Five-Brane Webs and Highest Weight Representations

We consider BPS-counting functions $\mathcal{Z}_{N,M}$ of $N$ parallel M5-branes probing a transverse $\mathbb{Z}_M$ orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over (affine) $A_{N-1}$. We make this symmetry of $\mathcal{Z}_{N,M}$ manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function. The resulting (simplified) $\mathcal{Z}_{N,M}$ can be written as a sum over weights forming a single irreducible representation of the Lie algebra $\mathfrak{a}_{N-1}$ (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of $(N,M)$ and generalise the arising pattern for generic configurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality $N\leftrightarrow M$ manifest.


Introduction
Six dimensional superconformal field theories (SCFTs) along with their compactifications to lower dimensions have attracted a lot of attention in recent years: on the one hand, the dynamics of these theories display very rich structures which are interesting to explore in their own right. On the other hand, these SCFTs have seen numerous applications in string-and field theories. Indeed, the fact that many of them can be engineered from string or M-theory through various brane constructions (see for example [1][2][3][4][5][6][7] for recent work on theories constructed from parallel M5branes (with M2-branes stretched between them)), has allowed to identify interesting structures in the latter and has provided an invaluable window into their inner workings [8]. Similarly, from the point of view of field theory, the recent years have brought to light interesting new dualities: for example, different types of compactifications of six dimensional SCFTs lead to various lower dimensional theories. The connection to a common higher dimensional parent theory gives rise to relations between certain quantities computed in these theories. The first example of this phenomenon was discussed in [9,10], relating the partition functions of four dimensional gauge theories to conformal blocks in Liouville theory. Since then, multiple other examples of this type have been found.
Describing these SCFT's, however, using traditional tools in field theory, is typically rather difficult, since in general no Lagrangian description is known. Therefore, different methods -many of them inspired by their relation to string-theory -have been developped. In particular, considering compactifications of F-theory [11] on elliptically fibered Calabi-Yau threefolds, a classification [12][13][14][15][16] (see also [17] for recent work in this direction) of six-dimensional SCFTs has been proposed. Those theories with N = (2, 0) supersymmetry allow an ADE classification and can be realised within type II string theory compactified on a R 4 /Γ singularity, with Γ a discrete ADE subgroup of SU (2). In the case of an A-type orbifold (i.e. Γ = Z N ) these theories have a dual description in terms of N parallel M5-branes probing a transverse R 4 ⊥ space. In this paper we study the A N symmetry in a series of mass-deformed theories that are described by N parallel M5-branes (separated along R or S 1 ) that probe a transverse R ⊥ /Z M singularity. The BPS partition functions Z N,M of this system have been computed explicitly in [1] for M = 1 and in [2,3] for generic M ∈ Z. There are various techniques to obtain Z N,M , which exploit different dual descriptions of the M-brane setup: • For general N, M one can associate a toric Calabi-Yau threefold 1 X N,M to the M-brane setup whose topological string partition function captures Z N,M .
• The M-brane setup is dual to a (p, q) 5-brane web in type II string theory [18]. The Nekrasov partition function on the world-volume of the D5-branes corresponds to Z N,M .
• Considering BPS M2-branes stretched between the M5-branes, the intersection of the two has been dubbed M-string in [1]. The partition function of the latter is computed by a N = (0, 2) sigma model, whose elliptic genus was shown in [1] to capture Z N,M .
Besides the mass parameter m, the partition function Z N,M needs to be regularised by the introduction of two deformation parameters 1,2 , which (from the perspective of the dual gauge theory) correspond to the introduction of the Ω-background [19,20]. For generic values of m, 1,2 , the Mstring world-sheet theory is described by a sigma model with N = (2, 0) supersymmetry. However, it was remarked in [1] that m = ± 1 − 2 2 the supersymmetry is enhanced to N = (2, 2) leading to Z N,M (m = ± 1 − 2 2 ) = 1 (after a suitable normalisation). In this paper we generalise this observation to make the A N −1 (or affine A N −1 ) symmetry of the partition function Z N,M manifest and organise it according to irreducible (integrable) representations of the associated Lie algebra a N −1 (or affine a N −1 ) for certain choices of the deformation parameters: for simplicity, we consider the unrefined partition functions (i.e. we choose Moreover, since the choice m = n is fully compatible with all symmetries of the elliptic fibration X N,M , notably A N −1 (or affine A N −1 in the case of a compact brane configuration), the latter are manifestly visible in Z N,M (m = n , ). Indeed, from the perspective of the Calabi-Yau manifold X N,M the t fa can be written as integrals of the Kähler form over a set of P 1 's that can be identified with the simple positive roots of the Lie algebra a N −1 (or affine a N −1 ) (see e.g. [5,21] for recent applications). Using this identification, specifically for the choice m = n we show in a large series of examples that Z N,M (m = n , ) can be written as a sum over weights that form a single irreducible (or integrable in the affine case) representation of the Lie algebra a N −1 (affine a N −1 ). In the basis of the fundamental weights, the highest weight of these representations is given by [M n 2 , . . . , M n 2 ]. Furthermore, each summand in the sum over weights is a quotient of Jacobi theta functions transforming with a well-defined index under an SL(2, Z) symmetry corresponding to the elliptic fibration of X N,M . Based on an extensive list of examples of different brane configurations (and choices for n ∈ N) we find a pattern for all these symmetries that allows us to formulate precise conjectures for generic values (N, M ) and n.
Finally, the compact M-brane configurations (i.e. where the M5-branes separated along S 1 rather than R) enjoy a duality upon exchanging M ↔ N as can be seen directly from the web diagram of X N,M . For the simplest 2 such configuration (i.e. N = 2 = M ) we show explicitly that the partition function can be written as a double sum over integrable representations of affine a N −1 and a M −1 respectively. The latter not only makes the algebraic structures but also the duality manifest. Since compact brane setups of the type (N, M ) capture [22][23][24] a class of little string theories (see [26][27][28][29][30][31] for various different approaches as well as [32,33] for reviews) with N = (1, 0) supersymmetry we expect that these findings will turn out useful for the further study of little string theories in general, in particular their symmetries and dualities (see e.g. [34] for a recent application).
The outline of this paper is as follows. In section 2 we describe the M-theory brane setup probing a transverse orbifold geometry. We introduce all necessary parameters to describe the configurations and discuss different approaches in the literature to compute the BPS counting functions Z N,M . Finally, we also discuss the supersymmetry preserved by these configurations (from the point of view of the M-string world-sheet theory) specifically focusing on their holonomy charges as a function of the deformation parameters (m, 1 = − 2 = ). In section 3 the expression for the topological string partition function is introduced. We furthermore motivate the choice m = n of deformation parameters by exhibit explicitly cancellations in Z N,M . In section 4 we present specific examples of non-compact brane setups and rewrite the corresponding partitions functions as sums over Weyl orbits of weights forming specific irreducible representations of a N −1 ∼ = sl(N, C). In section 5 we repeat a similar analysis for certain compact brane configuration and rewrite them in a similar manner as sums of Weyl orbits of weights forming integrable representation of the affine Lie algebra a N −1 ∼ = sl(N, C). Based on the examples of the previous two sections, in section 6 we give a general expression for the compact partition functions Z N,M (for generic (N, M )) as a sum over integrable representations of a N −1 . The non-compact partition functions in turn are obtained by an appropriate decompactification limit. Finally section 7 contains our conclusions. Several supplementary computations as well as additional information on simple and affine Lie algebras and their representations are relegated to 5 appendices.

M-Brane Configurations and Calabi-Yau Manifolds
In this paper we consider theories which can be described through particular BPS configurations of M-branes. In the following subsection we provide a review of these M-brane webs and relate them to a class of toric Calabi-Yau threefolds in section 2.2.3.

M-Brane Webs
In the following we describe configurations of parallel M5-branes with M2-branes stretched between them. Depending on whether the M5-branes are separated along S 1 or R, we call these configurations either compact or non-compact.

Non-Compact Brane Webs
We first discuss non-compact brane webs in M-theory compactified on T 2 × R 4 || × R × R 4 ⊥ (with coordinates x 0 , . . . , x 10 ) and consider a configuration of N M5-and K M2-branes as shown in table 1. Here the M5-branes are spread out along the x 6 direction and we denote their positions a a with a = 1, . . . , N (such that a a < a b for a < b). For explicit computations we introduce the x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 M5-branes = = = = = = M2-branes = = = Table 1: Non-compact BPS configuration of M5-and M2-branes.

Compact Brane Webs
By arranging the M5-branes on a circle rather than on R, we obtain compact M-brane configurations. Specifically, we replace the R along direction x 6 by S 1 6 with radius R 6 (i.e. x 6 ∼ x 6 + 2πR 6 ), as shown in table 2. As before, we denote the N positions of the M5-branes on S 1 6 by a a (with and introduce the N distance between adjacent branes as t fa = a a+1 − a a for a = 1, . . . , N − 1 , 2πR 6 − (a N − a 1 ) for a = N . (2.5) As in the non-compact case, we also introduce Q fa = e −t fa /R 0 , ∀a = 1, . . . , N , (2.6) along with the parameter 3 ρ := iR 6 /R 0 and Q ρ = e 2πiρ . (2.7) Notice the following relation (2.8) With this notation, the non-compact brane configurations are obtained in the limit ρ → 0.

Deformation Parameters
Computing the partition functions for the brane configurations introduced above, the latter are typically divergent. To circumvent this problem, one can introduce various deformation parameters [1]. Indeed, the underlying geometries allow for two different types of U (1) twists. Upon introducing the complex coordinates for R || and R ⊥ we can define • -deformation: As we go around the compact x 0 -direction (i.e. the circle S 1 0 ) we can twist by From the point of view of supersymmetric gauge theories which can be associated with the brane configurations described above (see [1,3]) this deformation introduces the Ωbackground [19,20] allowing to compute the partition functions in an efficient manner.
• mass deformation: As we go around the compact x 1 -direction (i.e. the circle S 1 1 ) we can twist by: As we shall briefly discuss further below, from the perspective of the gauge theories (that are engineered from a dual type II setup), this deformation parameter corresponds to a mass for certain hypermultiplet fields.
The action of the deformation parameters 1,2 and m can be schematically represented in table 3. The former regularise divergences in the partition function coming from contributions of the noncompact dimensions while at the same time breaking part of the supersymmetries, as we shall discuss in sections 3 and 2.3 respectively. Finally, we remark that in the later sections of this paper, the parameters 1,2 and m appear through (2.12) 3 We use the definition (2.3) also in the compact case.

Orbifold Action and Brane Web Parameters
A generalisation of the above M-brane configurations has been discussed in [2] (see also [3]). Indeed, upon considering M5-branes probing an orbifold geometry (rather than R 4 ⊥ ), the positions of the M2-branes can be separated in the transverse direction. Specifically, we generalise R 4 ⊥ to an Asymptotically Locally Euclidean space of type A M −1 (which we denote by ALE A M −1 ) for M ∈ N, which can be obtain as the following orbifold As explained in [2], the twists (2.10) and (2.11), which introduce the deformation parameters 1,2 and m, are compatible with the ALE A M −1 geometry. Indeed, when viewed as an S 1 fibration over R 3 , the latter posses two distinct U (1) isometries related to the fiber and base respectively. Therefore, the generalised M-brane configuration (including the deformation parameters 1,2 and m) can be represented by table 4, where we again allowed for the possibility of arranging the M5-branes along the x 6 -direction either on R or on S 1 6 . As in the case M = 1, the distances between the M5-branes along the direction x 6 give rise to N parameters t fa for a = 1, . . . , N (see eq. 2.5). The case |t f N | < ∞ corresponds to a compact brane configuration (i.e. the direction x 6 is compactified on S 1 6 with finite radius R 6 ), while the limit |t f N | → ∞ corresponds to a non-compact brane configuration (i.e. the direction x 6 is non-compact). As explained in [2], besides the (t fa , m, 1,2 ), the orbifolded configuration allows for another set of parameters, corresponding to the expectation values T i (for i = 1, . . . , M ) of the M-theory three-form along S 1 1 × C i , where C i is a basis of the 2-cycles of ALE A M −1 . In later computations, these parameters typically appear in the form (2.14) Furthermore, the parameters τ and ρ (see (2.3) and (2.7) respectively) are in this duality frame given by which is equivalent to The full orbifolded M-brane configuration is finally parametrised by (t f 1 , . . . , t f N , T 1 , . . . , T M , m, 1,2 ), which we denote more compactly by (t, T, m, 1,2 ).

Type II Description
The parameters introduced in the above M-brane configurations can be given a more geometric interpretation when dualising to the corresponding type II picture. Indeed, upon reducing the orbifold M-theory configuration along S 1 1 , it can be dualised into a web of intersecting D5-and NS5-branes as shown in table 5, where we represented the ALE A M −1 space as a (particular limit of a) fibration of S 1 7 over R 3 ⊥ (see [2] for more details).
While the parameters 1,2 can be introduced in the same fashion as in the M-theory case, the parameter m can no longer be interpreted as a U (1) deformation (since the corresponding circle S 1 1 is no longer present). The latter is introduced by giving mass m to the bifundamental hypermultiplets corresponding to strings stretched between the D5-and NS5-branes. At the level of the brane web, it corresponds to a deformation with (1, 1) branes in the (x 6 , x 7 )-plane, as shown in figure 1. This figure also shows the remaining parameters (t, T) as the distances of the D5-and NS5-branes in the x 6 and x 7 direction respectively. 4 As discussed in [2,24,25] choosing the deformation parameter m to be the same for all intersections of D5-NS5-branes is not the most general case since a generic such brane web has N M + 2 independent parameters. In the following, however, we focus on this simpler case, where all mass deformations are the same (as indicated in Figure 1).  x 7 Figure 1: Configuration of intersecting D5-branes (red) and NS5-branes (blue). The deformation parameter m introduced by the (1, 1)-branes (black) is chosen to be the same throughout the diagram. For |t f N | < ∞, the D5-branes are arranged on a circle (compact case), while the noncompact case corresponds to the limit t f N → ∞.

Toric Calabi-Yau Manifolds
There is a further description of the theories introduced above. Indeed, as explained in [1][2][3], one can associate a toric non-compact Calabi-Yau 3-fold (CY3fold) X N,M with the 5-brane web. More precisely, the web diagram shown in Figure 1 can be interpreted as the dual of the Newton polygon which encodes how X N,M is constructed from C 3 patches.
A generic X N,M can be described as a Z N × Z M orbifold of X 1,1 . The latter is a Calabi-Yau threefold that resembles the geometry of the resolved conifold at certain boundary-regions of its moduli space (i.e. upon sending τ, ρ → ∞). 5 More importantly, X N,M has the structure of a double elliptic fibration: it can be understood as an elliptic fibration over the affine A N −1 space, which (as already mentioned) itself is an elliptic fibration. The two elliptic parameters are ρ and τ , which were introduced in (2.15). The remaining parameters (t f 1 , . . . , t f N −1 ), (T 1 , . . . , T M −1 ) as well as m correspond to further Kähler parameters of X N,M . We shall further elaborate on the interpretation of the parameters 1,2 from the point of view of the Calabi-Yau manifold once we discuss the topological partition function on X N,M in section 3.
The double elliptic fibration structure of X N,M corresponds to the presence of two SL(2, Z) symmetries which act separately on the modular parameters τ and ρ. Particularly for the case M = 1 we have the following action on the various parameters [23] where a b c d ∈ SL(2, Z), i.e. a, b, c, d ∈ Z and ad − bc = 1.

Supersymmetry
In order to discuss the amount of supersymmetry preserved by the M-brane configurations described above, we adopt the point of view of the M-string [1]: for a configuration of parallel M5-branes probing a flat R 4 ⊥ with M2-branes stretched between them (i.e. configurations with M = 1), the M-string preserves N = (4, 4) supersymmetry with R-symmetry group Spin R (4). The latter acts on the space R 4 ⊥ transverse to the M5-branes. The supercharges [1] transform as the representations is the Lorentz group on the M5 world-volume (with Spin(1, 1) the Lorentz-group on the world-volume of the M-string) and the ± subscript denotes the chirality with respect to Spin(1, 1). As was explained in [1], upon introducing the simple roots of Spin(8) ⊃ Spin(4) R × Spin(4) (1, 2, 1, 2) − : Furthermore, as discussed in [2], the orbifold action (2.13) is not compatible with all 8 supercharges and indeed only (2, 1, 2, 1) + (i.e. the supercharge with positive chirality) is invariant. Therefore, for configurations with M > 1, supersymmetry is broken to N = (4, 0). The latter is in general further reduced by the deformations (2.10): while the mass deformation (2.11) (which acts in a similar manner on R 4 ⊥ as the Z M orbifold (2.13)) breaks the same supercharges as the orbifold action (and leaves invariant all of (2, 1, 2, 1) + ), the -deformation in general 6 only leaves the supercharges corresponding to e 1 + e 2 + e 3 + e 4 2 , and − e 1 + e 2 + e 3 + e 4 2 , (2.20) invariant. It therefore reduces the supersymmetry to N = (2, 0).

Compact and Non-Compact M-brane Configurations
An important quantity to describe the different M-brane configurations introduced above is the partition function Z N,M that counts BPS states. The latter can be weighted by fugacities related to the various symmetries described above. Concretely, the partition functions can be computed in various different manners, as explained in [1][2][3] • Topological string partition function The partition function Z N,M is captured by the (refined) topological string partition function on the toric Calabi-Yau threefold X N,M . The latter can efficiently be computed using the (refined) topological vertex [36][37][38] • M-string partition function Z N,M can also be computed as the M-string partition function. For configurations (N, 1) (i.e. for M = 1) it was shown in [1] that the latter can be obtained as the elliptic genus of a sigma model with N = (2, 0) supersymmetry whose target space is a product of Hilb[C 2 ], the Hilbert scheme of points in R 4 . This result was generalised in [3] to the case M > 1 where it was shown that Z N,M can be computed as the elliptic genus of a sigma model with N = (2, 0) supersymmetry whose target space is given by M(r, k), the moduli spaces of U (r) instantons of charge k.

• Nekrasov instanton calculus
The partition function can also be obtained from the 5-dimensional gauge theory that lives on the world-volume of the D5-branes in the type II brane-web description (see section 2.2.2). The non-perturbative partition function of the latter can be computed using Nekrasov's instanton calculus on the Ω-background [39].
• BPS scattering amplitudes in type II string theory As discussed in [3], certain of the partition functions Z N,M can also be obtained from a specific class of higher derivative scattering amplitudes in type II string theory.
Using either of these approaches, the partition function for a compact (i.e. Q ρ = 0) brane configuration (N, M ) can be written in the following manner [1][2][3] a |) and α N +1 Furthermore, for two integer partitions µ = (µ 1 , . . . , µ 1 ) and ν = (ν 1 , . . . , ν 2 ) of length 1,2 respectively, we have Here (i, j) denotes the position of a given box in the Young diagram of the partitions µ and ν respectively, µ t denotes the transposed partition of µ and , where θ 1 (τ ; x) (for x = e 2πiz ) is the Jacobi theta-function Finally, the factor W M (∅) in (3.1) is defined as and we also introduce the normalised partition function The latter was related in [2,5,22,23] to an U (N ) M gauge theory (which is dual to an U (M ) N gauge theory), as well as (five-dimensional) little string theory. For the explicit computations in the remainder of this work it is more convenient to rewrite the partiton function in the following form: . (3.8) Here we introduced 3.9) and the arguments of the Jacobi-theta functions in (3.8) are given by: Specifically, for M = 1 we have the following expression , (3.11) where we introduced the following shorthand notation for the arguments of the Jacobi thetafunctions The partition function for non-compact brane webs (which we denote Z line N,M ) can be obtained from (3.1) through the limit Q ρ → 0 (i.e. ρ → i∞): , (3.14) where the arguments (z

Particular Values of the Deformation Parameters
Viewed as a BPS counting function (3.8) (and its non-compact counterpart (3.13)) depend on the fugacities (T, t, m, 1 , 2 ) that refine various symmetries associated with the (N, M ) brane-web. We can summarise the latter in the following table parameter symmetry, compact case symmetry, non-compact case Here SL(2, Z) τ is a generalisation of (2.17) to the case M > 1 However, as remarked in e.g. [5], the structure of the M5-branes along this direction can be interpreted as the Dynkin diagram of a N −1 (or its affine extension a N −1 ) and the Q fa can be linked to the roots of these algebras respectively. Indeed, we will explain this connection in more detail in the following sections, when considering explicit examples of the partition functions Z N,M . Finally, we notice that in the compact case, the roles of T and t can be exchanged upon replacing (N, M ) −→ (M, N ). In the above table the parameters t have been singled out since we have decided to write Z N,M in (3.1) as a power series expansion in Q fa (rather thanQ i ). 7 Written as a function of all parameters mentioned above Z N,M is rather complicated and very difficult to analyse. In this paper we therefore consider particular values for some of the parameters, such that Z N,M simplifies and the various symmetries can be made more manifest. First, for simplicity, we choose to work in the unrefined case, i.e. we set which (as mentioned in section 2.3) leads to an enhancement of supersymmetry to N = (4, 0). Furthermore, (3.16) is fully compatible with the symmetries SL(2, Z) τ as well as A N −1 (or A N −1 ).
In order to further define regions in the parameter space in which the partition function simplifies, we first consider the case M = 1. In this case, the Spin(8) holonomy charges corresponding to the deformations (2.10) and (2.11) read where we recall that the first two entries (depending on ) correspond to a holonomy with respect to S 1 0 and the last two (depending on m) with respect to S 1 1 . For generic values of and m (in particular for m/ ∈ R/N) there is no cancellation between the corresponding holonomy phases.
Phrased differently, there is no mixing between states with distinct charges under U (1) m and U (1) 1 × U (1) 2 in the partition function. However, if we choose m = n , with n ∈ N , (3.18) the holonomy charges are no longer linear independent over Z and thus holonomy phases may cancel when we go multiple times around the circle S 1 0 . In this way, there may be non-trivial cancellations between the contributions of states with distinct charges under U (1) m and U (1) 1 × U (1) 2 in the partition function Z N,1 leading to possible simplifications of Z N,1 . 8 For M > 1, the same effect appears (at least) in the untwisted sector of the orbifold, such that we expect similar simplifications. Finally, we also remark that the choice (3.18) is still compatible with SL(2, Z) τ as well as A N −1 (or A N −1 ). Therefore, we can analyse the simplified partition functions Z N,M (T, t, m = n , , − ) with respect to these symmetries and write them in a fashion that makes them manifest.
Explicitly, at the level of the partition function, the reason for the above mentioned simplifications is the following: when choosing the parameters the arguments (3.10) of the theta-functions in (3.8) take the following form For specific partitions α (a) k these combinations may become zero even for generic , thereby (with θ 1 (τ ; 0) = 0) leading to a vanishing contribution to the partition function. We also notice that for (3.19) in general w  and − e1−e2−e3+e4 2 remain unbroken, thus leading to an enhancement of supersymmetry. This fact was already remarked in [1] for the more generic case m = 1− 2 2 .
Due to the previous constraint the only remaining partitions correspond to Young diagrams with a single row: For this particular box we have z such that all partitions with ν 1,1 ≥ 2 violate (4.1) and do not contribute to the partition function (3.14).
Combining these two constraints we find that the only possible choices are ν 1 = ∅ or ν 1 = and the partition function therefore is Notice that the right hand side is independent of τ and and only depends linearly on Q f 1 .
The partition function (4.4) can be rewritten in fashion that makes an a 1 symmetry manifest. Indeed, upon identifying where α 1 is the simple root of a 1 we can write with mult(α 1 ) = 1 and ∆ + (a 1 ) the space of positive roots of a 1 . Using the Weyl character formula, we can rewrite the product (4.6) as a sum over the Weyl group W( where ξ is the Weyl vector of a 1 and (w) is the length of w ∈ W(a 1 ) ∼ = Z 2 , i.e. the number of Weyl reflections that w is decomposed of.
While the re-writings (4.6) and (4.7) seem trivial (due to the fact that the root space of a 1 is one-dimensional, i.e. ∆ + (a 1 ) = {α 1 }), we shall see that both equations can be directly generalised for other choices m = n (with n > 1) and also N > 2 (as we shall discuss in section 4.2).

Choice
For the cases n > 1 we can repeat the above analysis to find all partitions that yield a non-vanishing contribution to the partition function (3.14). In doing so, we find a generic pattern, which can be summarised as follows: 9 only those partitions ν 1 = (ν 1,1 , ν 1,2 , . . . , ν 1, ) with ≤ n , and ν 1,a ≤ n , ∀a = 1, . . . , n , (4.8) satisfy (4.1). As a consequence, we propose that the partition function is a polynomial in Q f 1 and can be written as the finite sum For n > 1 the coefficients c (n) k depend explicitly on τ and and have the property are given by (we recall that relations (4.12) -(4.17) have in fact been checked explicitly up to n = 11): where for simplicity we have introduced the shorthand notation While not constant (as in the case of n = 1), the coefficients c (n) k display a clear pattern, which we propose to hold for generic (k, n) satisfying (4.11): every coefficient itself can be written in the form where the sum is over partitions µ(k, n) = (µ 1 (k, n) , µ 2 (k, n) , . . . , µ (k, n)) of length (with 0 ≤ ≤ k + 1) and f (µ) is a product of theta functions with r i ∈ N even and r i ≤ 4 that satisfy Here the first condition states that the number of θ 1 -functions in the numerator and denominator of (4.19) is the same, while the second condition ensures that each coefficient c (n) k (τ, ) transforms in an appropriate manner under SL(2, Z) τ transformations (see (2.17)). Specifically, we have Thus, we can assign an index under SL(2, Z) τ to each of the c  k (τ, ) in (4.19) are rather complicated, they are essentially determined by specifying all partitions µ(n, k) for which c(µ) = 0. These can be obtained from the partitions µ(n, k − 1) in an algorithmic fashion by increasing one of the µ a (k − 1, n) by either 1 or 2. The precise relation (along with explicit examples up to k = 5) is explained in appendix C and can be summarised by the fact that there is an operator R + such that (4.24) Schematically, the action of R + can be represented graphically in the following manner which also reflects the symmetry (4.10). These graphical representations are reminiscent of the highest-weight representation Γ n 2 of sl(2, C) where one can move between the various points (which represent certain one-dimensional functional spaces of theta-quotients) with the help of raising and lowering operators. In fact, we can make this connection more precise by writing where e −n 2 ξ = Q  while we have for the coefficients Finally, the O n λ in (4.25) can be understood as the (normalised) orbits of λ ∈ P + n 2 under the Weyl where we have used the identification (4.5) and the normalisation factor is given by Here |Orb λ (W(a 1 ))| is the order of the orbit of λ under the Weyl group of a 1 and |W(a 1 )| = |Z 2 | = 2.
To summarise, we propose that Z line 2,1 (τ, t f 1 , m = n , , − ) can be written as a sum over weights of sl(2, C), whose representatives fall into the fundamental Weyl chamber of the irreducible representation Γ n 2 . As we shall see in the following, this pattern continues to hold for the partition functions of other non-compact M-brane configurations (N, 1) for N > 2.

Configuration
The case (N, M ) = (3, 1) for the choice m = is analysed in detail in appendix E. Summarising the results, as above only finitely many partitions contribute to Z line N,1 in (3.14) which are given in the following table Combining these expressions, we find for the partition function Notice that this result is independent of τ and and only depends on Q f 1,2 in a polynomial fashion. Moreover, the partition function (4.30) can be rewritten in a fashion that makes an a 2 symmetry manifest. Indeed, upon defining where ∆ + (a 2 ) denotes the simple positive roots of a 2 , we can write Here we have used the fact that mult(α 1 ) = mult(α 2 ) = 1. Using the Weyl character formula, we can rewrite the product (4.32) as an orbit of the Weyl group where ξ = α 1 + α 2 is the Weyl vector and (w) is the length of w ∈ W(a 2 ), i.e. the number of Weyl reflections that w is decomposed of: the Weyl reflections of a 2 are defined as They are subject to the relations s 2 1 = s 2 2 = (s 1 s 2 ) 3 = 0. With this notation we can check (4.33) by working out all non-equivalent Weyl reflections Therefore, (using (4.31)), we have In view of generalising (4.33) to the cases m = n for n > 1, we prefer to write the action of the Weyl group W(a 2 ) ∼ = S 3 in a slightly different and more intuitive manner. To this end we introduce the simple weights (L 1 , L 2 , L 3 ) that span the dual of the Cartan subalgebra h * a 2 (as explained in appendix B) and identify which is compatible with (4.31). Furthermore, we introduce such that We note that the x r=1,2,3 are not independent, but satisfy x 1 x 2 x 3 = 1 due to the constraint L 1 + L 2 + L 3 = 0 (see (B.9)). Using the latter condition, we can write (4.34) in the following fashion . The action of the Weyl group in (4.38) can also be illustrated graphically by arranging all terms in the following weight diagram: where the blue numbers represent the factor sign(σ) in (4.38). This picture indeed illustrates the Finally, before continuing with further examples with m = n for n > 1 there are two comments we would like to make • The prefactor e −ξ in (4.38) simply serves to arrange the various terms in the expansion of Z line 3,1 (τ, t f 1 , t f 2 , m = , , − ) to be concentric with respect to the origin of the weight lattice spanned by (L 1 , L 2 , L 3 ).
• We can also add the 'central point' e −ξ (marked by a red circle in the above figure) to the partition function Z line Therefore, we can write the partition function in the more suggestive form where P + 1,1 is the fundamental Weyl chamber of the irreducible representation Γ 1,1 of a 2 As we shall discuss in the following, the form (4.41) can be generalised to the cases m = n for n > 1.

Case
Generalising the discussion of the previous subsection to the case 1 = − 2 = and m = 2 we find again specific conditions for the partitions ν 1,2 in (3.14) to yield a non-vanishing contribution to the partition function Z line 3,1 (τ, t f 1 , t f 2 , m = 2 , , − ). As a consequence, the latter is again polynomial in Q f 1 and Q f 2 with highest powers Q 8 f 1 and Q 8 f 2 respectively. However, the coefficient of each term in this polynomial is no longer a constant (i.e. ±1), but rather a quotient of Jacobi theta functions, i.e. schematically where the integers a r (i, j) and b r (i, j) implicitly depend on i, j. However, as we shall discuss presently, this expressions can still be written in a manner that makes the action of a 2 manifest.
To this end, we group together all terms corresponding to a given quotient of theta functions, however, rather than using the variables Q f i , we use the variables x r as introduced in (4.36). In terms of the monomials Q i The relation L 1 + L 2 + L 3 = 0 then implies x 1 x 2 x 3 = 1, which allows us to a generic monomial Q i f 1 Q j f 2 as a polynomial of x 1,2,3 with only positive powers. Specifically, for n = 2 we find : where the factors φ 2 [c 1 ,c 2 ] (τ, ) depend on τ and and are given as follows The subscripts 11 are chosen in such a way to make an action of the Weyl group W(a 2 ) ∼ = S 3 of sl(3, C) on Z line 3,1 (τ, t f 1 , t f 2 , m = 2 , , − ) (along the lines of (4.41) for n = 1) visible. They can be identified with the Dynkin labels of the irreducible representation Γ 4,4 , as we shall explain in the following: as in the case of n = 1 (see eq. (4.38)), the Weyl group W(a 2 ) acts as a permutation of the powers of a given monomial of the x 1,2,3 : which allows us to describe all monomials multiplying a given φ 2 [c 1 ,c 2 ] as the Weyl orbit of a single element. To describe the latter, we introduce the fundamental weights of a 2 and which serve as a basis for the weight lattice of a 2 and span the fundamental Weyl chamber. Concretely, every weight vector can be written as (4.49) 11 The superscript has been added as a reminder of the fact that we are dealing with the case n = 2.
Notice also |Orb [3,0] (W(a 2 ))| = 3. Before further generalising this discussion to generic m = n for n ∈ N, there are a few comments we would like to make • Comparing (4.50) to (4.41), both are structurally very similar in the sense that they are sums over Weyl orbits whose representatives are in the fundamental Weyl chamber of a certain irreducible representation of sl(3, C). However, in the case of (4.50), each orbit is still multiplied by a non-trivial function which depends on τ and . Another difference is the fact that the terms in each orbit in (4.51) come with the same relativ sign due to the absence of (−1) (w) which is present in (4.41).
Each such quotient has a well-defined index I τ under the action of SL(2, Z) τ (which was introduced in (2.17) 13 ) Specifically, I τ is given as Here (., .) stands for the inner product in the basis (ω 1 , ω 2 ). 12 In order to make contact with the Q f1 and Q f2 we recall that upon using (4.44), a given monomial in (4.43) can be written in the form Q i f1 Q j f2 = e (j−2i)ω1+(i−2j)ω2 . 13 Notice that SL(2, Z) τ remains a symmetry of the partition function even after the identification 1 = − 2 = and m = 2 . The results of the previous two subsections show an emergent pattern which can be generalised directly and which we conjecture 14 to hold for generic n ∈ N: for n a (finite) integer, only a finite number of partitions ν 1,2 can contribute to the partition function (3.8). Therefore Z line 3,1 (τ, t f 1 , t f 2 , m = n , , − ) is polynomial in the parameters Q f 1 and Q f 2 with the highest powers Q 2n 2 f 1 and Q 2n 2 f 2 . Each monomial Q i f 1 Q j f 2 is multiplied by a quotient of Jacobi-theta functions that depend on τ and . Specifically, we can write in a similar fashion as in (4.43) 59) Using the same notation as in the previous subsection, we propose that we can re-write the partition function in the following manner Here O n λ (t f 1 , t f 2 ) denotes the following normalised orbits of the Weyl group W( which agree with (4.46) and compensate the factor d λ for the cases |Orb λ (W(a 2 ))| < 6, in order to avoid overcounting. Furthermore, just as in the case n = 2 in (4.46), the functions φ n  The structure of (4.60) can be made more transparent by arranging all terms on the weight lattice of a 2 as shown in figure 2. Here the red lines indicate the fundamental Weyl chamber and we have · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · φ n    φ n [2,2] φ n [2,2] φ n [2,2] for n even · · · · · · · · · · · · · · · · · · · · · φ n [0,3n 2 for n odd · · · · · · · · · · · · · · · · · · · · · φ n [1,(3n 2 for n even  attached the coefficients for each weight respectively. In this way the symmetry under the Weyl group is made manifest. We notice, however, that for n odd, the weights [c 1 , c 2 ] for c 1 = 0 or c 2 = 0 do not contribute to the partition function. Indeed, in these cases we have due to the sign factors sign(σ) in the definition (4.61). 15 In order to further elucidate the connection between Z line 3,1 (τ, t f 1 , t f 2 , m = n , , − ) and the irreducible representation Γ n 2 ,n 2 of a 2 , we remark another property of the φ n λ (τ, ) in (D.1). As explained in appendix B.2, the weight diagram of the representation Γ n 2 ,n 2 is made from concentric hexagons whose weight spaces share the same multiplicity. Thus, one would expect that the quotients of the theta-functions φ n λ are elements of a vector space of functions whose dimension corresponds to the latter multiplicity. Concretely, we expect Comparing with the explicit expressions (D.1), we find that the functions φ n λ with weights λ = [c 1 , c 2 ] that are expected to be of multiplicity k ∈ N according to the above table, are indeed linear combinations of theta-quotients of the following type: 14 We have indeed verified the results further up to n = 6. 15 The vanishing is due to the same mechanism which leads to (4.40) for n = 1.
Thus, according to the grouping in (4.66), the φ n λ are indeed elements of a space of functions S k whose dimension k matches the expected mutliplicity. This is a further indication that the partition functions Z line 3,1 (τ, t f 1 , t f 2 , m = n , , − ) can be arranged according to the irreducible representation Γ n 2 ,n 2 of sl(3, C) for n ∈ N.
Finally, we would like to comment on the relation between Z line 3,1 (τ, m = n , t f 1 , t f 2 , , − ) and Z line 2,1 (τ, m = n , t f 1 , , − ) from the point of view of the representation theory of sl(3, C) and sl(2, C) respectively. Indeed, starting from the highest weight [n 2 , n 2 ] of Γ n 2 ,n 2 and acting with only a single root produces a highest weight representation of sl(2, C). Indeed, considering the functions

Configurations (N, M ) = (4, 1) and (5, 1)
We can repeat the above analysis for (N, M ) = (4, 1) and (5, 1). For simplicity, we restrict ourselves to the case m = In the former case, the partition function (3.14) contains a sum over non-trivial partitions (ν 1 , ν 2 , ν 3 ) and the relevant contributions are given by which are tabulated as follows Combining all these contributions, we find for the partition function which is polynomial in Q f 1,2,3 and invariant under the exchange Q f 1 ↔ Q f 3 . By making the following identifications with the simple roots of sl(4, C) we can write (4.69) As before, this can be rewritten, using the Weyl denominator formula, as a sum over the Weyl group for the corresponding root lattice e w(ξ)−ξ (4.72) where ξ = 3 2 α 1 + 2α 2 + 3 2 α 3 is the Weyl vector for sl(4, C). In a similar fashion as in the previous section we can give a graphical representation of the partition function by arranging its various terms on the weight lattice of a 3 (see figure 3). This presentation of the partition function indeed resembles a highest weight representation of a 3 ∼ = sl(4, C). We have also performed checks for n > 1: in all cases the partition function still has the structure of irreducible sl(4, C) representations. In the case (N, M ) = (5, 1), the partition function is a sum over four partitions (ν 1 , ν 2 , ν 3 , ν 4 ). Analysing the individual contributions, we find that the partition function can be written as where we used Q f i = e −α i , i = 1, 2, 3, 4 (4.74) As in the previous cases this can be rewritten as where ξ = 2α 1 + 3α 2 + 3α 3 + 2α 4 is the Weyl vector of a 4 .
Before generalising the above discussion to cases m = n for n ∈ N, we would like to make a further remark: in section 3.2 we argued that the simplification of the partition function Z N,M =1 for m = n and 1 = − 2 = is due to the fact that the Spin(8) holonomy charges are no longer linear independent over Z. Therefore, there are possible cancellation among contributions with different charges with respect to U (1) m and U (1) 1 × U (1) 2 . For M > 1, the same simplifications take place in the untwisted sector of the orbifold action (2.13), leading to similar simplifications of the partition function, as is indeed showcased in (4.77). However, along the same line of reasoning, identifyingT 1 = k for k ∈ N, should lead to further cancellations among different contributions in the partition function. Indeed, settingT 1 = in (4.77) we get φ 1,2 [0] (τ, , ) = 0, such that Z line 2,2 (τ,T 1 = , t f 1 , , − ) = 1 + Q 2 f 1 . Generalising the discussion of the previous subsubsection for m = n with n > 1 the partition function can schematically be written in the following form: 16 Analogously to the previous cases we propose that the partition function (4.84) can be written by summing the Weyl orbits for the weights in the fudamental Weyl chamber P + 2n 2 of the irreducible representation Γ 2n 2 of sl(2, C) (t f 1 ) = d λ w∈W(a 1 ) which is equivalent to (4.80) since (−1) M nl(w) = 1 for n ∈ Z. Furthermore, the first few coefficient Generalising (4.82) and using the notation (4.84), the index of the theta ratios is Finally, as for the case n = 1, there are additional cancellations in the partition function once we setT 1 = k (with k ∈ N) to be a(n integer) multiple of . Notice, however, when k < n the partition function Z line 2,2 (τ, T 1 = k , t f 1 , m = n , , − ) appears to diverge due to the fact that theta-functions in the denominator vanish. The choice k = n provides the simplest expression in the sense that certain φ n,M =2 λ vanish. Schematically, the vanishing coefficient functions can be shown in the following weight diagram of sl(2, C): The vanishing theta-quotients correspond to the following powers of Q f 1 in the partition function: , while for odd n also the power Q n 2 f 1 is vanishing as well.

Examples: Compact Brane Configuration
After having discussed examples of partition functions of non-compact brane configurations for the particular choice m = n (with n ∈ N), we now consider compact brane configurations. The non-compact case can be recovered in the limit n a=1 Q fa = Q ρ → 0, as we shall discuss in the following.
We also use the notation m 0 = ∅. With this notation, we only get the following three types of contributions to the partition function (n ∈ N) Thus, the normalised partition function (3.7) is This expression can also be written in the form 18 Following the discussion of the non-compact examples, we would like to Identify the Kähler parameters t f 1 and t f 2 with the affine roots α 0 and α 1 , which are introduced in appendix A.2. This involves choosing which t fa contains the null root δ . The final answer does not depend on this choice as the exchange Q f 1 ↔ Q f 2 does not change the partition function. Here we choose the following and using expression (A.15) for the positive roots of a 1 we can write (1 − e − α ) . Using the affine Weyl denominator formula (5.5) can be written ass a sum over elements of the affine Weyl group (with mult( α) = 1 for α ∈ ∆( a 1 )) We can work out the first few Weyl reflections to check (5.6) Therefore, using (5.6), we have which matches (5.3). While written as a sum of Weyl reflections of ξ, we can also interpret (5.8) as a sum over Weyl orbits of weights in the fundamental domain P + 1,1 of the highest weight representation Γ 1,1 of a 1 19 : following the discussion of appendix A.2, every affine weight of sl(2, C) can be decomposed into fundamental weights ( ω 0 , ω 1 ) as follows such that the affine root t f 1 and a generic monomial Q i f 1 Q j f 2 are decomposed as Therefore, we can write where we defined 13) and P + 1,1 is the fundamental Weyl chamber of the affine representation generated by the weight [1,1], i.e. P + 1,1 = {[1, 1, −l]|l ∈ N}. Thus (up to a prefactor e − ξ (1 − Q ρ )), the partition function Z 2,1 (τ, t f 1 , t f 2 , m = , , − ) can be written as a sum over the states contained in Γ 1,1 .
Finally, before discussing more general cases m = n with n > 1, we remark that in the limit Q ρ → 0 we reproduce the partition function Z line For m = n with n > 1, the partition function is an infinite sum of ratios of theta functions: To illustrate this expression, we first consider in some detail the case n = 2 and generic n later.
n=2 For n = 2 the first few terms of the partition function can be written in the following suggestive form where the notation is the same as in (5.9). Indeed, the φ 2 [c 0 ,c 1 ,l] are indexed by their Dynkin labels c 0 , c 1 and their grade l where the individual Weyl orbits are given by where the normalization is given by The weights of the affine [4,4] that are in the fundamental Weyl chamber P + 4,4 are those with positive Dynkin labels As for the non-compact cases the arguments of the φ 2 [c 0 ,c 1 ,l] can be related to their corresponding affine weights λ = [c 0 , c 1 , l] through where (.|.) stands for the inner product in the affine ω 1 basis.
Before continuing to the case of generic n, we consider the decompactification limit Q ρ → 0. In this case only those weights with = 0 survive, such that (with (5.10)) This expression indeed agrees with (4.9) as expected, since in the limit Q ρ → 0 the brane setup corresponds to the non-compact configuration (N, M ) = (2, 1).
generic n The above analysis can be extended for n > 2 with a pattern arising which allows us to conjecture the structure for generic n: Indeed, we propose that the partition function can be written as a sum over Weyl orbits of the representatives in the fundamental Weyl chamber P + where the Weyl orbits are given by 20 The fundamental Weyl chamber P + n 2 ,n 2 is given by  (N, 1) for N > 2

Configurations
We can generalise the discussion of the previous subsection to cases N > 2. For simplicity we restrict to n = 1 and show that the partition function can be written as a product over simple positive roots of a N −1 .
The first case corresponds to N = 3, i.e. three M5-branes. For the partition function, this requires to sum over three different partitions (ν 1 , ν 2 , ν 3 ). Analysing the configurations which lead to a non-trivial contribution, we summarise the first few in the following table (with g This expansion is matched by the expression 21 where ∆ + ( a 2 ) is the space of positive simple roots of a 2 . Notice the relation Repeating the computation for N = 4, we find up to order 6 in the expansion of Q f i that the partition function can be written as . (5.36)
where we have used the same notation as in section 5.
Here the Weyl orbits are given by We notice that the arguments of the φ 1 [c 0 ,c 1 ,l] are related to the affine weights by which directly generalises the cases M = 1 discussed above Before closing this section we would like to make a further remark: The brane configuration (N, M ) = (2, 2) is self-dual under the exchange of N and M . Furthermore, the appearance of the symmetry a N −1=1 is due to the expansion of Z 2,2 (τ, T 1 , ρ, t f 1 , t f 2 , m = , , − ) with respect to Q t f 1,2 and we would expect a similar structure with respect toQ 1,2 . It is therefore interesting to see whether the partition function can be written in a fashion that makes a symmetry a N −1=1 ⊗ a M −1=1 manifest. To this end, we first have to re-instate the normalisation factor (W M =2 (∅)) N =2 in (3.1). The latter can be read off from (5.5) Thus, multiplying the coefficient functions (5.41) -(5.43) with (W 2 (∅)) 2 1 =− 2 =m= , the non-trivial φ 1 λ are (up to integer coefficients) of the form which mirror (5.10) and (5.4) such that we can write for (5.46) where we denote e −2( κ 0 + κ 1 ) = e −2 ζ as the Weyl vector of a M −1=1 and where the non-vanishing coefficients p 1 λ 1 ,λ 2 ( ) are (with s, s ∈ N ∪ {0} and r, r ∈ N)

Generic Configuration (N, M ) and Representations
After the analysis of many specific cases we compile in this section generic relations that we conjecture to hold for arbitrary N , M and n. As the non-compact case is obtained as a limit of the compact case we start with the latter.
We propose that the normalised partition 3.7 can be written as a sum over the Weyl orbits for the representative weights in the fundamental Weyl chamber P + The partition functions of non-compact brane configurations are obtained by taking the limit where (., .) denotes the inner product in the basis of fundamental weights (ω 1 , . . . , ω N −1 ). 22 Notice that the transformation (6.3) is compatible with the decompactification limit.

Conclusions
In this paper we have studied the BPS partition functions of N parallel M5-branes probing a transverse ALE A M −1 space. We have distinguished cases of M5-branes separated along S 1 (with partition function Z N,M defined in (3.1)) and along R (with partition function Z line N,M defined in (3.13). The latter can be obtained from the former through the decompactification limit that sends one of the distances t fa of the branes to infinity.
To regularise the BPS partition functions, a set of deformation parameters, denoted by (m, 1 , 2 ) needs to be introduced. For simplicity, we have chosen to work in the so-called unrefined limit This presentation of the partition function also makes the structure of X 2,2 ∼ = X 1,1 /(Z 2 × Z 2 ) more tangible, which is dual to the M5-brane configuration.
These results make the algebraic properties of the BPS counting functions of specific M-brane configurations very tangible: indeed, in certain regions of the parameter space, the partition functions fall into the form of single highest weight representations of (affine) Lie algebras that are related to the geometric backgrounds of the M-brane configurations. While the results presented here are specific to the choice m = n , the a N symmetries are expected to be unbroken for generic deformations as well: indeed the dual Calabi-Yau manifolds X N,M can be understood as elliptic fibrations over A N −1 . Thus, our results have highlighted a region in the moduli space in which the latter are very manifest.
In view of the many other physical systems that are dual to the M-brane configurations that we have studied here, we expect our results to have many applications in the future: one of them is the study of little string theories (LSTs) [26][27][28][29][30][31] (see also [32,33] for reviews). Indeed, the compact brane configurations (N, M ) are related to a particular class of LSTs [22][23][24] with N = (1, 0) supersymmetry. It will be interesting in the future to find further regions in the moduli spaces of LSTs which make more of their symmetries manifest or possibly reveal new ones. Furthermore, our findings may also turn out useful to study algebraic properties of double-quantised Seiberg-Witten geometry related to the topological string partition function of X N,M and the definition of qq-characters (see [40][41][42][43] and [44][45][46] for recent progress respectively). .

Finally
, an interesting open question remains why in the limits we have discussed in this work, the partition function is governed by a single irreducible/integrable representation. While we have argued, based on the structure of the preserved supercharges, that the choice (3.16) and (3.18) leads to cancellations among different states in the partition function (and thus to massive simplifications) it does not fully explain why the remaining contributions have the structure of a single representation. As was pointed out to us by A. Iqbal, it would be interesting to study these results from the point of view of Chern Simons theory (see e.g. [36,37]) to see if one can find an interpretation from this side. We leave this possibility for future work.
An affine weight λ can thus be denoted by its eigenvalues under the Cartan subalgebra where λ is the corresponding weight in the finite Lie algebra g. The inner product between affine weights is defined as ( λ| µ) = (λ|µ) + k λ l µ + k µ l λ (A. 6) where the first term on the right hand side is the inner product between finite weights.
At the level of the root system the construction can be seen as follows. The root system ∆ of any finite dimensional Lie algebra g (whose basis is given by the simple positive roots α i ) contains a highest root θ ∈ ∆, such that We can use θ to extend the root lattice Λ g . To this end, we introduce the lattice Π 1,1 spanned by {β 1 , β 2 } whose inner product satisfies We now define the root lattice Λ g of the new algebra g by which is spanned by the new set of simple affine roots In complete analogy to the finite simple Lie algebra g the affine Weyl group W is defined to be the group generated by reflections with respect to the affine roots. As there is an infinity of the latter the Weyl group is infinite as well. We will give further details for the specific case α 1 .
The examples that we will mostly deal with in this work is the affine extension of a 1 , which we shall briefly discuss below.

A.2 Lie Algebra a 1
The affine counterpart to the highest root of a 1 is The null root δ is defined as δ = β 1 = (0; 0; 1) (A.12) The term null root comes from the fact (δ|δ) = 0. Thus, the simple positive roots of a 1 are α 0 = δ − α 1 = (−α 1 ; 0; 1) , and α 1 = (α 1 ; 0; 0) . (A.13) The root system of a 1 contains inifnitely many (imaginary) roots and the explicit expression can be found in [49] such that the positive roots are In analogy to the finite Lie algebras one can introduce the fundamental weights. In the case of a 1 they are given by Every affine weight λ can be decomposed as where λ 0 , λ 1 are the so called Dynkin labels. λ 1 corresponds to the finite Dynkin label corresponding to the associated finite weight λ. λ 0 is related to the level eigenvalue k by The Weyl group W( a 1 ) is generated by two elements s 0 , s 1 which correspond to the reflections with respect to α 0 and α 1 . Their action on affine weights is given as follows From this we immediately see that the action of s 0 changes the grade whereas the action of s 1 does not affect it.
B Representation Theory of sl(2, C) and sl(3, C) In this section we review representations of sl(2, C) and sl(3, C) which are relevant for the discussion in section 4. Our notation mainly follows [51] (see also [52]).

B.1 Irreducible Representations of sl(2, C)
We recall that the Lie algebra sl(2, C) is generated by (H, X, Y ) which satisfy the commutation relations As explained in [51], irreducible representation Γ n of sl(2, C) (with n ∈ N) can be decomposed as Here the one-dimensional eigenspaces V α are eigenspaces of H with weight α, i.e.
while the operators X and Y map from one eigenspace to another as well as Graphically, the structure of V (and the action of all generators) can be represented as follows Furthermore, for given n ∈ N the irreducible representation Γ n can be written as Explicitly, apart from the trivial representation Γ 0 , we present the first few irreducible representations by specifying the weights of the underlying subspaces

B.2 Irreducible Representations of sl(3, C)
Following [51], in order to describe the structure of representations of sl(3, C), we first recall the Cartan-Weyl decomposition where h is the Cartan subalgebra, which is defined as along with its dual (with i = 1, 2, 3) Furthermore we have and the (one-dimensional) root-space g L i −L j is generated by the 3 × 3 matrix E ij whose component (i, j) = 1, while all other entries are zero.
While each H ⊂ h maps each of the g α into itself, we have for the adjoint action ad(X)(Y ) = [X, Y ] (with X ∈ g α and Y ∈ g β ) ad(g α ) : g β −→ g α+β . (B.11) As in the case of sl(2, C), this action can be represented graphically in the form of 'translations' [51]. Indeed, while the subspaces g α can be graphically represented on a two-dimensional (hexagonal) lattice, the adjoint action of a given X ∈ g α acts through translation, e.g. for X ∈ g L 1 −L 3 we have schematically Irreducible representations of sl(3, C) follow a similar pattern: Indeed, as explained in [51], for any two integers n, m ∈ N there exists a finite dimensional irreducible representation V n,m which enjoys a weight decomposition V n,m = V α . The (one-dimensional) subspaces V α are characterised through their weights and are created from the heighest weight subspace V nL 1 −mL 3 through application of the generators E 2,1 , E 3,1 and E 3,2 .
Apart from the trivial representation (m = n = 0), we have the following weight diagrams for Γ 1,0 ∼ = C 3 and its dual Γ 0,1 Γ 1,0 : The multiplicity (i.e. the dimension of the corresponding subspace of Γ m,n ) is (i + 1) for the ith hexagon and min(m, n) for the triangles in the weight diagram. In the above picture we have indicated the double multiplicity of certain weights by .
In the case of m = n (which is the most important for us) the diagram consists of concentric regular hexagons (while for m = 0 or n = 0 it consists of equilateral triangles), e.g.

B.3 Integrable Representations of sl(2, C)
In order to describe the partition functions of compact M-brane configurations we also need (certain) irreducible representations of sl(2, C) (i.e. the affine extension of sl(2, C)). In this appendix we give a brief review of the specific representations required and we refer the reader to [50] for a more rigorous and complete discussion of the representation theory of sl(2, C).
The idea of constructing irreducible representation of affine algebras is similar to their finite counterparts: we start with a highest weight state λ and repeatedly subtract the positive roots, which act as ladder operators. In the notation introduced in appendix A.2, the latter can be written in the form However, we have to take care that the action of a single one of the two roots (i.e. either α 0 or α 1 ) creates a (finite dimensional) irreducible representation of sl(2, C) of the type explained in appendix B.1 and in particular truncates after a finite number of steps. Starting with the highest weight state 25 [n, n, 0] for n ∈ N , (B.14) 25 Notice that the grade of the highest weight state has been chosen to be zero for convenience.    [4,4,0] C Recursive Relation for the Configuration (N, M ) = (2, 1) In this appendix we provide more details on the recursive relation allowing to determine the coefficients c (n) k (τ, ) (introduced in (4.19)) from c (n) k−1 (τ, ) through the action of an operator R + as in (4.24). We also supply as an example the explicit coefficients for k = 1, . . . , 5 for generic n.