Quantum Vacua of 2d Maximally Supersymmetric Yang-Mills Theory

We analyze the classical and quantum vacua of 2d $\mathcal{N}=(8,8)$ supersymmetric Yang-Mills theory with $SU(N)$ and $U(N)$ gauge group, describing the worldvolume interactions of $N$ parallel D1-branes with flat transverse directions $\mathbb{R}^8$. We claim that the IR limit of the $SU(N)$ theory in the superselection sector labeled $M \pmod{N}$ --- identified with the internal dynamics of $(M,N)$-string bound states of Type IIB string theory --- is described by the symmetric orbifold $\mathcal{N}=(8,8)$ sigma model into $(\mathbb{R}^8)^{D-1}/\mathbb{S}_D$ when $D=\gcd(M,N)>1$, and by a single massive vacuum when $D=1$, generalizing the conjectures of E. Witten and others. The full worldvolume theory of the D1-branes is the $U(N)$ theory with an additional $U(1)$ 2-form gauge field $B$ coming from the string theory Kalb-Ramond field. This $U(N)+B$ theory has generalized field configurations, labeled by the $\mathbb{Z}$-valued generalized electric flux and an independent $\mathbb{Z}_N$-valued 't Hooft flux. We argue that in the quantum mechanical theory, the $(M,N)$-string sector with $M$ units of electric flux has a $\mathbb{Z}_N$-valued discrete $\theta$ angle specified by $M \pmod{N}$ dual to the 't Hooft flux. Adding the brane center-of-mass degrees of freedom to the $SU(N)$ theory, we claim that the IR limit of the $U(N) + B$ theory in the sector with $M$ bound F-strings is described by the $\mathcal{N}=(8,8)$ sigma model into ${\rm Sym}^{D} ( \mathbb{R}^8)$. We provide strong evidence for these claims by computing an $\mathcal{N}=(8,8)$ analog of the elliptic genus of the UV gauge theories and of their conjectured IR limit sigma models, and showing they agree. Agreement is established by noting that the elliptic genera are modular-invariant Abelian (multi-periodic and meromorphic) functions, which turns out to be very restrictive.

1 Introduction and summary Supersymmetric Yang-Mills theories (SYM) have been of central interest in string theory, especially since the advent of D-branes. In Type II string theories, the worldvolume interactions of BPS Dp-branes at low energies are described by maximally supersymmetric Yang-Mills theories in (p + 1)-dimensions (MSYM p+1 ). These theories have 16 supersymmetries, inherited from the target-space supersymmetries left unbroken by the half-BPS D-branes. For a stack of N D-branes, the gauge group of the MSYM is U (N ). The gauge field arises from the open strings that stretch between pairs of branes, which carry U (N ) Chan-Paton factors when the branes are coincident. The gauge theory is enhanced by the higher-form gauge fields and fluxes present in the string theory target place, which generalize the topological sectors of the theory. Properties of these gauge theories are intimately related to the interactions of D-branes. For example, topological sectors of the gauge theory are interpreted as the bound states of the branes with other objects in the string theory, including other D-branes of various dimensions and the fundamental string [1]. In fact, an entire non-perturbative formulation of M-theory was conjectured to arise from the N → ∞ limit of the N = 16 quantum mechanics MSYM 1 describing the interactions of D0-branes [2].
In this article, we will focus on the two-dimensional (2d) MSYM theories with gauge group U (N ) or SU (N ). In two dimensions, the weakly coupled gauge theory defined by the SYM Lagrangian is inherently the ultraviolet (UV) description, and such theories are asymptotically free. In the infrared (IR), the theory becomes strongly coupled. It is a difficult and interesting question to understand the infrared dynamics of MSYM 2 . Both of the closely related theories with U (N ) and SU (N ) gauge group have been extensively analyzed, and much has been conjectured about their infrared description and quantum vacua [1,[3][4][5][6]. For example, in [3,4], U (N ) MSYM 2 theory was developed into matrix string theory, describing matrix theory compactified on a circle. It was proposed that the N → ∞ limit of this theory should provide a non-perturbative formulation of Type IIA string theory. Using M-theory and string duality considerations, the authors of [3] related the IR limit of MSYM 2 with gauge group U (N ) to the supersymmetric sigma model into the symmetric orbifold Sym N R 8 , identified as the sector of second quantized free Type IIA strings with light-cone momentum p + = N . However, exact computations or quantitative evidence have been elusive -a situation we seek to remedy.
The Lagrangian of MSYM 2 can be obtained by dimensional reduction from 10d N = 1 SYM, and for U (N ) or SU (N ) gauge group it is given by [3,6] (1.1) The bosons X i , the left-moving fermions χα L , and the right moving fermions χ α R are in the 8 v , 8 c , and 8 s representations, respectively, of the Spin(8) R-symmetry. The fields are also in the adjoint representation of the gauge group, so they are valued in u(N ) (su(N )) and can be realized as N × N (traceless) Hermitian matrices for gauge group U (N ) (SU (N )). The theory has N = (8,8) supersymmetry generated by the transformations with 16 fermionic parameters (ǫ α L , ǫα R ). We take the worldsheet directions to be µ = 0, 9. The dimensional reduction of the Lagrangian and the supersymmetry transformations are reproduced in Appendix A.1.
The MSYM 2 theory was observed to have classical vacua determined by the zeroes of the bosonic potential V (X) = g 2 4 [X i , X j ] 2 , which are commuting matrices X i , modulo the Weyl group S N permuting the eigenvalues [1,3]. For the U (N ) theory on the worldsheet R t × S 1 , all the zero-energy configurations of the gauge field correspond to flat connections on the trivial U (N )-principal bundle, so in the quantum U (N ) theory, the gauge field contributes a single trivial zero-energy state to the vacuum wavefunction, as elaborated in [7]. Therefore, it seems natural to conjecture that in the infrared limit, as g → ∞, the theory flows to the supersymmetric sigma model into Sym N (R 8 ), parametrized by the N eigenvalues of the X i and fermionic partners [3]. Similar arguments could be made for the SU (N ) theory, by removing the contributions for the free diagonal U (1) factor of the U (N ) theory, leading to the supersymmetric sigma model into (R 8 ) N −1 /S N as the conjectural IR limit.
However, this is not all of the vacua and therefore not the end of the story. In his analysis of bound states of fundamental strings and D-branes in Type II string theories, Witten [1] argued that the existence of (M, N )-string bound states in Type IIB string theory requires the existence of various supersymmetric vacua for the SU (N ) MSYM 2 . For the worldvolume theory of N D1-branes, the sector with M bound fundamental strings corresponds to a "charge at infinity" in the form of a Wilson loop in the M th tensor power of the fundamental representation of SU (N ) [1]. Therefore, the (M, N )-string is naturally a superselection sector in the 2d quantum theory, and the vacuum in that sector is identified as the discrete θ vacuum [8]  We would like to analyze the classical and the quantum theory, and determine to what extent these predictions hold. The main feature of the MSYM 2 theory which gives rise to some important subtleties is that all the local fields are in the adjoint representation of the gauge group G. In particular, if G has a nontrivial center Z(G), then there are no fields charged under it, so the Z(G) charge cannot be screened, giving rise to superselection sectors labeled by the Z(G) charge. For example, for G = SU (N ), Z(G) = Z N , and there are N superselection sectors. Given a state in some sector, the emanation of a Wilson loop in some representation R of SU (N ) with charge N R under Z N will yield a state in another superselection sector, differing by N R units modulo N . Since there are no fields charged under the center, we can also define the G/Z(G) = SU (N )/Z N theory that has the same Lagrangian. The SU (N )/Z N theory has the θ angle parameter as additional discrete data, and for each of the N choices of θ, the spectrum is a restriction of the SU (N ) spectrum to one of the N superselection sectors. Likewise, one can define the SU (N )/Z K theory for K|N , which will have N/K superselection sectors with the same Z K ⊂ Z N charge for each of the K choices of the θ angle.
Interestingly, when we consider the classical vacua of the SU (N )/Z N theory, we recover a spectrum consistent with the spectrum of relative positions of the (M/D, N/D)-strings. This requires analyzing the topological sectors of the theory. Let's recall that the discrete θ vacua exist for the SU (N )/Z N theory because this gauge group has non-trivial fundamental group π 1 (SU (N )/Z N ) = Z N . Consequently, there are "instanton sectors" of the 2d theory corresponding to the topologically distinct SU (N )/Z N -principal bundles, labeled by elements in π 1 (SU (N )/Z N ) [8,9]. We denote the Z/N Z-valued instanton number by k. As usual, the effect of the θ angle in the path integral is to weigh the k-instanton sector by e iθk in the sum over the instanton sectors. Naturally, the θ angle takes values in the Pontryagin dual of the π 1 of the gauge group, which is Z N once again for π 1 (SU (N )/Z N ) = Z N . The theory at a given θ angle could be explicitly defined by including a surface operator constructed from the integral of a 2-form gauge field, as in [10]. When one puts the SU (N )/Z N theory on the two-torus T 2 , the SU (N )/Z N -principal bundle P N,k over T 2 with instanton number k admits flat connections, with moduli space M N,k , so there are classical zero-energy configurations of the gauge field in each instanton sector. As all of the fields are in one N = (8,8) vector multiplet, the modes supersymmetric to the zero-energy modes of the gauge field are also classically zero-energy field configurations. The moduli space of flat connections M N,k turns out to have complex dimension d − 1, where d = gcd(k, N ). Thus, one expects on general supersymmetry grounds to have a 8(d−1) real dimensional moduli space of vacua for the scalar fields, specifically (d − 1) real moduli for the eigenvalues of each of the scalars X i . Indeed, when d = 1, M N,k is a point, and there is a single classical zero-energy field configuration with all the scalars set to zero. When d > 1, the zero-energy scalar fields take the form with Tr X i = 0, (1.3) in the strong coupling limit g → ∞, and the eigenvalues parametrize (R 8 ) d−1 /S d . When d = N , we are in the trivial instanton sector with k = 0, with the classical vacua described by (R 8 ) N −1 /S N , in agreement with [1,3].
In the quantum theory, the wavefunction of a vacuum state spreads over all classical vacuum configurations, including the disconnected components. Although one expects that the quantum vacua should parallel the classical vacua in theories with high supersymmetry, one might be hesitant to reach this conclusion in our setting as it is a priori unclear how the sum over classical disconnected configurations reproduces the vacua wavefunctions. Nonetheless, the θ angle isolates superselection sectors corresponding to (M, N )-strings, which have string theoretic descriptions strikingly in parallel with the classical vacua, supporting this conclusion. Here, a few relevant studies are crucial in guiding one's intuition. First of all, the SO(8) R-symmetry anomalies vanish for MSYM 2 [1], so there are no anomaly arguments that rule out the existence of the various massive and massless vacua, unlike in theories with less supersymmetry. Also, in [5], it was argued that the IR description of MSYM 2 could not be a non-trivial superconformal field theory with N = (8,8) supersymmetry, as there is no extension of this N = (8,8) supersymmetry algebra to a linear superconformal algebra [11]. 1 This suggests that any scale invariant theories with massless excitations describing the IR fixed points should be free theories, or orbifolds thereof. Lastly, in [6], MSYM 2 was analyzed using discrete light-cone quantization (DLCQ). There, numerical results were obtained in finite resolution of light-cone momentum indicating the absence of normalizable massless states and supporting the existence of a vacuum with mass gap for the SU (N ) theory. By these considerations, the only possible choices for the IR limit of MSYM 2 are massive vacua or orbifolds of free N = (8, 8) sigma models. Given the favorable evidence, we conjecture that the quantum vacuum of the SU (N )/Z N theory with θ = 2πM/N corresponding to the (M, N )-string should be described by the sigma model into (R 8 ) D−1 /S D , and furthermore that the infrared fixed point of the theory with the given θ angle is this sigma model. We note that this description is invariant under the SL(2, Z) S-duality of the Type IIB string theory, which acts on the doublet (M, N ) but leaves D invariant. Also, the vacua of the related SU (N ) theory in one of its N superselection sectors is the vacuum of the SU (N )/Z N theory with the corresponding θ parameter.
We provide strong evidence in favor of our claim by computing the N = (8,8) analog of the elliptic genus -or, index for short -of MSYM 2 for SU (N ) and SU (N )/Z N gauge group, for the latter also including the surface operator specifying the θ-angle parameter. This index is a supersymmetric partition function on the Euclidean flat torus T 2 (with conformal class τ ), which counts states that are BPS with respect to a conjugate pair of right-moving supercharges. The choice of any such supercharge commutes with a Spin(6) subgroup of the Spin(8) R-symmetry, and we can refine the index with equivariant parameters a 1,2,3 = exp 2πiξ 1,2,3 coupling to the Spin(6) subgroup. This refinement keeps track of more information about the spectrum, as well as regulating the otherwise divergent sum over the infinitely many states contributed by the non-compact bosonic zero-modes. This index also agrees with the equivariant elliptic genus of the theory when viewed as a N = (0, 2) supersymmetric theory -from which perspective the Spin(6) symmetry is just a flavor symmetry. Concretely, the index of an SU (N )/Z K theory is defined as the following trace in the Ramond-Ramond (RR) Hilbert space H of the theory, which is a direct sum of K RR Hilbert spaces on the circle, H k , quantized in the given instanton background k: (1.4) Here, q = e 2πiτ , and H L and H R are the left-and right-moving Hamiltonians. We show that the index of the SU (N )/Z N theory with the θ(M ) = 2πM/N vacuum is where D = gcd(M, N ), and I D is the index of the supersymmetric sigma model into Sym D (R 8 ).
Of course, when D = 1, I θ SU (N )/Z N = 1, which is the index of a single massive supersymmetric vacuum. When D > 1, I D /I 1 is the index of the sigma model into (R 8 ) D−1 /S D , since by factoring the diagonal copy of The expressions for I θ SU (N )/Z N (τ |ξ) and I D I 1 (τ |ξ) are obtained through different methods, and it is non-trivial to show that they agree. Thankfully, both sets of functions enjoy multi-periodicity and SL(2, Z) modular invariance, and using these very restrictive properties we are able to establish (1.5) for N ≤ 7. Since the index is an invariant of the theory under renormalization group (RG) flow, which is furthermore a "strong" invariant in the sense that it contains data about the spectrum of the theory, matching the index computed in the UV with the index of our candidate IR fixed point is a powerful indication that the two theories are indeed related by RG flow.
From the SU (N )/Z N index, we infer the index of the SU (N )/Z K theory for any K|N , where the sum is over the N/K values of positive integers m between 1 and N equivalent to M modulo K. The terms being summed over are interpreted as the indices of the corresponding superselection sectors of the theory, and they are consistent with our earlier analysis of the superselection sectors.
Having understood the vacua SU (N ) MSYM 2 , we would like to analyze the U (N ) theory as well. Including the center of mass modes into our considerations of the SU (N ) theory, one can readily conjecture that the U (N ) theory also has vacua described by sigma models into Sym D (R 8 ) corresponding to the (M, N )-strings, as expected from string theory. However, the correct analysis of the full N D1-brane worldvolume theory is somewhat more complicated, and requires some discussion. For a standard 2d U (N ) gauge theory with only adjoint fields, the U (1) degrees of freedom decouple, and the index of the standard U (N ) MSYM 2 can be readily inferred from the SU (N ) index as (1.7) But, this theory is not accurately taking into account the full structure of the (M, N )-string bound states. The true gauge theory describing the full worldvolume theory of the N D1branes is not a standard U (N ) gauge theory, but also has the Kalb-Ramond 2-form gauge field B coming from the Neveu-Schwarz (NS) sector of the string theory. The B-field has an Abelian gauge symmetry generated by a 1-form gauge transformation, under which the trace mode of the U (N )-connection A is also charged. Due to this additional 1-form gauge symmetry, the theory has generalized field content roughly described by U (1) × SU (N )/Z N gauge bundles, and the structures of the classical and quantum vacua are different. Indeed, we find that the U (N ) MSYM 2 with the 2-form B-field has sectors corresponding to the (M, N )strings as sought. The M th sector has a net U (1) generalized electric flux of M units, which is interpreted as the flux of the M F-strings, as well as a θ angle 2πM/N in the SU (N )/Z N sector. When M = 0, the net flux is zero, with correspondingly zero Yang-Mills energy, so the index is readily interpreted as What about the other sectors with M = 0? Although the bundles with non-zero field strength have non-zero Yang-Mills action, these (M, N )-string configurations are still half-BPS in the the string theory target space, and must still preserve 16 supersymmetries! Explicitly, the D1-brane worldvolume theory has non-linearly realized supersymmetries acting on the U (1) center of mass modes, which are the goldstinos of the spontaneously broken translation symmetry in the presence of the D-branes [13][14][15]. The action or energy of this flux should be considered as part of the binding energy of the (M, N )-string, or as the difference in the central charge of the two BPS sectors of the target-space supersymmetry algebra. The binding energy should be attributed to the DBI action [16] in the same sense as the tension of the N D1-branes is, and should be excluded from the vacuum describing the fluctuations of the bound state. In particular, we can modify the definition of the elliptic genus to count states that are BPS with respect to the supercharges preserved by the bound state, essentially by shifting the Hamiltonian by the central charge of the superalgebra. The corresponding BPS states are exactly the configurations with fixed electric flux M and minimal energy. Since the U (1) factor is free, the fields that contribute to the index are unaffected by this modification. Thus, we obtain the index of the U (N ) theory for given sector with We note that this D1-brane index is invariant under the S-duality of the Type IIB string, which is generated by exchanging M and N and shifting M by a multiple of N , all the while leaving D invariant. By an S-duality followed by a T-duality on the circle wrapped by the D-string, the (M, N )-string is mapped to N F-strings bound to M D0-branes [3]. Thus, the index (1.10) is also an index of the N Type IIA F-strings bound to D0-branes. Our result suggests that the world sheet theory of N F-strings bound to M D0-branes in the free string limit g s = 0 is given by the supersymmetric sigma model into Sym D R 8 , and in particular, N/D F-strings bound to M/D D0-branes behave like free strings. The paper is organized as follows. In Section 2, we analyze the structure of topological sectors of MSYM 2 for SU (N ) and U (N ) gauge group, as well as the related SU (N )/Z K and U (N ) + B theories, and determine the moduli space of flat connections and the classical vacua when the spacetime is T 2 . In Section 3 we discuss how the elliptic genus generalizes for SU (N )/Z N gauge theories to include integration over the various components of the moduli space of flat connections. In Section 4, we compute the elliptic genus of SU (N )/Z K MSYM 2 , and infer the elliptic genus for the U (N ) theory with and without the B field. Finally, in Section 5, we compute the elliptic genus of the Sym N (R 8 ) sigma model, and establish some of its properties which allow us to match it to the gauge theory elliptic genus. We also include Appendix A, which spells out some details about the action and supersymmetry transformations of MSYM 2 .

The structure of vacua
Bound states of D1-branes with the F-strings in Type IIB string theory suggest that the MSYM 2 with SU (N ) gauge group should have N superselection sectors, and that the full worldvolume theory of the N D1-branes (with U (N ) gauge group) should have topological sectors labeled by Z [1]. A complete description of the vacua of the MSYM 2 should account for the vacua in these additional sectors as well. Therefore, we will now task ourselves with hunting for them. We will discover that a rich story underlies the various vacua.

Topological sectors
Let's start by focusing on the SU (N ) theory. It was shown in [1] that on a worldsheet with boundary, such as R 1,1 for concreteness, the sector with M F-strings attached to the stack of N D1-branes manifests itself as a Wilson loop "at infinity" in the M th tensor power of the fundamental representation of the gauge group. The vacua of superselection sectors of 2d non-abelian theories have been analyzed a long time ago by Witten [8]. Since MSYM 2 contains only adjoint fields, the center of the gauge group acts trivially on all fields. In particular, the net charge under the center cannot be screened by local fields. For G = SU (N ), the center is Z(G) = Z N . Therefore, we see that the N superselection sectors in the SU (N ) theory are labeled by the background Z N charge. More precisely, the theory has a Z(G) 1-form global symmetry, for which the charged objects are the Wilson loops in SU (N ) representations [10], and the corresponding conserved Z N charge labels the superselection sectors. The creation of a Wilson loop in representation R will act as a domain wall between two superselection sectors of Z N charge differing by the charge under the center (or N -ality) N R of the representation.
We would like to be able to identify and isolate the vacua. This is best done if one declares the gauge group to be G adj = SU (N )/Z N , which we can do since all the fields are uncharged under the Z N center. Indeed, the MSYM 2 Lagrangian (1.1) with the fields taken to be valued in su(N ) does not uniquely define a quantum field theory, since one can declare the gauge group to be any Lie group with Lie algebra su(N ). This choice does not affect the local physics, but determines which non-local operators and instanton sectors are present in the theory. For example, the theory with SU (N ) gauge group has Wilson loops in all SU (N ) representations, whereas the SU (N )/Z N theory only has Wilson loops in representations for which N R ≡ 0, but also has surface operators which have boundary Wilson loops in arbitrary SU (N ) representations (we will revisit these surface operators shortly). Moreover, because π 1 (SU (N )/Z N ) = Z(SU (N )) = Z N , the SU (N )/Z N theory has a total of N instanton sectors. When the worldsheet is R 1,1 , the instanton sectors were described in [8]. More generally, if one considers the SU (N )/Z N gauge theory on a closed Riemann surface Σ, the instanton sectors are the N SU (N )/Z N -principal bundles on Σ, labeled by discrete nonabelian 't Hooft electric flux [9] -or, mathematically, the second Stiefel-Whitney class of the bundle [17] w 2 (P ) ∈ H 2 (Σ, π 1 (G adj )) = H 2 (Σ, Z N ). (2.1) The G adj theory has additional data in the form of the discrete θ angle, which takes values in the Pontryagin dual Z N of π 1 (G adj ). For each of the N choices of the θ angle, the theory isolates a corresponding superselection sector of the SU (N ) theory, and the Hilbert space is a restriction of the SU (N ) Hilbert space to that sector. This structure mirrors the structure of vacua in the closely related pure Yang-Mills theories with SU (N ) and SU (N )/Z N gauge group [18]. The SU (N )/Z N and SU (N ) theories are of course closely related. One can obtain the SU (N )/Z N theory from the SU (N ) theory by gauging the 1-form symmetry generated by the center Z N = Z(SU (N )) [10]. The procedure is illuminating, as it allows one to explicitly construct the surface operator that detects w 2 . One can first enhance the SU (N ) gauge field to a U (N ) gauge field by adding in the trace componentÂ, and then impose the U (1) 1-form gauge symmetry generated byÂ which removes the field strength forÂ and also enhances the allowed gauge bundles to SU (N )/Z N bundles. In the resulting SU (N )/Z N theory, there are no Wilson loops in representations of SU (N ) that transform nontrivially under the center Z N , unless they are the boundary of a surface operator constructed from dÂ, which is now a 2-form gauge field. The closed surface operator e iM Σ dÂ/N (2.3) evaluates to e i2πM k/N for a bundle with 't Hooft flux Σ w 2 = k around the two-cycle represented by Σ. The integral here is schematic, as dÂ is not a globally-defined 2-form, instead one should integrate it as a Deligne-Belinson cocycle (see [10] and references therein). 2 This operator can be inserted into the path integral to obtain the SU (N )/Z N theory with the discrete θ angle equal to 2πM/N . The parameter M is quantized in integer units, as required by invariance under large gauge transformations. Even as classical theories, the G theory and the G adj theory are different. In particular, the G adj theory has additional classical field configurations corresponding to connections on G adj -bundles, even for those which are not G-bundles. Each of these bundles admit flat connections, so the moduli space of classical vacua of Yang-Mills theory on Riemann surfaces is enlarged to include flat connections of G adj -bundles on the Riemann surface. For theories with supersymmetry, one expects zero energy field configurations supersymmetric to flat connections for the non-trivial G adj -bundles. We will describe these configurations in Section 2.2.2, and find a pleasant parallel to the string theory predictions for the vacua.
It is perhaps good practice to say a few words about the definition of a gauge theory with gauge group G and solidify our footing. In accordance with the literature [19], we take a general G-gauge theory to satisfy the following properties: 1. All local fields are in representations of G.

Wilson lines in all representations of G are present.
3. The path integral sums over all G-bundles. There could be additional data that determines weights for the sum over G-bundles.
With these properties, the difference between a G and G/H theory where H ⊂ Z(G) is made explicit. We can go ahead and generalize our above analysis by also defining the SU (N )/Z K MSYM 2 theory with K|N accordingly. The 2d SU (N )/Z K theory has K instanton sectors, weighted by a Z K valued discrete θ angle. Since the theory contains only adjoint fields, the charge under the center Z N/K = Z N /Z K will not be screened, and for each choice of the θ angle the theory will have N/K superselection sectors corresponding to those superselection sectors of the SU (N ) theory with Z N charge congruent modulo K to a given value determined by the choice of θ. Let's return to the U (N ) MSYM 2 . The "standard" U (N ) MSYM 2 has superselection sectors analogous to the SU (N ) MSYM 2 The pure U (N ) Yang-Mills theory in 2d has N superselection sectors [18]. Similarly, a 2d U (N ) gauge theory without fields charged under the center of the gauge group also has N superselection sectors, thus so does U (N ) MSYM 2 . The U (N ) theory has instanton sectors labeled by the integers corresponding to the quantized electric flux (or vortex number) c 1 ∈ H 2 (Σ, Z). Although one might hope to identify these sectors with the (M, N )-string sectors, this turns out to be not quite right. The true theory describing the interactions of N D1-branes is not just the U (N ) MSYM 2 that we described above by the action (1.1), but also has a 2-form gauge field B coming from the restriction of the Kalb-Ramond field present in the NS-NS sector of the string theory target space to the brane worldvolume. The B-field plays a subtle and important role, primarily by enhancing the classical field configurations of the theory. The B-field, being a 2-form gauge field, has Abelian 1-form gauge transformations under which the U (N ) gauge field A also transforms [1], where λ is the 1-form gauge transformation parameter and 1 N is the N × N identity matrix generating the center of the u(N ) algebra. The correct gauge-invariant Lagrangian has the following kinetic term for the gauge field, and F = F + B1 N is the appropriately modified field strength. Writing the U (N ) gauge field as withÂ the U (1) gauge field corresponding to the trace and A ′ the leftover SU (N )/Z N gauge field, we note that the 1-form gauge transformation above acts only on the U (1) gauge field A. Since all of the scalar and fermion fields are in the adjoint,Â only appears in the gauge field kinetic term in the Lagrangian, and therefore none of the rest of the Lagrangian is modified with the inclusion of the B-field, as they are already gauge invariant under the 1form gauge symmetry. The N = (8, 8) supersymmetry remains intact once one modifies the supersymmetry transformations accordingly by replacing F with F. Now, let's consider what gauge bundles the theory has. As can be seem from the equation of motion forÂ, Tr F is constant, and has periods quantized in integer units when we impose the parameter λ generates the gauge group U (1) instead of R [1]. So the theory considered on a Riemann surface Σ has a topological quantum number labeled byc 1 = [Tr F/2π] ∈ H 2 (Σ, Z) corresponding to the generalized U (1) electric flux. For an honest U (N ) theory -without the B-field -the single Chern class c 1 = [Tr F/2π] ∈ H 2 (Σ, Z) would classify all U (N )-principal bundles. A U (N )-bundle can be thought of as the data of a U (1)-bundle and an SU (N )/Z N -bundle, such that the Stiefel-Whitney class of the SU (N )/Z N -bundle [20]. This can be seen at the level of the transition functions for the gauge field. However, in the theory with the B-field, the additional 1-form symmetry enhances the transition functions and generalizes the allowed bundles and connections, as detailed in [10,21]. The resulting generalized U (N )-connection admits an independent 't Hooft flux w 2 in addition to the electric fluxc 1 . This type of gauge bundle would be more accurately described in the language of gerbes or 2-bundles, but we will not need to go into such territory here. Due to the particularly simple 2-group structure, practically speaking we can think of the allowed gauge bundles as U (1) × SU (N )/Z N -bundles, with independently chosen characteristic classes (c 1 , w 2 ) ∈ H 2 (Σ, Z) × H 2 (Σ, Z N ). The classical configurations of the scalar and fermion fields in the theory mimic the configurations in a U (1) × SU (N )/Z N theory. It is important to emphasize that the theory is not a U (1) × SU (N )/Z N gauge theory; for example the operator content -such as Wilson lines and surface operators -is different.
Configurations with Σ Tr F/2π = M correspond to the binding of M F-strings [1]. The M units of flux is interpreted as the NS-NS charge carried by the F-string, and Tr F serves as a source for the B-field in the string target space. The generalized Yang-Mills action (or energy) of the flux is the binding energy of the (M, N )-string, measured as the difference from the mass of the N D-strings. If one considers the theory on the cylinder C = R t × S 1 , the presence of M units of Tr F flux implies that there is a Wilson loop at the boundary. However, this Wilson loop must also be complemented by the B-field to be gauge invariant. This can be seen by noting that the standard U (N ) Wilson loops are not gauge invariant in this theory, instead one has the following surface operators considered in [10], turned on in the sector with M units of electric flux. On a closed surface such as T 2 , the first factor measures the 't Hooft flux in the SU (N )/Z N sector, since T 2 dÂ = T 2 w 2 exactly as for the SU (N )/Z N theory discussed above. Once again, the presence of this term provides a discrete θ angle 2πM/N for the sum over the SU (N )/Z N -bundles. The second factor is simply the Wilson surface operator for the U (1) 1-form gauge symmetry. The "charge" M is nothing but the F-string winding number once again. This closed Wilson surface operator measures the U (1)-valued holonomy of the background B-field. We note that for the U (N ) theory with or without the B field, one can also add a continuous θ-angle term to the action proportional to Tr F or Tr F , or in general a supersymmetric FI parameter. For the theory with the B field, this θ angle is related to the axion of the Type IIB string theory [1]. However we will not consider including this term, as it does not affect the qualitative features of our discussion (or the elliptic genus).

Classical vacua on T 2
Motivated to perform a quantitative check of our conjectures regarding the structure and description of the vacua, we would like to compute the elliptic genera of the MSYM 2 theory with the various gauge groups discussed above. The elliptic genus is a certain supersymmetric partition function on the 2-torus T 2 [22], which counts (with a sign (−1) F ) states in the cohomology of a conjugate pair of right-moving supercharges Q ± R . 3 States in the cohomology correspond to right-moving vacua tensored with left-moving BPS states. Elliptic genera have been extensively used to study N = (2, 2) and more recently N = (0, 2) theories; for a very restricted set of examples see [23][24][25][26]. It is often useful to refine the elliptic genus by other conserved charges in the theory that commute with Q ± R , which allows more information about the spectrum of the theory to be captured. For a theory with at least N = (0, 2) supersymmetry, the elliptic genus can be schematically defined as where J L stands for the generators of left-moving R-symmetry, and f stands for the generators of bosonic flavor symmetries, all commuting with the Q ± R . With this philosophy, the definition of the elliptic genus can be extended to theories with higher supersymmetry, as we will do so for theories with N = (8, 8) supersymmetry in Sections 4 and 5. The trace can be taken in the Ramond or Neveu-Schwartz left-and right-moving Hilbert spaces of the theory on the spatial circle. We will specialize to the Ramond-Ramond sector. The elliptic genus is invariant under deformations of a theory preserving the right-moving supercharges, and therefore is a topological index of theories. In particular, it is invariant under RG flow, which allows it to be computed in the free UV limit of a theory. For example, for Landau-Ginzburg theories it is sufficient to know the contributions from the field content of the theory in the free limit and impose the restrictions on R-and flavor symmetries coming from the superpotential [23].
For gauge theories the elliptic genus can be computed in the free limit of the theory by introducing fugacities for the gauge charges, which amounts to doing the path integral in the presence of a fixed but arbitrary background flat gauge connection, and then imposing Gauss' Law to project onto physical states by integrating over the moduli space of flat connections [24,25,27]. As discussed, for gauge theories with only adjoint fields such as MSYM 2 , one has freedom in choosing the global form of the gauge group. For example, the theory with SU (N ) gauge group differs from the theory with SU (N )/Z K gauge group for any K|N , despite having the same Lagrangian. Since π 1 (SU (N )/Z K ) = Z K , the SU (N )/Z K theory has additional classical field configurations on T 2 , therefore both the moduli spaces of flat connections and the moduli space of classical vacua are enhanced to include various disconnected components. These additional components are crucial for the computation of the elliptic genus for such theories, as the path integral sums over them as well. We note that to compute the elliptic genus of the SU (N ) theory and the U (N ) theory without the B field, we only to integrate over the trivial moduli space of the SU (N ) bundle. However, to compute the elliptic genus of the U (N ) theory with the B field, we need to integrate over the full SU (N )/Z N moduli space. Also, once we have a description of the SU (N )/Z N moduli space, we can infer the SU (N )/Z K moduli space, and compute the elliptic genera for the SU (N )/Z K theories for free. To prime ourselves for computing the elliptic genera, we now turn to a description of the moduli space of flat SU (N )/Z N -connections on T 2 . As an added bonus, we will be able to understand the classical field configurations on T 2 for the various theories discussed, and discover the classical vacua.

Flat connections on SU (N )/Z N -bundles over T 2
A treatment of the moduli spaces of flat connections for SU (N )/Z N bundles was given in [28], where in particular it was shown that the moduli spaces for the topologically non-trivial bundles with structure group G are isomorphic to moduli spaces of trivial bundles for a different structure group G ω . Here, we will give a self-contained, very explicit, and somewhat pedestrian account of the moduli spaces of flat connections on T 2 , specializing to the structure group G adj = SU (N )/Z N .
Flat connections can be solved for by their holonomies, and the moduli space is given by  (2.14). To find the rest of the solutions, we can lift (2.14) to SU (N ), and find solutions there. In SU (N ), we have N equations, has the moduli space of flat connection precisely M N,k . Let's proceed to describe M N,k for given N and k. It will be useful to define d = gcd(N, k), as M N,k will turn out to have complex dimension d − 1. In fact, for given N and any two k 1 and k 2 such that d = gcd(N, k 1 ) = gcd(N, k 2 ), we will have the isomorphism M N,k 1 ∼ = M N,k 2 . This is not a surprise, since the bundles P N,k 1 and P N, To solve (2.15), we can diagonalize either A or B, and obtain the solutions (D m N , S n N ) or (S m N , D n N ), for some mn = k. We note that S N generates the Z N subgroup of the Weyl group S N , and therefore has the same eigenvalues as D N (up to an irrelevant determinant factor). So, S N and D N are conjugate and the solutions (S m N , D n N ) and (D m N , S n N ) are identified by gauge transformations. Also, since we necessarily have gcd(m, N ) = gcd(n, N ) = 1, the solutions for various m, n only reorder the eigenvalues of D N and S N up to an overall cyclic ordering, and are related by the action of the Weyl group. We can partially fix the gauge by choosing m = k and n = 1, and we are left with N 2 solutions in SU (N ) given by ( these N 2 points are also identified by gauge transformations generated by the simultaneous conjugation by D N and by S N ,  25) and the isomorphism M N,d=1 ∼ = M N,k is given by replacing the primitive N th root of unity ω N by its kth power.
Moduli space of bundles with d > 1: The essential observation for the d = 1 cases is that Since the d-dimensional factors commute, one can turn on arbitrary eigenvalues in the corresponding d-dimensional subgroup of the Cartan torus. Explicitly, the solutions are generalized to as one can easily check that (2.29) The unitarity condition fixes (θ s ) i and (θ t ) i to be real, and the determinant condition fixes their sums to zero. Assigning the two holonomies to the spatial (along 1) and temporal (along τ ) directions of the base torus, the moduli space inherits a natural complex structure, and is parametrized by complex coordinates In choosing this presentation of the holonomies, we have used part of the gauge symmetry to write them as products of factors of size N/d and d. We are left with a Z 2 N/d × S d subgroup of the gauge group. To see this, note that as far as the N/d by N/d factor is concerned, the situation is analogous to the d = 1 case, wherein we have used part of the gauge symmetry to order the eigenvalues of S N/d and D N/d up to a cyclic ordering, and there is a remaining Z 2 N/d , generated by simultaneous conjugation by S N/d ⊗ I d and by D N/d ⊗ I d , corresponding to the cyclic reordering of the eigenvalues, which acts on the solutions by identifying Here, T 2 is a copy of the base torus, with the same complex structure.
Once we project to SU (N )/Z N , the coordinates undergo the further identifications, u i ∼ u i + 1 N ∼ u i + τ N , so the solutions are fixed by the Z 2 N/d action above. The moduli space is then Once again, dependence on k is only through d, via the choice of an N/dth root of unity, and we can define where

Classical vacua in instanton sectors
The classical zero energy configurations in the SU (N )/Z N theory are gauge invariant solutions to the BPS equations, as can be seen from the fermionic supersymmetry variations, or directly from the action. In the IR limit as g → ∞, we can think of a particular solution as the data of a flat connection What about the other choices forc 1 ? In the brane picture, the U (N ) + B MSYM theory is the leading approximation to the brane effective action. One can identify the energy of the flux Tr F as the binding energy of F-strings to the D-strings [1,3,16]. These configurations are half-BPS in the target space, so the corresponding state in the MSYM theory should also preserve 16 supercharges. This is indeed the case, as the U (N ) + B MSYM 2 theory has nonlinearly realized supersymmetries, which are the goldstinos of the breaking of translation symmetry in the presence of D-branes [13][14][15], so the supersymmetry variation (A.10) of the fermions is corrected to Here, Θ = (χ, 0) T is the 10d Majorana spinor and 1 N is the generator of the center of the u(N ) algebra. In particular, the BPS equations are generalized to by choosing ǫ 2 = −2ΛΓ 09 ǫ 1 . So, there are BPS sectors with constant F 09 = 2π M N 1 N such that the minimal action configurations discussed above -with constant, commuting X i parametrizing Sym d (R 8 ), which are the solutions to the BPS equations for given bundle with w 2 -preserve 16 appropriately chosen supersymmetries. Therefore, these configurations are "supersymmetric vacua", but in a sector with a different central charge of the superalgebra.
We comment that it would be interesting to pursue the relation between the existence of the nonlinear supersymmetry to the presence of the B field.

Elliptic genera of SU(N)/Z N gauge theories
We now delve into the task set upon in 2.2 of generalizing the elliptic genus when there are additional bundles to consider, such as for SU (N )/Z N theories, or for the U (N ) theory with the B-field. Once again, as explored in [24,25,27], the elliptic genera of 2d gauge theories is a certain path integral on the torus, which due to localization can be calculated by integrating over the moduli space of flat connections. LetG be a simply-connected semi-simple Lie group with a discrete center Z(G). As discussed in the previous section, when one has a Lagrangian with gauge symmetryG and with all fields invariant under some subgroup H ′ of Z(G), one has several distinct choices of theories corresponding to a choice of the global form of the gauge group G =G/H, for each H ⊂ H ′ . These theories will generically have different choices of gauge bundles on the spacetime, and thus the choice of the gauge group will determine which bundles are being summed over by the path integral [19]. For such 2d theories, the elliptic genus is naturally also a sum over the path integrals for the sectors with different gauge bundles, each of which localizes to an integral over the moduli space of flat connections for that bundle. Furthermore, since π 1 (G) = H, each 2d G-gauge theory carries additional discrete data in the form of a θ angle dual to the relevant characteristic class w(P ) of the bundle P , which specifies a weight for the sum over components. So, the elliptic genus can be written schematically as where Z P is the result of the path integral for the sector of the gauge theory with gauge bundle P . Concretely, for 2d SU (N )/Z N theories, there are N SU (N )/Z N -bundles P N,k , and the relevant characteristic class is w 2 (P ) ∈ H 2 (T 2 , Z N ), with k = T 2 w 2 (P k ), so we write where θ takes values in For U (N ) MSYM 2 , there is an analogous but slightly more nuanced story. For a standard U (N ) theory without the B-field, the gauge bundles are U (N )-bundles, which are classified by a single integer characteristic class c 1 ∈ H 2 (Σ, Z). Only the trivial bundle with c 1 = 0 admits flat connections. Since the U (1) degrees of freedom are free and therefore decouple, the elliptic genus is computed as For the U (N ) theory with the 2-form gauge field B, recall that there are additional field configurations corresponding to connections on gauge bundles with G adj = U (1)×SU (N )/Z N structure group. On a Riemann surface, these bundles are characterized by two independent classes, (c 1 , w 2 ), however, only certain bundles will contribute to the elliptic genus. For the theory taken at face value, flat connections are only present when Tr F = 0, but there are still the SU (N )/Z N -bundles with flat connections to sum over, so we have the index where I 1 is the contribution of the free center of mass modes. As a check, note that for the U (1) theory, the sector with M strings attached, which is the (M, 1)-string, has index I 1 . The S-dual (1, M ) string indeed has the same index, if I θ SU (N ) = 1 for θ = 2π/N which we will show to be the case. We can also construct the elliptic genus that sums over each BPS sector (labeled by the M units of flux), To obtain each of the various indices, the crucial object we need to compute is I θ SU (N )/Z N . The computation requires some discussion, which we will now elaborate.

Integration over components of the moduli space of flat SU (N )/Z N -connections
To compute I θ SU (N )/Z N , we need to calculate the path integrals Z N,k for the SU (N )/Z N bundles, so let's analyze them. In general, Z P is the path integral over all connections for P , so we can write Z P = 1 Vol(G(P )) A∈Ω 1 (T 2 ,ad P ) DA Z(A).
The contour integral is determined by the Jeffrey-Kirwan residue operation JK-Res. The integrand is once again moduli space are also periodic on the SU (N )/Z N moduli space for a theory with no fields charged under the center. In particular, the contour integral overM N is just N 2 times the contour integral on M N,0 . So, (3.12) can be simplified as  14) where M N,d as given in (2.33). Once again, the contour is determined by the JK-Res operation. Collecting our results in equations (3.10) and (3.14), the elliptic genus (3.2) is computed by the formula 16) with W N,k = Z 2 N/d × S d . We will elaborate on the residue prescription JK-Res in Section 4.3 as part of the computation of the elliptic genus for MSYM 2 .

Adjoint fields in the presence of background flat connections
To evaluate the contribution to the index from each of the components of the moduli space, we need to analyze how fields behave in the presence of background flat connections, and determine what Z 1−loop (u) is for each component. In line with our end goal, here we will determine Z 1−loop (u) for a theory with all fields in the adjoint representation.
First off, as is well known, background flat connections on T 2 can be interchanged with boundary conditions around the two 1-cycles for fields charged under them. As a simple example, one could keep in mind that the choice of periodic or antiperiodic boundary conditions for fermions is equivalent to the choice of a background flat Z 2 -connection. Correspondingly, the boundary conditions determine the mode expansions of the fields into oscillators. Since the elliptic genus can be computed in the free field limit, the moding in the presence of arbitrary background flat connections can be easily determined by the charges of the fields. Let's start by considering adjoint fields in the presence of a flat connection for the bundle P N,1 over T 2 , described by a pair of SU (N )/Z N holonomies ([S N ], [D N ]). Although the two matrices S N and D N do not commute, their actions by conjugation on N × N matrices commute, since for N even, (3.18) but with the eigenvalue a = b = 0 excluded. We had to be careful in picking the sign of the exponent of q, since we would like our expression to be charge conjugation invariant. This will be necessary later for evaluating the elliptic genus, which is a trace in the Ramond sector. These choices are also invariant under the modular S transformation of the base torus, which amounts to exchanging a and b. To summarize, if the contribution to the path integral of modes with gauge fugacity z = e 2πiu is Ξ(u), the evaluation in (3.10) of Z 1−loop (u) at u ∈ M N,1 is (

3.19)
The result is identical for all bundles P N,k with k ⊥ N ; although the holonomies change to ([S k N ], [D N ]), the action on the Lie algebra is isomorphic -as expected, since they have isomorphic moduli spaces.
Next, we should consider the bundles with moduli spaces of positive dimension. We can study the holonomies ([S N/d ⊗ e ih(θs) ], [D N/d ⊗ e ih(θt) ]) ∈ M N,d , and the result will be the same for all k with gcd(k, N ) = d. Similar to our above discussion, conjugation by S N/d and D N/d furnish d 2 copies of a (N/d) 2 -dimensional representation of Z 2 N/d . Each of the d 2 copies has the usual gauge charges for the adjoint representation of SU (d). Explicitly, the matrices diagonalize the conjugation action, with eigenvalue where (E (d) ) i,j is the d × d matrix with a 1 in the (i, j)th entry and zeroes everywhere else. So, for a flat connection on the torus with these holonomies, the adjoint fields have gauge fugacities ω a N/d q One of the d modes with a = b = 0 and i = j corresponds to the identity matrix, and should be excluded, as above for the d = 1 case. Putting everything together, the contribution from a component of the moduli space isomorphic to M N,d is schematically where the 1/Ξ(0) term serves to remove from the product the mode corresponding to the identity element in the Lie algebra. As a check, we see that this formula reproduces our earlier expression (3.19) for d = 1, and reproduces 1/N times the expression for the integral over the SU (N ) moduli space obtained by [24,25,27] for the integral over the moduli space of the trivial bundle with d = N , as can be seen by lifting M d to SU (N ).

Setup
We are now sufficiently equipped to turn to the computation of the elliptic genus of MSYM 2 . To compute the elliptic genus of a N = (8, 8) supersymmetric theory, it is convenient to pick an N = (0, 2) subalgebra of the N = (8, 8) supersymmetry algebra and express the fields and the Lagrangian in representations of this N = (0, 2) superalgebra. As elaborated in Appendix A, a choice of an N = (0, 2) subalgebra is given by picking two right moving supercharges Q ± R that generate right-moving supersymmetry transformations ε ± R ⊂ ε α R , such that ε ± R (and thus Q ± R ) are eigenstates of a weight of the 8 s representation. To paraphrase the Appendix for convenience, this choice decomposes the R-symmetry group as Spin(8) → Spin(2) × Spin(6) ∼ = U (1) R × SU (4), such that ..,4 ⊂ h * be the weights of the 8 v representation of Spin (8), and let {K j } j=1,...,4 ⊂ h denote the Cartan generators with e i (K j ) = δ i j . A concrete choice of ε ± R is given by the 8 s weights ±r where r = 1 2 (e 1 + e 2 + e 3 + e 4 ), for which U (1) R is generated by the Cartan generator J R = 1 2 (K 1 + K 2 + K 3 + K 4 ). Under such a split, the SU (4) factor commutes with the supercharges Q ± R ; therefore it is a flavor symmetry from the perspective of the N = (0, 2) superalgebra. This allows us to define the index in the Ramond-Ramond sector via the N = (0, 2) flavored elliptic genus [24,25,27] Tr where f A are the Cartan generators of Spin(6) ∼ = SU (4). Generalizing the index to include the θ angle, we obtain Under the decomposition (4.1), the fields decompose into SU (4) representations as (4.5) The Fermi multiplet Λ is the N = (0, 2) vector multiplet, and carries the gauge field strength F 09 (or F 09 for the U (N ) + B theory).
There is also a J-term superpotential Perhaps the easiest way to derive these interactions is from the Lagrangian of 4d N = 4 SYM written in N = 1 supermultiplets. When dimensionally reduced to 2d, we get 2d N = (8, 8) SYM, expressed in N = (2, 2) vector and chiral superfields, denotedΣ andΦ 1,2,3 , respectively, with the N = 1 superpotential descending to the N = (2, 2) superpotential Now, we can decompose the N = (2, 2) multiplets and the N = (2, 2) superpotential into their N = (0, 2) counterparts as described in [30]. The vector multipletΣ decomposes into a chiral multiplet Φ 4 and the Fermi vector multiplet Λ. The chiral multipletΦ A of N = (2, 2) decomposes into a N = (0, 2) chiral multiplet Φ A and Fermi multiplet Ψ A4 , where the Fermi multiplet has E-term ∂Φ A , which reproduces our expression above. For the free U (1) theory, the index as defined vanishes due to the zero mode of λ − and its conjugate, as usual. This is because λ − and its conjugate are in the same eigenstate of bosonic symmetries as the N = (0, 2) supercharges, including the R-symmetry, and have opposite fermion number, so their contributions cancel. But, following [31,32], we can remove the contribution from the problematic zero modes by inserting a factor of J R into the definition of the trace, as we will discuss in detail in Section 5. Then the index is simply the product of the one loop partition functions for each of the superfields where ξ A are holonomies for flat background gauge fields for the SU (4) "flavor'' symmetry, coupling to fields via ρ(ξ) = ρ A ξ A , where ρ is a weight of the fundamental SU (4) representation. The holonomies ξ A satisfy A ξ A = 0, (4.9) which is the determinant constraint of SU (4), or equivalently the superpotential constraint. The Dedekind eta function is defined as (4.10) and the Jacobi theta function is defined as with q = e 2πiτ and z = e 2πiu . Let's recall that in order to compute the index for the interacting gauge theory, one also needs to introduce gauge fugacities, and then impose Gauss' Law, which takes the form of a contour integral. Since the theory is free in the UV, and the index is scale invariant, we can do the computation in the free UV limit, so we only need the contribution from each free field. The integrand of the contour integral for the gauge theory index is then where Ξ(τ |α(u); ξ) is the factor from the modes with charge α in the presence of a background flat gauge connection specified by u, with α(u) as discussed in Section 3.2 for the various components of the moduli space of flat connections. For MSYM 2 , the free field index is . (4.13) Note that we can recover the U (1) index as (4.14) The function Ξ(τ |u; ξ) inherits the following periodicity properties from the theta function θ 1 (τ |u), Ξ(τ |u + a + bτ ; ξ) = e −2πib(2ξ 4 ) Ξ(τ |u; ξ), Ξ(τ |u; ξ 1 + a + bτ, ξ 2 , ξ 3 ) = e 2πib(2u) Ξ(τ |u; ξ), (4.15) as well as the following modular transformation properties, (4.16) These properties imply that the integrand Z 1−loop (τ |u; ξ), and therefore the index is a modular invariant symmetric Abelian (multi-periodic) function of the variables ξ 1,2,3 with modular parameter τ . We will explore such functions in Section 5, and their uniqueness properties will help us match the gauge theory index to the symmetric orbifold index in Section 4.3.

Contribution from isolated flat connections
We are now ready to compute the various contributions to the SU (N )/Z N gauge theory index from the components of M flat . Let's start with the pointlike components, corresponding to isolated flat connections of the bundles P N,k with k ⊥ N . Applying our earlier result (3.19), we have We can now rewrite the contribution to the index as

Integral over flat connections on the trivial bundle
Let's move on to the contributions from components of M flat of positive dimension. We will start with the component corresponding to the trivial SU (N )/Z N -bundle P N,0 , which will be the bulk of our computation. As discussed in Section 3.1, we can lift the integral on the moduli space of flat connections M N,N of P N,0 to an integral on the moduli space of flat SU (N )-connectionsM/S N . This allows us to use the formula obtained by [25] (see also [24,27]) and write the integral in (3.12) as (4.20) where The authors of [25] give a detailed prescription for evaluating the JK-Res operation. Here, we will briefly recall parts of the prescription, and compute the residue. Let r denote the rank of the gauge group, so r = N − 1 here for SU (N ). The integrand Z 1−loop is naturally a meromorphic (r, 0)-form onM, which is the torus h C /(Q ∨ + τ Q ∨ ) ∼ = (C/Z + τ Z) r , where h is the Cartan subalgebra of SU (N ), and Q ∨ is the coroot lattice. We pick u 2 , . . . , u N as coordinates onM and solve for u 1 using the trace constraint i u i = 0. We observe that Z 1−loop is singular along the hyperplanes Let Q A ij ∈ h * denote the weight of the multiplet responsible for the hyperplane H A ij , which are the non-zero roots ij } denote the set of charges of the singular hyperplanes meeting at u * . The collection of points u * where at least r singular hyperplanes intersect is denoted byM sing * . When the charges Q(u * ) of all singular hyperplanes meeting at a point are contained in a half-space of h * , the arrangement of hyperplanes is termed "projective". When there are exactly r singular hyperplanes intersecting at a point, labeled say H j 1 , . . . , H jr , the arrangement is termed "non-degenerate". To evaluate the residue, we need to pick a covector η ∈ h * , which for theories with only adjoint fields specifies a Weyl chamber. For a projective and non-degenerate arrangement, the residue is determined by the operation otherwise. (4.23) Here, Cone(Q j 1 . . . Q jr ) stands for the positive cone generated by the charge rays Q j 1 , . . . , Q jr . When the arrangement is degenerate, so there are more than r singular hyperplanes intersecting, the JK-Res operation is more complicated, as one needs to specify the precise cycle to integrate on. However, for the case of interest for us, whenever the arrangement is degenerate, one can exploit the linearity of the JK-Res operation to determine the cycle relatively easily, as was pointed out in some examples in [25]. In any case, the JK-Res operation corresponds to a particular linear combination of iterated residues, and in our case we will be able express JK-Res explicitly as a somewhat simple prescription of iterated residues.
Let's analyze which poles give non-zero contributions to the sum in (4.20). It simplifies the classification of poles to note that non-zero residues are from points u * where s singular hyperplanes and s ′ zero hyperplanes intersect, such that s − s ′ = r. We see that Z 1−loop has zeroes along the hyperplanes defined by for i = j and B = 1, 2, 3. So, for example, at the N 2 points where the hyperplanes H A i+1,i with i = 1, . . . , N − 1 and some fixed A intersect, there are no other singular or zero hyperplanes intersecting (for generic ξ A ). These points therefore give non-zero contributions as long as η ∈ Cone({Q A i+1,i } i=1,...,N −1 ). However, whenever say H A i,j and H A i ′ ,j intersect, we have u i = u i ′ , at which point there is a double zero in the integrand, and such points don't contribute for generic ξ A .
We note that sets of hyperplanes that contribute a non-zero residue always intersect at N 2 points, and each of these points will contribute identical residues. This is coming from the fact that we have lifted the integral on the trivial SU (N )/Z N -bundle's moduli space to the SU (N ) moduli spaceM, which as we discussed in Section 3.1 is an N 2 → 1 covering. For concreteness, we will continue the integral onM to make direct contact with the literature, and observe that we will obtain N 2 times the integral over the SU (N )/Z N moduli space.
Let's return to the classification of poles. There are some points where a degenerate intersection occurs with the required number of zero hyperplanes for the residue to be nonzero. When this is the case, first of all, we need to determine what order of iterated residues JK-Res corresponds to. A second point that needs attention is as follows. We note that due to the constraint ξ A = 0, the second set of zero hyperplanes N B4 ij can be written as for a, b = 1, . . . , 4. A more suitable choice of coordinates is given by v i = Q i1 (u) = u i − u 1 for i = 2, 3, 4. The intersection points in these coordinates are at First of all, let's note that the integrand is doubly periodic in each of the variables v i under translations by Z + τ Z, so each of the poles contributes the same residue. Shifting the coordinates so that the intersection happens at v i = 0, we need to evaluate in coordinates dual to v i . We pick the convenient choice of η = (−1, −1, −1) in these coordinates. Now, we need to determine which cycle of integration JK-Res corresponds to for this η.
As discussed in (4.20), there could be various such cycles, depending on which sub-chamber η sits in; however, the results are equivalent. By linearity of the JK-Res operation, if we find some cycle of integration such that when applied to the 3-form defined by (4.30) gives the correct residue for each of the linear pieces, as according to (4.23), then it is the right prescription for the degenerate case. Noting that for the four subsets of charges, only Cone(Q 12 , Q 13 , Q 24 ) and Cone(Q 12 , Q 13 , Q 34 ) contain η, the correct cycles are determined as Res , as both evaluate to a + b when applied to ω 234 . Therefore, applying either of the iterated residues to (4.28), we see that it evaluates to ǫ(A, B)/4. Such poles generalize to N > 4 as Young tableaux along pairs (A, B) as one expects.
Another subtlety comes from poles containing "cubes", which starts occurring for N ≥ 8. Concretely, for N = 8, we have a pole at the point points outside of any half-space containing all the other charge vectors, so the arrangement is not projective. As was pointed out in [25], we can deal with this situation by relaxing the constraint on the R-symmetry fugacities (which resolves the intersection into a bunch of projective ones), computing the residues, and then taking the limit ǫ → 0. Relaxing the constraint on ξ A to ξ 1 + ξ 2 + ξ 3 + ξ 4 = ǫ, the singular point is resolved to two points, at v 8 = ξ 1 + ξ 2 + ξ 3 and at v 8 = −ξ 4 = −ξ 1 − ξ 2 − ξ 3 + ǫ with v 2 , . . . , v 7 as before. For η = (−1, . . . , −1), the second point does not contribute, and to obtain the contribution from the first point, we need to calculate (4.33) We can determine possible choices of a cycle of integration for this degenerate arrangement as above, and once again the residue is independent of this choice. One choice is given by with the problematic charge covector which made the arrangement non-projective in the first place, and is separate from the sign coming from zero-hyperplanes discussed above. So, in general we need to keep track of both sources of sign for the residue. Finally, we note that starting N ≥ 16, there are poles containing "hypercubes", with v 16 = ξ 1 + ξ 2 + ξ 3 + ξ 4 . Due to the constraint on ξ A , v 16 = 0 and there is a double zero from N 16,1 and N 1,16 , so such poles have vanishing residue.
We are now ready to compute the contour integral for general N . The contributing poles in any Weyl chamber are classified by certain 4d Young tableaux of size N . 5 A 4d Young tableau is a collection of N "nodes" Y = (y 1 , . . . , y N ) ∈ Z 4 ≥0 , subject to the "stacking" condition: if the node x = (x 1 , x 2 , x 3 , x 4 ) ∈ Y , then so do all the nodes y = (y 1 , y 2 , y 3 , y 4 ) with 0 ≤ y A ≤ x A for all A = 1, 2, 3, 4 [33]. We also require that each node y i have at most 3 non-zero coordinates y A i . We will denote the collection of such 4d Young tableaux of size N by Y N . Each such 4d Young tableau Y of size N describes N ! · N 2 poles of the integrand, at coordinates given by solutions to u i − u j = y A σ(i) ξ A , for some choice of j and the (N − 1)! orderings σ(i) of the remaining u i with i = j. 6 The choice of j is related to the choice of a Weyl chamber; for any choice of η only (N − 1)! · N 2 poles survive the JK-Res operation, corresponding to some fixed j. For concreteness, we fix j = 1 with the convenient choice of η = (−1, −1, . . . , −1) in coordinates (u 2 , u 3 , . . . , u N ). Since the integrand is symmetric in the u i , the (N − 1)! orderings σ(i) contribute identically, cancelling part of the factor coming from the order of the Weyl group. We define v i = Q i1 (u) = u i − u 1 for i = 2, . . . , N , noting the relation u i = 0. Contributing poles are at points v(Y ) given by coordinates v i = y A i ξ A + a + bτ , for a, b = 1, . . . , N . Due to the periodicity structure of the integrand, the sum over a, b is trivial and produces a factor of N 2 .
We introduce the following partial ordering on the nodes of 4d Young tableaux, which keeps track of the stacking of the nodes. The operation JK-Res for a pole Y = (y 1 , . . . , y N ), partially ordered such that y i y j if i < j, is given explicitly by the iterated residue JK-Res (4.36) 5 4d Young tableaux of size N also classify solid (3d) partitions of N , i,j,k n i,j,k = N, where for each nonzero n ijk , there are n ijk corresponding nodes (i − 1, j − 1, k − 1, l), with 0 ≤ l < n ijk . In [33], such partitions are denoted 4d partitions of N . 6 We have picked y A j = 0 which we are free to do for any Y .
The integral over the moduli space is then where we have introduced an auxiliary variable δ to simplify the expressions of the residues. The coefficient ǫ(Y ) is a sign due to degenerate and non-projective intersections, and is determined as follows. Let c 3 (Y ) be the number of nodes in Y with at least 2 nonzero entries in the first 3 coordinates, and let c 4 (Y ) be the number of nodes in Y with exactly 3 nonzero coordinates, or Then, the sign ǫ(Y ) is given by We conjecture that the sum over the residues greatly simplifies to the expression .
This is a highly nontrivial simplification to check analytically, as the summands on the lefthand side grow in number and complexity very quickly in N . Fortunately, the functions on both sides of this equation are very special, and they enjoy some very restrictive properties, which allows us to make some exact statements. Specifically, they are modular invariant symmetric Abelian (multi-periodic) functions of the variables ξ 1,2,3 with the modulus τ and period as in (5.14), of the kind explored in detail in Section 5. This follows from the periodicity and modular transformation properties of Ξ(τ |u; ξ) and I U (1) (τ |ξ); as the integrand (4.21) is such a function, so is the integral. We will explore some key properties of such functions in Section 5, leading up to Lemma 5.1 which states that such functions are completely determined by the rational function in variables a 1,2,3 = exp 2πiξ 1,2,3 obtained by setting τ = i∞ (or q = 0), corresponding to the constant term in the Fourier expansion in q. This dramatically simplifies the effort of checking (4.41), since if we can show the equality for q = 0, the full equality follows exactly! We were able to show this for N ≤ 7 by using Mathematica to simplify the sum over the residues with q = 0. For larger N , up to N ≤ 12, we checked that the pole structure of the rational functions obtained by setting q = 0 on both sides agrees, as well as by performing some numerical checks.

Integral over flat connections on generic bundles
Having computed the integral on the moduli space of the trivial bundle, turns out we can infer the integral on each of the other components of M flat . We first note that we can use the identity (4.18) to simplify the integrand in (3.22), (4.42) We recognize the first factor as the contribution from M N/d,1 . The integral is the same as the integral over M d,d , but with scaled flavor charges ξ → N d ξ. Quoting our result above, we have . (4.43)

Putting the pieces together
Adding up the contributions from each of the components of the moduli space of flat connections, we obtain the index . where D = gcd(M, N ). Thus we establish that the index for the SU (N )/Z N MSYM 2 theory at theta angle θ = 2πM N is equal to the index of the sigma model into (R 8 ) D−1 /S D , providing strong evidence that the IR limit of the gauge theory with the corresponding theta parameter is described by this sigma model.
We can also easily infer the index of the SU (N ) and SU (N )/Z K theories for each K|N with our results thus far. For each such theory, the contributing bundles are a subset of the SU (N )/Z N -bundles, with the moduli space of flat connections lifted appropriately. For the SU (N ) theory only the trivial bundle contributes, so we have the index For each of the K values of θ, the index is the sum of the indices of the N/K superselection sectors of the SU (N ) theory with the same Z K charge. As discussed, the index of the U (N ) theory can be inferred from that of the SU (N ) theory, and is We have used the fact that the U (1) factor is free, and since the field strength does not contribute to the index, Ic 1 U (1) = I 1 in the appropriate topological sector of the supersymmetry algebra. The index summing over all flux sectors (and therefore all BPS sectors) is (4.52) Once again, we note that the D1-brane index is invariant under the S-duality of the Type IIB string, which is generated by exchanging M and N and shifting M by a multiple of N , while leaving D invariant.

Elliptic genera of N = (8, 8) sigma models
We have thus far computed an N = (8,8) analog of the elliptic genus of the SU (N ) and the U (N ) MSYM 2 , and claimed that they are equal to the corresponding elliptic genus of some symmetric orbifolds of the supersymmetric sigma model into R 8 . In this section, we will compute the elliptic genus of the orbifold sigma model, and establish some of its key properties that allow us to match it with the gauge theory elliptic genus.

Elliptic genus of the R 8 sigma model
For brevity, we will denote the supersymmetric sigma model into R 8 by C. C is a free theory. When viewed as a non-supersymmetric theory, C carries 3 Spin(8) flavor symmetry groups, labeled K b , K l , and K r , each acting separately on the 8 real bosons, the 8 real left-moving fermions, and the 8 real right-moving fermions. When viewed as a N = (8,8) supersymmetric theory, these actions are combined into a single copy of Spin(8), K, which is the R-symmetry identified as the rotation symmetry of the target space, with the bosons, the left-moving fermions, and the right-moving fermions transforming in the 8 v , 8 s , and 8 c representations, respectively (up to Spin(8) triality). We can pick the representations of the fields under The philosophy for computing the flavored elliptic genus is to pick an N = (0, 2) supersymmetry, and insert into the trace fugacities for every bosonic charge which commutes with the chosen supersymmetry. We can think of C as an N = (0, 8) theory with R-symmetry K r , which has flavor symmetry K b × K l . Any choice of an N = (0, 2) subalgebra gives the free theory with 4 chiral and 4 Fermi complex N = (0, 2) superfields. The flavored elliptic genus in the RR sector of this theory is then where ξ A andζÃ are eigenvalues of flat background gauge fields for K b and K l corresponding to the Cartan generators K b,A and K l,Ã , with We have used the superscript tildes for the K l Cartan to denote the basis in which the 8 s weights are diagonal. The transformation to the basis in which the 8 v weights are diagonal is given by (5.4) However, this K b × K l flavor symmetry only commutes with the action of a N = (0, 8) superalgebra, and does not respect the full N = (8, 8) supersymmetry of the theory. If we insist that C is indeed an N = (8, 8) supersymmetric theory, there is a single K = Spin(8) R-symmetry, which is not respected by the backgrounds considered above. As described in Section 4.1 and Appendix A.2, once an N = (0, 2) subalgebra of the N = (8, 8) algebra is chosen, the supersymmetry generators Q ± are eigenstates of a corresponding Spin(2) subgroup of K, and there is only a Spin(6) ∼ = SU (4) symmetry commuting with it. In this case, we can define an index with fugacities for the SU (4) ''flavor'' symmetry, which we label K ′ , with K ′ B the Cartan generators of K ′ . But the left-moving fermions and the right-moving supersymmetry generators transform in the same representation, 8 s , of K. So, for any choice of an N = (0, 2) subalgebra, there will be left-moving fermions which are eigenstates of the Spin(2) R-symmetry, and therefore uncharged under the SU (4) flavor symmetry. The index as defined in (5.5) vanishes due to the contributions of these fermion zero modes, as was the case for the free U (1) multiplet as discussed in the paragraph leading up to equation (4.8).
Once again, as is commonly done in the literature, we can remove the contributions from the uncharged fermion zero modes by slightly modifying the index (5.5). This is done by (re)introducing fugacities for symmetries the fermions with problematic fermion zero modes are charged under (so that the modified index has a zero when the fugacities are turned off), taking appropriate derivatives to get rid of the zero, and then turning off the fugacities, as in [32] (see also [31]). We can do this by relating (5.2) to (5.5). First, we identify K b and K l diagonally, and write the reduced N = (0, 8) index .
• (Modularity.) I 1 is modular invariant, i.e. under SL(2, Z) transformations τ → aτ +b cτ +d , we have, It follows from these properties that I 1 is an honest map (H/SL(2, Z))×(C/Z+τ Z) 3 → C, and also a 3 variable Jacobi form (function) of weight 0 and index (0, 0, 0). The periodicity in τ → τ + 1 and ξ A → ξ A + 1 allows for a Fourier expansion, of the form Since the function is holomorphic in q, the coefficients f m (ξ) of q m are unique and well-defined.
But since the f m are meromorphic functions themselves, they might have multiple Fourier expansions. For example, we can easily determine (5.17) The function I 1 | q=0 (ξ) has different Fourier expansions in different regions of convergence of the a A . Now, we can use the periodicity in ξ A → ξ A + τ to find a recursion relation for c(m, l A ), which, when combined with modular invariance, determines I 1 completely given I 1 | q=0 := I 1 (τ = i∞|ξ). Explicitly, we have where χ(ξ) is the SL(4, C) character To see this, note that the periodicity of I 1 under ξ 1 → ξ 1 + τ implies the identity, The case with l A < 0 such that m + l A = 0 should be handled with more care. In that case, for say A = 1, we have The only coefficients that are not determined by these relations are those of the form c(m, 0, 0, 0), implying that the function is determined up to a holomorphic function of q. Requiring the function to be modular invariant fixes this ambiguity, since the only holomorphic modular invariant functions are constants. For I 1 , c(m, 0, 0, 0) = 0 for m > 0, and we obtain (5.18).
It is important to note that our discussion above proves that if any Abelian, modular invariant function f (τ |ξ) with the same period matrix Ω as I 1 (τ |ξ) agrees with I 1 at q = 0, then it must equal I 1 . More generally, we have the following result.
A particularly useful class of such functions for us turn out to be I 1 (τ |N ξ), which satisfy the same properties as I 1 (τ |ξ).

Elliptic genus of the Sym N (R 8 ) sigma model
There are various equivalent methods of computing the partition function Z N of a symmetric product theory given the partition function of the base theory Z 1 . We list three prominent methods here.
• Summing over S N connections and twisted sectors • The DMVV formula [34] Z := 1 + The Hecke operators turn out to be the most straightforward to extract a closed-form expression for Z N , given one for Z 1 . For the index we are interested in, we need to perform the "index operation" to remove zero-mode contributions, like we did to obtain I 1 . Analogous to the case in [32], only the term linear in Z 1 survives this operation, as all the other terms have zeroes of order greater than 1 at b 1 = 1. Thus, Specializing to the sigma model into Sym N (R 8 ), turns out we can simplify further, This last simplification is nontrivial, but can be seen in two ways. One can notice that the q = 0 piece of the two expressions in (5.30) and (5.31) agree, and they are both periodic functions on (C/Z + τ Z) 3 ; therefore they are equal by Lemma 5.1. Alternatively, one can directly compute from the Fourier expansion: (5.32)

Conclusions and future directions
We have computed the elliptic genera of the SU (N )/Z K MSYM 2 and U (N ) MSYM 2 with and without the B field, with each corresponding choice of the discrete θ angle, and matched it with the elliptic genus of a corresponding N = (8, 8) sigma model into a symmetric orbifold of R 8 , which we claim describes the IR fixed point in that sector. While the main focus of this work was in answering questions about the vacua of MSYM 2 , the elliptic genera we have computed as part of our analysis are interesting objects in their own rights. For example, they are related to the supersymmetric partition function of the free second quantized Type IIA string as explored in [34], if one performs the sum over the string winding number N ; where p = e 2πiσ . One needs to modify this expression with an appropriate factor to obtain the T-duality invariant partition function Z(τ, σ|ξ) [34]. T-duality exchanges sting winding number and oscillator number, so acts by interchanging p and q, which can be used to determine Z. One could try to extract information about the strongly coupled limit of the string, which is M-theory, using the topological invariance of this function. It would also be an interesting question to understand the automorphic properties of Z, a la [35]. One might also consider replacing I N with the full D1-brane index I U (N )+B , which in the Type IIA picture sums over the bound states with D0-branes as well. This work was inspired by the 4d-2d correspondence explored in [30], as well as by recent developments in the computation of flavored elliptic genera for 2d gauge theories. In particular, MSYM 2 can be obtained by considering M5-branes on a four-dimensional torus T 4 and letting the volume of the T 4 shrink to zero. On the other hand, considering M5-branes on T 6 = T 2 × T 4 , and compactifying first on the T 2 factor taken to be the worldvolume of the MSYM 2 , one obtains 4d N = 4 SYM. Following the general idea of [30], the elliptic genus of MSYM 2 is then related to the Vafa-Witten partition function of the 4d N = 4 theory on T 4 , as well as to an appropriate supersymmetric partition function of the 6d N = (2, 0) theory on T 6 . We will be exploring this relation in upcoming work. The dimensionally reduced Lagrangian is [3,6] We will summarize the derivation presented in [6], but adopt a "mostly plus" metric signature in contrast. We use the 10 dimensional metric where µ, ν = 0, 9, and i, j = 1, 2, . . . , 8, and η µν = diag(−1, +1). We can write the following where the σ a are the usual Pauli matrices, and the γ i are 16 × 16 SO(8) gamma matrices of the reducible 8 s ⊕ 8 c representation, with the β i satisfying {β i , β T j } = 2δ ij . The 10d spinor Θ is Majorana, and has real components in the Majorana basis we have chosen above, thus we can identify the charge conjugation matrix C = −Γ 0 . Θ also satisfies the Weyl condition Θ = Γ 11 Θ, where Γ 11 = Γ 0 · · · Γ 9 = σ 3 ⊗ I 16 is the 10d chirality matrix, which allows us to write Θ = (χ, 0) T . The 8d chirality matrix γ 9 allows us to decompose further as χ = (χ L , χ R ).
Dimensionally reducing on the 1, 2, . . . , 8 directions, we define scalars X i := A i , and obtain the action After dimensional reduction, they are given by Note that ±r are eigenvalues for the action of the Cartan generator J = 1 2 (K 1 +K 2 +K 3 +K 4 ) on the weightspaces of ±r, where e i (K k ) = δ ik . With this choice, the Spin(8) representations reduce as 16) under the decomposition U (1) R × SU (4) ∼ = Spin(2) × Spin(6) ⊂ Spin (8), where U (1) R is generated by J. The supersymmetry generators are now (εα L , ε α R ) = (ε A L , (ε L ) A , ε ± R , ε AB R ), where A, B = 1, 2, 3, 4 are SU (4) indices for the fundamental representation 4. The field content of the theory is organized into N = (0, 2) superfields as in (4.5) in the main text, with the Lagrangian given by the standard D-terms and the superpotential (4.6).
To get an N = (2, 2) subalgebra, one can to pick l := 1 2 (e 1 + e 2 + e 3 − e 4 ). Then, the vector and axial R-symmetries are determined by The supersymmetries are generated by (ε ± L , ε A L , (ε L ) A , ε ± R , ε A R , (ε R ) A ). In N = (2, 2) superspace, the SU (3) singlets correspond to the components of the vector multipletΣ and its conjugate, and the 3 ⊕3 correspond to the compontents of the chiral fieldsΦ B and its conjugate, with the superpotential as in (4.7).