013)Google Scholar
- Huybrechts, D., Macrì, E., Stellari, P.: Derived equivalences of $$K3$$ K 3 surfaces and orientation. Duke Math. J. 149(3), 461–507 (2009)Google Scholar
- Kac, V.: Vertex algebras for beginners, vol. 10 of University Lecture Series, 2nd edn. American Mathematical Society, Providence (1998)Google Scholar
- Kac, V.G.: Infinite-dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)Google Scholar
- Kac, V.G., Peterson, D.H.: Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. Math. 53(2), 125–264 (1984)Google Scholar
- Knapp, A.W.: Elliptic curves. Mathematical Notes, vol. 40. Princeton University Press, Princeton (1992)Google Scholar
- Kondo, T.: The automorphism group of Leech lattice and elliptic modular functions. J. Math. Soc. Japan 37(2), 337–362 (1985)Google Scholar
- Krauel, M., Mason, G.: Vertex operator algebras and weak Jacobi forms. Int. J. Math. 23(6), 1250024 (2012)Google Scholar
- Landweber, P.S. (ed.): Elliptic curves and modular forms in algebraic topology. Lecture Notes in Mathematics, vol. 1326. Springer, Berlin (1988)Google Scholar
- Leech, J.: Some sphere packings in higher space. Can. J. Math. 16, 657–682 (1964)Google Scholar
- Leech, J.: Notes on sphere packings. Can. J. Math. 19, 251–267 (1967)Google Scholar
- Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Progress in Mathematics, vol. 227. Birkhäuser Boston Inc., Boston (2004)Google Scholar
- Li, H.S.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules. In: Moonshine, the Monster, and related topics (South Hadley, MA, 1994), vol. 193 of Contemp. Math., pp. 203–236. American Mathematical Society, Providence (1996)Google Scholar
- McKay, J.: Graphs, singularities, and finite groups. In: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), vol. 37 of Proc. Sympos. Pure Math., pp. 183–186. American Mathematical Society, Providence (1980)Google Scholar
- Miyamoto, M.: A modular invariance on the theta functions defined on vertex operator algebras. Duke Math. J. 101(2), 221–236 (2000)Google Scholar
- Miyamoto, M.: Modular invariance of vertex operator algebras satisfying $$C_2$$ C 2 -cofiniteness. Duke Math. J. 122(1), 51–91 (2004)Google Scholar
- Nahm, W., Wendland, K.: A Hiker’s Guide to $$K3$$ K 3 . Commun. Math. Phys. 216 (2001)Google Scholar
- Niemeier, H.V.: Definite quadratische Formen der Dimension $$24$$ 24 . J. Number Theory 5, 142–178 (1973)Google Scholar
- Ono, K., Rolen, L., Trebat-Leder, S.: Classical and Umbral moonshine: Connections and $$p$$ p -adic Properties. ArXiv e-prints (2014)Google Scholar
- Ono, K.: Unearthing the visions of a master: harmonic Maass forms and number theory. In: Current developments in mathematics, 2008, pp. 347–454. Int. Press, Somerville (2009)Google Scholar
- Queen, L.: Modular functions arising from some finite groups. Math. Comp. 37(156), 547–580 (1981)Google Scholar
- Ramanujan, S.: The lost notebook and other unpublished papers. Springer, Berlin (1988) (with an introduction by George E. Andrews)Google Scholar
- Ramanujan, S.: Collected papers of Srinivasa Ramanujan. In: Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M. (eds.) Third printing of the 1927 original, with a new preface and commentary by Bruce C, Berndt. AMS Chelsea Publishing, Providence (2000)Google Scholar
- Serre, J.P.: A course in arithmetic. Springer, New York (1973) (translated from the French. Graduate Texts in Mathematics, No. 7)Google Scholar
- Shimura, G.: Introduction to the arithmetic theory of automorphic functions, vol. 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton (1994) (Reprint of the 1971 original, Kanô Memorial Lectures, 1)Google Scholar
- Sims, C.C.: On the isomorphism of two groups of order $$44,352,000$$ 44 , 352 , 000 . In: Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), pp. 101–108. Benjamin, New York (1969)Google Scholar
- Thompson, J.G.: Finite groups and modular functions. Bull. Lond. Math. Soc. 11(3), 347–351 (1979)Google Scholar
- Thompson, J.G.: Some numerology between the Fischer-Griess Monster and the elliptic modular function. Bull. Lond. Math. Soc.
**11**(3), 352–353 (1979)View ArticleMATHGoogle Scholar - Van Ekeren, J.: Modular invariance for twisted modules over a vertex operator superalgebra. Commun. Math. Phys. 322(2), 333–371 (2013)Google Scholar
- Van Ekeren, J.: Vertex operator superalgebras and odd trace functions. In: Advances in Lie superalgebras, vol. 7 of Springer INdAM Ser., pp. 223–234. Springer, Cham (2014)Google Scholar
- Venkov, B.B.: On the classification of integral even unimodular $$24$$ 24 -dimensional quadratic forms. Trudy Mat. Inst. Steklov. 148, 65–76 (1978) (Algebra, number theory and their applications)Google Scholar
- Volpato, R.: On symmetries of N = (4, 4) sigma models on T $$^4$$ 4 . ArXiv e-prints (2014)Google Scholar
- Wendland, K.: Orbifold Constructions of $$K3$$ K 3 : a Link between Conformal Field Theory and Geometry. “Orbifolds in Mathematics and Physics”. Contemp. Math. 310, 333–358 (2002)Google Scholar
- Witten, E.: Elliptic genera and quantum field theory. Comm. Math. Phys. 109(4), 525–536 (1987)Google Scholar
- Zagier, D.: Elliptic modular forms and their applications. The 1-2-3 of modular forms. Universitext, pp. 1–103. Springer, Berlin (2008)Google Scholar
- Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann). Astérisque, (326):Exp. No. 986, vii–viii, 143–164, 2009. Séminaire Bourbaki. vol. 2007/2008 (2010)Google Scholar
- Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)Google Scholar
- Zwegers, S.: Mock Theta Functions. Ph.D. thesis, Utrecht University (2002)Google Scholar