Twining Genera of (0,4) Supersymmetric Sigma Models on K3

Conformal field theories with (0,4) worldsheet supersymmetry and K3 target can be used to compactify the E8xE8 heterotic string to six dimensions in a supersymmetric manner. The data specifying such a model includes an appropriate configuration of 24 gauge instantons in the E8xE8 gauge group to satisfy the constraints of anomaly cancellation. In this note, we compute twining genera - elliptic genera with appropriate insertions of discrete symmetry generators in the trace - for (0,4) theories with various instanton embeddings. We do this by constructing linear sigma models which flow to the desired conformal field theories, and using the techniques of localization. We present several examples of such twining genera which are consistent with a moonshine relating these (0,4) models to the finite simple sporadic group M24.


Introduction
To obtain a (1, 0) supersymmetric compactification of the heterotic string to six dimensions, one should choose an internal CFT with (0,4) supersymmetry and right-moving central charge c R = 6. In the realm of geometry, such CFTs arise as non-linear sigma models with K3 target. In order to satisfy the Bianchi identify for the three-form field strength H of the heterotic string dH = Tr(R ∧ R) − Tr(F ∧ F ) , (1.1) one should further embed 24 instantons into the E 8 × E 8 gauge group. If one chooses bundles V 1,2 of rank r 1,2 in the two E 8 s (which should be stable and holomorphic, and have vanishing first Chern class c 1 = 0, in the simplest case), then the left-moving fermions in the sigma model couple to the gauge connections on these bundles, and c L = 4 + r 1 + r 2 .
The explicit construction of such (0,4) CFTs is a difficult task, and computations of observables in such intricate theories are in general complicated to perform. Ideally, one would like to be able to compute the partition function of the internal conformal field theory. But more generally, one has to settle for obtaining coarser index information. One such compromise is given by the elliptic genus, Z(τ, z) = Tr(−1) F y J L q L 0 q L 0 , y = e 2πiz , q = e 2πiτ . (

1.2)
This is a graded trace over the Hilbert space of the left movers, containing further information about quantum numbers under a left-moving U (1) current algebra whose generator is J L .
In this paper, our focus will be on explicit examples of (0,4) models and their twining genera, which are close relatives of the elliptic genus. They can be defined as follows. Consider a (0,4) theory with discrete symmetry g. Then, one can modify (1.2) to Z g (τ, z) = Tr(−1) F g y J L q L 0 q L 0 , (1 .3) that is, one can take the trace with an insertion of the action of g on the physical states.
We will construct (0,4) models as gauged linear sigma models with K3 target. The basic ideas involved in constructing such sigma models with Calabi-Yau target were developed in the beautiful paper [1], and the extension to (0,2) models was discussed in detail in [2]. As (0,4) models are a simpler subset of (0,2) models, our models will be simple examples of the constructions in [2].
We will compute the twining genera by using the techniques of localization. Localization was recently used to give a very explicit formula for the elliptic genus of linear sigma models with rank one gauge groups in [3], with an extension to higher ranks appearing in [4]; a small modification of that formula suffices to compute the twining genera (1.3). Earlier results on the elliptic genera of (0,2) gauged linear sigma models appeared in [5], which also anticipated (without derivation) aspects of the residue formula of [3].
While one justification for computing the observables (1.3) is that they contain valuable information about the spectrum of an interacting conformal field theory, we had a more specific motivation for undertaking this study. There is a Mathieu moonshine relating the (4,4) sigma models with K3 target to the Mathieu group M 24 [6]. The key first piece of evidence for this moonshine was a decomposition of the coefficients of the q-expanded elliptic genus of K3, in terms of dimensions of representations of M 24 . Given such a decomposition, one can make predictions for the twining genera (1.3) for the (4,4) theories, if one inserts any element of M 24 . By finding explicit realizations of symmetries of K3 sigma models, and computing (1.3) explicitly, one can check whether these symmetries correspond to (conjugacy classes of) elements of the hypothetical M 24 . Such checks were carried out in [7,8,9] with impressive results. The existence of a graded M 24 module with the desired properties has since been proved at a rigorous level [10].
Possible extensions of this moonshine to theories with only half as much supersymmetry, including (0,4) heterotic string compactifications, were discussed in [11]. As a logical extension of that work, it is desirable to find explicit symmetries of (0,4) K3 conformal field theories and check if the twining genera (1.3) match with those of suitable M 24 conjugacy classes. This note, as well as the companion [12] to [11] which studies twining genera of (0,4) supersymmetric K3 orbifold conformal field theories, will present examples where exactly such matching can be seen.

Basic multiplets and terms in the action
We will write down (0,4) linear sigma models by working in (0,2) superspace and using vector bundles constructed as the cohomology of an exact sequence, as in [2]. The enhanced worldsheet supersymmetry is not manifest, but should be expected to emerge in the IR on general grounds when we construct models which have a large-radius interpretation as K3 sigma models.

Chiral superfields Φ satisfy
and have a component expansion with ψ a right-moving fermion. Fermi superfields Λ also satisfyD + Λ = 0, but have component expansion instead, with λ a left-moving fermion and an auxiliary field.
We will be considering (0,2) gauge theories with U (1) gauge group, so we also need to discuss the (0,2) gauge multiplet. It consists of a pair of superfields V, A whose expansion, in Wess-Zumino gauge, is given by with a,ā the left/right moving pieces of the gauge field, α,ᾱ left-moving gauginos, and D an auxiliary field. The field strength supermultiplets are The basic terms which appear in a supersymmetric action will be the following. A gauge invariant kinetic term for a charged chiral multiplet Φ with charge Q is while a gauge invariant kinetic term for a charged Fermi multiplet Λ of charge Q is The gauge kinetic term is The Fayet-Iliopoulos D-term and θ angle are (where t = θ 2π + ir plays the role of a Kähler parameter in large radius geometric phases of the theories to come). The (0,2) superpotential takes the form Here, F needs to be chosen to be a homogeneous polynomial of the appropriate degree in the charged field Φ so that (2.12) is gauge invariant.

The class of models of interest
Our interest is to describe stable, holomorphic vector bundles V with c 1 (V ) = 0 and c 2 (V ) = c 2 (T M ) over K3 surfaces M . A simple class of models which admits a gauged linear sigma model description is the following. We choose for M the Calabi-Yau hypersurface in the W P 3 with weights w i (i = 1, ..., 4), described by the equation We define V as the cohomology of the exact sequence 14) The conditions that c 1 (T M ) = c 1 (V ) = 0 and c 2 (T M ) = c 2 (V ) are captured by the Diophantine equations with d being the degree of the defining polynomial W (φ) of the K3 surface; these equations follow simply from the adjunction formula for Chern classes. The second equation in (2.15) just imposes the requirement of worldsheet gauge anomaly cancellation for the abelian gauge field.
These theories can be represented as gauged linear sigma models in the following way. Let us consider the (0,2) supersymmetric abelian gauge theory with the matter content shown in Table 1. Table 1: Field content of our (0,2) sigma models 2) chiral multiplets, while Λ a and Γ are Fermi multiplets. For our (0,2) superpotential we choose with W , F a coinciding with the data in the definition of the K3 hypersurface and the bundle V above. One can verify, as in [2], that in the limit of large r, this theory flows to the sigma model governed by the geometric objects (2.13) and (2.14), with the scalars living on the hypersurface (2.13) while the left-moving fermions transform as sections of the bundle (2.14). Of course, as one varies the Fayet-Iliopoulos parameters in such a gauge theory, other interesting phases can arise (with Landau-Ginzburg orbifold phases being a prototypical such phase).
We will need to generalize this construction in a trivial way, in order to capture the geometry of two non-trivial bundles V 1,2 which we embed into the two E 8 s. The appropriate generalization is to introduce two chiral analogues of the P field P 1,2 , with charges m 1,2 , and two sets of Fermi multipets Λ a 1 and Λ α 2 of charges n a and q α , with a = 1, ..., r 1 + 1 and α = 1, ..., r 2 + 1. The superpotential is now with F a and G α defining the bundles V 1,2 through exact sequences as in (2.14). The constraints on the Chern classes now become Again, the second equation in (2.18) is required for gauge anomaly cancellation, and is interpreted in space-time as implementing the condition which is required to satisfy the Bianchi identity (1.1).
Intuitively, the equation (2.19) means that in perturbative supersymmetric heterotic models on K3, one should choose non-negative integers n (1) , n (2) with and place n (1) and n (2) gauge instantons in the two E 8 s. Our goal in the next section will be to show that in a variety of examples constructed as above, reflecting distinct choices of n 1,2 , one can find (0,4) sigma models with discrete symmetries g whose twining genera (1.3) are consistent with the properties expected from Mathieu moonshine for (0,4) models. This strengthens the case made in [11] that moonshine extends to a portion of the web of 4d N = 2 (or 6d N = 1) supersymmetric heterotic string theories, as well as their type II (or F-theory) Calabi-Yau duals.

Specific examples of models and discrete symmetries
We will focus on four classes of specific models with different values of n (1) and n (2) , but it should be clear that many other models exist and could be fruitfully analyzed in this way. In each case, we just discuss some simple symmetries which arise for easy choices of the defining data; we are not exhaustive. We label the models by the instanton numbers (n (1) , n (2) ) chosen in each. The four models we will study are:

Model 1: A (24,0) model
For our first example, we will study the theory with d = 4 and w i = 1, 1, 1, 1. To begin with, we can choose the defining data of the target manifold to be i.e. the Fermat point in the moduli space of this K3 hypersurface. The bundle is defined by choosing For generic defining data, this model simply defines the (0, 4) model obtained by deforming the tangent bundle of K3 away from the (4,4) supersymmetric locus, while extending the rank from SU (2) to SU (3) (by partially Higgsing the E 7 space-time gauge group with a 56 of E 7 ). It has instanton numbers (n (1) = 24, n (2) = 0). The (4,4) theory was studied in more detail in [14]; we simply discuss this model here to provide a warm-up on more or less familiar territory.
We can study several simple symmetries in the Fermat K3. We will study three: 1. The Z 2 symmetry which acts as The Z 3 symmetry which acts as a permutation of cycle shape (123) on Λ 1,2,3 and Φ 1,2,3 .
3. The Z 4 symmetry We can also obtain more elaborate symmetries by choosing slightly different data. For instance, if we choose a complex structure then we can find a Z 5 symmetry: 4. Z 5 symmetry: Defining data for the vector bundle which respects this symmetry could include e.g. F a (Φ) = ∂W ∂Φa or suitable variants.
Another K3 which admits an interesting symmetry has the complex structure This surface admits the Z 7 symmetry: 5. Z 7 symmetry: , Again suitable defining data for the bundle could be F a (Φ) = ∂W ∂Φa with other choices also possible.
Again, these fermions are from V 1 , and one should choose data F 1,2 (φ) consistent with the symmetry.

Model 3: A (14,10) model
Now, we work on the K3 hypersurface embedded in W P We consider two symmetries in this model: 1. A representative Z 2 symmetry is, for instance, with the Λs being fermions involved in the construction of V 1 . Simple choices of the F a (Φ) are consistent with such a symmetry.
2. We can consider a Z 3 symmetry as follows: with the two charge 1 fermions in V 1 , Λ 2,3 , rotating as

Computation of the twining genera
In this section, we compute the twining genera under the various model symmetries described in §2.3. We begin by outlining the general strategy and formulae that are relevant, and then simply present the results of applying these formulae to the various cases. Our work relies heavily on the elegant residue formula derived recently in [3].

Residue formula for elliptic genus
The elliptic genus was first discussed in [15,16,17]. Its application to string compactification was pioneered in [18], and it was first computed by localization in (2,2) supersymmetric Landau-Ginzburg models in [19] and for (0,2) models in [5]. It has recently been the focus of attention in, for instance, [3,20,4] The formalism we discuss only assumes N = (0, 2) supersymmetry, though our application will be to (0, 4) theories . Although in many discussions of the elliptic genus in theories with (2, 2) supersymmetry the left-moving R-symmetry plays a crucial role, here there is no longer a left-moving R-charge. However, the models we consider will have an extra U (1) global current J L , and we will grade by the quantum number under the associated charge in the elliptic genus. In the models described in the previous section, J L = 0 for Γ and Φ i , and for the Λ a,α , J L = −1, whereas for the P a,α , J L = +1.
We follow the discussion of (0, 2) abelian gauge theory in [3]. Let us define u to be the holonomy of the U (1) gauge field around the cycles of the torus with t, s the temporal and spatial directions, and τ the modular parameters of the torus. The elliptic genus is given by the graded trace Obtaining a formula for (3.2) via localization involves doing an integral over the Wilson lines u of the abelian gauge field.
This integral localizes to a sum of contour integrals around loci (in the moduli space of flat connections) where some of the fields become massless; we refer to these as singular points. Let us consider a general (0, 2) U (1) gauge theory, with a number of gauge charged chiral and Fermi multiplets Φ i and Λ a , as well as one vector multiplet. Suppose that the charges of the chiral and Fermi multiplets under the gauge and U (1) global symmetry are Q i,a and J i,a respectively. Then, defining the expression that has been obtained for the elliptic genus is [3] Z(τ, z) = −η(q) 2 where M + is the relevant set of singular points. 1 These points are defined as the solutions to the equation .
That of a Fermi multiplet with global U (1) charge J is Finally, the contribution of a (0,2) vector multiplet is independent of u. The product of these expressions over all multiplets present in a given theory, integrated over the u-plane, can be reduced to the formula (3.4).

K3 elliptic genus
The standard results for the elliptic genus of K3 (or in the language of quantum field theory, for the N = (4, 4) sigma model with K3 target) is [18] which has the expansion Z K3 ∼ 2 y + 20 + 2y + 20 For a (0, 4) model on K3 with rank r gauge bundle, the elliptic genus is given by as derived in [5]. It is easy to check that applying (3.4) to our models of §2.3 agrees with the result (3.11), with r = r 1 + r 2 the sum of the ranks of the bundles embedded in the two E 8 s.

Residue formula for twining genera
Our real interest is to compute the elliptic genus with the insertion of a symmetry operator, g, into the path integral for various particular (n 1 , n 2 ) instanton embeddings. We can do this with a slight modification to the computation of the untwined elliptic genus.
Consider an operator g which acts on chiral and Fermi multiplets as 13) and is a symmetry of the action. When inserting this operator into the path integral, it modifies the contribution due to the chiral and Fermi multiplets. The contribution of a (0, 2) chiral multiplet Φ i to the integrand in (3.4) becomes (3.14) while one obtains iθ 1 (q, e 2πiβa y Ja x Qa ) e iπβa η(q) (3.15) from the twined Fermi multiplet Λ a . One then sums over the (now shifted) poles that previously contributed to the elliptic genus -the detailed locations of the poles in M + on the u-plane, as well as their orders, can be modified depending on the g charges of the fields involved.
Denote the elliptic genus of the (4, 4) theory twined by a conjugacy class g of M 24 by Z g . 2 Then we expect the twined elliptic genus of an (n 1 , n 2 ) model to decompose as i.e. a product of twined (4, 4) genera and twined SO(2r − 4) characters.
In writing (3.16), we are making two important assumptions: 1) We assume that the M 24 module which is relevant in the moonshine for (0, 4) models with arbitrary instanton embeddings, has the same representations at each level as the one which arises in the (4, 4) theory. Evidence for this was presented already in the new supersymmetric index computations of [11], which are valid for all instanton embeddings.
2) We are assuming that the factor of in the elliptic genus of a (0, 4) theory with rank r bundle transforms as an element of the spinor minus conjugate spinor representation of SO(2r − 4). This is motivated by the results to appear in the companion paper about (0, 4) orbifolds [12]. Heuristically, the SO(2r − 4) symmetry could appear manifestly in a field theory where one deformed the bundle V 1 ⊕ V 2 to be an SU (2) bundle with instanton number n (1) + n (2) . As the elliptic genus is invariant under such smooth deformations, this may explain the appearance of such factors (related to further 'hidden symmetries') in the twining genera of (0, 4) sigma models.
We now show that our results for the set of models discussed in §2.3 satisfy the assumption (3.16). We view this as a check of M 24 moonshine for (0, 4) theories with a variety of instanton embeddings.

Model 1
Here, we considered five symmetries in §2.3.1: a Z 2 symmetry, a Z 3 symmetry, a Z 4 symmetry, a Z 5 symmetry and a Z 7 symmetry. The results for the twining genera are: Here, Z 2A , Z 3A , Z 4A , Z 5A and Z 7A are the corresponding twining genera of the (4, 4) elliptic genus with an insertion in those M 24 conjugacy classes (see appendix B). The first argument of the theta function has been suppressed here and below.

Model 2
We considered two symmetries in §2.3.2: a Z 2 symmetry and a Z 4 symmetry. The results for the twining genera are:

Model 3
Here, we also considered two symmetries in §2.3.3 -a Z 2 and a Z 3 . The results are: (3.20)

Model 4
We considered a Z 3 symmetry in §2.3.4. The result is

Discussion
In this note, we used the recently derived localization formula for the elliptic genus of (0, 2) supersymmetric rank one two-dimensional gauge theories [3] to compute twining genera of (0,4) gauged linear sigma models with K3 target. We did this for a variety of discrete symmetries in (0, 4) models with four different sets of instanton numbers (n (1) , n (2) ).
In several cases, we found that the simple discrete symmetries give twining genera which are consistent with those of M 24 elements of the same order, with the trace in the elliptic genus taken over the M 24 module conjectured to exist in [6] and constructed in [10]. These direct computations are an analogue, for a conjectural (0,4) moonshine with various instanton numbers, of the twining calculations in [7,8,9]. Interestingly, the 3B conjugacy class of M 24 , which does not descend from the classical symmetries of K3 surfaces (as they lie in M 23 [21]) and which has been elusive, appears here in one of the first cases we examined.
It should not be difficult to find linear sigma models which admit relatively elaborate discrete symmetries. The Z 5 and Z 7 examples of §2.3.1 were found by using a strategy developed in [22], and it seems quite plausible that one can write down examples which show twining in higher order M 24 conjugacy classes in this way. It should also be instructional to go through the list of e.g. the 'famous 95' weighted projective K3 hypersurfaces of Reid [23], and see which of them admit interesting symmetries; this may lead to interesting new examples even in the (4, 4) theory.
A major question which remains is the proper interpretation of the evidence presented here, as well as in [11,12], for a moonshine relating heterotic (0, 4) theories (and their type II Calabi-Yau duals) to M 24 . The observations of [14] indicate that M 24 does not play a canonical role as an embedding group for symmetries of (4, 4) superconformal theories with K3 target. The symmetries available in (0, 4) theories will of course only be richer; developing a classification would be very interesting. Failing a complete classification, a detailed study of particular families with large symmetry groups (extending the philosophy of [24] from the (4, 4) case) could also prove illuminating. It is even within the realm of possibility that some (0, 4) superconformal theory, or perhaps a non-perturbative heterotic vacuum with small instantons replacing the gauge bundles V 1,2 , could manifest the full symmetry and 'explain' the appearance of M 24 in the elliptic genus. But other interpretations of the moonshine, in terms of Rademacher sums arising naturally in AdS/CFT [25], or in terms of supersymmetric indices of NS5 branes [26], are also quite promising. Related directions to explore are discussed in [27,28].

(B.4)
In practice, one can find simple closed-form expressions for Z g as discussed in detail in e.g. [7,8,9]. The first few terms in the q-expansions of the Z g for various conjugacy classes are shown in Table 3.