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K3 string theory, lattices and moonshine
 Miranda C. N. Cheng^{1, 2},
 Sarah M. Harrison^{3},
 Roberto Volpato^{4, 5, 6} and
 Max Zimet^{5}Email authorView ORCID ID profile
https://doi.org/10.1007/s4068701801504
© The Author(s) 2018
 Received: 4 April 2017
 Accepted: 9 July 2018
 Published: 24 July 2018
Abstract
In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on \(K3 \times {\mathbb {R}}^6\), where Niemeier lattices play an important role. This extends earlier results by including points in the moduli space with enhanced gauge symmetries in spacetime, or, equivalently, where the worldsheet CFT becomes singular. After classifying the symmetries as abstract groups, we study how they act on the BPS states of the theory. In particular, we classify the conjugacy classes in the Tduality group \(O^+(\Gamma ^{4,20})\) which represent physically distinct symmetries. Subsequently, we make two conjectures regarding the connection between the corresponding twining genera of K3 CFTs and Conway and umbral moonshine, building upon earlier work on the relation between moonshine and the K3 elliptic genus.
1 Introduction
In this paper, we study discrete symmetry groups of K3 string theory and their action on the BPS spectrum. K3 surfaces play an important role in various aspects of mathematics and string theory. For instance, type II string compactifications on \(K3\times ~T^d \times ~{\mathbb {R}}^{5d,1}\) preserve 16 supersymmetries, leading to various exact results regarding the spectrum of BPS states from both the spacetime and worldsheet points of view. In addition, they provide some of the first instances of both holographic duality and a microscopic description of black hole entropy. Geometrically, the Torelli theorem allows for an exact description of the geometric moduli space and makes it possible to analyze the discrete groups of symplectomorphisms in terms of lattices. In particular, there is an intriguing connection between K3 symmetries and sporadic groups which constitutes the first topic of the current work.
Recall that the sporadic groups are the 26 finite simple groups that do not belong to any of the infinite families of finite simple groups. Their exceptional character raises the following questions: Why do they exist? What geometrical and physical objects do they naturally act on? This is one of the reasons why the discovery of (monstrous) moonshine—relating the representation theory of the largest sporadic simple group and a set of canonical modular functions attached to a chiral 2d CFT—is such a fascinating and important chapter in the study of sporadic groups. On the other hand, the relation of other sporadic groups to the ubiquitous K3 surface is a surprising result that provides another hint about their true raison d’être. In this work, we will relate two properties of sporadic groups: moonshine and K3 symmetries.
The connection between K3 surfaces and sporadic groups first manifested itself in a celebrated theorem by Mukai [73], which was further elucidated by Kondo [69]. Mukai’s theorem established a close relation between the Mathieu group \(M_{23}\), one of the 26 sporadic groups, and the symmetries of K3 surfaces, in terms of a bijection between (isomorphism classes of) \(M_{23}\) subgroups with at least five orbits and (isomorphism classes of) finite groups of K3 symplectomorphisms. A generalization of this classical result to “stringy K3 geometry” was initiated by Gaberdiel, Hohenegger and Volpato in [54], using lattice techniques in a method closely following Kondo’s proof of the Mukai theorem. More precisely, the symmetry groups of any nonlinear sigma model (NLSM) on K3, corresponding to any point in the moduli space (2.2) excepting loci corresponding to singular NLSMs, have been classified in [54]. From the spacetime (Dbranes) point of view, the results of [54] can be viewed as classifying symplectic autoequivalences (symmetries) of derived categories on K3 surfaces [66]. See also [15] for related discussion on symmetries of appropriately defined moduli spaces relevant for curve counting on K3. The embedding of relevant sublattices of the K3 cohomology lattice into the Leech lattice plays an important role in the analysis, and as a result the classification is phrased in terms of subgroups of the automorphism group \(\textit{Co}_0\) (“Conway zero”) of Leech lattice. Recall that there are 24 equivalence classes of 24dimensional negativedefinite even unimodular lattices, called the 24 Niemeier lattices.^{1} All but one of them have root systems of rank 24; these are generated by the lattice vectors of length squared two. The only exception is the Leech lattice, which has no root vectors.
The first part of the results of the present paper, consisting in a corollary (Corollary 4) of two mathematical theorems (Theorems 1 and 2), extends this classification to theories corresponding to singular loci in the moduli space of K3 NLSMs. It is necessary to make use of Niemeier lattices other than the Leech lattice (conjecturally all 24 of them, in analogy with [78]) in order to generalize the analysis to include these singular loci. Despite the fact that the type IIA worldsheet theory behaves badly along these loci [5], the full type IIA string theory is not only completely well defined but also possesses special physical relevance in connection with nonAbelian gauge symmetries. Recall that the spacetime gauge group is enhanced from \(U(1)^{24}\) to some nonabelian group at these loci, and the ADEtype gauge group is given by the ADEtype singularity of the K3 surface [5, 95]. The existence of such loci with enhanced gauge symmetries in the moduli space, though not immediately manifest from the worldsheet analysis in type IIA, is clear from the point of view of the dual heterotic \(T^4\) compactification. In this work, we are interested in finite group symmetries which preserve the \(\mathcal{N}=(1,1)\) spacetime supersymmetry from the point of view of type IIA compactifications.
Apart from these physical considerations, another important motivation to understand the discrete symmetries of general type IIA compactifications on K3 surfaces is the following. The K3 surface–sporadic group connection has recently entered the spotlight due to the discovery of new moonshine phenomena, initiated by an observation of Eguchi, Ooguri and Tachikawa (EOT) [46]. The K3 elliptic genus (3.1) is a function which counts BPS states of K3 NLSMs and a loopspace index generalizing the Euler characteristic and the Aroof genus, and the observation in [46] suggests that it encodes an infinitedimensional graded representation of the largest Mathieu sporadic group \(M_{24}\). (Note that the group featured in Mukai’s theorem, \(M_{23}\), is a subgroup of \(M_{24}\) as the name suggests.) The existence of a suitable \(M_{24}\)module has been eventually proved by Gannon [56], building on previous work in [19, 43, 52, 53]. On the other hand, a “natural” construction of this module, in the context, for example, of conformal field theory, string theory or geometry, is still an open problem. A natural guess is that there exists a K3 NLSM with \(M_{24}\) acting as its symmetry group. However, the classification result of [54] precludes this solution, and one must find an alternative way to explain Mathieu moonshine. See Sect. 5 for further discussion on this point.
The observation of EOT was truly surprising and led to a surge in activity in the study of (new) moonshine phenomena. Two of the subsequent developments, regarding umbral and Conway moonshines and their relation to K3 NLSMs, motivated the second part of our results which are encapsulated by two conjectures (Conjectures 5 and 6) and further detailed in “Appendix D.”
The first development is the discovery of umbral moonshine and its proposed relation to stringy K3 geometry. A succinct and arguably the most natural way to describe Mathieu moonshine is in terms of the relation between a certain set of mock modular forms and \(M_{24}\). See, for instance, [28] for an introduction on mock modular forms. Studying Mathieu moonshine from this point of view [21], it was realized in [16, 24] that it is but one case of a larger structure, dubbed umbral moonshine. Umbral moonshine consists of a family of 23 moonshine relations corresponding to the 23 Niemeier lattices N with nontrivial root systems: While the automorphism group of a Niemeier lattice dictates the relevant finite group \(G_N\) (cf. (2.12)), the root system of the lattice helps determine a unique (vectorvalued) mock modular form associated with each conjugacy class of \(G_N\). See Sect. 3.3 for more detail. One of the umbral moonshine conjectures then states that there exists a natural way to associate a graded infinitedimensional module with the finite group \(G_N\) such that its graded character coincides with the specified mock modular forms. So far, these modules have been shown to exist [38, 56], although, with the exception of a special case [39], a “natural” construction, providing an intuitive interpretation of this vector space and of the action of the corresponding group, is still lacking. Although it is not yet clear whether the structure of vertex operator algebra (VOA; or chiral CFT) is as relevant here as in the classical case of monstrous moonshine, the existence of the generalized umbral moonshine [13, 55] suggests that certain key features of VOA should be present in the modules underlying umbral moonshine. Subsequently, motivated by previous work [78, 79], the relation between all 23 instances of umbral moonshine and symmetries of K3 NLSMs was suggested in [22] in the form of a proposed relation (3.17) between the umbral moonshine mock modular forms and the K3 elliptic genus twined by certain symmetries (3.5).
The second important development, inspired by the close relation between the Conway group \(\textit{Co}_0\) and stringy K3 symmetries [54], relates Conway moonshine also to the twined K3 elliptic genus [42]. The Conway moonshine module is a chiral superconformal field theory with \(c=12\) and symmetry group \(\textit{Co}_0\), which was first discussed in [49] and further studied in [40, 41]. Using the Conway module, the authors of [42] associate two (possibly coinciding) Jacobi forms to each conjugacy class of \(\textit{Co}_0\) and conjecture that this set constitutes a complete list of possible K3 twining genera. In particular, it was conjectured that one of the two such Jacobi forms arising from Conway moonshine is attached to each symmetry of any nonsingular K3 NLSM. Note that many, but not all, of the functions arising from umbral moonshine [22] and Conway moonshine [42] coincide.
As the first part of our results establishes the importance of several (possibly all) Niemeier lattices in the study of symmetries of K3 string theory, it is natural to suspect that both umbral and Conway moonshine might play a role in describing the action of these symmetry groups on the (BPS) spectrum of K3 string theory. Note that the CFT is not well defined at the singular loci of the module space, and hence we restrict our attention to the nonsingular NLSMs when we discuss the (twined) elliptic genus. Motivated by the connection between the stringy K3 symmetries and moonshine, our analysis of worldsheet parity symmetries of NLSMs (see Sect. 3.2) and results regarding Landau–Ginzburg orbifolds [17], in this paper we conjecture (Conjecture 5) that the proposed twining genera arising from umbral and Conway moonshine as defined in [22] and [42] capture all of the possible discrete stringy symmetries of any NLSM in the K3 CFT moduli space. Moreover, we conjecture (Conjecture 6) that each of the umbral and Conway moonshine functions satisfying certain basic assumptions (that the symmetry preserves at least four planes in the defining 24dimensional representation) is realized as the physical twining genus of a certain K3 NLSM. These conjectures pass a few nontrivial tests. In particular, in this paper we also obtain an almost complete classification of conjugacy classes of the discrete Tduality group \(O^+(\Gamma ^{4,20})\), as well as a partial classification of the twined K3 elliptic genus using methods independent of moonshine. These classification results, summarized in Table 4, are not only of interest on their own but also provide strong evidence for these conjectures which consolidate our understanding of stringy K3 symmetries and the relation between K3 BPS states and moonshine.
Note that the computation of twined elliptic genus determines the twined BPS states counting of black hole states in the 4d theory of type IIA string theory compactified on \(K3\times T^2\) via the second quantization formula [30, 35, 36, 67, 84, 86]. The twined BPS states counting functions uniquely determine the representations of the symmetry group underlying the BPS indices. Indeed, the space of BPS states splits into a direct sum of finitedimensional representations and the twining functions provide the complete list of characters for each such representation. In particular, the group action on the part of the elliptic genus corresponding to the RamondRamond ground states determines its action on the BPS states of D2 branes wrapping cycles on K3 and hence its action on enumerative K3 invariants [15]. Moreover, the twined elliptic genera also determine the BPS spectrum of the corresponding CHL models [20, 29–32, 67].
The rest of the paper is organized as follows. In Sect. 2, we classify the symmetry groups which arise in type IIA string theory on \(K3\times {\mathbb {R}}^6\) and preserve the worldsheet \({\mathcal {N}}=(4,4)\) superconformal algebra in terms of two theorems. This extends the result of [54] to singular points in the moduli space of K3 NLSMs. In Sect. 3, we discuss how these symmetry groups act on the BPS spectrum of the theory. In particular, we present two conjectures relating the twining genera of NLSMs to the functions which feature in umbral and Conway moonshine. In Sect. 4, we summarize all the computations of twining genera in physical models that are known so far, including torus orbifolds and Landau–Ginzburg orbifolds, and explain how these data provide evidence for our conjectures. Finally, we conclude with a discussion in Sect. 5. A number of appendices include useful information which complements the main text. In “Appendix A,” we summarize some basic facts about lattice theory. The proofs of our main theorems discussed in Sect. 2 can be found in “Appendix B.” In “Appendix C,” we present the arguments that we employ in Sect. 3 to determine the modular properties of certain twining genera. In “Appendix D,” we discuss the method we use to classify distinct \(O^+(\Gamma ^{4,20})\) conjugacy classes. The result of the classification, as well as the data of the twining genera, is recorded in Table 4.
2 Symmetries
In this section, we classify subgroups of \(O^+(\Gamma ^{4,20})\) that pointwise fix a positive fourplane, a fourdimensional oriented positivedefinite subspace of \(\Gamma ^{4,20} \otimes _{\mathbb {Z}} \mathbb {R}\). They have the physical interpretation as groups of supersymmetrypreserving discrete symmetries of type IIA string theory on \(K3\times \mathbb {R}^6\). Alternatively, they can be viewed as the symmetry groups of NLSMs on K3 surfaces that commute with the \(\mathcal {N}=(4,4)\) superconformal algebra and leave invariant the four RR ground states corresponding to the spectral flow generators.
Our result extends [54] by allowing the coinvariant lattice to contain root vectors. Namely, we include those subgroups of fourplane preserving type such that there exists a \(v\in \Gamma _G\) with \(\langle v,v \rangle =2\), where \(\langle \cdot , \cdot \rangle \) denotes the bilinear form of the lattice \(\Gamma ^{4,20}\). We say that a positive fourplane is a singular positive fourplane if it is orthogonal to some root vector. Physically, they correspond to type IIA compactifications with enhanced gauge symmetry, or to singular NLSMs. The 23 Niemeier lattices with roots play an important role in the analysis of these singular cases.
2.1 The moduli space
Note that in the existing literature, the moduli space is often defined as the quotient of the Grassmannian by the full automorphism group \(O(\Gamma ^{4,20})\) instead of \(O^+(\Gamma ^{4,20})\). As we explain in more detail in Sect. 3.2, dividing by \(O^+(\Gamma ^{4,20})\) amounts to distinguishing between NLSMs that are related by worldsheet parity [75]. Due to the existence of symmetries that act differently on the right and leftmoving states of the NLSM, it is crucial for us to identify \(O^+(\Gamma ^{4,20})\) instead of \(O(\Gamma ^{4,20})\) as the relevant group of duality.
2.2 Symmetry groups
Let us denote by \(\mathcal {T}(\Pi )\) the NLSM associated with a given nonsingular positive fourplane \(\Pi \). With some abuse of notation, we will use the same letter for the lattice automorphism \(h\in O^+(\Gamma ^{4,20})\) and the corresponding duality between the two CFTs \(\mathcal {T}(\Pi )\) and \(\mathcal {T}(\Pi ')\), where \(\Pi ' := h(\Pi )\). Let G be the group of symmetries of a nonsingular NLSM \(\mathcal {T}(\Pi )\) preserving the \(\mathcal {N}=(4,4)\) superconformal algebra and the four spectral flow generators. It is shown in [54] that G is given by the largest \(O(\Gamma ^{4,20})\)subgroup whose induced action on \(\Gamma ^{4,20}\otimes _\mathbb {Z}\mathbb {R}\) fixes \(\Pi \) pointwise, and hence is always a subgroup of \(O^+(\Gamma ^{4,20})\subset O(\Gamma ^{4,20})\). From the spacetime point of view, the group G admits the alternative interpretation as the spacetimesupersymmetrypreserving discrete symmetry group of a sixdimensional type IIA string theory with halfmaximal supersymmetry, away from the gauge symmetry enhancement points in the moduli space. More precisely, G is the group of symmetries commuting with all spacetime supersymmetries, quotiented by its continuous (gauge) normal subgroup \(U(1)^{24}\).
Let us now consider the problem of classifying all \(O^+(\Gamma ^{4,20})\) subgroups of fourplane fixing type, including those involving singular four planes. Notice that by definition the invariant lattice \(\Gamma ^G\) has signature (4, d) for some \(0\le d\le 20\), and hence the coinvariant lattice \(\Gamma _G\) is negative definite of rank \(20d\).
In order to generalize the classification of the symmetry groups G to singular four planes, we have to consider the case where \(\Gamma _G\) contains a root. It is clear that in this case, lattices with nontrivial root systems—i.e., Niemeier lattices other than the Leech lattice—are necessary for the embedding. In fact, in this case, the coinvariant lattices can be always embedded into one of the Niemeier lattices, as we show with the following theorem.
Theorem 1
Proof
See “Appendix B.1.” \(\square \)
Note that the embedding is generically far from unique, and often \(\Gamma _G\) can be embedded in more than one Niemeier lattice N. At the same time, we believe that all Niemeier lattices are necessary in order to embed all \(\Gamma _G\) as in (2.10). In particular, in a geometric context it was conjectured in [78] that for each of the 24 Niemeier lattices N there exists a (nonalgebraic) K3 surface X whose Picard lattice P(X) can be primitively embedded only in N. This conjecture has been proven for all but two Niemeier lattices: those with root systems \(A_{24}\) and \(2A_{12}\). It is possible to find an appropriate choice of the Bfield such that the orthogonal complement lattice \(\Gamma _G\) contains the Picard lattice. Therefore, we expect all Niemeier lattices (and not just the Leech lattice) play a role in the study of physical symmetries of type IIA string theory on K3.
By Theorem 1, every group of symmetries G is isomorphic to a subgroup \(\hat{G}\subset O(N)\) of the group of automorphisms of some Niemeier lattice N, fixing a sublattice of N of rank at least 4. In fact, the converse is also true by the following theorem.
Theorem 2
Proof
See “Appendix B.2.” \(\square \)
As we will discuss in the next subsection, for many G arising in the way described above, there exist continuous families of \(\Pi \) such that the above statement is true, while for those groups with invariant sublattice of rank exactly four, the family consists of isolated points.
Proposition 3
 1.
\(\hat{G}\) has nontrivial intersection with the Weyl group \(W_N\) if and only if \(\Gamma _G\) contains some root.
 2.
if \(\Gamma _G\) has no roots, then G is isomorphic to a subgroup of \(G_N:=O(N)/W(N)\).
Proof
See “Appendix B.3.” \(\square \)
From the above theorems and proposition, we led to the following corollary for the stringy K3 symmetries:
Corollary 4
 1.
The supersymmetrypreserving discrete symmetry groups \(G_{\mathrm{IIA}}\) that are realized somewhere in \(\mathcal{M}\) are in bijection with the fourplane preserving subgroups of the Niemeier groups \(G_N\).
 2.
Consider the sublattice of the Dbrane lattice \(H^*(X,\mathbb {Z})\cong \Gamma ^{4,20}\) orthogonal to the \(G_{\mathrm{IIA}}\)invariant subspace of \(H^*(X,\mathbb {R})\). The isomorphism classes of lattices that arise in this way somewhere in \(\mathcal{M}\) are in bijection with the isomorphism classes of coinvariant lattice \(N_{\hat{G}}\), with N a Niemeier lattice and \(\hat{G}\subseteq G_N\) a fourplane preserving subgroup of the corresponding Niemeier group.
2.3 Gfamilies
 (1)
\(\Gamma _G\) contains no roots
 (2)
\(\Gamma _G\) contains roots

if G is a group of geometric symmetries (i.e., if G arises as a group of hyperKähler preserving symmetries of a K3 surface), then the corresponding family \(\mathcal {F}_G\) contains some singular models. To see this, first recall that a necessary and sufficient condition for G to be geometric is that the invariant lattice \(\Gamma ^G\) contains an even unimodular \(\Gamma ^{1,1}\subset \Gamma ^G\). In this case, one can take any root \(v\in \Gamma ^{1,1}\) and notice that \(v^\perp \cap (\Gamma ^G\otimes _\mathbb {Z}\mathbb {R})\) has signature \((4,d1)\), so it contains some \(\Pi \) of signature (4, 0) that is by definition singular.

If \(\Gamma ^G\) has rank exactly four, then \(\mathcal {F}_G\) consists of a single point, and if we are in case 1 above, this is by definition nonsingular.

If the defining 24dimensional representation of G is not a permutation representation, then all four planes in \(\mathcal {F}_G\) are nonsingular. This can be seen as follows. For each \(\Pi \) in the family \(\mathcal {F}_G\), one can show, using techniques analogous to the proof of Theorem 2, that the orthogonal sublattice \(\Gamma _\Pi := \Pi ^\perp \cap \Gamma ^{4,20}\) can be primitively embedded in some Niemeier lattice N (possibly depending on \(\Pi \)). This implies that also \(\Gamma _G\subset \Gamma _\Pi \) can be primitively embedded in N. Recall that the defining 24dimensional representation is a permutation representation for all subgroups of the Niemeier group \(G_N\) unless N is the Leech lattice; indeed, when N is not the Leech lattice, \(G_N\) is just the group of automorphisms of the Dynkin diagram of the corresponding root lattice ([26], chapter 16) and therefore acts by permutations on a basis of 24 simple roots. By hypothesis, \(\Gamma _G\) has no roots, so that by Proposition 3 G must be isomorphic to a subgroup of \(G_N\). The only N such that the 24dimensional representation of \(G_N\) is not a permutation representation is the Leech lattice. We conclude that, for all \(\Pi \) in \(\mathcal {F}_G\), \(\Gamma _\Pi \) can be embedded in the Leech lattice, and therefore it cannot contain any root.

If \(\mathcal {F}_G\) contains some singular fourplane \(\Pi \), then \(\Gamma _G\) can be embedded in some Niemeier lattice N with roots, so that G is isomorphic to a subgroup of the Niemeier group \(O(N)/W_N\). The argument for this is analogous to the previous statement. The sublattice \(\Gamma _\Pi := \Pi ^\perp \cap \Gamma ^{4,20}\) orthogonal to a singular fourplane \(\Pi \) can be primitively embedded in some Niemeier lattice N. By definition, \(\Gamma _\Pi \) contains some root and hence N cannot be the Leech lattice. Furthermore, \(\Gamma _G\) is a primitive sublattice of \(\Gamma _\Pi \), so it can also be primitively embedded in N.^{4}
3 Twining genera
In this section, we investigate how the symmetry groups discussed in the previous section act on the BPS spectrum of the theory. In particular, in Sect. 3.3 we will present two conjectures relating the twining genera of NLSMs and the functions featured in umbral and Conway moonshine. In this section, we restrict our attention to nonsingular NLSMs as the elliptic genus is otherwise not well defined.
Let \(g\in O^+(\Gamma ^{4,20})\) be a group element fixing pointwise a sublattice \(\Gamma ^g\subseteq \Gamma ^{4,20}\) of signature (4, d) and such that the coinvariant lattice \(\Gamma _g \)contains no roots. Then, the family \(\mathcal {F}_g^{ns}:=\mathcal {F}_{\langle g\rangle }^{ns}\) of nonsingular four planes with symmetry g is nonempty and connected. Furthermore, if we assume that the operators \(L_0,\bar{L}_0,J_0,\bar{J}_0\) and g vary continuously under deformations within the family of NLSM corresponding to \(\mathcal {F}_g^{ns}\), then the twining genus \(\mathcal{Z}_g\) is constant on \(\mathcal {F}_g^{ns}\).
The proof is an obvious generalization of the arguments showing that the elliptic genus is independent of the moduli. One first defines the twining genus \(\mathcal{Z}_g\) along any connected path within the family \(\mathcal {F}^{ns}_g\) and then uses continuity of \(L_0,\bar{L}_0,J_0,\bar{J}_0\) as well as the discreteness of their spectrum within the relevant space of states to show that \(\mathcal{Z}_g\) must be actually constant along this path. An even simpler proof can be given if one adopts the equivalent definition of the twining genus \(\mathcal{Z}_g\) as an equivariant index in the Qcohomology of a halftwisted topological model. In this case, it is sufficient to use the fact that a ginvariant and Qexact deformation cannot change the index.
Finally, a twining genus \(\mathcal{Z}_g\) is invariant under conjugation by any duality \(h\in O^+(\Gamma ^{4,20})\). More precisely, suppose h is a duality between the models \(\mathcal {T}\) and \(\mathcal {T}'\), i.e., an isomorphism between the fields and the states of the two theories that maps the superconformal generators into each other and is compatible with the OPE. Then, the twining genus \(\mathcal{Z}_g\) defined in the model \(\mathcal {T}\) equals the twining genus \(\mathcal{Z}_{hgh^{1}}\) defined in the model \(\mathcal {T}'\). This follows after noticing that h maps \((L_0,J_0)\)eigenspaces in \(\mathcal {T}\) into the corresponding eigenspaces in \(\mathcal {T}'\) and using the cyclic properties of the trace. The effect of a conjugation under a duality in \(O(\Gamma ^{4,20})\setminus O^+(\Gamma ^{4,20})\) is much more subtle and will be discussed in Sect. 3.2.
Using the above results, one can assign a twining genus \(\mathcal{Z}_g\) to any conjugacy class [g] of \(O^+(\Gamma ^{4,20})\) such that \(\langle g \rangle \) is a subgroup of fourplane fixing type and that the coinvariant sublattice \(\Gamma _g\) contains no roots. In principle, \(\mathcal{Z}_g\) and \(\mathcal{Z}_{g'}\) are distinct if \(g'\) is conjugate to neither g nor \(g^{1}\) as elements of \(O^+(\Gamma ^{4,20})\), unless accidental coincidences occur.^{7} In the next subsection we will classify the conjugacy classes of \(O^+(\Gamma ^{4,20})\).
3.1 Classification
While many examples of twining genera have been computed in specific sigma models, a full classification of the corresponding conjugacy classes in \(O^+(\Gamma ^{4,20})\) and a complete list of all corresponding twining genera are still an open problem. In this work, we solve the first problem for all but one of the 42 possibilities (labeled by conjugacy classes of \(Co_0\)).
A summary of these results is the following. Out of the 42 distinct fourplane preserving Frame shapes of \(O^+(\Gamma ^{4,20})\), there was only one (with Frame shape \(1^{4}2^53^46^1\)) for which we were unable to determine the number of its \(O(\Gamma ^{4,20})\) (and thus \(O^+(\Gamma ^{4,20})\)) classes. For this Frame shape, we can only prove that either a) there is one class or b) there are two classes for which one is the inverse of the other. The remaining 41 \(Co_0\) conjugacy classes give rise to 58 distinct \(O(\Gamma ^{4,20})\) conjugacy classes and 80 distinct \(O^+(\Gamma ^{4,20})\) conjugacy classes.
3.2 Worldsheet parity
We have argued earlier that the twining genera \(\mathcal{Z}_g\) are invariant under conjugation by \(O^+(\Gamma ^{4,20})\) dualities. In many physical applications, however, the larger group \(O(\Gamma ^{4,20})\) is taken to be the relevant duality group. As will be discussed below, the elements of \(O(\Gamma ^{4,20})\setminus O^+(\Gamma ^{4,20})\) correspond to dualities between NLSMs that reverse the worldsheet orientation. In particular, we will show that the twining genera \(\mathcal{Z}_g\), on the other hand, are in general different unless the two theories are related by an element of \(O^+(\Gamma ^{4,20})\subset O(\Gamma ^{4,20})\). For this reason, it is necessary, for our purposes, to consider two NLSMs related by dualities \(h\in O(\Gamma ^{4,20})\setminus O^+(\Gamma ^{4,20})\) as distinct, although we will still refer to such h as (orientationreversing) dualities.
Let us first explain why only elements of \(O^+(\Gamma ^{4,20})\subset O(\Gamma ^{4,20})\), which by definition preserve the orientation of any positive fourplane, preserve the orientation of the worldsheet of NLSM [75]. To understand this, let us first recall some known facts about NLSM on K3 (see, e.g., [75] for a discussion and more details). The tangent space to \((SO(4)\times O(20))\backslash O^+(4,20)\cong (O(4)\times O(20))\backslash O(4,20)\) at the point corresponding to a four plane \(\Pi \) can be identified with the 80dimensional space of \(\mathcal {N}=(4,4)\)preserving exactly marginal operators of the corresponding NLSM \(\mathcal {T}(\Pi )\). The latter have schematically the form \(G_{1/2}\overline{G}_{1/2}\chi _i\), where \(\chi _i\), \(i=1,\ldots ,20\), are fields of weight (1 / 2, 1 / 2) belonging to 20 different \(\mathcal {N}=(4,4)\) irreducible representations. In particular, the \(\mathcal {N}=(4,4)\)preserving marginal operators are the descendants of conformal weights (1, 1) that are singlets under the internal Rsymmetries \(SU(2)^{\textit{susy}}_L\times SU(2)_R^{\textit{susy}}\), generated by the zero modes of the holomorphic and antiholomorphic \(\hat{su}(2)_1\) current algebras within \(\mathcal {N}=(4,4)\). On the other hand, they transform as \((\mathbf{2 },\mathbf{2 })\) under the group \(SU(2)_L^{\textit{out}}\times SU(2)_R^{\textit{out}}\) of outer automorphisms acting on the left and rightmoving supercharges. The space \((SO(4)\times O(20))\backslash O^+(4,20)\) has an holonomy group \(SO(4)\times SO(20)\), where the SO(4) factor is the group \(SO(\Pi )\) of rotations of the fourplane \(\Pi \subset \mathbb {R}^{4,20}\), while SO(20) rotates its orthogonal complement \(\Pi ^\perp \subset \mathbb {R}^{4,20}\). This holonomy group must act on the space of exactly marginal operators \(G_{1/2}\overline{G}_{1/2}\chi _i\); in particular, the SO(20) factor rotates the fields \(\chi _i\), while the \(SO(\Pi )\cong SO(4)\) factor acts on the supercharges \(G_{1/2}\) and \(\bar{G}_{1/2}\) by automorphisms of the \(\mathcal {N}=(4,4)\) superconformal algebra. Therefore, the double cover \(Spin(\Pi )\cong Spin(4)\) of \(SO(\Pi )\) can be identified with the group \(SU(2)_L^{\textit{out}}\times SU(2)_R^{\textit{out}}\) of algebra automorphisms. Now, let \(h\in O(\Gamma ^{4,20})\) be a duality mapping \(\mathcal {T}(\Pi )\) to \(\mathcal {T}(h(\Pi ))\). The group \(SO(h(\Pi ))\) rotating the dual 4plane \(h(\Pi )\) and its double cover \(Spin(h(\Pi ))\) are the conjugate by h of the group \(SO(\Pi )\subset O^+(4,20)\) rotating \(\Pi \) and its cover \(Spin(\Pi )\cong SU(2)_L^{\textit{out}}\times SU(2)_R^{\textit{out}}\), respectively. Recall that, given an oriented Euclidean fourdimensional space, conjugation by (a lift of) an element \(h\in SO(4)\) maps each SU(2) factor in the spin cover \(Spin(4)\cong SU(2)\times SU(2)\) into itself, while conjugation by \(h\in O(4)\setminus SO(4)\) exchanges the two SU(2) factors. Therefore, if a duality h is in \(O^+(\Gamma ^{4,20})\), then conjugation by h will send the factor \( SU(2)_L^{\textit{out}}\) (respectively, \( SU(2)_R^{\textit{out}}\)) of \(Spin(\Pi )\) to the group of outer automorphisms of the leftmoving (resp., rightmoving) \(\mathcal {N}=4\) superconformal algebra in the dual model \(\mathcal {T}(h(\Pi ))\). On the other hand, if \(h\in O(\Gamma ^{4,20})\setminus O^+(\Gamma ^{4,20})\), conjugation by h will map the group of leftmoving outer automorphisms \( SU(2)_L^{\textit{out}}\) in the model \(\mathcal {T}(\Pi )\) to the group of rightmoving outer automorphisms in the dual model \(\mathcal {T}(h(\Pi ))\), and vice versa. By consistency, in this case h must also send the holomorphic \(\mathcal {N}=4\) superconformal algebra in \(\mathcal {T}(\Pi )\) to the antiholomorphic \(\mathcal {N}=4\) algebra in \(\mathcal {T}(h(\Pi ))\). This means that a duality \(h\in O(\Gamma ^{4,20})\) preserves the worldsheet orientation if and only if \(h\in O^+(\Gamma ^{4,20})\subset O(\Gamma ^{4,20})\). Since the definition of the twining genera effectively only focuses on the action of g on the leftmovers (i.e. on the rightmoving ground states), and in general g acts on the rightmovers differently, one expects \(\mathcal{Z}_g\) to be invariant only under \(O^+(\Gamma ^{4,20})\) duality transformations.
Finally, recall that \(Z_{g'}(\mathcal {T}';\tau ,0,u)\) and \(Z_{g'}(\mathcal {T}';\tau ,z,0)=\mathcal{Z}_{g'}(\mathcal {T}';\tau ,z)\) necessarily have the same multiplier, since they both coincide with that of \(Z_{g'}(\mathcal {T}';\tau ,u,z)\), and thus we conclude that the twining genera \(\mathcal{Z}_{g'}(\mathcal {T}';\tau ,z)\) and \(\mathcal{Z}_{g}(\mathcal {T};\tau ,z)\) have multiplier systems that are the inverse (equivalently, complex conjugate) of each other. In particular, \(\mathcal{Z}_{g'}(\mathcal {T}';\tau ,z)\) and \(\mathcal{Z}_{g}(\mathcal {T};\tau ,z)\) cannot be the same unless \(\psi _g = \overline{\psi _g}\). As a result, symmetries g leading to a twining genus with a complex multiplier system necessarily act differently on left and rightmoving states. Note, however, that it can happen that a symmetry acting asymmetrically on left and rightmovers leads to a twining genus with a multiplier system of order one or two. In what follows, we will refer to a symmetry g of a NLSM a complex symmetry if the resulting twining genus has complex multiplier system.
3.3 Conway and umbral moonshine
Once the possible \(O^+(\Gamma ^{4,20})\) classes of symmetries have been determined, it remains to calculate the corresponding twining genera. As we will see in Sect. 4, many examples have been computed in specific NLSMs. However, the list of such functions is still incomplete. After reviewing the earlier work [22, 42], in this subsection we present two conjectures relating physical twining genera to functions arising from umbral and Conway moonshine, as well as some evidence for their validity.
There are a few motivations for us to modify this conjecture and to make the conjecture in [22] more concrete. Firstly, the classification theorems of Sect. 2 suggest that if one does not exclude the loci in the moduli space (2.2) corresponding to singular four planes, one should treat the Leech lattice and the other 23 Niemeier lattices with nontrivial root systems on an equal footing when discussing the fourplane preserving symmetry groups. As a consequence, one might expect both Conway and umbral moonshine to play a role in describing the twining genera. Secondly, UV descriptions of K3 NLSMs given by Landau–Ginzburg (LG) orbifolds furnish evidence that suggests that the Conway functions alone are not sufficient to capture all the twining genera [17] (see also Sect. 4.3). To be more precise, there are twining genera arising from symmetries of UV theories that flow to K3 NLSMs in the IR that can be reproduced from the set \(\Phi (N)\) for some N with roots, but do not coincide with anything in \(\Phi (\Lambda )\). One caveat preventing this result from being a definitive argument is that the action of the corresponding symmetry on the IR \({\mathcal {N}}=(4,4)\) superconformal algebra is not accessible in the UV analysis.
The third and arguably most convincing argument to include functions arising from both Conway and umbral moonshine is the following. As we have seen in Sect. 3.2, a pair of theories related by a flip of worldsheet parity gives rise to twining genera with inverse multiplier systems. At the same time, \(\Phi (\Lambda )\) contains some twining functions with a complex multiplier system and no functions with the inverse multiplier. Such functions can always be recovered from \(\Phi (N)\) for some other Niemeier lattice N. As a result, no single \(\Phi (N)\) (not even for N the Leech lattice) is sufficient to reproduce both a physical twining function \(\mathcal{Z}_g(\mathcal {T})\) with complex multiplier and its parityflipped counterpart \(\mathcal{Z}_{g'}(\mathcal {T}')\).
These observations lead us to formulate the following conjecture:
Conjecture 5
Let \(\mathcal {T}(\Pi )\) be a K3 NLSM, and let G be its symmetry group. Then, there exists at least one Niemeier lattice N such that \(\Gamma _G\) can be embedded in N, \(G \subseteq G_N\), and for any \(g\in G\) the twining genus \(\mathcal{Z}_g\) coincides with an element of \(\Phi (N)\).
In other words, we conjecture that for each K3 NLSM \(\mathcal {T}\), the set \(\tilde{\Phi }(\mathcal {T}):=\{\mathcal{Z}_g(\mathcal {T}(\Pi ))g\in O^+(\Gamma ^{4,20}), ~g~\text {fixes }\Pi \text { pointwise}\}\) of physical twining genera is a subset of the \(\Phi (N)\) for some Niemeier lattice N. Clearly, for most theories, the Niemeier lattice N satisfying the above properties is not unique. In particular, recall that there are many coincidences among the functions associated with different Niemeier lattices. In other words, there exist \(\phi \in \Phi (N)\), \(\phi ' \in \Phi (N')\) with \(N\ne N'\) such that \(\phi =\phi '\).
Conversely, we conjecture that all elements of \(\Phi (N)\) play a role in capturing the symmetries of BPS states of K3 NLSMs:
Conjecture 6
For any element \(\phi \) of any of the 24 \(\Phi (N)\), there exists a NLSM \(\mathcal {T}\) with a symmetry g such that \(\phi =\mathcal{Z}_g(\mathcal {T})\).

If a given function in \(\Phi (N)\) has complex multiplier system, then Conjecture 6 implies that it has to coincide with a twining genus arising from a complex symmetry acting differently on the left and rightmoving Hilbert spaces.

As we argued in Sect. 3.2, if a theory \(\mathcal {T}\) leads to the twining function \(\mathcal{Z}_g(\mathcal {T})\) with a complex multiplier system, the parityflipped theory \(\mathcal {T}'\) has a twining genus \(\mathcal{Z}_{g'}(\mathcal {T}')\) with the inverse multiplier system. As a result, the following observations constitute consistency checks and circumstantial evidence for Conjecture 5 and Conjecture 6. Namely, whenever there exist a Niemeier lattice N and a function \(\phi \in \Phi (N)\) with a complex multiplier system, arising from a group element with a given Frame shape \(\pi \), then there exists at least one other Niemeier lattice \(N'\) such that there exists a \(\phi '\in \Phi (N')\) with the inverse complex multiplier system, which moreover arises from a group element with the same Frame shape \(\pi \). See Table 3 for the pairs \((N',g')\) with the above properties.

In fact, by inspection one can check that there are never two functions \(\phi ,\phi '\in \Phi (N)\) arising from the same Niemeier lattice that have inverse complex multiplier systems. As a result, Conjecture 5 predicts that a theory corresponding to the fourplane \(\Pi \) must have its orthogonal sublattice \(\Gamma ^{4,20}\cap \Pi ^\perp \) embeddable into more than one Niemeier lattice in the event that it has a complex symmetry.

Recall that a theory in the NLSM moduli space (2.2) on a torus orbifold locus—one of the few types of exactly solvable models—always contains symmetries which can only be embedded using the Leech lattice (in the sense of Theorem 1) [51]. As a result, assuming the veracity of Conjecture 5, complex symmetries can never arise in such a model. This makes it particularly difficult to find examples of K3 NLSMs with complex symmetries and probably explains why we have seen no such examples so far. In Sect. 4.3, we will discuss results of the aforementioned investigation of LG orbifolds [17], while in Sect. 4.4 we will analyze the constraints on such genera coming from modularity.
4 Examples
In this section, we collect all known explicit calculations of twining genera in NLSMs on K3. Most of these results have appeared earlier in the literature, the only exceptions being certain genera appearing in Sects. 4.2 and 4.4. See Table 4 for the data. While these examples do not cover the complete set of all possible twining genera, the fact that these partial results fit nicely with the general properties described in the previous sections represents strong evidence in favor of our conjectures.
4.1 Geometric symmetries
4.2 Torus orbifolds
When a CFT has a discrete symmetry, it is also useful to discuss the twisted sectors of the symmetry (modules of the invariant subalgebra), labeled by the twisting group element g. For any element h of the discrete symmetry group that commutes with the twisting element g, one can consider the graded trace of h over the gtwisted sector, analogous to the way in which a twined partition function or twined elliptic genus is defined. Such a character is often called the twistedtwining partition function/elliptic genus. As usual in the literature, we use \(\mathcal{Z}_{h,g}\) to denote the gtwining function in the htwisted sector. In particular, the twining function of the original unorbifolded theory is given by \( \mathcal{Z}_{g} := \mathcal{Z}_{e,g}\).
Frame shapes corresponding to quantum symmetries Q of torus orbifolds
\(r_L\)  \(r_R\)  \(\pi _Q\)  ws parity 

1 / 2  1 / 2  \(1^{8}2^{16}\)  \(\circ \) 
1 / 3  1 / 3  \(1^{3}3^{9}\)  \(\circ \) 
1 / 4  1 / 4  \(1^{4}2^{6}4^4\)  \(\circ \) 
1 / 6  1 / 6  \(1^{4}2^{5}3^46^1\)  \(\circ \) 
1 / 5  2 / 5  \(1^{1}5^{5}\)  \(\updownarrow \) 
2 / 5  1 / 5  
1 / 4  1 / 2  \(2^{4}4^{8}\)  \(\updownarrow \) 
1 / 2  1 / 4  
1 / 6  1 / 2  \(1^{2}2^{4}3^{2}6^4\)  \(\updownarrow \) 
1 / 2  1 / 6  
1 / 6  1 / 3  \(1^{1}2^{1}3^36^3\)  \(\updownarrow \) 
1 / 3  1 / 6  
1 / 8  5 / 8  \(1^{2}2^{3}4^18^2\)  \(\updownarrow \) 
5 / 8  1 / 8  
1 / 10  3 / 10  \(1^{2}2^{3}5^210^1\)  \(\updownarrow \) 
3 / 10  1 / 10  
1 / 12  5 / 12  \(1^{2}2^{2}3^24^112^1\)  \(\updownarrow \) 
5 / 12  1 / 12 
4.3 Landau–Ginzburg orbifolds
Though these minimal models all have central charge less than 3, LG theories prove to have geometric applications through the orbifold construction. Namely, one can construct theories which flow in the IR to a NLSM on a CY dfold by taking superpotentials of multiple chiral multiplets, such that the sum of their charges equals 3d, along with an orbifold which projects the Hilbert space onto states with integer U(1) charges. This connection between CY geometry and LG orbifolds was further elucidated by Witten [93] using the framework of gauged linear sigma models.
Symmetries of torus orbifolds whose twining genera are given by (4.12)
\(r_L\)  \(r_R\)  o(Q)  \(o(g')\)  \(\pi _{Qg'}\) 

1 / 4  1 / 4  2  2  \(2^{12}\) 
1 / 6  1 / 6  2  3  \(1^4 2^1 3^{4} 6^5\) 
3  2  \( 1^5 2^{4} 3^1 6^4\)  
1 / 6  1 / 2  2  3  \(1^{2} 2^4 3^{2} 6^4\) 
1 / 6  1 / 3  3  2  \( 1^{1} 2^{1} 3^3 6^3\) 
1 / 8  3 / 8  2  4  \(2^44^4\) 
4  2  \(2^44^4\)  
1 / 10  3 / 10  2  5  \(1^22^15^{2}10^3 \) 
5  2  \(1^3 2^{2} 5^1 10^2 \)  
1 / 12  5 / 12  6  2  \(2^36^3\) 
4  3  \(1^23^{2}4^{1}6^212^1\)  
3  4  \(1^12^23^14^{2}12^2\)  
2  6  \(2^36^3\) 
The symmetries of order 11, 15 and 14 all have a unique Frame shape (\(1^211^2\), 1.3.5.15 and 1.2.7.14, respectively), and each occurs in two nonConway Niemeier groups, corresponding to Niemeier lattices \(N_1, N_2\) with root lattices \(\{A_1^{24}, A_2^{12}\}\), \(\{A_1^{24}, D_4^6\}\) and \(\{A_1^{24}, A_3^8\}\), respectively. Since these symmetries preserve exactly a fourplane, the Conway module associates two different twinings functions with these Frame shapes. In each of these three cases, the two umbral moonshine twinings given corresponding to two Niemeier lattices yield two different results \(\phi ^{N_1}_{g_1}\) and \(\phi ^{N_2}_{g_2}\), coinciding with the two twinings \(\phi ^\Lambda _{g,+}\) and \(\phi ^\Lambda _{g,}\) arising from Conway module.
The twinings of order 11, 15 and 14 computed in the abovementioned LG models match those associated with root systems \(A_2^{12}\), \(D_4^6\) and \(A_3^8\), respectively. This can be viewed as evidence for the connection between (non\(M_{24}\) instances of) umbral moonshine and Conway moonshine, to the symmetries of K3 NLSMs.^{10} We refer to [17] for more examples and details.
4.4 Modularity
The approach described above is particularly effective in constraining twining genera with nontrivial multiplier \(\psi \), since the space \(M_2({\mathfrak G}_g;\psi )\) is often quite small. We illustrate our arguments with the following example. Consider g with Frame shape \(3^8\). The possible multipliers can be determined using the methods described in “Appendix C.” In particular, \({{\mathrm{Tr}}}_{V_{ 24}}(g)=0\) and \({\mathfrak G}_g=\Gamma _0(3)\), and hence the order of the multiplier system is either 1 or 3. The Witten index of a putative orbifold by g is 8, which is different from 0 or 24. We can therefore conclude that the orbifold is inconsistent and hence the multiplier has order \(n=3\). (See “Appendix C” for the detailed argument.) Thus, \(F(\tau )=2+O(q)\) is modular form of weight 2 for \(\Gamma _0(3)\) with multiplier of order 3. It turns out that there are two possible multipliers \(\psi \) and \(\bar{\psi }\) of order 3, with the property \(\dim M_2(\Gamma _0(3);\psi )=\dim M_2(\Gamma _0(3);\bar{\psi })=1\). Hence, in both cases there are a unique weight 2 form F and therefore a unique weak Jacobi form \(\mathcal{Z}_g\), with the required normalization (4.21), giving the umbral twining function corresponding to the root systems \({A_1^{24}}\) and \({A_2^{12}}\).
Using similar arguments, one can determine the twining genera for the Frame shape \(4^6\) for both possible choices of multipliers, and the twining genera for the Frame shapes \(6^4\) and \(4^28^2\) for one of the two possible multipliers. In all such cases, the resulting twining genera coincide with some umbral functions, i.e., some \(\Phi (N)\) (see “Appendix D.2”), offering support for our Conjecture 5.
4.5 Evidence for the conjectures

As reported in Appendix D.2, there are either 81 or 82 distinct \(O^+(\Gamma ^{4,20})\) classes of symmetries. In particular, for the Frame shape \(1^{4} 2^5 3^4 6^1\), there is either a single \(O^+(\Gamma ^{4,20})\) class or two classes that are the inverse of each other and hence must have the same twining genus. Therefore, there are potentially 81 distinct twining genera \(\mathcal{Z}_g\). Only 56 have been computed using the methods described in Sects. 4.1–4.4. In all such cases, one has \(\mathcal{Z}_g\in \Phi (N)\) for at least one Niemeier lattice N.

Whenever there is an umbral or Conway twining genus \(\phi _g\in \Phi (N)\) which has a complex multiplier \(\psi \), there exists another \(\phi '_{g'} \in \Phi (N')\) corresponding to the same Frame shape \(\pi _g=\pi _{g'}\) and with the conjugate multiplier \(\bar{\psi }\). Furthermore, \(\pi _g\) has distinct \(O^+(\Gamma ^{4,20}) \) conjugacy classes which are related by worldsheet parity. Note that in all cases we have \(N\ne N'\). Table 3 shows the pairs of \(N,N'\), denoted in terms of their root systems in the case \(N\ne \Lambda \), leading to Jacobi forms with complex conjugate multipliers.

Fix a fourplane preserving Frame shape \(\pi _g\). Denote by K the number of distinct twining functions \(\phi _g^N\) associated with \(\pi _g\) arising from either Conway or umbral moonshine, and denote by \(K'\) the number of \(O^+(\Gamma ^{4,20}) \) conjugacy classes associated with \(\pi _g\). In all cases, \(K' \ge K\), and for a vast majority (35 out of 42) of the fourplane preserving Frame shapes this inequality is saturated.
Frame shapes with complex multiplier and corresponding Niemeier lattices
Frame shape  \(\psi \)  \(\overline{\psi }\) 

\(3^8\)  \(A_2^{12}, D_4^6, A_8^3, E_8^3, \Lambda \)  \(A_1^{24}, A_4^6, D_8^3\) 
\(4^6\)  \(A_3^8, A_4^6, A_{12}^2, \Lambda \)  \(A_1^{24}, A_2^{12}, A_6^4, D_6^4\) 
\(6^4\)  \(A_2^{12}, D_4^6, A_8^3, \Lambda \)  \(A_1^{24}, A_4^6\) 
\(4^28^2\)  \(A_3^8, E_6^4, \Lambda \)  \(A_2^{12}\) 
5 Discussion

Apart from classifying the symmetry groups of K3 NLSMs as abstract groups, it is also important to know what their actions are on the (BPS) spectrum. In particular, the twining genus can differ for two K3 NLSM symmetries with the same embedding into \(\textit{Co}_0\) [17, 22, 54]. This motivated us to classify the distinct conjugacy classes in \(O^+(\Gamma ^{4,20})\) and \(O(\Gamma ^{4,20})\) for a given fourplane preserving Frame shape.
Given this consideration and given our Conjectures 5 and 6 relating twining genera and moonshine functions, an important natural question is the following: Given a particular K3 NLSM, how do we understand which case(s) of umbral moonshine govern its symmetries?

In this paper, we extend the classification of symmetry groups to singular points in the moduli space of K3 NLSMs. These singular points correspond to perfectly welldefined string compactifications where the physics in the sixdimensional noncompact spacetime involves enhanced nonabelian gauge symmetries. It will be interesting to study the BPScounting functions arising in these compactifications.
Moreover, as these points are Tdual to type IIB compactifications on K3 in the presence of an NS5brane [91], it would be interesting to explore the symmetries of these special points from this spacetime point of view. Furthermore, it may also be interesting to classify the symmetry groups in more general fivebrane spacetimes, such as those studied in [61, 62] in connection with umbral moonshine.

More generally, one can try to classify the discrete symmetry groups which arise in other supersymmetric string compactifications, in varying dimensions and with differing numbers of supersymmetries. For example, one case of particular interest is the symmetries of theories preserving only eight supercharges. One difficulty in studying such theories is the global form of the moduli space is often not known, so one does not have the power of lattice embedding theorems used to study theories with 16 supercharges. However, it may be possible to get partial results in certain examples. The connection between sporadic groups, geometry and automorphic forms in theories with eight supercharges has only somewhat been studied (see, e.g., [23, 60]), and it would be interesting to explore it further.

Twining genera of K3 NLSMs can be lifted to twining genera of the Nth symmetric product CFT \(Sym^N(K3)\) through a generalization [19] of the formula for the symmetric product elliptic genus of [36]. It can happen that a symmetry which is not a geometric symmetry of any K3 surface can be a geometric symmetry for a hyperKähler manifold that is deformation equivalent to the Nth Hilbert scheme of a K3 surface for \(N\ge 2\). The symmetries of such hyperKähler manifolds of \(K3^{[N]}\) type were classified in [64] for \(N=2\) in terms of their embedding into \(Co_0\). This includes Frame shapes corresponding to elements of order 3, 6, 9, 11, 12, 14 and 15 which are not geometric symmetries of any K3 surface. Each of these elements has at least two distinct twining functions associated with it via umbral and Conway moonshine as presented in Table 4. We noticed that for the elements of order 11, 14 and 15, these distinct twining functions lift to the same twined elliptic genus for \(Sym^N(K3)\) for \(N=2,3,4.\) It would be interesting to understand when this general phenomenon occurs, and more generally the structure of symmetries of string theory on \(K3 \times S^1\).

The compactification of type IIA on \(K3\times T^2\) gives rise to a fourdimensional model with halfmaximal supersymmetry (16 supercharges). When the internal NLSM has a symmetry g, one can construct a new fourdimensional model (CHL model) with the same number of supersymmetries [10–12, 83]. The CHL model is defined as the orbifold of type IIA on K3\(\times T^2\) by a fixedpointfree symmetry acting as g on the K3 sigma model and, simultaneously, by a shift along a circle \(S^1\) in the \(T^2\). The twining genus \(\mathcal{Z}_g\) is directly related to the generating function \(1/\Phi _g\) of the degeneracies of 1/4 BPS dyons in this CHL model [20, 29–32, 35, 36, 67, 84]. Up to dualities, the CHL model only depends on the Frame shape of g [80]. This is apparently puzzling for those Frame shapes that correspond to multiple \(O^+(\Gamma ^{4,20})\)classes and therefore to multiple twining genera \(\mathcal{Z}_g\): In these cases, there are different candidates \(1/\Phi _g\) for the 1/4 BPScounting function, one for each distinct twining genus \(\mathcal{Z}_g\). Since \(O^+(\Gamma ^{4,20})\) is part of the Tduality group of the fourdimensional model, a natural interpretation of this phenomenon is that the different \(1/\Phi _g\) functions count 1/4 BPS dyons related to different Tduality orbits of charges in the same CHL model. In view of this interpretation, it would be interesting to understand the precise correspondence between \(O^+(\Gamma ^{4,20})\)classes and Tduality orbits of charges.

One piece of supporting evidence for our conjectures concerns twining genera with complex multiplier systems. However, so far we have not been able to directly obtain these proposed twining genera from K3 NLSMs. Nevertheless, we argue that this is unsurprising and does not constitute discouraging counterevidence for our conjectures for the following reason. Recall that the argument in Sect. 3.2 indicates that these functions must arise from a symmetry acting differently on left and rightmovers. Then, our Conjecture 5, together with the observation that such twining functions always arise from multiple instances of umbral and Conway moonshine (see Sect. 4.5, 2nd bullet point), predicts that these theories correspond to lattices embeddable into multiple Niemeier lattices. This precludes most of the exactly solvable models that have been studied so far, in particular all torus orbifolds and some Gepner models, since these always contain a quantum symmetry which can only arise from a Leech embedding. So far most of the NLSM analysis has focussed on these exactly solvable models, and this explains why we have not observed these proposed twining genera yet.
On the other hand, a number of the proposed twining genera with complex multipliers (as well as many with real multipliers) were found by twining certain LG orbifold theories [17]. These include functions arising from symmetries of order 3, 4, 6 and 8 and with Frame shapes \(3^8, 4^6,6^4\) and \(4^28^2\)—the four Frame shapes which both preserve a fourplane in \(Co_0\) and correspond to twining genera with complex multiplier. In order to obtain these twining genera, one has to consider symmetries which act asymmetrically on the left and rightmoving fermions in the chiral multiplets, such that the UV Lagrangian, the rightmoving \({\mathcal {N}}=2\) algebra and the four charged Ramond ground states are preserved. In general, however, the leftmoving \({\mathcal {N}}=2\) algebra is not preserved. Though \(H_L\) and \(J_L\) must remain invariant for the twining genus to be well defined, \(G_\) and \(\overline{G}_\) are transformed under these symmetries, such that the symmetry maps the leftmoving \({\mathcal {N}}=2\) to a different but isomorphic copy. See [17] for more details. It is important to note that though these symmetries do not preserve the full UV supersymmetry algebra, it does not preclude the possibility that they preserve a copy of the IR \({\mathcal {N}}=(4,4)\) SCA. After all, there is only an \({\mathcal {N}}=(2,2)\) supersymmetry algebra apparent in the UV, and only after a nontrivial RG flow involving a complicated renormalization of the Kähler potential does the symmetry get enhanced to \({\mathcal {N}}=(4,4)\) at the conformal point. A clarification of the IR aspects of these UV symmetries would be helpful in unravelling the nature of these leftright asymmetric symmetries.

While our Conjecture 6 states that all umbral and Conway moonshine functions corresponding to fourplane preserving group elements play a role in the twining genera of K3 NLSMs, the physical relevance of the umbral (including Mathieu) moonshine functions corresponding to group elements preserving only a twoplane remains unclear. We highlight a number of approaches to this problem here.
One possible approach to the problem is to find a way to combine symmetries realized at different points in moduli space and in this way generate a larger group which also contains twoplane preserving elements. This approach is motivated by the fact that the elliptic genus receives only contributions from BPS states and is invariant across the moduli space. This possibility was first raised as a question “Is it possible that these automorphism groups at isolated points in the moduli space of K3 surface are enhanced to \(M_{24}\) over the whole of moduli space when we consider the elliptic genus?” in [46]. Concrete steps toward realizing this idea in the context of Kummer surfaces were taken in [17, 87, 88]. See also [50] for recent progress in the direction.
A second approach is to consider string compactifications where larger groups are realized at given points in moduli space as symmetry groups of the full theory (and not just the BPS sector). For theories with 16 supercharges, this is only possible for compactifications with less than six noncompact dimensions. For example, it was shown that there are points in the moduli space of string theory compactifications to three dimensions which admit the Niemeier groups as discrete symmetry groups [68]. In the type IIA frame, these are given by compactifications on \(K3\times T^3\). The action of these symmetry groups on the 1/2BPS states of the theory has been analyzed [68], and it would be interesting to understand the action on the 1/4BPS states.
A third approach stems from the vertex operator algebra (VOA) perspective. In [14], a close variant of the Conway module is shown to exhibit an action of a variety of twoplane preserving subgroups of \(\textit{Co}_0\), including \(M_{23}\), and yields as twining genera a set of weak Jacobi forms of weight zero and index two.^{11} In addition, the mock modular forms which display \(M_{23}\) representations appear to be very closely related to the mock modular forms which play a role in \(M_{24}\) moonshine. However, the physical relevance of this module is still unclear. A better understanding of the connection between the Conway module and K3 NLSMs could help explain Mathieu and umbral moonshine.
Finally, yet another approach is to consider compactifications preserving less supersymmetry [23, 60]. It is not unlikely that the ultimate explanation of umbral moonshine will require a combination of the above approaches.
Note that this is different from the terminology used in [16], where the name “Niemeier lattice” is reserved for the twentythree 24dimensional negativedefinite even unimodular lattices with nontrivial root systems, and hence excludes the Leech lattice.
Here and in the following, by “orientation of positive fourplanes,” we mean each of the two equivalence classes of oriented positive four planes in \(\mathbb {R}^{4,20}\) modulo O(4, 20) transformations connected to the identity.
While the Lie algebra of \(\mathcal {G}\) is uniquely determined by this description, it is an interesting problem to determine the precise group \(\mathcal {G}\) acting faithfully on the states of the string theory. We will not try to answer this question here, since it is not important for our subsequent discussion.
However, the converse is not true: It can happen that \(\Gamma _G\) admits a primitive embedding into a Niemeier lattice while \(\mathcal {F}_G\) contains no singular model, as exemplified by certain examples when \(\Gamma ^G\) is exactly fourdimensional and \(\mathcal {F}_G\) contains only an isolated point.
This is true except at points in moduli space where a noncompact direction opens up and the CFT is singular.
In the present paper, we make the standard assumption in theoretical physics that the path integral formulation of NLSM yields correct answers.
Coincidences like this occur, for example, when the dimension of the relevant space of modular forms is small. See Sects. 4.4 and 4.5 for more details.
Note that the holomorphic part of the elliptic genus in question is well defined both from a physical and mathematical point of view. From the physics perspective, the holomorphic part corresponds to the contribution from the discrete part of the spectrum [1, 44, 59, 89]. From the mathematical point of view, the holomorphic part corresponds to the holomorphic part of the harmonic Maass form [9].
Note that, in [40], the Jacobi form attached to a \(Co_0\) conjugacy class with representative g depends on the choice of an orientation for the space \(\tilde{H}(X,\mathbb {Z})\otimes \mathbb {R}\) (see the beginning of section 9 in [40]). In particular, in the explicit formulae for \(\phi _g\) (eq.(9.14) of [40]), this choice of orientation determines the sign of the parameter \(D_g\), while its absolute value depends on the \(Co_0\) conjugacy class. Therefore, \(\phi _{g,+}^\Lambda (\tau ,z)\) and \( \phi _{g,}^\Lambda (\tau ,z)\) coincide only for those classes for which \(D_g=0\). See also [14] for further discussions about this point.
It is intriguing to note that the forms of \( \mathcal {W}^c_1\), \( \mathcal {W}^c_2\) and \({\mathcal W}^q\) are closely related to the superpotentials which flow to the \(A_2\), \(D_4\) and \(A_3\) \({\mathcal {N}}=2\) minimal models, where the Atype case is given above, and the \(D_4\) case is \( W_{D_4}(\Phi _1, \Phi _2)\sim \Phi _1^3+\Phi _1\Phi _2^2. \) It would be interesting to understand whether this is connected to the fact that the twinings correspond to cases of umbral moonshine whose root systems contain copies of \(A_2\), \(D_4\) and \(A_3\), respectively.
Coincidentally, in [7], and as further discussed in [18], it was shown that this module also admits an \(M_{24}\) action; although the twining genera are no longer weak Jacobi forms, the representations are less closely related to those of Mathieu moonshine.
Similarly, a negative sign structure is given by a choice of orientation of a maximal negativedefinite subspace in \(L\otimes _\mathbb {Z}\mathbb {R}\). In the following, we will only consider positive sign structures and simply refer to them as sign structures.
Notes
Declarations
Acknowlegements
We thank John Duncan, Francesca Ferrari, Matthias Gaberdiel, Gerald Höhn, Shamit Kachru and Natalie Paquette for discussions about related subjects. The work of M. C. is supported by ERC starting Grant H2020 ERC StG 2014. S. M. H. is supported by a Harvard University Golub Fellowship in the Physical Sciences and DOE Grant DESC0007870. R. V. is supported by a grant from “Programma per giovani ricercatori Rita Levi Montalcini.” M. Z. is supported by the Mellam Family Fellowship at the Stanford Institute for Theoretical Physics.
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