${\cal N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$

We study ${\cal N}=2$ compactifications of $E_8\times E_8$ heterotic string theory on orbifolds of $K3 \times T^2$ by $g'$ which acts as an $\mathbb{Z}_N$ automorphism of $K3$ together with a$1/N$ shift on a circle of $T^2$. The orbifold action $g'$ corresponds to the $26$ conjugacy classes of the Mathieu group $M_{24}$. We show that for the standard embedding the new supersymmetric index for these compactifications can always be decomposed into the elliptic genus of $K3$ twisted by $g'$. The difference in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of $K3$ twisted by $g'$. We work out in detail the case for which $g'$ belongs to the equivalence class $2B$. We then investigate all the non-standard embeddings for$K3$ realized as a $T^4/\mathbb{Z}_\nu$ orbifold with $\nu = 2, 4$ and $g'$ the $2A$ involution. We show that for non-standard embeddings the new supersymmetric index as well as the difference in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the difference in number of hypermultiplets and vector multiplets in the spectrum.


Introduction
String compactifications with N = 2 supersymmetry has been extensively investigated as an important testing ground for string dualities. The canonical example of such a compactification is the heterotic string on K3 × T 2 . In the context of string dualities this theory was first investigated in [1]. The various theories studied differed on how the spin connection was embedded in the gauge connection. A simple method of explicitly constructing these compactifications is to realize K3 as a T 4 /Z ν orbifold with ν = 2, 3, 4, 6. A comprehensive list of these orbifold compactifications together with all possible embeddings of the spin connection in the gauge connection is given in [2,3]. Supersymmetric observables like the new supersymmetric index or the difference in one loop gauge threshold corrections can be shown to be independent of the orbifold realization [2,4,5].
An important observable in these compactifications is the new supersymmetric index [4][5][6][7][8][9] which is defined by (1.1) Here the trace is performed over the Ramond sector in the internal CFT with central charges (c,c) = (22,9). F refers to the world sheet fermion number of the right moving N = 2 supersymmetric internal CFT. Recently it has been observed that the new supersymmetric index of K3 × T 2 which enumerates BPS states in these compactifications admits Mathieu moonshine symmetry [10], see [11] for a review of aspects of moonshine. This observation was generalized in [12] which considered orbifolds of K3 × T 2 by g acted as a Z N automorphism in K3 and and 1/N shift on one of the circles of T 2 . It was observed that for the standard embedding the new supersymmetric index admits a decomposition in terms the elliptic genus of K3 twisted by g . This ensures that the new supersymmetric index admits an expansion in terms of the McKay Thompson series associated with g embedded in the Mathieu group M 24 . It was also observed in [12] that the difference in one loop gauge corrections to gauge couplings with Wilson lines for these compactifications can be written in terms of Siegel modular forms corresponding to the theta lift of the twisted elliptic genus of K3.
The g considered in these compactifications of [12] were restricted in the conjugacy class pA of M 24 with p = 2, 3, 5, 7. In fact only the class 2A was explicitly constructed 1 , and the analysis was restricted to the standard embedding. In this paper we study compactifications of the E 8 × E 8 heterotic string theory on orbifolds of K3 × T 2 by g in more detail.
We show that for all g corresponding to the 26 conjugacy classes of M 24 and for compactifications which involve the standard embedding of the spin connection of K3 into one of the E 8 's the resultant new supersymmetric index always can be written in terms of the elliptic genus of K3 but twisted by g . The standard embedding breaks the gauge group to E 7 × U (1) × E 8 . The difference in one loop corrections of the gauge groups E 7 and E 8 are automorphic forms of SO(2 + s, s; Z) with s = 0, 1. For s = 0, the automorphic forms are functions of Kähler, complex structure of the torus T 2 while for s = 1 they are also functions of the Wilson line embedding in either of the gauge groups. We show that these automorphic forms are obtained as theta lifts of the elliptic genus of K3 twisted by g . We demonstrate these statements explicitly for 2 examples. We first consider the situation when K3 is realized as T 4 /Z 4 and then construct the corresponding g action corresponding to the 2A conjugacy class. We show the new supersymmetric index is determined by the corresponding twisted elliptic genus. This result is identical to that obtained in [12] when K3 is realized as the orbifold T 4 /Z 2 which illustrates that the new supersymmetric index is independent of the realization of K3. In the second example we consider the situation when K3 is realized as a rational conformal field theory based on the affine algebra su(2) 6 and for g belonging to the conjugacy class 2B studied in [14]. For this situation we show that that the new supersymmetric index is determined by the elliptic genus of K3 twisted by the 2B action.
We then examine non-standard embeddings of K3 × T 2 compactifications. This is done by considering all the non-standard embeddings in which K3 is realized as a T 4 /Z 2 as well as T 4 /Z 4 orbifold and the action of g in the conjugacy class 2A. We study the spectrum and then evaluate the new supersymmetric index for these compactifications. The results for the spectrum are summarized in tables 6,7,8,9,10. We show that the new supersymmetric index classifies all the models into 4 distinct types depending on the difference of the number of hypermultiplets and vector multiplets, N h − N v of the model. The result can be read off using the table 13 and equation (4.7) In each case we see that the new supersymmetric index again admits a decomposition in terms of the elliptic genus of K3 twisted by g . However there is also a dependence in N h − N v . We then evaluate the difference in one loop gauge coupling corrections for all these models with the Wilson line and show that they result in SO(3, 2; Z) automorphic forms. The automorphic forms for all the models are entirely determined by the instanton numbers of the embeddings as well as N h − N v of these models. The result can be read off using the tables 14, 15 and equation (4.19) The organization of the paper is as follows. In section 2 we prove that for the standard embedding , compactifications on orbifolds of K3 × T 2 result in a new supersymmetric which can always be written in terms of the elliptic genus of K3 twisted by g . Section 3 works out in detail for the situation when K3 is realized as T 4 /Z 4 with g ∈ 2A and when K3 is realized as a rational conformal field theory based on the su(2) 6 affine algebra with g ∈ 2B. In section 4 we first introduce all the embeddings in which K3 is realized as a T 4 /Z ν orbifold with ν = 2, 4 and g ∈ 2A and evaluate the spectrum, the new supersymmetric index and the difference in one loop gauge thresholds. Section 5 contains our conclusions. Appendix A contains the notations, conventions and a list of identities used in the paper, appendix B contains the details of evaluating one loop threshold integrals. Finally the appendix C summarises the content of mathematica files which were used to arrive at some of the results in the paper.

Standard embedding
In this section we first define N = 2 supersymmetric compactifications of the E 8 ×E 8 heterotic string theory on orbifolds of K3 × T 2 by g in which the spin connection of K3 is embedded in one of the E 8 's in the standard manner. g acts as a Z N automorphism of K3 together with a 1/N shift along one of the circles of T 2 . The automorphism g corresponds to any of the 26 conjugacy classes associated with the Mathieu group M 24 by which one can twist the elliptic genus of K3 [ [15][16][17].
We define the standard embedding as follows. Let the current algebra of one of the E 8 's be realized in terms of left moving fermions λ I , I = 1, · · · 16. The other E 8 can be realized in terms of its bosonic lattice or the fermions λ I . The gauge connection is assumed to have the structure Here the second and third Hilbert spaces refer to the D6 lattice and the E 8 lattice respectively and the the last refer to the CFT on T 2 . With this decomposition, we can now specify the action of g . The g acts as a Z N automorphism on the (6,6) CFT H together with a 1/N shift on one of the circles in H (2,3)

New supersymmetric index and twisted elliptic genus of K3
Let us now evaluate the new supersymmetric index on the internal CFT given in (2.2).
The right moving Fermion number F can be written as the sum of the Fermion number on T 2 together with the Fermion number on K3 Then it is easy to see that the because of the right moving Fermion zero modes on T 2 , the only contribution to the index arises from Again examining the trace we can see that the contributions from left moving bosonic and fermionic oscillators on T 2 cancel. Thus it is only the zero modes on T 2 and the left moving bosonic oscillators on T 2 which contribute to the index. With these arguments we see that the trace reduces to The sum over the sectors (r, s) is implied and r, s run from 0 to N − 1. The origin and the definition of each term in the index is as follows.
1. The term Γ (r,s) 2,2 η 2 arises from the lattice sum on T 2 together with the left moving bosonic oscillators. The lattice sum is defined as T, U are the Kähler and complex structure of the T 2 . Note that the lattice sum is the only part of the index that contains anti-holomorphic dependence. Furthermore the insertion of g and the twisted sectors of g are taken care of by the phase e 2πim 1 s/N and the fact the winding modes are shifted from integers by r N . 2. The terms in the square bracket arises from evaluating the index on the lattice D6 together with the combined D2K3. Note that the partition function on the D6 lattice in the various sectors are given by While the indices on the combined D2K3, (6, 6) conformal field theory are given by We will relate them to the twisted elliptic genus of K3 below.
3. Finally the term E 4 (q) η 8 (τ ) arises from the partition function of the second E 8 which is untouched in the standard embedding. E 4 is the Eisenstein series of weight 4.
We now show that the indices in (2.9) are related to the twisted elliptic genus of K3 by g . In indices given in (2.9) note that the spin connection of the K3 is coupled to the fermions in D2 conformal field theory and therefore trace can be thought of as a trace in the K3 super conformal field theory with central charge (6,6). Let us examine the twisted elliptic genus of K3 which is defined as (2.10) Here g belongs to to automorphism related to the 26 conjugacy classes of M 24 . Since this theory admits a N = 2 spectral flow we can relate the trace over the various sectors in (2.9) by the following equations .
From (2.6) and (2.11) we see that the new supersymmetric index for compactifications which involve the standard embedding admits a decomposition in terms of the elliptic genus of K3 twisted by g . This decomposition then can be used to show that the new supersymmetric index can be expanded in terms of the MacKay-Thompson associated with g embedded in M 24 following the arguments of [10,12].

New supersymmetric index in terms Eisenstein series
Let us further simplify the expression the expression for the new supersymmetric index for the standard embedding. The elliptic genus of K3 twisted by g in general can be written as where the Jacobi forms A(τ, z) and B(τ, z) are given by , B(τ, z) = θ 2 1 (τ, z) η 6 (τ ) . (2.13) The numerical coefficients α , g β g and the form f (0,1) g (τ ) depend on the twist g . For example, for the conjugacy class pA with p = 2, 3, 5, 7 of M 24 we find and A comprehensive list of the twisted elliptic genus for all the 26 conjugacy classes of M 24 can be found in [16]. All the remaining elements of the twisted elliptic genus F (r,s) (τ, z) can be obtained by modular transformations using the relation In (2.16) the indices cs + ar and ds + br are taken to be mod N where N is the order of g . Using this information of the twisted elliptic genus we can write the new supersymmetric index for the standard embedding given in (2.6) in terms of Eisenstein series. Substituting the following identities in (2.6) and using (2.11) we obtain Recall that only the lattice sum is dependent on both (τ,τ ) while the Eisenstein series E 6 , E 4 as well as f (r,s) are holomorphic in τ . Furthermore in the (2.19) sum over r, s from 0, · · · N − 1 is understood.

Difference of one loop gauge thresholds
Now let us evaluate the gauge threshold corrections with Wilson line turned on in the untouched E 8 lattice, we call this gauge group G and the broken E 8 , G . From the discussion in [5], [2] and [12], we see that the new supersymmetric index with Wilson line becomes (2.20) The presence of the Wilson line introduces an additional moduli V and with T, U . The lattices sums now are given by Γ (r,s) 3,2 (q,q) = m 1 ,m 2 ,n 2 ,b∈Z, The product ⊗ and function E 4,1 are defined in the appendix A. The one loop corrections to the gauge coupling G is defined by the following integral over the fundamental domain where B can be written in terms of the new supersymmetric index with the Wilson line as follows (2.24) The constant b(G) in (2.22) can be fixed by demanding that the integral is well defined in the limit τ 2 → ∞. The details which are involved in arriving at the integrand (2.23) are given in [12] where the class 2A was discussed in detail. Essentially the action of B G is to convert the lattice sum with the Wilson line E 4,1 →Ẽ 2 E 4,1 − E 6,1 . This occurs because of is summing over the lattice weighted with the charge vectors.
Similarly the one loop corrections to the gauge coupling G is defined by an integral of the same form in (2.22), with the integrand given by (2.25) Here note that E 6 →Ẽ 2 E 6 − E 2 4 . Using the identities 1 we evaluate the difference in the one loop thresholds integrands which results in Thus the difference in the one loop corrections to gauge couplings is given by There is a constant term that we have ignored in the integrand which is necessary to make the integral well defined in the τ 2 → ∞ limit. From (2.28) we conclude that for compactifications on the orbifold (K3 × T 2 ) by g involving the standard embedding, the difference in the one loop thresholds is the automorphic form of SO(3, 2; Z) which is obtained by the theta lift of the elliptic genus of K3 twisted by g . To obtain the threshold correction without the Wilson line one can take the limit V → 0 in (2.28). Then the automorphic form SO(3, 2; Z) reduces to SO(2, 2; Z modular forms.

Standard embedding: 2 examples
In this section we will discuss in detail 2 examples that demonstrate the for standard embeddings, the new supersymmetric index can be written in terms of the twisted elliptic index. The first example deals with the 2A orbifold of K3 in which K3 is at its T 4 /Z 4 limit. The second example deals with the recent construction of the 2B orbifold of K3 [14]. In this section we will construct the orbifold of K3 by g where g belongs to the class 2A. The well studied method of obtaining this orbifold is to realize the K3 CFT as a T 4 /Z 2 orbifold as discussed in [18]. Here we will consider the 2A orbifold when K3 is at the orbifold limit T 4 /Z 4 . As far as we are aware the construction is new. This will enable us to investigate the spectrum and the threshold corrections of all the non-standard embeddings of heterotic string at the orbifold T 4 /Z 4 discussed in [2] after the g action.
We define the orbifold of K3 by g as follows. Let us first consider T 4 × T 2 with co-ordinates x 1 , x 2 parameterizing T 2 and y 1 , y 2 , y 3 , y 4 labelling T 4 . Then K3 is realized by the Z 4 which is action given by g s : (x 1 , x 2 , y 1 + iy 2 , y 3 , +iy 4 ) ∼ (x 1 , x 2 , e 2πi s/4 (y 1 + iy 2 ), e −2πi s/4 (y 3 + iy 4 )), s = 0, 1, 2, 3. (3.1) This orbifold limit of K3 is well known and discussed in [19]. We now consider the g orbifold which is a Z 2 action given by g : (x 1 , x 2 , y 1 , y 2 , y 3 , y 4 ) ∼ (x 1 + π, x 2 , y 1 + π, y 2 + π, y 3 + π, y 4 + π). (3.2) We will first show that the twisted elliptic genus remains the same as that when K3 is realized as a T 4 /Z 2 orbifold. This result in fact a test that the orbifold action given in (3.1) and (3.2) in fact K3 twisted by the element 2A. We will then evaluate the spectrum of heterotic string compactified on this orbifold K3 × T 2 for the standard embedding. Using the orbifold action we will explicitly show that the new supersymmetric index admits a decomposition in terms of the twisted elliptic genus. Therefore this is a verification of the result in the previous section that the new supersymmetric index for compactifications on orbifolds of K3 in any standard embedding just depends on the twisted elliptic genus of K3. We then evaluate the difference in one loop gauge thresholds and show that indeed the resulting modular form is the theta lift of the elliptic genus of K3 twisted by the element 2A.

Twisted elliptic genus
The twisted elliptic genus under under the orbifold (3.1) and (3.2) is given by the index Here the trace is taken over theory of 4 free bosonic coordinates y 1 , y 2 , y 3 , y 4 and 4 free fermions which form their superpartners, F L , F R are the left and right moving fermion numbers respectively. We have suppressed the shifts L 0 − 1/4,L 0 − 1/4 in the definition of the index. Let us further define the trace To evaluate each sector of the above twisted elliptic genus we will need the fixed point under the elements g a g r and what elements preserve these fixed points. This information is summarized in table 1.
Fixed points g g g 2 g 3 g g g g 2 g g 3 Let us discuss the twisted elliptic genus for each of the sectors. The sector (0, 0) is easiest to deal with. Since there are no twists in g or insertions of g to deal with we see that the trace reduces to where Z K3 is the elliptic genus of K3.
Let us now examine the sector (0, 1). We see from table 1, that a single insertion of g does not preserve any of the fixed points. Thus we have Therefore we need to look at F(0, 0; b, 1) and F(2, 0; b, 1). Evaluating the trace in the untwisted sector we see the contributions are , .
The vanishing of the first set of equations in (3.7) is due to the fact that the fixed points in the relevant traces are not invariant under g or g 2 g insertions as can be seen from the table 1. The numerical factors in the last line equations in (3.7) is due to presence of 4 fixed points in these twisted sectors. Now summing up the contributions we obtain The equality in the second line of the above equation is due to identities involving the theta functions. Thus we see that the twisted elliptic genus of the orbifold given in (3.1), (3.2) belongs to the class 2A.
Though the other sectors of the twisted elliptic genus can be obtained by modular transformations, for completeness we provide some of the details. Lets examine contributions to F (1,0) . Due to the presence of right moving Fermionic zero modes we obtain F(0, 1; 0, 0) = 0. Now the following vanish This is because due to the insertions of powers of g the trace can contribute only if there are zero modes in the winding sector. However since this sector is twisted in g , the winding modes are all half integer modded and cannot vanish. The only non-trivial contributions arise from the following , for a = 1, 3, b = 0, 2, (3.10) .
The rest of the indices vanish due to the fact that the fixed points in those sectors are not invariant with the relevant insertions of g, g in the trace. Summing up the contributions it can be seen that Finally due to the same reasons we see that the only contributions to F (1,1) arise from , .
Again summing up the contributions leads to (3.13) To conclude, from (3.4), (3.8), (3.11) and (3.13) we see that the twisted elliptic genus is identical to the class 2A first evaluated in [18] using K3 in the T 4 /Z 2 orbifold limit.

Massless spectrum
In this section we will derive the massless spectrum of heterotic string theory compactified on the orbifold given in g in (3.1) and g (3. 2) with standard embedding. In orbifold language the standard embedding of is achieved by accompanying the Z 4 action (3.1) together with the shift (3.14) in the E 8 × E 8 lattice. The spectrum of the T 4 /Z 4 with the standard shift was first studied in [20]. We will follow the discussion of [21] which set up the general discussion for studying orbifold compactifications of heterotic string theory which preserve N = 2 supersymmetry. The orbifold action g (3.2) does not produce any fixed points and therefore preserves N = 2 supersymmetry. Thus the massless spectrum organizes into the 4 dimensional N = 2 gravity multiplet coupled to N v vectors and N h hypers. The massless states of the theory in the g n twisted sector is determined by setting left and right masses to zero Here P is the E 8 × E 8 lattice vector which is generically of the form The 8 dimensional lattice vector P E 8 can belong to either the vector or the spinor conjugacy class which we denote by E n is the shift in the zero point energy on the ground state due to the twisting and is given by where ν = 4 for the T 4 /Z 4 orbifold and n = 0, 1, 3, 4. r is a SO(8) weight vector with v is a 4 dimensional vector given by Further conditions on r, v, P so that we obtain massless states m L = m R = 0 will be discussed below. The degeneracy of the massless states can be obtained from [21] and χ(n, m) refers to the number of fixed points in the g n twisted sector which are invariant under the action of g m . ρ is the phase by which the oscillators in the T 4 are rotated by the Z 4 action. In the untwisted sector n = 0 we have Our goal is to obtain the spectrum when there is a further action by the Z 2 group g given in (3.2). The first thing to note is that there are no massless states arising from the twisted sectors of g . This is because all these states have half integer Kaluza-Klein modes on T 4 and therefore they are massive. Thus the only change in obtaining the massless spectrum is that the degeneracy given in (3.23) changes to where χ (g ) is the number for fixed points in the g n twisted sector invariant under the action of g m g . Essentially we have inserted the projection over g . In the untwisted sector and again the phases in (3.26) just implement the projection of the spectrum under g m . For the twisted sector, from the tabel 1 we obtain We are now ready to obtain the spectrum of the model.

Untwisted sector
It is clear from (3.24), (3.27) and (3.26) we see that there is no change in the spectrum for the untwisted sector. Thus the untwisted sector remains the same as that worked out earlier in [21]. This sector contains the N = 2 gravity multiplet and the N = 2 vectors. The gauge group breaks from Thus the Non-Abelian N = 2 vector multiplets are in the 133 of E 7 and the 248 of E 8 . In the untwisted sector there are 2 singlet hypers under E 7 × E 8 which we denote as (1, 1) and 2 hypers charged as (56, 1). The twisted sector consists of only hypermultiplets Twisted by g and g 3 From (3.25), (3.28) and (3.26) we see that the degeneracies in the g 2 and g 3 twisted sector becomes half of the theory on the orbifold (T 4 /Z 2 ) × T 2 worked out in [21]. In fact the states in the g 3 twisted sector form the anti-particles of the states in the g twisted sector. The hypers for the g orbifold are 2(56, 1) + 16(1, 1) 3 .

Twisted by g 2
It in only in this sector we really need to explicitly work out the details of the states and using the formula (3.26). For massless states in the twisted sector we have the conditions Using the equations (3.20), (3.22 and (3.29) we see that p R given in (3.15) indeed vanishes for N R = 0. Lets examine the condition p L = 0.
1. For N L = 0 in the g 2 twisted sector we see p L = 0 results in the condition This condition can only be satisfied by two ways. Firstly we can take the lattice vectors in both the E 8 's in the vector conjugacy class. Thus we have which in turn can be satisfied by n 1 = 0, n 2 = 1 or n 1 = −1, n 2 = 0 with one of the n j = ±1, j = 3, 4, 5, 6, 7, 8. The restriction that these are in the first lattice comes from the condition in the last line of (3.18). All together this results in 24 solutions. Now the second choice of lattice vectors is, in which we have the spinor conjugacy class in the first E 8 and the vector class in the second E 8 . Therefore (3.30) reduces to Here we can have n 1 = −1, n 2 = 0 and any odd number of the 6 n j s as 0 or -1 which can be achieved by 32 ways ( 6 C 1 + 6 C 3 + 6 C 5 = 32). The 24 + 32 = 56 solutions of (3.31) and (3.32) form the (56, 1) dimensional representation of E 7 ×E 8 . Let us now evaluate the degeneracy of these states. They are solutions to the mass shell condition and satisfy P · V = −1/4, and have ρ = 0. Using (3.29) and the values of v and V from (3.22) and (3.14) respectively We find that ∆(2, 1) = 1. Then from (3.26) we see that the degeneracy of these states is D(2, g ) = 3, where we need to divide by 2 to account for the anti-particles. This equation admits two solutions: n 1 = n 2 = n j = 0 and n 1 = −1, n 2 = 1, n j = 0 which have P · V = 0. Evaluating the phase ∆(2, 1) for ρ = ±1/4 we obtain ∆(2, 1) = ±1. The degeneracy from (3.26) for these states is given by 2×(3+1) = 8, here again we are not counting anti-particles. The 2 factor arises due to the 2 solutions for (3.34) Finally since we have two pairs of oscillators with ρ = ±1/4 the total number of states is given by have 2 × 8 = 16 These states are singlets with respect to the E 7 × E 8 , therefore 5 .
To summarize the spectrum of the g orbifold of T 4 /Z 4 with the standard shift of (3.14) consists of a N = 2 gravity multiplet with a gauge multiplet in the (133, 248) of E 7 × E 8 and a U (1). The hypermultiplet content is summarized in table 2. Evaluating N h − N v = −12. For comparison we have also summarized the hypermultiplet content of the same model without the g model in table 3. The vector multiplet content is the same. N h − N v = −244 for this model which is dictated by anomaly cancellation since this model admits a lift to a chiral 6d theory unlike the g orbifold. This phenomenon of the vector multiplet being invariant but the reduction of the number of hypers by the action of g was also observed in [12]. In the subsequent section we will verify that the N h − N v = −12 for the g orbifold by evaluating the new supersymmetric index.   Table 3: Hypermultiplet content of T 4 /Z 4 × T 2 with the standard embedding.

The new supersymmetric index
In this section we will evaluate the new supersymmetric index for the orbifold defined by the actions (3.1), (3.2) with the shift in (3.14) in E 8 × E 8 . We adapt the method developed in [2] to incorporate the additional g orbifolding action. Evaluating the trace, the new supersymmetric index given in (2.3) splits into the following sectors First note that the anti-holomorphic dependence in q occurs only in the lattice sum Γ (r,s) 2,2 (q,q) Let us define each of the component in (3.35). The trace over the T 4 directions is given by Here the left moving CFT consists of 4 free bosons with c = 4 and the right movers consists of 4 free bosons and 4 free Fermions which is in the Ramond sector. The F R is the fermion number of the right moving states. The explicit expressions for this trace using the orbifold action in (3.1), (3.2) is given by . (3.37) The coefficients k (a,r,b,s) for the various values of (r, s) are given by the following matrices Note that rows and columns are labelled by a and b respectively. The coefficients for (r, s) = (0, 0) are identical to the situation without the g orbifolding. The remaining coefficients can be easily obtained by using the same arguments discussed in section while evaluating the twisted elliptic genus of this orbifold. The Eisenstein series E 4 (q) in (3.35) results from the partition function of the untouched E 8 lattice which is not coupled to the spin connection of K3. The partition function of the first E 8 lattice with the shifts are given by Also in the Z 2 subgroup sector we have The definition of the generalized Jacobi theta functions is given by Note that θ 1 (τ, z) = θ [ 1 1 ] (τ, z) In the above equation when the argument of the θ-function is not explicitly mentioned, it is understood that it is evaluated at z = 0 and at τ .

We can now sum over (a, b) in the equation (3.35). After using (3.36) and (3.39) we obtain the expected results
We performed the sum over (a, b) in (3.35) for each of the (r, s) sectors using Mathematica to arrive at the result (3.42). From (2.14) we see that the new supersymmetric index of the orbifold of T 4 /Z 4 × T 2 by g agrees with that of the 2A orbifold of K3 × T 2 . This result was expected since we have seen in section 3.1.1, that the twisted elliptic genus of the orbifold in (3.1), (3.2) agrees with the 2A class. Then the general arguments in section 2.1 show that for standard embeddings the new supersymmetric index can be written in terms of the twisted elliptic genus. However it is indeed nice to see this using explicit computations.
As a consistency check of our calculations we will evaluate the N h − N v from the new supersymmetric index. From the general arguments of [4] the q 1/6 coefficient of the following expression which is related to the new supersymmetric index evaluates where Z (0,s) new is the corresponding sector of the new supersymmetric index without the lattice factor Γ (0,2) 2,2 . We focus on these terms to extract out the massless states contributing to the new supersymmetric index. The 1 4 factor is introduced to take into account the normalizations of the new supersymmetric index used in this paper. Substituting the new supersymmetric index for the standard embedding of the 2A orbifold of K3 × T 2 evaluated in (3.42) we obtain (3.44) Note that this agrees with the explicit computation of the spectrum in table 2 6 . Now turning on Wilson line in the unbroken E 8 and evaluating the thresholds proceeds identically to that discussed in section 2.2. We thus obtain the result that the difference in one loop gauge thresholds for this orbifold compactification is the theta lift of the twisted elliptic genus of K3 belonging to the class 2A.
Recently in [14], the K3 sigma model has been studied in terms of a rational conformal field theory based on the affine algebra su(2) 6 . In this model of K3 the action of g 7 an element of order 4, which belongs to the conjugacy class 2B of M 24 was explicitly constructed and the twisted elliptic genus was evaluated. In this section we will use this realization of K3 to evaluate the new supersymmetric index of heterotic compactified on K3 × T 2 orbifolded by the order 4 element g . We will show that indeed as demonstrated by the general analysis of section 2.1, that new supersymmetric index can be written in terms of the twisted elliptic genus of K3 twisted by g . Furthermore as discussed in section 2.2, this implies that the difference in one loop gauge thresholds is determined by the theta lift of the corresponding twisted elliptic genus.

Twisted elliptic genus
Let us evaluate the twisted elliptic genus as defined by the trace in (2.10). From the definition of the trace we need the characters of the su(2) 6 model in the Ramond section. These were listed in [14], here we present them in the table 4.
while 1 represent the spinorial su(2) character given by The comma in the list of table 4 separates the left moving su(2) characters and the right moving ones. The SU (2) L × SU (2) R R-symmetry of K3 is carried by the first su(2) character among the left and right moving characters respectively. As shown in [14], the elliptic genus with the characters given in the table reduces to that of K3.
The g orbifold on K3 is implemented by the action Where ρ L refers to the fact that the action of g is restricted to the left moving characters. The SU (2) rotation matrices of g on the su(2) characters is given by The F (0,0) component of the elliptic genus is easy to evaluate and we see that it is given by On evaluating the trace, the right movers contribute a factor of 2 since the zero modes form a SU (2) doublet. Note that the F (0,0) , component differs from the elliptic genus of K3 by a 1/4 factor. Using the action of g on the characters we evaluate the following components of the twisted elliptic genus to be All the remaining components of the twisted elliptic genus can be obtained from modular transform given in (2.16). Note that the twisted elliptic genus falls into the form given in (2.12) with the identifications

New Supersymmetric Index
From the discussion in section 3.2.1 in which K3 is realized as a rational su (2)  These sectors couple to the corresponding sectors of the D6 lattice realized in terms of fermions. Table 5 lists the characters the R + , N S + and N S − of the su(2) 6 CFT. Comparing tables (5) and (4) Here we have used Riemann's bilinear identities to simplify the resulting expressions and obtain the result in terms of theta functions with argument τ . We can now multiply these along with the characters of the D6 lattice in the corresponding sectors as given in (2.6) and we obtain the following result for the (0, 0) sector of the new supersymmetric index Note that this is 1 4 of the result expected for compactifications of heterotic on K3×T 2 . Lets move now to the (0, 1) sector which represents a single insertion of g . For Φ In the Φ Finally the characters which survive the g insertion in Φ Now combining this along with the corresponding D6 characters as in (2.6) we obtain Here there we have used identities which relate the θ functions to Eisenstein series which are provided in the appendix. Using the action of g 2 which is given by Again combining these with the corresponding D6 characters and after using identities (A.21) which relate the theta functions to Eisenstein series we obtain All the remaining terms in the new supersymmetric index can be obtained by performing modular transformations.
On comparing the coefficients of the twisted elliptic genus of the 2B orbifold given in (3.52) with new supersymmetric index given in (3.54), (3.58), (3.61) we see that it agrees with the expression derived in (2.19) using general arguments for the standard embedding. It is important to realize that this agreement was due to non-trivial identities relating the theta functions to Eisenstein series together with the function E 2 and E 4 . Using the expression (3.43) we obtain N h − N v = −380 for this model. Now that we have shown the new supersymmetric index admits a decomposition in terms of the twisted elliptic genus for standard embeddings, the rest of the analysis in section 2.2 can be applied. Therefore we conclude that the difference in one loop gauge thresholds when the Wilson line is embedded in the unbroken E 8 is the theta lift of twisted elliptic genus.

Non-standard embeddings
In this section we study the non-standard embeddings of heterotic compactifications of K3 × T 2 orbifolded by g belonging to the conjugacy class 2A. We first realize K3 as the Z 2 orbifold of T 4 and consider the 2 non-standard embedding studied in [2]. We then move one to the situation in which K3 is realized as the Z 4 orbifold of T 4 and g is implemented as given in equations (3.1) and (3.2). We consider all the 12 non-standard embeddings studied in [2]. In these orbifold limits, the various embeddings are implemented by different lattice shifts in the E 8 × E 8 . From the spectrum of these embeddings we show that the they can be organized into 4 types depending on the difference N h − N v which take values −12, 52, 84, 116 for these types. The value −12 as we have seen corresponds to the standard type. The new supersymmetric index for all the embeddings also depends only on N h − N v . After turning on the Wilson line we show that the new supersymmetric index as well as the difference in one loop gauge thresholds depends on N h − N v and the instanton numbers of the embedding.

Massless spectrum
We can evaluate the massless spectrum of the non-standard embeddings by following the same method as discussed in section 3.1.2. The spectrum for various nonstandard embeddings of K3 × T 2 without the g orbifold were obtained in [3]. Essentially the orbifold by g changes the degeneracy formula given in (3.23) by changing the number of fixed points of the various twisted sectors as discussed around (3.26) for the orbifold in (3.1), (3.2). The various embeddings are determined by the lattice shifts in E 8 × E 8 . In table 6, we first tabulate the spectrum for embeddings when K3 is realized as the T 4 /Z 2 orbifold and g as half shift given by following orbifold actions g : (x 1 , x 2 , y 1 , y 2 , y 3 , y 4 ) ∼ (x 1 , x 2 , −y 1 , −y 2 , −y 3 , −y 4 ), (4.1) g : (x 1 , x 2 , y 1 , y 2 , y 3 ) ∼ (x 1 + π, x 2 , y 1 + π, y 2 , y 3 , y 4 ).

New supersymmetric index
In this section we evaluate the new supersymmetric index for all the embeddings discussed in section 4.1. We will show that for the when the Wilson line is not turned on, the index Z new for the 2A orbifold of K3 × T 2 depends only on the 4 types of the lattice shifts organized in tables (11) and (12). Z new is invariant for any lattice shift belonging to a given type. When the Wilson line is turned on, then the index depends both on the type as well as the instanton number corresponding to the lattice shift. Let us first discuss the case without the Wilson line. Evaluating the trace defined in (2.3) we see that it reduces to where ν = 2, 4 depending on the whether K3 is realized as a T 4 /Z 2 or T 4 /Z 4 orbifold. The partition function over the shifted E 8 lattices are defined by where γ,γ are the shifts in the two E 8 lattices. The trace over the T 4 directions is as defined in (3.36). However the g, g correspond to the actions in (4.1) for the Z 2 orbifold limit of K3 and to actions (3.1) and (3.2) for the Z 4 orbifold limit of K3. This trace is given by , (4.5) where the k's are read out from the following matrices.
= 64 where Γ 2 = γ 2 +γ 2 . Using all this inputs we evaluate the new supersymmetric index for the list of lattice shifts given in tables 11 and 12. This results in following general result The value ofb for each of type of embeddings is given in  Thus the valuesb takes are discrete and just depends on the type of embedding or lattice shift. In fact since N h − N v remains constant in each type of embedding we can relate it tob. This relation can be found by using the equation in (3.43) and is given by Note that standard embedding belongs the caseb = 0, also note that the only non-standard embedding of the 2A orbifold when K3 is realized as T 4 /Z 2 as seen in table 11 belongs to type 3. One important point to emphasize is that the new supersymmetric index in (4.7) still can be decomposed in terms of the twisted elliptic genus of K3. Comparing (2.19) for the 2A orbifold with (4.7) the only difference is that the lattice sum E 4 has been replaced by b E 2 2 (τ ) + ( 2 3 −b)E 4 for the (0, 1) sector. The lattice sum E 6 − E 2 ( τ 2 )E 4 associated by the 2A orbifold remains the same. Similar statements can be made for all the other sectors.
Let us now turn on the Wilson line in the E 8 lattice and evaluate the new supersymmetric index. To do this we follow the procedure in [2]. First the partition function in the E 8 lattice is evaluated with a chemical potential along one of U (1) directions. The lattice sum then becomes (τ, z) is then coupled to the Γ 3,2 lattice using the ⊗ product defined in the appendix. It was shown in [2] that for all orbifold realizations of K3, the new supersymmetric index just depends on instanton numbers of the embedding or the lattice shifts. The result is given by the expression Here the parametersâ,ĉ depend on the instanton numbers n 1 , n 2 of the embedding and the value ofb byâ The [ · ] denotes the corresponding term obtained by modular transformation of the (0, 1) sector. For example in the (1, 0) sector, we replace the terms with E 2 (τ ) of the (0, 1) sector to − 1 2 E 2 ( τ 2 ). Similarly in the (1, 1) we have − 1 2 E 2 ( τ +1 2 ). We summarize the values ofâ,b, n 1 for each of the shifts considered in the tables 14 and 15. Using these tables and equation (4.11), the result for the new supersymmetric index with the Wilson line for these orbifolds can be read out.

Difference of one loop gauge thresholds
We now evaluate the difference in one loop gauge thresholds for all models whose new supersymmetric index is given by (4.11). The one loop threshold for the group G is given by (2.22). We take the G to be the group the Wilson line is embedded in. Then using (4.11) we obtain where the terms in the [ · ] can be obtained by modular transformation from the corresponding term in the (0, 1) sector. Note that we have used the identity in the terms proportional tob. Similarly the terms for the gauge group G we obtain We now evaluate the difference in the threshold integrals. To simplify the expressions we use the following identities together with (4.14) and This results in the following expression for the threshold integral where we have used the relations (2.26). Note that the integrands for all the embeddings in table (14) and (15) just depend on the instanton number and theb which is related to the difference N h − N v . One simple check of our result is that on setting b = 0, n 1 = 24, the equation in (4.18) reduces to the standard embedding result for the 2A orbifold of K3.
The threshold integral in (4.18) over the fundamental domain can be performed using the methods developed in [22]. The details are provided in the appendix B.
Here we quote the final result.
) log(det(Im(Ω)) 6 |Φ 6 (U, T, V)| 2 ) (4.19) Here Φ 10 is the unique cusp form of weight 10 under Sp(2, Z), while Φ 6 is the Siegel modular form of weight 6 which is obtained from the theta lift of the elliptic genus of K3 twisted by the 2A orbifold action. Φ 6 was first constructed as a theta lift in [18]. As expected for the standard embeddingb = 0, n 1 = 24 the threshold integral reduces to only Φ 6 .

Conclusions
We have explored N = 2 compactifications of heterotic string theory on orbifolds of K3 × T 2 by g which acts as a Z N automorphism on K3 together with a 1/N shift on one of the circles of T 2 . g can correspond to any of the 26 conjugacy classes of the Mathieu group M 24 . We showed that for the standard embedding of the spin connection in one of the E 8 the new supersymmetric index can be written in terms of the elliptic genus of K3 twisted by g . The difference in gauge thresholds are shown to be theta lifts of the twisted elliptic genus of these compactifications. This generalizes the observation in [12] as well as [23,24] who observed similar behaviour for nonsupersymmetric compactifications 8 We demonstrated this by explicitly studying 2 examples. The first one considered the 2A orbifold of K3 when K3 is realized as T 4 /Z 4 . The result is same as that obtained in [12] where the 2A orbifold of K3 is obtained by taking K3 to be T 4 /Z 2 . We also studied the recently constructed [14] 2B orbifold of K3 when K3 is realized as su(2) 6 rational conformal field theory. Finally we considered non-standard embeddings for the 2A orbifold of K3 and showed that the new supersymmetric index depends only on the difference N h − N v of the model and the gauge threshold correction depends on the instanton number of the embedding as well as N h − N v . The detailed spectrum of these compactifications has also be obtained.
There are a number of directions which are worth exploring. One is to generalize the study of non-standard embedding to all the orbifold limits of K3, here we considered only the limits T 4 /Z 2 and T 4 /Z 4 . Another direction is to study the type II duals of these theories. Not only this will teach us more about S-duality, but it will also involve the study of new Calabi-Yau manifolds. However perhaps the most interesting extrapolation of the observations of this paper is the fact that it is also possible to consider compactifications of string theory of type II on (K3 × T 2 )/g where g corresponds to any of the 26 conjugacy classes of M 24 . These compactifications preserve N = 4 supersymmetry. The theta lifts of the twisted elliptic genus for all these cases should capture degeneracies of 1/4 BPS dyons. The case of g in the conjugacy class pA, p = 1, 2, 3, 5, 7 was studied in [18,[25][26][27][28][29][30][31]. It will be certainly interesting to generalize the results regarding dyon partition functions to all the conjugacy classes of M 24 . This will possibly will teach us about black hole degeneracies in N = 4 string theory and its relation to the symmetry M 24 . For simplifications in the section 3.2 dealing with the 2B orbifold we need to relate theta functions and E 4 . This is given by

B Threshold Integrals
In this appendix we detail the steps in performing the integral in (4.18). First we write the integrand in a from so that we can identity integrals which has already been performed. Adding and subtracting terms in the integrand we obtain where  To evaluate this integral we can use the general result in [22] for integrals of this form which we will now state. Given the integral of the form

C Mathematica files
There are 2 Mathematica files included in the supplementary attachments. Both the Mathematica files begin with definitions of the generalized theta functions, Dedekind eta function, Jacobi forms of index 1 and Eisenstein series.
1. z4wilson.nb: The partition function of the shifted E 8 × E 8 lattice together with the left moving bosonic partition function on K3 is written in terms of generalized theta functions and compared with the the (0, 1) sector of (4.11).
2. relations.nb: Different relations given in the appendix A and used in the main text are checked by q expansions. The formula for N h − N v as a function ofb given in (4.8) is checked against the general expression (3.43). N h − N v is also evaluated for the 2B model.