The Superconformal Index of the (2,0) Theory with Defects

We compute the supersymmetric partition function of the six-dimensional $(2,0)$ theory of type $A_{N-1}$ on $S^1 \times S^5$ in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of $W_N$ algebras and affine Lie algebras of type $A_{N-1}$.


Introduction
A remarkable prediction of string theory is the existence of interacting N = (2, 0) superconformal quantum field theories in six dimensions, which underpins many recent developments in mathematical physics. For example, compactifications on Riemann surfaces lead to a rich class S of N = 2 supersymmetric quantum field theories in four dimensions [1,2]. A consequence of the six-dimensional perspective is that supersymmetric partition functions of these theories on the manifolds S 4 and S 1 ×S 3 are related to conformal field theories [3] and topological field theories [4,5] respectively. Furthermore, compactifications on three-manifolds [6,7] and four-manifolds [8] also lead to interesting classes of supersymmetric theories in lower dimensions and new connections between mathematics and physics.
In this paper, we will consider the supersymmetric partition function of the N = (2, 0) theory itself on S 1 × S 5 , which is closely related to the 6d superconformal index. Although this partition function cannot be computed directly in six dimensions, due to the absence of a useful lagrangian formulation, it has been computed recently using maximally supersymmetric Yang-Mills theory in five dimensions on S 5 [9][10][11] 1 . It relies on the conjecture that the nonperturbative physics of 5d SYM allows us to extract non-trivial dynamics of the 6d (2,0) theory with circle compactification [17,18]. The most important part of the dictionary is that the five-dimensional gauge coupling g 2 is related to the radius β of S 1 by g 2 = 2πβ .
(1.1) so that strong coupling corresponds to large radius. Although this five-dimensional theory is non-renormalizable, it is conjectured that by including non-perturbative contributions in five dimensions, one can capture all of the protected states contributing to the 6d superconformal index. This is remarkable given that the 5d partition function is computed as an instanton expansion in powers of e −4π 2 /β , while the 6d superconformal index is naturally an expansion at large radius in powers of e −β with integer coefficients.
In this paper, we will extend these results to compute the partition function of the (2, 0) theory on S 1 × S 5 , or superconformal index, in the presence of extended defect operators.
One motivation for this is that extended defect operators play an indispensable role in engineering defects of various kinds in compactifications to lower dimensions. As one example, compactification on a Riemann surface with codimension 2 defects sitting at punctures leads to flavor symmetries in the resulting four dimensional N = 2 theory [1,2].
We will concentrate solely on the six dimensional (2, 0) theory of type g = A N −1 , which arises on the worldvolume of N coincident M5 branes. There are codimension 4 defects from intersecting M2 branes and codimension 2 defects from intersecting M5 branes. Our working assumption is that the defects have the following classification: 1. Codimension 4 defects are labelled by a dominant integral weight λ of g. 1 See also [12][13][14][15][16] for closely related works on S 5 partition functions.
3. Codimension 4 defects coincident with a codimension 2 defect of type ρ are labelled by a dominant integral weight of the stabilizer of Im(ρ) ⊂ g.
For our computations, we will assume that defects wrapping S 1 have an effective description in 5d N = 2 SYM theory as: 1. A supersymmetric Wilson line in the irreducible representation of SU (N ) with highest weight λ.

2.
A surface defect obtained by coupling to the three-dimensional N = 4 theory T ρ (SU (N )).
Alternatively, a monodromy defect whose monodromy is labelled by ρ.

A supersymmetric Wilson line in an irreducible representation of the unbroken gauge
group in the presence of the monodromy ρ.
Let us now provide a some more details about the computation. The most general superconformal index or partition function on S 1 × S 5 preserves a single supercharge and its conjugate. It depends on five parameters corresponding to combinations of bosonic charges that commute with the supercharge. In the language of five-dimensional gauge theory on S 5 , these parameters can be understood as follows. Firstly, there is the radius β of S 1 which is related to the five-dimensional gauge coupling g 2 by the formula g 2 = 2πβ. Secondly, there are three squashing parameters ω = (ω 1 , ω 2 , ω 3 ) for the geometry of S 5 . Finally, there is a real mass parameter µ for the adjoint hypermultiplet inside the N = 2 vectormultiplet.
We will furthermore concentrate on a limit where we tune the real mass parameter as follows µ → 1 2 (ω 1 + ω 2 − ω 3 ) . (1. 2) The partition function preserves an additional supercharge in this background and leads to dramatic simplifications in the answer [9,11,19]. In the absence of defects, the partition function has been shown to coincide with the character of the vacuum module of the W Nalgebra with central charge c = (N −1)+N (N 2 −1) (ω 1 +ω 1 ) 2 ω 1 ω 2 . This result has been interpreted recently in the context of chiral algebras [19]. For this reason, we will refer to this background as the 'chiral algebra' limit.
To picture where we can add supersymmetric defects to the calculation it is convenient to picture S 5 as a (S 1 ) 3 fibration over a triangle -see figure 1(a). This is described further in the main text. There are three distinguished circles S 1 (a) that may support supersymmetric Wilson loops and three distinguished squashed spheres S 3 (a) which can support supersymmetric surface defects. In the chiral algebra limit, the circle S 1 (3) plays a distinguished role. We expect quantitatively different results depending on whether or not the supersymmetric defects wrap the particular circle S 1 (3) . The configurations preserving the supercharges of the chiral algebra limit are shown in figure 1(b) and 1(c). Let us discuss each case in turn.
The most general configurations of defects that are supported away from S 1 (3) are shown in figure 1 (b). The summary of our results is as follows: 1. Adding supersymmetric Wilson loops in representations of highest weights λ 1 and λ 2 supported on S 1 (1) and S 1 (2) we find the characters of fully degenerate modules of the W N -algebra.
2. Adding a supersymmetric surface defect of type ρ supported on S 3 (3) we find the character of a semi-degenerate module of the W N -algebra. The non-degenerate modules correspond to the maximal case ρ = [1 N ].
3. Adding a supersymmetric surface defect of type ρ together with supersymmetric Wilson loops in representations of the stabilizer of Im(ρ) ⊂ g with highest weights λ 1 and λ 2 , we find further semi-degenerate modules with a more intricate structure of null states.
This completely exhausts the spectrum of irreducible modules of the W N -algebra described in [20]. From perspective of chiral algebras, our computations provide evidence that combinations of supersymmetric defects orthogonal to the chiral algebra plane are realized by chiral vertex operators for the W N -algebra, as speculated in [19].
Let us now summarize what happens when a codimension two defect is supported on either S 3 (1) or S 3 (2) and hence wraps the distinguished circle S 1 (3) . We will present the result for S 3 (1) as shown in figure 1(c). The result for S 3 (2) is obtained by simply interchanging ω 1 ↔ ω 2 . Furthermore, we focus on supersymmetric surface defects labelled by the partition ρ = [1 N ].
In this case we find 1. For a supersymmetric surface defect supported on S 3 (1) we find the character of the vacuum module of affine su(N ) at level k = −N − ω 1 /ω 2 .
2. Adding a supersymmetric Wilson loop supported on S 1 (1) in a representation of highest weight λ we find the character of an irreducible module of the above with highest affine This is a small generalization of the conjecture for the chiral algebra associated to a codimension 2 defect in [19]. For supersymmetric surface defects of generic type ρ, we would expect to find characters of modules of W (ρ) -algebras, which are obtained from the affine algebra by Drinfeld-Sokolov reduction. From this point of view the W N -algebra is the special case ρ = [N ] corresponding to the absence of a defect. However, we could not evaluate the matrix integrals arising from localization in the generic case.
In summary, it is remarkable that the combinatorics and characters of irreducible modules of a large class of chiral algebras are in 1-1 correspondence with supersymmetric configurations of M2 and M5 branes on S 1 × S 5 .
We now summarize the contents of the paper. In section 2 we explain how to compute the general superconformal index in the presence of defects using localization on S 5 and perform some computations relevant for the chiral algebra limit. In section 3 we evaluate the partition functions with configurations of defects relevant for W N -algebras, as in figure 1 In section 4 we evaluate the partition functions with configurations of defects relevant for affine algebras, as shown in figure 1(c). We conclude in section 5 with a discussion of some interesting directions for further study.

Computational Method
In this section, we fix our notation for the paper and explain the method for computing the superconformal index of interacting (2, 0) theories in the presence of codimension 2 and 4 defects. The reader interested only in the final results and the connections to characters of vertex operator modules of chiral algebras can safely turn to sections 3 and 4.
The maximal bosonic subalgebra is so(2, 6) ⊕ usp(4) and we denote the corresponding Cartan generators by (E, h 1 , h 2 , h 3 , R 1 , R 2 ). In particular, the generator E corresponds to dilatations, (h 1 , h 2 , h 3 ) to rotations in three orthogonal planes of R 6 and (R 1 , R 2 ) to R-symmetry generators or equivalently rotations in two orthogonal two-planes of the transverse R 5 .
In addition, there are supersymmetry generators Q R 2 ,R 2 h 1 ,h 2 ,h 3 labelled by their charges under the bosonic subalgebra. The indices may take the values ± 1 2 but for brevity we denote these by ± in what follows. There are sixteen Poincaré supercharges with h 1 h 2 h 3 < 0 and sixteen conformal supercharges with h 1 h 2 h 3 > 0. In radial quantization, conjugation reverses h 1 , h 2 , h 3 , R 1 , R 2 and hence interchanges Poincaré and conformal supercharges.
The superconformal index can be defined as a trace over the Hilbert space of the theory in radial quantization [21]. For 6d SCFTs, it was first introduced in [22]. Here, we define the superconformal index using the supercharge Q ≡ Q ++ −−− . Although all choices lead to an equivalent superconformal index, this choice has the feature that it is symmetric in the generators h 1 , h 2 , h 3 . The superalgebra generated by this supercharge is with the conjugate supercharge Q † ≡ Q −− +++ . The superconformal index counts states in short representations annihilated by Q and Q † and therefore saturating the bound The superconformal index is then defined by where H Q is the subspace of the Hilbert space in radial quantization that is annihilated by Q and its conjugate Q † . The chemical potentials β, a 1 , a 2 , a 3 , µ (together with the constraint a 1 +a 2 +a 3 = 0) are introduced for the combinations of Cartan generators that commute with Q. F is the fermion number operator and we can take F = 2h 1 by the spin statistics theorem.
This index at generic chemical potentials respects only a su(1|1) subalgebra generated by Q and Q † .
It is often convenient to rephrase the superconformal index as where we have defined together with ω j = 1 + a j . In moving between the two expressions, we have used that states in H Q saturate the bound (2.2). For convergence, we will assume that |p| < 0, |q j | < 0.
In this paper, we will consider an unrefined limit of the superconformal index obtained by specializing the parameters as follows or equivalently p → (q 1 q 2 /q 3 ) 1/2 . This limit was first considered in [23] (see also [19]).
In this limit the index has an enhanced supersymmetry by a second supercharge that we denote Q ≡ Q +− ++− and its conjugate. The remaining combinations of Cartan generators appearing in the superconformal index commute with the extra supercharges. This leads to dramatic simplifications due to additional cancellations between bosons and fermions. It is straightforward to see that the index simplifies to where we have defined q ≡ q 3 and s ≡ q 1 /q 2 . The trace is now over the Hilbert space H Q,Q of states annihilated by the supercharges Q and Q and their conjugates in radial quantization.
It is clear that the plane rotated by h 3 now plays a distinguished role. Indeed, as shown in reference [19] the superconformal index in this limit can be interpreted as a vacuum character of a chiral algebra on this plane. For this reason, we will refer to it as the 'chiral algebra' limit.
Let us consider a simple example: the free tensormultiplet in six dimensions. The superconformal index in this case can be evaluated by first enumerating the single letter contributions, which are summarized in table 1. The superconformal index is then given by where we use the standard definition of the Plethystic exponential. The denominator factors inside the Plethystic exponential come from summing the action of holomorphic derivatives on the single letter contributions. In the chiral algebra limit, the result simplifies to Table 1. The abelian tensormultiplet has a scalar φ in fundamental representation of so(5) R , 16 fermions ψ R1R2 h1,h2,h3 with h 1 h 2 h 2 < 0 and a self-dual three-form flux H. Recalling that E(φ) = 2, E(ψ) = 5/2 and E(H) = 3, the fields commuting with the supercharges Q and Q † and their contributions to the index are shown above. There is also a contribution from a fermionic equation of motion.
In this limit, the index receives contributions only from the scalar φ corresponding to highest weight in the fundamental of so(5) R i.e. (R 1 , R 2 ) = (1, 0) and its holomorphic derivatives in the plane rotated by h 3 . In particular, the index is independent of s. The index is proportional to the vacuum character of the of a free boson in two dimensions, or the vacuum character of u(1).

S 5 Partition Function
For the interacting (2, 0) superconformal theories we do not have a free quantum field theory description and another method must be found to compute the superconformal index. In this subsection, we will summarize the conjecture that the superconformal index of the (2, 0) theory of type A N −1 is captured exactly by a path integral of the maximal supersymmetric Yang-Mills theory on S 5 .
The first claim is that the superconformal index in six-dimensions can be expressed as a path integral on S 1 × S 5 with periodic boundary conditions. The chemical potentials {ω j , µ} of the superconformal index are now reinterpreted as squashing parameters for the geometry of S 5 and expectation values of background R-symmetry gauge fields on S 5 . The radius of The second claim is that this supersymmetric partition function on S 1 × S 5 is captured exactly by the partition function of N = 2 SYM on S 5 with gauge group SU (N ) and gauge coupling g 2 = 2πβ. This theory is non-renormalizable in five dimensions but is expected to have a UV completion by the (2, 0) theory of type A N −1 on a circle of radius β. Although the UV completion may involve new and unknown degrees of freedom, the claim is that the perturbative and non-perturbative spectrum of N = 2 SYM theory on S 5 is sufficient to capture all of the protected states contributing to the superconformal index.
Let us explain in more detail how the chemical potentials {β, ω j , µ} of the superconformal index are identified with parameters of the S 5 partition function.
• ω j are squashing parameters for the S 5 metric -see the final equation of appendix (B.2).
• µ is a real mass parameter for the adjoint hypermultiplet inside the N = 2 vectormultiplet.
For generic values of the parameters, the S 5 partition function preserves two supercharges, Q and Q † , which are identified with those used in the 6d superconformal index.
Finally, we note that the transformation between the superconformal index and the partition function on S 1 × S 5 will likely involve an anomalous background coordinate and Rsymmetry gauge transformation. Thus we can expect them to agree up to a multiplicative factor determined by global anomalies. More precisely, we will find that where F is a finite Laurent polynomial in the parameters βω 1 , . . . , βω 3 and βm that can be determined from the anomaly polynomial of the six-dimensional theory 2 . We will confirm this structure in examples.

Computation of S 5 Partition Function
The partition function Z S 5 can be evaluated exactly using the method of supersymmetric localization or alternatively the refined topological string partition function [9][10][11][12][13][14][15][16]. Here we focus on the former approach. A short review of the localization computation is given in The path integral localizes to a matrix integral over a set of saddle points. The saddle points are classified as follows. Firstly, one of the 5 adjoint scalars has a constant non-zero vacuum expectation value φ = ia. This is the real scalar in the N = 1 vectormultiplet.
In addition, there are non-perturbative instanton saddle points on top of this background.
The instanton saddle points are the self-dual Yang-Mills instantons on CP 2 base of the Hopf fibration S 5 → CP 2 . The 1-loop determinant of the fluctuations around these saddle points 2 It will be demonstrated in forthcoming work that the finite Laurent polynomial F is an equivariant integral of the anomaly polynomial [24].
gives rise to perturbative and non-perturbative contributions to the measure of the matrix integral.
Our crucial assumption is that the one-loop and non-perturbative contributions factorize into contributions from 3 fixed circles of the U (1) 3 isometry group of the squashed S 5 . We will denote these fixed circles by S 1 (i) with i = 1, 2, 3. This factorization can be verified explicitly for the perturbative contributions but has not been demonstrated conclusively for the non-perturbative contributions. Under this assumption, the partition function has the form [10,11]: where as above φ = ia is the N = 1 vectormultiplet scalar expectation value and β = g 2 /2π.
The measure of integration is 12) and the integral domain is over iR N −1 . The contributions Z (i) are copies of the 5d Nekrasov partition function Z Nek (a, m, 1 , 2 ) on S 1 × R 4 1 , 2 where the circle S 1 has radius r. The equivariant parameters (r, m, 1 , 2 ) at each fixed circle are replaced as shown in table 2. 2π/ω 1 ω 2 ω 3 µ + 3 2 ω 1 S 1 (2) 2π/ω 2 ω 3 ω 1 µ + 3 2 ω 2 S 1 (3) 2π/ω 3 ω 1 ω 2 µ + 3 2 ω 3 Table 2. The S 5 partition function is constructed from three copies of the 5d Nekrasov partition function with the parameters identified as above. The arguments in the final three columns may be taken modulo ω j in the row corresponding to S 1 (j) The Nekrasov partition function Z Nek (a, m, 1 , 2 ) on S 1 × R 4 1 , 2 can be expressed as a supersymmetric index with both perturbative contributions from fundamental BPS particles and non-perturbative contributions from BPS instanton particles [25,26]. It is constructed from moduli space of k U (N ) instantons M k,N which carries an action of U (1) 2 × U (N ) corresponding to rotations in two orthogonal planes of R 4 and gauge transformations. The parameters 1 , 2 and a = (a 1 , . . . , a N ) are equivariant parameters for these symmetries. It is well-known that the fixed points with respect to this action are labelled by an N -tuple of Young tableaux Y = (Y 1 , . . . , Y N ) such that the total number of boxes is given by | Y | = k.
The starting point for computing the partition function is the equivariant Chern character of the universal bundle E evaluated at the fixed point corresponding to Y [27,28]: where we define + ≡ ( 1 + 2 )/2 and (2.14) Then the equivariant index of the tangent bundle T M k,N at the critical point labelled by Y is given by [27,28] Ind where m is the equivariant parameter for the flavor symmetry of the hypermultiplet. As for the vector multiplet, this index can also be divided into the perturbative contribution, the term without V, and the instanton contribution, the other terms.
We use the conversion rule from the equivariant index to the partition function at the fixed point where the integer n i denotes the degeneracy for given weight ω i . Also, we should keep in mind the momentum factor t∈Z e 2π r t along the temporal circle in five-dimensions. It should be put on each equivariant index. The instanton partition function then becomes where s denotes a position of a box in the Young tableau Y i and h i (s) and v j (s) are the distance from s to the right and bottom end of i-th and j-th Young is the instanton fugacity.
The perturbative contribution can be computed in the similar manner. However, there is an ambiguity in computation of the perturbative partition function associated to boundary conditions of R 4 at infinity. Studying the correct boundary conditions goes beyond the scope of this paper. Here we simply take the average of the equivariant index with its charge conjugation and compute the perturbative contribution, which makes the equivariant index invariant under the charge conjugation. This choice is rather convenient for seeing the simplifications that occurs in the chiral algebra limit at the level of the Nekrasov partition function.
We also need to regularize the infinite products involved in the perturbative contribution. To regularize them, we shall use Barnes' multiple gamma functions defined in [29] as follows: and Barnes' zeta functions defined by the series Therefore the regularized perturbative partition function is given by and put the prime on it to deal with the zero modes at z = 0 such thatΓ (0) = lim z→0 zΓ(z).
It is often convenient to rewrite the perturbative contribution in the more concise expres- is the multiple q-Pochhammer symbol and p ≡ e −r 1 , q ≡ e −r 2 and the prime denotes the zero modes at x = 0 are absent.
As explained above, the chiral algebra limit of the 6d superconformal index is approached by tuning the real mass parameter µ → 1 2 (ω 1 + ω 2 − ω 3 ) [19,23]. Using that the contributions from the circle S 1 (j) are periodic in ω j we can deduce that from the point of view of each fixed point this is equivalent to the limits where ± = (± 1 + 2 )/2. These points correspond to an enhancement of supersymmetry and lead to additional cancellations.
First consider the limit m = ± − . We note that there is a universal center of mass factor multiplied to each non-zero instanton number contribution taking the form of and it vanishes in both cases m = ± − . Therefore the instanton contribution is unity in this limit. Furthermore, the perturbative contributions simplifies to a product of sine functions, so that which we get from (2.22) using the identities of Γ 3 . This formula is derived for U (N ). For U (1) the perturbative and instanton contributions are trivial. Thus we expect that the answer for SU (N ) is simply obtained by imposing the condition a 1 + · · · + a N = 0 on the above formula.
Second consider the limit m = ± + . In this case, the contributions to the index from the vectormultiplet and adjoint hypermultiplet cancel out exactly. Thus the perturbative contribution is unity and the instanton contribution simply counts the number of fixed points.
Therefore we have (2.27) In this simple limit, we expect that the SU (N ) answer is obtained by dividing by the partition (2.28)

Codimension 4 Defects
It is expected that the six-dimensional (2, 0) theories have codimension four surface defects where n ∈ Z is the abelian charge and dτ and ds j are line elements along S 1 and S 1 (j) respectively. B denotes the two-form gauge field with self-dual curvature H = dB and Φ is one of the five scalars in the abelian tensormultiplet. The scalar Φ is characterized as a singlet under the generators R 1 and R 2 . Note that the Dirac quantization in 6d CFT forces the charge n to be an integer.
Turning off all of the chemical potentials in the superconformal index, it is straightforward to check that the supersymmetric Wilson surface supported at the fixed point S 1 (j) always preserves the two supercharges Turning on the general chemical potentials in the superconformal index, the Wilson surface respects only Q = Q ++ −−− and its conjugate. This means we can still define and compute the superconformal index with generic chemical potentials in the presence a Wilson surface on any S 1 × S 1 (j) . However, the Wilson surface commutes with the second supercharge Q ≡ Q +− ++− preserved in the chiral algebra limit only if it wraps S 1 (1) or S 1 (2) . From the six-dimensional perspective, these cases correspond to codimension four defects transverse to the chiral algebra plane. We will concentrate only on these cases in the present work.
The dimensional reduction of the Wilson surface operators along the S 1 gives rise to the supersymmetric Wilson loops in the five-dimensional U (1) gauge theory on S 5 . In particular, the two-form B provides the five-dimensional gauge field A µ ≡ βB τ µ where µ = 1, . . . , 5 and similarly we define the five-dimensional scalar φ = βΦ. Taking these fields independent of τ and integrating over the M-theory circle S 1 , we find a supersymmetric Wilson loop in five (2.32) As in 6d abelian case, the charge n is quantized, which naively seems not to be the case since the abelian theory has no charged object perturbatively. However, non-perturbative objects, for example the singular instantons which we assume to be involved in this paper, can carry nontrivial charge and thus the quantization condition is required.
In the case of the six-dimensional (2, 0) theory of type g = A N −1 we cannot formulate the codimension four defect directly on S 1 × S 1 (j) . Nevertheless, taking inspiration from the abelian tensormultiplet, we can conjecture that it is computed by inserting a supersymmetric where the trace is taken in the representation of SU (N ) of highest weight λ.
On the saddle points we have A = 0 and φ = ia and hence integrating over S 1 (j) of length 2π/ω j the supersymmetric Wilson loop will make a classical contribution Tr λ (e 2πia/ω j ) to the normalized by the bare partition function. To insert a Wilson loop on the circle S 1 (j) , we insert this factor with parameters at S 1 (j) to the partition function. Let us again consider the special limits m = + and m = − . Firstly the Wilson loop in the limit m = + receives a rather simple instanton correction. After the huge cancellations, we obtain However, we do not discuss Wilson loops in this limit further in this work since it corresponds to inserting the Wilson loop in the chiral algebra plane.
On the other hand when m = − , since the center of mass factor multiplied to each instanton sector vanishes, the Wilson loop reduces to its classical value We conjecture that the U (N ) fundamental Wilson loop involves the U (1) Wilson loop factor and thus the SU (N ) Wilson loop expectation value is given by where W U (1) is the abelian Wilson loop of unit charge under the overall U (1) gauge group.
One can also identify the abelian Wilson loop expectation value from the U (1) gauge theory computation. The answer is where a is the equivariant parameter of the U (1) gauge symmetry and z U (1) is the letter index of the U (1) instanton partition function given by The abelian Wilson loop can be interpreted as a heavy tensor multiplet in the 6d theory.
The U (1) instanton partition function is the Plethystic exponential of the letter index z U (1) and agrees with the index of a single tensor multiplet on T 2 × R 4 [32]. The index of the heavy tensor fields can be obtained from the index of the single tensor multiplet by removing the factors corresponding to the motion along R 4 . We can achieve it by multiplying the factor . Thus the index of a heavy 6d tensor multiplet is the precisely the index of the abelian Wilson loop in (2.38).
The Wilson loop expectation value in the other representations can be computed in the similar manner. We have to insert the corresponding Chern character to the instanton partition function. One can construct the equivariant Chern character in the general representation of the gauge group using that of the universal bundle E. For instance, the equivariant characters in the symmetric and anti-symmetric representations are given by [28] Ch sym = 1 2 On the other hand, it exhibits a rather complicated expression when we take the limit m = + of Wilson loop in a general representation, which we do not discuss in this paper.

Codimension 2 Defects
The non-abelian (2, 0) theories are expected to have codimension 2 surface defects whose study was initiated in [1,2] in the context of the construction of four dimensional theories of class S. For a detailed discussion of the classification and properties of codimension two defects see [33][34][35][36][37].
For g = A N −1 the codimension 2 defects are in 1-1 correspondence with homomorphisms ρ : su(2) → g. They can be labelled by a partition [n 1 , . . . , n s ] with s j=1 n j = N and by convention we take n i ≤ n j if i < j. This data encodes how the fundamental representation decomposes N → n 1 + · · · n s into representations of the image of ρ. An important property of codimension two defects is that they support a flavor symmetry. Let j be the number of times that the number j appears in the partition [n 1 , . . . , n s ] so that j j j = N . Then the flavor symmetry supported by the defect is s(⊕ j u( j )).
Codimension 2 defects can be understood as transverse M5 branes intersecting with the primary stack of N coincident M5 branes. However, there is an alternative description of codimension 2 defects discussed in [38,39] whose connection to our computations is more transparent. This involves the primary stack of N M5-branes probing a multi-centered Taub-NUT (TN) space with s singularities. The M5-branes wrap the circle fiber and extend along a radial direction of the base R 3 of TN, with a number n i M5 branes ending on i-th singularity.
Thus the data classifying such configurations is a partition [n 1 , · · · , n s ] of N . Each set of n i M5-branes is supported on a cigar in the TN geometry. When all s centers coincide, TN develops an Z s orbifold singularity. Such configuration generates a codimension 2 defect at the tip of the cigar spanned by the M5 branes. Later in this section, we will compute the partition function in the presence of the codimension 2 defects using the instanton calculus of the 5d SYM on the Z s orbifold plane.
The brane descriptions indicate the symmetries that are preserved by a codimension 2 defect. Firstly, the six-dimensional conformal and R-symmetries are broken to Let us consider in detail the case where the codimension 2 defect spans the plane rotated by (h 2 , h 3 ) (we could also consider (h 1 , h 3 ) with similar results) so that in particular it fills the chiral algebra plane rotated by h 3 . In this case it is illuminating to write the supersymmetry algebra generated by the charges Q and Q in terms of the su(2, 2|2) ⊕ u(1) generators as (2.43) The chiral algebra limit of the superconformal index can be written where we have introduced additional fugacities t j for the Cartan generators f j of the s(⊕ j u( j )) flavor symmetry of the defect. Note that s becomes a fugacity for the additional u(1) d flavor symmetry. In what follows, we find that all states contributing to the superconformal index have d = 0 and therefore the index is independent of s. The index then coincides with the Schur limit of the N = 2 superconformal index for the su(2, 2|2) algebra in four dimensions [5,40,41].
As before, we cannot compute the superconformal index in the presence of a codimension 2 defect directly in six dimensions. However, it is expected that for a codimension 2 defect wrapping S 1 , there is an equivalent description as a surface defect in 5d N = 2 SYM. For the codimension 2 defect labelled by the partition ρ = [n 1 , . . . , n s ], the relevant surface defect can be obtained by coupling 5d N = 2 SYM to the three-dimensional N = 4 theory called This theory has a description as a linear quiver shown in figure 2. In particular, there is a sequence gauge groups U (r i ) where r i = n 1 + · · · + n i for i = 1, . . . , s − 1 (so that r s = N ).
There is an su(N ) symmetry acting on the N hypermultiplets at the final node, and on At this point, the most rigorous way to proceed would be to attempt an exact localization computation for N = 2 SYM on S 5 coupled to T ρ (g) supported on S 3 ⊂ S 5 . This computation is beyond the scope of the present paper. Instead, we will employ an effective description of these surface defects as monodromy defects of Levi type l = s(u(n 1 ) ⊕ · · · ⊕ u(n s )) and factorize the computation of the partition function into contributions from three fixed circles of S 5 . The validity of our procedure will be tested a posteriori by reproducing the S 3 partition function of T ρ (g) by sending the bulk coupling g 2 → 0.
Let us now describe the computational scheme. Our first assumption is that in the presence of a surface defect of type ρ, the partition function can once again be expressed as a matrix integral, whose integrand is factorized into contributions localized at the three fixed circles S 1 (a) , a = 1, 2, 3. For definiteness, suppose that the surface defect is supported on the three-sphere S 3 (3) ⊂ S 5 containing S 1 (1) and S 1 (2) as Hopf linked circles but supported away from S 1 (3) . Thus contributions localized at S 1 (3) should not be changed by the presence of the defect. On the other hand, from the perspective of S 1 (1) and S 1 (2) the defect is supported on subspaces S 1 × C 1 and S 1 × C 2 of S 1 × C 2 1 , 2 with the equivariant parameters identified as in Table 2.3.
Thus it is reasonable that the partition function of the combined system can be expressed where the contribution from the third fixed point Z (3) is the same as in the absence of the defect. The measure now becomes as the gauge group is broken to the subgroup L. The summation σ = 1, . . . , N ρ runs over the supersymmetric vacua of the three-dimensional theory T ρ (g) on S 1 × C. The number of these vacua is in general N ρ = N !/(n 1 !n 2 ! . . . n s !). Note that the classical action has an additional contribution with the monodromy parameter m whose derivation on round S 5 is given in the appendix B.
Our second assumption is that the contributions Z ρ,σ (j) at fixed points S 1 (1) and S 1 (2) are given by the 5d Nekrasov partition function in the presence of a monodromy defect. The partition ρ = [n 1 , n 2 , . . . , n s ] labelling the surface defect determines the Levi subgroup L = S (U (n 1 ) × · · · × U (n s )) left unbroken by the defect. Given a Levi subgroup L, the additional label σ specifies the nonequivalent choices for how L can be embedded into SU (N ). A particular choice can be denoted by L σ . The monodromy defect labelled by σ = 1 corresponds to the singularity Note that m can be characterized by the property ( m, ρ l ) = 0 where ρ l = ρ n 1 ⊕ . . . ⊕ ρ ns is the Weyl vector of the subalgebra l with the embedding σ = 1. The remaining supersymmetric Thus σ correspond to permutations that are not simply permutations within each block.
The number of such permutations is clearly N ρ = N !/(n 1 ! . . . n s !) which matches the number obtained from the quiver description of T ρ . Thus, the supersymmetric vacua are elements σ ∈ W/W l where W l is the Weyl group of l.
Let us now explain how to compute the 5d Nekrasov partition function in the presence of a monodromy defect using the ramified instantons computations of [36,39]. It is known that the moduli space of the ramified instantons of U (N ) gauge theory is equivalently described by the moduli space of the instantons on the orbifold space C × C/Z s where Z s acts on the complex coordinates as (z, w) → (z, ωw) with ω = e 2πi s . The defect is filling the plane of z. The geometric orbifold action is accompanied by the non-trivial U (1) s holonomy action on the gauge group which will be explained momentarily. We also twist the rotation symmetry so(2) 1 of the coordinate w with the so(2) 2 R-symmetry subgroup in so(5) R to have unbroken supersymmetries. The Z s then acts on the diagonal combination u(1) d .
This allows us to construct the instanton moduli space with a monodromy defect from the usual ADHM construction simply by applying the Z s action. The standard orbifolding procedure leads to an ADHM construction whose quiver, called the chain-saw quiver, is shown in Figure 3. After the orbifolding, we have vector spaces V i and W i of complex dimensions for the nodes in the quiver diagram. Here the index i is defined modulo s so that V s+1 = V 1 and W s+1 = W 1 . The associated ADHM data are given by matrices . As a complex manifold, the moduli space of the ramified instantons M ρ,k 1 ,··· ,ks is obtained by setting to zero the complex moment map and performing a quotient by the complexified gauge group ⊗ i GL(k i , C).
The localization of the ramified instanton partition function was explained in [36,39,43].
The saddle points of the localization are again classified by the standard N -tuple of Young tableaux Y . The Young tableaux are now labelled by Y = {Y j,α } , (j = 1, · · · , s , α = 1, · · · , n s ) . The Z s holonomy effectively shifts the these parameters such as a j,α → a j,α − j 2 . (2.53) For the tangent bundle T M ρ of the ramified instanton moduli space, the equivariant index at the fixed point Y is given by and, for the adjoint hypermultiplet, we get the index Remember that the Z s orbifold acts on the u(1) d which simultaneously rotates the coordinate w and the so(2) 2 R-symmetry, and also on the Cartans of u(N ). In the above indices we have implemented the Z s orbifold as the action only on 2 by shifting the mass parameters a and m. In addition, we need to multiply the momentum factor t∈Z e 2π r t along the temporal circle in five-dimensions.
The partition function computation is straightforward using the conversion rule. The ramified instanton partition function is given by is the product of weights in the equivariant indices at the saddle point Y and q i are the instanton fugacities. q ≡ q 1 q 2 · · · q s is related to the dynamical coupling of the bulk gauge theory. We identify the instanton fugacities with the monodromy parameters as follows: Similarly, we can compute the perturbative contribution under the Z s orbifold using the above equivariant indices. It is given by where · denotes the ceiling function. It turns out that the perturbative partition function factorizes into the contributions from the 5d theory and from the 3d theory supported on the defect. For instance, for the full defect of type ρ = [1, 1, · · · , 1], the perturbative partition function factorizes as with the double sine function defined as the following regularized infinite product The primed function is defined as S 2 (0) ≡ lim z→0 S 2 (z)/z. Z 3d,pert is the perturbative contribution from the 3d theory on the defect. Indeed, this 3d factor agrees with the perturbative contribution in the holomorphic block of the three dimensional T [U (N )] theory, which is believed to be the 3d theory living on the defect. We find that the product of two 3d factors, from S 3 (3) for example, with the physical parameters µ, ω 1,2,3 constructs the 1-loop contribution to the S 3 partition function of the T [U (N )] theory in the Higgs branch expression [44]: a); ω 1 , ω 2 ) S 2 ((e, a) + µ + ω 1 +ω 2 +ω 3
(2.61) Furthermore we find that the ramified instanton partition function in the decoupling limit Let us now discuss the limits m = ± which are needed for the chiral algebra limit of the superconformal index. Due to the mass shift m → m − 2 2 , these limits become the limits m = ± 1 /2. For simplicity we shall consider a particular embedding σ = 1 of the Levi subgroup.
In the limit m = − 1 /2, the perturbative contribution (2.58) reduces to unity due to the cancellation between contributions from vector and hypermultiplets lim m→ 1 /2 Z ρ pert = 1 .

(2.63)
A similar cancellation happens in the instanton calculus and we find that the contribution at each instanton fixed point becomes unity. Thus the instanton series is significantly simplified so that it simply counts the number of fixed points characterized by the same instanton numbers (k 1 , k 2 , · · · , k s ). For generic ρ, we claim that where the q-Pochhammer symbol is defined as (x; q) ∞ = ∞ i=0 (1 − xq i ). The first equality was given in [39]. Here the index a is taken to be modulo s. We have checked the second equality for N = 2, 3, 4, 5, 6, 7 with arbitrary ρ at some lower orders in q i expansions.
In the second limit m = 1 /2, after some algebra and using identities of S 2 , we find that the perturbative contribution simplifies to

W N -Algebra Characters
We will now combine the results of the previous section to compute of the chiral algebra limit of the 6d superconformal index in the presence of supersymmetry preserving configurations of defects. In this limit, we can evaluate the Coulomb branch integral of the S 5 partition function explicitly and express the result manifestly as a 6d superconformal index.
In the absence of defects this superconformal index has been shown to coincide with the vacuum character of u(1) in the case of the abelian tensormultiplet and the W N algebra for the non-abelian theory of type A N −1 [9,11,19]. In this section, we consider combinations defects that do not intersect the fixed circle S 1 (3) ⊂ S 5 that is distinguished by the chiral algebra limit -as shown in figure 4. From the perspective of the superconformal index, this means that the defects are point-like in the chiral algebra plane and are expected to correspond to chiral vertex operators. Indeed, we will reproduce the characters of irreducible modules of u(1) and the W N -algebras found in [20].

Vacuum Module
Let us first review the computation of the chiral algebra limit of the 6d superconformal index in the absence of any defects [9,11].
Tuning the mass parameter in the 5d partition function to µ → 1 2 (ω 1 + ω 2 − ω 3 ), the computation of the five-sphere partition function simplifies dramatically. In particular, the instanton partition functions at fixed points (1) and (2) are one and at fixed point (3) becomes  (1) and (2). Let us consider the abelian tensormultiplet and the non-abelian theories in turn.
It is convenient to introduce the notation 2πiτ = −βω 3 so that q = e 2πiτ . We also set ω 1 ω 2 = 1 since the final results do not depend on it (i.e. only depend on the ratio b 2 ≡ ω 1 /ω 2 ).
For the U (1) theory, the contributions from each fixed point are We have multiplied to Z (3) an overall factor e πi 12τ to make it as a modular form. As studied in [11] this factor is related to the leading high temperature behavior of the 6d abelian index, which cannot be observed from the 5d partition function because we have assumed the index to be smooth in 5d limit and regularized it. The detailed discussion will be left for a later work [24]. Combining these contributions with the classical contribution e πia 2 τ we have a gaussian integral .

(3.4)
For compactness we have introduced the shorthand notation . We have also introduced the standard notation Q = (b + 1/b)ρ where ρ = N j=1 ω j is the Weyl vector and we have a summation over the Weyl group W of g = A N −1 . The integration over a in the Cartan subalgebra was again gaussian and performed by systematically completing the square in the exponential. Now, using the Weyl denominator formula

5)
we can express the partition function on squashed S 5 in a form that is manifestly a superconformal index or partition function on S 1 × S 5 , −l |0 are null if 0 < l < n and that the vacuum module is freely generated by W

Degenerate Modules
In this subsection, we enrich the above computation by adding supersymmetric Wilson loops wrapping S 1 (1) and S 1 (2) . We expect to find non-vacuum modules of the relevant chiral algebra. In the abelian tensormultiplet theory, we find non-vacuum modules whose dimension depends on a pair of integers n 1 and n 2 . For the non-abelian theory of type A N −1 , we will find the characters of the so-called completely degenerate modules of the W N -algebra.
Let us first consider the abelian tensormultiplet theory and add supersymmetric Wilson loops of integer charge n 1 and n 2 on the circles S 1 (1) and S 1 (2) respectively. As described section 2.4, the presence of the Wilson loops modifies the instanton partition functions localized at fixed points (1) and (2). However, in the special limit the instantons at these fixed points decouple and the contribution is simply the classical expectation values. In summary we have Combining with the classical contribution, we again have a gaussian integral This expression is the character of an irreducible non-vacuum module of u(1) with dimension ∆ = − 1 2 (n 1 /b + n 2 b) 2 . In the non-abelian case, we can add supersymmetric Wilson loops supported on the circles S 1 (1) and S 1 (2) and labelled by irreducible representations of A N −1 with highest weights λ 1 and λ 2 respectively. Let us first consider the case where λ 2 = 0 in some detail. As above, the instanton contributions at fixed points (1) and (2)  which is precisely the character of a completely degenerate representation of the W N -algebra with momentum µ = −λ/b and dimension ∆(µ) = (Q, µ) − 1 2 (µ, µ). It is again illuminating to express this result in terms of the Plethystic exponential. Using the formula This demonstrates that we have a null state at level (ρ + λ, e) for each positive root e > 0. For instance, in the case N = 2 we find the character of the degenerate module of the Virasoro algebra with a null vector at level r where now c = 1 + 6(b + 1/b) 2 .
The most general completely degenerate module is found by placing two codimension 4 defects labelled by λ 1 and λ 2 wrapping the circles S 1 (1) and S 1 (2) respectively. The partition function now evaluates to corresponding to a simple module with momentum α = −λ 1 /b − λ 2 b. This exhausts the spectrum of fully degenerate modules.

Semi-Degenerate Modules
In this subsection, we consider the a surface defect supported on the three-sphere S 3 (3) . From the perspective of the superconformal index this corresponds to a codimension 2 defect orthogonal to the chiral algebra plane.
Let us briefly consider the abelian tensormultiplet theory. As we have argued in section, the presence of a monodromy defect does not change the instanton contributions in this case.
The only modification comes from a classical contribution e −2πima where m is the monodromy parameter. The S 5 partition function is nothing but the character of an irreducible module of u(1) of dimension ∆ = −m 2 /2. We remind the reader that m is imaginary in our notation.
There are N ! supersymmetric vacua labelled by an element σ ∈ W and corresponding to a permutation of the monodromy parameters m = (m 1 , . . . , m N ). In what follows, we denote the monodromy parameters m by simply m (or µ = Q + m), which should not be confused with the N = 1 * mass parameter that we have already tuned to the special value.
In the chiral algebra limit, we find that the contributions from the fixed points are independent of the permutation σ. The contributions from the fixed circles S 1 (1) and S 1 (2) are one: (3.17) where we have defined µ = Q + m and as before ∆(µ) = (µ, Q) − 1 2 (µ, µ). This expression is precisely the character Tr Vµ q L 0 −c/24 of a non-degenerate irreducible module with momentum µ = Q + m. . (3.19) This is precisely the squashed S 3 partition function of the 3d N = 4 theory T (U (N )) in the chiral algebra limit. This is an important evidence that our computation in terms of monodromy defects is correctly reproducing the surface defect that we intended. Note that the combination S µ,α = µ(a)Z(a, m) can be identified with the modular transformation matrix for non-degenerate W N -characters.
Let us now consider a generic codimension 2 defect labelled by the partition ρ = [n 1 , n 2 , · · · , n s ] where n 1 ≤ n 2 ≤ · · · ≤ n s and s i=1 n i = N . In the presence of the defect, the gauge symmetry of the five-dimensional theory is broken to the Levi type l = s[u(n 1 ) × · · · × u(n s )].
The supersymmetric vacua are labelled by a permutation σ ∈ W/W l . Due to the presence of non-abelian factors in l, there are now non-trivial perturbative contributions where ∆ + j corresponds to the positive root space generated by {e r j , . . . , e r j+1 −1 } where r j = n 1 +. . .+n j . These are the roots whose non-zero elements lie entirely within the n j ×n j block.
The instanton contributions to the the fixed point S 1 (3) remain unaffected by the presence of the defect, so Z (3) = 1/η(−1/τ ) N −1 . As before, there is an additional classical contribution e −2πi(σ(m),a) -see appendix B.
Putting the contributions together and summing over the supersymmetric vacua we have We can also introduce the codimension 4 defects supported on the fixed circles S 1 (1) and S 1 (2) . As these circles are inside S 3 (3) we may only introduce supersymmetric Wilson loops in the unbroken gauge symmetry. For the abelian theory, we can introduce supersymmetric Wilson loops of any integer charges n 1 and n 2 , and the result is the character of an irreducible For the non-abelian theories, the supersymmetric Wilson loops at the fixed circles S 1 (1) and S 1 (2) are characterized by the dominant integral weights λ 1 and λ 2 of the the Levi subalgebra l ⊂ g, respectively. The weights obey the constraints (λ 1 , e) ≥ 0 and (λ 2 , e) ≥ 0 for all e ∈ s j=1 ∆ + j . Plugging these Wilson loop contributions into the partition function we obtain Here W l is the Weyl group and ρ l is the Weyl vector of l.

Affine Characters
In this section, we will consider adding a surface defect wrapping one of the circles S 3 (1) or S 3 (2) . From the perspective of the 6d superconformal index this corresponds to adding a codimension 2 defect wrapping the chiral algebra plane. In the case of a codimension 2 defect labelled by the partition ρ = [1 N ] we will now find characters of irreducible modules of the affine algebra su(N ) at level k = −N − b ±2 . We will leave exploration of general type ρ defects for the future.

Vacuum Module
Let us first consider a surface defect of type ρ supported on S 3 (1) . The same formulae will apply for S 3 (2) by interchanging b ↔ 1/b. These correspond to codimension 2 defects wrapping the chiral algebra plane. In the chiral algebra limit, the contributions localized at the first two fixed points are = s j=1 e∈∆ + j 2 sin πb(e, σ(a)) , (4.1) and, from (2.64), at the third fixed point is Combining all three fixed contributions, the full partition function is where µ = Q + m − b −1 ρ − bρ l . This is not yet the superconformal index as the function Z ρ In what follows we shall focus on a surface defect of maximal type ρ = [1 N ] in the chiral algebra limit for which Z (3) has a nice modular property and can be re-expanded by q. We with the theta function the latter factor appears to be ambiguity which is not fixed in the 5d partition function.
The partition function in the presence of the defect is given by where we write z j = e 2πim j . The partition function can be recognized as the character of the vacuum module of the affine algebra su(N ) at level k = −N − b 2 and with associated Sugawara central charge c = (N/b 2 + 1)(N 2 − 1). More precisely we have where the vacuum module V 0 has highest affine weightλ = kω 0 . A summary of our conventions and a derivation of this result can be found in appendix (A. 3). Note that in this case, we reproduce the vacuum character up to a small prefactor e − iπk τ (m,m) depending only on the level.

Highest Weight Modules
Let us now introduce a supersymmetric Wilson loop wrapping S 1 (1) . It is important that the support of this supersymmetric Wilson loop does not intersect the support S 3 (1) of the surface defect. This means that the gauge symmetry is unbroken near the fixed point (1) where now the dimension is ∆ λ = (λ,λ+2ρ) 2(k+N ) and χ λ (z) denotes the character of the finite dimensional representation of A N −1 with highest weight λ. This is exactly the character an irreducible module of the affine lie algebra at level k = −N − b 2 with affine highest weight λ = kω 0 + λ. As before, we refer the reader to appendix (A.3) for a summary of this result.

Discussion
Let us conclude with a number of speculations and promising directions for further research.
Firstly, we believe that our computations provide an important step towards deriving the connection between the superconformal index of 4d N = 2 theories of class S and topological quantum field theory [4,5]. Let us recall from [5] that in the Schur limit the superconformal index of the class S theory corresponding to a Riemann surface with n maximal punctures of type ρ = [1 N ] and genus g can be expressed as where the summation is over the finite dimensional irreducible representations of g and the expression for the structure constants C λ and wavefunctions ψ λ (a) can be found in [5]. Promoting the 4d superconformal index to the S 1 × S 3 partition function we observe in the case Presumably this can be extended to non-maximal punctures of generic type ρ.
The above picture also suggests a concrete proposal for how to compute the wavefunctions ψ ρ,λ (a i , p, q, t) appearing in the superconformal index of class S theories with general fugacities turned on. They should correspond to the superconformal index on S 1 ×S 5 with a codimension 2 defect of type ρ wrapping say S 1 × S 3 (1) and and codimension 4 defect of type λ wrapping S 1 × S 1 (1) . Note that the two defects coincide only along S 1 : S 3 (1) and S 1 (1) are Hopf-linked inside S 5 . The 4d parameters are identified with the parameters of the 6d superconformal index as {a i , p, q, t} 4d = {z i , q 2 , q 3 , (q 1 q 2 q 3 /p) 1/2 } 6d (5.2) Some initial checks of this proposal are performed in [44]. This observation is of mathematical interest as these wavefunctions should provide a complete set of eigenfunctions the elliptic Ruijsenaars-Schneider integral system for the codimension 2 defect of maximal type ρ and more generally its degenerations.
It is natural to identify the wavefunction ψ λ with the contribution to the 4d superconformal index of class S theories from a disk with puncture. This can be understood by reformulating the 6d superconformal index in terms of the 4d superconformal index on S 1 × S 3 (1) together with a topologically twist along the two transverse directions involving S 1 (1) , which are identified with a disk with puncture. The puncture corresponds to the insertion of the codimension 2 defect of type ρ. The boundary condition along the S 1 (1) may be specified by the codimension 4 defect of type λ.
Note that there is an additional parameter in the 6d superconformal index conjugate to h 1 + R 2 , which is turned off in the identification (5.2). This reflects the topological twisting along the Riemann surface. One can also check the chiral algebra limit p → (q 1 q 2 /q 3 ) 1/2 of the 6d superconformal index corresponds precisely to the Schur limit t → q of the 4d N = 2 superconformal index of the theory on the S 1 × S 3 (1) . This suggests that it is possible to enumerate the states contributing to the superconformal index of 4d theories of class S and identify their six-dimensional origin. We hope to return to this question in future work.
For codimension 2 defects wrapping S 1 × S 3 (1) , we could express the partition function manifestly as a 6d superconformal index only in the case of a maximal puncture ρ = [1 N ].
For more generic punctures, although we could find an integral expression and perform the integral explicitly, we could not transform the instanton contribution Z ρ (3) from the third fixed point from a weak-coupling expansion inq = e −2πi/τ to an expansion in the 6d fugacity q = e 2πiτ . It will be interesting to find the modular property of Z ρ (3) for a generic ρ and compute the superconformal index. Based on previous work, we would expect to find the characters of modules of W (ρ) -algebras, which are obtained from the affine algebra su(N ) by Drinfeld-Sokolov reduction [46]. As shown in [41], one can also reduce the flavor symmetry Finally, a complementary approach to computing the 6d superconformal index is to use the 5d gauge theory on S 1 × CP 2 by reducing the 6d (2,0) theory on S 1 × S 5 along the Hopf fiber of S 5 . In this case, the partition function including the non-perturbative instanton contributions is expressed manifestly in the form of a 6d index, without the need for performing a modular transformation on τ . The partition function with codimension 2 defects of generic type ρ could be computed by this method and compared to the characters of W (ρ) -algebra.
This could also allow the 6d superconformal index with defects to be computed in the case of general fugacities.

Acknowledgements
It is a pleasure to thank Nikolay Bobev, Davide Gaiotto, Jaume Gomis  We will often represent elements of g by traceless anti-hermitian matrices with (a, b) = −Tr(ab) on h. For an element a ∈ h we define its components by a j = (a, h j ) so that N j=1 a j = 0 and (a, a) = Tr(a 2 ) = a 1 1 + . . . + a 2 N .

A.2 W N -algebra Characters
In this appendix, we will summarize the spectrum of simple modules of the W -algebra of type g = A N −1 , which we have called the W N -algebra in the main text, following closely reference [20].
The W N -algebra is generated by holomorphic currents W j (z) of spin j = 2, . . . , N . The holomorphic current W 2 (x) = T (x) is identified with the stress energy tensor and generates a Virasoro subalgebra with central charge c that can be parametrized by The simple modules V µ are highest weight modules labelled by an element µ ∈ h called the momentum. They are constructed from a Verma module with chiral primary of dimension by subtracting the descendants of any null vectors. The simple modules are sometimes classified crudely as non-degenerate, semi-degenerate or fully degenerate, depending the structure of null vectors appearing in the Verma module.
To construct simple modules we first choose a homomorphism ρ : su(2) → g. This can be labelled by a partition [n 1 , . . . , n s ] with s j=1 n j = N and by convention n i ≤ n j if i < j. This specifies how the fundamental representation of g decomposes N → n 1 + · · · + n s under the image of the homomorphism Im(ρ) ⊂ g. The stabilizer of Im(ρ) in g is the subalgebra where ∆ j is generated by the subset of simple roots {e r j , . . . , e r j+1 −1 } with r j = n 1 + · · · + n j .
The corresponding spaces of positive roots are denoted by ∆ + l and ∆ + j with Weyl vectors ρ l and ρ n j defined as half the sum of the positive roots therein.
Given a homomorphism ρ : su(2) → g, a simple module is constructed by starting from a Verma module with momentum where m is an imaginary element of h obeying (m, ρ l ) = 0 and λ 1 and λ 2 are dominant integral weights of l ⊂ g. The latter obey the conditions (λ 1 , e) ≥ 0 for all e ∈ ∆ + l . There is in general an intricate structure of overlapping Verma modules generated by the null vectors. The character of the simple module obtained by subtracting the descendants of the null vectors is where W l is the Weyl group of l. The term in this formula with w the identity element is the contribution from the full Verma module. The terms where w is a reflection by a simple root where we have the momentum µ = −λ 1 /b − bλ 2 . In this case, we have the maximum number of null vectors and the simple modules are called fully degenerate. In particular, the vacuum module corresponds to the case λ 1 = λ 2 = 0. All other simple modules are broadly referred to as semi-degenerate.

A.3 Affine Characters
We now consider some simple modules of the affine algebra g with level in the regime k = −N − with > 0. This is generated by spin-1 holomorphic currents J a (x) with a = 1, . . . , N 2 − 1.
The Sugawara construction provides a Virasoro subalgebra with central charge We will consider simple highest weight modules Vλ labelled by a highest affine weightλ.
The components of an affine weight are denoted byλ = (λ, k, n) where λ ∈ h is a finite weight, k is the level and n is the component dual to the generator −L 0 . There is a metric denoted by (λ,λ ) = (λ, λ )+kn +nk . We use the common abuse of notation and write λ = (λ, 0, 0). It is convenient to introduce the fundamental affine weights which have componentsω 0 = (0, 1, 0) andω j = (ω j , 1, 0) for j = 1, . . . , N − 1. An affine weight that is a linear combination of the fundamental affine weights can be written These roots generates a Coxeter group with associated Kazdan-Lusztig polynomials. In the case that k = −N − with > 0 it has two components (ρ + λ, α) ∈ Z for α ∈ ∆ + (ρ + λ, α) − n ∈ Z for α ∈ ∆, n > 0 . (A.16) Here we want to consider generic > 0 so that the second component can only be non-empty by tuning λ in a way that depends on . We will not consider this scenario. Instead, we take where I have assumed that only non-affine weight λ appears in the numerator. In passing to the third line we have used the Weyl denominator formula, and χ λ denotes the character of the simple finite dimensional module of g with highest weight λ. In particular, the vacuum module of g with level k corresponds to λ = 0 andλ = kω 0 .
To compute the physical character we replace the formal expression e −α−nδ where α = j j h j by the monomial q n j µ j j . Recall that h j are the weights of the fundamental representation and so j components of α in the orthogonal basis. Therefore, we have (1 − q n µ i /µ j ) = q − c

B S 5 partition function and codimension 2 defects
One can construct the 5d maximal SYM theory on (squashed) S 5 from the 6d (2,0) theory on S 5 × S 1 by dimensional reduction along the S 1 . We first reduce the abelian (2,0) theory to five-dimensions and find its non-abelian generalization. Let us consider the 6d theory defined on the curved metric ds 2 6 = e − 2 3 Φ ds 2 5 + e Here w i = 1+a i are chemical potentials for the U (1) 3 rotation of the holomorphic coordinates z i = n i e iφ , (n 2 1 + n 2 2 + n 2 3 = 1). The dimensional reduction along the time circle gives rise to the 5d theory on the squashed five-sphere with squashing parameters ω j . The background 'dilaton' Φ and 'RR gauge field' C µ are also turned on.
We restrict the 5d reduction such that it preserve two supercharges Q and Q † used to define the 6d superconformal index. The 5d gauge theory action is uniquely determined under this reduction. The explicit action can be found in [11]. The 5d supercharge Q satisfies the following algebra: (pω 1 + qω 2 + rω 3 + (e, a)) ((p + 1)ω 1 + (q + 1)ω 2 + (r + 1)ω 3 + (e, a)) (pω 1 + qω 2 + rω 3 +μ + (e, a))((p + 1)ω 1 + (q + 1)ω 2 + (r + 1)ω 3 −μ + (e, a)) = lim x→0 S 3 (x)/x S 3 (μ) Let us now turn to the codimension 2 defects on S 5 , which are related to the codimension 2 operators in the 6d (2,0) theory. The BPS defects can be supported on S 3 ⊂ S 5 . For simplicity, let us stick to the maximal SYM theory with U (N ) (or SU (N )) gauge group on round S 5 . These defects are defined by specifying a singular behavior of the gauge field as one approaches their location. Near the defects, we parametrize two normal directions by a complex coordinate z = re iθ where θ is one of the angle coordinates in S 5 . Then the defect is defined with a gauge field which behaves around the defect as A µ dx µ ∼ mdθ ≡ diag(m 1 , m 2 , · · · , m N )dθ . where * dΩ S 3 is the Hodge dual of the three-sphere volume form.
Let us now derive the classical action in the presence of the defect. We use the off-shell supersymmetry formulation of the 5d SYM studied in [9,13]. We focus on the round S 5 background. It turns out that the codimension 2 defects preserve the supercharge Q used in the localization. The BPS condition from the gaugino variation is given by The Killing spinor for the supercharge Q satisfies the following conditions † γ µ = v µ , † γ µν = iJ µν , σ 3 = , (B.9) where v µ is the Killing vector along the Hopf fiber of S 5 and J µν is the Kähler form of In the main context, we would use the convention a → ia by analytic continuation.
We would not perform an explicit localization computation in the presence of codimension 2 defect. However, turning on the squashing parameters, we expect that the path integral again localizes to three fixed points and the full partition function takes the form of products of three fixed point contributions. The contributions at the fixed points can be computed using the results on the local S 1 × C 2 , which are explained in sections 2.3 ∼ 2.5.