Elliptic genera of monopole strings

We obtain the elliptic genera of monopole strings in 5d MSYM. We find agreement with corresponding TST-dual dyonic-instanton single particle indices in 1110.2175. We make use of (2,2) superconformal algebra and its spectral flow, and the agreement can therefore be taken as evidence that monopole string (4,4) sigma models are exactly quantum superconformal.


Introduction
In the recent development of M5 brane theory, the proposal of the M5/D4 correspondence [1,2] plays a central role despite some critical difficulties. The proposal says that a system of M5 branes compactified on the M-theory circle is dual to the dimensionally reduced system of D4 branes whose worldvolume dynamics is described by 5d maximally supersymmetric YM (MSYM) theory. The KK modes after the dimensional reduction might get recovered as solitonic instanton particle states of the 5d MSYM theory. These correspond to D0 branes bound to the D4 branes. Their mass spectrum agrees with the KK momentum where k is the instanton number and R 5 is the M-theory circle radius. For U(1) gauge group, the partition functions of M5 on T 6 and of the dimensionally reduced D4 on T 5 have been computed explicitly. For U(1) gauge group, dimensional reduction does in fact truncates the KK modes and they are not recovered [3]. However if we make an infinitesimal noncommutative deformation of the 5d MSYM theory, the KK modes are recovered as noncommutative instanton particles [4,13].
But the generalization to the nonabelian case is not so straightforward. The gauge coupling constant of 5d MSYM theory is dimensionful and is related to the circle radius R 5 by and, hence, the theory is perturbatively nonrenormalizable. Even with the maximal number of supersymmetries in 5d, it turns out that the theory involves infinities beginning at six-loop order [5]. Therefore, the 5d MSYM theory as a definition of the M5 brane theory has some difficulties at the moment 1 . Nevertheless, in this paper we will find that this 5d MSYM description of M5/D4 is useful at least for BPS states with some remaining supersymmetries.
Below we shall be concerned with the Coulomb branch dynamics of N D4 (M5) branes. The U(N) gauge symmetry is maximally broken down to U(1) N −1 by the vev of the one of the scalar field 2 where v i represents the location of i-th D4 (M5) brane in the 6th direction and we shall order without loss of generality. 1 However, localization computations in 5d SYM theory have produced expected results of parallel M5 branes in flat Euclidean space. See for instance [6,7,8,9,10]. 2 The overall U(1) gauge symmety will not be broken by the vev. In this paper we will not study S-duality for this overall U(1) gauge group, as this will require a separate treatment.
One candidate definition of the M5 brane theory is the DLCQ N =8 quantum mechanics of k instantons, which may be used to compute physical quantities (including net non-BPS contributions) within the k instanton sector. The DLCQ limit of the k D0 branes is described by the N =8 quantum mechanics over the moduli space of k instantons whose metric can be obtained by the ADHM construction of the 5d MSYM theory. Our N =8 quantum mechanics also involves a potential of the form where G r is the triholomorphic Killing vector whose form is determined by the vev of the scalar field. The N =8 supersymmetric completion is uniquely fixed by the moduli space metric g rs and the triholomorphic Killing vector G r . Also in the DLCQ limit, this N =8 quantum mechanics becomes exact, which is argued in [11,12]. Using this DLCQ description, we have computed the 1/4-BPS index partition functions of one (k=1) dyonic instanton [12] and found that the result agrees with that from the 5d MSYM theory based on the localization method [13]. Thus the N =8 quantum mechanics may be used to deal with the KK sector of M5 brane compactified on the M-theory circle if we accept the M5/D4 correspondence. However a direct check of the validity of the proposal for the nonabelian case is not possible since we do not know any direct formulation of the nonabelian M5 brane theory.
An indirect test of the M5/D4 correspondence is the test of duality in 5d MSYM compactified on a circle. If we compactify N parallel M5 branes on a two-torus with radii R 4 and R 5 , we have a large diffeomorphism group SL(2, Z) that acts on the coordinates x 4 and x 5 of this two-torus.
If we dimensionally reduce along x 5 we get a 5d MSYM on a circle with radius R 4 , Yang-Mills coupling constant g 2 Y M = 4π 2 R 5 and gauge group U(N). Let us refer to this as theory A. If we instead dimensionally reduce the M5 brane system along x 4 we get 5d MSYM theory on a circle with radius R 5 and Yang-Mills coupling constant g 2 Y M = 4π 2 R 4 and again the gauge group is U(N) (See [14] for furher details on how corresponding Lie algebras are transformed under this duality.). Let us refer to this as theory B. Now if the M5/D4 correspondence is correct, it would not matter along which circle we dimensionally reduce. Both theory A and theory B would be dual to the same M5 brane theory, and so they would also be dual to each other.
As a check of this duality, we will show how certain 1/4-BPS states are mapped into each other under duality. In theory A we will consider 1/4-BPS dyonic instanton states. We will be mostly interested in the 1/4 BPS states that are associated with the singly connected maximal string F1 (06) from D4 1 to D4 N along the 6th direction. (The digits in parentheses, here (06), will represent the worldvolume directions of branes or strings.) In a more general situation we have a singly connected string from D4 i to D4 j with i < j, whose 1/4-BPS index works as a basic building block of the general 1/4-BPS multi-particle index of dyonic instantons (see Eq. (2.1)). In the decompactification limit R 4 → ∞, the 1/4 BPS index of a singly connected string has been computed in [13] from the 5d MSYM theory based on the localization method.
For definiteness, let us consider the single-particle 1/4-BPS index which corresponds to a singly connected maximal string for U(N) gauge group. With the compact x 4 direction, we may also consider a D2 (046) which will have the finite mass where v is the difference of the vev's of the two associated D4 branes. The number of such D2 branes cannot be fixed due to the finite mass. However in the decompactification limit, the mass becomes infinite, and therefore we can put the number of D2 branes to be zero consistenly with the dynamics in the decompactification limit. We may also compare the mass of D2 with the mass of the F1 (06) connecting the same two D4 branes. This will have the mass and we see that we need R 4 R 5 in order for the D2 to be much heavier than the F1.
On the other hand, if R 4 R 5 , then no matter how large we take R 4 , we will always find that D2 branes are lighter than F1 strings, and then we can no longer consistently put the number of D2 to be zero. One must now consider more general 1/4-BPS states of the singly connected maximal string, which involve relative charges of D2's and k (> 0) D0's at the same time. The dynamics of these generalized 1/4 BPS states has not been fully understood up to now and our study below gives some prediction of their multiplet structures with nontrivial dependence on the relative D2 charges.
In theory A we will consider a dyonic instanton configuration which involves D4 (01234), D0 (0) and F1 (06). But as we argued above, if x 4 is compactified, we must to this system also add an unspecified number of D2 (046) 3 .
To see that theory A is dual to theory B (without resorting to the M5 brane theory), we first perform T-duality along x 4 which maps D4 into D3 of IIB string theory. We then make S-duality. We finally perform T-duality again, now along x 5 to get the D4 brane of theory B.
We will describe these duality maps in more detail in Section 9.2. By TST duality the brane configuration turns into one with D4 (01235), W (05) and D2 (056) where W represents a BPSwave carrying momentum along the 5th circle direction. The decompactification limit R 4 → ∞ of theory A becomes the strong coupling limit of theory B where the M-theory circle is along x 4 .
In theory B, we describe the system from the viewpoint of D2's which correspond to monopole strings wrapped around the 5th circle direction. To this system we also find the added F1 (06) which are TST duals of the added D2 (046). The low-energy dynamics of the monopole strings is governed by a 2d (4,4) nonlinear sigma model whose target space is given by the moduli space of monopoles. In theory B the singly connected maximal D2 (056) (which is the TST dual of the singly connected maximal F1 (06)), the 4(N − 1) dimensional monopole moduli space is described by the Lee-Weinberg-Yi (LWY) metric whose explicit form is known explicitly [16].
In particular for the U(3) gauge group, the relative moduli space corresponds to the Taub-NUT (TN) space which we shall discuss in detail below.
This (4,4) nonlinear sigma model description becomes precise in the DLCQ limit of the 5th circle direction together with the weak coupling limit R 5 R 4 . The monopole moduli space involves U(1) N −1 isometry and the corresponding Noether charges are interpreted as electric charges of F1's connecting from D4 i to D4 i+1 (i = 1, 2, . . . , N − 1). The charge of the overall part will be denoted by Q overall while their relative charges by Q (m) relative (m = 1, 2, . . . , N − 2). Both the left-moving and right-moving Hamiltonians of the 2d sigma model commute with these electric charges, which means that we may refine the elliptic genus by introducing chemical potentials for these charges. We thus defined the refined elliptic genus Here J L is a particular combination of the fermionic R-charge generators of the (4,4) sigma model that has an interpretation of the (2,2) supersymmetries. The (unrefined) elliptic genus Z(q, y) = Z(q, y, 1) is defined by omitting the factor N −2 m=1 x Q (m) relative m inside the trace. We may further refine by inserting x Q overall tot inside the trace. But the elliptic genus will be independent of x tot because states with nonvanishing Q overall give no net contribution to the elliptic genus.
By the projection to zero relative electric charges, we may hope to compare our result with the 1/4-BPS index of dyonic instantons in [13]. In the decompactification limit, we may consistently put number of added D2 to be zero, which on the TST dual side corresponds to projecting down to the zero-charge sector of the elliptic genus. However, the decompactification limit of theory A is the strong coupling limit of theory B. But our moduli space approximation of monopole string requires weak coupling. Nevertheless, we will successfully find a match with the result in [13]. The reason for this, is that our elliptic genus is really independent of the coupling constant R 4 of theory B so we can reliably compute it at weak coupling and then make the comparison with [13] by going to strong coupling.
For U(3) gauge group, our monopole moduli space has the form [17,18] The (refined) elliptic genus of the overall part, R 3 × R 1 , is denoted as Z com (q, y), which is the same as the elliptic genus of R 4 and is independent of x tot under the refinement. The remaining is the (refined) elliptic genus of the TN space. In this note, we shall explicitly compute the (refined) elliptic genus of the TN space and show that where Z EH (q, y) and Z AH (q, y) denote the elliptic genera of the (4,4) sigma model over the Eguchi-Hanson [19] (EH) and the Atiyah-Hitchin [20] (AH) space respectively. For the EH space, there has been a proposal [21] of its elliptic genus which agrees with our explicit computation. We shall further carry out the computation of the refined elliptic genus of the TN space, Z TN (q, y, x).
The projection of the refined elliptic genus to the zero charge sector where we attach the letter B to indicate this is computed in theory B, is shown to agree with the 1/4-BPS index which was obtained for the dyonic instanton [13] in theory A in the decom- For the U(2) gauge group, the elliptic genus of the AH space is relevant for the dynamics of two identical monopole strings. With an appropriate refinement, we shall verify the agreement of the two sides. For the case of U(N) with N > 3, we shall compute the zero charge projection of the elliptic genus of the LWY space by turning on potential related to the vev of another scalar field, which allows a localization in the moduli space, and again find an agreement of the two sides confirming the TST duality.
The charged 1/4-BPS sector of the monopole strings is of interest. The quantum mechanics of the zero mode part does not have this 1/4-BPS generalization. In addition, the overall part, even including its oscillator contribution, does not receive a net contribution from the charged sector. Hence the generalization purely comes from the oscillator modes of the relative part of the 2d sigma model. Hence the relative F1's (06) are coming from the oscillator contribution, where the conventional rigid string interpretation of F1 is broken down. This is something that has a genuinely 2d character in the sense that it is not found in the corresponding moduli space dynamics of monopoles. In the TST dual side, the relative F1's correspond to the relative D2's and the corresponding 1/4-BPS generalization requires the presence of nonzero number of D0's.
Further understanding in this direction will be of interest.

Review of the 1/4-BPS dyonic instanton index
In this section we will review the result that was obtained in [13]. But we will also make a small note on a refinement of the result that they presented by including the dependence of one chemical potential that we denote by y.
In the Coulomb branch of theory A with vev's v i (i = 1, 2, · · · , N ) ordered as described in the introduction, we break the gauge group maximally down to U(1) N −1 (times the center of mass U(1)). We have sectors labeled by the instanton number k = 1, 2, .... In each instanton sector an index is defined as 4 Here H denotes the Hamiltonian of the D4-D0 system, Π i is the U(1) i gauge group generator, with a corresponding chemical potential X i = e −µ i , and J is one of the Cartan generators of the little symmetry group SO(4) × SO(4) ⊂ SO(1, 4) × SO(5) of a massive object in presence of a vev, with corresponding chemical potential y = e 2πiz . To match with the notation in reference [13] we shall take 2πz = γ 2 and J = −2J 2L . We view the other two chemical potentials γ 1 and γ R in [13] as regulators that we take towards zero.
As usual, the index does not directly depend on β which we may take towards zero. This localizes the path integral to a set of saddle points, which enables explicit computation of the index. This has been done in [13] where it was found that the index in the sector (k, N ) is a sum of contributions I {Y 1 ,...,Y N } where {Y 1 , ...., Y N } denotes a set of N Young diagrams, some of which can be empty, and where k boxes are distributed over all these Young diagrams. The contribution from one set of such Young diagrams is given by 5 and to get the index I k,N we shall sum over all such sets. In this formula, the index i is taken from a set Y ⊂ {1, ..., N } which corresponds to Young diagrams that are not empty. We can then pick a box s = (m, n) ∈ Y i at row m and column n, and to it assign and ν im denotes the number of boxes in row m in Y i and ν jn denotes number of boxes in column n in Y j . We define ν im = 0 if there are no boxes at that row, or if the whole Young diagram Y i is empty.
The multi-particle index is given by where we put I 0,N = 1. The expansion parameter is given by q k = e −S(k) where S(k) denotes the Euclidean 5d MSYM classical action evaluated at instanton number k, If we include a graviphoton (which is the up-lift to 5d of the theta parameter in 4d SYM) in the action, this will complexify the Euclidean action. We may then define where τ = i β R 5 In theory B after TST-duality, we have monopole strings. The energy of a BPS wave on the monopole string is fixed by the BPS equation that we derive from the 2d (4,4) sigma model. This BPS equation is given by H = P where H is the sigma model Hamiltonian, and P is the momentum along the monopole string. By the fact that the monopole string is circlecompactified with radius R 5 , it follows that, for BPS states (from the 2d sigma model viewpoint), The expansion parameter in the elliptic genus will again be given by (2.2) with the same τ = i β R 5 . In a more general situation we can in both theory A and theory B also have a real part of the complex parameter τ . From the M5 brane viewpoint this is the τ -parameter of the two-torus spanned by Euclidean time x 0 and x 5 . Note that the S-duality we consider acts on the two-torus which is spanned by x 4 and x 5 and therefore S-duality does not act on the above τ -parameter. From the multi-particle index I N we can extract the single particle index z sp (which is expected to correspond to the index of monopole strings, in a way that we will clarify a bit further below) from the plethystic exponential Following [13], we factor out the divergent factor I com (γ) from z sp and define For generic N we define X ij = e −(µ i −µ j ) which is the chemical potential of an M2 brane (that is, an F1 (06) in the theory A description and a D2 (056) in the theory B description) stretching between M5 branes (D4 branes) i and j. If we have n M2 branes stretched from M5 i to M5 j this will come with chemical potential X n ij . In other words, by expanding z sp in powers of X ij we extract the contribution from n M2 branes between M5 i and M5 j by reading off the coefficient of X n ij .
In the Appendix E we extract the following single particle indices from the general formula z N =3,n=1 (q, y) = 1 + 10 − 6(y + y −1 ) + y 2 + y −2 q + O(q 2 ) (2. 3) The series expansion for z N =2,n=1 matches with what one gets when one expands out the following proposed closed formula [13] z N =2,n=1 = − θ 1 (q, yu)θ 1 (q, yu −1 ) We summarize the theta functions and their modular transformations in the Appendix A. We also can see that Here we have only verified (2.4) up to order q 2 and (2.6) up to order q and for the case when N = 3. But these series expansions were presented at y = −1 to higher orders in q in [13], In this reference the closed formulas (2.4) and (2.5) were verified up to these orders at y = −1, and a series expansion for Z A 0 (q, −1) was extracted as

Some basics of the elliptic genus
Before going into detailed computations of elliptic genera, we recall some basics [23]. The elliptic genus of a (2,2) superconformal 6 2d sigma model can be defined as where we define the left-moving and right-moving Hamiltonians as where H and P are the time translation and space translation generators of the 2d sigma model.
The trace is over all states in the Hilbert space. Here J L is acting non-trivially only on the leftmoving sector. Then due to the insertion of (−1) F R all states with non-vanishing H R mutually cancel out in the elliptic genus, so that only states which saturate the BPS bound contribute. We may think on P as a central charge which appears in the superalgebra. This means that although we insertq H R into the trace, the elliptic genus will only depend holomorphically on q. The BPS bound of H can actually be higher than P if there are other central charges present in the 2d sigma model supersymmetry algebra (see footnote 10). So the bound (3.1) may be lower than the BPS bound. We therefore like to avoid referring to states saturating the bound (3.1) as BPS states. We will instead refer to states which saturate the bound (3.1) as left-moving BPS states.
4 The elliptic genus for R 3 × S 1 e sigma model Let us begin with U(2) gauge group and a single fundamental monopole string. The 2d sigma model that lives on this monopole string has (4,4) supersymmetry and the target space R 3 × S 1 e , which is the same as the moduli space of a single fundamental SU(2) monopole. For a generic radius on the gauge circle S 1 e (in relation to the radius R 5 of the monopole string), there are no left-moving states that carry non-vanishing momentum Q overall along S 1 e . 7 Therefore the elliptic genus can equally well be computed with the target space being replaced with R 4 . The elliptic genus for 2d (2,2) sigma model with flat target space C = R 2 has been obtained in [23]. The result is Here y is the chemical potential associated to the R-charge and u is associated to the global U(1) symmetry of the target space C of the (2,2) sigma model.
For our 2d (4,4) sigma model on R 4 the result becomes The target space R 4 has SO(4) = SU (2) × SU (2) rotation symmetry. The SU(2) R-symmetry of the (4,4) sigma model is generated by the three Kahler forms J I+ . On C 2 these can be realized as selfdual 't Hooft matrices η I+ ij . The other commuting global U(1) symmetry must therefore be generated by η I− ij which acts on both fermions and bosons. Thus the elliptic genus is defined as We notice that the elliptic genus (4.2) agrees with the corresponding index (2.4) in theory A.
We also notice that (4.2) satisfies the spectral flow equation forĉ = c/3 = 2, Existence of left-moving states with non-vanishing momentum Q overall is possible if we also have for example a winding number also equal to Q overall and if the gauge circle radius equals R5. We will not consider such special situations in this paper, but will assume the radii are generic. In that case no winding strings will be left-moving.
Here c denotes the central charge, which for our (4,4) sigma model on 4d target space is given by 4 · 1 + 4 · 1 2 = 6 for four bosons and four fermions. We will return to this spectral flow equation in more detail in the next section. To check that (4.2) satisfies (4.3), it is enough to take m = 1 and use Eq. (A.1).
We notice that two data points are particular simple. Namely at the two points y = 1 and y = −1. Here the elliptic genus reduces to where we have noticed that θ 1 (q, u −1 ) = −θ 1 (q, u). At these data points, the elliptic genus becomes the square of the elliptic genus for the (2,2) sigma model. We can thus be confident about the correctness of the result at these two data points. But we can then uniquely deduce the full elliptic genus for any y just using these two data points and the fact that it shall satisfy the spectral flow equation.
Of course, in this simple example we could just compute the full elliptic genus explicitly.
But for more complicated noncompact target spaces, a direct computation remains an unsolved problem. In this case we will instead rely on the spectral flow, once we can calculate the elliptic genus at just two data points. As we will explain shortly, these two data points will be chosen as y = 1 and y = −1.

Spectral flow and elliptic genus for TN sigma model
We would now like to compute the elliptic genus of 2d (4,4) TN sigma model, as a first step towards our goal of obtaining the elliptic genus associated to two distinct fundamental monopole strings for U(3) gauge group, carrying electric charges as well. For now, we will ignore the refinement associated to these electric charges, and just focus on the TN elliptic genus.
The elliptic genus at y = 1 is the Euler characteristic of TN. Also the elliptic genus at y = −1 and in the limit q → 0 gives us the Hirzebruch signature of TN. These two numbers are known for the TN space, and reflect the fact that there is one unique normalizable selfdual harmonic two-form on TN space [17,18]. In the Appendix C we show that if we use the path-integral to compute the elliptic genus, we encounter regularization problems at all points except y = 1 and y = −1. If we use the Hamiltonian quantization for the oscillator modes (Appendix D), we can circumvent that regularization problem for all values on y. Nevertheless the result we get 8 does not satisfy the spectral flow equation. One might think this could be taken as an 8 Our solution agrees with the proposed solution in [24] Z(q, y) = where xa are the Chern roots associated to the curvature of M2n. Here we like to stress that this formula is indication that the (4,4) sigma model with TN target is not superconformal at quantum level.
However it is superconformal at one-loop since the Ricci curvature of TN space vanishes. What we will show in this paper, is that by using spectral flow and the two data points at y = 1 and y = −1, we are able to find agreement with the single particle indices we reviewed previously for the dyonic instanton, for any value of y. We would like to interpret our result as evidence that the (4,4) sigma model is exactly superconformal, including nonperturbative effects. We thus believe that the formula (5.1) is not applicable for TN target space and that this is related to the non-compactness of TN. We need to be careful with the boundary conditions at infinity.
Further understanding of this point would be interesting.
We will thus assume that the 2d sigma model on TN is exactly superconformal. We will then use the (2,2) superconformal part of the orginal (4,4) superconformal symmetry of the TN sigma model. In particular, one can show that the spectral flow [26] L n → L n + αJ n + α 2 2ĉ δ n,0 is the inner automorphism of the superconformal theory, where α is an arbitrary real parameter.
Since J 0 , which is identified with J L , is integral or half-integral quantized depending onĉ, for n ∈ Z. Using the spectral flow and taking α = m ∈ Z, the RR sector remains to be RR after the spectral flow and it is straightforward to show that The most general solution of this constraint is given by an arbitrary linear combination of whose coefficients are functions of only τ and Namely the solution is applicable for compact spaces even when their first Chern class is non-vanishing. We have in particular reproduced the elliptic genus for CP 2 which was obtained in [25] by using this formula. 9 We define q = e 2πiτ and y = e 2πiz . When it is more convenient, we will view the elliptic genus as a function of τ and z instead of q and y.
The consistency with the spectral flow requires that To show this, we note that To be consistent with the RR spectra, D(l) should be chosen as and then the consistency with the spectral flow requires since the combination L 0 − 1 2ĉ J 2 0 is invariant under the spectral flow. For the elliptic genus of the (4,4) TN sigma model,ĉ = 2 and hence (5.6) involves two independent functions of τ , h 0 (τ ), h −1 (τ ). We also have the boundary data There will be a unique solution to the spectral flow equation which satisfies these boundary data. To show the uniqueness, we need to show that the system of equations has a unique solution (h −1 (τ ), h 0 (τ )). To this end, we note that which can be seen from the following product realizations, In principle then, we can proceed and express the solutions in a rather complicated by solving the above equations for h −1 (τ ) and h 0 (τ ). However, the form of this solution will be rather complicated. It turns out that the solution to the spectral flow equation and the above boundary data can be expressed in a simple form as To show this, it is enough to check that this solutions satisfies the boundary data (which is obvious) and the spectral flow equation. It then follows from uniqueness that this must be identical with the solution that we find by solving for h −1 and h 0 and plugging this back into To show that the solution (5.12) satisfies the spectral flow equation we use Eqs (A.1), (A.2). One may also check explicitly that the first few terms in the series expansions agree for the two different forms of this solution.
One can see that the Witten index Z(τ, 0) = 1 is invariant under the SL(2,Z) modular transformation as required by the RR sector with the fermion boundary condition required by (−1) F term along the time circle direction.
Including z dependence, one finds that Hence we do not have a covariance under the full SL(2,Z) modular transformation.
But under the modular transformation by an element of Γ 0 (2), the above elliptic genus is covariant as Note that the usual SL(2,Z) is defined by where ad−bc = 1 and a, b, c, d ∈ Z. Then Γ 0 (N ) is a subgroup of SL(2,Z) with further conditions a, d = 1 mod N and c = 0 mod N .
To show this modular property, we note where the first and the second are respectively T and ST 2 S which are generators of Γ 0 (2).
Since the computation for the (4,4) EH sigma model is precisely the same (and the Euler number and the Hirzebruch signature are the same), we also conclude that 6 Refined elliptic genus for TN sigma model The (4,4) TN sigma model has a SU(2)×U(1) global symmetries in addition to the SU(2) R symmetry discussed previously. Let us here denote the generator of the U(1) isometry by J U(1) , which is proportional to the momentum Q relative which is conjugate to the 4π periodic fibercoordinate of TN and hence is half-integer quantized. The proportionality constant is yet to be determined. We shall compute the refined index defined by where Z(τ, z) is the elliptic genus for TN given in (5.12).
We note that the action of J U (1) is similar to J L but it acts both fermions and bosons at the same time whereas J acts only on fermions. Since J L and J U (1) commute with each other, they are simultaneously diagonalized. Also within each BPS multiplet, any positively charged state should be paired with a negatively charged state with the same magnitude of the charge.
With this consideration, basically there are two possibilities of the refined elliptic genus: One is Z(τ, z, µ) = 1 2 or the other is Below we shall show that the former is the correct one. Let us first note that, for either possibilities, and Since J U (1) is integral or half-integral quantized for any boundary conditions, J U (1) is independent of the spectral flow. Therefore, Z(τ, z + mτ + n, µ) = (−1)ĉ (m+n) e −πiĉ(m 2 τ +2mz) Z(τ, z, µ) (6.7) Then withĉ = 2, the most general solution reads Using the data (6.5) and (6.6), H −1 and H 0 can be determined as We have then confirmed that the first few terms in a series expansion agree with (6.3). We may also verify that only this solution solves the spectral flow equation since this is the only option where the µ dependence among the phase factors is canceled.
The refined elliptic genus has the transformation rule which defines a weak Jacobi form of weight zero and index one under Γ 0 (2).
We may Fourier decompose the refined elliptic genus as where the first few Fourier coefficients have the q-expansions It now seems to us that we shall identify J U(1) = 2Q relative and then we read off the bound |Q relative | ≤ k (6.9) on the possible relative charges on these left-moving BPS states for a given momentum k/R 5 .
With this choice of normalization, all left-moving BPS states carry integer values on Q relative . This is what we like to have when Q overall = 0 and v 12 = v 23 . We will return to this point (see around the eq's (9.3) and (9.4)) when we further interpret our results.
We see that the expansion of the zero-charge component, Z B 0 , is in agreement with the single particle index Z A 0 in theory A which we presented in eq (2.6) and which was extracted from the U(3) dyonic instanton index.
The sum of all nonzero (positive as well as negative) charge contributions (at x = 1) is Multiplying this by the center of mass contribution (2.4) we find an agreement with the expansion 3). This agreement should not be expected here for the TN space, and in the next section we will explain that this agreement is rather accidental and that it really belongs to the AH space.

Two identical SU(2) monopole strings
The moduli space of two identical massive fundamental monopole strings in U(2) 5d MSYM is We now note the Z 2 projection on the total moduli space M 2 . This Z 2 acts on the 2π ranged The momentum conjugate to this coordinate is the total electric charge. The Z 2 acts simultaneously on the AH space [27] (Eq. (15)) as a translation of a 2π ranged relative angle in AH space, ψ → ψ + π Also note that the latter transformation alone is the discrete isometry of AH space, for which one may refine the elliptic genus. We shall denote this symmetry of AH space by Z AH 2 . The story for the zero mode part is well known [27]. Since the monopole charge quantum number is 2 as we consider two monopole strings, the electric charge quantum number for any threshold bound state is an odd integer 2n + 1 which is relative prime to the magnetic charge.
The wave function for a dyonic string with electric charge quantum number 2n + 1 on the center of mass part of the moduli space is ψ com = e i(2n+1)χ and it picks up the factor of −1 under the above Z 2 transformation. The total wave function shall be invariant under Z 2 so the relative part of the wave function ψ rel which lives on the AH space, must likewise pick up a factor of −1 under Z 2 . In [27] it was shown that there exist such a wave function. It was also shown that there does not exist a zero-mode wave function on AH which is even under Z AH 2 . There will be a corresponding separation of the elliptic genus on AH into odd and even parts under the Z AH 2 , Z AH = Z AH,even + Z AH,odd and the refined elliptic genus will be defined as where µ AH is the chemical potential for the Z AH It is thus the odd part Z AH,odd which contains a contribution from a threshold bound state at zero momentum q k=0 . In other words, this part of the elliptic genus starts as and corresponds to electrically charged monopole strings. Furthermore, there is no threshold bound state which is even under Z AH 2 so we shall have In this paper we are concerned with monopole strings that carry no electric charge. Hence what we are after is Z AH,even .
From the dyonic instanton index, we could extract from the single-particle index, the contribution that comes from two identical SU(2) monopole strings as z N =2,n=2 (q, y) = 8 − 4(y + y −1 ) q + 112 − 72(y + y −1 ) + 16(y 2 + y −2 ) q 2 + O(q 3 ) We now recall that this expression also agrees with z N =2,n=1 (q, y)Z B charged (q, y) that we obtained from the elliptic genus on TN. (The factor z N =2,n=1 corresponds to the center of mass contribution coming from R 3 × S 1 part of the monopole moduli space.). We already said that we expect the elliptic genus on TN to be the same as the elliptic genus on AH. But here we can also see that this correspondence is generalized also for the refined elliptic genera. We see that Z charged (T N ) corresponds to Z even (AH) which in turn corresponds to the elliptic genus of two identical, and electrically neutral SU(2) monopole strings.

Higher rank U(N) gauge groups
We may deform the (4, 4) sigma model by a potential without breaking any supersymmetry.
However such a potential will modify the superalgebra by extra central charge terms 10 . These central charge terms will correspond to U(1) charges which are Lie derivatives along U(1) isometries on the moduli space. If a state has non-vanishing central charge it will not give a net contribution to the elliptic genus. Only right-movers satisfying H = P contribute. Thus if we deform the sigma model so that some states have non-vanishing central charge, this deformation will deform our elliptic genus. Our first priority would be to find a deformation which does not deform our elliptic genus. This turned out to be a difficult task. We were unable to find any such deformation. However if we are interested in the zero U(1) charge sector of the elliptic genus only, then we may indeed tolerate that the elliptic genus can be deformed, since the zero charge sector will be kept intact by such a deformation. This is because the BPS equation will remain H = P on those zero charged BPS states.
The bosonic part of the potential produced by our deformation [28,29] is given by Its eigenvalues are ± √ P 2 + Z 2 and therefore the BPS bound is given by is a tri-holomorphic Kiling vector field on the monopole moduli space. Let us begin with U (3) and consider only Taub-NUT part of the moduli space with metric Here we have a tri-holomorphic vector field K = a∂ ψ where σ 3 = dψ + cos θdφ and ψ is 4π periodic. The potential is V = a 2 g ψψ = a 2 r r + 1 Near the minimum r = 0 the Taub-NUT space is flat R 4 with the flat metric We may define r = ρ 2 /4 for which this metric becomes The potential is that of the harmonic oscillator near ρ = 0, By taking a large the wave function localizes near ρ = 0 where Taub-NUT space can be replaced with R 4 .
Once we have localized to the origin of TN, we may turn off the potential again since the zero charge sector of the elliptic genus does not depend on a, no matter the potential is turned on for sigma model on R 4 or TN. We thus expect that the zero charge sector of the elliptic genus for TN and for R 4 coincide. Indeed we have confirmed that this is true up to order q 3 using Mathematica to expand the corresponding elliptic genera and picking out the zero charge sectors, and x=0 dx 2πix For higher rank U(N) gauge group we again project onto the sector where all the relative charges are zero. The full elliptic genus factorizes as This full elliptic genus is not known for N = 4, 5, · · ·. But we may again introduce a potential picking up contributions for each relative electric charge in the relative part of the moduli space.
Taking the vev's to be large, the wave function localizes to the origin of the moduli space where it is locally on the form R 4(N −2) which we can think of as N − 2 copies of harmonic oscillators on R 4 . From each R 4 we pick up the contribution Z B 0 . Hence the full contribution from the zero charge sector to the elliptic genus becomes Indeed this is the structure we find from the dyonic-instanton index computation given in Eq. (2.5).

The duality between theory A and theory B
We have argued that duality explains why we may identify the elliptic genus of the monopole string in the zero charge sector with the dyonic instanton index. We would now like to explain this duality in more detail. First this duality can be understood quite intuitively from the M5 brane viewpoint as two different dimenional reductions. Unfortunately we do not know a direct formulation of the M5 brane theory and therefore this argument cannot be made very precise.
Nevertheless, this provides a quite simple geometrical picture of the duality. We will therefore begin by describing the duality as dimensional reductions of M5. We will then also describe the duality using string theory S-and T-dualities.

Duality viewed from M5 brane
Let us first consider the U(2) case (N = 2) since this involves only the overall part of the moduli space of the corresponding monopole string. The overall part of the dynamics is described by the 2d sigma model with a flat target space R 3 × S 1 e . Note that the S 1 e here is the gauge circle direction and there is a discrete target-space momentum Q overall along the S 1 e direction. From the viewpoint of the 5d MSYM theory, this target-space momentum Q overall implies the presence of an electric field transverse to the monopole string and along the 123 directions, which is generated by the electric charge distributed along the monopole string location 11 . This is a dyonic string where Q overall represents the total charge of F1 (06) stretched between D4 branes.
The target-space momentum Q overall is carried only by the zero mode part of the 2d (4,4) sigma model, which is clear from the definition where X e is the target-space coordinate of the S 1 e direction. Namely the target-space momentum Q overall does not receive any contribution from the oscillator part of the sigma model.  If Q overall = 0 this is the usual mass formula of a 1/4-BPS dyonic instanton (M2-W threshold bound state), where n is the electric charge and k the instanton number. It is also the energy of a wave with momentum P = k R 5 along the monopole strings in theory B. For the 2d sigma model effective field theory, to get the energy of the wave on the monopole string, we need to subtract the mass M monopole = nM 056 of the static monopole strings. We are then left with the mass of the wave along the monopole strings, which will correspond to the Hamiltonian of the 2d sigma model. If we expand this out, we get 2nM 056 in the non-relativistic limit (which is the limit in which the 2d sigma model is defined). This mass exceeds the momentum P = k R 5 if Q overall is non-zero. States which contribute to our elliptic genus have H = P and therefore states with Q overall = 0 do not contribute. We note that this is true for both k = 0 and k > 0.
For U(N) gauge group when N > 2 we have also a relative part of the moduli space. Let us consider N = 3 to be specific, and where the relative part of the moduli space is the TN space [17,18]. Let us assume generic electric charges q 12 and q 23 . Here q 12 counts the number of oriented F1 stretched between D4 1 and D4 2 which can be positive or negative integer number or zero. Likewise for q 23 . The momentum along the overall gauge direction (which generically is no longer a circle) is given by The quantization we see in Q overall corresponds to the Z-identification on the monopole moduli space (1.9). The momentum along the TN fiber-circle is half-integer quantized due to the 4πidentification of the fiber-circle and is given by The mass of the F1 strings is given by Just as for the U(2) case, here again states with nonvanishing Q overall can not be left-moving, and therefore they do not contribute to the elliptic genus. When v 12 = v 23 and Q overall = 0, we find that Q relative = q 12 is integer quantized. This is what we like to read off from our elliptic genus, and this motivates our choice of normalization of the U(1) charge operator as J U(1) = 2Q relative , which then leads to the bound |Q relative | ≤ k in eq (6.9). 12 Momentum Q relative along the TN space leads to an increased kinetic energy H of the 2d sigma model Hamiltonian. When k = 0 the momentum Q relative does not induce an increased longitudinal momentum P and therefore these states are not left-moving. Therefore states with M 046 > 0 and k = 0 do not contribute to the elliptic genus.
On the other hand, when k > 0 the TN space momentum Q relative does induce a corresponding increased momentum P . So the bound of the elliptic genus H = P can be still satisfied.
We can perform an explicit computation for a sigma model on R 4 = C 2 which is how TN space looks near its origin. There momentum in TN corresponds to common phase rotation in the both factors C 2 × C 2 . We may thus consider sigma model with target C since the two factors are decoupled. Here it is straightforward to verify that H = P can hold for states with a nonvanishing U(1) charge (where the U(1) acts on the phase of C), as can also be seen explicitly by expanding out the final result that we presented in Eq (4.1).
More generally, for N > 2, we have in place of TN the general LWY moduli space metric, which has U(1) N −2 isometries. (It also has an SU(2) isometry, just like the TN space has.) The corresponding momenta along these U(1) directions are relative electric charges, and, for k > 0, we may excite these relative U(1) charges while preserving the equation H = P . So such charged states do contribute to our elliptic genus.

Duality viewed from D3 brane
We have no direct access to the M5 brane theory, and therefore the argument we presented in the previous section cannot be made very precise. Instead we can use TST-duality. We use T-duality along the 4th direction and map theory A on the D4 branes to D3 branes in type IIB string theory. Here we can use S-duality to get another D3 brane theory. Finally we may T-dualize back to get the other D4 brane theory B.
In theory A we have dyonic-instanton BPS states which can be realized in IIA string theory as the brane configuration of N D4 (01234), k D0 (0) and F1 (06). The projection operators of surviving supersymmetries are 12 When v12 = v23 it is generically not possible to put Q overall = 0 unless also Q relative = 0 which means that generically only states with Q overall = Q relative = 0 contribute to the elliptic genus. However when v12 = v23 states with nonvanishing Q relative will contribute. Now when we computed the elliptic genus on TN we did not take into account the Z-identification on the full monopole moduli space. We then got an elliptic genus on TN where some states appear to carry nonvanishing Q relative . To obtain the elliptic genus of the full monopole moduli space, we have to project out states which are killed by the Z-identification. For generic vev's they are all killed, although for exceptional cases, some such states may survive, and if v12 = v23 they will all survive.
where Γ 5 is the 10d chirality matrix Γ 5 = Γ 012346789 (10) but we may also think on this in the M-theory context as the gamma matrix of a 5th direction which will be the M-theory circle. Since all three projections (9.5) mutually commute, we have 4 real remaining supersymmetries after the projection down to the 1/4-BPS states 13 . We assume that the 4th direction is circle compactified. Still the instanton configuration may carry angular momentum in the 123 plane. Including the F1, we have dyonic instantons, which again may carry angular momentum in the 123 plane. When the 4th direction is circle compactified, we shall also include D2 (046).
Let us now apply the TST duality. First we apply a T-duality along the 4th circle direction. Then We now apply the S-duality. The D3 (0123) remains, but as we will explain more fully in a moment, since S-duality permutes the 4th and 5th directions, it takes D1 (04) into F1 (05), while and F1 (06) turns into D1 (06). This configuration corresponds to 1/4-BPS dyons. Again these dyons may involve the angular momentum along the 123 plane, and, if there are D1's (06) in addition, they become F1's (06). S-duality maps the U(N) gauge group into its Langlands dual, which is again U(N). It maps the coupling constant into its inverse , then S-duality maps τ → −1/τ . Thus we obtain after S-duality the coupling constant In this paper we take the passive viewpoint that S-duality acts on τ while we keep the torus and Finally we apply again T-duality along the 5th circle direction. Then D3 becomes D4 (01235), D1 becomes D2 (056), while, for F1 (05), the winding and the momentum along the 5th circle 13 With D4, we have 16 supersymmetries and one 1/4 refers to the fraction of preserved supersymmetries starting from the 16.
direction will be exchanged. Namely the winding (corresponding to the instanton number k in our original configuration) becomes the momentum P (= k) along the 5th circle direction. The corresponding projections read which commute one another leading to 1/4-BPS configurations. In case there are F1's (06) in addition, they remain to be F1's (06) under the second T-duality. The coupling constant of this 5d MSYM theory (theory B) becomes g 2 Y M = 4π 2 R 4 . This was to be expected from the M5 brane viewpoint where we dimensionally reduce along the 4th direction to get theory B. The gauge group remains U(N).
The decompactification limit R 4 → ∞ of theory A corresponds to the strong coupling limit of theory B. But to compute the elliptic genus we need to take the weak coupling of theory B.
However, the elliptic genus depends on q = exp 2πiτ where τ = iβ/R 5 . Hence q is independent of the SYM coupling R 4 in theory B. Furthermore, the elliptic genus is a topological invariant, so it will not change at all if we continuously change from strong coupling (R 4 R 5 ) to weak coupling (R 4 R 5 ). We can thus compute the elliptic genus at weak coupling, and then compare with the dyonic instanton index by a trivial extrapolation of our elliptic genus to strong coupling.
In the decompactification limit of theory A the number of D2 (046) correspond to various superselection sectors. In [13] the sector with no D2 (046) was considered in the decompactification limit.
In the dual theory B this corresponds to the projection of the elliptic genus down to the zero charge sectors. For the overall part we do not have the concept of relative charge. Thus the projection on the elliptic genus becomes on the form This is then the quantity which we have matched with the result in [13] of theory A.

Discussion
We have seen that duality between theory A and theory B can be argued to be a TST-duality, and one may think that we have added nothing new to already well-known dualities in string theory.
However, we used T-duality to relate D4 to D3 and this does not correspond to dimensional reduction of 5d MSYM to 4d MSYM, but rather to dimensional reduction with all the KK modes kept [31]. Therefore our S-duality lives in 5d rather than in 4d since T-duality is mainly a reformulation of the 5d theory. The 5d S-duality from the field theory point of view, has only quite recently been studied [14]. Our result provides one further evidence of 5d S-duality. It may be seen as an extension to include the KK modes, of the corresponding S-duality checks in 4d MSYM for 1/2-BPS dyon states [17,18] for U(3) case, and also of [27] for the U(2) case with two identical monopole strings. 14 One may try to extend our S-duality check by also including the 1/4-BPS dyon states [29] which also will have corresponding dyonic string up-lifts.
In this paper we did not construct any states explicitly. Their quantum numbers and degeneracy are encoded in our elliptic genera and index on the dual side. But an explicit construction of states would be a nice confirmation of our result. For low values of instanton number k one may hope to be able to explicitly construct such periodic dyonic-instanton-monopole-string bound states.
It might be possible to extend our exact computation of the elliptic genus for TN to higher ranks N = 4, 5, · · · since the corresponding LWY moduli space metrics are known exactly.
However, as the central charge c increases as we increase the rank, just knowing two data points y = 1 and y = −1 will not be enough to uniquely determine the elliptic genus from spectral flow.
It would also be interesting to understand the breakdown of the general formula (5.1) for the elliptic genus when the target space is non-compact. For TN this formula gives the correct answer only at the points y = 1 and y = −1 and otherwise it disagrees with the result we get from spectral flow. We believe that the fact that spectral flow gives agreement with the dyonic instanton index, can be taken as evidence that monopole string (4,4) sigma models are quantum superconformal.

Acknowledgement
We would like to thank Stefan Hohenegger, Seok Kim, Kimyeong Lee and Filippo Passerini for discussions. This work was supported in part by NRF Mid-career Researcher Program 2011-

A Theta functions
In this appendix, we present the theta functions and their transformation properties under SL(2,Z). Their basic definitions are as follows: To be more precise, we checked the S-duality on the zero charge sector Q overall = Q relative = 0 only. It will be very interesting to include non-vanishing charged states in our S-duality check. The S-duality of charged states can be tested once we can compute the index of a periodic dyonic-instanton bound to monopole strings.

= 2q
and They have the periodicity properties and Their modular transformation are and Let us first compute Z(τ, 0). We follow the computation in [32]. The elliptic genus at this point is nothing but the Witten index which is independent of β due to its topological nature. (β was introduced in section 2.) In the β → 0 limit, the functional integral splits into an integral over the zero modes and an integral over non constant configurations. The latter can be evaluated in the perturbation theory in β, and its leading term is given by the ratio of the fermion and bosonic determinants, which comes from the Gaussian approximation of the action of the non constant modes. Due to the supersymmetry, the leading zero point energy contributions of non-constant boson and fermions are canceling with each other, leading to Thus in the β → 0 limit, the index has the path integral representation This can be written as where we introduced This then becomes where χ denotes the Euler number of the manifold M d . For case of d = 4, explicitly C Derivation of Z(τ, 1/2) For this computation we use path-integral and we closely follow [33], with emphasis on the regularization problem, which can be avoided at z = 1/2.
For z = 1/2, we do not have the fermion zero mode for the right moving sector. Hence the expansion of the Lagrangian reads to the leading order contribution of τ 2 . The partition function becomes where N d is the standard normalization given by and AP and P denotes respectively the anti-periodic and periodic boundary condition along the Euclidean time circle direction. Let us first note that This follows from the zeta function regularization where we used ζ(0) = −1/2. Then Let us compute the determinant Then one finds The eigenvalue of ∂ + can be evaluated as with the eigenfunction where m, n ∈ Z and we used the periodic boundary condition for the time circle direction. For the antiperiodic boundary condition for the time circle direction, we get Since R +ab is an antisymmetric matrix transforming covariantly under the SO(4) rotation of the tangent space, this can be block diagonalized into the form by an appropriate SO(4) rotation. We would like to evaluate then Since one can use e +imt−ins as a basis instead of e −imt+ins , we have Thus we need to evaluate where ω = m + nτ . The computation of the former is fairly standard. We use (n + a) = −2i sin πa (C. 13) leading to We now compute the second factor. Let us introduce the function σ(τ, z) defined by where Λ(z) is defined by We also introduce κ(z) = (ln σ(z)) = 1 z 16) and the Weierstrass function We also introduce where κ 1 = κ(1/2), e 1 = P(1/2) and It is well known that σ(z) can be expressed in terms of the theta functions by where e − Λ(z) is the regularization term which can be potentially problematic, for instance, with symmetries of the system. Hence, Similarly for the fermion determinant with the anti-periodic boundary condition along the time circle direction, we need to evaluate With the zeta function regularization similar to (C.14), one finds where we use the cancellation of the regularization terms as a mathematical identity. Therefore, and where z k = ir k τ 2 . Integration of the fermion zero modes leads to where the signature is defined by Finally it is well known that τ (M TN ) = 1 for TN space.

D Derivation of Eq (5.1) using Hamiltonian quantization
Here we will quantize the oscillator modes using the Hamiltonian quantization but for the zero modes we use the path integral. For the path integral of the zero modes, we do not distinguish between space and time directions. We can have a constant fermionic zero mode in the path integral only if the fermion is periodic in both space and time directions. This is the case for the right-moving fermions only since the left-moving fermions are twisted by y J L . So only right-moving fermions ψ i + have zero modes. Let us assume the world-sheet metric is Euclidean with complex coordinate w. Then the sigma model action becomes where the world-sheet is the torus and we have the eigenvalues where m and the RR integer mode numbers, and n are integer mode numbers for the time direction to be used in the path integral. For the reduction to quantum mechanics we pick the RR mode number m = 0 sector (rigid string).
Expaning the action in Riemann normal coordinates and defining For the oscillator modes we will now obtain the Hamiltonian. We begin with the fermions for which we find the left and right moving Hamiltonians where the canonical commutation relations imply that for any matrix M ij and operator we have the following commutation relation To find the eigen-energies of the Hamiltonians we thus just need to find the eigenvalues of the corresponding matrices. We off-diagonalize R ij with off diagonal eigenvalues x a . Let us assume target space is a four-manifold. Then a = 1, 2 and We also have The energy eigenvalues now obtained as where N L,R n = 0, 1 are the fermionic number operators in left and right sectors. (We ignore the zero point energy contribution, which in the end will cancel against the bosons' zero point energies.) We then get the fermionic oscillators contribution where we rescaled x a to −2πτ 2 x a . By supersymmetric pairing of all the oscillator modes it is immediately clear that the oscillator mode contribution to the elliptic genus, by including the bosons, is given by For the zero modes we consider the rigid string sector for which we have the quantum mechanics The contribution from the zero modes can be computed from the path integral Z 0 (y) = DψDXe −S 0 with twisted boundary conditions We then get We now use the zeta function regularization to get The full elliptic genus, by including both oscillators and zero modes can be expressed as Z(q, y) = M 2n a=1,2 x a θ 1 (q, ye 2xa ) θ 1 (q, e 2xa ) As a special case, we find the Euler characteristic at y = 1, We see that the limit y → 1 is smooth. A direct computation at y = 1 requires additional fermionic zero modes to be taken into account because there is twisting of fermionic boundary conditions at this point.
Also in the limit q → 0 and at the point y = −1 we find which is the Hirzebruch signature.
We may also notice that in the flat space limit x a → 0 the integrand formally corresponds to the flat space elliptic genus Eq (4.1) with chemical potential u = 1, although this limit is singular. In fact, the elliptic genus at the point u = 1 on R 4 is zero due to extra left-moving fermionic zero modes.

E Application of formula (2.1)
Let us first apply the formula (2.1) to k = 1 for generic N . Let us assume that we have a Young diagram at i = 1 with one box, and no Young diagrams at i = 2, · · · , N . Then We then get where we introduce the quantities following the notation of [13]. We see that I com diverges as γ 1 , γ R → 0.

E.1 Series expansions for N = 3
The relation between the multi-particle and single-particle indices is quite simple if we just expand up to first order in q, the complication starts at quadratic order. After we have extracted the divergent I com , we can put γ 1 = γ R = 0. In terms of the basic building block From the O(x 0 12 ) term we read off the index coming from having no M2 branes stretching between the two M5's. Yet we can have a wave with momentum k in each M5 brane. This amounts to the index of two copies of U (1) MSYM theory. For each copy, this index was found to be 1 for each k = 1, 2, 3, · · ·. Thus we shall have z N =2 = 2(q + q 2 + q 3 + · · ·) + O(x 12 ) At the first few orders we find z N =2,n=0 = 2q + · · · z N =2,n=1 = 1 + 2(2 − y − y −1 )q + · · · z N =2,n≥2 = 0 + 2n(2 − y − y −1 )q + · · · For n = 1 we have added 1 by hand. This corresponds to a single M2 brane (or W-boson) stretched between the two M5's which carries zero longitudinal momentum (or zero instanton number). For n = 1 there is a closed form of the index given by [13] z N =2,n=1 = − θ 1 (q, yu 1 )θ 1 (q, yu −1 1 ) θ 1 (q, u 1 ) 2 Here we keep the chemical potential u 1 = e iγ 1 before we have extracted the divergent I com (y, u 1 ) out of it. If we extract this divergent piece, then we can take γ 1 → 0 in the remaining piece, which then will have the series expansion For N = 3 we may extract the single particle index associated to an M2 brane stretching between M5 1 and M5 3 by extracting the coefficient of x 13 keeping in mind that x 12 x 23 = x 13 .
E.2 Expansion up to quadratic order in q for N = 2 Going to the next higher order in q enables us to make a more convincing check of our result and of TST duality. In this subsection we will therefore expand the index to quadratic order in q but for simplicity we will just consider the case N = 2. We will extract a single particle index to the same order in q. Explicit expressions for the indices at these first few orders in q can be inferred from the general formula (2.1) as I 0 = 1 I 1 I com (γ) = I(µ, γ) + I(−µ, γ) I 2 I com (γ) = 4I com (γ R )I(µ, γ)I(µ − iγ R , γ) +I com (γ)I(µ + iγ R , γ)I(−µ + iγ R , γ) Here we use the short-hand notation γ for (γ 2 , γ R ) and we write µ in place of µ 12 and we define x = e −µ < 1.

+O(q 3 )
It should be noted that the series expansion for z N =2,n=1 agrees with the expansion of the proposed closed formula (2.4).