S-duality wall of SQCD from Toda braiding

S-duality domain walls couple quarks of a four-dimensional $N=2$ theory on one side of the wall to monopoles on the other side. This paper describes the S-duality wall of SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional $N=2$ SQCD with gauge group U(N-1) and 2N fundamental and 2N antifundamental chiral multiplets. The theory is found through the AGT correspondence by computing the braiding kernel of two semi-degenerate vertex operators in the Toda CFT. This three-dimensional theory is a limit of a USp(2N-2) gauge theory whose four-dimensional lift is self-dual, a situation reminiscent of how the theory T[G] on the S-duality wall of $N=4$ super Yang-Mills is self-mirror.

We will consider three-dimensional domain walls which interpolate between S-dual descriptions of four-dimensional N = 2 SU (N ) gauge theories. These S-duality domain walls are constructed in two steps 1 . First one defines a Janus domain wall by letting the gauge coupling vary near the wall from one constant value to another. In general, the theories on each side of the wall are physically distinct, but when they are S-dual applying S-duality to one side of the wall makes coupling constants equal. The resulting S-duality domain wall interfaces between identical theories and couples electric degrees of freedom on one side to magnetic degrees of freedom on the other side. The construction can be made to preserve half of the supersymmetry of the bulk theory.
Domain walls (and boundary conditions) of 4d N = 4 SU (N ) super Yang-Mills (SYM) have been extensively studied [GW09b,GW10a,GW09a]. The S-duality wall of this theory is alternatively described as a certain 3d N = 4 theory T [SU (N )] on the interface, coupled to fields on both sides of the wall. The 3d theory can be determined as the limit of the 4d/3d system when the coupling of the 4d theory goes to zero. Consider first the Janus domain wall interpolating between a weakly coupled N = 4 SYM and its S-dual. In the zero-coupling limit, the degrees of freedom on one half-space decouple and we are left with a Dirichlet boundary condition for the other half-space. The half-space theory and its boundary are realized as D3 branes ending on D5 branes in IIB string theory. Next, note that S-duality of N = 4 SYM is realized as S-duality in IIB string theory. The dual brane setup consists of D3 branes ending on NS5 branes. The 4d fields, now weakly coupled, can be explicitly decoupled by placing the other end of the D3 branes on additional D5 branes. Based on this brane diagram (directions of branes are indicated below), the 3d N = 4 theory T [SU (N )] on the domain wall is the infrared limit of the following quiver gauge theory: This U (N −1)×U (N −2)×· · ·×U (1) gauge theory has hypermultiplets in the bifundamental representation of each pair U (k) × U (k − 1), as well as N hypermultiplets in the fundamental representation of U (N − 1), which have an SU (N ) flavour symmetry. While the symmetry can be gauged using the projection of one of the two 4d vector fields, coupling to the other 4d theory is more subtle: since IIB S-duality interchanges D5 and NS5 branes, T [SU (N )] is self-mirror; the manifest SU (N ) symmetry of the mirror description is gauged by the other 4d vector field. In this brane construction of T [SU (N )], it is crucial that S-duality of 4d N = 4 SYM coincides with S-duality in IIB string theory. Unfortunately, this does not generalize to 4d N = 2 theories 2 . For the large class of 4d N = 2 SU (N ) gauge theories obtained by twisted dimensional reduction of 6d (2, 0) A N −1 superconformal theory on a punctured Riemann surface Σ, the AGT correspondence provides another approach [AGT10] (see also [Wyl09,HH12,NT13]). It relates observables of the 4d theory on an ellipsoid S 4 b to observables of the Toda CFT on Σ, a generalization of the Liouville CFT with a larger symmetry algebra W N . In particular, an S-duality domain wall placed along the equator (or a parallel) S 3 b of S 4 b corresponds to a certain W N braiding transformation [DGG11].
2 In fact, the world-volume theory of fractional D3 branes at the center of an orbifold R 4 /Z k is a 4d N = 2 necklace quiver. However, IIB string theory only provides SL(2, Z) S-dualities acting on all gauge couplings at once, while there exists a much richer set of dualities acting "locally" on the quiver.
Section 2 explains this relation between the expectation value of an S-duality wall placed along the equator S 3 b of S 4 b , the partition function of a 3d theory on S 3 b coupled to 4d fields on both sides of the wall, and an integral kernel which braids W N primary operators of the Toda CFT. The same approach was applied in [HLP10] to the mass deformation of 4d N = 4 and in [TV14] to 4d N = 2 SU (2) SQCD with 4 fundamental hypermultiplets.
This paper focuses on 4d N = 2 SU (N ) SQCD with 2N fundamental hypermultiplets. The relevant braiding kernel is determined in later sections using 2d CFT techniques. Comparing it to explicit expressions of S 3 b partition functions [Jaf12, HHL11a, HHL11b] we deduce a description of the S-duality wall of 4d N = 2 SU (N ) SQCD as 3d N = 2 U (N − 1) SQCD on the wall coupled to the 4d theories on both sides of the wall. We then find evidence that the 3d theory can also be obtained as a certain limit of an USp(2N − 2) theory, in which some symmetries are more manifest. With notations explained below, (1.2) In these quiver, the 4d theories the upper and lower round nodes labelled N denote SU (N ) gauge groups of the 4d theories on both sides of the wall, the diagonal edges stemming from them are 2N hypermultiplets (again on both sides of the wall) that share a common SU (2N ) flavour symmetry across the wall. The rest of the quiver describes the 3d N = 2 theory on the wall: a U (N − 1) or USp(2N − 2) vector multiplet coupled to 2N fundamental and N + N antifundamental chiral multiplets. The labels ±µ will be explained shortly. Additionally, the 3d and 4d matter multiplets are coupled by a cubic superpotential W along the defect: Here, q s and q s denote the 2N antifundamental chiral multiplets and q f the 2N fundamental ones, while Φ| 3d and Φ | 3d are limits of 4d hypermultiplets at the interface. In isolation, the U (N −1) 3d theory has an SU (2N )×SU (2N )×U (1) flavour symmetry. The superpotential identifies the subgroup SU (N ) × SU (N ) × U (1) × SU (2N ) of 3d flavour symmetries to 4d symmetries: the two SU (N ) gauge groups, the shared SU (2N ) flavour symmetry and the baryonic flavour symmetry U (1) which is inverted by S-duality. Furthermore, W must have R-charge 2 under the SO(2) R symmetry of 3d N = 2. This is a subgroup of the SU (2) R symmetry of 4d N = 2, so SO(2) R -charges of the 4d fields Φ and Φ are integers. By continuity these charges on S 4 b must be equal to those in flat space, which are 1 since hypermultiplet scalars are in a doublet of SU (2) R . As a result, the R-charges of q and q must sum to 1, as do those of q and q . Up to a diagonal gauge redundancy, this sets R-charges of all 3d chiral multiplets to their canonical value 1 2 . The USp(2N − 2) theory is similar. The flavour symmetry is U (4N ) because the fundamental and antifundamental representations of USp are isomorphic. The superpotential breaks this symmetry as before, but an additional U (1) µ symmetry remains, under which the 2N chiral multiplets marked +µ have charge +1 and the others charge −1. The limit (1.2) means that we add ±µ to the masses of chiral multiplets according to their U (1) µ charges, then take µ → ±∞. We find evidence in Section 5.2 that despite giving masses to all chiral multiplets, the µ → ±∞ limit does not decouple them all. The gauge group seems to be reduced to U (N − 1), a subgroup under which the (anti)fundamental representations of USp(2N − 2) each split as sums of a fundamental and an antifundamental representations. Giving a vacuum expectation value of µ to the U (1) ⊂ U (N − 1) vector multiplet scalar cancels the masses of half of these chiral multiplets while doubling masses of the others. The µ → ±∞ limit is thus the U (N − 1) 3d theory depicted in (1.2). A separate concern caused by U (1) µ is that the R-symmetry of 3d N = 2, which is abelian, can mix with U (1) µ along the RG flow, and one should perform F -extremization [Jaf12] for each value of µ to determine the IR R-charges. Very limited numerical tests suggests that R-charges remain bounded and their µ dependence does not affect the limit.
Discrete symmetries of the pair of 4d theories are also symmetries of the 3d theory. In particular, applying charge conjugation to both symmetry groups SU (N ) × U (2N ) of the 4d SQCD theory leaves it invariant. We prove in Section 5.3 that the partition function of the USp(2N − 2) theory is invariant. This relies on hyperbolic hypergeometric integral identities [Rai06] which descend from identities between indices of Seiberg-dual 4d N = 1 USp(2N − 2) theories (given the number of flavours, the rank N − 1 is unchanged). The proposal is checked in at the level of partition functions, for which the contribution from mesons introduced by Seiberg duality combines nicely with the contribution of hypermultiplets of each 4d theory, simply changing the sign of their masses. Given that the limits µ → ±∞ are equivalent, the symmetry should extend to the U (N − 1) 3d theory. This self-duality property of the domain wall, required by a symmetry of 4d theories, is analoguous to how the theory T [SU (N )] on the S-duality wall of 4d N = 4 SYM must be self-mirror: in the brane realization, exchanging the two 4d theories corresponds to applying IIB S-duality, which acts on T [SU (N )] as mirror symmetry. In our case, it is not clear how to interpret these equalities of partition functions in terms of 3d dualities: the Aharony dual [Aha97] of a USp(2N − 2) theory with 4N fundamental chiral multiplets has a different gauge group USp(2N ). However, the integral identities we use reduce to Aharony duality in the limit where one flavour becomes massive.
Our description of the duality wall of SU (N ) SQCD with 2N hypermultiplets as U (N − 1) SQCD with 4N chiral multiplets may seem at odds with previous results [TV14] in the SU (2) case. From a known Virasoro braiding kernel, the theory on the duality wall was found to be 3d N = 2 SU (2) SQCD with 6 doublet chiral multiplets. Along the steps of the derivation, a possible description as U (1) SQED with 4 positively and 4 negatively charged chiral multiplets was discarded due to a constraint on R-charges and masses of the 8 chiral multiplets. In fact, this is precisely our U (N − 1) theory with 2N + 2N chiral multiplets, and R-charges and masses are constrained by the superpotential (1.3). Recall now that our U (N − 1) = U (1) description is a limit of an USp(2N − 2) = SU (2) theory with 8 doublets of masses ∼ ±µ. In Section 5.3 we use integral identities between indexes of Seiberg-dual SU (2) theories (in four dimensions) to find that the S 3 b partition functions of SU (2) SQCD with 8 doublets is invariant under a certain mapping of the masses. The mapping depends on how the 8 doublets are grouped into 4+4 fundamental/antifundamental chiral multiplets. For one such grouping, two doublets acquire a mass ±2µ while other masses do not depend on µ. The limit µ → ±∞ then precisely reproduces SU (2) SQCD with 6 doublets. Again, it is not clear how to give an interpretation in terms of 3d dualities because the dual of an SU (N ) theory is expected to be a U (1) × U ( N ) theory [PP13]. The rest of this paper is organized as follows. Section 2 explains the AGT correspondence relating the S-duality domain wall of 4d N = 2 SU (N ) SQCD to a Toda CFT braiding kernel, then gives an introduction to the Toda CFT. Section 3 evaluates some braiding matrices which are special cases of the braiding kernel when an operator is fully degenerate. Section 4 generalizes these discrete results to a continuous braiding kernel. It describes how this integral kernel reduces to Liouville CFT braiding kernels [PT99] and proves that the kernel obeys a shift relation deduced from Moore-Seiberg pentagon identities [MS88]. Section 5 extracts the gauge theory description of the S-duality wall from the braiding kernel. Using an hyperbolic hypergeometric integral identity [Rai06], it checks that the kernel has the appropriate discrete symmetries. We conclude with some remarks in Section 6.

AGT relation
This section describes supersymmetric localization for 4d N = 2 theories, then translates S-duality domain walls of SU (N ) SQCD to the 2d CFT language using the AGT correspondence.

Localization
For a given choice of supercharge Q under which a path integral is invariant, localization reduces the path integral to a simpler integral over Q-invariant field configurations only 3 . As an IR cutoff we place the theory on the ellipsoid S 4 b given in coordinates as Preserving supersymmetry requires non-trivial background fields [HH12]. There exists another deformation of the sphere in which the parameter b is complex rather than real [NT13]. Domain walls will be placed at constant x 0 , such as the equator S 3 b at x 0 = 0. The partition function of 4d N = 2 theories on S 4 b was computed [Pes12, HH12] using a supercharge Q whose square rotates the ellipsoid in the planes (x 1 , x 2 ) and (x 3 , x 4 ) and leave two poles x 0 = ±1 fixed. In Q-invariant configurations one vector multiplet scalar takes a constant value ia and other fields vanish, except at the North and South poles where there can be instantons and anti-instantons. The exact S 4 b partition function is then More precisely, localization relies on a choice of Q-closed and Q-exact (and positive) deformation term δS for the action: the deformed path integral including a factor exp(−tδS) is then independent of t ≥ 0. At large t the saddle point approximation is exact. For an appropriate δS, saddle points are Q-invariant configurations.
where the real scalar a runs over the gauge Lie algebra, x stands for exponentiated gauge coupling constants and m for the masses. The integrand consists of a classical contribution exp(−S cl ) evaluated at the saddle point (away from the poles), a one-loop determinant due to fluctuations around the saddle point, and (anti)instanton contributions at the poles, which depend (anti)holomorphically on x. These instanton partition functions are power series which normally converge in a finite region near the weakly-coupled limit x → 0. After reducing the integral to the Cartan algebra, combining the resulting Vandermonde determinant with Z 1-loop , and factorizing the classical contribution into holomorphic functions of x andx, one gets (2.3) below. S-dual theories have equal ellipsoid partition functions, hence On the other hand, consider the 4d N = 2 theory on S 4 b with coupling x throughout the ellipsoid, coupled to 3d fields on the equator. Q-invariant configurations of such a 4d/3d system have constant vector multiplet scalars ia and ia on the two halves of S 4 b , instantons at each pole, and possibly additional non-trivial field configurations on the 3d equator. The contribution from 3d fields is simply the S 3 partition function of the theory obtained by freezing the 4d fields to their values (ia, ia ). The 4d classical and instanton contributions combine as before into f (m, a, x) and f D (m, a ,x) (provided an appropriate Chern-Simons term is included on the wall), and finally 4d one-loop determinants on each half-ellipsoid are functions of (m, a) and of (m, a ). Altogether, (2.7) can be reproduced provided one finds a 3d theory whose ellipsoid partition function is essentially B(m, a, a ): To find this 3d gauge theory description we must evaluate the right-hand side of (2.8). The relevant structure constants C(m, a) are known. The one-loop determinants of 4d vector and hypermultiplets are only known on the full ellipsoid and not on a half-ellipsoid. We take as inspiration the analoguous situation in two dimensions: sphere one-loop determinants involve a combination Γ(x)/Γ(1 − x) while hemisphere ones involve Γ(x) or 1/Γ(1 − x) depending on boundary conditions [ST13,HO15,HR13]. In four dimensions, we note that the one-loop determinant of a hypermultiplet of mass m is Γ b q 2 + im Γ b q 2 − im 4 . It is thus natural to propose that the one-loop deteminant on a half-ellipsoid is a single one of these two Γ b functions. The situation is similar for vector multiplets. Even if the proposal turns out to be incorrect, our main statements about how the 4d theories are coupled by a 3d theory will hold, since one-loop determinants depend on the two theories separately. The most important ingredient in (2.8) is the S-duality kernel B.

Toda CFT
To determine the kernel B we will use the AGT correspondence [AGT10], found by remarking that the various factorizations (2.3) and (2.4) of Z into holomorphic factors are reminiscent of conformal block decompositions in 2d CFT. Observables of 4d N = 2 SU (N ) gauge theories (of class S) on S 4 b are equal to observables in the A N −1 -type Toda CFT. This generalization of the Liouville CFT (the case N = 2) has a symmetry algebra W N (for N = 2, the Virasoro algebra). We use standard notations 5 . The vertex 4 We denote q = b + b −1 . Besides the Barnes double-Gamma function Γ b = Γ 1/b , normalized by Γ b ( q 2 ) = 1, we will use the double-Sine function . All are meromorphic in x and obey Here m, n ≥ 0 are integers, γ(y) = Γ(y)/Γ(1 − y), and we used the Euler identity Γ(y)Γ(1 − y) = π/ sin(πy). 5 Let h1, . . . , hN denote the weights of the fundamental representation of AN−1, which sum to 0. These form an overcomplete basis of the Cartan algebra h, identified to h * by the Killing form defined by hi, hj = δij −1/N . The highest weight of any representation of AN−1 can be written Ω = n1h1 +· · ·+nN hN with integers n1 ≥ · · · ≥ nN = 0. We also let ρ = 1 hi be the half-sum of positive roots, q = b + b −1 and Q = qρ. operators V α , whose momentum α depends on N − 1 parameters, are W N primaries of dimension ∆(α) = 1 2 α, 2Q − α . They are invariant under Weyl transformations, which permute their components α − Q, h i 6 . Among these primary operators we call the oneparameter class of momenta α = κh 1 semi-degenerate, or simple 7 , and the discrete set α = −Kbh 1 (for integer K ≥ 0) fully degenerate 8 . The S 4 b partition function of 4d N = 2 SQCD is then equal to a sphere correlator of two generic and two simple vertex operators: The cross-ratio x of their positions is the exponentiated gauge coupling, mapped to 1/x by S-duality. The unimportant exponents γ i can be fixed by matching Toda CFT and gauge theory asymptotics as x → 0, ∞. The momenta α 1 and κ 2 h 1 encode N hypermultiplet masses and α 3 and κ 4 h 1 the other N . The Toda CFT correlator thus does not make all gauge theory symmetries explicit, which leads to various sign asymetries. Momenta are (2.10) From the Toda CFT point of view, the two S-dual decompositions (2.3) and (2.4) of the partition function are conformal block decompositions obtained by taking the operator product expansion (OPE) of V κ 2 h 1 (x,x) with either V α 1 (0) ("s-channel decomposition") or with V α 3 (∞) ("u-channel decomposition"). Considering the s-channel for definiteness, the integration variable a parametrizes the primary operator resulting from the OPE of V α 1 with V κ 2 h 1 . The one-loop contributions C(m, a) are structure constants of the Toda CFT, and the (anti)instanton partition functions f (m, a, x) and f (m, a,x) are (anti)holomorphic s-channel conformal blocks.
Four-point function of W N primary operators do not decompose so simply into (anti)holomorphic conformal blocks in general. Inserting a complete set of states (both primaries and their descendants) in a generic four-point function each three-point function feature a semi-degenerate vertex operator, its null-vectors can be used to convert the action of W and W into that of Virasoro generators only, which are known to act as differential operators. While this fails for the t-channel decomposition of (2.9), it holds for the s-channel and the u-channel, and it guarantees that the contributions from descendants of a given primary operator factorize into conformal blocks. The s-channel and u-channel decompositions converge in different regions |x| ≶ 1. In both cases, every conformal block is a fractional power of x multiplied by a power series which converges in the unit disc. They can in fact be analytically continued to the whole plane minus branch cuts joining 0, 1, and ∞. A convenient choice will be to cut along [0, ∞), in other words normalize conformal blocks so that their leading term is a fractional power of (−x). We define the braiding kernel as the integral kernel expressing s-channel blocks in terms of u-channel ones after analytic continuation. This is the Toda CFT translation of the S-duality kernel defined in (2.5). For comparison we write formulas next to each other: The α 32 integral runs over imaginary values for α 32 − Q. The blocks F(x) and f (x) differ by the same factor (− In the normalization where the leading term of f (x) is a power of x, the S-duality kernel is changed by a phase due to altered branch cuts. Using (2.13) We will interpret these phases as a Chern-Simons term on the wall. The braiding kernel receives similar phases. Our goal is to find the kernel B α 12 α 32 . We determine it in Section 3 when κ 2 h 1 is fully degenerate, namely κ 2 = −Kb with K ≥ 0 an integer, then generalize to all κ 2 in Section 4.

Braiding matrices
In this section we study braiding for conformal blocks of two generic, one semi-degenerate, and one fully degenerate operators in the Toda CFT 9 . We will consider fully degenerate momenta labelled by the fundamental representation (so α = −bh 1 ) in Section 3.1 and the K-th symmetric representation (so α = −Kbh 1 ) in Section 3.2. The case of the Kth antisymmetric representation (so α = −bh 1 − · · · − bh K ) was worked out in [GLF14,Appendix A.3].
Due to the fully degenerate operator, the four-point function is not an integral of conformal blocks but simply a finite sum of holomorphic times antiholomorphic blocks. The number of terms is equal to the dimension N +K−1 K of the representation of A N −1 . Braiding is thus given by (square) matrices of this size.

Braiding a fundamental degenerate
We focus here on , a four-point correlation function with two generic momenta α ∞ and α 0 , one semi-degenerate (κ + b)h 1 and a degenerate −bh 1 , labelled by the highest weight h 1 of the fundamental representation of A N −1 . The shift by b in κ simplifies some expressions.
This correlator can be expressed in terms of hypergeometric series. This was initially obtained by solving the null-vector differential equations for N = 3 and writing the natural generalization of these results [FL07]. We bypass the differential equation (null-vectors are not tractable for N > 3) and use monodromy properties of conformal blocks instead. These only fix conformal blocks up to an overall multiplication by a function Λ(x) that has no branch cut, namely is meromorphic on the whole Riemann sphere. The indeterminacy does not affect braiding matrices that we are interested in.
Our derivation only relies on two fusion rules which can be proven for the Toda CFT using the Coulomb gas formalism: where [ V ··· ] denotes contributions from W N descendants of a primary operator with the given momentum. As usual in 2d CFT, the four-point function can be expanded in three different channels by taking the OPE of V −bh 1 with any of the three other operators. The fusion rules restrict the internal momentum to α 0 − bh s for 1 ≤ s ≤ N in the s-channel, α ∞ − bh s in the u-channel, and κh 1 or (κ + b)h 1 − bh 2 in the t-channel.
In the s-channel, the four-point function is a sum of N factorized terms: where we have absorbed all of the position dependence in the conformal blocks F. A useful property of conformal blocks is their where (1 + · · · ) is a series in non-negative integer powers of x, and similarly for F (s) j (x). Because of radial ordering, the functions F (s) j are a priori only defined on the unit disc (with a branch point at 0), but since V V V V is smooth away from 0, 1, and ∞ the functions can be analytically continued to any simply connected domain avoiding these points. Two natural choices that we will use are the complex plane minus cuts on (−∞, 0] ∪ [1, ∞), and the complex plane minus cuts on [0, 1] ∪ [1, ∞).
The u-channel decomposition is similar, and conformal blocks have a simple x → ∞ expansion in terms of a series 1 + · · · with non-negative integer powers of 1/x: (3.7) j (x) can be extended to C minus some cuts, for instance along [0, 1] ∪ [1, ∞). The t-channel is more subtle, as it involves three-point functions of V α∞ , V α 0 , and a descendant of the primary V κh 1 or V (κ+b)h 1 −bh 2 . Contributions from primaries with momentum κh 1 factorize because this momentum is semidegenerate. Contributions from primaries with momentum (κ + b)h 1 − bh 2 do not factorize (except for N = 2). We deduce with the following x → 1 expansions, where 1 + · · · denote series in non-negative integer powers of (1 − x) and (1 −x), (3.10) Away from the cuts, the equality j C implies that the sets form another basis of the same space. The expansions of conformal blocks at 0, 1, and ∞ above imply certain monodromy properties when analytically continuing the functions through cuts. The monodromy M (0) around x = 0 is diagonal in the basis F (s) and eigenvalues can be read off from the expansion (3.5). While the monodromies M (1) and M (∞) are non-diagonal N × N matrices in this s-channel basis, their eigenvalues can be read off from the expansions above. Namely, eigenvalues of the monodromy M (1) around x = 1 are seen from (3.9) and (3.10), the latter having multiplicity N − 1, and the monodromy M (∞) has N eigenvalues known from (3.7). Finally, , 1, ∞} are the only singular points.
The s-channel conformal blocks proposed in [FL07] are j denote the hypergeometric function in (3.11). We now derive the braiding matrix for the functions j (x), then convert it to the braiding matrix (3.19) for the blocks G by introducing phases.
To compute their braiding matrix it is convenient to introduce notations. Let im p and i m p be 2N complex numbers such that j (x) can be expressed as the Mellin-Barnes integral given below, which converges away from the positive real axis. For |x| ≶ 1 we can close the contour integral towards κ → ±∞, and enclose either the poles at κ + im j ∈ Z ≥0 or the N families of poles at κ − i m k ∈ Z ≤0 labelled by 1 ≤ k ≤ N . The first choice yields a single s-channel factor, while the second yields a sum of N u-channel factors: (3.14) We will not need the explicit expression for g (u) k , which are series 1 + · · · in non-positive integer powers of x. The coefficients D,B 0 and D are given by . (3.16) Here we have included a parameter ∈ {0, ±1}. It is sometimes convenient to consider s-channel factors j (x) analytically continued with branch cuts on (−∞, 0] ∪ [1, +∞), and u-channel factors (3.17) The braiding matrix for G (s) . (3.20) The s-channel blocks G (s) j (x) have the expected power of x (hence the expected monodromies) around 0. The eigenvalues of the monodromy M (∞) around ∞ are also correct, as can be checked by combining the u-channel factors in (3.14) with the additional powers of x and 1 − x in (3.11) and comparing to the u-channel asymptotics (3.7) expected from CFT. Finally, from the braiding matrix above we deduce the monodromy (1) has the eigenvalue e 2πiγ 1 with multiplicity N − 1. Its last eigenvalue is fixed by det e −2πiγ 1 M (1) = detB + / detB − . These eigenvalues coincide with those expected of t-channel blocks. In fact the pecise exponents of (1 − x) also match [Nør55].
Let us prove that in this setup having the correct eigenvalues of monodromy matrices is in fact enough to show that the proposed conformal blocks G  (x 1 , . . . , x N ) is diagonal, and for some vector u. Such a basis is obtained in two steps: first make M (0) diagonal and note that as M (1) − y 2 has rank 1 it can be written as (3.21) but with v i w j instead of u i u j , next rescale the i-th basis vector by v i /w i and let . On the one hand the matrix is the sum of a diagonal matrix y 2 (1 − x p M −1 (0) ) and a rank 1 matrix hence its determinant is very simple (we have assumed momenta are generic hence the x are distinct). On the other hand the matrix is equal to (M (∞) − y 2 x p )M −1 (0) . We get Therefore, . (3.24) This fixes M (1) up to signs which can be absorbed in a choice of basis, and concludes this straightforward proof of uniqueness. We now know that the two sets of functions G i (x), then expressing the result in terms of Λ and G must give the same result as doing these steps in the opposite order, hence (3.25) Importantly the monodromy matrices for F and G are the same. Since the monodromies around 0 of the G (s) j are all distinct while the functions Λ have no monodromy around 0, the equality (3.25) is true term by term, and thus Λ i = Λ j . Therefore, F (s) where Λ(x) does not depend on i. Considering again a monodromy around 1 we find that Λ has no discontinuity along [1, ∞): it is meromorphic on the whole Riemann sphere as announced 10 . Regardless of Λ, the braiding matrix is given by (3.19).

Braiding a symmetric degenerate
We now generalize the discussion above to a degenerate vertex operator V −Kbh 1 labelled by the K-th symmetric representation R(Kh 1 ) of A N −1 . Namely we consider its fourpoint function with two generic operators V α∞ and V α 0 and one semi-degenerate V (κ+Kb)h 1 , including a shift by Kb for later convenience. Contrarily to the braiding for K = 1 determined in the previous section, the braiding found here is mostly meant to motivate the generalization to continuous values of K in Section 4. The braiding (3.44) we find is new.
where γ(y) = Γ(y)/Γ(1 − y) and we introduced the notations . (3.30) For a given weight h [n] of R(Kh 1 ), and a choice of 1 ≤ p ≤ N we consider the following analogue of the Mellin-Barnes integral (3.14) used for K = 1. . (3.31) The contours lie between poles of all the Γ(−i m s + iσ j ) and poles of all the Γ(−im s − iσ j ). The integral converges for x ∈ [0, ∞). The omission of some Γ functions (those with s = p) in the denominator is crucial for convergence, but this arbitrary choice of p will complicate calculations.
For |x| ≶ 1 we can close contours towards iσ j → ∓i∞, enclosing some poles. The first case yields a linear combination of s-channel factors (3.30): (3.32) . (3.33) -15 - The sum ranges over weights of R(Kh 1 ), but the matrix T p is "triangular" in the sense that its component T p [n][k] vanishes if n s < k s for any s = p. The second case yields a linear combination of u-channel factors (3.34) . (3.35) This leads to the braiding (3.36) We thus need to invert the matrix T p then multiply the result by U p . A consistency check will be that (T p ) −1 U p must not depend on p.
A proposal for (Ť p ) −1 is found by trial and error: 0 if n t < k t for any t = p, and otherwise N s<t 1 π sin π(im sks −im tk t ) 1 π sin π(imtn t −imsn s ) N t =p 1 π sin π(im tk t −im pkp ) 1≤s≤N,0≤µ≤ns (s,µ) =(t,k t ) 1 π sin π(imsµ−im tk t ) . (3.39) We must prove that [ . This is the sum of residues of N s<t s,t =p 1 π sin π(iτ t − iτ s ) 1 π sin π(im tnt − im sns ) Each iτ t appears in N − 2 + j p − n p − 1 sines in the numerator, and s =p (1 + n s − j s ) = N − 1 + j p − n p in the denominator, in other words, two more. Thus the function is 1-periodic in each variable iτ t , and decays exponentially as iτ t → ±∞. The sum of residues thus vanishes, because it is the sum of all residues in a fundamental domain of the periodicity, and there is no contribution from infinity. This establishes (3.39).
The braiding matrix (3.36) is then The result is a sum of residues of some function of N − 1 variables τ t for t = p. Relabelling the variables τ t using a permutation of 1, N so that they are numbered from 1 to N − 1 and φ(N ) = p, we obtain (3.42) where (−1) φ is the signature of φ. This expression does not change if we replace φ by another permutation such that φ(N ) = p and we permute the τ j accordingly: indeed, the sign coming from sin π(iτ j − iτ i ) is compensated by the change in (−1) φ .
Let us show that B φ does not depend on the arbitrary choice of p either, hence is independent of φ. Choose an index 1 ≤ j ≤ N −1. The variable τ j appears in N −2+K sines in the numerator and N + K sines in the denominator of (3.42). We thus have iτ j → iτ j + 1 periodicity, and no residue at infinity, hence the sum of residues at iτ j = −im φ(j)k j is equal to minus the sum of all other residues in a strip of width 1. This yields a sum over iτ j = −im φ(i)k for all 1 ≤ i ≤ N with i = j and 0 ≤ k ≤ n φ(i) . The contribution from a given i with i < N (and i = j) vanishes by antisymmetry under the exchange τ i ↔ τ j , thus only the poles at −im φ(N )k = −im pk contribute. All in all, we obtain the same expression as (3.42), with φ(j) and φ(N ) exchanged. The sign coming from reversing the contour is absorbed into a change of the signature (−1) φ .
As for K = 1, the braiding matrix for F (s) [n] (x) is obtained by including a phase e iπ γ 1 , and another phase comes from using factors x ··· instead of (−x) ··· . Putting everything together yields The explicit expression of B φ involves a permutation φ, but is independent of it. To translate this expression explicitly back from the {im, i m} notation to momenta, replace im sµ = b Q−α 0 , h s +µb 2 + 1 N N t=1 im t and i m sµ = bκ N +b Q−α ∞ , h s +µb 2 −1− 1 N N t=1 im t , then shift the variables iτ j to absorb 1 N N t=1 im t . Note that the starting point of this calculation, namely the explicit expression (3.26) for the four-point function, is not proven. However, we prove in Section 4 that the braiding (3.44), or rather its continuous generalization, obeys a shift relation which expresses the braiding for a given V −Kbh 1 in terms of that for V −(K−1)bh 1 and that for V −bh 1 . By recursion on K this proves that the braiding matrix given here is correct.

Braiding kernel
This section gives the braiding kernel (4.5) of two semi-degenerate vertex operators, which generalizes the braiding/fusion kernel for Virasoro (N = 2) conformal blocks [PT99]. We show in Section 4.1 how it reduces to the discrete results of Section 3.2 and in Section 4.2 that it obeys a very constraining shift relation deduced from a Moore-Seiberg pentagon identity. Symmetries are investigated in Section 5.3.
The four-point function V α 3 (∞) V α 4 (1) V α 2 (x,x) V α 1 (0) with two generic momenta α 1 , α 3 and two semi-degenerate momenta α 2 = κ 2 h 1 , α 4 = κ 4 h 1 has an s-channel decomposition where | · · · | 2 involves conjugating x but not momenta. Note that the internal momentum α 12 is continuous rather than discrete because there is no fully degenerate vertex operator. The s-channel conformal blocks are in principle fixed by W N symmetry. In practice, closed forms are only known thanks to the AGT relation with instanton partition functions, and we will not need them. In this section we again normalize conformal blocks as F (s) α 12 (x) = (−x) ∆(α 12 )−∆(α 1 )−∆(κ 2 h 1 ) (1 + · · · ): the use of −x instead of x avoids phases. The u-channel counterpart of (4.1) has κ 2 ↔ κ 4 : (4.2) Again, we normalize these u-channel conformal blocks so that their leading term is a power of (−x), namely F (u) . Both sets of conformal blocks are analytic on C \ [0, ∞). The two decompositions are related by a braiding transformation Our goal is to find the braiding kernel B α 12 α 32 . From Section 3.2 we know this braiding kernel in the limit κ 2 h 1 → −Kbh 1 , in other words when one of the semi-degenerate operators turns into a degenerate operator. Then it is a sum of residues (hence an integral) of a product of sines (3.42) which involve various multiples of b 2 in their arguments. This product of sines can be recast in terms of a the double Sine function S b defined in footnote 4, which obeys The braiding kernel for generic κ 2 should thus be an integral of some S b functions. Writing all generic momenta as α = Q − ia, we propose (4.5) up to a constant factor that does not depend on any momentum. The integration contours go from −∞ to ∞ with poles of the numerator S b functions above the contours, and zeros of the denominator below them. For instance, if all components ia 1 , h s and ia 3 , h 2 are purely imaginary and 0 < Re κ i N < q, then contours can be taken to be horizontal lines with For other values of momenta, the contour is deformed to keep the same set of poles on each side. Another remark is that has no pole: it simplifies to a product of sines (4.14). In a normalization of conformal blocks where the leading term is a power of x, the braiding kernel includes phases, depending on the sign of Im x: A preliminary check of (4.5) is that it reproduces known results [PT99] for the Liouville theory (N = 2). In their equation (48) replace their Q by our q, shift the integration variable s → s − α 21 + α 4 − q/2, then map α 2 → q − α 2 (for N = 2 this is a Weyl symmetry). The factors with U 3,4 become S b ±(α 3 − q/2) + s . The factors with U 1,2 become S b α 4 + α 2 − q ± (α 1 − q/2) + s . The denominator factors with V 1,2 become S b α 2 ± (α 32 − q/2) + s . The denominator factors with V 3,4 become S b α 4 ± (α 21 − q/2) + s . Thus, the integrand from [PT99] coincides with that of (4.5) for N = 2. It is straightforward to check that prefactors also coincide.

Reduction to fully degenerate
We now describe how to take the limit κ 2 h 1 → −Kbh 1 in (4.5), and retrieve the sum of residues from Section 3.2.
When a contour integral is pinched by poles getting close together from the two sides of the contour, the integral is singular. Indeed, if f (z) is holomorphic in a neighborhood of a, and a L and a R are points in this neighborhood, then where the initial contour goes between the two points, with a L on its left and a R on its right, and where the second contour is moved through a L . As a L , a R → a, the second term is regular, so the residue is 2πif (a). This residue is obtained from the original integrand by taking the limit a L , a R → a then considering the second residue of the result f (z)/(z − a) 2 . We denote this operation of taking the second residue as res 2 . The integrand in (4.5) has poles at for integers m, n ≥ 0. As mentionned before, the contour for iτ is chosen with poles (4.8) on the left and poles (4.9) on the right. This is possible as long as the two sets of poles are disjoint. Otherwise, the contour is pinched between the two sets and the integral diverges. Whenever one of the (4.8) is equal to q− κ 4 N − ia 12 , h s +mb+n/b, the contour is pinched, but the prefactors in (4.5) (more precisely, the denominator Γ b functions) cancel the singularity. On the other hand, if one of the (4.8) is equal to q − κ 2 N + ia 32 , h s + mb + n/b, then prefactors do not cancel the singularity, and the braiding kernel is genuinely singular. These singularity, together with those of numerator Γ b functions in (4.5), precisely reproduce singularities of the u-channel three-point functions, at least for the Toda CFT: . (4.10) It may be interesting to pursue further the analysis by considering multiple singularities, keeping in mind the constraints t ia j , h t = 0 for each momentum. We now focus on the limit κ 2 = −Kb + N iε for ε → 0 (and ε > 0). The OPE of V −Kbh 1 with a generic vertex operator constrains α 12 and α 32 , so we further focus on α 21 = α 1 − bh [n] and α 32 = α 3 − bh [ n] for some weights h [n] and h [ n] in R(Kh 1 ). To simplify some later expressions we set κ 4 = κ + Kb. The poles (4.8) are now at and (4.9) are at (4.12) As ε → 0, the contour is thus pinched whenever iτ j = q − κ N − ia 1 , h s − bl for any 1 ≤ j ≤ N − 1, 1 ≤ s ≤ N and 0 ≤ l ≤ n s . The most singular contribution, of order 1/ε N −1 , comes from values of iτ where all iτ j take this form.
We will only describe the contour integral part of the braiding matrix (4.5), as prefactors only make computations more tedious. The term of order 1/ε N −1 in this integral is . (4.13) Note that (4.14) The shift relations for Γ b and S b yield .
(4.15) This expression differs from the desired sum of residues (3.42) in the following respects: iτ j → biτ j + bκ N − b 2 − 1, a sum over choices of the p j , and additional factors of the form sin π b (· · · ). These factors are independent of the k j except for a sign. After extracting a sign and taking the residue, these factors are where p N denotes the (single) element of 1, N \ {p i | i < N } so that p is a permutation of 1, N . Then these factors are independent of the permutation p, except for a sign: the signature of p. For each permutation p we get a sum of residues times the signature of p, and this structure coincides with that of (3.42). Below that equation we had proven that it is independent of the permutation, hence summing over permutation simply introduces a trivial factor. We have thus reproduced qualitatively the structure of the braiding matrix of V −Kbh 1 by taking the appropriate limit of the braiding kernel. This is confirmed by a more detailed calculation.

Shift relation from pentagon identity
Braiding and fusion kernels (or matrices) obey Moore-Seiberg pentagon and hexagon relations. Here we consider a particular pentagon relation shown in Figure 1. Going through Figure 1. Pentagon identity. 1 → 2 and 4 → 5 are braidings of two semi-degenerates, 2 → 3 and 1 → 4 are known fusions of V −bh1 and a semi-degenerate, 5 ↔ 3 is a known braiding of V −bh1 and a semi-degenerate. the moves 1 → 2 → 3 we find On the other hand, going through the moves 1 → 4 → 5 → 3 yields (4.21) The coefficients of each conformal block F[3] (these are labelled by the choice of 1 ≤ s ≤ N and α 32 = α 32 − bh s ) must be the same in (4.18) and (4.21).
To check that the proposal (4.5) obeys the pentagon identity, we will need the braiding matrix obtained from (3.19) using α 1 = Q − ia 1 and α 32 = Q − ia 32 + bh s : . (4.22) We will also need coefficients of the fusion of (κ 2 + b)h 1 and −bh 1 into κ 2 h 1 , which can be deduced from the braiding matrix (3.19), as done in [GLF11, equation (B.14)]. . (4.23) We now write down (4.21) explicitly for a fixed choice of α 32 and of 1 ≤ s ≤ N , and simplify it in order to find (4.18). All generic momenta are written as α = Q − ia and we denote ia u = ia, h u for conciseness. Note that α 32 = Q − ia 32 + bh s . Let us start! N p=1 . (4.24) We collect factors which do not depend on p, s using shift relations of Γ b and S b . Factors of √ 2π and powers of b cancel, and we combine many Gamma as Γ(x)Γ(1 − x) = π/ sin πx.
The last line is a sum of residues at υ = bia p 1 π sin π(υ − bia t 1 ) , which is equal to minus its residue at the last pole υ = b 2 − bκ 4 N − bia s 32 . That residue turns out to cancel most factors in the second to last line. Together, these last two lines of (4.25) are equal to (4.26) In particular, this does not depend on iτ j and can be pulled out of the integral. The first two lines of (4.25) then reproduce precisely the braiding matrix (4.5) of two semi-degenerate vertex operators. This concludes our check of the pentagon relation (4.18) = (4.21). This pentagon relation expresses the braiding kernel B α 12 α 32 as a sum of N braiding kernels with κ 2 → κ 2 + b, α 1 → α 1 − bh p and α 32 → α 32 = α 32 + bh s . Thus, if the braiding kernel is known for some value κ 2 = λ, it can be deduced for κ 2 = λ − Kb for integer K ≥ 0. The pentagon identity (1 → 2 → 3 → 5) = (1 → 4 → 5) is checked through very similar computations. It allows the opposite shifts: from the κ 2 = λ braiding kernel one gets the κ 2 = λ + Kb braiding kernel. By symmetry, identical shift relations exist with b → 1 b , thus fixing braiding kernels for κ 2 = λ + Kb + L/b for all integers K, L. For generic real b 2 , continuity then determines the braiding kernel uniquely. Unfortunately, it is not clear to the author whether κ 2 = 0 (or any κ 2 = −Kb) is a valid starting point for this reasoning, as fusion rules contrain other momenta too. Nevertheless, the shift relations we have proven are at least very strong evidence that the proposed braiding kernel is correct.

Domain wall and its symmetries
Recall from Section 2 that the S-duality domain wall of 4d N = 2 SU (N ) SQCD with 2N flavours should be described by a 3d N = 2 theory whose partition function on the ellipsoid S 3 b obeys (2.8) namely We have computed the S-duality kernel B as the Toda CFT braiding kernel (4.5). In this section we find that 3d N = 2 U (N − 1) SQCD with 2N flavours has the partition function (5.1). We rewrite the partition function as a limit of the partition function of USp(2N − 2) SQCD with 2N flavours, and show that self-duality of a 4d lift of this USp theory (under Seiberg duality) lets the explicit braiding kernel be invariant under an expected symmetry. Finally, we discuss peculiarities for N = 2 due to USp(2) = SU (2).

Domain wall theory
In (4.5), Toda CFT momenta are converted to gauge theory parameters a and a and masses m f (f = 1, . . . , 2N ) using the dictionary (2.10), and α 12 = Q + N j=1 ia j h j and Reassuringly, despite the asymetry between masses m 1 , . . . , m N and m N +1 , . . . , m 2N in Toda CFT expressions, the final expression is invariant under permutations of these masses. The first line in (5.2) gives perfect candidates for the one-loop determinant of a hypermultiplet on a half-ellipsoid and that of a vector muliplet, where e|a is the usual scalar product of roots with elements of the Cartan algebra. These candidates appear to be consistent with results on the full ellipsoid: indeed, The need to divide by the one-loop determinant of a vector multiplet on S 3 b is not surprising since we would otherwise be overcounting degrees of freedom.
We are left with the task of finding a 3d N = 2 gauge theory whose S 3 b partition function is the integral in (5.2). Such partition functions are known through supersymmetric localization [Jaf12,HHL11a,HHL11b]: the path integral is localized to field configurations where a real vector multiplet scalar takes an arbitrary constant value, which can be reduced to the Cartan algebra of the gauge group G by a gauge transformation. The partition function takes the form (5.6) where the exponentials are classical values of the action, with k the Chern-Simons level (one per simple factor of G) and λ the Fayet-Iliopoulos parameter (one per abelian factor of G), the product over roots e of G is the one-loop contribution of the vector multiplet, and finally each chiral multiplet transforming in a representation R I of G contributes a product over weights w I (we write w I ∈ R I for lack of a better notation), which involves the R-charge r I and mass m I of the chiral. The integration contour agrees with R away from a compact set and is chosen so that poles of each S b are all on the same side. Note that the vector multiplet contribution has no pole since S b (y)S b (−y) −1 = −4 sin(πby) sin(πb −1 y).
We now find what 3d N = 2 theory reproduces (5.2) as follows. From the product of S b (iτ i − iτ j ) which does not depend on masses we deduce that the gauge group is U (N − 1). After using 1/S b (x) = S b (q − x), all remaining S b functions are one-loop determinants of chiral multiplets with canonical R-charge 1 2 (since the arguments take the form q 4 + · · · ). The multiplets are 2N fundamentals of U (N − 1) with masses −m f for f = 1, . . . , 2N , and 2N antifundamentals with masses a s − m and a s + m for s = 1, . . . , N . As discussed in the introduction (Section 1) these masses and R-charges are nicely explained (and fixed) by a cubic superpotential (1.3) coupling the 3d and 4d matter multiplets along the defect: Here, q s and q s denote the 2N antifundamental chiral multiplets and q f the 2N fundamental ones, while Φ| 3d and Φ | 3d are limits of 4d hypermultiplets at the interface. The R-charge of these fields originating from 4d must be an integer since the 3d SO(2) R symmetry is embedded in the non-Abelian SU (2) R , and is precisely 1 because the scalars in 4d hypermultiplets transform in a doublet of SU (2) R . Therefore every term in the superpotential has R-charge 2 provided the 3d chiral multiplets have their canonical R-charge 1 2 . It is immediate to check that other charges sum to zero for each term: As explained at the end of Section 2.2, if instanton partition functions (conformal blocks) are normalized to have a leading term x ··· rather than (−x) ··· , then the S-duality kernel is changed by phases (2.13) exp iπ 1 2 j a 2 j + 1 2 j a 2 j depending on the half-plane ( = ±1 is the sign of Im x). These phases are reproduced by a Chern-Simons term of level 1 2 for each 4d gauge group SU (N ).

Limit of USp
The U (N − 1) theory that we have just derived and its superpotential coupling are conveniently depicted as the quiver below. We now describe a USp(2N − 2) theory which has the U (N − 1) theory as a limit, up to a factor e N (N −1)πqµ omitted here: The round nodes labelled N are 4d N = 2 SU (N ) vector multiplets on each side of the wall, the diagonal edges attached to them are 2N fundamental hypermultiplets on each side of the wall sharing a common SU (2N ) flavour symmetry, and remaining edges denote 3d N = 2 chiral multiplets charged under one of the 4d groups and under a 3d N = 2 vector multiplet. For each triangle in these two quivers, there is also a cubic superpotential term (5.7). The limit µ → ±∞ is taken after adding µ to masses of half of the chiral multiplets, and −µ to the others: this is compatible with the superpotential. The S 3 b partition function of the second 3d N = 2 theory in (5.8) is (5.9) We wish to take µ → ∞. From the asymptotics of Γ b [Spr09, Proposition 8.11] we work out that for |χ| → ±∞ away from the imaginary axis, (5.10) We will always take χ real so χ sign(Re χ) = |χ|. Let us apply (5.10) to pairs of S b functions in (5.9) which have opposite dependence on µ and τ j , taking A and B to be everything apart from µ and τ j . We ignore for now factors that are uniformly bounded functions of µ and τ j (and have uniformly bounded inverse), and will denote them by (finite). This allows us to keep only the first exponential in (5.10). In fact, Now |µ − τ j | + |µ + τ j | = max(2|µ|, 2|τ j |) and |τ i − τ j | + |τ i + τ j | = max(2|τ i |, 2|τ j |). Sorting the parameters as |τ 1 | < . . . < |τ I | < |µ| < |τ I+1 | < · · · < |τ N −1 |, the exponential is (5.12) The second and third terms are negative. Hence the dominant contribution to the integral (5.9) as µ → ∞ is when |τ j | − |µ| are finite, and away from these regions the integrand decays exponentially.
We can now go back and keep finite factors when expanding the integrand in the region |τ j | ∼ |µ|. Given symmetries under τ j → −τ j , we focus on the case τ j = µ +τ j withτ j finite. Half of the S b factors remain finite and form the partition function of the U (N − 1) theory, while the other half can be paired and turned into exponentials through (5.10). One gets lim µ→±∞ E (µ, b, m,m, a, a ) As announced, the partition function of the U (N − 1) theory is a limit of the partition function of the USp (2N − 2) theory. It would be interesting to understand to what extent this statement holds for the theories themselves. One would need to carefully account for the factor c, which consists of Chern-Simons terms for all flavour symmetry groups of the 3d theories.

Symmetries
We now have all the necessary tools to study symmetries, from the point of view of the duality wall, of its 3d gauge theory description, and of the Toda CFT. Symmetries of Toda CFT correlators (hence of their braiding kernel) may be least familiar to the reader, so we will describe them and give their gauge theory interpretation.
Any vertex operator V α is invariant under Weyl transformations, namely permutations of the components α − Q, h s . Applying this symmetry to V α 1 permutes the first N masses, and applying it to V α 3 permutes the last N . These are manifest symmetries of the S-duality wall and of its 3d gauge theory description. In fact, permuting all of the 2N masses is also a symmetry, but the Toda CFT description does not make all permutations manifest.
Conformal blocks are invariant under conjugating all momenta, which maps (up to a Weyl transformation) α → 2Q − α and κh 1 → (N q − κ)h 1 . It maps α → 2Q − α and κh 1 → (N q − κ)h 1 hence flips the sign of all m f , a s and a s . This is the effect of charge conjugation in the 4d theories on both sides of the wall, which is an expected symmetry of the S-duality wall. However, the effect is harder to describe in the 3d gauge theory description, because this theory is chiral, hence not invariant a priori under charge conjugation. Correspondingly, the explicit form of the braiding kernel does not appear invariant under charge conjugation. Most of this section will be spent proving the invariance, which will turn out to be a limit of USp-type Seiberg duality of 4d N = 1 indices. Before proceeding let us describe two more symmetries.
Conformal blocks are also invariant under some permutations of the operators. In particular, the braiding kernel is thus invariant under a permutation obtained by rotating the diagrams by 180 • in the plane: 14) The map α 12 → 2Q − α 12 is due to the arrow being reversed by the rotation. Composing with conjugation of all momenta yields the transformation α 1 ↔ 2Q − α 3 and κ 2 ↔ N q − κ 4 , which given the dictionary (2.10) simply exchanges the first N and the last N masses. Lastly, the S-duality wall and its 3d gauge theory description are invariant under exchanges of the two hemispheres. This maps a s ↔ a s and m → −m. Momenta are mapped as κ 2 ↔ κ 4 and α 12 ↔ α 32 , and the braiding kernel becomes the kernel for the opposite braiding, which expresses u-channel blocks in terms of s-channel ones. These braiding kernel are inverses of each other. But they are also equal, up to some structure constants. To prove this, write the s-channel and u-channel decompositions of the four-point function, then use braiding kernels to get an expression with a holomorphic s-channel block and an antiholomorphic u-channel block, and match coefficients: Now that we have described all of the manifest symmetries of the explicit braiding kernel (4.5), we must tackle invariance under charge conjugation.
We will use identities of hyperbolic hypergeometric integrals [Rai06]. The hyperbolic Gamma function Γ (2) h of that paper reduces to our S b upon taking ω 1 /ω 2 = b 2 . For definiteness, we take ω 1 = ib and ω 2 = i/b and note S b (x) = Γ (2) h (ix; ib, i/b). The BC n hyperbolic hypergeometric integral is defined by (to avoid factors of i we let µ r = iν r ) ν r = (m + 1)q, where as usual q = b + b −1 . Corollary 4.2 of [Rai06] states the invariance under m ↔ n and ν → q/2 − ν: The S 3 b partition function (5.9) of the USp(2N − 2) SQCD with 4N chiral multiplets which we studied earlier is such a hyperbolic hypergeometric integral, with m = n = N − 1 and 4N parameters ν r summing to N q: N ) .
The identity (5.18) states that Z S 3 b is invariant up to a factor r<s S b (ν r + ν s ) under changing the signs of all µ,m f , m, a s , a s .
We now take the limit µ → ±∞ in (5.18), after multiplying by E(µ) and E(−µ) respectively as explained in (5.13). On each side, we obtain partition functions of U (N − 1) theories, and the product of S b (ν r + ν s ) reads (5.20) The phases are exactly E(−µ)/E(µ). The remaining S b functions are one-loop determinants of mesons formed as the product of a fundamental and an antifundamental chiral multiplet under U (N − 1). The full S 4 b partition function (5.2) of the 4d/3d coupled system also includes one-loop determinants of hypermultiplets on half-ellipsoids, which are Γ b functions with the same arguments as the S b functions in (5.20) up to signs. Since S b q 2 + x = Γ b q 2 + x /Γ b q 2 − x , the S b functions coming from mesons in the 3d duality convert between Γ b q 2 + x . This is fully consistent with 4d charge conjugation. Identical calculations show that the braiding kernel is invariant under Toda CFT charge conjugation.
The identity (5.18) was proven in [Rai06] as a hyperbolic limit of an elliptic hypergeometric integral identity. In physics terms, the elliptic identity states that two Seiberg-dual 4d N = 1 theories with USp(2N − 2) gauge group and 4N fundamental chiral multiplets have the same S 3 b × S 1 partition function (supersymmetric index). Note that while the dual of 4d N = 1 USp(2N c ) SQCD with 2N f fundamental chiral multiplets has N c = N f −N c −2 colors, the analoguous 3d N = 2 Aharony duality has N c = N f − N c − 1 colors instead [Aha97]. For our case N c = N − 1 and N f = 2N the 4d theory is self-dual but the 3d theory is not. On the other hand, Aharony duality can be retrieved as a limit of (5.18) when ν 2m+2n+2 , −ν 2m+2n+3 → i∞. It would be interesting to clarify whether such a limit from the 4d to the 3d USp theories can be taken directly in gauge theory, following the lines of [ARSW13a,ARSW13b].
The limit µ → ∞ then takes the S 3 b partition function of SU (2) SQCD with 8 doublets to that of SU (2) SQCD with 6 doublets. This reproduces the 3d N = 2 description found in [TV14] for the S-duality domain wall of 4d N = 2 SU (2) SQCD with 4 flavours.

Conclusions
In this work we have determined the integral kernel (4.5) which braids two semi-degenerate vertex operators of the Toda CFT. Through the AGT relation, we have deduced the ellipsoid expectation value of an S-duality domain wall in 4d N = 2 SU (N ) SQCD with 2N flavours. We have then described the wall by coupling 3d N = 2 U (N − 1) SQCD with 2N + 2N chiral multiplets on the wall to the 4d theories on both sides of the wall.
The shift relations found in Section 4.2 are not sufficient to prove that the braiding kernel is correct. An obvious question would be to fill in this gap by checking additional Moore-Seiberg relations. It may be interesting to relate the braiding kernel to Racah-Wigner coefficients for the modular double of U q (sl N ), as was done for N = 2 in [PT99].
It would be valuable to evaluate one-loop determinants of hypermultiplets and vector multiplets on the half-ellipsoid with appropriate boundary conditions, and clarify whether the result indeed consists of half of the Γ b factors in the full-ellipsoid results.
S-duality is expected to map Wilson loops to 't Hooft loops. Expectation values of Wilson loop and 't Hooft loop observables on the ellipsoid are known exactly [Pes12,GOP12] and we now know the explicit S-duality kernel. Conjugating the Wilson loop by this kernel should thus yield the 't Hooft loop, namely if one considers the 4d theory on S 4 b with two S-duality walls near the equator and a Wilson loop in between, then the collision limit should yield a 't Hooft loop. In fact, a preliminary question is to understand how the collision of two domain walls which perform S-duality and its inverse yields a trivial operator.
The S-duality wall generalizes to the case where part (or all) of the SU (2N ) flavour symmetry shared by the two 4d theories is gauged by 4d vector multiplets. Class S theories, constructed by twisted dimensional reduction of the 6d (2, 0) SU (N ) superconformal theory on a punctured Riemann surface Σ, provide interesting examples such as linear quivers of SU (N ) gauge groups. The 3d description of an S-duality wall in such a quiver is again 3d N = 2 U (N − 1) SQCD coupled with 4d fields through cubic superpotentials: with a cubic superpotential term coupling 3d and 4d fields for each triangle and a quartic superpotential for 3d chiral multiplets in the central paralellogram.
From the Toda CFT point of view, each of these products of duality walls corresponds to a product of two braiding kernels. It may be interesting to translate Moore-Seiberg relations of braiding kernels into the gauge theory language and understand their implications for duality walls.
Note that all of this work focused on gauge theories with a Lagrangian description, or equivalently Toda CFT correlators with "enough" degeneracy. In particular, we have avoided the limit x → 1 of 4d N = 2 SQCD, which involves a strongly coupled matter theory instead of hypermultiplets, coupled to a vector multiplet. For N = 3 this theory includes T 3 . The corresponding crossing symmetry on the Toda CFT side consists of the fusion of two simple punctures into a less degenerate operator V α . Conformal blocks in this limit are not uniquely characterized by α (and external operators) and one needs a label for the (continuous) multiplicity with which V α appears in the fusion of the two full punctures. These conformal blocks are eigenfunctions of (the square of) the braiding kernel, and it is tempting to try and diagonalize this kernel. Unfortunately, we only succeeded to tame multiplicities in the simplest discrete versions of the kernel, and could not generalize.
Another direction worth pursuing is to consider S-duality walls in 4d N = 2 theories with gauge groups such as Sp(N ) (see for instance [LY98]). Too little is known at present about braiding kernels of D-type and E-type Toda CFT in order to apply the techniques used here. Understanding whether the U (N − 1) 3d theory found in this paper can be derived through brane constructions may help in generalizing to other gauge groups by orbifolding.