Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums

We continue our study of symmetries of a class of little string theories of A-type, which are engineered by N parallel M5-branes probing a flat transverse space. Extending the analysis of the companion paper [1], we discuss the part of the free energy that is sensitive to the details of the 𝔞N−1 gauge structure, by computing explicit series expansions for the cases N = 2, 3, 4. Based on these examples, we find a class of functions that we conjecture to resum whole sectors in the instanton expansion of the free energy and which combine in a natural manner its modular properties as well as the gauge symmetry. These functions have previously been introduced in the literature as the generating functions of multi-divisor sums and in the case N = 2 can also be cast into the form of a generalised Eisenstein series. We use these resummed contributions to the free energy to perform a number of non-trivial consistency checks for our results.


JHEP03(2020)016
where ω and ω are related by a suitable duality transformation. This was shown explicitly for M = 1 in [11] and for generic (N, M ) (but in the limit of vanishing 1,2 ) in [18], using a more geometric approach. Focusing on M = 1, it was furthermore argued in [7] that this duality also leads to additional symmetries for any single such theory. More concretely, it was demonstrated (and verified explicitly in a number of examples) that Z N,1 (ω, 1,2 ) is invariant under the symmetry group G(N ) ∼ = G(N ) × S N where S N ∼ = Dih N ⊂ S N acts on the gauge parameters, while (1.1) Here Dih ∞ can also be characterised as the group that is freely generated by two elements of order 2, with no additional braid relations. The group G(N ) has a natural action as a matrix group on the Fourier coefficients appearing in the expansion of the free energy F N,1 (ω, 1,2 ) after a suitable choice of basis.
In [1], we have further analysed consequences of G(N ) on the so-called reduced free energy, which captures a subsector of the BPS spectrum that is uncharged with respect to the Cartan subalgebra of the maximal a N −1 gauge algebra. 3 In previous works [22] it has already been argued, that these BPS states in fact constitute a symmetric orbifold CFT. In [1], we furthermore argued that G(N ) acting on this sector, along with two SL(2, Z) symmetries, is in fact promoted to a paramodular group Σ N , which in the Nekrasov-Shatashvili-limit [23,24] (corresponding to 2 → 0), is further extended to Σ * N ⊂ Sp(4, R). The latter is indeed the hallmark of a symmetric orbifold CFT, thus corroborating the observation of [22].
In the current paper we are focusing on the complement of this 'orbifold sector' within the spectrum of the LSTs, i.e. all BPS states that are not captured by the above mentioned reduced free energy. Our strategy is to analyse explicitly the cases N = 2 and N = 3 (as well as N = 4, albeit to a lesser degree of detail), from where we shall observe several emerging patterns (for which we also provide non-trivial checks). The latter allow us to reformulate the free energy in terms of a class of generating functions of multiple divisor sums, first introduced in [25], as well as, for N = 2 in terms of a generalised Eisenstein series introduced by G. Eisenstein and reviewed by A. Weil in [26]. The former presentation, in turn can generically be cast into a form which combines modular properties with the transformations under an a N −1 gauge algebra. In the simplest case N = 2, the basic building block can be written akin to an Eisenstein series where the usual divisor sigma is replaced by a sum over the root lattice of a 1 . Similar structures also appear for N > 2, however, they are in general accompanied by more complicated terms containing more involved sums probing the structure of the U(N ) gauge group. The latter, however, can still JHEP03(2020)016 be expressed in terms of the generating functions of multiple divisor sums. The appearance of these latter functions might hint to a better understanding of the full non-perturbative partition function and free energy of the LSTs, potentially also their algebraic properties. This paper is organised as follows: section 2 provides a review of the partition function Z N,1 as well as the free energy F N,1 . After having established our notation, in an attempt to render this paper more readable, section 3 provides a summary of the results obtained or conjectured in the remainder of this paper. These shall be worked out explicitly in the subsequent sections: section 4 discusses the case N = 2, which (due to the moderate complexity of the partition function) allows to study the free energy up to order Q 3 R (and partially even Q 4 R ), which from a field theoretic perspective corresponds to the instanton level. We particularly argue that the free energy can be re-written in terms of the generating functions of multiple divisor sums mentioned above. Sections 5 and 6 repeat the same analysis (albeit to less orders in Q R owing to the increased complexity of the free energy) for N = 3 and partially for N = 4 respectively. Section 7 contains our conclusions and an outlook to future work. Finally, additional reviews on (quasi)modular forms, the generating functionsà la [25] as well as several lists of expansion parameters, which have been deemed too lengthy for the main text, have been relegated to three appendices.

M-brane partition functions and its symmetries
In an attempt to make this paper minimally self-contained, we shall provide a lightning review of some crucial concepts, as well as our notation for the BPS partition function of Z N,1 of a system of N parallel M5-branes on a circle probing a transverse R 4 . The latter can also be computed as the topological string partition function on a class of toric Calabi-Yau manifolds X N,1 . For more information, we refer the reader to the companion paper [1] as well as the original literature, which we shall reference throughout this section.

Partition function and free energy
Following the notation of [9], the partition function can be expressed as Here ( a 1 . . . , a N , S, R) is a basis of independent Kähler parameters of X N,1 (see e.g. [1] for the precise definition) and m i = m i (R , S, a i1,...,N ) is a function that notably depends on R and Q m i = e 2πim i . Therefore (2.1) is essentially a series expansion in Q R = e 2πiR : indeed {α} stands for a sum over N integer partitions α 1,...,N . Furthermore, the precise expression for the building blocks W α 1 ,...,α N α 1 ,...,α N ( a 1,...,N , S; 1,2 ) can be found in [1,7,9]. Rather than working directly with Z N,1 , in this paper we prefer the free energy where PLog denotes the plethystic logarithm which only receives contributions from single particle BPS states [28]. Following the notation of [1], it can be expanded in the following JHEP03(2020)016 introducing the coefficients f (s 1 ,s 2 ) i 1 ,...,i N ,k,r , along with the notation Extending the notation of [1], we introduce the expansions In the following we will focus on the cases of H i 1 ,...,i N ,k,r are invariant under cyclic rotations of (i 1 , . . . , i N ), we shall in most cases assume (without restriction of generality) that i N = 0.
On the one hand, from the perspective of the system of parallel M5-branes with M2branes stretched between them, the functions G (i 1 ,...,i N ) (s 1 ,s 2 ) (R, S) count BPS states for a configuration with fixed numbers (i 1 , . . . , i N ) of M2-branes. These functions have previously been studied in the literature and several properties have been discussed (see the following subsubsection). On the other hand, the functions H (i 1 ,...,i N ,r) (s 1 ,s 2 ) (ρ, S) sum configurations of (i 1 + , . . . , i N + ) M2-branes stretched between the M5-branes. From the little string sector, they count the multi string expansion of a specific charge sector. Resumming the latter, we can define

Symmetries and modular properties
The function G (i 1 ,...,i N ) (s 1 ,s 2 ) (R, S) in (2.5) transforms like a quasi-Jacobi form with respect to a congruence subgroup 4 of SL(2, Z) R that acts in the following manner [1]: is a quasi-Jacobi form of index K = N a=1 i a and weight s 1 + s 2 − 2 so it can be written in the following manner 5 (2.9) JHEP03(2020)016 Here g (s 1 ,s 2 ) u (R) is a quasi-modular form with weight s 1 +s 2 +2(u−1), i.e. it is a polynomial in the Eisenstein series {E 2 (p i R), E 4 (p i R), E 6 (p i R)} for all the prime factors p i appearing in the prime factorisation of K. Notice that the presence of E 2 (ρ) leads to the fact the G do not only transform with a weight factor under modular transformations, but in general also produce a shift term (see appendix A.1.2 for more information on quasi-Jacobi forms).
Furthermore, a priori, the function H (2.10) Here S N ⊂ S N acts by shuffling the Q 1,...,N in (2.3). Since the M5-brane setup can be thought of as a regular N -gone, whose symmetry group is S N , we have S N ∼ = Dih N (which can be indeed checked by studying explicit expressions for f (s 1 ,s 2 ) i 1 ,...,i N ,k,r ). In the following we will mostly focus on studying the functions H (i 1 ,...,i N ,r) (s 1 ,s 2 ) (ρ, S) for the values N = 2, 3, 4, which will reveal a number of interesting patterns that shall allow us to make a conjecture of their generic form.

JHEP03(2020)016 3 Summary of results
After having introduced the main actors of this paper in the previous section, we are now ready to analyse in detail the H (i 1 ,...,i N ,r) (s 1 ,s 2 ) (ρ, S). However, due to the complexity of some of the results obtained or conjectured below, we shall summarise the most important patterns that we have found before presenting the technical analysis in the subsequent sections. We hope that this increases the readability of this work. Furthermore, while these patterns appear in their cleanest and most tangible form (which shall also allow us to make contact to many other structures previously encountered in the literature) for the case N = 2, they also extend to N > 2. However, in these cases all expressions tend to become more complicated and involved.
Moreover, we stress that the above mentioned patterns shall appear from studying expansions of the free energy in various parameters. Since these, in most cases, do not enjoy modular properties, most results do not lend themselves to (simple) rigorous proofs to all orders and therefore have to be considered conjectures. We note, however, that below we present an overwhelming number of different cases, all following the same type of pattern, thus giving us reasonable confidence that our conjectures actually hold true.
As mentioned above, the main object we shall study throughout this work is the sector of the free energy H where various contributions have been summed up. While not strictly speaking modular objects (with respect to a group SL(2, Z) ρ acting on ρ), explicit expansions (shown below) suggest that they can be written in the form (ρ) can be characterised as series expansions in Q ρ = e 2πiρ . While in general not (quasi)modular forms, they nevertheless exhibit interesting patterns: in general, they are not constrained by G(N ), however, the group S N ⊂ G(N ) (together with the fact that at least one of the i 1,...,N has to be chosen to be 0) leaves a finite number of distinct classes of H (i 1 ,...,i N ,r) (s 1 ,s 2 ) (ρ, S) for fixed N (see e.g. (6.2) below for N = 4). While their explicit form strongly depends on the indices (i 1 , . . . , i N ) (or more precisely the class under consideration), they can schematically be written in the following form Here p ∈ N, the q ,a and t are linear combinations of i 1,...,N , while the p are homogeneous polynomials in (differences of) the i 1,...,N and polynomials of the Eisenstein series with fixed weight. Based on the explicit expressions studied in the subsequent sections, we find the following pattern: let w be the (combined) weight of all Eisenstein series in p and d JHEP03(2020)016 the degree of the polynomial in i 1,...,N , then The functions H (ρ, S) are only certain terms appearing in the expansion of the free energy in a Fourier series. They can be partially resummed in the sense of (2.7) to form the B N,(r) (s 1 ,s 2 ) ( a 1 , . . . , a N , S). The form of (3.2), in particular the denominator, suggests that the the latter can be expressed as combinations of generating functions of multiple divisor sums, called T (X 1 , . . . , X p ; ρ), which were first introduced in [25] and whose definition is given in (B.3). Here X 1,...,p are suitable linear combinations of a 1,...,N −1 and ρ. This becomes intuitively clear, since the Q ρ -expansion of B N,(r) (s 1 ,s 2 ) ( a 1 , . . . , a N , S) can in general be written as (multiple) divisor sums (decorated sums over (part of the) root lattices of the gauge algebra a N −1 ). While this is most transparent in the case of N = 2 (see e.g. section 4.1.3), similar patterns also appear for cases N > 2. However, they generically involve summations over several copies of the root lattices. The generating functions T (X 1 , . . . , X p ; ρ) are not Jacobi forms themselves, but form a natural generalisation thereof (see appendix B). In the case of N = 2, we can furthermore show that the appearing combinations naturally combine into the generalised Eisenstein series defined in (A.15) (see [26] for further information). For N > 2, the explicit combinations of T (X 1 , . . . , X p ; ρ) that appear seem more complicated and it is not obvious that they can be recombined into a simpler object.
Furthermore, the explicit expressions obtained for B N,(r) (s 1 ,s 2 ) ( a 1 , . . . , a N , S) lend themselves for further analysis. Indeed, combining the latter with H (0,...,0,r) (s 1 ,s 2 ) (ρ, S), which is a quasi-Jacobi form, certain cancellations among them occur: these reduce the depth of the combined expression, i.e. the maximal power of E 2 (ρ) that is appearing. Finally, we have also verified the relation which was first conjectured in [29]. This not only serves as a strong check of the correctness of our results presented here, but also the work available in the literature. In the following we shall substantiate the above conjectures (in particular the explicit form (3.2)) for the examples N = 2, 3, 4 and for low orders in Q R .

Case N = 2
The simplest non-trivial example to study is the case N = 2. 6 In the following we discuss different orders of Q R (i.e. different values of r ≥ 1 in eq. (2.6)) and specify the functions appearing in (3.1) and (3.2) respectively. JHEP03(2020)016

Explicit contributions to the free energy
For N = 2 and at order Q 1 R , the distinct functions (2.6) (for given (s 1 , s 2 )) are characterised by a single integer n: While the case n = 0 is also well defined, it is in fact part of the reduced free energy that was studied in [1] and we shall therefore not discuss it in detail at present. As remarked above, while the object (4.1) is not in general modular, the coefficients f (s 1 ,s 2 ) n+ , ,k,1 are still symmetric with respect to G(2) (see appendix A). The latter imposes in fact infinitely many relations among the Fourier coefficients: e.g. acting on f Here G 2 (2) is one of the generators of G(2). Indeed, for generic N , we have Specifically, as (N + 2) × (N + 2) matrices, we have and (4.4) Furthermore, B is defined as which realises the symmetry f (s 1 ,s 2 ) Using relation (4.2) as well as studying the explicit coefficients stemming from (2.2) we find the following form 7 (see (3.1)).
Since in this way we are only capable of checking a finite number of coefficients and since H (n,0,1) (s 1 ,s 2 ) (ρ, S) does not enjoy modular properties, eq. (4.6) should be considered a conjecture. We checked the latter up to order Q 15 ρ in eq. (4.7).

JHEP03(2020)016
where the functions in (3.2) are a priori generic series in Q ρ . Expansion up to order Q 20 ρ suggests that for (s 1 , s 2 ) = (0, 0) the latter are in fact given by 8 and g 2,(n,1) such that We can treat other values of (s 1 , s 2 ) in the same fashion, leading to the following conjectures of closed form expressions of H (n,0,1) 8 In the following we shall assume that |Qρ| < 1.

(Generalised) Eisenstein series
In order to compute B N =2,(r) (s 1 ,s 2 ) ( a 1 , a 2 , S) (which was defined in (2.7)) we have to sum the contributions in (4.10) over n ∈ N, weighted by (Q a 1 ) n + Qρ Q a 1 n : 9 Taking into account the form of the g i,(n,1) (s 1 ,s 2 ) in (4.10), the relevant sum to study is of the form the generating function I 0 is (4.14) The first term can be rewritten as follows where we used the notation of [26], reviewed in appendix A.2, specifically eq. (A.12). We therefore find with eq.

JHEP03(2020)016
Here E 2 is a generalised Eisenstein series, as defined in (A.15). Following [26], the latter can also be expressed in terms of the Weierstrass function ℘ (see (A.20)), such that where G 2 (ρ) is the Eisenstein series defined in (A.7). Furthermore, using the recursive relation (A.18), we have Using this result, we can write for example 19) and similarly for the remaining examples (for simplicity, we do not display explicitly all arguments) -11 -JHEP03(2020)016

Root lattices
Finally, another way of writing I α (which will generalise to later cases) is to exchanged the order of the two summations in the last term of (4.14) While the first term has a second order pole at a 1 = 0, the limit of the second term in fact yields the holomorphic Eisenstein series E 2 (ρ) lim a 1 →0 are the Bernoulli numbers. The form (4.21) of the generating function I α (which is the building block for the free energy) can be written in another suggestive fashion: where ∆ a is the set of all roots of the Lie algebra a 1 , i.e. the set {− a 1 , a 1 } and ∆ + a the set of positive roots of a 1 (i.e. the set { a 1 }. While a seemingly trivial rewriting of (4.21), the presentation (4.22) is very similar to the Fourier expansion of the Eisenstein series in (A.7), except that the divisor sigma σ 2k−1 (n) is replaced by a summation over the root lattice of a 1 . Notice in this regards that also can in a sense be thought of as a generating function of Riemann zeta functions as can be seen from (B.6). From this perspective, it is clear that the generating functions I α encode how the modular parameter ρ is coupled together with the roots of the su(2) gauge algebra in the free energy, to make both the modular and gauge structure manifest. As we shall see below, similar patterns appear also for N > 2. Furthermore, in the following subsubsection, we shall give another perspective on the I α , which also focuses on generating functions of divisor sigmas, as advocated in [25].

Generating functions of multiple zeta values
In view of generalising to the cases N > 2, we provide another way of organising the B N =2,1 (s 1 ,s 2 ) , which formalises (4.21) and introduces generating functions of multiple-divisor sums and multiple zeta values, as reviewed in appendix B. Indeed, using the generating JHEP03(2020)016 function T ( a 1 , . . . , a ; ρ) (first defined in [25] and reviewed in (B.3)) of brackets of length = 1, we can write for I 0 in (4.12) where for later convenience, we have introduced the combination such that e.g.
and similarly for all other terms in (4.50). Following the discussion of (4.17) and the form (4.24), T can be understood as a primitive of the Weierstrass elliptic function. While (4.24) is evident by comparing the form of (4.10) to the definition (B.3), the relation between T ( a 1 ; ρ) and the generalised Eisenstein series discussed previously can also be understood in a different fashion. To this end, we consider where W is a two-dimensional lattice with generators (u, v) = (1, ρ), as in appendix A.2.
Using the notation of [27] we write Using the Lipschitz summation formula we can also write ∞ m=1 n∈Z Here we have differentiated under the summation in the last expression (assuming convergence of the sum 10 ). Using the presentation (B.4) we have Therefore (4.28) can be written as and with (A.14) we have for c ∈ C a constant. Finally, with (A.13) as well as 2πi 4π 2 π cos π a 1 sin π a 1 = 1 the behaviour of T ( a 1 ; ρ) under elliptic transformations (B.5) finally implies which indeed yields (4.24). The crucial step in the above computation is (4.28): intuitively, (4.27) corresponds to a summation over the whole lattice W , shifted by x (notice that x / ∈ W in general) as shown in figure 2(a). The expression (4.28) corresponds to a separation of this summation into three different contributions, as in figure 2(b). The summation over the blue and green points ultimately combine into D a 1 T ( a 1 − ρ; ρ), while the summation over the orange lattice points translates into D a 1 T (− a 1 ; ρ). In the following, we will see that the separation into various different sublattices according to (4.26) continues to higher orders in Q R , as well as higher values of N , however, with increased complexity. JHEP03(2020)016

Free energy and self-similarity
So far we have only considered the contributions n > 0 in (4.11). Indeed, the sector n = 0 behaves rather differently (for one, it transforms under a modular symmetry acting on ρ) and was discussed in detail in [1]. It is, however, an interesting question to combine it with the present results (4.11).
To leading order in 1,2 the contribution H which does not depend on E 2 (ρ). This expression is therefore invariant under modular transformations (2.8), while at the same time holomorphic. Similar cancellations also take place for higher orders in 1,2 , however, in general only the highest power in E 2 (ρ) cancels exactly. Various contributions H (0,0,1) (s 1 ,s 2 ) (ρ, S) for low values of s 1,2 are given in appendix C.1. Indeed, we can therefore extract the following terms containing the highest power in E 2 (ρ) Here the dots indicate terms containing powers of E 2 (ρ) that are lower than and which generically do not cancel between H (0,0,1) (s 1 ,s 2 ) (ρ, a 1 , S) we can in fact perform a check to confirm (4.20) for s 2 = 0. In [29] it was observed that in a particular region in the moduli space and in the Nekrasov-Shatashvili limit (i.e. in the limit of vanishing 2 ), the free energy enjoys a so-called self-similarity. In our notation, the latter amounts to the JHEP03(2020)016 The coefficients H (0,1) (s 1 ,0) (ρ, S) are given in appendix C.3 and the particular region in the moduli space corresponds to a 1 = a 2 = ρ 2 . To compute the left hand side of (4.36), we recall the definition of I α in (4.12), which for a 1 = ρ 2 takes the following form where we have introduced Q 2 = Q ρ and also define ρ = 2ρ (such that Q = e 2πiρ ). Using the relation (for z ∈ C) where θ 4 is a Jacobi theta function, we can write With this, we indeed find (4.40) Here the last equality can be established by comparing the first few coefficients in an expansion in Q: indeed, (4.39) implies that each term on the left-hand side of (4.40) is a quasi-Jacobi form of weight −2 and index 1 of the congruence subgroup Γ 0 (2) ⊂ SL(2, Z) ρ , which acts in the following fashion The precise combination appearing in (4.40) is in fact a Jacobi form of weight −2 and index 1 of SL(2, Z) ρ . Comparing with the coefficient H (0,1) (0,0) ( ρ 2 , S) given in (C.5), eq. (4.40) indeed agrees with (4.36). In the same fashion we have also verified (4.36) for s 1 ∈ {2, 4, 6}. This is a highly non-trivial check indicating that our results are compatible with [29].

Non-holomorphicity and differential equations
Before moving to cases N > 2, we stop to comment on a further detail with regards to the explicit expressions (4.10). Indeed, upon close examination, the same structures JHEP03(2020)016 (i.e. combinations of powers of n and Eisenstein series) appear in various different g i,(n,1) (s 1 ,s 2 ) . Furthermore, we also have (with x 1,2,3 , y 1,2 ∈ N) This suggests certain (differential) relations between g i,(n,1) (s 1 ,s 2 ) . For example, based on the expressions we find evidence for the following relations  where the dots indicate lower powers in n, while where g i,(n,r=2) (s 1 ,s 2 ) (ρ) have an integer series expansion in Q ρ . For simplicity, we shall only discuss the case (s 1 , s 2 ) = (0, 0) in the following. 11 Repeating the computation of the previous subsection to order O(Q 2 R ), we find a pattern, which suggests the following expressions   Here we have used the notation which is in fact a weight 2 Eisenstein series of the congruence subgroup Γ 0 (2), 12 i.e. despite being composed from E 2 , it transforms without shift-term under modular transformations with respect to the latter (see [30][31][32]). Notice, we prefer to distinguish the cases gcd(n, 2) = 1 and gcd(n, 2) > 1, rather than n ∈ N even or n ∈ N odd . We will provide non-trivial evidence that the former is the correct prescription that generalises to higher orders in Q R as well 11 Explicit computations become very difficult for higher orders in 1,2. However, we still expect similar results to also hold in the cases (s1, s2) = (0, 0): for example, the case (2, 0) is exhibited in appendix C.2. 12 See [30,31] (see also [32]) as well as citations therein for a basis of modular forms for congruence subgroups of the type Γ0(N ).
Finally, we remark that with (4.25) and (4.18), we can re-write (4.50) in terms of generalised Eisenstein series. However, in order to keep our presentation compact, we refrain from writing the explicit expressions.

Free energy and self-similarity
Finally, for completeness, we shall also attempt to combine the results above in (4.50) with the contribution H (0,0,2) (0,0) (ρ, S), which is explicitly given in (C.2). To order Q R , we have seen that there are certain cancellations that reduce the depth of the complete free energy (to this order in Q R ) as a quasi-Jacobi form. In the following we shall see, that similar cancellations also occur to order Q 2 R : indeed, expanding (C.2) in powers of a 1 (using (4.24) along with (4.17)) we can write where the dots denote terms depending explicitly on a 1 that are Jacobi forms of the congruence subgroup Γ 0 (2). 13 Comparing with (C.2), we see that these terms do not completely JHEP03(2020)016 cancel, however, add up to which is a Jacobi form of Γ 0 (2). Furthermore, we have verified that where the contribution H (0,2) (0,0) (ρ, S) is given in (C.6), thus confirming (4.36) also to order Q 2 R .

R
We can provide further support for the fact that similar structures that we found in the previous subsections in fact persist to all orders in Q R , by analysing the case Q 3 R . Since computations are exceedingly difficult in this case, we once more focus on (s 1 , s 2 ) = (0, 0).
Expanding the free energy to order Q 3 R , the results are compatible with the following presentation of H

Free energy and self-similarity
Following the approach of the previous sections, we combine the results above in (4.60) with the contribution H (0,0,3) (0,0) (ρ, S), as exhibited in (C.3). Expanding the latter in powers of a 1 (using (4.24) along with (4.17)) we find where as before the dots denote terms depending explicitly on a 1 that are Jacobi forms of the congruence subgroup Γ 0 (3). Comparing with (C.3), we see that these terms combine to which is a Jacobi form of Γ 0 (3). Finally, we have verified that where the contribution H

Higher orders in Q R
While in general, orders higher than Q 3 R are extremely difficult to analyse directly by expanding the free energy (2.3), there are other methods that we can use to extract certain information. Indeed, as discussed in [6] and reviewed briefly in section 2.2, the G (i 1 ,i 2 ) (0,0) (R, S) (as introduced in (2.5) are quasi-Jacobi forms of weight −2 and index i 1 + i 2 under the congruence subgroup Γ 0 (k(gcd(i 1 , i 2 ))) ⊂ SL(2, Z) R . Here we are using the notation introduced in [1]: (4.63) In order to determine the explicit expansion (2.9) only a finite number of coefficients are required, which (at least for low values of i 1,2 ) are provided by the results up to order Q 3 R . We exhibit a few explicit expressions in appendix C.4. In turn, upon assuming the general decomposition H (n,0,r) the former allow us to determine generically 2 coefficients in a series expansion of g u,(n,0,r) (0,0) (ρ) in Q ρ . While by far insufficient for generic u, there are two 14 classes of coefficients for which we can make non-trivial statements: • coefficients g 2r−1,(n,r) (0,0) (ρ) These are the simplest non-vanishing coefficients, which, following the pattern arising 14 We do not count the trivial vanishing g (1−Q n ρ ) up to some numerical prefactor. 15 Analysing the latter up to order r = 50 and n = 4 we conjecture g 2r−1,(n,r) (0,0) where we defined (4.66) Notice that the appearance of (parts of) divisor sums might indicate that similar structures that we have found when analysing the free energy as a function of ρ, may also apply when we think of F 2,1 (ρ, a 1 , S, R; 1,2 ) as a function of R. It would therefore be interesting to analyse the latter from the point of view of genus two generalisations of the generating functions (B.3).

Case N = 3
After analysing the free energy for N = 2 to various orders in Q R , we now switch to N = 3. The main difference compared to the previous case is the fact that the H Due to the complexity of the computations, we content ourselves with a discussion to order Q 1 R (i.e. r = 1) and partially Q 2 R (i.e. r = 2) , as well as leading order in 1,2 (i.e. (s 1 , s 2 ) = (0, 0)). Within the limits of this restriction, we shall find similar structures as in the case N = 2, albeit much more complicated.

Explicit contributions to the free energy
Analysing the the coefficients f (0,0) i 1 ,i 2 ,i 3 ,k,1 up to order i 1 + i 2 + i 3 = 30, suggests the following expressions These objects are not modular forms in any sense, such that the analysis of a finite number of coefficients (i.e. such as i 1 + i 2 + i 3 = 30) is not sufficient to proof the relations above. However, since we find structures that (at least superficially) look very similar to those we have encountered in the previous section, we conjecture that (5.2)-(5.4) are in fact correct.

Root lattices
After having established a conjecture for the H (i 1 ,i 2 ,i 3 ,1) (0,0) (ρ, S), the next step is to compute B 3,1 (0,0) (ρ, a 1 , a 2 , S) defined in (2.7). We shall present two different approaches, generalising the computations for N = 2 in sections 4.1.3 and 4.1.4 respectively. We shall start with the former, using summations over the root lattice of a 2 .
Here we choose to regroup B 3,1 (0,0) into four different contributions Following similar steps as in (4.13), the summation appearing in L 1 can be re-arranged as 16 Here ∆ a 2 is the set of roots of a 2 , while ∆ a 2 is the set of positive roots of a 2 , i.e. ∆ a 2 = { a 1 , a 2 , a 1 + a 2 , − a 1 , − a 2 , − a 1 − a 2 } , and ∆ a 2 = { a 1 , a 2 , a 1 + a 2 } .

JHEP03(2020)016
In order to treat the summation in L 2 , we use the following identities (with d : such that we have where we have used similar steps as in (4.13). We can further simplify the expression to The summation appearing in L 3 and L 4 is more complicated and we shall not write the complete explicit expression for reasons of brevity. We remark however, that they can no longer be written as summations generalising the divisor sigma appearing in conventional Eisenstein series. Instead, applying similar methods as before, we find structures which are schematically of the form u>1 Q u ρ n 1 >n 2 >0 n 1 k 1 +n 2 k 2 =u Notice in particular the appearance of two summations over (part of) the root lattice of a 2 . Terms of this form are more akin to sigmas of length 2, as introduced in (B.1) for = 2. The full free energy N = 3 at order Q 1 R can therefore be treated more compactly using the generating functions defined in (B.3), as we shall do in the following subsubsection. It remains, however, an interesting question to speculate, whether the structures (5.12) are a a 2 -generalisation of a large class of modular objects than the Eisenstein series. We will leave such speculations for future work. JHEP03(2020)016

JHEP03(2020)016
Here we have used the generating function (B.3) along with the identities (5.9). Combining the terms in (5.13) we can write them as where we have defined This expression can be understood as a generating functional of derivatives of Eisenstein series Notice furthermore that e.g. 20) where the prime signifies the derivative with respect to the first argument.
Finally, we have also verified explicitly thus confirming the self-similarity first observed in [29].

R
To order Q 2 R , the expansion of the free energy is much more challenging to work out and we were only able to determine the first coefficients. Indeed, upon writing g we have found the following patterns
The results and conjectures presented in this work serve as another indication of the importance of symmetries in the study of the network of dual gauge theories engineered from the toric Calabi-Yau threefolds X N,1 . In the companion paper [1], we have focused on the so-called reduced free energy H (0,...,0,r) (s 1 ,0) (ρ, S) and have unraveled more of the group structure acting on the particular sector of the BPS spectrum captured by it. The current work deals mostly with the complement of this part of the spectrum, i.e. the remaining part of the free energy. We have shown that the latter also has numerous interesting new and unexpected structures and we have shown a class of functions (namely the generating functions of multiple divisor sigmas), which is adapted to capture the latter in a natural fashion. In the simplest case N = 2, the latter is related to so-called generalised Eisenstein series which in turn are related to the elliptic Weierstrass function.
The appearance of these functions is very interesting when one considers the case N = 2 as a generalisation of N = 1. As was argued in [7], the latter has as automorphism group Sp(4, Z), which contains two SL(2, Z) groups (called SL(2, Z) R and SL(2, Z) ρ , since they act on R and ρ as modular parameters respectively) in a natural fashion. In the case of N = 2, one of these (SL(2, Z) R in our conventions) remains more or less intact, while the second one gets modified due to the introduction of the root a 1 of an SU(2) gauge group 18 in addition to the original modular parameter ρ. The functions we found in section 4, notably the Eisenstein series E 2k combine the two, in a natural fashion, making both the modular properties, as well as the SU(2) character of the free energy manifest.
For the future it would be interesting to study the patterns we have found in this work from a more group theoretic perspective and analyse the full automorphism group of the BPS spectrum of the theories engineered from X N,1 . Notably the relation (4.54), which connects the coefficients that appear in the building blocks of the free energy to different orders in Q R , is very similar in spirit to a Hecke type of relation appearing in the reduced free energy H (0,...,0,r) (s 1 ,0) (ρ, S). This could be seen as an indication that the whole free energy at order Q r R can be understood as some operator acting on a certain seed function. In this regard, it would also be interesting to understand, what physical role is played by the algebra of the generating functions T ( a 1 , . . . , a N ; ρ) (which form the general building blocks of the free energy) that was uncovered in [25]. We leave these questions for future work.
Furthermore, the generating functions of multiple divisor sums are also intimately related to multiple zeta values [25,27]. The latter have in recent years appeared in many other branches of physics, notably the study of scattering amplitudes in (supersymmetric) string theories (see [33][34][35][36][37][38][39][40][41][42][43][44][45] for recent work). While the latter are in general perturbative JHEP03(2020)016 quantities, the appearance of the same type of objects in the instanton partition function (and the corresponding free energy), which is an inherently non-perturbative object, might indicate a more profound role of multiple zeta values in gauge theories also at the nonperturbative level. It will be interesting in the future to study, what type of lessons we can take away from this connection [46].
Finally, based on earlier results [7] in this paper we have restricted ourselves to theories engineered from the toric Calabi-Yau manifolds X N,1 . The latter are part of the larger class of manifolds X N,M , for which the free energy has also been studied in detail. It would be interesting to repeat the current analysis for these cases. Similarly it would be even more interesting to analyse theories that are engineered by parallel M5-branes probing various different orbifold backgrounds. Due to the similar nature of these theories one might expect to find the same type of patterns as described in the current work.
JHEP03(2020)016 t in µ. A quasi-Jacobi form ψ(ρ; z) of depth (s, t) is the constant term of an almost meromorphic Jacobi form of index zero and depth (s, t) when considered as a polynomial in λ and µ. Under modular and elliptic transformations, it schematically behaves in the following manner (for a b c d ∈ SL(2, Z) and 1,2 ∈ Z) As discussed in [47], the algebra of quasi-Jacobi forms is the algebra of functions on H × C generated by functions E n (z; ρ) and E 2 , where E n is a class of generalised Eisenstein series that we shall discuss in the following subsection.

A.2 Generalised Eisenstein series
In this appendix we recall parts of the review [26]. Let z ∈ C and n ∈ N e n (z) = The functions (A.12) satisfy the following recursive relation de n dz = −n e n+1 , ∀n ≥ 1 , (A.14) such that in particular e 2 (z) = π 2 sin 2 (πz) . The definition (A.12) can be generalised according to [26] in the following manner: let W be a lattice in the complex plane, generated by two complex numbers (u, v) that satisfy v/u = δρ (with δ = ±1). The generalised Eisenstein series is defined as E n (z; ρ) = w∈W (z + w) −n .
(A. 15) We remark that throughout the main body of this paper, we prefer to work with the specific choice of generators u = 1 , v = ρ , δ = +1 . (A.16)

JHEP03(2020)016
Originally studied by Eisenstein, (A.15) is absolutely convergent for n ≥ 3, while for n ∈ {1, 2}, the summation is supposed to be understood as Throughout the main body of this paper, we will always assume that this Eisenstein summation is used whenever appropriate. The E n satisfy the following recursive relation where M has to be chosen such that |Q M +1 ρ e 2πiz | < 1 and |Q M +1 ρ e −2πiz | < 1. Focusing on the case |Q ρ | < |e 2πiz | < |Q ρ | −1 , one can choose M = 0. Furthermore, E 2 can also be expressed in terms of the Weierstrass function ℘ (defined in (A.8)) where the summation does not include the origin of W .

B Generating functions of multiple divisor sums
Following [25] we define the multiple divisor sum The last term can be rewritten as (B.8) Here the last term is in fact part of the generating function of brackets of length 1. We can furthermore write it in the following fashion Summarising we can therefore write Notice, that the relation (B.5) as well as (B.11) allow to treat all cases with N = 3 in section 5.

C Coefficients
In this appendix we collect several explicit coefficients in the decomposition of the free energy that were too lengthy to be displayed in the main body of the paper. For completeness in the following we list the coefficients g i,(n,2) (2,0) , which constitute the nextto-leading order in 1 for the free energy of N = 2 to order Q 2 R . if gcd(n, 2) > 1 . As we can see, these coefficients display similar characteristics as their counterparts g i,(n,2) (0,0) . To keep the presentation short, we shall not present the resummed free energy to this order.