Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the $\mathbb{Z}_2$ Model

Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $\mathcal{N}=4$ string-string duality is examined in a reduced rank theory on a less studied BPS sector. For this we identify a candidate partition function of ``untwisted'' quarter-BPS dyons in the heterotic $\mathbb{Z}_2$ CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields a meromorphic Siegel modular form for the Iwahori subgroup $B(2) \subset \text{Sp}_4 (\mathbb{Z})$ which generates BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition function is shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our result also agrees with a recently conjectured formula of Bryan and Oberdieck for the partition function of primitive, untwisted DT invariants of the CHL orbifold $X=( \text{K3} \times T^2 )/ \mathbb{Z}_2$, as demanded by string duality with type IIA theory on $X$.


Introduction
Extremal black hole entropy from counting quarter-BPS dyons in N = 4 string theory A benchmark test of any theory of quantum gravity is to provide a microscopic, statistical explanation of the entropy carried by a black hole, which is semi-classically given by the Bekenstein-Hawking entropy [1,2]. Within string theory such an explanation was provided in [3,4], where the statistical entropy of a D-brane system indeed matches the Bekenstein-Hawking entropy of the corresponding five-dimensional extremal black hole in the large charge limit. As shown in [5], the entropy can also be derived in four-dimensional N = 4 string theory (heterotic strings on T 6 or type IIA strings on K3 × T 2 ) from an exact microscopic index formula for the quarter-BPS dyons of the theory. According to [5] the dyon degeneracies 1 are given by a countour integral of the reciprocal Igusa cusp form χ −1 10 , extracting Fourier coefficients of this Sp 4 (Z)-Siegel modular form. Their growth, as an asymptotic expansion in large charges, is estimated by a saddle-point approximation [5,11] that picks up the dominant pole of the integrand, again reproducing the macroscopic entropy formula of the extremal dyonic black hole [12,13] along with a series of power suppressed and exponentially suppressed contributions (from the dominant, respectively, sub-dominant pole) [14]. Here the leading correction to the entropy can, on the macroscopic supergravity side, be attributed to the inclusion of the Gauss-Bonnet term in the effective action [7,15] by following Wald's generalization [16] of the black hole entropy [17,18]. Apart from such higher-derivative corrections there are also quantum corrections to the dyonic extremal black hole entropy. Based on the AdS/CFT correspondence the proposed quantum entropy function of [19,20] captures both kinds of corrections and accounts for exponentially suppressed contributions 2 as demanded by the microscopic index formula [9,14,25]. See [10,26,27] for reviews and [28][29][30][31][32][33] for more recent studies of the (quarter-BPS) quantum entropy that rely on localization of the supergravity path integral.

Dyon counting in CHL models from dual perspectives
The quarter-BPS index formula of [5] was generalized to dyons in N = 4 CHL orbifolds [34][35][36][37][38] in [39][40][41][42] with an appropriate Siegel modular form taking the role of χ 10 . 3 These theories are obtained upon orbifolding heterotic strings on T 2 × T 4 by a 1/N shift along a circle in T 2 and a supersymmetry-preserving order N action on the internal CFT describing heterotic strings on T 4 . By string-string duality [49][50][51][52][53] these are dual to type IIA theory on orbifolds (K3 × T 2 )/Z N , where the Z N action is a shift on T 2 combined with an order N holomorphic-symplectic automorphism of the K3.
Clearly, apart from its eminent role in understanding black hole microstate entropy versus its macroscopic counterpart, the counting of quarter-BPS states provides a non-perturbative window to various string dualities. This manifests also in the approaches taken to physically derive the proposed counting functions. In the type IIB frame these are computed at weak coupling as partition functions of a rotating D1-D5-sytem in a Taub-NUT background [54], similar to the proposal of [55]. 4 Another derivation was given in [42,58], representing the dyons as string webs lifted to M-theory where the dyon partition function can eventually be related to a genus two partition function of a heterotic string. More recently the BPS 1 These degeneracies are actually indices, i.e., sixth helicity supertraces [6][7][8], making them invariant under (small) variations of the moduli. An interpretation of this microscopic BPS index as a macroscopic black hole degeneracy was justified in [9,10]. 2 Logarithmic corrections consistently vanish micro-and macroscopically for N = 4 theories [10,[21][22][23][24]. 3 As was pointed out in [43], the dyon partition functions of [5,39] actually only capture dyons satisfying a primitivity constraint, namely that the discrete invariant I = gcd(Q ∧ P ) built from the dyon charge (Q, P ) is unity. Partition functions for I > 1 have subsequently been worked out in [44][45][46][47][48]. 4 See [26,56,57] for reviews.
indices have also been extracted from four-and six-derivative couplings in the low energy effective action of three-dimensional heterotic CHL vacua with 16 supersymmetries. Upon circle decompactification to four dimensions [59][60][61] the index formula is found with the correct choice of moduli-dependent integration contour proposed in [62]. This contour prescription renders the quarter-BPS index duality-invariant and captures the (dis-)appearance of two-centered bound states at walls of marginal stability [47,[62][63][64][65]. In the configurations contributing to the quarter-BPS index the two centers are each half-BPS [66]. 5 (Mock) Jacobi forms, moonshine and dyon counting At fixed magnetic charge invariant, quarter-BPS dyons are captured by meromorphic Jacobi forms arising in the Fourier-Jacobi decomposition of the respective Siegel modular form. 6 A decomposition of the former into a finite part (a mock Jacobi form) and a polar part (an Appell-Lerch sum) separates the counting of single-centered black holes, which are stable across walls of marginal stability, from the two-centered black hole bound states [69]. Modular invariance can be restored upon addition of a non-holomorphic completion, and the completion of the polar part has recently been interpreted physically as the continuum contribution in the supersymmetric index of the quantum mechanical bound states [70]. Understanding mock modularity from the physics perspective is an active research area [28,31,71,72].
Recent interest in strings on K3, N = 4 CHL models and their BPS counting has also arisen in moonshine contexts and we shall briefly sketch some connections. Quarter-BPS partition functions can be constructed for more general CHL compactifications involving orbifolds by any -not necessarily geometric -symmetry of the K3 non-linear sigma model (NLSM) [73,74] that commutes with the worldsheet N = (4, 4) superconformal algebra and the half-integral spectral flows. Such K3 NLSM symmetries 7 belong to elements of the largest Conway group Co 0 and were classified in [76], partially extended in [77] to full type IIA theory at singular loci in moduli space, while [78] classify the resulting CHL models and observe that the Fricke involution acts as S-duality in the self-dual models (i.e., when the symmetry has balanced frame-shape, as is the case with geometric symmetries). 8 The dyon partition functions are obtained by multiplicative lifts [80][81][82][83] of the twining genera associated to symmetry conjugacy classes [77], which map (vector valued) weak Jacobi forms to Siegel modular forms for some congruence subgroup of Sp 4 (Z) [84][85][86]. 9 This includes the original case of type IIA theory on K3 × T 2 and χ 10 [5,91,92] as a special case. In combination with constraints from modularity and wall-crossing this construction has recently [93] been used to explicitly derive (almost) all of the twining genera. These K3 twining genera 5 Three-center BPS bound states are conjecturally enumerated by a degree three Siegel modular form [67]. 6 For an introduction to the theory of Jacobi forms and their connection to Siegel modular forms see [68]. 7 K3 NLSM symmetries can also be interpreted in terms of derived equivalences of K3 surfaces [75]. 8 A yet broader notion of "CHL models" was proposed in [79], generically exhibiting Atkin-Lehner dualities. 9 Similarly quarter-BPS partition functions for twisted BPS indices in the sense of [87][88][89] can be constructed from twisted-twining genera, see [90] for an overview.

Quarter-BPS indices, Donaldson-Thomas invariants and the lost chapters
Finally, the problem of counting quarter-BPS states has an avatar in the enumerative geometry of holomorphic curves in the Calabi-Yau threefold X = K3 × T 2 as was first pointed out for the type IIA theory on X in [110]. The reduced Gopakumar-Vafa invariants on X were given an interpretation in terms of the cohomology of the moduli space M n β associated to D0-D2-brane bound states inside X. Given a D2-brane wrapping a holomophic genus g curve C g in the class β ∈ H 2 (X, Z), the moduli space M n β was constructed in [110] as a singular Jacobian fibration Jac(g) over the deformation space of the curve C g in X. Roughly the integer n can be thought of as the number of D0-branes and its relation to the degenerations of Jac(g) maps it to the genus counting parameter g. The decomposition of the cohomology of the moduli space M n β with respect to an sl(2) r × sl(2) l Lefshetz action constructed using the Abel-Jacobi map allows explicit computations, if the singularities of the Jacobian fibration are not too bad, and was used to conjecturally identify the quarter-BPS states of the heterotic string with the reduced Gopakumar-Vafa invariants. These symplectic invariants have been mathematically rigorously defined in terms of stable pair invariants and DT invariants in [111]. For unit-torsion dyons the proposal of [110] leads to the Igusa cusp form conjecture [112,113] for the primitive DT invariants, proven in [114,115].
Given the success of this highly non-trivial physics prediction, one is immediately lead to the question of how this generalizes to the CHL orbifolds. This question has recently been addressed in [116], focussing on orbifolds by a symplectic automorphism g of the K3 (i.e., geometric K3 NLSM symmetries in the above terminology) and in particular the g = Z 2 orbifold. Conjecture A of [116] proposes that the primitive DT partition function of the Z 2 CHL orbifold is given by the multiplicative lift of the twisted-twined elliptic genera belonging to the [g] = 2A conjugacy class as in [40] -but only for DT invariants from "twisted" curve classes. There is a binary distinction 10 between twisted and untwisted curve classes and for the latter Conjecture B of [116] proposes an alternative primitive DT partition function.
So far this new Siegel modular form, somewhat surprisingly, does not seem to have made any appearance in physics, where the twisted partition function has (almost) exclusively been considered. Does it have a physical (dyon counting) interpretation? Regarding stringstring duality, we should be able to provide a derivation from the heterotic perspective.
Apart from that, there are stringent constraints coming from wall-crossing and S-duality invariance. Moreover, as in the twisted case, for large charge invariants the asymptotic 10 Physically this distinction does not apply for the unorbifolded case, for which the electric and magnetic charge lattices are isomorphic and the U-duality group acts transitively on the unit-torsion dyon charges. Due to duality invariance of the BPS index we hence expect only one quarter-BPS partition function.
growth of the Fourier coefficients should also reproduce the correct black hole entropy. From what is known about quarter-BPS black holes in four dimensions, we expect not only to see the leading Bekenstein-Hawking term, but also a subleading term that can be associated with the (model-dependent) Gauss-Bonnet term in the effective action. This will be the content of the lost chapters.
Before proceeding, we shall explain where to fit them in in the CHL story, so let us comment on the distinction between twistedness and untwistedness on the physics side.
It is known (though mentioned less frequently) that the partition function of [40] counts unit-torsion dyons whose electric charge Q in the heterotic frame belongs to the "twisted sector", i.e., for perturbative half-BPS states of charge (Q, 0) the component corresponding to the string winding number around the CHL circle is half-integral. This is in contrast to untwisted sector charges corresponding to integral winding along the CHL circle. It has been argued [56] that S-and T-transformations (i.e., those inherited from the parent theory compatible with the orbifolding procedure) do not mix dyon charges with twisted and untwisted sector charges in the winding number sense. It is hence natural to expect a connection between the untwisted DT partition function and a quarter-BPS partition function for dyons with untwisted sector electric charge, as also remarked in [116, app. A]. 11 Guided by this hypothesis, we will independently derive a quarter-BPS partition function for dyons with untwisted sector electric charge lying in the sublattice Q ∈ U 6 ⊕ E 8 (−2).
Following the M-theory lift of string webs approach of [42], this candidate BPS partition function is deduced from a chiral genus two orbifold partition function in the heterotic Z 2 CHL model. 12 It then remains to check the above mentioned constraints coming from charge quantization, Γ 1 (2) S-duality, wall-crossing and black hole entropy. As we will see, these modular and polar constraints 13 are strong enough to (almost) guess the untwisted partition function once the appropriate ring of Siegel modular forms has been identified.
First steps in that direction have been presented in [44] quite some time ago, though the analysis in [44] was not carried through and remains limited to a subsector of the untwisted sector addressed here. 14 As the original motivation for finding a complementary quarter-BPS partition function in the Z 2 model comes from DT theory of (K3×T 2 )/Z 2 , we shall show how our computation provides additional evidence for the Conjecture B of [116]. Our analysis physically explains why a Siegel modular form for the Iwahori subgroup B(2) ⊂ Sp 4 (Z) appears as counting function in the untwisted case and it also gives an alternative (but equivalent) expression for the latter in terms of the multiplicative lift of twining elliptic 11 The twisted/untwisted nomenclature employed here does not seem fully equivalent to the one of [60,61]. 12 The result of [42] has recently been extended in the appendix of [61], where the remaining genus two orbifold blocks are presented along with the Siegel-Narain theta functions omitted in [42]. These results turn out to be very helpful for identifying the quarter-BPS partition function of untwisted dyons similar to [42]. 13 Modular constraints mean that the partition function is expected to transform as a Siegel modular form under certain Sp 4 (Z) elements, while polar constraints give the singular behaviour near certain divisors associated with walls of marginal stability.
14 It is worth mentioning that in [44] these constraints were also used to propose the correct partition function of dyons with torsion greater than one for the unorbifolded theory. genera and two of its modular transforms. 15 The paper is organized as follows. In section 2 we review N = 4 CHL models with a focus on quarter-BPS dyon counting, largely following [44]. Section 3 reviews the half-BPS counting functions specific to the Z 2 model, which appear as wall-crossing data. The genus two derivation of our candidate partition function in section 4 then proceeds in a similar fashion. Modular and polar constraints on the latter are checked in section 5, with the black hole entropy being treated separately in section 6. We compare our results to the DT results in section 7 and conclude. Background material on Siegel modular forms is collected in the appendix.

Construction of N = 4 CHL models
By virtue of N = 4 string duality these models have dual descriptions as freely acting orbifolds of heterotic string theory on T 6 or IIA string theory on K3 × T 2 . Such models have been classified in [78]. We will mainly be interested in the simplest and most studied case where the orbifolding group G orb = Z N is a cyclic group of order N ∈ {1, 2, 3, 5, 7} and the rank of the resulting gauge group in the four non-compact spacetime dimensions is r = 2k + 4, the integer k = 24/(N + 1) being determined by N .
As is well-known, the maximal rank case (i.e., the trivial orbifold) gives a gauge group U(1) 28 at a generic point of the moduli space  16 The discrete groups acting from the left are the T- 15 We stress that eq. (2.14) of [61] gives an expression for the quarter-BPS index in (conjecturally) arbitrary charge sector and reproduces the result of [40] as a special case. Nevertheless, it is far from obvious to match the index formula to Conjecture B of [116], which is a main objective of the present work. In addition, our analysis also demonstrates an extended range of applicability of the approach in [42] and, as mentioned in the text, also provides a physical interpretation of the Iwahori subgroup together with an alternative expression for the untwisted DT partition function of [116]. 16 Recall that locally the Narain moduli space is parametrized by the metric and the antisymmetric B-field on T 6 as well as by the 16 Wilson lines for the Cartan-torus of the E8 × E8 or Spin(32)/Z2 gauge group. This provides an embedding of the abstract lattice E ⊕2 8 ⊕U ⊕6 ∼ = Λ22,6, which is the unique even unimodular lattice and S-duality group of that theory. where Λ 20,4 ∼ = H * (K3, Z) is the integral cohomology lattice of the K3 surface, while Λ 2,2 is the winding-momentum lattice for S 1 ×Ŝ 1 . As an abstract lattice, the latter is given by the direct sum of two hyperbolic lattices, i.e., Λ 2,2 ∼ = U ⊕2 . 17 Let us turn to the reduced rank theories. In the type IIA theory the cyclic orbifold group is generated by a pair (g, δ), consisting of an order N action g on the N = (4, 4) K3 non-linear sigma model (NLSM) and a simultaneous order N λ shift in the direction δ on S 1 , where δ ∈ Λ 2,2 has square zero in order to satisfy level matching. The condition on g is to fix the superconformal algebra on the worldsheet and the spectral flow generators, see [76] for a precise characterization. Indeed, one can choose λ = 1 for symmetries g that are geometric in the sense that g describes an automorphism of the K3 surface that fixes the holomorphic-symplectic (2,0)-form (and thus keeps the SU(2) holonomy). Such symmetries are uniquely determined by their induced action on the lattice H 2 (K3, Z). They are in fact, up to lattice automorphisms, already determined by the order 1 ≤ N ≤ 8 of g and symplectic automorphisms of any order in that range do actually exist. 18 In this way we only consider CHL models associated to a symplectic automorphism of a K3 surface that has prime order.
The middle cohomology lattice of the K3 decomposes into an invariant Λ g and a coinvariant (Λ g = (Λ g ) ⊥ ) lattice with respect to g, i.e., We illustrate the case N = 2, where g is called a Nikulin involution. The induced action on Λ exchanges the E 8 (−1) sublattices and fixes U ⊕3 pointwise. Equation (2.4) becomes
On the heterotic side, the Z N orbifold action is asymmetric, i.e., acts by a Z N cyclic permutation and a shift on the left-moving coordinates while the right-moving coordinates are invariant (up to shifts) [36]. For the N = 2 case this gives an exchange of the internal E 8 × E 8 factors and an order two shift along a circle of T 6 . The one-loop partition function of this heterotic orbifold is reproduced in section 3 as it will be needed later.
Moduli of a CHL model are given by the g-invariant moduli of the parent theory and take values in for some discrete U-duality group G 4 (Z) in four non-compact dimensions, which includes a T-duality group T acting (only) on the first factor and an S-duality group S acting on the second factor (via Möbius transformations on the heterotic axio-dilaton [132,133]), The S-duality group turns out to be [118] for the Z N CHL models, while the T-duality group T should at least contain 19 the centralizer As it has been argued in [78], there should also be a Fricke involution acting as S het → −1/(N S het ) on the axio-dilaton and by an orthogonal, not necessarily integral, transformation on the other moduli, see for instance [60,61] for further discussion in that 19 In practice, we will take this to be an equality and do not rigorously draw distinctions. 20 Equivalently we can write direction. For simplicity we will mostly neglect possible Fricke type dualities. Here we think of elements in the T-duality group T always as automorphisms of the electric charge lattice defined next, T ⊂ O(Λ e ) .

(2.12)
Electric-magnetic charges. Electric charges take values in a lattice of signature (2k − 2, 6) (and rank r = 2k + 4) while the magnetic charges take values in the dual lattice, which again has the same rank and signature, (2.14) Their direct sum gives the electric-magnetic lattice For N > 1 the lattices are no longer self-dual (unimodular). Rather, they are N -modular, , they agree with their dual upon rotation and rescaling (see [60, eq. (2.10)] for a concrete example): (2.16) The notation Λ m ( 1 N ) means that the bilinear form is rescaled by 1/N . Multiplying (2.16) by N from the left and using the natural inclusion Λ m ⊂ Λ * m it follows that 21 For later reference we give the electric and magnetic lattice for the N = 2 orbifold explicitly,  21 The inclusion N Λ * m ⊂ Λm is claimed in [60,61], equivalent to N v, w ∈ Z for all v, w ∈ Λe.
The T-duality group T ∋ O fixes S het but acts on the remaining moduli and the charges We denote the quadratic T-invariants as The S-action of Γ 1 (N ) on these follows from (2.19). For later convenience let us also introduce the map 23 There are further, discrete T-duality invariants characterizing the duality orbit of a charge (Q, P ). Following [43], take some basis of the lattice Λ em and denote the integer coordinates of a charge (Q, P ) with respect to this basis by Q i and P i , the greatest common divisor of the integers (Q i P j − Q j P i ), denoted as will then be a T-duality 24 and S-duality invariant, sometimes called torsion. 25 It has been shown that for Het[T 6 ] the quantity I and the above quadratic T-invariants are sufficient to uniquely determine a duality orbit under S-and T-transformations in G 4 (Z). If Stransformations are left out, apart from I and the quadratic T-invariants three further discrete T-invariants (on which the S-duality group acts non-trivially) are needed to characterize a T-orbit unambigously, see [134,135] and [56, section 5.3] for details. Just in the special case I = 1, which fixes the remaining three discrete T-invariants to unity, there is a single T-orbit.
As was also pointed out in [116, app. A], the precise duality group G 4 (Z) of a fourdimensional Z N CHL model with N > 1 is not yet determined, nor is a complete set of duality invariants that uniquely specifies the distinct charge orbits in Λ em with respect to G 4 (Z). There a "residue" of a charge (Q, P ) ∈ Λ em was defined as the class in the 22 We use the notation O −⊺ = (O ⊺ ) −1 . 23 Because of (2.13), (2.14) (2.15), P 2 /2 and Q · P are actually integral. 24 As shown in [134, section 2] a change of basis given by an SLr(Z) matrix leaves the gcd invariant (there r = 22 + 6 was considered). If T ⊂ O(Λe) ⊂ SLr(Z) this argument also holds for ZN CHL orbifolds of Het[T 6 ]. 25 We give some remarks. (1.) First note that (Q, P ) being primitive in Λem does not imply that Q ∈ Λe or P ∈ Λm is primitive. In turn, if Q or P is primitive, then (Q, P ) is primitive as well. (2.) If Q or P is non-primitive then I > 1. On the other hand, I > 1 does not imply that Q or P are non-primitive, as the example in [44, subsection 6.3] with I = 2 shows: there both Q and P are primitive (and Q ± P are both twice a primitive vector). So I = 1 is a sufficient, but not necessary condition for having both Q and P primitive. discriminant group This quantity was shown to be invariant under Γ 1 (N ) × C (g,δ) and distinguishes between the twisted and untwisted curve classes of the CHL orbifold. In any case, we expect that again finitely many duality invariants suffice to uniquely determine a duality orbit.
Having several distinct duality orbits of charges means we should also expect several a priori distinct degeneracies associated to states with charge in the respective orbits. In this work we will elaborate on this in the case of counting certain dyonic quarter-BPS states in the Z 2 CHL model, whose charges satisfy I = 1 and further belong to a different duality orbit than those mostly considered in the literature. We come back to this point after recalling some facts about quarter-BPS dyons and their counting functions.

Structure of quarter-BPS partition functions
In this subsection we briefly review the structure of partition functions of quarter-BPS dyons in four-dimensional N = 4 string theories, closely following the discussion in [44].
Many details will be omitted and can be found in the reference.
BPS multiplets and indices. Recall that quarter-BPS states transform in 2 6 -dimensional intermediate multiplets. It also follows from the N = 4 superalgebra that quarter-BPS states with electric-magnetic charge (Q, P ) ∈ Λ em must satisfy Q ∦ P (i.e., the charges are not collinear as vectors in R r ). Half-BPS states in turn transform in 2 4 -dimensional short multiplets and obey the opposite charge condition, Q P . 26 Since a quarter-BPS dyon breaks 12 out of 16 supercharges, an appropriate, i.e., nontrivial, index to "count" such states of a given charge (Q, P ) ∈ Λ em is the sixth helicity supertrace 27 , denoted by Ω 6 (Q, P ; · ). Here the dot represents the moduli of the theory.
Locally this index is constant, but it changes discontinously once the asymptotic moduli of the theory are variied across certain real codimension one subspaces, called walls of marginal stability. Each wall is associated to a specific decay of the quarter-BPS dyon into a pair of half-BPS dyons. This wall-crossing phenomenon [137][138][139][140][141][142] is best understood in the case where the decay products carry primitive charges and for simplicity we restrict us to this case. Considering a quarter-BPS dyon with charge (Q, P ) ∈ Λ em that decays at a certain (generically present) wall into two half-BPS states via it is clear that we should restrict us to dyons where both Q ∈ Λ e and P ∈ Λ m are primitive lattice vectors. Furthermore we restrict to the case I = 1. According to [43] this is also 26 See, for instance, the references [57,136] for explanation. 27 See, for instance, [8,27,57] for explanation or [6,7] and references therein. a necessary condition for the dyon partition function to be related to a chiral genus two partition function of the heterotic string, as we will discuss later.
In principle there can also be decays where at least one decay product is quarter-BPS, however [143], if Q and P are both primitive charges these occur in the moduli space at codimension two or higher. Thus generic points in this space can be connected by paths that do not cross these loci and the BPS index is not affected by such decay channels.
BPS charge sets. For the purpose of analyzing or constraining a (quarter-BPS) dyon partition function it may be convenient to reduce the problem to analyzing charge subsectors, for which the counting problem simplifies. Let us introduce some notation. For a set of electric-magnetic charges Q ⊂ Λ em we define the following conditions: For all (Q, P ) ∈ Q we have Q ∦ P .
(Q3) T-closure condition: For any given triplet (q 1 , q 2 , q 3 ) of the quadratic T-invariants the set if not empty, maps to itself under the action of the T-duality group T .
(Q4) T-transitivity condition: Any two elements of subsets of the form (2.26) are related via T .
(Q5) Unboundedness condition: Any of the quadratic T-invariants takes arbitrarily large absolute values on Q.
(Q6) Quantization condition: There are (maximal) rational 28 numbers q i ∈ Q + such that for any (Q, P ) ∈ Q we can find integers ν i ∈ Z satisfying examples in [44] are constructed such that already the T-representatives form an affine rankthree lattice L Q ⊂ Λ em which then bijects to its T-invariants t(Q) = L and Q is obtained by simply taking all T-images, Q = T L Q . In this way (Q1)-(Q6) are satisfied simultaneously.
BPS partition functions. We make the standard assumption that the sixth helicity supertrace Ω 6 (Q, P ; ·) (or simply BPS index in the following) is invariant under S-and T-transformations, i.e., at a given generic point in the moduli space it only depends on the duality orbit of (Q, P ) ∈ Λ em . Given Q satisfying (Q1), (Q3) and (Q4), because of T-invariance the BPS index of dyons with charge (Q, P ) ∈ Q will already be uniquely determined by specifying the quadratic T-invariants of the charge and for some appropriate One can also introduce a partition function for these numbers via 30 where a sign factor has been introduced to follow conventions in [44] and the sum runs over all quadratic values belonging to charge vectors (Q, P ) ∈ Q.
Under the condition (Q5) the partition function is expected to have infinitely many nonzero terms. 31 Typically the generalized chemical potentials σ, τ, z conjugate to Q 2 /2, P 2 /2, Q· P must lie in a suitable domain of the Siegel upper half plane H 2 for this series to converge (see appendix A for a defintion) and we will assume that this is the case. Different domains of convergence admit different Fourier expansions, which in turn give BPS indices valid for different regions of the moduli space. If Q satisfies (Q6), the partition function will be periodic: Recall that affine means that it is given by some lattice (including the zero vector), shifted by a non-zero vector. 30 The fractional notation here is adopted from [44], which is reminiscient of the original DVV result 1/χ10 and the CHL orbifold analogs considered by Sen et al. 31 Eventually we want ZQ to be a Siegel modular form (for some congruence subgroup) and we expect that this requires infinitely many non-zero "Fourier modes" exp(2πıkx), for each x ∈ {τ, σ, z}.
BPS indices can be extracted from Z Q by taking an appropriate contour integral over a (minimal) period in each direction at some fixed, large imaginary part. 32 As mentioned before, we are mainly concerned with quarter-BPS dyons of unit-torsion. We expect that a finite number of discrete T-invariants provides a partition of the set into a finite number of pairwise disjoint subsets Q i , each obeying (Q1) to (Q4). The important point is that this yields a finite set of (a priori different) quarter-BPS partition functions  .32)). We will verify this subsector result in section 5.

Constraints from S-duality symmetry and charge quantization
Generically a subset Q i in (2.32) will not be preserved (setwise) under the full S-duality group S but only under a subgroup S Q i ⊂ S and transformations in S\S Q i map to other subsets Q j . This is in line with the discussion after (2.23) and further examples can be found in [44]. In any case, the invariance under S Q ⊂ S has important consequences for Z Q , as we will now discuss. 32 In this work we will stay schematic with regard to the choice of integration contour, which could in principle be analyzed more carefully as in [62], see also [44,63]. 33 More explanation is given in section 3. For the untwisted/twisted distinction of curve classes in DT theory see section 7.
Recall that the S-duality group acts on the charges via (2.19). Those transformations which map Q to itself form a subgroup S Q and for such transformations a b c d S-duality invariance of the BPS indices can be recasted into the (suggestive) form (see [44] for a derivation) and suitable periods r 1 , r 2 , r 3 subject to the choice Q. This is also a special case of a symplectic matrix, see eq. (A.4) with S = ( r 1 r 2 r 2 r 3 ).

Constraints from wall-crossing
Let us now explain how wall-crossing puts additional modular constraints on Φ Q . A general parametrization for the decay of a quarter-BPS dyon into a pair of half-BPS dyons is given by The decay products on the right hand side of (2.36), where we have set again have to belong to the charge lattice Λ em . Note that a charge set Q ⊂ Λ em always comes along with its allowed decays (2.36) and thus determines charges (Q ′ , P ′ ) and (Q i , P i ).
Following the ansatz that the jump in the BPS index due the decay (2.36) is determined by a second order pole of Φ −1 Q at the contour integral (2.31) for the Fourier coefficient of (2.29) needs to pick up a residue 34 up to a sign. In this expression d h (Q,P ) = Ω 4 (Q, P ) denotes the fourth helicity supertrace, an index only sensitive to half-BPS multiplets of dyonic charge (Q,P ) (often simply called half-BPS index). As in [44] we want to restrict to those cases where the half-BPS indices again can be written as Fourier coefficients of a suitable partition function, Here the integration contour lies parallel to the real axis and extends over a unit period (Q7) For any (Q ′ , P ′ ) appearing as above, the values (Q ′ ) 2 takes for fixed (P ′ ) 2 are independent of the latter. The same holds for their roles reversed. 36 (Q8) For fixed "decay code" a 0 b 0 c 0 d 0 , all the decay products (Q 1 , P 1 ) obtained from letting (Q, P ) run over Q need to fall into a single T-orbit for each value of Q ′2 .
The same holds for (Q 2 , P 2 ) and P ′2 . Without (Q8), i.e., if there were several orbits, the half-BPS indices would not be functions of the mere quadratic T-invariants.
A sufficient condition for the jump is that near z ′ = 0 the function Φ Q behaves as in the transformed variables Note that (Q7) is generically required for the factorization in (2.44).
Given that the functions φ m (τ ; b 0 , d 0 ) and φ e (τ ; a 0 , c 0 ) transform as weight k + 2 modular forms 37 under fractional linear transformations (a.k.a. Möbius transformations) of τ encoded by SL 2 (Z)-matrices α 1 β 1 γ 1 δ 1 and ( p 1 q 1 r 1 s 1 ), respectively, we can map these to symplectic transformations of the form respectively. These in turn act as Z → (AZ + B)(CZ + D) −1 when written in the usual block form. Typically such a half-BPS partition function is a modular form for some congruence subgroup of SL 2 (Z). In some cases (2.47) and (2.48) lift to modular symmetries of Φ Q in the sense of (2.33). Hence, wall-crossing determines the location and coefficients of quadratic poles in our quarter-BPS partition function together with candidate Siegel modular symmetries. 38 We remark that the middle matrix in each (2.47), (2.48) preserves the locus z = 0, while 37 Hence determining the weight of φe,m fixes the weight of ΦQ and vice versa. 38 There might be additional ("accidental") modular symmetries as in [44, subsection 6.4] or some of the (genus one) modular symmetries do not lift to the full quarter-BPS partition function, see, for instance, the example in [44, subsection 6.2]. the conjugated matrix preserves the locus z ′ = 0. all decays are related to the one at z = 0 by an SL 2 (Z) transformation, which is known to be the S-duality group of that theory. However, in CHL orbifolds we may find inequivalent walls after modding out the mentioned redundancies.
As was multiply exemplified in [44], the expected properties of Φ Q just described lead to a heuristics for finding quarter-BPS counting functions subject to a charge set Q. By the same token, they provide a set of highly non-trivial tests for any given candidate counting function. Since the half-BPS partition functions form a key ingredient of this approach, we will now recall some facts about the latter in case of the heterotic Z 2 CHL model.

Half-BPS spectra from Dabholkar-Harvey states in the Z 2 model
In this section we reproduce from [144] the computation of electric half-BPS partition functions in the heterotic Z 2 CHL orbifold 39 that appear in wall-crossing relations for quarter-BPS partition functions. Doing so we set the notation and collect relevant wall-crossing data for section 5. The genus two analysis of section 4 will eventually go along similar lines, so this review section also serves as a warm-up exercise.  39 See also [117] and [60, app. A.1] for closely related results. For the prime order CHL models these half-BPS partition functions, or rather those of the singly twisted sector, have recently been revisited in [145] from a macroscopic point of view. 40 When we speak of DH states in the following, we will always mean the perturbative heterotic half-BPS states. Otherwise, DH states are not always half-BPS [69, f.n. 6]. 41 Here the convention is made to call the superconformal side of the heterotic string right-moving. 42 Helicity supertraces are reviewed in appendix E and G of [8], see also [148] for a short textbook chapter.
Consider the generating function [8] where J 3 denotes the Cartan generator of the massive little group in four non-compact dimensions 43 and F denotes the spacetime fermion number. Helicity supertraces can be obtained by taking appropriate derivatives with respect to the generalized chemical potentials v andv coupling to the left and right helicity, respectively: Note that Z(q,q; 0, 0) is just the ordinary one-loop partition function of the heterotic string (or its orbifold) including the GSO projection. Oscillators that contribute to the right helicity are the right-moving light-cone bosons ∂X ± = ∂X 3 ± i∂X 4 contributing helicity ±1, respectively, and the light-cone fermions ψ ± again contributing ±1 to the right helicity.
On the other hand, only ∂X ± contribute to the left helicity. For instance, the 2+2 chiral light-cone bosons contribute a factor of .
Taking all together the generating function of helicity supertraces for the Z 2 CHL model is 44 Here α, β = 0, 1 run over the four spin structures, h = 0, 1 indicates the untwisted or twisted sector and g = 0, 1 indicates an insertion of the orbifold involution into the trace. In the above expression we have the partition function of the (shifted) Narain-lattice
where the subscript L/R denotes the left-and right-part of the lattice vectors 45 and is the Narain-lattice associated to T 6 , possibly shifted by the null vector δ = (0 6 ; 0 6−1 , 1) such that the CHL action on T 6 is given by a half-translation along the last circle in T 6 (the CHL circle). Thus h = 0 means summation over untwisted sector charges, Q ∈ Λ 6,6 ∼ = U ⊕6 , the winding number along the CHL circle taking integral values. On the other hand, h = 1 gives twisted sector charges Q ∈ Λ 6,6 + δ 2 with the winding number along the CHL circle taking values in Z + 1 2 . The factor (−1) δ·Q , for g = 1, then becomes (−1) for an odd number of momentum quanta along the CHL circle and (+1) for an even number. Furthermore, in and . We now want to obtain the fourth helicity supertrace B 4 . The fermion terms in the first line of (3.4) can be rewritten using the Riemann identity to giveθ 4 1 (v/2). This implies that the only combination of v-andv-derivatives that does not vanish when evaluated at v =v = 0 is taking fourv-derivatives, since θ 1 (0|τ ) = 0. Using further and ξ(0) =ξ(0) = 1 we obtain 47 (3.10) Inserting the identities (3.7) and (3.8) we can also write Before interpreting the result (3.11), we interlude with a reminder of the unorbifolded case. The contribution with Z 6,6 [ 0 0 ] corresponds (up to the factor 1/2) to helicity supertraces of perturbative states in the unorbifolded theory Het[T 6 ]: . The level number N (not to be confused with the order of the CHL orbifold group) is related to the charge Q via the level matching condition 49 (3.14) 47 The factor 3/2 arises as 24 × (1/2) 4 coming from the 4! = 24 permutations ofv-derivatives and the inner derivative, c.f. the argumentṽ =v/2. 48 Recall that p24(N ) is the number of ways of writing the non-negative integer N ∈ as a sum of 24 nonnegative integers. This is also the Fourier coefficient of q N−1 in η −24 (τ ). For any τ ∈ H the Fourier series of the latter converges, so there is no ambiguity, i.e., no wall-crossing for these half-BPS states and no moduli dependence in Ω4. 49 This holds also in the Z2 CHL case [144, eq. (3.15)].
Also recall that in the unorbifolded case the discriminant function ∆(σ) = η 24 (σ) appears on the diagonal divisor limit of χ −1 10 (Z), which is the (complete I = 1) quarter-BPS partition function of heterotic strings on T 6 (c. f. the discussion in 2.2.2). Historically the appearance of this perturbative half-BPS partition function and its magnetic counterpart, together with manifest electric-magnetic (S-)duality between them, was a crucial point in the proposal of [5].
We return to the CHL orbifold and apply the same logic to B 4 (q,q) in eq. (3.11). As we are eventually interested in the quarter-BPS partition function for dyons with electric charge in the untwisted sector, more precisely let us identify the half-BPS counting function for purely electric DH states of that charge.
According to our wall-crossing discussion this function should re-appear in the quarter-BPS partition function of interest on the diagonal locus z = 0. To this end, the (P 1 , P 2 ) ∈ E ⊕2 8 Narain-lattice vectors are decomposed 50 with respect to the sum -which is invariant under Z 2 and hence a physical charge -and the difference, i.e., P 1 ± P 2 = 2P ± + P for some root lattice vectors P + , P − ∈ E 8 and P ∈ E 8 /(2E 8 ) ∼ = E 8 (2) * /E 8 (2) (an element of a finite group of rank 2 8 ). In terms of E 8 (2) theta functions with characteristics P the theta function for E ⊕2 8 may be expressed as Here it has been used that θ E 8 (2),P only depends on the orbit of P under the Weyl group of With θ E 8 (2),1 = E 4 (2τ ) being a lattice sum over the (anti-)diagonal in E ⊕2 8 , we have a subset of untwisted sector DH states whose electric half-BPS partition function yields (after renaming τ → σ, where σ is conjugate to Q 2 /2). In the corresponding lattice sum, i.e., the second group of factors in (3.18), summation is over charge vectors Q ∈ Λ 6,6 ⊕ E 8 (−2) as desired. This function plays an important role for interpreting the untwisted sector quarter-BPS partition function (see section 5). Two remarks are in order.
CHL momentum parity. For later reference we also remark that B 4 can be written in terms of twisted and untwisted parts as with the (complete) untwisted sector contribution reading 51 zero-mode cancel out to give an invariant state. As can be seen, e.g., from the explicit form of the T-transformations on charges in [56], this "momentum parity" along the CHL circle is also invariant under T-transformations, so we have a splitting into two disjoint T-orbits.
Correspondingly we find that odd (even) momentum parity DH states possess a separate half-BPS partition function 1 2 . Using this (3.11) can be re-expressed as , (3.24) where the notation [60] was introduced. Pairs of (Narain) theta functions multiplying the same eta-quotient have been recasted into a single lattice sum for the electric lattice Λ e or magnetic lattice Λ * e ⊂ Λ e , as defined in (2.18). This also nicely demonstrates the assertion that the DH states are electrically charged with respect to Λ e as given in (2.18). In section 4 a genus two analog of (3.23) will now become important.
4 Quarter-BPS spectra from genus two partition function in the Z 2 model Our analysis in section 2 mostly concerned generic quarter-BPS partition functions. We now turn specifically to unit-torsion quarter-BPS dyons in the Z 2 CHL model whose electric charge in the heterotic frame belongs to the untwisted sector (untwisted sector dyon partition function for short). The goal of this section is to obtain a closed expression for this function by relating it to a genus two chiral partition function for the four-dimensional heterotic Z 2 CHL model. 52 Properties of the candidate dyon partition function thus obtained will be addressed in section 5.
According to [41,42,58,150] quarter-BPS dyons can be represented as string webs [151,152], which via an M-theory lift are related to a chiral genus two partition function of the heterotic string. As was argued in [43], the genus g of the M-theory lift of the string web is actually given by g = I + 1, so the genus two partition function is expected to only capture unit-torsion dyons (I = 1). Indeed, in [42] the twisted sector dyon partition function of [39,40] was re-derived by identifying appropriate contributions to the genus two orbifold partition function that can be interpreted as arising from states of the relevant charge type. 53 Our untwisted sector quarter-BPS partition function should in a similar fashion be found in this heterotic genus two partition function. The latter was recently revisited in [61, section B.2], expanding the results of [42] by, e.g., also writing down the remaining orbifold blocks.
For the sake of a clear and coherent presentation, we will reproduce parts of [61] and collect the relevant formulae that are needed in the subsequent analysis.
Genus two orbifold blocks. As in the one-loop case, the chiral partition function is given by a sum of orbifold blocks, each associated to a choice of periodicity conditions [h 1 , h 2 ] and [g 1 , g 2 ] along the A-and B-cycles of a genus two surface with period matrix Ω = ( τ z z σ ) = Ω 11 Ω 12 Ω 21 Ω 22 = Ω 1 + iΩ 2 ∈ H 2 , i.e., At least on the locus of the moduli space where the Narain-lattice splits as E 8 ⊕ E 8 ⊕ Λ 6,6 we may factorize the orbifold blocks into a contribution of the ten-dimensional E 8 × E 8 string and the contribution of the bosonic zero-modes of the chiral bosons on T 6 , Here we have the genus two analog of the Narain-lattice associated with T 6 , shifted by the null vector δ = (0 6 ; 0 6−1 , 1). 54 In view of the twisted sector dyon states, the authors of [42] computed the orbifold block building on the results of [154]. Here we have the genus two theta series for the E 8 root lattice, Θ agreeing with the Siegel-Eisenstein series E 4 (Ω), as well as the weight six Siegel modular forms Φ 6,k defined in appendix A (one of which is given by a multiplicative lift of the K3 twining genera of class 2A). Rescalings and shifts in the arguments of the E 8 theta series can be rewritten in terms of the theta series for 2-modular lattices and insertions of sign factors, for instance: 55 Further expressions of this kind we will encounter below are moved to appendix A. The third and second term in (4.5) are modular images of the first under the Petersson slash operator, where we adopted the notation P e iπQ r,L ΩrsQ s,L −iπQ r,RΩrs Q s,R (4.12) for the case Λ 0 = U ⊕6 ⊕ E 8 (2), P = (−1) δ·Q 2 (the E 8 charges being only "left-moving", as consistent with (4.6)).
Further modular transformations on the above block with γ ∈ Sp 4 (Z)/Γ (2) e 1 (2) generate 55 Here we present the theta series that are needed later. The ones in (4.5) are related to these by an obvious exchange in the roles of (τ, Q1), (σ, Q2). the remaining 14 of the 2 4 − 1 = 15 orbifold blocks with non-trivial boundary conditions: 56 (4.14) As in the one-loop partition function there is a modular identity arising from the equivalence of the E 8 × E 8 theory with its orbifold obtained by exchange of the E 8 factors (without any shift along T 6 ): This is the genus two analog of (3.23). Combining this with (4.13) yields Explicit expressions for the other blocks Z 8 h 1 h 2 g 1 g 2 can be found in [61, Table 1], we will only need a subset of them in the following: Using the behaviour of E 4 (Ω) and the genus two Thetanullwerte θ a 1 a 2 b 1 b 2 (Ω) (which appear in Φ 6,k ) in the diagonal limit z → 0 together with some simple theta identities (see appendix A), it is straightforward to verify that each orbifold block factorizes into two genus one orbifold blocks: This limiting behaviour mirrors the wall-crossing constraints that we address in section 5.
Identification of quarter-BPS partition functions. In the following we will identify the genus two period matrix Ω ∈ H 2 with the chemical potentials conjugate to the quadratic T-duality invariants obtained from the electric and magnetic components of a dyonic charge,  Let us see which vectors appear in the lattice sums of these expressions, first addressing the toroidal part Z 6,6 h 1 h 2 g 1 g 2 . ForZ u , we have h 1 = h 2 = 0 in the torus blocks, so both lattice vectors Q 1 , Q 2 run over the (non-shifted) momentum-winding-lattice Λ 1,1 ∼ = U of the CHL circle (recall (4.3) and (4.4)). In comparison to that, forZ t we have h 2 = 1 and the projection of Q 2 along the CHL direction runs over the coset U + δ 2 ⊂ U ( 1 2 ) with half integral winding. 58 Furthermore g 1 = 0, 1 takes both values in the torus blocks, so effectively the presence of a projector in the toroidal lattice sum restricts further to the level two sublattice U (2) ⊂ U with respect to the component of Q 1 that counts charge along the CHL circle, e. g.Z (4.25) 57 However, in the sequel paper [43] the authors also use the convention employed here. 58 In the convention of [44] this means an odd number of winding quanta along the CHL circle.
Recalling our identification of the chemical potentials this restriction is necessary for regarding Q 1 as a magnetic component of a dyon charge (Q, P ), whose projection to the sublattice associated with T 6 becomes (Q 2 , Q 1 ). On the other hand, since g 2 = 0 in the above torus blocks, the component of Q 2 along the CHL direction runs over U , not just U (2) ⊂ U . So Q 2 represents an untwisted sector electric charge where the number of momentum quanta along the CHL direction takes both even and odd values.
Next we address the E 8 part. ForZ u we return to (4.17) as well as the identities (4.7)-(4.9) and (A.42)-(A.47). Since Q 2 2 ∈ 2Z for Q 2 ∈ E 8 (2) ⊂ E 8 ( 1 2 ) we may split off a theta series for this sublattice as follows: The according contribution of these charges to the orbifold block (4.17) reads .
For comparison, we find inZ t a contribution Z t (Ω) = 1 2 which gives the well-known twisted sector quarter-BPS counting function in parentheses. 59 In what follows, we will test our proposal (4.30) for the untwisted sector quarter-BPS partition function by checking wall-crossing and S-duality constraints as reviewed in subsection 2.2, by black hole entropy considerations and also by comparing it to recent results from Donaldson-Thomas theory.

Modular and polar constraints in the untwisted sector
Our working hypothesis so far has been that the new, untwisted quarter-BPS partition function counts unit-torsion dyons with electric charge in the untwisted sector sublattice . 60 In light of subsection 2.2 there are non-trivial constraints especially from S-duality symmetry and wall-crossing such a partition function must satisfy. These constraints will be addressed in the following.
In fact, this analysis already highly constrains Z untw Het . With only few assumptions one might already "guess" the form of the latter, as we will see shortly.

Quantization of the charge invariants
As a simple, first constraint we remark that for dyon charges (Q, P ) ∈ Λ u e ⊕Λ m the quadratic T-invariants ( 1 2 Q 2 , 1 2 P 2 , Q · P ) take integral values, while 1 2 Q 2 also takes half-integral values for generic charges Q ∈ Λ e . According to (2.35), all symplectic matrices of the form 59 Note that Φ6,3(τ, σ, z) = Φ6,1(σ, τ, z) upon swapping the diagonal elements, so this is the same Siegel modular form as in [42] once the meaning of the chemical potentials is properly matched. Also the leading factor of 1/2 should match to the factor 1/N (N = 2) in [155, eq. (4.28)] (also see [61, eq. (2.8)]). 60 Care has to be taken with various notions of "untwisted" charges. Our untwisted sector dyon charges (Q, P ) ∈ Λ u e ⊕ Λm are not the same as the "untwisted" sector dyon charges considered in [60,61], which refer to charges in (Q, P ) ∈ Λm ⊕ N Λe ⊂ Λe ⊕ Λm (such that Q 2 ∈ 2Z but P 2 /2, Q · P ∈ N Z). Similarly electric charges Q in [60,61] are called untwisted iff Q ∈ Λm ⊂ Λe, while we refer to Q as untwisted (in the winding number sense) iff Q ∈ Λ U e := U ⊕ U 5 ⊕ E8(− 1 2 ) with the first lattice summand corresponding to the momentum-winding lattice of the CHL circle. Note that this is the Λ U e defined in [78, eq. (4.14)] and we do draw a distinction between Λ U e and Λ u e = U ⊕ U 5 ⊕ E8(−2) Λ U e .
acting on Z ∈ H 2 in the usual way should then leave the untwisted quarter-BPS counting function Z untw Het (Z) invariant. Indeed, this is the case for the Z untw Het derived above, since (5.1) lies in B(2).

Wall-crossing relations
We now apply the general lessons from 2.2.2 and study the implications of wall-crossing.
For the magnetic component we will make the assumption that the half-BPS partition function of the purely magnetic states (0, P ) that can arise in the above decay agrees with the function in the corresponding twisted sector decays, .
Though appearing ad hoc at this point, the assumption can be motivated by arguing that we expect the quarter-BPS dyons to be counted here to differ only in the electric component Q from the twisted sector states (compare the lattice sums of (4.29) and (4.32)). We remark that in [60], in accordance with Γ 1 (2) S-duality and Fricke symmetry, it was shown that the half-BPS index (fourth helicity supertrace) for primitive charges (Q, P ) ∈ (Λ e ⊕ Λ m )\(Λ m ⊕ 2Λ e ) is given by while for charges in the complement (Q, P ) ∈ (Λ m ⊕ 2Λ e ) it is given by The numbers c 8 (...) are the Fourier coefficients of the Γ 0 (2) modular form So P ∈ Λ m \2Λ e is a sufficient condition for the absence of a second contribution (in (5.8)) and for (0, P ) being counted by (5.6) and (5.7). Comparing this to the well-known twisted sector quarter-BPS dyons, the latter condition is satisfied, as can easily be seen from the charge representatives given, e.g., in [44,56], and the function (5.6) is indeed found at the quadratic pole z = 0 of the twisted sector quarter-BPS counting function. For the untwisted case at hand, we will see shortly that (5.6) is consistent with explicit charge vectors representing a subsector of the untwisted sector of interest.
For the electric component, i.e., primitive half-BPS states of charge (0, Q) ∈ Λ u e , our review of electric DH states in section 3 leads us to the counting function (3.19), which is why we already used the same symbol φ −1 e in (5.5). Since f e (σ) := E 4 (2σ)/∆(σ) and η −8 (τ )η −8 (2τ ) transform as weight −8 modular forms for Γ 0 (2) = Γ 1 (2), the weight of Z untw Het is fixed to be −6. It should also transform as a Siegel modular form under modular transformations given by Indeed, using the expression (4.30) found for Z untw Het and the behaviour of the genus two theta constants for z → 0 (appendix A), the proposed untwisted sector quarter-BPS partition function obeys (5.5) with the correct electric and magnetic half-BPS partition functions.
Second wall. Next we want to investigate the decay into half-BPS states (Q, P ) → (Q − P, 0) + (P, P ) . (5.11) This decay is now encoded by the matrix ( 1 1 0 1 ) and demands that Z untw Het exhibits a quadratic pole at z ′ = 0 (recall eq. (2.40)), with coefficient given by The variables for this decay are related via (2.45), explicitly Even though this decay is related to the previous one by an S-duality transformation in Γ 1 (2), we shall briefly analyze it to further illustrate the appearance of the Iwahori subgroup B (2) on physical grounds. Furthermore, it allows to better test our proposed partition function against the analysis presented in [44, section 6.5], which concerns a subsector of the quarter-BPS sector with untwisted electric charge. Now note that adding any vector from Λ m to Q ∈ Λ u e cannot change the residue [Q] ∈ Λ * m \Λ m and especially the winding number along the CHL circle (quantized in 1 2 Z in our convention 61 ) modulo one and the momentum quanta along the CHL circle (quantized in Z in our convention) modulo two will be preserved. The former is integral and the latter takes both even and odd values. Thus the half-BPS partition function for decay products (Q − P, 0) ∈ Λ u e here again becomes (5.14) Also since S-duality relates (P, P ) ⊺ = (( 1 1 0 1 )(0, P )) ⊺ to (0, P ), we know that .
Both (5.14) and (5.15) also follow from S-duality invariance for elements (5.2) by combining (5.3), (5.5) and (5.13). Since the partition function φ −1 e (σ) belongs to primitive, half-BPS states of charge (Q, 0) ∈ Λ u e , consistency requires (Q − P, 0) ∈ Λ u e to be primitive as well. 62 We have already mentioned that the functions appearing here are modular forms for the  [44,56]. 62 IfQ = (Q − P ) was an even multiple of some vector, the momentum parity would be even, which would lead to eq. (3.22). Also note that in the new duality frame obtained by the S-duality transformation ( 1 1 0 1 ) we still count dyons of torsion one, soQ must be primitive in Λe. is required to transform as a Siegel modular form with respect to where γ 1 and r 1 are even, while α 1 , δ 1 , p 1 and s 1 are odd.
Again we compare these constraints to the explicit form of Z untw Het proposed before. Since (2.45) describes a B(2) transformation, Z untw Het (Z) = Z untw Het (Z ′ ) holds. This immediately translates (5.5) into and therefore matches (5.12).
Third wall. There is one decay channel only possible for dyons with untwisted sector charge (Q, P ) ∈ Λ u e ⊕ Λ m subject to the extra condition Q ∈ Λ m ⊂ Λ u e , namely (Q, P ) → (Q, Q) + (0, −Q + P ) .
Recall that Q is primitive in Λ e since we consider unit-torsion, so the first decay product , (5.21) in accordance with (5.7). Note that (Q, Q) ⊺ = ( 1 1 0 1 ) 0 Q is related via an S-transformation in Γ 0 (2) to a purely magnetic charge of the form (0,P ) ∈ Λ m \2Λ e . 63 It was mentioned before that Q − P is primitive for the dyons captured by Z untw Het , so the 63 So the subscript "e" in φ −1 e (σ ′ ; 1, 1) is a notational artifact inherited from [44].
second decay product, a purely magnetic half-BPS state of charge (0, −Q + P ) ∈ Λ m \2Λ e , also corresponds to the eta-quotient .
Combining the ingredients we infer that wall-crossing demands that for z ′ → 0 the quadratic pole in Z untw Het becomes with the given eta-quotients. As a consequence, by (2.47) and (2.48) the partition function Z untw Het should also transform as a Siegel modular form with respect to the embedded transformations . Let us see whether (5.23) is also satisfied for the concrete Z untw Het proposed before. Starting from (4.31), we consider the tautology Z untw (5. 25) This means that and thus .
Now use the behaviour of the theta constants θ a 1 a 2 b 1 b 2 (Z ′ ) under z ′ → 0 (again see appendix A) to find that the second term in (5.27), being proportional to θ 2 1111 (Z ′ ), vanishes quadratically in z ′ for z ′ → 0. Only the first term contributes to the quadratic pole in z ′ which is relevant for the BPS indices. More precisely, for z ′ → 0 we have nicely matching our wall-crossing expectations.
Modular symmetry group. In close parallel to [44, section 6.5] the question emerges, whether the symplectic matrices (5.1), (5.2), (5.10), (5.16) and (5.24) fit into a subgroup of Sp 4 (Z) defined by some congruence relation. Affirmative answer can be given for the group where the group on the right hand side is the Iwahori subgroup B(2). This group is defined in eq. (A.9) and indeed turns out to be the modular symmetry group of Z untw Het in eq.
As this restriction is only preserved for S-transformations in Γ(2) ⊂ Γ 1 (2), the partition function for this subsector does not need to be invariant under all elements in (5.2), but only under those where b is even. In this case the second entry of the first row and the third entry in the last row of (5.29) can consistenly be restricted to elements in 2Z, recovering the group proposed in [44].
Second, the above parity restrictions on the quadratic T-invariants can be implemented in the (full untwisted) partition function by applying suitable projections, for odd Q 2 /2, for instance, one has the lower sign in Third, the odd CHL momentum restriction corresponds to choosing ǫ = −1 in (3.22) for the partition function of half-BPS decay products (Q, 0) in [44], which is why our Z untw Het is allowed to deviate in the counting of (Q, 0) half-BPS decay products on the divisor z = 0.
In other words, our Z untw Het counts also those dyons corresponding to ǫ = +1 and the terms cancel as explained in section 3.
Last, regarding magnetic half-BPS states (0, P ) counted at z = 0, our assumption (5.6) is compatible with the explicit representatives P chosen in [44, section 6.5]. These are primitive vectors P ∈ Λ m \2Λ e . Indeed, these are also the same magnetic charges as occuring in the twisted sector quarter-BPS states [44, section 6.4] (up to restriction to odd K quantum number there, causing P 2 /2 to be odd for the untwisted case).
Taking these points into account, our results are compatible with the predictions in [44, section 6.5]. Affirmative consistency checks starting from charge representatives in other subsectors can be performed in complete analogy to [44, section 6.5], however, these are mostly straightforward and in light of our preceeding analysis rather redundant so we will not display them here.
Making an ansatz. In summary, the constraints from quantization laws, S-duality and wall-crossing suggest that Z untw Het transforms as a Siegel modular form for the Iwahori subgroup B(2) with weight −6. Since explicit generators for the ring of even (positive) weight Siegel modular forms for Γ  (2), which has index two in SL 2 (Z). We can therefore demand that Z untw Het (Z) must exhibit a quadratic pole at all images of the diagonal locus ( τ 0 0 σ ) under the group generated by SL 2 (Z)-transformations (2.34) and integer translations (5.1). 64 The arguably simplest compatible ansatz one might choose for Z untw Het (Z) is F (Z)/χ 10 (Z), where the Igusa cusp form χ 10 , i.e., the product of the square of the ten even genus two Thetanullwerte, vanishes quadratically at all Sp 4 (Z)-images of the diagonal. The latter is also the partition function for unit-torsion quarter-BPS dyons in the parent theory and our dyons of interest might be regarded as an invariant subset thereof. In this ansatz F (Z) is a weight four Siegel modular form for B (2), which is expected to be holomorpic in H 2 such that there are no additional, spurious poles. Zeroes in F (Z) however might cancel any additional, spurious poles in χ −1 10 (if there are such). Working, for instance, with the ring generators given in (A.36) and the properties of the theta constants, the behaviour of Z untw Het at the wall-crossing divisors fix F (Z) eventually to F = 1 2 (Y + 16T ), as in (4.31) (or eq. (7.10) below). Hence, as announced earlier, the modular and polar constraints alone are (almost) restrictive enough to guess Z untw Het in closed form. Of course, the analysis of section 4 or 7 already provides an explicit expression, which we have shown to satisfy all constraints. 64 As an aside, motivated by CHL dyon counting functions Cléry and Gritsenko [156] classified and constructed all so-called dd-modular forms, i.e., Siegel modular forms for the Hecke congruence subgroups Γ (2) 0 (N ) which vanish precisely along the Γ (2) 0 (N )-translates of the diagonal divisor z = 0 (with vanishing order one; possibly with a multiplier system). Especially, this includes the square roots of the Igusa cusp form and the Siegel modular form Φ6,0 appearing in the N = 1, 2 CHL models. However, this does not characterize our untwisted sector partition function.

Black hole entropy
There is one more physical constraint on a quarter-BPS partition function in four-dimensional N = 4 theories. For large dyon charges (Q, P ) the microscopic BPS index should yield the macroscopic entropy of an extremal black hole carrying these charges, e.g., as computed in the supergravity approximation. The following analysis will focus on the least intricate features, namely the Bekenstein-Hawking term and the first correction in inverse powers of the charges. We simply quote the mathematical consequences for Z untw Het , paralleling the discussion for its twisted sector counterpart [5,11,39,44,54,56].
Generic for all CHL models, the leading term in the entropy of an extremal black hole carrying large charges (Q, P ) is the just mentioned Bekenstein-Hawking area term. Together with the leading correction in inverse powers of the charge this gives an entropy This correction has been determined in [18] from the entropy function [17] by including the Gauss-Bonnet term in the effective supergravity action [7,157], where τ = a + iS denotes the axio-dilaton modulus. We also have a model dependent function φ which for our Z 2 model becomes φ(a, S) = − 1 64π 2 [8 log S + log g(a + iS) + log g(−a + iS)] + const. In order to formulate the resulting constraint for our microscopic Z untw Het we follow [5,11,39,54,56]

Let us thus address what behaviour of Z untw
Het is expected near D = 0. Using a symplectic transformation to introduce new coordinates around the divisor, the partition function should behave near D = 0 (now z ′ = 0) like As noted in [44, section 6.5], if this is the case the macroscopic entropy (6.1) will be reproduced (upon standard procedure executed already for the twisted sector case).
We shall now check whether the partition function (4.31) satisfies (6.8). Since the symplectic transformation M A is not an element of Γ For instance, we have Making use of (A.26) and (A.27) we find for the limit z ′ → 0 that only the second term in the first line of (4.31) (with Y ′ in the numerator, formally the same as the twisted sector partition function) contributes to the double pole. The calculation gives which indeed reproduces the expectation (6.8) by virtue of the first eta-product identity in (A.29). In other words, our untwisted sector partition function Z untw Het gives rise not just to the leading Bekenstein-Hawking term, but also to the correct subleading correction in inverses powers of the charges (6.1), very similar to the twisted sector quarter-BPS partition function. We leave it as an open problem to perform more careful, extensive analyses as, e.g., in [14,25] and to check whether a difference in the entropy of twisted sector and untwisted sector (quarter-BPS unit-torsion) dyons can be found in further subleading terms (say at exponentially suppressed orders). If so, one might ask for a macroscopic explanation in the quantum entropy function (say as certain sub-leading saddles to the supergravity path integral), see the references in the introduction for similar research.
Having successfully passed the test of black hole entropy, we finally check whether Z untw Het also passes the test of matching to the DT results.

Comparison to results from Donaldson-Thomas theory
The spectrum of quarter-BPS states in four-dimensional N = 4 string theories has been linked to the enumerative geometry of algebraic curves in Calabi-Yau threefolds. Predictions from string duality have thus led to precise mathematical conjectures [112,116], some of which have been proven in recent years [114,115]. Here we utilize the connection between quarter-BPS indices and reduced Donaldson-Thomas (DT) invariants in order to test our proposal for the partition function of untwisted sector quarter-BPS dyons in the Z 2 CHL model against a recently conjectured formula for its (tentative) DT counterpart [116].
In brief, DT invariants can be given for two different kinds of curve classes (twisted and untwisted) on the Z 2 CHL orbifold. As has been explained in [116], one set of invariants By the divisibility of a curve class γ ∈ Image(Π| N 1 (S) ) one means the biggest integer m ∈ N >0 for which γ m ∈ Image(Π| N 1 (S) ) ⊂ 1 2 is satisfied, where N 1 (·) denotes the group of algebraic one-cycles. If its divisibility is 1, γ is called a primitive class, which is further called untwisted if γ ∈ H 2 (S, Z), or twisted if Z). For some primitive, non-zero γ with self-intersection we consider the curve class The reduced Donaldson-Thomas invariant DT X n,(γ,d) only depends on n, s, d and whether γ is untwisted or twisted, so one may also write DT untw n,s,d and DT tw n,s,d for the two cases. Introducing respective partition functions and writing q = e 2πiτ , t = e 2πiσ , p = e 2πiz , and Z = τ z z σ ∈ H 2 (7.8) one obtains tentative Siegel modular forms.
The partition function for the twisted primitive DT invariants on X is conjecturally given by the negative reciprocal of the Borcherds lift of the corresponding twisted-twined elliptic genera, and thus agrees with the quarter-BPS counting function obtained in [39,40], which is (possibly up to a multiplicative constant) the factor in parentheses in (4.32).
On the other hand, the untwisted primitive invariants are determined by 4 (2Z) χ 10 (Z) , (7.10) where χ 10 is the weight ten Igusa cusp form appearing already in the partition function of the unorbifolded model (DT theory on S × E, physically IIA[S × E] or Het[T 6 ]). In the numerator we have two Siegel modular forms G 4 (Z) and E (2) 4 (2Z), both of weight four for the level two congruence subgroup Γ (2) 0 (2) ⊂ Sp 4 (Z). The function F 4 (Z) is a weight four Siegel paramodular form of degree two for the paramodular group K(2). All of them can be expressed within the ring of even genus two theta constants, see appendix A. Thus, Z untw is a weight −6 Siegel modular form for the level two Iwahori subgroup B(2) = K(2) ∩ Γ DT invariants as BPS indices. A connection to physics was already outlined in the appendix of [116], which we shall reproduce and build on.

Conclusion
In this paper we have studied the spectrum of quarter-BPS states in the Z 2 CHL model.
A novel partition function for unit-torsion dyons with untwisted sector electric charge in ⊂ Λ e has been derived by from a chiral genus two orbifold partition function in the heterotic frame, paralleling the original derivation for well-known twisted sector dyons.
Stringent constraints coming from charge quantization, wall-crossing and S-duality symmetry are shown to be satisfied. As we have argued, these do not only elucidate the role of the Siegel modular symmetry group underlying the untwisted sector counting function, but also allow for a "modular bootstrap derivation" of the latter. As a partition function of black hole microstates it also reproduces the correct macroscopic entropy of a dyonic extremal black hole in the large charge limit, including the Bekenstein-Hawking term and the first powersuppressed correction that can be accounted for by the inclusion of the Gauss-Bonnet term in the 4D effective action. A further non-trivial check of the proposed partition function comes from N = 4 heterotic-type IIA duality: regarding the BPS indices as Donaldson-Thomas invariants of the CHL orbifold X = (K3 × T 2 )/Z 2 , we arrive at an alternative, but equivalent, expression for the primitive untwisted DT partition function that has recently been conjectured by Bryan and Oberdieck [116].
Let us first comment on possible extensions of the Z 2 model analysis presented here. In many points this means performing checks and derivations that were hitherto only done with the twisted sector counting function (possibly because a handy closed formula was not known Furthermore, it has been conjectured in [158] that for single-centered black holes with regular horizon, i.e., quarter-BPS states states subject to a certain restriction on their quadratic charge invariants, the quarter-BPS index should be positive. Numerical evidence was given in [158] for low lying (twisted sector) charges in the order N ∈ {1, 2, 3, 5, 7} orbifolds, and for N non-prime in [159]. For the unorbifolded theory (i.e., N = 1), this conjecture has rigorously been proven for a subset of the corresponding charges [160]. 68 We leave it to future work to check the conjecture for the untwisted sector quarter-BPS states considered here.
Clearly, there is also room for extending the present analysis to untwisted sector dyons in other N = 4 CHL models. On one side, this includes models for symplectic K3 automorphisms for higher (prime or composite) order N as well as non-cyclic orbifold groups. On the other side, we may consider CHL models arising from "non-geometric" symmetries of a K3 non-linear sigma model [76,78]. This could lead to interesting enumerative predictions for the symplectic invariants of these theories, as was also remarked in [116]. 69 Since one of the biggest challenges in counting stable BPS states in different regions of the moduli space and relating them to the black hole entropies is to consider less supersymmetric -in particular N = 2 -compactifications, let us conclude here with a comparison of the formalism used in this paper with recent BPS counts on related N = 2 Calabi-Yau geometries. The closest N = 2 cousin of the CHL (K3 × T 2 )/Z N compactification is the Ferrara-Harvey-Strominger-Vafa (FHSV) compactification [162], which is likewise a (K3 × T 2 )/Z 2 , with the difference that the Z 2 action is the Enriques involution on the K3 and the hyperelliptic involution of T 2 leaving the four branch points fixed. The threefold has a K3 fibration with four Enriques fibers, SU(2) × Z 2 holonomy, Euler number χ = 0 and a considerably milder scale dependence than generic N = 2 compactifications. In particular it has no genus zero world-sheet instantons and the D2-D0-brane states that are counted by the topological string theory at higher genus are related to the direct integration of the holomorphic anomaly [163], starting with the automorphic function constructed by a Borcherds lift as product formula in [80] for the K3 fiber and the Dedekind eta function for the base [164,165]. In N = 2 compactifications the duality symmetries are realized in a much more complicated way, which makes it hard to extract information about different brane states from the same modular object. However, in [163] it was observed that the automorphic form of [80] has two cusp expansions: one yielding the D2-D0-brane states and one involving light D4-brane states wrapping the K3 fiber. It would be very important to understand the non-perturbative completion of the FHSV model by an extension of the 68 In the case of N = 4 type II toroidal orbifolds the conjecture, however, seems to be violated [71]. 69 In addition, [79] proposed a yet broader notion of CHL compactifications. Speculatively, there might be CHL versions of heterotic strings on T 8 whose BPS spectrum could have an enumerative geometric meaning on appropriate Calabi-Yau duals, say on K3×T 4 or orbifolds thereof. This was also speculated in [161], where for the case of heterotic strings on T 8 a BPS counting function was written down in terms of the Borcherds automorphic form Φ12. It is natural to guess the appearance of similar automorphic forms in the corresponding CHL orbifolds. duality and wall-crossing arguments used here for the CHL orbifold model.
Heterotic strings on CHL Z N orbifolds are dual to type IIA compactifications on elliptically fibered K3 with N sections, completed by the T 2 modded out by the shift symmetry [116]. As mentioned the primitive DT partition functions Z N of [116] are related to the inverse of Siegel modular forms of (Iwahori) congruence subgroups of Sp 4 (Z). These forms generalize the weight ten Igusa cusp form of Sp 4 (Z), whose inverse Z 1 describes the BPS states of K3 × T 2 . Moreover, the Fourier expansions of Z N specialize in the non-perturbative limit to inverses of the cusp forms ∆ N (τ ) of Γ(N ) of weight ⌈ 24 N +1 ⌉ times theta factors. For N prime these are the same cusp forms that have been obtained in the leading nonperturbative expansion in [39], see eq. (2.18) there, and they are associated to the action of Γ 1 (N ) on the perturbative CHL string. On the N = 2 side it has been realized in [166] that on elliptically fibered Calabi-Yau threefolds with one section and no singularities from the fiber, the topological string has an expansion in terms of Jacobi forms that bears similarity to the expansion of Z 1 in terms of Jacobi forms of SL 2 (Z) as discussed in section 5 of [69], with the difference that the index of the Jacobi form grows cubic with the exponent of the expansion parameter Q β rather than linear as in [69]. This fact has been used to check the microscopic entropies of spinning black holes [167]. In [168] the work of [166] has been extended to genus one fibrations with N -sections as well as to elliptic fibrations to N sections in the limit of the Kähler parameters that corresponds to the additional sections. In the former case Jacobi forms of Γ 1 (N ) occur in the BPS expansion related to D2-D0-branes also for non-prime N , while in the latter case these BPS expansions are expressible in terms of Jacobi forms of Γ(N ) and in particular the cusp forms ∆ N (τ ) occur in the denominators.
This indicates that some features of the analysis of BPS degeneracies related to the more complicated fibration structures carry over from the N = 4 to the N = 2 case.
Acknowledgements: We would like to thank Georg Oberdieck for useful explanations concerning his work and Cesar Fierro Cota for discussion on Jacobi forms of congruence subgroups of the modular group. F. F. and C. N. also thank the Bonn-Cologne Graduate School of Physics and Astronomy (BCGS) for generous support.

A Siegel modular forms
In this appendix we collect basic definitions and useful formulae for the Siegel modular forms appearing in the main text. Our main references are [169,chapter VII], [116, section 2] and [61, appendix A], also see [69] for a review that emphasizes the relation between the theory of Siegel modular forms, mock modular forms and quantum black holes.
Preliminaries. By Sp 4 (Z) we denote the symplectic group of integer 4 × 4 matrices M = A B C D that satisfy which is equivalent to for the 2 × 2 block matrices in M . The groups Sp 4 (Q) and Sp 4 (R) are defined analogously.
If M ∈ Sp 4 (Z) as above then the inverse of M is given by and by using this in (A.1) we see that also M ⊺ ∈ Sp 4 (Z). Taking the Pfaffian and using For a prime number p ≥ 1 the group K(p) is defined by [170,171] while the Iwahori subgroup is defined by the intersection By conjugation in GL 4 (Q) (see [170] for references) the group K(p) is related to the Siegel paramodular group Γ para (p), formed by integer 4×4 matrices that obey (A.1) with J replaced by J 2 (p) = 0 P −P 0 with P = diag(1, p). Let respectively. From these expressions it follows that the diagonal locus z = 0 is preserved under the two embedded subgroups, where they operate componentwise on τ ∈ H 1 and σ ∈ H 1 , respectively. Another symplectic transformation preserving the diagonal locus is given by (A.5) with U = ( 0 1 1 0 ), which exchanges the diagonal entries of Z. Now let f : H 2 → C be a holomorphic function, k be an integer and Γ ⊂ Sp 4 (Z) be a congruence subgroup (or a discrete subgroup Γ ⊂ Sp 4 (R) with finite covolume [170,172]). If with shorthand e(z) = exp(2πiz) for z ∈ C. This is also written as θ[ a b ] = θ a 1 a 2 b 1 b 2 . The theta constants vanish identically iff a ⊺ b mod 2 is odd. For genus two there are precisely ten "even" non-trivial theta constants. There is a useful transformation formula under As was proven in [172] (see also [173]), the functions X, Y, Z, W are Siegel modular forms for Γ (2) 0 (2) of respective weight 2, 4, 4 and 6 and they generate the ring of even weight modular forms for Γ The function W agrees with the function "K" defined in [156]. On the other hand, the function T is a weight four modular form for the Iwahori subgroup B(2) and by [172] the structure of the ring of even weight modular forms for B(2) is known to be Mod For the structure of the ring of modular forms for K(2) ⊃ B(2) we refer to the results given in [170] and just mention that the function F 4 (Z) appearing in the untwisted sector quarter-BPS partition function is the unique weight four Siegel modular form for K (2), which may be defined as Also the Siegel modular form G 4 (Z) appears in the untwisted sector partition function, which satisfies G 4 (Z) = 1 120 4 (Γ    The Igusa cusp form χ 10 ∈ Mod 10 (Sp 4 (Z)), whose reciprocal counts unit-torsion dyons in Het[T 6 ], is given by the well-known product of the squares of all even genus two theta constants χ 10 (Z) = Y W .
(A. 48) In the Z 2 orbifold we also encounter the Γ  There are many quadratic relations that the squares of the theta constants satisfy and which have, for instance, been reviewed in [174]. One particular identity important for our untwisted partition function is the relation [174, eq. (5.1)] θ 2 0100 θ 2 0110 = θ 2 0000 θ 2 0010 − θ 2 0001 θ 2 0011 , (A.59) 72 The minus sign in (A.52) and (A.54) is imporant for reproducing the result for the orbifold block Z8[ 0 0 0 1 ] obtained in [42, eq. (4.38)], c. f. the relative signs between the terms in (4.5) and ultimately also for matching the DT result for Z untw to the heterotic result for Z untw Het .
which implies for the above Siegel modular forms Finally, we remark that the quadratic divisors, on which χ 10 and its orbifold analog Φ 6,1 (or Φ 6,3 with the roles of the diagonal entries swapped) vanish quadratically, can, for instance, be found in [54, section 4] or [56, appendix D]. By an appropriate Sp 4 (Z)-transformation they can be mapped to the standard diagonal divisor, as was used in [25].