Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting

The degeneracies of $1/4$ BPS states with unit torsion in heterotic string theory compactified on a six-torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $\Phi_{10}$ of weight $10$. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of $1/\Phi_{10}$. The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rm Sp}(2, \mathbb{Z})$ which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of $1/\eta^{24}$ by means of a continued fraction structure.

Modular and Jacobi forms, as shown in the first two examples, have automorphic properties under SL(2, Z) and are defined in the σ upper half plane. Both of these forms have a finite number of Fourier coefficients associated with a negative power of q. These coefficients are referred to as a polar coefficients. The modular symmetries of these functions are powerful enough to constrain the Fourier coefficients so that each non-polar coefficient can be exactly expressed in terms of the polar ones in a Rademacher expansion [6]. As an illustrative example, we present the well-known Rademacher expansion for 1/η 24 , which has been used to extract asymptotic degeneracies for 1/2 BPS states in toroidally compactified heterotic string theory [7], d(n) = +∞ γ=1 d(−1) 2π γ n 13/2 Kl(n, −1, γ)I 13 4π √ n γ .
(1. 2) We see that the polar coefficient d(−1) is sufficient to reconstruct the modular form. In this note we will focus on the third of the above examples, the reciprocal of the Igusa cusp Siegel modular form Φ 10 of weight 10. It is defined on the Siegel upper half plane, with each of its Fourier coefficients defined in terms of three integers corresponding to the three variables defining the Siegel upper half plane. Further, each of its polar terms satisfies ∆ < 0, where ∆ = 4mn − 2 , and is determined in terms of the coefficients of 1/η 24 and the continued fraction representation of /2m [8][9][10]. Using the automorphic properties of Φ 10 with respect to Sp(2, Z) 1 , we will demonstrate that this symmetry group constraints the Fourier coefficients of 1/Φ 10 even more powerfully than in the SL(2, Z) case, resulting in a fine-grained Rademacher expansion that not only reconstructs each element of the infinite set of its Fourier coefficients that satisfy ∆ > 0 from the finite polar data, whose elements satisfy ∆ < 0, but also encodes the continued fraction structure underlying the polar terms. Thus, the result of this paper can be summarised as follows: We use the Sp(2, Z) symmetries of 1/Φ 10 to construct a fine-grained Rademacher expansion which expresses its Fourier coefficients as a regularized sum over residues of its poles. The construction uses two distinct SL(2, Z) subgroups of Sp(2, Z) which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of 1/η 24 by means of a continued fraction structure.
The Rademacher expansion for d(m, n, ) was derived in [11] by viewing d(m, n, ) as Fourier coefficients of a Mock Jacobi form ψ F m [12,13] using the mixed Mock Rademacher expansion developed in [14]. However, just as in the case of modular forms, the Rademacher expansion in [11] does not contain information about the explicit values of the polar coefficients of ψ F m , as computed in [9,10].
2 Set up of the calculation and outline of the paper In order to define a Rademacher expansion for the reciprocal of the Igusa cusp Φ 10 of weight 10, we adopt the following sequential program elucidated below: 1. Identify polar data in 1/Φ 10 . In contrast to the modular and Jacobi forms displayed above, the Siegel modular form Φ 10 has a countably infinite set of zeroes defined in the Siegel upper half plane by the loci [15], D(n 2 , n 1 , m 1 , m 2 , j) : n 2 (ρσ − v 2 ) + jv + n 1 σ − m 1 ρ + m 2 = 0 (2.1) m 1 , n 1 , m 2 , n 2 ∈ Z, j ∈ 2Z + 1, m 1 n 1 + m 2 n 2 = 1 − j 2 4 .
Viewing non-polar data as the complementary information to its polar counterpart, one is motivated to identify the non-polar data in 1/Φ 10 as the contribution to g(m, n, ) with ∆ ≥ 0. Generically, these Fourier coefficients receive contributions from both linear and quadratic poles. where ρ = ρ 1 + iρ 2 , σ = σ 1 + iσ 2 , v = v 1 + iv 2 , and where the imaginary parts of ρ, σ and v are held fixed in the Siegel upper half plane, which is defined by the conditions ρ 2 > 0, σ 2 > 0 and ρ 2 σ 2 − v 2 2 > 0. The n 2 = 0 poles of 1/Φ 10 correspond to co-dimension one surfaces in the Siegel upper half plane. Therefore, in order to identify the non-polar data in terms of the Fourier coefficients, we first need to isolate a chamber in the Siegel upper half plane where we perform a Fourier expansion. Following [8] we will define this region, which is referred to as the R-chamber, by We will write down the Rademacher expansion for the Fourier coefficients with ∆ ≥ 0 in the R-chamber.
3. Setting up the Rademacher expansion. Viewing (2.3) as a ρ integral, we define: where 'reg' refers to a regularized sum of residues, (a) Resf is the residue of f at an n 2 = 0 pole D in the ρ plane; Further, the condition that D lies in the Siegel upper half plane constraints the range of v 1 ; (c) Γ σ (D) lies in the projection of D in the σ upper half plane with the restriction that σ 1 ∈ [0, 1).
This defines our starting point. Our objective is to express the right hand side of (2.5) in terms of the polar data. This implies that the sum over residues of quadratic poles must be rewritten as one over linear pole contributions.
In section 3 we use the Sp(2, Z) symmetries to map each quadratic pole to the simplest linear pole v = 0. We will reparametrize the five numbers defining the quadratic poles by entries of the Sp(2, Z) matrix that performs this map and consequently rewrite the summation on the right hand side of (2.5) as a sum over these matrix entries. In sections 4 and 5 we explicitly evaluate the summands by performing the v and σ integrals respectively. We isolate multiplier systems, Kloosterman sums and error functions giving rise to Eichler integrals and to the continued fraction structure. These elements serve as building blocks for the Rademacher expansion. We identify a symmetry that is crucial to obtain the Rademacher expansion by enabling massive cancellations between various terms. The resulting expansion is given by (5.118) in the case when ∆ > 0. Additionally, we write down the Rademacher expansion for the case when ∆ = m = = 0 in (6.31). In section 7 we conclude with comments on the implications of this expansion for defining exact quantum entropy functions from a gravity path integral. In the appendices we review and discuss various useful relations and calculations.
3 Sp(2, Z) symmetries, poles and residues of 1/Φ 10 In this section, we map each quadratic pole of 1/Φ 10 to the simplest linear pole v = 0. In order to so, we will use the Sp(2, Z) symmetries of Φ 10 to reparametrize the five numbers defining the quadratic poles by entries of the Sp(2, Z) matrix that performs this map. Subsequently, we rewrite the summation on the right hand side of (2.5) as a sum over the entries of these mapping matrices.
The Igusa cusp form Φ 10 transforms as follows under Sp(2, Z) transformations, where The elements of Sp(2, Z) satisfy Thus, under S-duality, we infer the transformation laws They leave Φ 10 invariant, Next, we consider a different set of Sp(2, Z) transformations, which we denote by SL(2, Z) σ . We denote its group elements by α β γ δ , αδ − βγ = 1. (3.8) They operate on Ω through the Sp(2, Z) transformations which results in the transformation laws Under this set of transformations, Φ 10 transforms as Furthermore, Φ 10 is also invariant under the integer shifts which implies that it possesses a Fourier expansion. Performing first an SL(2, Z) σ transformation, then an S-duality transformation, and finally an integer shift of v by −Σ, Φ 10 changes as 14) The reciprocal of Φ 10 transforms accordingly.

Reparametrization of the poles of 1/Φ 10
The Siegel modular form Φ −1 10 (ρ, σ, v) has the following behaviour near v = 0, Therefore, it has poles at all the Sp(2, Z) images of the divisor v = 0 in the Siegel upper half plane. The location of the poles is determined by The poles are labelled by five integers, (m 1 , m 2 , j, n 1 , n 2 ) that satisfy the constraint Since (3.16) and (3.17) are invariant under (m 1 , m 2 , j, n 1 , n 2 ) → (−m 1 , −m 2 , −j, −n 1 , −n 2 ), we may restrict to n 2 ≥ 0 [16]. A specific parametrization of these poles was given in [17]. We will use an equivalent parametrization, as follows.
Proposition 3.1. Any pole (3.16) with n 2 ≥ 1 can be reparametrized in terms of nine integers as where eight of the integers can be arranged into two SL(2, Z) matrices, 19) with entries that satisfy a > 0, c < 0, γ > 0, α ∈ Z/γZ, while Σ ∈ Z. Moreover, any pole (3.16) with n 2 ≥ 1 can be mapped to the pole v = 0 by the following sequence of Sp(2, Z) transformations, namely first an SL(2, Z) σ transformation, then an S-duality transformation, and finally an integer shift of v.
Using the parametrization (3.18), each of the poles (3.16) with n 2 ≥ 1 corresponds to an element of the set P ∪ {Σ ∈ Z}, where P is defined by Defining the sets of matrices, we will be summing over elements in Γ ∞ \S Γ , S G , and in {Σ ∈ Z}, where We can therefore parametrize the sum over poles in the set P as 3.3 Residues at v = 0 As shown above, any pole (3.16) with n 2 ≥ 1 can be mapped to the pole v = 0 under the Sp(2, Z) transformation with ρ , σ , v given by (3.14). Defining From the above we see that for a Λ satisfying the constraint ρ = Λ, a translation of Σ in integral units of ac will modify Λ such that it will no longer satisfy the constraint in the given range of ρ 1 . Hence we restrict Σ to take values Σ ∈ Z/|ac|Z. Next we evaluate the ρ-integral in (2.3). We only consider the contributions from the residues associated with the poles of 1/Φ 10 with n 2 ≥ 1. Since any such pole can be mapped to the pole v = 0, we compute the residue associated with ρ = Λ(σ, v), for fixed σ and v by noting that in the neighbourhood of v = 0, 1/Φ 10 behaves as .
Therefore, for fixed σ and v, evaluating the residue at ρ = Λ(σ, v) using (3.46), we obtain we obtain Integrating over v Next, we perform the integration over v in (2.5). We will first define the contour Γ v (D) for a given n 2 ≥ 1 pole. For notational simplicity we will henceforth refer to this contour by Γ v . Recall that we are considering poles with n 2 ≥ 1. The pole v * = 0 specified in (3.50) will be in the Siegel upper half plane provided Imρ * > 0, Imσ * > 0 . (4.1) Evaluating Imρ * = 1 n 2 |γσ + δ| 2 γabσ 2 − γδa 2 v 2 + γ 2 a 2 (v 1 σ 2 − v 2 σ 1 ) , and imposing (4.1) results in Using the value for v 2 /σ 2 given in (2.4), we obtain The above defines a contour Γ v of integration for the v-integral that goes from Since n 2 ≥ 1, the range of integration specified by (4.4) lies in the original unit interval length integration contour for v 1 , as required.

The v integral
The residue associated with the pole v * = 0 was given in (3.49). We now turn to the v-integral over the sum of the residues associated with the poles that belong to the set P ∪ {Σ ∈ Z} given in (3.38). We follow the prescription given in (2.5), (4.7) Since ρ * and σ * satisfy (4.1), we may Fourier expand 1/η 24 and E 2 using and Then, (4.7) becomes P Σ∈Z Substituting the values for ρ * and σ * given in (3.50) results in This leads us to define the combinations Notice that (4.11) contains the following sum over Σ, which is only non-vanishing provided that the combination L is an integer, in which case this sum equals −ac. In other words, the only contributions to (4.11) will come from poles that satisfy the divisibility condition Thus, restricting to L ∈ Z, (4.11) becomes Using (4.13), we obtain We note that the triplets (m,ñ,˜ ) and (M, N, L) are related by the following SL Therefore, since the triplet (M, N, L) consists of integers, alsoñ and˜ have to be integers. Thus, we have two triplets of integers that are related by the SL(2, Z) matrices G given in (3.19).
Note that∆ is invariant under the SL(2, Z) transformation G given in (3.19), Now we recall that we are considering BPS dyons with ∆ = 4mn − 2 > 0, and hence m > 0, in which case we may perform the following rewriting of the exponent in (4.16), Using this, (4.16) can be written as (4.23)

T -shifts
In the following, we rewrite the expression (4.23) by recasting the sum over b ∈ Z as a sum over a new integer T , as follows. Let T ∈ Z and consider the matrix while leaving a and c invariant. Therefore, expressions such as L that only depend on a and c are invariant under this change. Hence, also∆ is invariant under this change. On the other hand,˜ andñ transform as follows, Next, note that˜ can be written as and hence the condition b ∈ Z/aZ translates to˜ ∈ Z/2mZ. Thus we can express (4.23) as Using (4.25), the integration contour (4.6) goes from 29) and the range of integration specified by (4.4) requires restricting the values of T to T ∈ Z/γZ.

Performing the v-integral
We now perform the v-integral in (4.28) along the contour Γ v specified by (4.29). To do so, we assume the legitimacy of interchanging the integration with the summation over M, N in (4.28). Note that the dependence on v is contained in the last line of (4.28), only, (4.31) Using the expression for the error function we obtain for (4.31), where we take the principal branch of the square roots. Hence, integrating (4.28) over v results in (4.34)

Splitting the error functions
The error functions in (4.34) can be split into three different terms. We refer to Appendix A for the details on this. Here we briefly summarise its salient features. Using the property Erf(−x) = −Erf(x), we write the difference of the error functions in (4.34) as Next we define the following functions, and and consider the product Depending on the sign of XY , the expression for the sum (4.37) will take a different form, as follows. If XY > 0, (4.37) becomes If XY = 0, (4.37) has the form where we used that when X = 0, Y is given by Y = −1/ac, and when Y = 0, X is given by X = −1/ac. Finally, when XY < 0 , (4.37) takes the form Since −1/ac > 0, we can re-express the condition XY ≥ 0 as This condition [10] is satisfied by all the convergents of the continued fraction of 2m . (4.46) The above shows that the error functions in (4.34) give rise to three distinct types of terms, namely constants, terms involving E m,γ(γσ+δ) and terms involving I m,γ(γσ+δ) . As we will see below, upon performing the σ-integral, these three distinct contributions will give rise to terms involving the Bessel function I 23/2 , the Bessel function I 12 and the integral of the Bessel function I 25/2 , respectively.

A symmetry
The expression (4.34) possesses a symmetry that can be used to simplify the terms that involve E m,γ(γσ+δ) (x) and I m,γ(γσ+δ) (x), as follows. Let us first consider the case when XY > 0 or XY < 0. Then, (4.34) will contain terms of the form (c.f. (4.42) and (4.44)) (4.47) We observe that under the mapping, the property a , −c > 0 as well as the unit determinant property are preserved. Further, the following quantities remain invariant: Thus, we can write (4.47) as and hence we can express the combination which occurs in (4.34) when XY > 0 or XY < 0, as On the other hand, when XY = 0, (4.34) will contain terms of the form (4.43), which, using the mapping (4.48), can be brought to the form (4.51). Hence, the combined contributions from X = 0 (in which case Y = −1/ac) and from Y = 0 (in which case X = −1/ac) take again the form (4.53). Thus, irrespective of whether XY > 0, XY = 0 or XY < 0, the contributions to (4.34) from the terms involving E m,γ(γσ+δ) (x) and I m,γ(γσ+δ) (x) takes the form (4.54) 5 Integrating over σ Next, we turn to the σ-integral of (4.34). We first explicate our choice of integration contour Γ σ (D) in (2.5) for a given n 2 ≥ 1 pole D in P ∪ {Σ ∈ Z}.

Contour of integration
The contour Γ σ (D) defined in the σ upper half plane restricted to σ 1 ∈ [0, 1) crosses the locus (3.16) associated with quadratic poles. Writing out the real and imaginary parts of (3.16) gives Solving the second equation for ρ 1 yields Now we recall that v 1 lies in the range (4.3), which we write as where x ∈ (0, −1/acγ). Inserting (5.3) as well as (5.2) into the first equation of (5.1) gives This describes an ellipse in the (σ 1 , x)-plane provided the right hand side of this equation is non-vanishing and positive, The equation (5.4) depends on the parameters ρ 2 , σ 2 and v 2 , where we recall that ρ 2 σ 2 − v 2 2 > 0. We now keep ρ 2 fixed and decrease σ 2 continuously to 0, while ensuring that Then, the condition (5.5) becomes When = 0, this yields while when = 0, using the R-chamber constraint ρ 2 σ 2 implies The condition (5.7) will not be satisfied for large values of σ 2 , but it will be satisfied for small values of σ 2 . When (5.7) is satisfied, a variation of σ 2 will induce a change of the ellipse (5.4). Combining the σ 2 -direction with (5.4) results in an ellipsoid in three dimensions, with σ 2 taking values in the range specified by (5.7). Now rewrite (5.4) as and observe that when σ 2 → 0, this equation degenerates to σ 1 + δ γ = 0, for any x ∈ (0, −1/acγ). The cross-sectional slice of the ellipsoid at fixed x is a closed contour with one point removed, namely the point (σ 1 = −δ/γ, x, σ 2 = 0). At fixed x, this is homotopic to a Ford circle C(−δ, γ) in the complex σ-plane anchored on the real axis at σ 1 = −δ/γ (see Appendix C.1 for details). The chosen range σ 1 ∈ [0, 1) constrains the poles contributing to (4.34) to those associated with 0 ≤ − δ γ < 1. Since this holds for any x ∈ (0, −1/acγ), our integration contour over σ for a given pole is Γ σ (D) = C(−δ, γ), which for notational simplicity we will denote by Γ σ .
The interpretation of this construction is the one given in [21]. When σ 2 is large, the integration contour does not intersect the ellipsoid described above. When lowering the value of σ 2 , the integration contour will cross some of the poles in the Siegel upper half plane described by (5.10). This will cease to be the case when σ 2 reaches the boundary σ 2 = 0 of the Siegel upper half plane. We note that as we decrease σ 2 , we continue to remain in the R-chamber, ensuring that the integration contour does not cross any n 2 = 0 pole.
We will now perform the σ-integral of (4.34) over the Ford circle Γ σ described above, following the prescription given in (2.5), which results in (5.11) To perform this integral, we will use the decomposition of the error functions given above.

Bessel function I 23/2
We first focus on the constant terms in the decompositions (4.42) and (4.43).
Firstly we show that the two cases corresponding to X = 0 and Y = 0 give rise to the same contribution, as follows. The condition X = 0 yields m − a 2 M = −c 2 N , and hence L = 2cN/a. We will show later that only terms with∆ < 0 contribute. Therefore combining L = 2cN/a with∆ < 0 results in L > 0, which in turn implies N = −1. The latter implies a|2. Consequently˜ = −b 2m/a = km, k ∈ Z. Therefore˜ = −˜ mod 2m and hence ψ(Γ) −˜ = ψ(Γ)˜ . Using the mapping (4.48), we obtain To proceed, we interchange the integration with the summation over M, N . This is allowed by the following arguments. First we note that the condition 0 ≤ b a +˜ 2m < − 1 ac in the summation above can be written as − m < a 2 M − c 2 N ≤ m. (5.13) Then, using the expression for L given in (4.13), we write out∆ = 4M N − L 2 and obtaiñ (5.14) Now let us consider terms that satisfy∆ ≤ 0, in which case we obtain from (5.14), 1 + e iθ 2 (5.19) or, equivalently, by where θ ∈ [0, π) ∪ (π, 2π), we infer that on the contour Γ σ , is exponentially suppressed for large M, N , the sum over M, N in (5.12) is uniformly convergent on Γ by the Weierstrass M test, and since each summand is integrable, we conclude that interchanging the integration with the summation over M, N is justified also when∆ > 0.
Thus, interchanging the integration with the summation over M, N results in
The essential singularity is now located at the originσ = 0. We choose the branch cut, which originates atσ = 0, to lie along the negative imaginary axis of theσ-plane. Next, we change the integration variable once more, Now the branch cut originates at w = 0 and lies along the negative real axis of the w-plane.
The integration contour now runs along along a line parallel to the imaginary axis, with˜ > 0. Now recall that ∆ > 0. When∆ ≥ 0, the coefficient∆∆/(4mγ) in the exponent is ≥ 0, and hence the integration contour can be closed in the half plane Re w > 0, where the integrand is analytic and hence the integral vanishes. Thus, we now take∆ < 0. Then, performing the redefinition t = π|∆|∆ 2mγ w . where I ν (z) denotes the modified Bessel function of first kind of index ν, where > 0. Then, (5.22) becomes Note that in (5.31) the dependence on α and δ is entirely encoded in multiplier system ψ(Γ)˜ and in the phase Since 0 ≤ −δ < γ and α ∈ Z/γZ, and since α is the modular inverse of δ, i.e. αδ = 1 mod γ, each δ uniquely specifies one α. Thus, the sum over δ yields the generalized Kloosterman sum Kl( ∆ 4m ,∆ 4m ; γ, ψ) ˜ , Thus, (5.31) can be written as where∆ = 4M N − L 2 , with L given in (4.13). Note that the sum over the allowed M, N is finite. Next, using (4.17), we express the triplet (N, L, M ) in terms of the triplet (ñ, m,˜ ). We then trade the sum over M, N for a sum overñ,˜ . In section 5.3 we will show thatñ is bounded byñ ≥ −1. Writing∆ as∆ = 4mñ − 2 , we rewrite (5.34) as which we write as (5.37) Note that the above sum includes two subsets of matrices in S G . The first subset contains matrices satisfying˜ /2m = −b/a, while the second subset contains matrices that correspond to the continued fraction expansion of˜ /2m. The latter subset is finite by definition, while the bounds M, N ≥ −1 and (5.13) can be used to show the finiteness of the first subset. This is consistent with the proofs of finiteness of [9, 10].

Lower bound onñ
We next show thatñ ≥ −1 whenever the condition is true. To this end, we first recall that (5.38) can be written as 2. bc < 0: In this case, we obtain, Using the lower bound in (5.39), we get,

Bessel function I 12
Now we focus on the terms E m,γ(γσ+δ) in the decompositions (4.42), (4.43) and (4.44). In section 5.5, we will show that these terms only give a non-vanishing contribution when XY = 0, whereas when XY = 0 they cancel out. Thus, we will assume XY = 0 in the following. We proceed as in section 5.2.1, namely, we restrict the sum over δ to the range 0 ≤ −δ < γ and perform the integration over the Ford circle Γ σ in (5.23). Collecting the terms proportional E m,γ(γσ+δ) in (5.11), and changing the integration variable toσ given in (5.25), we obtain where L and˜ are given in (4.13), and where the generalized Kloosterman sum Kl( ∆ 4m ,∆ 4m ; γ, ψ) ˜ is given in (5.33). The set P 0 is given by The integration contourΓ denotes a Ford circle centered atσ = i/2 that skirts the origiñ σ = 0. Using the symmetry property (4.54), we write (5.47) as we obtain Note that the integrand does not exhibit a branch cut. Performing the variable change given in (5.26), the integral over the Ford circleΓ takes a form similar to (5.27), with ∆/4m replaced by N/a 2 . The integral will be non-vanishing provided N/a 2 < 0 . This in turn implies N = −1. As shown in section 5.5, the only contributions to the sum come from the terms in set the P 0 satisfying a = 1 and M = m. Using the expression for L and given in (4.13), we infer that in this case, so that (5.51) yields (using d(−1) = 1) and using the multiplier system property ψ (j+2mk) (Γ) = ψ j (Γ) (with k ∈ Z) [20], we write (5.57) as Since the dependence on c is only contained in (5.56), the sum over c becomes which is divergent. We now describe a regularization procedure that enables us to extract a finite contribution.

Bessel integral
Now we return to (5.54), which we regularize using the expression (5.68),

Isolating non-vanishing contributions
We return to (4.54), show that the sum exhibits cancellations between various terms, thereby identifying non-vanishing contributions. Taking into account the form of E m,γ(γσ+δ) (X) and I m,γ(γσ+δ) (X) given in (4.38) and (4.39), we write (4.54) as where F m,γ(γσ+δ) is a function that only depends on X 2 . Consider changing the variable σ toσ = γ(γσ + δ), and integrating along a Ford circleΓ, centered atσ = i/2, that skirts the originσ = 0. Then, as described in section 5.4, only when N = −1 do we get a non-vanishing contribution. Thus, in the following, we set N = −1.
We evaluate We recall that the condition L ∈ Z in ( .

Bessel function I 25/2
Finally, we focus on the terms I m,γ(γσ+δ) in the decompositions (4.42), (4.43) and (4.44). In the previous subsection we showed that these terms only give a non-vanishing contribution when XY = 0. Thus, we will assume XY = 0 in the following.
We proceed as in section 5.2.1, namely, we restrict the sum over δ to the range 0 ≤ −δ < γ and perform the integration over the Ford circle Γ σ in (5.23). Collecting the terms proportional I m,γ(γσ+δ) in (5.11), and changing the integration variable toσ given in (5.25), we obtain where L and˜ are given in (4.13), and where the generalized Kloosterman sum Kl( ∆ 4m ,∆ 4m ; γ, ψ) ˜ is given in (5.33). The set P 0 is given in (5.48). The integration contourΓ denotes a Ford circle centered atσ = i/2 that skirts the originσ = 0.
Using the symmetry property (4.54), we write (5.92) as (5.93) Using the expression for I m,σ given in (4.39) and the relation (5.50), we obtain where s ∈ R\{0}, and where t ∈ C with Re t > 0. Setting u = −imz and t = im/σ (note that Re t = mσ 2 /|σ| 2 > 0), the integral (5.95) can be expressed as Interchanging the two integrations in (5.94), and performing the integration overσ along the Ford circleΓ as described in section 5.4, the latter will only be non-vanishing provided that N a 2 + mx 2 < 0, which in turn implies N = −1. Then, as shown in the previous subsection, the only non-vanishing contribution to (5.94) stems from N = −1, a = 1, M = m.
Then, setting N = −1, a = 1, M = m, we havẽ Using (5.58) and the multiplier system property ψ (j+2mk) (Γ) = ψ j (Γ) (with k ∈ Z) [20], we write (5.94) as Noticing that the dependence on c is quadratic, we write Thus we rewrite (5.100) as Note that the sum over p and g builds up the standard weight 1/2 index m Jacobi theta function, with the exception of the term 2mγp + g = 0, Next, we follow [11,14]. Using the results reviewed in Appendix B, we rewrite (5.105) as Interchanging the two integrations and performing the integration over w first, we only get a non-vanishing Bessel integral provided that we restrict the range of integration over x to 1 − mx 2 > 0, that is toˆR Performing another change of variables, (5.114) which, using (5.30), equals (5.116) The above expression involves two Kloosterman sums. Using the property f γ,−g,m (−x ) = f γ,g,m (x ), we can show that both Kloosterman sums give rise to the same contribution, because the sign change of j in the multiplier system can be compensated by a sign change in g and x . Thus, we arrive at the expression γ>0 j∈Z/2mZ g∈Z/2mγZ g=j mod 2m (5.117)

Complete result for ∆ > 0
Combining the contributions (5.36), (5.72) and (5.117), and relabelling j by˜ , we obtain the following expression for the degeneracy d(m, n, ) defined in (2.5): Note that the continued fraction structure is encoded in a subset of S G , while the Kloosterman sums Kl and Bessel functions are built up from Γ ∞ \S Γ and S G .
The expression (5.118), without the continued fraction structure of c F m (ñ,˜ ), was first derived in [11] by viewing d(m, n, ) as Fourier coefficients of a mixed Mock Jacobi form ψ F m [13]. The above result is an exact expression for d(m, n, ) which can be viewed as the non-perturbative completion of previous results in [16,17,23]. 6 The case ∆ = m = = 0 Now we consider the case m = = 0, which implies ∆ = 0.
We return to (4.16) and set m = = 0, where the integers L,˜ andñ are given by We apply the T -shift transformation (4.25) to b and to d, which leaves L,˜ invariant and changesñ intoñ →ñ −˜ T.
Then, as before, we split the sum over b into a sum over b ∈ Z/aZ and a sum over T ∈ Z. We pick the following integration contour Γ v . We set v 2 = 0. This choice is motivated by noting that setting = 0 in (2.4) one obtains v 2 = 0. Then, setting v 2 = 0 in the expressions (4.2), we infer that the range of v 1 is restricted to where we have made use of the T -shift. Demanding that the range of integration over v 1 is contained in an interval of length 1 constrains T to take values in T ∈ Z/γZ. Then, (6.1) takes the form We interchange the integration over v 1 with the summation over M, N . The integration over v 1 will produce a different result depending on whether˜ vanishes or not. Therefore, we discuss both cases separately. The integral over v 1 is trivial and independent of T . The sum over T yields a factor γ. Thus, we obtain for (6.5), as well as (6.6), we get 1 ac Next, we integrate over σ 1 . We proceed as in section 5.2.1, namely, we restrict the sum over δ to the range 0 ≤ −δ < γ and perform the integration over the Ford circle Γ σ in (5.23). We interchange the integration over σ with the sum over M and N . The resulting integral is similar to the one in (5.24), but without a branch cut, and will only be non-vanishing provided that N < 0, i.e. when N = −1. Inserting N = −1 into the expression for L −˜ given in (6.2), and recalling that gcd(a, c) = 1, fixes a = 1. Then, from (6.6), we infer M = −c 2 , which in turn implies M = −1 and c = −1, and hence L = 2. Moreover, since b ∈ Z/aZ with a = 1, this fixes b to a single value. Thus, integrating (6.9) over σ, we obtain, using d(−1) = 1, Now we take˜ = 0 in (6.5), and obtain (6.19) and the expressions forñ,˜ given in (6.2), we get We will integrate over σ 1 following the procedure described below (6.9). The integration over the Ford circle Γ will select the values M = −1 and N = −1 in (6.20), respectively. Let us then focus on these terms in (6.20). We first consider the contribution from M = −1. Inserting M = −1 into the expression for L and˜ ∈ Z given in (6.2), we see immediately that the integrality of these two variables, combined with gcd(a, c) = 1, fixes c = −1. Then, we obtain the contribution (−1) 2πi P b∈Z/aZ, T ∈Z/γZ (γσ + δ) 11 where a|N , since L,˜ ∈ Z. If N > 0, then for each divisor a of N , there will be another divisor a = N a > 0 which contributes Since˜ =˜ , this contribution will cancel against the contribution (6.22) in (6.21). Thus, (6.21) does not receive contributions from N > 0. When N = 0, we have L/˜ = 1,˜ = a,ñ = −b. Now we recall that the Ttransformation (4.25) can be viewed as either imposing the restriction b ∈ Z/aZ or d ∈ Z/(−cZ). We choose the latter viewpoint, in which case we may set d = 0, since c = −1. Then ad − bc = 1 implies b = 1, and we obtain from (6.21), Summing over T restricts a to Therefore, only when a = k γ, with k ∈ N, do we get a contribution to (6.24), The sum over k > 0 is divergent. We regularize this sum by replacing it with where ζ(s) denotes the Riemann zeta function. Next, performing the σ-integral over the Ford circle Γ σ following the procedure described below (6.9), we obtain Finally, we note that the extra sign stemming from the zeta function regularization (6.27) is crucial for obtaining agreement with the expression of the Rademacher expansion (6.34).

Conclusions
By using two distinct SL(2, Z) subgroups of Sp(2, Z) we have obtained a Rademacher type expansion for the exact degeneracies of 1 4 BPS states with unit torsion in heterotic string theory compactified on a six-torus. This expansion is obtained by summing over the contributions from the quadratic poles (3.16) of 1/Φ 10 . The resulting expansion is given by (5.118) in the case when ∆ > 0, and by (6.31) in the case when ∆ = m = = 0. When ∆ > 0, the exact expression for the degeneracies exhibits a fine-grained structure that is tied to the presence of the two sets S G and S Γ of SL(2, Z) matrices. The result (5.118) reproduces the expression that was obtained in [11] by a very different approach, namely by the Rademacher expansion of the Fourier coefficients of the mixed Mock Jacobi forms ψ F m . The latter were shown in [13] to encode the coefficients d(m, n, ) computed in a region [25] where there are no linear pole contributions when n > m. These coefficients are called immortal degeneracies. The Rademacher expansion in this case involved information about the shadow which repairs the modular behaviour of ψ F m . The Mock behaviour manifests itself through the I 12 and the I 25/2 terms in (5.118), both of which would be absent in the case of a modular form. In our approach, the v-integral yields error functions, each of which can be split into three distinct types of terms, namely a constant, a term involving E m,γ(γσ+δ) and a term involving I m,γ(γσ+δ) (c.f. (4.38) and (4.39)). These respectively yield the I 23/2 terms, the I 12 term and the I 25/2 term. Further, the sum over S G matrices which have a non-vanishing contribution to the I 25/2 term yields an integral of a Jacobi theta function minus a constant term. This, when added to the regularized contribution from the E m,γ(γσ+δ) terms, yields precisely the Eichler integral of the shadow for ψ F m . Further, the restriction on the range of v 1 , which arises by demanding that the poles lie in the σ upper half plane, coupled with the constraint v 2 /σ 2 = − /2m, reveals the continued fraction structure underlying the polar coefficients c F m (ñ,˜ ). This structure is not revealed in the approach of [11].
If the microscopic result (5.118) is to be obtained from a suitably defined gravity path integral, such as the quantum entropy function [26], then the latter has to make use of the two SL(2, Z) subsets S G and S Γ . How these two subsets are built into the quantum entropy function is an interesting research question worth exploring [17]. Earlier work [16,27] on the quantum entropy function for this heterotic model did not identify the distinct roles played by S G and S Γ . Moreover, inspection of (5.118) shows that the series associated with the Bessel function I 23/2 is organized in powers of 1/γ, and not in powers of 1/n 2 which appears in the semi-classical asymptotic expansion [16,23,28]. Understanding how the Rademacher expansion can be used to rigorously write down the quantum entropy function for immortal degeneracies associated with single centre 1 4 BPS black holes in this heterotic model is a challenging question.
The error function Erf and the complementary error function Erfc are defined by When Re x > 0, the complementary error function can be brought to the form by performing the change of variables t = t 2 and subsequently integrating by parts. Using (3.10) we infer since Im σ > 0. Then where we take the principal branch of the square root. Defining the quantities x = On the other hand, when X < 0, we use the relations and B Deriving (5.108) We briefly describe how to obtain (5.108), following [11,22]. Performing the change of variables z = iz in (5.105), we obtain 2 z dz .
(B.1) Using (5.96) and setting we infer Next, we perform the sum over p ∈ Z in (5.105). Noting that p only enters in the denominator of (B.4), we evaluate (B.5) Using the Mittag-Leffler formula we infer, in the case when g = 0 mod 2mγ, while when g = 0 mod 2mγ, there is one term in the sum that gets excluded and hence π 2 The Fourier coefficients c F 0 (n) of have the following well-known exact expression. Namely, using the Fourier coefficients c F 0 (n) can be written as and hence, (C.5) may also be written as Kl(n, −1, γ) 4π γ n 11/2 I 11 4π √ n γ − 24 n 6 γI 12 4π √ n γ . (C.8) The Rademacher expression (C.8) can also be obtained by deforming the integration contour along Ford circles and subsequently subjecting (C.1) to a modular transformation to re-express the integrand on each Ford circle. In the following, we will use this approach to obtain the expression (C.8).
(C.9) Therefore, 2 E 2 (σ) η 24 (σ) = (γσ + δ) 10  To obtain the Rademacher expansion for its Fourier coefficients, we will apply Rademacher's method (see [29][30][31]), which was originally developed for modular forms, to this quasi-modular form, as follows. The contour C is any contour in the σ upper half plane that starts at some point, σ 0 , and ends at σ 0 + 1. Rademacher's approach consists in deforming the contour C to a new contour that is the union of upper arcs along Ford circles defined by the Farey sequence of order N , and subsequently use the modular properties of the integrand to re-express it on each of these arcs. Then, in the limit N → +∞, the integral becomes expressed as an infinite sum of integrals over Ford circles, yielding a convergent series called the Rademacher expansion. We now review this construction following [29][30][31].
Definition: The set of Farey fractions of order N , denoted by F N , is the set of reduced fractions in the interval [0, 1] with denominators ≤ N , listed in increasing order of magnitude.
Definition: For a given rational number δ/γ with gcd(δ, γ) = 1, the Ford circle C(δ, γ) is defined as the circle in the complex upper half plane with radius 1/(2γ 2 ) that is tangent to the point δ/γ.