Central Charges of 2d Superconformal Defects

In conformal field theories (CFTs) of dimension $d>3$, two-dimensional (2d) conformal defects are characterised in part by central charges defined via the defect's contribution to the trace anomaly. However, in general for interacting CFTs these central charges are difficult to calculate. For superconformal 2d defects in supersymmetric (SUSY) CFTs (SCFTs), we show how to compute these defect central charges from the SUSY partition function either on $S^d$ with defect along $S^2$, or on $S^1 \times S^{d-1}$ with defect along $S^1 \times S^1$. In the latter case we propose that defect central charges appear in an overall normalisation factor, as part of the SUSY Casimir energy. For 2d half-BPS defects in 4d ${\mathcal N}=2$ SCFTs and in the 6d ${\mathcal N}=(2,0)$ SCFT we obtain novel, exact results for defect central charges using existing results for partition functions computed using SUSY localisation, SUSY indices, and correspondences to 2d Liouville, Toda, and $q$-deformed Yang-Mills theories. Some of our results for defect central charges agree with those obtained previously via holography, showing that the latter are not just large-$N$ and/or strong-coupling limits, but are exact. Our methods can be straightforwardly extended to other superconformal defects, of various codimension, as we demonstrate for a 4d defect in the 6d ${\mathcal N}=(2,0)$ SCFT.


Introduction
Defect operators are essential for classifying Quantum Field Theories (QFTs) [1][2][3][4]. For example, two gauge theories with the same gauge algebra but different gauge groups can have identical correlators of all local operators, but different spectra of 1d (line) operators, such as Wilson and 't Hooft lines [2]. Furthermore, such 1d operators are the order parameters classifying vacua as confining, Higgs, Coulomb, etc. Similarly, higher-dimensional defect operators are order parameters classifying phases in which higher-dimensional objects, such as strings, condense [3,4].
A fundamental question in QFT is therefore how to characterise and classify defect operators. A formidable obstacle to answering this question is in dealing with (strongly) interacting degrees of freedom of the ambient QFT and/or on the defect. To overcome this obstacle we will employ a common strategy: impose highly restrictive symmetries. Specifically, we will require both conformal symmetry and SUperSYmmetry (SUSY). Furthermore, we will focus exclusively on 2d defects, which in 4d are also called surface operators. In short, we will focus on 2d superconformal defects in SuperConformal Field Theories (SCFTs).
We use conformal symmetry to provide order in the space of QFTs. In particular, Conformal Field Theories (CFTs) occupy privileged places in the space of QFTs, as fixed points of Renormalisation Group (RG) flows, and c-theorems then imply irreversibility along those RG flows, providing a hierarchy among QFTs. More specifically, c-theorems state that certain central charges must decrease monotonically along RG flows. As a result, these central charges can count degrees of freedom, which we expect to decrease along RG flows as the UltraViolet (UV) physics becomes more coarse grained and massive modes decouple. Proofs exist for c-theorems in 2d [5][6][7][8], 3d (F -theorem) [9,10], and 4d (atheorem) [8,[10][11][12][13][14][15].
How to extend c-theorems to defect CFTs remains an open question. Currently only two defect c-theorems have been proven. The first is for RG flows along 1d interfaces separating 2d CFTs. In these systems the "g-theorem" [16][17][18] requires the interface entropy, denoted ln(g), to decrease monotonically along the RG flow. Intuitively, ln(g) measures the ground state degeneracy of the 1d quantum system and thus counts the number of degrees of freedom localised at the interface. Often ln(g) can be computed, even in (strongly) interacting systems, using powerful methods available to 2d CFT. 1 The second defect c-theorem is for 2d defects in CFTs of dimension d ≥ 3, with an RG flow on the defect [19,20]. To be precise, let M denote the background manifold for a d-dimensional Euclidean CFT with coordinates {x µ }, with µ = 1, 2, . . . , d, and let Σ → M with coordinates {ξ a }, with a = 1, 2, be the 2d submanifold on which the defect has support. In these cases the trace anomaly includes the usual contribution from the ambient CFT, which can be non-zero only if d is even, plus a contribution delta-function localised at Σ [21][22][23][24], where we use the sign and normalisation conventions of refs. [25,26]. 2 In eq. (1.1), E 2 andII µ ab are the Euler density and traceless second fundamental form of Σ, respectively, and W abcd is the pullback of the ambient Weyl tensor to Σ. In general, a 2d defect thus has three possible central charges, b, d 1 , and d 2 . In the classification of ref. [28], b is type A while d 1 and d 2 are both type B. Among other things, the classification means b cannot depend on defect marginal couplings, while d 1 and d 2 can. In general, all three can depend on marginal couplings of the ambient CFT [29,30], unless the defect preserves 2d N = (2, 0) SUSY, in which case b cannot depend on ambient marginal couplings either [31].
The central charge b obeys a c-theorem for defect RG flows [19,20], sometimes called the "b-theorem." Whether d 1 and d 2 obey c-theorems still remains unknown. However, proofs exist that in reflection-positive theories d 1 ≥ 0 [29,32], and if the average null energy condition is obeyed in the presence of the defect then d 2 ≥ 0 [26]. Free field computations show that b can be negative, but whether b obeys a lower bound has not been proven. For d = 3, a lower bound b ≥ − 2 3 d 1 was conjectured in ref. [32]. In general, b, d 1 , and d 2 are difficult to calculate, and indeed to our knowledge they have been calculated only in free-field CFTs [22,23,33,34] and holographic CFTs [21,25,26, The g-theorem also applies to a 2d CFT with a boundary, interpreted as an interface with an "empty" CFT on one side, and to a 2d CFT with a point-like impurity. 2 For a surface defect in d = 4 that breaks parity, additional parity-odd terms may also appear in the trace anomaly [26,27]. These will play no role in this paper. 45]. This is why we turn to SUSY, to provide tractable examples of interacting theories in which we can calculate b, d 1 , and d 2 without using holography. In particular, we will consider 1/2-BPS 2d superconformal defects in 4d and 6d SCFTs. In 4d we focus on N = 2 SCFTs of class S, which are obtained generically by wrapping a stack of N coincident M5-branes on a genus-g n-punctured Riemann surface C g,n [46]. The torus, C 1,0 , is a special case where the 4d SUSY is enhanced, producing the maximally SUSY SCFT in 4d, N = 4 SU (N ) super Yang-Mills (SYM) theory. We consider superconformal defects preserving 2d N = (2, 2) SUSY (enhanced to N = (4,4) in N = 4 SYM theory) arising as either M2-branes ending on the M5-branes, which sit at a point on C g,n , or a second stack of M5-branes intersecting the first stack over a codimension 2 surface, and which wrap all of C g,n [47][48][49][50].
Our goal is to use SUSY methods to compute b, d 1 , and d 2 for these classes of 2d superconformal defects. In fact, in a 4d SCFT with a 2d N = (2, 0) superconformal defect, ref. [52] proved that the SUSY algebra requires d 1 = d 2 . Ref. [52] further provided compelling evidence for the conjecture that d 1 ∝ d 2 for 2d N = (2, 2) superconformal defects in any d. All of these results extend equally well to 2d N = (4, 4) superconformal defects. We will thus only explicitly compute b and d 2 .
We will compute b only for 2d superconformal defects in 4d N ≥ 2 SCFTs. To do so, we will compute the partition function of the Euclidean CFT on M = S 4 , with the defect wrapping an equatorial sphere Σ = S 2 , and then perform a re-scaling of the S 4 radius, with all other scales held fixed. The trace anomaly is the statement that the partition function changes by an overall power of the sphere radius under Weyl re-scaling, fixed by the central charge of the SCFT. We obtain b by calculating that power and subtracting any contribution from the ambient SCFT's type A central charge.
We will make a general argument for how to extract d 2 from the SUSY partition function on M = S 1 R × S d−1 . These partition functions turn out to be a product of two factors. One factor is the Schur index [81,82]. By appealing to a growing body of evidence from various perspectives [52,71,74,[83][84][85], we claim that the other factor is e −REc , where E c is the SUSY Casimir Energy (SCE). We propose that introducing the defect shifts E c by a term ∝ d 2 , and provide compelling evidence from our two examples.
Our first example is N = 4 SU (N ) SYM, where we will use the fact that the Schur limit of the 4d SUSY partition function is equivalent to the partition function of 2d SU (N ) q-deformed Yang-Mills (qYM) theory on a Riemann surface (C g,n ) in the zero area limit [75]. In this correspondence, the insertion of p 2d superconformal defects labelled by representations R i of SU (N ) deforms the 4d SUSY partition function [86] in a way that is captured in 2d qYM as a p-point correlation function O R 1 . . . O Rp [86,87]. Our second example is the Wilson surface in the M5-brane theory, for which a form for the SUSY partition function on S 1 R × S 5 was proposed in ref. [73]. In fact, we will obtain a more general result: we will compute the shift in E c due to two intersecting Wilson surfaces, which turns out to be more than just a sum of the contributions from two individual Wilson surfaces, possibly because of additional degrees of freedom arising at the 1d intersections along S 1 R . Holographic results for b and d 2 exist for the 2d N = (4, 4) superconformal defects in N = 4 SYM theory, in the 't Hooft large-N limit with large 't Hooft coupling [21,26,36,88], and for Wilson surfaces in the M5-brane SCFT, in the large-N limit [21,25,26,35]. Our results using SUSY methods agree perfectly with the holographic results, whenever they overlap. However, the SUSY methods involve no approximations and are valid at all couplings and for any N : they provide exact results for b and d 2 . We thus find that the holographic results are exact, and not merely large-N or strong coupling limits. Furthermore, the agreement with holography provides compelling evidence that E c ∝ d 2 , especially in the Wilson surface case, as we will discuss.
Ultimately, our main message is the methods themselves. For the classes of superconformal defects that we study, we find practical ways to obtain exact results for b and d 2 . The various methods that we present also provide different perspectives on what b and d 2 are counting. Furthermore, these methods can be straightforwardly generalised to superconformal defects of other (co-)dimension, such as SUSY interfaces or domain walls between SCFTs [58,89]. In fact, we further apply the method of computing d 2 via the change in SCE to compute a putative central charge for 4d superconformal defects as the character of a semi-degenerate module in a W N -algebra [73,78]. In this case the form of the trace anomaly remains unknown, so we cannot say exactly which (linear combination of) defect central charges we obtain. Nevertheless, we believe the methods we develop in this paper can play a crucial role in characterising and classifying defects quite generally. This paper is organised as follows. In section 2 we review key facts we will need about 2d superconformal defects of SCFTs of class S. In section 3 we present our calculations of b using SUSY localisation and the AGT correspondence. In section 4 we present our calculations of d 2 for superconformal defects in N = 4 SYM theory, using q-deformed YM, and for Wilson surfaces in the M5-brane theory, using the S 1 × S 5 partition function. In section 5 we conclude with a summary, and discuss possible directions for future research. We collect in two appendices various technical results that we will use along the way.

Review: 2d Superconformal Defects
In this section, we will provide a short overview of the relevant features of 2d superconformal defects that will be useful for our computations below. In particular, all of our work will focus on deforming 4d and 6d SCFTs by the addition of such defects. Even for these narrowly focused applications, there is a vast amount of extant literature, the surface of which we will only scratch.
In general, surface operators in a 4d theory can be characterised in two distinct but sometimes related ways [90]. One may either (i) assign singular behaviour to ambient 4d fields at the 2d submanifold Σ, or (ii) introduce an auxiliary 2d theory at Σ and couple it to the ambient 4d theory. This is a broad partitioning of defects according to whether we are using only the behaviour of ambient fields to describe a surface defect, or adding new degrees of freedom supported only on Σ. They are sometimes related because, for example, integrating out degrees of freedom on Σ may produce singular behaviour of the ambient fields at Σ.

2d Levi Type-L Defects
Approach (i) to describing surface defects is quite powerful and, by now, well-studied [4,90,91]. Here we briefly review the construction for N = 2 and N = 4 gauge theories.
Consider N = 4 SYM theory on M = R 4 with coordinates {x µ }, with µ = 1, . . . , 4, and a 1/2-BPS surface operator supported on Σ = R 2 with coordinates {x 1 , x 2 }. Let us write the coordinates on the normal bundle N Σ = C as x 3 + ix 4 = z = re iθ . To define the surface defect, one needs to prescribe a singularity in the normal component of the gauge field A = A z dz and the 1-form scalar in the adjoint N = 2 hypermultiplet ϕ = (ϕ 1 + iϕ 2 ) dz. In preserving 2d N = (4, 4) SUSY along Σ, A and ϕ have to satisfy a set of simple BPS conditions that take the form of Hitchin's equations [90,92]. The leading singular behaviour of the BPS solutions that additionally preserve defect conformal symmetry is given by for constants (α, β, γ). Relaxing the constraint that the defect preserves conformal symmetry would allow for non-trivial dependence on the radial coordinate r.
If the ambient 4d theory has only N = 2 SUSY rather than N = 4, one constructs a 1/2-BPS surface defect by prescribing a singularity in the 4d gauge field only. In these cases the BPS conditions do not allow for a singularity in any scalar fields, so no analogue of β or γ exists -those are special to N = 4 SYM.
The data (α, β, γ) describing the 1/2-BPS defect are valued in T × t × t, where T is the maximal torus of the gauge group G and t is the associated Cartan subalgebra. Thus, quantisation of the 2d-4d system requires the preserved gauge symmetry consistent with solving the BPS equations to be a subgroup of G containing T, called the Levi subgroup L ⊂ G. There are a number of ways to construct L, and choosing a particular L ⊂ G is part of defining the defect. Thus, 2d defects of the types that we have been describing are called Levi type-L. Unless otherwise specified, we will only consider G = U (N ) or SU (N ) and There are two types of 2d Levi type-L defects commonly encountered in the literature that are given special names and will be considered below. For gauge group G = SU (N ), if L = S[U (N − 1) × U (1)] then the surface defect is called simple, and if L = T = U (1) N −1 then the surface defect is called full.
Lastly, in addition to L and (α, β, γ), one can turn on a quantum 2d theta angle parameter, η, along the defect. The importance of η can be seen in studying the behaviour of Levi type-L defects under dualities. Under, say, S-duality (α, η) → (η, −α), and so for a generic 2d N = (4, 4) superconformal defect in 4d N = 4 SYM theory, specifying (L; α, β, γ, η) completely describes the defect. The parameters (β, γ) are together valued in the L-invariant part of t, while α is valued in T and η is valued in the maximal torus L T of the Langlands dual L G of G. All of the parameters grouped together transform in the part of (T × t × t × L T) invariant under the Weyl group of L [90,91].
Unless otherwise stated, the Levi type-L surface defect examples considered below will have β = γ = 0. This is particularly relevant for the computation of superconformal indices or twisted partition functions for N ≥ (2, 2) defects. The parameter β + iγ being non-zero is generally incompatible with the necessary symmetries for computing the defect index. In particular, non-zero β and/or γ breaks rotational symmetry in the plane normal to Σ. 3 Having set the basis to describe Levi type-L defects, it is useful to understand the physical meaning of the parameters (α, β, γ, η). 2d N = (4, 4) superconformal defects defined by eq. (2.1) are non-local probes of the Higgs branch of the moduli space M H . As mentioned, the η parameter is a 2d theta angle, but the other "classical" parameters (α, β, γ) encode geometric information about M H . M H is constructed by a hyper-Kähler quotient [93], and as such there are both complex structure -one of three labelled I, J, K -and Kähler parameters that describe the local geometry. By making a choice of which parameters go into the solution for ϕ in eq. (2.1), we are in effect picking a complex structure, while the parameter controlling the singular behaviour of the 4d gauge field A is the Kähler parameter. In ref. [90], the combination β + iγ was identified with complex structure I, and in this complex structure α was the Kähler parameter. Cyclicly permuting the roles of the parameters, one may identify γ + iα and α + iβ with complex structures J and K with Kähler parameters β and γ, respectively.

2d Defects from 2d QFTs
In approach (ii), we begin with a 4d theory and add 2d degrees of freedom localised on Σ. We will consider cases where the latter are a Gauged Linear Sigma Model (GLSM) or a Non-Linear Sigma Model (NLSM). The 4d and 2d degrees of freedom can be coupled in Figure 1: Linear quiver diagram corresponding to a 2d N = (2, 2) GLSM. Its field content consists of U (K i ) 2d vector multiplets for i = 1, . . . , n, N fundamental φ fund n and N anti-fundamental φ anti-fund n chiral multiplets coupled to the U (K n ) vector, one chiral multiplet φ bif i(i+1) in the bifundamental representation of (K i , K i+1 ), and one chiral multiplet φ bif (i+1)i in the bifundamental representation (K i , K i+1 ) for each 1 ≤ i ≤ n − 1. Additionally, depending on the particular details of the 2d N = (2, 2) gauge theory, there can be one adjoint chiral X i of U (K i ) for each node. This quiver diagram can be used to construct a surface operator whenever the 4d N = 2 gauge theory has at least an S[U (N ) × U (N )] flavour or gauge symmetry group. various ways, for example by superpotential couplings and/or by gauging a shared symmetry group [62,65,90,91,94].
We will consider 4d SCFTs that enjoy at least N ≥ 2 SUSY. To engineer a 1/2-BPS surface defect, consider the GLSM with N ≥ (2, 2) SUSY and gauge group G 2d described by the quiver in figure 1. The i th circular node denotes a 2d gauge multiplet with gauge group U (K i ), the arrows connecting the i th and (i + 1) th node represent chiral multiplets in the bifundamental representation , depending on the direction of the arrow. We collectively denote the fields in the bifundamental by φ bif i(i+1) and φ bif (i+1)i , respectively. The dashed arrows starting and ending on the same node are adjoint chiral multiplets X i . In what follows our quivers will always have the bifundamental fields, but may or may not have the adjoint chirals, depending on the type of defect we wish to study. For each gauge node, we may also turn on an FI parameter and a 2d theta angle for its U (1) factor. The square nodes on the left indicate the number of flavours of the (anti-)fundamental chiral multiplets under the U (K n ) gauge group. We denote these fundamental and anti-fundamental chirals by φ fund n andφ anti-fund n , respectively.
If we set the (real) twisted masses of all matter fields to zero, and provided that the FI parameters do not run, then such a GLSM may flow to an interacting InfraRed (IR) fixed point [95]. To the best of our knowledge, it is unknown whether such an IR fixed point exists, but whenever it does the 2d SCFT has central charge c 2d given by where R are the 2d fields' representations of G 2d and q R are their R-charges. The representation data and dim G 2d can be expressed in terms of the ranks of the gauge groups, K i . We will compute several explicit examples in section 3, but an important illustrative example is 1/2-BPS surface defects in N = 4 SU (N ) SYM. In this case c 2d can be written more usefully in terms of the difference of adjacent ranks, N i = K i − K i−1 , with K 0 ≡ 0 and K n+1 ≡ N . In particular, for an N = (4, 4) GLSM, a result that we will find again in several different ways in the following.
To obtain a 1/2-BPS superconformal defect, one couples the ambient 4d theory to a GLSM, and flows to the IR fixed point, if it exists. Typically, the Vacuum Expectation Values (VEVs) of the 4d fields enter as twisted mass parameters in the 2d partition function [65][66][67]. A planar 1/2-BPS superconformal defect then breaks the 4d superconformal algebra to a subalgebra: An alternative description of a 1/2-BPS surface defect can be obtained by coupling a 2d NLSM to the ambient field theory [4,90,91,94]. The NLSM description is obtained from the GLSM above by a defect RG flow: the gauge group is generically Higgsed, and the 2d vector multiplets become massive. By taking the gauge coupling in the GLSM to be parametrically large, the massive modes decouple and one obtains the NLSM as an effective theory. The moduli space of the GLSM becomes the target space of the NLSM.
As mentioned above, under certain conditions the two ways (i) and (ii) of introducing a surface defect are equivalent. One may suspect that integrating out the 2d degrees of freedom produces the delta-function singularities in the 4d fields on the support of the defect. Indeed, this is the case for a 1/2-BPS surface defect in 4d N = 4 SYM theory with gauge group G = SU (N ) [90,91,94]. In order to obtain a Levi type-L defect from a GLSM, the latter needs to have N = (4, 4) SUSY and global symmetry G, which we gauge to couple the GLSM to the 4d theory. The Levi subgroup L is captured by the gauge symmetry in the linear quiver: Consider the linear quiver of figure 1 whose gauge group U (K 1 ) × . . . × U (K n ) is such that K i > K i−1 for all i = 2, . . . , n. Then, the Levi subgroup The parameters (α, β, γ, η) are encoded in the GLSM as follows. The linear combination α k + iη k , with k = 1, . . . , n + 1, corresponds to the complexified FI parameters of the GLSM, and the complex structure moduli β k + iγ k characterise the 2d superpotentials.
In the NLSM description, the requirement of N = (4, 4) SUSY and global symmetry G translate to requiring the target space to be hyper-Kähler and to admit a G-action. In terms of the NLSM description, (α, β, γ) are encoded in the moduli of the target space, whereas η is associated with the 2-form B-field. The authors of ref. [94] conjecture that the NLSM target space is T * (G/L), which agrees with the moduli space M H of the Levi type-L defect. The complex dimension of the target space is which holds for general values of the parameters (α, β, γ, η). Note that this agrees with c 2d /3 of the associated N = (4, 4) GLSM in eq. (2.3).
In the case of an ambient N = 2 theory, there is a similar but weaker statement. A 2d GLSM with N = (2, 2) SUSY -or NLSM whose target space admits a Kähler structure -coupled to a 4d N = 2 theory, is equivalent in the IR to the N = 2 theory with prescribed singularities in the gauge field on the support of the defect [96].

2d Defects in Theories of Class S
A large class of 4d N = 2 SCFTs are the theories of ref. [97], often called class S. These come from the 6d N = (2, 0) SCFT of type A N −1 on a product manifold M 4 × C g,n where M 4 is a four-manifold and C g,n is a genus-g Riemann surface with n punctures. A SUSY twist makes the corresponding partition function independent of the size of both M 4 and C g,n (though still dependent on their shape). Thus, one can shrink either M 4 or C g,n to zero without affecting the value of the partition function on M 4 × C g,n . By definition, class S theories are those obtained by shrinking C g,n . When M 4 = S 4 and the class S theory has a known Lagrangian description, then its partition function can be computed exactly via SUSY localisation [57]. The AGT correspondence is the statement that the S 4 partition function is equivalent to a Liouville/Toda correlator on C g,n [68,69].
Within M-theory, these 4d theories can be found by wrapping M5-branes on M 4 × C g,n and shrinking C g,n to zero size. SUSY defects in the 4d SCFT can then be engineered by introducing a stack of either M2-or M5-branes ending on or intersecting the initial stack of M5-branes. In the 6d SCFT these M2-or M5-branes describe a 2d or 4d defect, respectively. To obtain a 2d defect in the 4d SCFT obtained by reducing on C g,n , we must either place the M2-branes at a point on C g,n or let the M5-branes wrap all of C g,n .
The M2-branes localised at a point on C g,n were discussed in detail in ref. [62] and refined in ref. [65]. In the 4d SCFT, the 2d defect arising from M2-branes is described by the n-node quiver GLSM of figure 1 with K i < K i+1 , an adjoint chiral multiplet on every node except the n th one, and non-vanishing FI parameter only for the n th node. The information encoded in the quiver can be summarised by a Young tableau of width n which labels a representation of A N −1 . The length of the j th column is the difference in the ranks of the j th and (j − 1) th node, i.e. K j − K j−1 . Furthermore, the authors of refs. [62,65] show that in the AGT correspondence to Liouville/Toda theory on C g,n , this surface defect corresponds to the insertion of a degenerate Toda primary labelled by the Young tableau. Its position on C g,n is specified by the FI parameter of the n th node. The FI parameters of all other nodes of the quiver are turned off.
The M5-branes wrapping C g,n were studied in refs. [50,98]. After compactifying on C g,n , one obtains a different type of surface defect in a 4d N = 2 SCFT which can be described by a Wess-Zumino-Novikov-Witten (WZNW) model on C g,n .
In the 4d SCFT, the authors of refs. [90,91,96] proposed a Seiberg-like duality between the 2d defects that arise from these M2-and M5-branes in 6d. More specifically, the duality is a particular type of integral transform between the partition functions of the corresponding Liouville/Toda and WZNW theories living on C g,n . We will not need any details of this duality except that, like any duality, it is a mapping between physical observables of the two cases. Of importance to us is the fact that under the duality the metric on M 4 and the submanifold Σ are invariant, and the stress tensor maps to itself. As a result, the Weyl anomaly is invariant under the duality, and hence the defect central charges are also. We will see that this is the case in our examples below.

Holographic Results
Refs. [26,99] showed that for a 2d conformal defect in a higher-d CFT, the entanglement entropy of a sphere centred on the defect includes a logarithmic term with a universal coefficient given by a linear combination of three central charges: the ambient CFT's type A central charge (when d is even), b, and d 2 . Furthermore, d 2 determines the stress tensor onepoint function in the presence of the defect. By calculating this entanglement entropy and stress tensor one-point function, holographic calculations of b and d 2 have been performed for Levi type-L defects in 4d N = 4 SU (N ) SYM [26,88,100] and for Wilson surfaces in the 6d N = (2, 0) A N −1 SCFT [25,26,100,101]. One of our goals is to reproduce these results using purely field theory means, so let us review them in detail.
For the Levi type-L surface defect in 4d N = 4 SU (N ) SYM theory, the holographic results for b and d 2 are where λ is the 't Hooft coupling of N = 4 SYM. As mentioned in section 1, in this case ref. [52] proved that d 2 = d 1 , so in fact the holographic calculations provide all three defect central charges. As also mentioned in section 1, these defects preserve enough SUSY that b cannot depend on defect or ambient marginal couplings, while d 1 and d 2 can. The b in eq. (2.5a) indeed does not depend on defect or ambient marginal couplings, and in fact depends only on the choice of Levi subgroup L. On the other hand, d 2 = d 1 manifestly depends on the defect marginal parameters β i and γ i and on the ambient marginal coupling λ.
The S-duality of N = 4 SYM sends N/λ → λ/N , under which d 2 = d 1 appears to change. However, as mentioned at the end of the previous subsection, Weyl anomaly coefficients are invariant under any duality that leaves the metric on M 4 and the submanifold Σ invariant, and maps the stress tensor to itself. This includes the S-duality of 4d N = 4 SYM. Indeed, after accounting for the S-duality transformations of β i and γ i , described in ref. [90], the combination of N , λ, β i , and γ i in eq. (2.5b) is invariant under S-duality.
Our first result is simply the observation that b/3 from eq. (2.5a) agrees exactly with c 2d /3 from eq. (2.3) for the GLSM construction of the 2d Levi type-L defect, and thus also with the complex dimension of the target space of the NLSM construction, eq. (2.4). Moreover, the expression in eq. (2.4) was conjectured to hold for arbitrary values of the parameters (α, β, γ, η), which strongly suggests that we can uniquely identify c 2d /3 = dim C X with b/3 and not d 2 /3 = d 1 /3, since the latter depend on β i and γ i .
As mentioned above, however, in all that follows we will take β i = 0 and γ i = 0. In that case, the holographic results of eq. (2.5) have b = d 2 = d 1 , so we will not be able to distinguish these three central charges from one another. This will be important in section 4, where we will make a proposal for how to extract a defect central charge from the SUSY partition function of N = 4 SYM on S 1 R × S 3 with the Levi type-L defect on S 1 R × S 1 . We will only perform an explicit calculation with β i = 0 and γ i = 0, so strictly speaking we will not be able to identify uniquely which central charge we calculate, although we will provide multiple arguments that we almost certainly compute d 2 = d 1 .
For Wilson surfaces in the 6d N = (2, 0) A N −1 SCFT we will be able to distinguish b from d 2 , since in that case generically b = d 2 . A Wilson surface defect is labelled by a Young tableau corresponding to a representation of su(N ) with highest weight ω. The holographic results for b and d 2 for a Wilson surface are [25,26,100,101] where ρ is the Weyl vector of su(N ). Clearly in these cases In section 4 we will extract a defect central charge from the SUSY partition function of the 6d we can unambiguously say the defect central charge we obtain is ∝ d 2 . However, in this case ref. [52] provided compelling evidence, though not a rigorous proof, that d 2 = d 1 , so the defect central charge we obtain could in fact be a linear combination of d 2 and d 1 .

Partition Function on S 4
In this section, we extract defect central charges from partition functions of N ≥ 2 SCFTs on M = S 4 with 1/2-BPS superconformal defects along an equatorial Σ = S 2 . For arbitrary M and Σ an infinitesimal Weyl transformation δg µν = 2g µν δω of the partition function Z gives rise to an integrated Weyl anomaly of the general form, including the defect contribution eq. (1.1), where E 4 and E 2 are the Euler densities for M and Σ, respectively, γ ab is the induced metric on Σ, and a 4d and c 4d are the central charges of the 4d CFT. When M = S 4 and the defect wraps an equatorial Σ = S 2 , all the type B terms above vanish. In particular, M = S 4 is conformally flat, so its Weyl tensor vanishes, hence W µνρσ W µνρσ = 0 and W ab ab = 0. The second fundamental form for S 2 → S 4 is pure trace, soII µ abI I ab µ = 0 as well. Thus, the full integrated Weyl anomaly reduces to a linear combination of the A-type anomaly coefficients a and b, In other words, under a global Weyl re-scaling of M = S 4 , Z → e (−4a 4d + b 3 ) ω Z. Hence, we may extract the linear combination of central charges in eq. (3.2) from the transformation of the partition function Z under a global Weyl re-scaling, and if we know a 4d for the 4d SCFT, then we can identify b. Below we exploit two methods for computing Z that make use of this idea to obtain b if the CFT enjoys enough SUSY, namely SUSY localisation and the AGT correspondence.

SUSY Localisation
In this subsection, we use existing results for Z computed via SUSY localisation [56,57] to extract novel results for b.
SUSY localisation is usually performed on the Ω-background, R 4 1 , 2 , or on an S 4 deformed by the ratio of equivariant parameters 2 / 1 ≡ b 2 . The two ultimately give equivalent results, and we will follow the latter approach. The dimensionless parameter b determines how the sphere is "squashed," which we denote as S 4 b . Viewed as a hypersurface in R 5 , S 4 b is defined by where {x i }, with i = 0, . . . , 4, are the Euclidean coordinates on R 5 , and r is the equatorial radius. Note that the mass dimensions of 1,2 are 1, which we denote by [ 1,2 ] = 1. The round S 4 of radius r is recovered in the limit 1 = 2 = 1 r . The deformation parameters 1,2 break the isometry group of the 4-sphere to U (1) × U (1). An N = 2 theory on this background preserves an su(1|1) ⊂ osp(2|4) SUSY subalgebra of the round S 4 .
Generically, the localised partition function of a 4d N ≥ 2 gauge theory without a defect factorises into three contributions [57]: a classical part Z class , a 1-loop part Z 1-loop , and an instanton part Z inst . Each of these is parametrised by the VEV of the adjoint scalar Φ = a which is valued in the Cartan subalgebra h ⊂ g. The full partition function is obtained by integrating a over h. Schematically, We implement global Weyl re-scalings by taking 1 = 2 = 1 r and then re-scaling r. If the theory is a SCFT, the 4d Weyl anomaly implies Z S 4 → r −4a 4d Z S 4 . The only contributions to a 4d come from the integration measure da and Z 1-loop , since the other factors are Weyl-invariant. More specifically, Z 1-loop is a product of one-loop determinants of Laplacians for fields of different spins. Each such one-loop determinant is an infinite product of eigenvalues that diverges, and needs to be regulated. As explained in appendix A, we use zeta-function regularisation, which, crudely speaking, means that via analytic continuation we replace each infinite product with special functions, usually combinations of (multiple) Gamma functions. From that point of view, the "quantum" contribution to the Weyl anomaly of Z S 4 b comes from the "anomalous" scaling properties of these special functions, while the "classical" contribution comes from da. We provide more details of this in appendix A, and we will see explicit examples below.

Now consider a surface defect wrapping
A 2d N = (2, 2) theory on Σ preserves the same su(1|1) ⊂ osp(2|2) SUSY subalgebra as above. Thus one can introduce couplings between the 2d N = (2, 2) theory to the ambient 4d N = 2 theory on Σ without breaking any further SUSY. As mentioned in section 2.2, we can couple the ambient 4d fields to the 2d fields on Σ by introducing superpotential couplings on Σ to couple the 2d and 4d matter multiplets and/or by gauging a global symmetry on Σ and identifying it with an ambient 4d global/gauge symmetry [62,65]. Some of the examples of 2d superconformal defects considered below are constructed from 2d N = (2, 2) GLSMs in the UV before flowing to the putative IR superconformal fixed point. The S 2 1 partition function Z S 2 of a purely 2d GLSM with gauge group G 2d can also be computed through SUSY localisation. This is most conveniently done on the Coulomb branch of the moduli space [60,61]. The field configurations on the locus are parametrised by a GNO-quantised 2d gauge flux m = 1 2π F on S 2 1 and the VEV of a real vector multiplet scalar σ. Schematically, combining the 1-loop and non-perturbative partition functions yields where W g 2d is the Weyl group of the associated gauge Lie algebra g 2d , and h Z 2d is the GNOlattice. 4 Note that the kinetic term of σ in the vector multiplet action is normalised such that [σ] = 1. We emphasise that the localised partition function is independent of the 2d Yang-Mills coupling g, and only depends on the S 2 (or S 2 1 ) through its equatorial radius. We will again implement a global Weyl re-scaling by re-scaling r, in which case the 2d Weyl anomaly implies Z S 2 → r c 2d /3 Z S 2 . Similarly to Z S 4 , the quantum contribution to the 2d Weyl anomaly comes from zeta-function regularisation of the infinite products in Z gauge 1-loop Z matter 1-loop , while the classical contribution comes from dσ. For more details, see appendix A. We will also see explicit examples in the following.
As explained in section 1 and above, our aim in this subsection is to extract b from the Weyl anomaly of the localised partition function of 2d-4d coupled systems. For such systems many SUSY localised partition functions have been computed, but we will focus on cases where the 4d ambient theory is conformal, namely N 2 free massless hypermultiplets, N = 4 SU (N ) SYM theory, and N = 2 SU (N ) SQCD with 2N flavours with 1/2-BPS N = (2, 2) surface operators, enhanced to N = (4, 4) for N = 4 SYM.

Free Massless Hypermultiplets with a Generic Surface Defect
To start, we consider the theory of N 2 free massless hypermultiplets on S 4 b . This theory enjoys global U Sp(2N 2 ) flavour symmetry. To this ambient theory we couple the 2d GLSM in figure 1, which we put on Σ = S 2 1 . The GLSM enjoys an SU (N ) × SU (N ) symmetry acting on the (anti-)fundamental chirals, whereas the bifundamental and adjoint chirals enjoy a U (1) symmetry. We couple the ambient free hypers to the GLSM via cubic and quintic superpotential couplings that identify the shared 2d-4d flavour symmetry Absent an ambient 4d vector multiplet to couple to the GLSM, the saddle points of the 2d-4d theory are parametrised by independent contributions from decoupled 2d and 4d loci, and so the SUSY localised partition function of this theory factorises [62,65] We denote by Z Σ the partition function of the GLSM on S 2 1 , and is the partition function of the N 2 free massless hypers in zeta-function regularisation. The Upsilon function is defined as where Γ 2 (z|a 1 , a 2 ) is the double Gamma function. For more details about these special functions, see appendix A. However, the only information we currently need about the Upsilon function is its behaviour under re-scaling of its arguments, eq. (A.17), where ζ 2 (s; z|a 1 , a 2 ) is the Barnes double zeta-function defined in eq. (A.1).
Since the 2d-4d partition function factorises, it is sufficient to just consider the scaling of Z Σ in order to compute b. Hence, b is identified with c 2d .
However, to be clear, we hasten to add that this c 2d is not (necessarily) the central charge of a 2d CFT, because the 2d stress tensor of our defect degrees of freedom is not necessarily conserved, due to the coupling to the ambient 4d fields. This implies various differences from a 2d CFT: no lower bound on our b = c 2d is currently known, the usual 2d c-theorem does not necessarily apply (although the b-theorem does), and so on. In practical terms, however, the upshot is that we still compute c 2d from eq. (2.2), which in particular requires identifying the representations and R-charges of the 2d fields.
Let us consider the three contributions to eq. (3.6) separately and explicitly study their scaling behaviour. If the 2d gauge group G 2d has n U (1) factors, the classical part of the localised partition function takes the form where z j ≡ e −2πξ j +iθ j , ξ j are FI parameters, θ j are theta-angles, and Tr j denotes a projection to the j th U (1) factor [95]. Writing the measure in eq. (3.6) as h 2d dσ = , it is clear that rσ in eq. (3.11) is just a dimensionless dummy variable in the integration over the locus. Hence, the classical part Z class is trivial under Weyl rescalings, as advertised. The measure, however, contributes factors of the S 4 radius r, which transform under Weyl re-sclaings with weight −rank G 2d .
The 1-loop contribution coming from the gauge sector takes the form where ∆ + is the set of positive roots and ρ 2d is the Weyl vector of g 2d , the Lie algebra associated to G 2d . Collecting the overall factors of r, we see that under Weyl re-scalings eq. (3.12) transforms with weight −dim G 2d + rank G 2d . After zeta-function regularisation, the 1-loop partition function of the matter sectorcomposed of massless chiral multiplets in the R representation of G 2d -becomes where q R is the 2d R-charge of the multiplet, and {h R } denotes the set of weights of R.
Counting the factors of r that appear in eq. (3.13), we see that Z matter 1−loop transforms with weight R (1 − q R )dim R under Weyl re-scaling. For g 2d a direct sum of semi-simple Lie algebras and u(1)'s, the remaining term in the Weyl re-scaling R {h R } 2ih R (rσ) reduces to a sum over the charges under the U (1) factors of G 2d , which vanishes.
Combining the weights from the gauge sector, the matter sector, and the measure, one finds that the partition function is scale-invariant up to an overall factor of r c 2d /3 , with c 2d given in eq. (2.2). Hence, b = c 2d in the round sphere limit 1 = 2 = 1 r , as argued above. Even though the 4d theory is very simple, considering the case of N 2 free massless hypers with a surface defect illustrates the important point that the non-trivial contribution to the central charge comes from the scaling of the 1-loop partition function. The simplicity of the above example stems from the factorisation in eq. (3.7), which immediately led to identifying b = c 2d .
For a more generic 2d-4d system, one might suspect that matter charged under both 2d and 4d gauge groups would spoil the factorisation of Z Σ →S 4 and possibly alter b. However, that is not the case, if the system enjoys enough SUSY. It is now understood that a 1/2-BPS surface defect engineered in a generic 4d N = 2 gauge theory by gauging symmetries [67] or through Higgsing [66] mixes ambient and defect degrees of freedom in only two ways. Firstly, any 4d adjoint hypermultiplet scalars frozen at their VEVs enter as twisted mass parameters in Z Σ , while keeping its functional form unchanged. Secondly, any coupling of 2d and 4d degrees of freedom leads to an extra factor in the partition function that is entirely non-perturbative: it arises from the interactions of instantons and vortices. The 1-loop part of the partition function receives no modifications. Hence, central charges extracted from Weyl re-scalings of the partition function are unchanged. In other words, we expect b = c 2d to be the case always. Indeed, we will see examples of this below.
Let us point out that the scaling behaviour of the partition function can often be obtained in a more straightforward, yet ad hoc way by using three facts: only the 1loop partition function (and the measure of the integration over the VEV of the adjoint scalar) contributes, factors of 1/r arise in the evaluation of 1-loop determinants, and special functions are the result of zeta-function regularisation.
The explicit dependence on the scale r is often left implicit. If one was given the partition function Z Σ without any scale factors, one could still deduce the scaling behaviour by re-instating the correct r-dependence, dealing with special functions appropriately and accounting for the measure. For example, if one encounters the Euler Gamma-function , one first needs to insert appropriate factors of r to make its argument dimensionless, i.e. σ → rσ. The natural function that appears in the zeta-function regularisation of the matter sector 1-loop partition function is the Barnes single Gammafunction Γ 1 (z|a, b) defined in eq. (A.2). To obtain the scaling behaviour one should interpret the Euler Gamma-function as Γ 1 one correctly recovers the partition function Z Σ with appropriate scale factors. We refer to appendix A for more details and definitions of these special functions.

Examples of Defects Coupled to Free Massless Hypermultiplets
Having determined b = c 2d for superconformal surface defects coupled to 4d free massless hypers, we can now consider some specific defect models. All that needs to be done to compute b is to determine the 2d R-charges q R of the matter fields. Due to the su(1|1)-invariant coupling between the 2d N = (2, 2) GLSM and the hypermultiplets, the 2d R-symmetry generators are linear combinations of the U (1) N generator of rotations in the normal bundle to Σ and U (1) R ⊂ SU (2) R of the ambient R-symmetry [65]. The coefficients determining the exact 2d R-symmetry depend on b. The 2d R-charges of the 4d hypermultiplet scalars restricted to Σ can be found in terms of their 4d charges under U (1) N × U (1) R . Requiring that the 2d-4d superpotentials have 2d R-charge q = 2 together with constraints from the identified flavour symmetry then fixes the R-charges of the 2d fields and sets their twisted masses to zero.
The precise superpotential terms depend on the particular quiver diagram, and were found in refs. [62,65]. In particular, they depend on whether the j th node has an adjoint chiral X j . If it does, we define η j ≡ +1, and if it does not, η j ≡ −1. Further let us define ε i ≡ n j=i η j . One finds that the hypermultiplet scalars restricted to Σ have 2d R-charge q hyper = 1 + b 2 , the fundamental and anti-fundamental chirals have q fund n + q anti-fund n = 1 − b 2 , the adjoint chirals have 15) and the bifundamentals have R-charges where we have a total of n nodes and ε n+1 ≡ +1. Notice that in a 2d SCFT, with a conserved stress tensor, unitarity and the BPS bound require positive R-charges. In contrast, our 2d defect fields do not have a conserved 2d stress tensor, and so can have negative R-charges.
Example 1: N = (2, 2) SQCD. As a first example consider N = (2, 2) SQCD with gauge group G 2d = U (K) and N fundamental and N anti-fundamental chiral multiplets coupled to N 2 ambient free massless hypers. Note that q fund n + q anti-fund n = 0 in the round sphere limit b = 1. Thus, using eq. (2.2) we find Example 2: N = (2, 2) SQCDA. We now add an adjoint chiral to the previous example, where q X = −2 in the limit b = 1. Using eq. (2.2), this "extra" field thus contributes an additional These two examples clearly obey the b-theorem [19]. If we start in the UV with SQCDA, with b in eq. (3.18), and deform the theory by a mass term for the adjoint chiral, then in the IR we will find SQCD [62], with b in eq. (3.17). In this case, b UV − b IR = 9K 2 ≥ 0.
Example 3: N = (2, 2) quiver with n adjoint chirals. We can also consider more general quiver gauge theories. For example, consider the n-node quiver depicted in figure 1 with gauge group G 2d = U (K 1 ) × . . . × U (K n ), N fundamental and N anti-fundamental chirals of U (K n ) and adjoint chirals on each node, coupled to N 2 free hypers. Using q X j = −2, q bif j(j−1) + q bif (j−1)j = 4 and q fund n + q anti-fund n = 0, and eq. (2.2) we find after a bit of algebra that where we have defined K 0 ≡ 0.
Example 4: N = (2, 2) quiver with (n − 1) adjoint chirals. Consider the same quiver as the previous example, but with adjoint chirals on all nodes but the n th one. In this case, q X j = 4, q bif j(j−1) + q bif (j−1)j = −2, and q fund n + q anti-fund n = 0, so that These two examples also obey the b-theorem [19]. If we start in the UV with an N = (2, 2) quiver with n adjoint chirals, with b in eq. (3.19) and deform by a mass term for the n th adjoint chiral, then in the IR we will find an N = (2, 2) quiver with n−1 adjoint chirals [62], with b in eq. (3.20). We thus have

N = 4 SYM with a Generic Surface Defect
To construct a 1/2-BPS superconformal surface defect in N = 4 SYM, one can couple a 2d N = (4, 4) GLSM to the ambient theory [94]. N = (4, 4) SUSY requires the i th node in figure 1 to have an adjoint chiral multiplet X i for all i. The N = (2, 2) adjoint chiral recombines with the N = (2, 2) vector multiplet into an N = (4, 4) vector multiplet. Similarly, the bifundamentals φ bif i(i+1) and φ bif (i+1)i regroup into bifundamental hypers, and the N (anti-)fundamental chirals φ fund n andφ anti-fund n recombine into N fundamental hypermultiplets. The N hypers enjoy SU (N ) flavour symmetry such that the GLSM can be coupled to 4d N = 4 SU (N ) SYM theory by gauging the 2d flavour group. As argued in the previous subsection, b = c 2d as there is no perturbative 2d-4d contribution to the partition function. We may thus calculate b through yet another counting exercise. 5 Assuming the GLSM flows to an IR fixed point, the central charge of the 2d SCFT is given by eq. (2.2). To determine the 2d R-charges, one considers the allowed superpotential terms which schematically look like W = φXφ in N = (2, 2) language. The R-charge assignments are easily deduced by looking at the U (1) R action on the mesons built from fundamental chirals, which combine into non-compact scalars at the IR fixed point. The exact low-energy U (1) R symmetry cannot act as a rotation on the mesons due to chiral factorisation of the R-symmetry in a CFT. This gives the assignment that matter sector chiral multiplets in the (anti-)fundamental and bifundamental representations have q = 0, while the adjoint chiral multiplets carry q = 2. Thus, eq. (2.2) gives as quoted in eq. (2.3), which is in agreement with the complex dimension of the moduli space of the Levi type-L defect in eq. (2.4), and with the holographic result in eq. (2.5a), thus proving that the latter is not merely the large-N limiting value.

N = 4 SYM with a Full Levi Defect
A useful check of the previous result eq. (3.22) can be performed in special cases. In refs. [59,63], the authors consider N = 2 * SYM with gauge group G = SU (N ) and a full surface defect (L = T) engineered by putting the theory on the orbifold C × C/Z N . 6 By taking the mass of the 4d adjoint hyper to zero, the N = 2 * SUSY enhances to N = 4.
Let us now compute b for this system. The non-trivial contribution comes from the 1-loop partition function, where a i are the components of the VEV of the 4d adjoint scalar Φ = diag(a 1 , . . . , a N ), and x denotes the ceiling of x.
The arguments of the Upsilon-functions in eq. (3.23) have mass dimension one. To make the overall scale factor explicit, we should factor out 1/r from their arguments. Define the dimensionless quantities˜ 1,2 ≡ r 1,2 and Q ≡˜ 1 +˜ 2 . Under a re-scaling, the Upsilonfunction transforms according to eq. (3.10), which means eq. (3.23) becomes where In other words, the 1-loop determinant in the presence of the full Levi defect is scaleinvariant. Hence, 6 In ref. [59], the authors found the following equivalence: instanton moduli space of a 4d N = 2 gauge theory with full surface defect ⇐⇒ instanton moduli space without defect but on the orbifold C × C/ZN . This allows one to compute the instanton partition function of the coupled 2d-4d system by instead working on the orbifold. It was then conjectured in ref. [63] that this equivalence should hold more generally for the full partition function. The author of ref. [63] computes the 1-loop determinants on the orbifold and goes on to check that the partition function obtained in this way correctly encodes the coupled 2d-4d and 2d degrees of freedom.
where the right-hand side is the contribution of the measure in eq. (3.4). The central charge a 4d for N = 4 SU (N ) SYM theory is well-known, 4a 4d = N 2 − 1, and so we find b 3 = N 2 − N , (3.28) which agrees with eq. (3.22) in this special case, as advertised. This agreement may seem surprising, given the different M-theory origins of this surface defect and the defects that lead to eq. (3.22). This full surface defect comes from the compactification on a torus of the 6d N = (2, 0) theory with a codimension-two defect [59,63]. This arises in M-theory from a stack of coincident M5-branes wrapping the torus and a second stack of M5-branes intersecting the first stack and also wrapping the torus, thus producing the orbifold in 4d, i.e. a codimension-two singular surface. On the other hand, the surface defects that lead to eq. (3.22), namely the GLSM quivers reviewed in section 2, come from a codimension-four defect in the 6d theory. This arises in M-theory from M2branes ending on M5-branes, localised at a point on the torus. In these two descriptions b agrees because of the duality of ref. [96], mentioned in section 2, which leaves invariant the S 4 , the Σ = S 2 wrapped by the defect, and the stress tensor, and hence leaves invariant b. Of course, also crucial is the fact that b depends only on the Levi subgroup of each defect: if b depended on more detailed information, then the equivalence would not be possible.

N = 2 SQCD with 2N Flavours and a Full Levi Defect
A large class of SCFTs are the theories of class S introduced in ref. [97]. One of the simplest, yet non-trivial examples of such theories is massless N = 2 SQCD with 2N flavours. A full surface defect in this theory is considered in ref. [63].
The 1-loop determinant with a full surface defect is where h i are the weights of the fundamental representation of SU (N ). Following the same strategy as above, we factor out 1/r to write the arguments of the special functions in terms of dimensionless quantities˜ 1,2 . Let us consider the numerator first. The scaling behaviour of the Upsilon-functions in eq. (3.10) in the b → 1 limit gives a scaling weight of the numerator of the form where ρ is the Weyl vector, and ∆ is the set of all (positive and negative) roots. The denominator contributes to the overall scaling weight a factor A vanishing beta function implies (see e.g. ref. [57]) Hence, upon summing the contributions of the numerator and denominator one finds that all terms that depend on the VEV a cancel, giving an overall scaling weight for eq. (3.29) of the form Finally, to account for a 4d , we normalise by the partition function without the defect: which scales with weight where we have again used eq. (3.32). Thus subtracting eq. (3.35) from eq. (3.33), we find that b for a full Levi type-L defect in N = 2 conformal SQCD is given by

AGT Correspondence
In the seminal work [68], Alday, Gaiotto and Tachikawa (AGT) proposed a remarkable correspondence between the partition function of a class of 4d asymptotically conformal N = 2 SUSY SU (2) quiver gauge theories on S 4 b and Liouville theory on a genus-g Riemann surface with n punctures, C g,n . This AGT correspondence has been further extended to SU (N ) quiver gauge theories and A N −1 Toda field theories [69].
In this section, we will employ an extension of the methods originally used in ref. [102] to compute the type A anomaly coefficients in a number of 4d class S theories in order to extract the central charge b of a certain type of surface operators, via the AGT correspondence. To begin with, we give a very brief review of the AGT correspondence, and then discuss its modification for taking into account the insertion of a certain class of surface operators in the 4d theory. In appendix B, we establish our notation, briefly review A N −1 Toda field theories, and report all the formulae that we will need in this section.
The object of interest on the 4d side of the AGT correspondence is the SUSY localised partition function Z S 4 b of an N = 2 gauge theory on S 4 b discussed in some detail above in section 3.1 and displayed in eq. (3.4). In general, Z S 4 b depends on some complexified couplings denoted by {q}, some masses {m} and VEVs of the adjoint scalars {a} in the vector multiplets. Hence, in most of the examples considered below the ambient theories are not strictly 4d SCFTs but rather are SUSY gauge theories that are conformal in certain limiting regimes. However, we will explicitly compute b only for 4d massless free hypermultiplets, which are a CFT. We will of course reproduce the results of section 3.1.1, but now from the perspective of the Liouville/Toda theory. In order to discuss the 2d side of the AGT correspondence, consider the class S theory constructed by an n-node linear quiver [97] where the gauge sector of the i th node is described by the gauge group SU (2) i , the coupling τ i , the "sewing parameter" q i = e 2πiτ i , and adjoint scalar VEVs a i . The matter sector of the quiver is described by two antifundamental hypermultiplets of mass µ 1,2 at the first node, bifundamental hypermultiplets of mass m i between the i th and (i + 1) th nodes, and two fundamental hypermultiplets of mass µ 3,4 on the n th and final node (see e.g. the left side of figure 2 for the case n = 1). The UV curve of the kind of quiver just described is given by the (n + 3)-punctured sphere C 0,n+3 . Through the AGT correspondence, the partition function eq. (3.4) is equivalent to the (n + 3)-point correlation function of the Liouville field theory on S 2 , namely and V α is the suitably normalised Liouville exponential (see eq. (B.8)). Since all Liouville/Toda correlation functions going forward will be of the form V αa (∞) V rm 0 (1) V α b (0) . . . , we will omit position unless clarity is needed. Expanding the right hand side of eq. (3.37) using OPEs, the resulting conformal blocks correspond to the classical and instanton partition functions in eq. (3.4), while the one-loop contribution corresponds to the coefficient of the three-point function and structure constants.
As an example, let us consider the four-punctured sphere in Liouville theory corresponding to the SU (2) quiver diagram in figure 2 (and to eq. (3.37) with n = 1). The correlator can be decomposed in terms of s-channel conformal blocks F (s) as [103,104] where the integral is taken along the Q/2 + iR line, C(·, ·, ·) denotes the coefficient of the Liouville three-point function and C · ·,· the structure constants which correspond to the fusion of two primaries. Both C(·, ·, ·) and C · ·,· are defined in terms of the primary operators normalised as in eqs. (B.8) and (B.9). In ref. [68], the authors showed by explicit computations that the combination C(·, ·, ·) C · ·,· provides the one-loop part of the partition function together with the Vandermonde determinant, while the conformal blocks F α give the instanton and classical contributions. Finally, the integration over the internal Liouville/Toda momentum α corresponds to the integration over the VEV of the adjoint scalar of the SU (2) gauge group. When the rank of the gauge group N > 2 the situation is slightly different. In particular, many types of punctures are possible on C g,n , each of which corresponds to a hypermultiplet with some flavour symmetry: see for example ref. [105]. We will only consider punctures corresponding to SU (N ) flavour nodes, called full punctures, and U (1) flavour nodes, called simple punctures [69]. In the Toda theory picture, a full puncture corresponds to the insertion of a Toda primary operator with unconstrained momentum α. A simple puncture corresponds to a semi-degenerate Toda primary operator, i.e. with momentum either of the form α = κh 1 or α = −κh N with h 1 (−h N ) being the highest weight of the (anti-) fundamental representation of SU (N ) and κ a numerical factor. The standard example is 4d N = 2 conformal SQCD whose quiver is depicted in figure 2. The partition function corresponding to this quiver can be expressed as the Toda four-point function on the sphere with two full and two simple punctures.

Surface operators and Toda degenerate primaries
In refs. [47,58,106] the Toda/gauge theory dictionary was enlarged to describe the addition of 1/2-BPS line and surface operators. It has been shown that 1/2-BPS surface operators in 4d class S theories descending from 2d superconformal defect operators in the 6d N = (2, 0) SCFT correspond to the insertion of one or more degenerate Toda primary operators in eq. (3.37) [47,61,70,[107][108][109]. 7 As opposed to the punctures giving rise to semidegenerate primary insertions described above, a generic degenerate Toda primary operator has momentum α = −bω 1 − 1/bω 2 where ω 1 and ω 2 are highest weight vectors of two representations R 1 and R 2 of A N −1 . Note that the parameter b is mapped to the squashing parameter of S 4 b in the AGT correspondence. In ref. [62] the authors found that inserting degenerate operators of the type α = −bω, i.e. R 2 is a trivial rep, corresponds to engineering an N = (2, 2) surface operator whose field content is described by the quiver in figure 1. 8 The representation R is described by a Young tableau consisting of n columns with N j boxes for 1 ≤ j ≤ n, which corresponds to an n-node quiver with adjoint chiral multiplets on every node except the n th node. The number of boxes of each column is related to the ranks of the 2d gauge nodes as N j ≡ K j − K j−1 with K 0 = 0. The surface defect in the 4d theory sits at a marked point (x, x) on C g,n , which corresponds to a non-trivial FI parameter and theta angle only in the 7 For surface operators descending from 4d defect operators in 6d, the correspondence is a bit different in that the legs of the 4d defect wrapping the Riemann surface deform the Toda theory on Cg,n to another 2d CFT, e.g. a WZNW model. We will not consider such operators in this section. 8 Even though in the present work we will focus on degenerate insertions parametrised by only one representation of AN−1, we mention that a generic degenerate operator, with α = −bω1−1/bω2, corresponds to two surface defects supported on two (squashed) 2-spheres that intersect at the north and south pole of S 4 b . These can be engineered by intersecting M2-branes ending on N M5-branes wrapping the Riemann surface Cg,n [65]. n th node. This kind of surface operator has an M-theory realisation in terms of M2-branes labelled by the representation R of A N −1 ending on the wrapped M5-branes. Considering multiple degenerate primary insertions with appropriate representations, one could turn on additional FI parameters or, alternatively, add an adjoint chiral on the n th node or remove them from other nodes [62]. 9 In the following, we will focus only on rank-k totally (anti-)symmetric representations. We will denote by R N j ,ε j the rank-N j totally symmetric (ε j = +) or anti-symmetric (ε j = −) representation of A N −1 . The type of representation determines if an adjoint chiral occurs on a given node or not: For R Nn,+ there is an adjoint chiral on U (K n ), while for R Nn,− there is not. Then, if ε j = ε j+1 there is an adjoint chiral on U (K j ), otherwise not.
Consider coupling N 2 free 4d hypermultiplets to a surface operator through a cubic superpotential involving the ambient hypermultiplets and the chiral multiplets in the (anti-) fundamental representation of U (K n ). As discussed in section 3.1, the partition function for this 2d-4d system factorises as Z Σ →S 4 = Z S 4 Z Σ , and the precise correspondence with the Toda theory reads [62] where ω (N j ,ε j ) is the highest weight of the representation R N j ,ε j , and (1 + im s + im s ) . Let r 1 = 1/ 1 be the equatorial radius of Σ = S 2 1 , and denote the complexified twisted masses of the fundamental chiral multiplets m fund ≡ m = r 1m + iq/2 wherem is the (real) twisted mass and q is its 2d R-charge. Similarly, let m anti-fund ≡m be the complexified twisted masses of the anti-fundamental chirals. The R-charges are constrained by the superpotentials and are given in eqs. (3.15) and (3.16) and the text above them. Similarly, the superpotential and the global symmetry SU (N ) × SU (N ) × U (1) relate the hypermultiplet complexified twisted masses of the 4d theory to the (anti-)fundamental chiral masses as [62,65]. Finally, the normalisation of the semi-degenerate and degenerate primaries is given in eq. (B.9).

b from Toda correlators
Before considering specific examples, let us explain the strategy for computing b from the 2d CFT correlators through the AGT correspondence. First of all, as observed above, b is computed by the scaling weight of Z 1-loop and the measure of the integral over the locus.
This means that, via the AGT dictionary, we should be able to extract b solely from the scaling behaviour of the three-point function coefficient and structure constants appearing in the Toda correlators and, in principle, the integration measure over internal Toda momenta in the conformal block expansion. In particular, we will not need the explicit form of the conformal blocks since, as summarised above, they correspond to the instanton part of the partition function, which does not contribute to the Weyl anomaly.
To isolate b from the scaling behaviour of the Toda correlator, we remove the contribution from the 4d ambient theory by dividing the Toda correlator with degenerate insertions by the same correlator without them. Crucially, since the fusion rule eq. (B.7) prevents the degenerate insertion from adding more integrals, the normalisation by the correlator without the degenerate operator insertion eliminates the contribution from the integration measure over internal Toda momenta. Thus, the only contributions to b should come from the scaling behaviour of three-point function coefficients and structure constants themselves.
From the gauge theory side we know that the special functions in the structure constants arise from the zeta-function regularisation of some infinite product. In particular, the structure constants can be expressed in terms of γ 1 (x|1/b), which is defined in eq. (A.7), and Υ(x) ≡ Υ(x|b, 1/b). Following the same logic as above eq. (3.14), one re-introduces the factors of r by interpreting γ 1 (x|1/b) as γ 1 (x/r|1/r) where we set b = 1. From eqs. (A. 16) and (A.17), we arrive at the following substitution rules under Weyl re-scaling: We stress that this approach can be applied whenever the special functions come from the zeta-function regularisation of infinite products. Before moving on to more complicated examples, let us illustrate the method in the simplest non-trivial case: one degenerate insertion of the type V −bω (N 1 ,ε) in the theory of N 2 free hypermultiplets, i.e. the correlator eq. (3.40) with n = 1. In the case of a single degenerate primary insertion, the fusion rule eq. (B.7) allows us to write the four-point function in the s-channel decomposition as [104,110] where |F| 2 = F(x)F(x) denotes the conformal blocks, the sum runs over the weights {h (N 1 ,ε 1 ) } of the representation R N 1 ,ε , and C(·, ·, ·) and C · ·,· are the appropriately normalised coefficient of the Toda three-point function eq. (B.6) and the structure constants, respectively. Now, we need the structure constant part of the Toda correlator without any degenerate insertion. This is given by the three-point function [62] C(α ∞ , α 0 , , (3.45) which can easily be found by using eq. (B.6) together with the normalisations eqs. (B.8) and (B.9). Notice that this corresponds to the partition function of the N 2 free hypermultiplets with the masses M st defined in the text below eq. (3.42). Thus, the quantity to study is (3.46) which in this case reduces to a combination of γ-functions. Note that the function C N 1 ,ε 1 depends on the particular weight h (N 1 ,ε 1 ) . However, its weight under Weyl re-scaling is independent of it. This is important because it ensures that the correlator (3.44) has a well-defined scaling behaviour from which the charge b can be extracted.

Examples of Defects Coupled to Free Massless Hypermultiplets
We will now compute b for defects coupled to 4d free massless hypers, reproducing our SUSY localisation results from section 3.
where {p} labels the weights of the representation as explained in appendix B. The constant A 1 contains factors of b coming from the form of the structure constants eq. (B.10). We observe that those factors, being independent of the Toda momenta, are unaffected by the Weyl re-scaling, so they can be ignored. By performing the replacement eq. (3.43) first, then setting the (anti-)fundamental chiral twisted masses to zero and leaving the R-charges generic, we find an overall prefactor expressible as a power of r. The exponent is given by (q s +q s ) . (3.48) Recall from above eq. (3.15) that q s +q s = 0 (q ≡ q anti-fund ) for b = 1. Thus, the last term vanishes, and one finds agreement with eq. (3.17).
Example 2: N = (2, 2) SQCDA. Now consider a single symmetric insertion that corresponds to 2d N = (2, 2) SQCD with gauge group U (K 1 ) (K 1 = N 1 ) and an adjoint chiral on U (K 1 ). The structure constant part of the Toda correlator is [62] where n s are non-negative integers that label the weights h (N 1 ,+), [n] as explained in appendix B. Also in this case, the constant A 1 contains factors independent of the momenta, and it does not play any role in the scaling behaviour of the structure constants. Thus, by applying again the substitution eq. (3.43) and setting the twisted masses to zero, we find that the exponent of the overall factor of r in the structure constants is which agrees with b given in eq. (3.18).
Example 3: Multiple degenerate insertions. Finally, we consider the insertion of n degenerate primary fields in totally symmetric or totally anti-symmetric representations. In this case, we can use the fusion rule eq. (B.7) to determine the allowed momenta which run between the degenerate insertions. The non-trivial part in writing the (n+3)-point function in terms of three-point functions is finding the conformal blocks. This has been discussed in ref. [62] by employing the correspondence with the four-dimensional gauge theory and the localisation results. However, to obtain b it suffices to consider the following quantity . (3.51) In the equation above, h [n j ] = N t=1 n j t h t for integers n j t ≥ 0 such that h [n j ] − h [n j−1 ] is a weight of the representation R N j ,ε j . For R N j ,+ (symmetric), n j t − n j−1 t ∈ Z >0 , while for R N j ,− (anti-symmetric), n j t − n j−1 t ∈ {0, 1}. In both cases, we have N t=1 n j t − n j−1 t = N j , from which it follows N t=1 n j t = j i=1 N i . The form of the first contribution does not depend on the type of representation (symmetric or anti-symmetric), and it can be found from eq. (3.45): where we employed the shift property eq. (A.8).
Let us start with the symmetric case where we need the structure constants in eq. (B.11). It is convenient to compute the scaling weight of the structure constants (denoted in the following with square brackets) involving only a single symmetric degenerate insertion. By setting b = 1 and the (anti-)fundamental chiral masses to zero, we obtain (3.53) Then, by adding the contribution of eq. (3.52) to eq. (3.53) we get where in the second step we used that Again, this result reproduces eq. (3.19) found by employing the formula eq. (2.2).
Before we move on to study a generic combination of symmetric and anti-symmetric degenerate insertions let us first obtain the contribution due to a single anti-symmetric insertion. The structure constants for this case are given by eq. (B.10). By using repetitively the fact that n j t − n j−1 t ∈ {0, 1} (since the j th rep is anti-symmetric), we obtain the corresponding contribution to the central charge b, namely (3.55) By comparing the above with eq. (3.53), we find that the difference between the contributions of a single symmetric and anti-symmetric insertion is simply given by With this result, we can compute b for free massless hypermultiplets coupled to the GLSM of figure 1 with any combination of ε = ± (and thus any number of adjoint chirals). In particular, we can compute b when all the insertions are anti-symmetric. We find

SUSY Partition Function on S 1 × S d−1
In an SCFT with at least a single U (1) R symmetry, the twisted partition function Z of the theory on M ≡ S 1 R × S d−1 is identified with the superconformal index [81], up to a normalisation factor (that will be important in what follows!). Here S 1 R is a temporal circle of radius R. The superconformal index is, like other indices, functionally a count of a certain set of degrees of freedom obeying specific shortening conditions. After affecting the sum over multiplets obeying the shortening condition, the remaining non-trivial contributions to the superconformal index come from multiplets that are protected under RG flow. Further, the superconformal index is itself invariant under continuous deformations that preserve the supercharge used to define the index. Thus, Z is a likely candidate to capture putative defect central charges. Indeed, as we will demonstrate below in certain cases, an overall prefactor in Z depends explicitly on defect Weyl anomaly coefficients.
Specifically, in this section we will first argue for the appearance of the central charge d 2 in Z in the presence of a 2d superconformal defect wrapping Σ = S 1 R × S 1 . After setting the general framework in section 4.1, we will examine two models where d 2 has been calculated holographically [26]: Levi type-L defects in N = 4 U (N ) SYM theory (section 4.2) and the Wilson surface operator in the A N −1 6d N = (2, 0) SCFT (section 4.3). We will see that upon deformation of Z by these specific 2d defects the exponent in the normalisation, i.e. the SUSY Casimir energy (SCE) 10 [72,74,84], changes by a factor proportional to d 2 .
Actually, in the 6d N = (2, 0) SCFT we obtain a more general result: we compute the change in the SCE due to a pair of Wilson surfaces wrapping Σ 1 = S 1 R × S 1 1 and Σ 2 = S 1 R × S 1 2 , where S 1 1 and S 1 2 intersect only at the poles of the S 5 . Taking the limit of a single Wilson surface on, say, Σ 1 , we will find that the change in the SCE reduces to a term ∝ d 2 . Interestingly, for two intersecting Wilson surfaces the change in the SCE is not merely the sum of results for two lone Wilson surfaces, but also contains contributions purely from their intersections at the poles. A detailed interpretation of this result would require knowledge of the algebra of intersecting surface operators and the degrees of freedom intrinsic to the 1d submanifold they share, which is beyond the scope of this paper.
In the final part of this section, we will propose the form of a type B anomaly coefficient (or possibly a linear combination thereof) of a 1/2-BPS codimension two defect wrapping S 1 R × S 3 in the A N −1 6d N = (2, 0) SCFT, our eq. (4.34) below. Despite the lack of an explicit form of the defect Weyl anomaly for a 4d defect in a 6d CFT, the logic of the construction that computed d 2 in the two examples considered below is straightforward to extend to superconformal defects of arbitrary codimension. We thus claim that eq. (4.34) must be proportional to type B defect central charges.

Anomalies and SUSY Casimir Energy
In this subsection we are concerned with setting up a general framework for arguing that the change of the SCE due to a defect on Σ is proportional to d 2 . To begin, consider the twisted partition function of a SCFT, Z(R, µ j ), on M = S 1 R × S d−1 for even d, where µ j are chemical potentials for superconformal Cartan generators that commute with the supercharge used to define the index. The main argument in refs. [72,74,83] is that by utilising SUSY localisation to compute Z(R, µ j ), one finds a general form proportional to the superconformal index I: where E c is the SCE. The understanding here is that, as an object counting protected operators starting from the identity operator, I is an ascending polynomial in non-negative powers of fugacities, q j , start at one, i.e. I = 1 + q # j + . . ., i.e. # > 0. In the presence of a defect preserving the supercharge used to define the index, I will generically pick up negative powers in an expansion in q, which will need to be compensated in order to maintain the normalisation that the index begins counting with the identity operator [80,86]. That is, the superconformal index in the presence of a surface defect is still counting states, in a similar sense as in the ambient theory, but now including defect states in radial quantisation around the defect.
A form of the SCE as an integrated anomaly has been conjectured, but to our knowledge not rigorously proven. Here we will briefly present the existing pieces of evidence for this conjecture, which we will then use to motivate the appearance of defect type B anomaly coefficients in the SCE. We will subsequently show in the examples of sections 4.2 and 4.3 that d 2 indeed appears in the SCE, providing compelling evidence for our arguments.
In ref. [72], for 4d SCFTs on S 1 R × S 3 , E c was computed by SUSY localisation to be proportional to the Weyl anomaly coefficients a and c as where µ 1,2 are chemical potentials for SO(2) 1,2 rotations preserved in squashing the S 3 . This connection between E c and Weyl anomaly coefficients was refined in ref. [84]. The authors of ref. [84] draw a direct relationship between the Weyl anomaly and the SCE explicitly by reducing Z(R, µ i ) on the squashed three-sphere S 3 1 , 2 , with squashing parameters 1 and 2 , to SUSY Quantum Mechanics (SQM) on S 1 R . The expectation value of the 1d theory's Hamiltonian, H SQM , in the limit R → ∞ was identified with the SCE in a manifestly scheme independent way, unlike directly computing S d−1 √ g T τ τ , which is scheme dependent. In the end, H SQM was found to be given by eq. (4.2) where the role of the chemical potentials µ 1,2 is played explicitly by the squashing parameters 1,2 .
In ref. [85] the above arguments of ref. [72,83] were extended to more general backgrounds, of the form M = S 1 R × M d−1 , where for example in d = 4 M d−1 is a circle bundle over a Riemann surface. In d = 4 examples, it was demonstrated that in terms of U (1) f "flavour parameters" collectively referred to as ν and the "geometric parameter"τ , which is related to complex structure moduli, the twisted partition function has a Casimir contribution of the form where A abc and A a are cubic U (1) f and mixed gravitational anomalies respectively. The authors of ref. [74] made a more general conjecture, that the SCE in a SCFT is given by the equivariant integration of the anomaly polynomial, A d+2 (M), which, if true, obviously means E c can depend on Weyl anomaly central charges. However, we reiterate that eq. (4.4) remains a conjecture, albeit one strongly supported by evidence from a number of examples in various dimensions [74]. Now, we would like to outline how we conjecture a 2d surface defect wrapping Σ → M modifies E c . One line of reasoning starts from eq. (4.4), and requires that we make two assumptions from the start: (i) the deformed anomaly polynomial factorises into ambient and defect localised contributions (ii) there is a sufficient amount of superconformal symmetry preserved by the defect such that the defect Weyl anomaly sits in a multiplet with other global defect localised anomalies, e.g. defect chiral anomalies. In addition to finding a general proof of E c being given by A d+2 (M), proving the validity of these assumptions is the focus of on-going work. If both assumptions (i) and (ii) hold, then the result of the equivariant integration of A 4 (Σ) is related to the integrated defect Weyl anomaly. That is, the anomaly coefficients that can appear in A 4 (Σ) are controlled by coefficients appearing in the non-vanishing contributions to the integrated defect Weyl anomaly.
From the form of the defect Weyl anomaly reviewed in section 1, it is immediately clear in eq. (1.1) that the type A term will not contribute to the integrated anomaly: the Euler character of Σ = S 1 R × S 1 , and its squashings, vanishes. However, the integrated type B contributions coming fromII 2 and W ab ab do not necessarily vanish on a squashed sphere. Moreover, for our 2d superconformal defects d 1 = d 2 has been proven in d = 4 and conjectured in other d > 4 [52]. Thus, if assumptions (i) and (ii) hold, then the change in E c due to the presence of a superconformal defect wrapping Σ must be proportional to d 2 when d = 4, and, supported by evidence in the following subsections, we conjecture that it is proportional to d 2 in other d as well.
It should be mentioned that there could be another, possibly more direct, way to show that the defect induced change in E c is related to d 2 following the logic in ref. [84]. Since S 1 R ⊂ Σ, the reduction of the defect on S d−1 will change the SUSY Hamiltonian H SQM on S 1 R by additional chiral and Fermi multiplets, as well as possible superpotential deformations due to the reduction of couplings between defect and ambient degrees of freedom. In the cases where there is an explicit defect action, the computation would amount to the regulated counting done in ref. [84]. However, this would not in general be a constructive proof of the connection between defect central charges and the SCE, but if demonstrated to hold in a number of examples, could provide a useful computational tool to try to predict d 2 in novel models.
Finally, while not directly related to anomalies, a different line of reasoning also suggests the appearance of d 2 in the SCE. From the point of view of constructing VOAs from 4d SCFTs [77,79] and 6d SCFTs [78], the Schur limit of the SUSY partition function of an N ≥ 2 SCFT on S 1 × S 3 or N ≥ (1, 0) SCFT on S 1 × S 5 is the character of the vacuum module of the VOA, see e.g. ref. [111]. As shown in ref. [80], the SUSY partition function in the presence of a superconformal surface defect inserted normal to the VOA plane instead computes in the Schur limit the character of some non-vacuum module. Crucially, the dimension of the defect identity in the module is given by −d 2 [52]. This is precisely the statement that introducing the defect shifts E c by a term ∝ d 2 . This VOA perspective will be especially useful in our 6d computation below.

4d SUSY Casimir Energy
In this subsection we consider the SUSY partition function of N = 4 SU (N ) SYM theory on M = S 1 R × S 3 with a Levi type-L defect along Σ = S 1 R × S 1 . Crucially, for the reasons mentioned in section 2, we need to set β i = γ i = 0 in eq. (2.5b), so we can only study cases where b = d 2 . Nevertheless, from the arguments above and our evidence in 6d in section 4.3, we believe the SCE obtained from the SUSY partition function is proportional to d 2 alone.
To compute the twisted partition function, Z, on M = S 1 R × S 3 , we will use the correspondence between its Schur limit and correlators in 2d q-deformed YM theory (qYM) on a genus-g Riemann surface with n-punctures in the zero-area limit, C g,n [75]. Ref. [87] showed that the deformation of the 4d twisted partition function by surface defects corresponds in the qYM theory to local operator insertions O R i , where R i is a label descending from the representation data of the i th defect. In the description of qYM as Chern-Simons theory on S 1 × C g,n , these local operators arise from Wilson lines in representations R i extended along the S 1 , and hence localised at a point on C g,n .
While the authors of ref. [87] consider an arbitrary number of defect insertions, for us it will suffice to consider the 2d qYM one-point function corresponding to a single surface operator in representation R of the 4d gauge symmetry. This takes the form Explicitly, the Schur polynomial is computed as a ratio of determinants where the matrix A has components A ij ( a, ) = a i +N −i j . The modular-S matrix, S, appearing in eq. (4.6) is defined by S R,R = S 0,0 χR(q ρ+κ )χ R (q ρ ) . (4.8) Here, we are using the form of the Weyl vector ρ = (ρ 1 , . . . , ρ N ) in a particular orthogonal basis The original partition data = [ 1 , . . . , N −1 , 0] re-expressed in the orthogonal basis is denoted κ = [κ 1 , . . . , κ N ], where (4.10) The notation adopted in the arguments of the Schur polynomials in eq. (4.8) is then to be interpreted as, e.g. q ρ+κ = (q ρ 1 +κ 1 , . . . , q ρ N +κ N ).
The case that we are interested in is a single N = (4, 4) Levi type-L defect labelled by a representation R, specified by a partition of N , inserted in 4d N = 4 SU (N ) SYM theory. This corresponds to computing the one-point function of O R on a torus with no punctures (g = 1, n = 0), in which case eq. (4.6) completely collapses to (4.11) From eq. (4.11), we can read off the defect SCE from the overall power of q that needs to be stripped off in order to match the "start-at-one" normalisation of the index. Explicitly, the Schur polynomial χ R (q ρ ) can be expanded in q as an overall prefactor multiplying an ascending polynomial in non-negative powers of q, (4.14) 11 Formally, one starts with a partition = [ 1, . . . , N ] and constructs the transpose partition as Put more plainly,˜ i is given by the number of entries in that are greater than or equal to i. The Young tableau of˜ is obtained from the tableau of by exchanging columns and rows. Thus, if is a partition of N , then so is˜ .
Comparing to the holographic result for d 2 in eq. (2.5b), with β i = γ i = 0, we thus identify the Levi type-L defect contribution to the SCE encoded in the 2d qYM one-point function: Crucially, this calculation involved no approximations, relying only on SUSY and the equivalence of the twisted partition function with qYM correlators. This calculation thus strongly suggests that the holographic results in eq. (2.5b) are in fact exact, and not merely large-N and/or strong-coupling limits.
To repeat once again, in these cases where β i = γ i = 0 the Levi-type defect in N = 4 SYM has b = d 2 , so that our identification of d 2 alone in eq. (4.15a) is ambiguous. In other words, how do we know we obtain d 2 alone, rather than b alone, or a linear combination of b and d 2 ? If our arguments in section 4.1 for the connection between the integrated defect Weyl anomaly and the defect contribution to the SCE hold, then the exponent in eq. (4.15a) is d 2 and not b. Moreover, in the following subsection we will study Wilson surface defect indices in 6d, and in that case b and d 2 in eq. (2.6a) are distinct, allowing us to identify d 2 unambiguously, which will provide compelling evidence for our arguments more generally.

6d SUSY Casimir Energy
In this subsection, we are concerned with the twisted partition function of the 6d N = (2, 0) A N −1 SCFT on the squashed S 1 R × S 5 in the presence of 2d or 4d superconformal defects. 12 In M-theory this SCFT arises as the low-energy theory on the worldvolume of N coincident M5-branes, and we are interested in the defects arising from either M2-or M5-branes that end on this initial stack of M5-branes. The M2-branes give rise to a 2d defect (codimension four), namely a Wilson surface operator, which we place along Σ = S 1 R ×S 1 . The M5-branes give rise to a 4d defect (codimension two), which we place along S 1 R × S 3 . Ref. [73] carried out a systematic study of the twisted partition function of this 6d SCFT with both types of defects. Using the results of ref. [73] and our arguments from section 4.1, we will calculate central charges for both types of defects. For the Wilson surfaces, we will unambiguously find d 2 in the SCE. For the 4d defects, we do not yet know their contribution to the trace anomaly, so we cannot say exactly which central charge(s) we are computing. Our result serves as a prediction for such putative central charge(s).
The generators h i rotate the planes R 2 i ⊂ R 6 into which the squashed S 5 is embedded.
Among the SUSY generators Q R 1 R 2 h 1 h 2 h 3 , where the indices are all ± 1 2 , the privileged supercharge used to construct the index is Q ≡ Q ++ −−− . The states contributing to the superconformal index obey the shortening condition in saturating the bound (4.16) Assuming saturation of eq. (4.16), the index can be expressed as where H Q is the subspace of the Hilbert space annihilated by Q and Q † . The fugacities are q i ≡ e −R i and p ≡ e −Rµ , where µ is the chemical potential for the R-symmetry generator R 1 − R 2 . The unrefined limit of I, defined by µ → 1 2 ( 1 + 2 − 3 ), has an additional supercharge Q ≡ Q +− ++− that commutes with the Cartan generators, and so the unrefined index collapses to where q ≡ q 3 and s ≡ q 1 /q 2 . Note the privileged status of rotations in the plane R 2 3 , which is identified with the VOA plane of the 6d theory [78]. The index I unref is then interpreted as the character of the vacuum module of the VOA.
Due to the lack of a Lagrangian description of this 6d SCFT, the authors of ref. [73] compute its twisted partition function by dimensionally reducing on S 1 R and computing the twisted partition function of the 5d U (N ) N = 2 SYM theory with coupling g 2 = 2πR [112] on the squashed S 5 , Z S 5 . The codimension four and two defects wrapping S 1 R in 6d reduce to Wilson lines or certain 3d defects in the 5d SYM theory on the squashed S 5 . Further, the authors of ref. [73] argue that both the perturbative and non-perturbative contributions to the partition function of the localised theory on the squashed S 5 are sufficient to count the states contributing to the 6d index, and hence to defect indices. Although this is far from a proven fact about the dimensional reduction to 5d, we will adopt the same working assumption. The fact that for Wilson surfaces we will recover precisely the holographic result for d 2 provides some evidence for this assumption.
In the absence of defects, the localised partition function of the 5d U (N ) N = 2 SYM theory on a squashed S 5 takes the form where Z 1 is the Nekrasov partition function [56] on S 1 1 × R 4 2 , 3 , with Z 2 and Z 3 obtained from Z 1 by cyclic permutation of the labels {1, 2, 3}, and a is a constant adjoint-valued scalar parametrising the locus.
Without any defects, the localised partition function of the free non-Abelian tensor multiplet in the 6d N = (2, 0) A N −1 theory in the unrefined limit computes the character of the vacuum module in the W N algebra. Defining 2πiτ = −R 3 so that q = e 2πiτ , and defining 1 2 = 1 and b 2 = 1 / 2 , 13 the partition function sees contributions in the unrefined limit from the three fixed points on the S 1 i of the form where η(·) is the Dedekind η function. Let Q = ρ(b + b −1 ) with ρ being the Weyl vector of su(N ), and let W g be the Weyl group of g = A N −1 . After integrating over a, eq. (4.19) becomes where ε(σ) = (−1) (σ) and (σ) is the length of the Weyl group element σ. The exponent of the prefactor is related to the central charge c of the VOA as q − c 24 . Recalling that η(τ ) ∝ q 1/24 , we thus have where we identify c/24 as the chiral limit of the 6d SCE found in ref. [74].

2d Defects
Adding two surface operators wrapping S 1 R × S 1 1 or S 1 R × S 1 2 will deform the index to compute the character of degenerate modules of the associated W N -algebra in the VOA plane. The reduction to 5d yields Wilson loop operators with winding n 1 and n 2 on S 1 1 and S 1 2 , respectively, and carrying irreducible representations of su(N ) with highest weights ω 1 and ω 2 , respectively. The fixed point contributions on S 1 1 and S 1 2 are modified from those in eq. (4.20) to where Tr ω is a trace over the representation specified by ω. Again, the plane R 2 3 is designated as the VOA plane and so the Wilson lines cannot wrap S 1 3 and also preserve the necessary nilpotent charge needed to define the VOA, 14 hence Z 3 remains unchanged compared to eq. (4.20). Plugging Z 1 and Z 2 from eq. (4.23) into the partition function and integrating over a gives 13 Note that in this section we adopt a different convention for b compared to section 3. Moreover, in contrast to section 3, the squashing parameters 1,2,3 of the S 5 are chosen to be dimensionless, i.e. they come with appropriate factors of the equatorial radius of the S 5 , which we take to be the identity in this section. 14 It is also true for 2d N = (4, 4) defects in 4d N = 4 SYM theory that the surface operators must be inserted orthogonal to the chiral algebra plane. Note, though, that a 2d chiral (e.g. N = (0, 8) or (0, 4)) superconformal defect could be inserted along the chiral algebra plane while preserving the nilpotent supercharge used to define the VOA. We thank W. Peelaers for pointing this out.
where the new "central charge" is To isolate the defect contribution to the partition function, we divide eq. (4.24) by the ambient theory result in eq. (4.21), which gives the change in the central charge, Eq. (4.26) is our most general result for 2d defects in the N = (2, 0) 6d SCFT, for two intersecting Wilson surfaces. However, to compare to the holographic result for a single Wilson surface in section 2, we restrict to a single defect wrapping, say, S 1 1 , in which case eq. (4.26) becomes Taking b → 1, so that Q = ρ(b + b −1 ) → 2ρ, and using d 2 = 24(ρ, ω) + 6(ω, ω) from eq. (2.6b), we find C ω 1 − c = 48 (ρ, ω 1 ) + 12 (ω 1 , ω 1 ) = 2 d 2 . (4.28) We have thus shown that a single defect changes the normalisation factor from q −c/24 to q −Cω 1 /24 , or recalling that q = e 2πiτ = e −R 3 , the defect shifts the SCE from E c = − c 24 3 to E c = − Cω 1 24 3 . Our result eq. (4.28) then shows that the change in E c is ∝ d 2 , as advertised. Crucially, as mentioned above, for a Wilson surface in the 6d N = (2, 0) theory we can distinguish d 2 and b, namely d 2 = b + 3(ω, ω), as opposed to the Levi defect in 4d. The comparison thus leaves no doubt: d 2 controls the defect contribution to the SCE.
However, similar to the Levi defect of section 4.2, the calculation here involved no approximations, relying only on SUSY and the assumptions about the reduction on S 1 R mentioned above. Our result eq. (4.28) thus provides strong evidence that the holographic result for d 2 in eq. (2.6b) is in fact exact, and not just the leading large-N limiting value.
Our result for two intersecting Wilson surfaces, eq. (4.26), is not merely the sum of two copies of the result for the single Wilson surface, eq. (4.28), due to a cross term 2(ω 1 , ω 2 ). This difference could potentially arise for various reasons: some special contributions to the Weyl anomaly from the intersections, some 1d degrees of freedom at the intersection points, and so on. We leave this as an important question for future research.

4d Defects
In 6d N = (2, 0) A N −1 SCFTs, there is another class of superconformal defects that one could construct: 4d defects. In the M-theory description, these types of defects arise from 1/2-BPS M5-M5-brane intersections. The authors of ref. [73] also constructed the index for these 4d defects, using arguments similar to the 2d case.
Codimension 2 operators in 6d N = (2, 0) A N −1 SCFTs, in particular, are in one-to-one correspondence with homomorphisms : su(2) → A N −1 , and in the unrefined limit correspond to a deformation of the VOA by the insertion of a semi-degenerate operator labelled by a partition of N, i.e. [N 1 , . . . , N n+1 ] where n+1 i=1 N i = N . That is, 4d superconformal operators preserve the Levi subalgebra l = s n+1 i=1 u(N i ) . In the reduction along S 1 R , which the intersecting M5-branes wrap, such a codimension 2 defect has an equivalent description as a prescribed singularity in the gauge field of the resulting 5d N = 2 SYM theory. Given a Levi subalgebra l, the monodromy parameters are m = ⊕ n+1 i=1 m i with each m i being a rank N i vector whose components are all identically m i , and the Weyl vector of l is ρ l = ⊕ n+1 i=1 ρ N i with each ρ N i being the Weyl vector of su(N i ). The SUSY vacua of the localised theory are labelled by σ ∈ W g /W l -where W g and W l are the Weyl groups of g = A N −1 and l, respectively -which also labels a permutation of the monodromies, i.e. different inequivalent choices of embeddings of l in A N −1 .
To compute the index in the presence of the defect, we need to use the form of the localised partition function in eq. (4.19) supplemented by the classical action from the monodromies given by e −2πi(σ( m),a) and the Nekrasov partition functions corresponding to the particular and choice of σ where ∆ + i is the space of positives roots of the i th summand of l and as above Z 3 = η(−τ −1 ) 1−N . Summing over all σ and integrating over the locus parametrised by a gives We can easily compute (ρ l , ρ l ) = 1 12 ( n+1 i=1 (N 3 i − N i )) by considering each individual su(N i ) summand in l. In the limit b → 1 we thus find As mentioned above, we do not have a sufficient understanding of the form of the Weyl anomaly of a 4d defect in a 6d CFT to state definitively which central charge(s) the above expression might be. For now, eq. (4.34) serves as a prediction for 4d superconformal defects in the 6d N = (2, 0) SCFT. Our result in eq. (4.34) bears a resemblance, modulo overall sign and powers of N and N i , to d 2 for the N = (4, 4) Levi type-L surface operator in 4d N = 4 SYM theory in eq. (2.5b). Given the connection between the two constructions via dimensional reduction, this superficial resemblance is perhaps not surprising. Beyond the scope of the current work, but the focus of on-going investigation, is finding the behaviour of the defect Weyl anomaly of the 4d Levi type-L defect in 6d under dimensional reduction to a 2d Levi type-L defect in 4d N ≥ 2 SCFTs.

Summary and Outlook
We have illustrated a variety of techniques for computing the central charges b and d 2 of 2d superconformal defects in SCFTs. These techniques rely only on a sufficient amount of SUSY, with no approximations. In particular, we used existing results for SUSY localisation, the AGT correspondence, and superconformal indices to extract new results for b and d 2 . Some of these results agreed perfectly with existing holographic results, proving that the latter were not merely large-N or strong-coupling limits, but were in fact exact.
Our results pave the way for many fruitful generalisations. Obviously, a variety of other existing results for SUSY partition functions on S d and S 1 × S d−1 could be mined for further novel results for b and d 2 . Additionally, to our knowledge a variety of 2d superconformal defects have yet to be described using any of the SUSY methods we have discussed. A prominent example is chiral defects, such as defects with 2d N = (0, 4) SUSY. Chiral defects break parity, producing parity-odd terms in the trace anomaly that define two parity-odd central charges [26,27]. These could in principle be calculated using the methods we have described. Furthermore, as deformations of the superconformal index, 2d N = (0, 4) defects can preserve the nilpotent supercharge used in the cohomological construction of chiral algebras from 4d SCFTs [77], and so their central charges may appear in the vacuum character of a deformed chiral algebra.
Other approaches to computing SUSY partition functions on S d and S 1 × S d−1 could also be developed along similar lines as useful tools to extract defect central charges in novel systems. Examples include geometric engineering [70,113], or computing a 5d SUSY partition function on S 1 × S 4 with a 3d SUSY defect along S 1 × S 2 [114] and then reducing on the common S 1 to obtain a 4d SUSY partition function on S 4 with a 2d defect along S 2 → S 4 [4]. More importantly, studying how the defect trace anomalies and associated central charges behave under dimensional reduction could provide a new window into how defect physics changes under RG flows across dimensions [115].
All the above methods could also be straightforwardly generalised to defects of other dimensions. For example, in 4d SCFTs various 1/2-BPS interfaces and domain walls have been studied using holography [116][117][118], SUSY localisation [58], and other methods [119]. In these cases the interface contribution to the trace anomaly defines two central charges [120][121][122][123][124] that could in principle be calculated from existing results. In 5d and 6d SCFTs, higher-dimensional defects are possible, such as the 4d defect in the M5-brane theory that we discussed at the end of section 4. However, in these cases the defect contribution to the trace anomaly, and in fact many other quantities are unknown, so what (linear combination) of central charges the SUSY methods could compute is unclear.
Indeed, more generally the contributions of defects to trace anomalies as in eq. (1.1), entanglement entropy [41], and other quantities are clearly crucial for characterising and classifying defects, including for proving c-theorems [19,99], positivity [26] or other lower bounds [32], and other constraints on defect central charges. We hope our methods provide useful tools for addressing such issues. which frequently appears in Liouville/Toda theory. This Upsilon-function obeys Υ(z + a 2 |a 1 , a 2 ) = γ 1 (z|a 1 )Υ(z|a 1 , a 2 ) , (A. 6) where γ 1 (z|a) ≡ Γ 1 (z|a) Γ 1 (a − z|a) , and a similar relation for the shift Υ(z + a 1 |a 1 , a 2 ) replacing a 1 → a 2 . This can be recast in the more familiar form Υ(z + a 2 |a 1 , a 2 ) = a 2z/a 1 −1 1 γ(z/a 1 )Υ(z|a 1 , a 2 ) , . (A.9) The special functions above appear in the evaluation of 1-loop determinants as alluded to in section 3. One usually encounters infinite products that diverge and require regularisation. Zeta-function regularisation instructs us to replace a diverging product In this appendix we review the essential features of the A N −1 Toda field theory needed for the computations in section 3.2. For more details see for example refs. [110,127]. The action for A N −1 Toda field theory is given by where g µν is the metric of the Riemann surface, R the corresponding scalar curvature, b is the dimensionless coupling constant, e i and ρ are respectively the simple roots and Weyl vector of the A N −1 Lie algebra, and (·, ·) denotes the scalar product on the weight space. The requirement of conformal invariance fixes Q = (b + 1/b)ρ. In terms of Q, the central charge is given by Besides the conformal symmetry, the theory of eq. (B.1) is invariant under higher-spin symmetry transformations generated by the (n − 1) holomorphic currents W k (z) with spins k = 2, . . . , N , whose algebra is called W N -algebra. These currents can be written in terms of the field φ as where h i = h 1 − e 1 − · · · − e k with k = 1, . . . , i are the weights of the fundamental representation of the A N −1 with highest weight h 1 . Their scalar product reads (h i , h j ) = δ ij − 1/N . We observe that W 2 (z) = T (z). It is not difficult to see that for N = 2, the Toda field theory reduces to the Liouville theory. Primary fields with respect to the W N -algebra are the vertex operators V α = e (α,φ) , (B.4) with quantum numbers ω (k) (α) and conformal dimension ∆(α) = ω (2) (α) = (α,2Q−α) 2 . The three-point functions of W N primaries can generically be expressed as While the z-dependence is fixed by conformal symmetry, all the non-trivial information about the three-point function is encoded in the coefficient C(α 1 , α 2 , α 3 ). In the Liouville case, this coefficient has been found for generic values of the momenta α [104,128]. On the other hand, for N > 2 the structure of the three-point function is more complicated and a general expression is not available. However, there are useful limiting cases where analytic results can be obtained. For example, a simplification occurs if one of the primaries is semi-degenerate, i.e. it satisfies the special condition α = κh 1 where κ is a real number. In this case, the coefficient in eq. (B.5) can be expressed in a closed form where the function Υ(x) ≡ Υ(x|b, 1/b) (see eq. (3.9)). Another remarkable case is when the field is fully degenerate. Degenerate fields are parametrised by α = −bω 1 − 1/bω 2 where ω 1 and ω 2 are highest weights of two representations R 1 and R 2 of A N −1 . The operator product expansion of a degenerate primary with a generic primary field V α consists of a finite number of primaries. Precisely, we have where the square brackets denotes all the descendants and α s,p = α − bh (ω 1 ),s − 1/bh (ω 2 ),p , h (ω 1 ),s and h (ω 2 ),s being the weights of the representation R 1 and R 2 , respectively. In this work we only consider the case α = −bω 1 .
In the computations performed in section 3.2, we will employ the following normalisation for the generic W N -primary fields [62]: while for semi-degenerate and degenerate vertex operators we define With these choices of normalisation, the three-point function eq. (B.6) simplifies to eq. (3.45) of the main text with α 1 = α ∞ , α 2 = α 0 . For reference and use in section 3.2, we list here the results for the structure constants in the cases that R 1 is the rank-k totally antisymmetric or totally symmetric representation of A N −1 .
The rank-k antisymmetric representation denoted as R k,− has highest weight given by ω (k,−) = k s=1 h s while all the other weights can be expressed in terms of the weights of the fundamental representation as h (k,−),{p} = {p} h p where the set {p} consists of k numbers such that 1 ≤ p 1 < · · · < p k ≤ N . The number of distinct weights is given by the number of ways in which one can choose such a set {p}. Using the normalisations in eqs. (B.8) and (B.9), the structure constants take the form [110]  where in the last step we employed that γ(bx) = b 2bx−1 γ 1 (x|1/b), which can be deduced from the definitions in eqs. (A.7), (A.9) and the property in eq. (A.4). The rank-k symmetric representation is denoted R k,+ , its highest weight is ω (k,+) = kh 1 , and all of its other weights are h (k,+),[n] = N s n s h s with N s=1 n s = k. The corresponding structure constants have been found in refs. [62,110]: (B.11)