Superconformal index of low-rank gauge theories via the Bethe Ansatz
aa r X i v : . [ h e p - t h ] F e b SISSA 07/2021/FISI
Superconformal index of low-rank gaugetheories via the Bethe Ansatz
Francesco Benini , , and Giovanni Rizi SISSA, Via Bonomea 265, 34136 Trieste, Italy INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy ICTP, Strada Costiera 11, 34151 Trieste, Italy
We study the Bethe Ansatz formula for the superconformal index, in the case of 4d N = 4 super-Yang-Mills with gauge group SU ( N ). We observe that not all solutions tothe Bethe Ansatz Equations (BAEs) contribute to the index, and thus formulate “reducedBAEs” such that all and only their solutions contribute. We then propose, sharpening aconjecture of Arabi Ardehali et al. [1], that there is a one-to-one correspondence betweenbranches of solutions to the reduced BAEs and vacua of the 4d N = 1 ∗ theory. We testthe proposal in the case of SU (2) and SU (3). In the case of SU (3), we confirm that thereis a continuous family of solutions, whose contribution to the index is non-vanishing. Introduction and results
The AdS/CFT correspondence [2–4] provides us with a non-perturbative definition of quan-tum gravity in anti-de-Sitter (AdS) space, in terms of an ordinary quantum field theory(QFT) living on its conformal boundary. In particular, ensembles of states in the bound-ary theory capture the physics of black holes in the bulk [5]. This means that a countingof quantum microstates in the boundary theory, performed by studying suitable partitionfunctions in the large N limit, can reproduce and explain the Bekenstein-Hawking entropyof black holes in AdS [6]. The last few years have seen a lot of activity in this direction, inthe context of supersymmetric black holes in which explicit and precise computations canbe performed, see for instance [7–46].An interesting case is that of type IIB string theory (and its supergravity low-energylimit) in AdS × S , whose physics is captured by the four-dimensional N = 4 super-Yang-Mills (SYM) theory with gauge group SU ( N ) [2]. The microstates of 1/16 BPS blackholes in AdS are captured by the superconformal index [47, 48], a supersymmetric partitionfunction that can be computed exactly. The large N limit can be studied with a variety ofmethods [21, 22, 36, 43, 48–50]. From it, one can extract the Bekenstein-Hawking entropy,but also find hints of new physics. For instance, one can compute perturbative correctionsexpected to come from higher-derivative terms [42], as well as non-perturbative correctionscoming from Euclidean complex saddles of the gravitational action [51,52] (thus highlightingthe importance of Euclidean complex saddles [20]) and corrections expected from D-branes.A particularly effective way to compute the large N limit is to use the Bethe Ansatz (BA)approach [22], based on the Bethe Ansatz formulation of the superconformal index [53, 54].The standard integral formula for the index [47–50] can be recast as a sum over the solutionset to certain transcendental equations, dubbed Bethe Ansatz Equations (BAEs) because oftheir similarity with the ones appearing in the context of integrable systems. One particularsolution to the BAEs reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS[22]. On the other hand, other solutions can be associated with complex semiclassical saddlesof the gravitational path-integral [51, 52]. A comprehensive picture of all the contributionsfrom BA solutions and the physics they predict is an interesting open problem.Even for finite N , the complete list of solutions to the BAEs is not known. A subclass ofexact solutions was found in [10, 55], and we will refer to them as Hong-Liu (HL) solutions.These are discrete solutions, in one-to-one correspondence with subgroups of Z N × Z N oforder N , and are parametrized by three integers { m, n, r } with N = mn and r ∈ Z n . On theother hand, a very interesting conjecture was put forward in [1]: that there could be a sort ofcorrespondence between some solutions to the BAEs and the vacua of the 4d N = 1 ∗ theory— which is N = 4 SYM deformed by N = 1 preserving mass terms — compactified on S .1he set of vacua of the N = 1 ∗ theory on R , [56] (that we will review below) include bothdiscrete massiva vacua, which are easily seen to be in one-to-one correspondence with theHL solutions, as well as massless Coulomb vacua, which give rise to continuous families ofvacua upon compactification on S . Indeed, the authors of [1] gave evidence for the existenceof continuous families of solutions to the BAEs. On the other hand, they also pointed outthat there are many more solutions to the BAEs (both discrete and continuous) that do notfit into the pattern.In this Letter we will make the conjectural correspondence sharper, by studying in detailthe low-rank cases of gauge groups SU (2) and SU (3).Our first result (Section 2) is that, for generic N , most of the solutions to the BAEs leadto contributions that cancel out. This means that we can formulate a more restrictive setof equations, that we call “reduced Bethe Ansatz Equations”, such that only solutions tothe latter actually contribute. We then conjecture that there is a one-to-one correspondencebetween branches of solutions to the reduced BAEs and vacua of the N = 1 ∗ theory on R , .In particular, isolated solutions to the reduced BAEs are in correspondence with massivevacua of the N = 1 ∗ theory, while complex k -dimensional manifolds of solutions are incorrespondence with massless Coulomb vacua with k photons.We test our conjecture in the cases of SU (2) and SU (3) gauge group. In the case of SU (2) (Section 3) we show that the reduced BAEs have only 3 solutions, which indeed areprecisely the discrete HL solutions. We then show, by expanding the index order by order inthe operator dimension and computing many terms, that those 3 solutions exactly reproducethe superconformal index (as it follows from the integral formula).In the case of SU (3) (Section 4), we are able to analytically solve the reduced BAEs andfind all solutions. They comprise the 4 HL solutions, as well as a continuous family whichhas complex dimension 1 and is made of a single connected component. This is precisely incorrespondence with the vacua of the N = 1 ∗ theory. We also show that, in this case, the HLsolutions alone do not reproduce the index, and thus the contribution from the continuousfamily must be non-trivial. We leave the interesting issue of evaluating such a contributionfor future work.The method we use, in the SU (3) case, to solve the reduced BAEs and to exhibit thestructure of discrete and continuous solutions sprouts from the observation that the equationsfactorize into multiple hyperplanes and curves, at whose intersections lie the solutions. Weare hopeful that this method can be generalized to higher N . Note added : While this work was completed, its results had been announced in [57],and we were in the process of writing up this Letter, we received the paper [58] which hasconsiderable overlap with our work. 2
Bethe Ansatz formula for the superconformal index
We study the superconformal index of 4d N = 4 SYM [47, 48], which counts (with sign)operators in short representations (1/16 BPS) of the superconformal algebra, preserving acomplex supercharge Q . Using N = 1 notation, the field content of N = 4 SYM is given bya vector multiplet and three chiral multiplets X , X , X , all in the adjoint representationof the gauge group, with superpotential W = Tr (cid:0) X [ X , X ] (cid:1) . The R-symmetry is SU (4) R ,and we choose the Cartan generators R , , such that each one of them assigns charge 2 toone of the chiral multiplets and charge 0 to the other two. States are further labeled by twoangular momenta J , with semi-integer eigenvalues, generating rotations of two orthogonalplanes in R , and we define the fermion number as F = 2 J . All fields in the theory haveinteger charges under R , , , moreover F = 2 J , = R , , (mod 2) . (2.1)The superconformal index is defined as I ( p, q, y , y ) = Tr (cid:20) ( − F e − β {Q , Q † } p J + R q J + R y ( R − R )1 y ( R − R )2 (cid:21) . (2.2)Here p, q, y , y are fugacities, and it is convenient to introduce chemical potentials σ, τ, ∆ , ∆ such that p = e πiσ , q = e πiτ , y a = e πi ∆ a . (2.3)The index is well-defined for | p | , | q | <
1. It is a single-valued function of the fugacities,therefore periodic under integer shifts of the chemical potentials, because all states haveinteger charges with respect to the exponents in (2.2). Besides, it will be convenient tointroduce an auxiliary fugacity y and chemical potential ∆ such that ∆ + ∆ + ∆ − σ − τ ∈ Z . (2.4)By standard arguments [59], the index only counts states annihilated by Q and Q † and istherefore independent of β .When reading off the (weighted) multiplicities of BPS operators from the index in a “low-temperature” expansion, it may be convenient to use the original parametrization of [48] interms of fugacities ( t, y, v , v ). They are related to our fugacities by p = t y , q = t /y , y = t v , y = t v , y = t /v v , (2.5) This will restore the permutation symmetry acting on the index a = 1 , , R a , y a and ∆ a . Such asymmetry is the Weyl group of the global symmetry SU (3) ⊂ SU (4) R that commutes with Q . y is an auxiliary variable defined by y y y = pq . In terms of these fugacities,the index reads I = Tr BPS ( − F t J + ) y J − v q v q . (2.6)Here J ± = ( J ± J ) / SU (2) × SU (2) Lorentz group, ∆ isthe dimension of the operators, the trace is over BPS states — all and only states satisfying∆ = 2 J + + ( R + R + R ) / q , = ( R , − R ) / SU (3) ⊂ SU (4) R that commutes with Q . Integral formulation.
The index is independent of the gauge coupling, and thus can becomputed exactly [47–50] in terms of a certain contour integral. Specializing to the case ofgauge group SU ( N ), the integral formula reads I SU ( N ) = κ N Z T N − Z ( u ; ∆ , σ, τ ) N − Y j =1 dz j πiz j , (2.7)where z j = e πiu j , and u j with j = 1 , . . . , N are gauge holonomies along the Cartan gen-erators. We regard u j with j = 1 , . . . , N − u N , is fixed by the SU ( N ) constraint N X j =1 u j = 0 . (2.8)The prefactor is κ N = 1 N ! (cid:16) I U (1) (cid:17) N − , (2.9)written in terms of the index I U (1) ( p, q, y , y ) = ( p ; p ) ∞ ( q ; q ) ∞ Y a =1 e Γ(∆ a ; σ, τ ) (2.10)of the free N = 4 vector multiplet. Here ( z ; q ) ∞ is the q -Pochhammer symbol while e Γ( u ; σ, τ )is the elliptic gamma function, both defined in Appendix A. The integrand is Z ( u ; ∆ , σ, τ ) = N Y i = j Q a =1 e Γ (cid:0) u ij + ∆ a ; σ, τ (cid:1)e Γ (cid:0) u ij ; σ, τ (cid:1) , (2.11)where u ij = u i − u j . This expression can be slightly simplified using the identity N Y i = j e Γ( u ij ; σ, τ ) = N Y i In order to compute the large N limit, it is convenient toconsider the alternative Bethe Ansatz formula [53, 54]. Such a formula applies whenever theangular chemical potentials can be brought, with suitable integer shifts, to have rationalratio, σ/τ ∈ Q . This means that σ = aω , τ = bω for positive coprime integers a, b and achemical potential ω with I m ω > 0. Our discussion of the Bethe Ansatz equations will begeneral, however, for simplicity when computing the index we will restrict to the case of twoequal angular fugacities: p = q ⇔ σ = τ . (2.14)In this case the Bethe Ansatz formula simplifies to I ( q, y , y ) = κ N X ˆ u ∈ BAEs Z (ˆ u ; ∆ , τ ) H (ˆ u ; ∆ , τ ) − . (2.15)The sum is over the solution set { ˆ u } to a system of transcendental equations, dubbed BetheAnsatz Equations. Defining the U ( N ) Bethe operators as Q j ( u ; ∆ , τ ) = e − πi P Nk =1 u jk Y a =1 N Y k =1 θ ( u kj + ∆ a ; τ ) θ ( u jk + ∆ a ; τ ) (2.16)for j = 1 , . . . , N , the SU ( N ) BA equations are given by1 = Q i Q N for i = 1 , . . . , N − . (2.17)The unknowns are the “complexified SU ( N ) holonomies” u i living on a torus of modularparameter τ , namely with identifications u i ∼ u i + 1 ∼ u i + τ for i = 1 , . . . , N − , (2.18)while u N is fixed by the constraint (2.8). The BAEs (2.17) are invariant under such shifts.In fact, a stronger property holds: Q i are invariant under shifts of the components of theantisymmetric tensor u ij by 1 or τ , even relaxing the condition that u ij = u i − u j . This willbe used later. It was proven in [54] that only the solutions that are not invariant under any Such a set is dense in the space of angular chemical potentials. SU ( N ) (namely, only solutions with all u i differenton the torus) actually contribute to the sum in (2.15). In the general case σ/τ ∈ Q , the BAformula takes a more complicated form, however the BAEs are the same as in (2.17) withthe only difference that τ is replaced by ω . Therefore the analysis we will make of the BAEsand their solutions will be valid in the general case.The prefactor κ N and the integrand Z in (2.15) are the same as in the integral formuladescribed before. On the other hand, H is a Jacobian defined as H = det (cid:20) πi ∂ log( Q i /Q N ) ∂u j (cid:21) i,j =1 ,...,N − . (2.19)Notice that Q i , κ N , Z and H are all invariant under integer shifts of τ , ∆ and ∆ , inaccord with the fact that the superconformal index (2.15) is a single-valued function of thefugacities. Reduced Bethe Ansatz equations. It turns out that not all solutions (faithfully actedupon by the Weyl group) of (2.17) actually contribute to the sum in (2.15). In order tounderstand this point, notice that Q Ni =1 Q i = 1 identically. It follows that the solutions tothe BAEs (2.17) break into N “sectors” parametrized by λ ∈ Z N and given by Q j = e πiλ/N for j = 1 , . . . , N , (2.20)each sector corresponding to a different integer value of λ = 0 , , . . . , N − Z the shifts of complexified SU ( N ) holonomies on the torus [53, 54]: Z ( u − δ k τ ) = Q k ( u ) Q N ( u ) Z ( u ) for k = 1 , . . . , N − , (2.21)where u − δ k τ = ( u , . . . , u k − τ, . . . , u N − , u N + τ ) denotes a shift of the k -th and N -thcomponents of u , so as to preserve the SU ( N ) constraint. On the other hand, if we regard Z ( u ) as function of N independent holonomies u i of U ( N ), we find Z (cid:0) u − ¯ δ k τ (cid:1) = ( − N − Q k ( u ) Z ( u ) for k = 1 , . . . , N , (2.22)where u − ¯ δ k τ = ( u , . . . , u k − τ, . . . , u N ) denotes a shift of the k -th component only. TheBAEs only depend on the differences u ij and, as already noted, are invariant under shiftsof u ij by multiples of 1 and τ . It follows that one solution for u ij on the torus gives riseto multiple solutions for u i , of the form u i = u (0) i + ( α + βτ ) /N for i = 1 , . . . , N − u N = u (0) N + (1 − N )( α + βτ ) /N , where α, β = 0 , . . . , N − The factor ( − N − did not appear in [54] because that paper only dealt with semi-simple gauge groups. 6e equivalent up to the Weyl group action). The function Z , to be evaluated in (2.15)on those solutions, only depends on u ij , thus the dependence on α, β is the same as if u N were shifted by − ( α + βτ ). There is no dependence on α , while the dependence on β is aphase (cid:0) ( − N − Q N ( u ) (cid:1) β described by (2.22). We conclude that in all but one sector of BAsolutions, the sum over β leads to a cancelation because the sum of phases vanishes. Theexception is the sector in which ( − N − Q N ( u ) = 1.Therefore, in the Bethe Ansatz formula (2.15) we can restrict to BA solutions solely inthe sector λ = N ( N − mod N , namely to solutions of Q i = ( − N − for i = 1 , . . . , N (2.23)with Q i given in (2.16), which moreover are faithfully acted upon by the Weyl group. We callthese the “reduced Bethe Ansatz Equations”. Recall that the product of the N equations isidentically equal to 1, therefore one of them could be removed from the set. Hong-Liu (HL) solutions. The full set of solutions to (2.23) is not known, however, alarge set was found in [10, 55] and we will refer to them as HL solutions. They are labelledby three positive integers: { m, n, r } such that N = m · n , r ∈ Z n . (2.24)The solutions are u j ≡ u ˆ ˆ k = ¯ u + ˆ m + ˆ kn (cid:16) τ + rm (cid:17) . (2.25)Here we have decomposed the index j = 0 , . . . , N − = 0 , . . . , m − k = 0 , . . . , n − 1. Moreover, ¯ u is a constant chosen in such a way to solve the SU ( N )constraint. Since what enters in all formulas are the differences u ij = u i − u j , to eachHL solution is associated a multiplicity that we will discuss below. Notice that the HLsolutions are in one-to-one correspondence with subgroups of Z N × Z N of order N , that iswith sublattices of index N in generic two-dimensional lattices. It is worth remarking that,surprisingly enough, the HL solutions (2.25) do not depend on the flavor chemical potentials∆ a , although the BAEs do.The BAEs (2.23) are invariant under SL (2 , Z ) modular transformations of the torus,namely under the generators T : τ τ + 1 u u S : τ 7→ − /τu u/τ C : τ τu 7→ − u . (2.26)It follows that the HL solutions form orbits under P SL (2 , Z ), completely classified by the7nteger d = gcd( m, n, r ). The action of P SL (2 , Z ) is given by T : { m, n, r } 7→ { m, n, r + m } , S : { m, n, r } 7→ (cid:26) gcd( n, r ) , m n gcd( n, r ) , m ( n − r )gcd( n, r ) (cid:27) . (2.27)For given N , the total number of HL solutions is given by the divisor function σ ( N ) = X k | N k . (2.28)For each integer d such that d divides N , there exist a separate P SL (2 , Z ) orbit of HLsolutions generated by (cid:8) d, Nd , (cid:9) , which is isomorphic to the orbit generated by { , N/d , } in the case of SU ( N/d ). The number of elements in such an orbit is given by ψ (cid:0) N/d (cid:1) ,expressed in terms of the Dedekind psi function ψ ( n ) = n Y p | n (cid:18) p (cid:19) (2.29)where the product is over all prime numbers that divide n .The contributions to the BA formula (2.15) from HL solutions connected by the actionof T have a simple relation. Indeed, T acts on solutions to the BAEs by q → e πi q , andthere is no extra effect of this shift on the summand in (2.15). This implies that the actionof T on the contributions to the BA formula is also given by q → e πi q . In terms of thealternative set of fugacities ( t, v , v ) defined in (2.5) (we are setting y = 1 here), this is t → e πi t , v a → e πi v a . (2.30)Since the action of T on HL solutions forms finite cyclic orbits, this can pose constraints onthe contributions from those solutions (although notice that, as opposed to the full index, theseparate contributions to (2.15) in general are not single-valued functions of the fugacities).We will see this in the examples in Section 3 and 4. Multiplicities. From each solution in terms of u ij on the torus, one gets up to N solutionsin terms of u i , related by a shift of the “center of mass” of the first N − N , namely it gives rise to N inequivalent solutions in terms of u i that cannot be identified by the action of the Weylgroup. Notice that, consistently, X d | N ψ (cid:18) Nd (cid:19) = σ ( N ). N ! related to the action ofthe Weyl group. Therefore, each HL solution has total multiplicity N · N !. N = 1 ∗ theory It has been pointed out in [1] that the Hong-Liu ones are not the only solutions to the BAEs.Furthermore, by a combination of analytical and numerical work, evidence was given thatfor N ≥ N = 1 ∗ SU ( N ) theory on S . This, in particular, allowed them topredict the appearance of continuous branches of BA solutions for various values of N . Onthe other hand, it was already noticed in that work that, even for N = 2, the BAEs admitsolutions that do not have a counterpart in the N = 1 ∗ theory, therefore the problem ofunderstanding in which terms the correspondence could be correct and precise remainedopen.In this Letter we present a sharper version of the conjecture: we propose that thereis a one-to-one correspondence between branches of solutions to the reduced Bethe Ansatzequations, and vacua of the N = 1 ∗ SU ( N ) theory on R , . In particular, isolated solutionsto the reduced BAEs are in correspondence with massive vacua of the N = 1 ∗ theory, whilecomplex k -dimensional manifolds of solutions are in correspondence with massless Coulombvacua with k photons. If the conjecture is correct, it implies that the HL solutions are theonly discrete solutions to the reduced BAEs. It also implies a precise characterization of thenumber of continuous families and their dimensionality, for each value of N . In Sections 3and 4 we provide some evidence of the conjecture.Let us review the N = 1 ∗ SU ( N ) theory, which is obtained from N = 4 SYM by an N = 1 preserving mass deformation W def = X a =1 m a X a . (2.31)Its vacua have been analyzed in [56] (see also [61, 62]). F- and D-term equations reduceto the classification of isomorphism classes of homomorphisms ρ : su (2) → su ( N ) modulo The compactification on S poses another problem, if one wants to make the correspondence precise.Indeed, certain gapped vacua of the N = 1 ∗ theory on R , give rise to multiple vacua on R , × S , due tothe presence of a residual discrete gauge symmetry [60, 61]. For instance, the SU (2) theory has 3 vacua on R , , but 4 vacua on R , × S , while the reduced BAEs have 3 isolated solutions. We are grateful to JanTroost for making us appreciate this point. L ( N, C ), which correspond to partitions of N . Since the chiral superfields X a all have baremasses, possible massless fields come from the gauge sector. At the classical level, the SU ( N )gauge group is broken to the subgroup H that commutes with the image of ρ . Quantummechanically, all simple SU ( n ) factors of H confine leading to a number n of massive sectors;on the other hand, if H contains U (1) factors, then one obtains a massless Coulomb phase.The net result can be summarized as follows. Consider all integer partitions of N . Foreach partition, let n j be the number of times the integer j appears in it, and so label thepartition by a string ( n , . . . , n j ) such that N X j =1 j n j = N . (2.32)Then the subgroup H is given by H = " N Y j s.t. n j =0 U ( n j ) /U (1) . (2.33)Thus, such a partition contributes to the moduli space with a certain number of vacua thatcan be either gapped or contain massless photons: • number of photons = (number of non-zero n j ’s) − • number of vacua = product of non-zero n j ’s.It turns out [56] that the massive vacua are in one-to-one correspondence with subgroups oforder N of Z N × Z N , and thus in correspondence with the HL solutions to the BAEs. Onthe other hand, we put each Coulomb vacuum in correspondence with a continuous familyof solutions to the BAEs, with complex dimension equal to the number of photons. SU (2) Let us analyze in detail the case of gauge group SU (2). Defining u ≡ u = u − u , therefined Bethe Ansatz equation (2.23) can be rewritten as θ (∆ + u ) θ (∆ + u ) θ ( − ∆ − ∆ + u ) θ (∆ − u ) θ (∆ − u ) θ ( − ∆ − ∆ − u ) = − , (3.1) More precisely, H can contain extra discrete factors [61]. The latter would affect the number of vacuaupon compactification on S . 10n terms of the standard Jacobi theta function θ (see Appendix A). For the sake of clarity,here and in the following we leave the dependence of the Jacobi theta functions θ r on τ implicit. Let us define the function f ( u ; ∆ , τ ) = θ ( u ) θ (∆ + u ) θ (∆ + u ) θ ( − ∆ − ∆ + u ) . (3.2)Then, multiplying and dividing (3.1) by θ ( u ) and using that θ is an odd function of u (while θ , , are even functions of u ), we can rewrite the reduced BAE as f ( u ; ∆ , τ ) − f ( − u ; ∆ , τ ) f ( − u ; ∆ , τ ) = 0 . (3.3)Using standard identities among the theta functions, see for instance [63], we obtain f ( u ; ∆ , τ ) = c θ (2 u ) + c θ (2 u ) + c θ (2 u ) + c θ (2 u ) (3.4)where the coefficients are given by c = − θ (∆ ) θ (∆ ) θ (∆ + ∆ ) c = − θ (∆ ) θ (∆ ) θ (∆ + ∆ ) c = + 12 θ (∆ ) θ (∆ ) θ (∆ + ∆ ) c = − θ (∆ ) θ (∆ ) θ (∆ + ∆ ) . (3.5)From the parity properties of the theta functions, the reduced BAE becomes c (∆ , τ ) θ (2 u ) f ( − u ; ∆ , τ ) = 0 . (3.6)For generic values of ∆ a and τ , the coefficient c is non-zero and thus the full set of solutionsis given by u = ( m + nτ ) / m, n that are not both even. On the torus thereare 3 solutions: u = 12 , τ , τ + 12 . (3.7)These are precisely the 3 Hong-Liu solutions { , , } , { , , } , { , , } , respectively, as itfollows from (2.25). Therefore, for N = 2, the HL solutions exhaust the full set of solutionsto the reduced BAEs, in agreement with the proposed correspondence with vacua of the N = 1 ∗ theory. When m, n are both even, the zero of θ in the numerator cancels with the zero of f in the denominator. Notice also that u = 0 would lead to configurations ( u , u ) that are fixed by the non-trivial element ofthe Weyl group, and thus would have to be excluded even if it was a solution. valuation of the index. According to the BA formula (in the special case σ = τ forsimplicity), we have the identity I SU (2) ( q, y , y ) = I { , , } SU (2) + I { , , } SU (2) + I { , , } SU (2) . (3.8)On the left-hand side of this identity, I SU (2) is full index computed with the integral formula(2.7). On the right-hand side, instead, each of the three terms I { m,n,r } SU (2) is the contributionto the Bethe Ansatz formula (2.15) from one of the HL solutions: I { m,n,r } SU ( N ) = N · N ! κ N Z (ˆ u ; ∆ , τ ) H (ˆ u ; ∆ , τ ) − (cid:12)(cid:12)(cid:12) ˆ u ∈{ m,n,r } , (3.9)where ˆ u j are as in the HL solution { m, n, r } given in (2.25), while the factor N · N ! comes fromthe multiplicity. We have verified (3.8) numerically for many values of the fugacities. Thisconfirms that, for N = 2, the contributions from the three HL solutions exactly reproducethe superconformal index.To gain a more interesting physical understanding, we can perform a “low-temperature”expansion of both sides of (3.8), corresponding to an expansion of the index into BPS op-erators, order by order in the operator dimension and starting from the identity. In orderto do that, it is convenient to use the set of fugacities ( t, y, v , v ) in (2.6). The restrictionto p = q corresponds to y = 1. On the other hand, the dependence on v , v organizes intocharacters of SU (3) that we indicate by χ dim . For instance χ = 1 χ = v + v + v − v − χ = v + v + v v + v − + v − + v − v − , (3.10)and so on. With computer assistance, we perform a Taylor or Laurent expansion of bothsides of (3.8) around t = 0.Expanding the integral expression, left-hand side of (3.8), we obtain the following. Switch-ing off also the flavor fugacities ( v = v = 1) we find I SU (2) = 1+6 t − t − t +18 t +6 t − t +6 t +84 t − t − t +309 t + O ( t ) . (3.12)Notice that the expansion only contains integer powers of t , and the coefficients are well-defined because the number of operators with bounded dimension is finite. Including the A much longer list of coefficients can be found in [39] for gauge group U (2), and then the coefficients for SU (2) can be derived using (2.13). Switching off flavor fugacities, one finds I U (1) = 1 + 3 t − t + 3 t + 6 t − t + 12 t − t + 27 t − t − t + O ( t ) . (3.11) I SU (2) = 1 + χ t − χ t + (cid:0) − χ (cid:1) t + 2 (cid:0) χ ¯3 + χ (cid:1) t + (cid:0) χ − χ (cid:1) t − (cid:0) χ + χ (cid:1) t + (cid:0) χ + 4 χ ¯3 − χ (cid:1) t + O ( t ) . (3.13)In terms of Dynkin labels: = [3 , = [4 , ′ = [2 , = [5 , 0] and = [3 , I { , , } SU (2) = − − t − t − t − t − t − t − t − t + O ( t ) I { , , } SU (2) = − t − − t − + 32 − t + 6 t − t + 21 t − t + 62 t − t + 177 t − t + 447 t − t + 21352 t − t + 2439 t + O (cid:0) t (cid:1) I { , , } SU (2) = + 14 t − + 34 t − + 32 + 3 t + 6 t + 434 t + 21 t + 1534 t + 62 t + 105 t + 177 t + 11314 t + 447 t + 27754 t + 21352 t + 1635 t + 2439 t + O (cid:0) t (cid:1) . (3.14)Interestingly, the contributions from the two { , , r } solutions contain many unwanted fea-tures: fractional coefficients, and fractional as well as negative powers of t . However theunwanted powers of t cancel out in the sum, and the coefficients sum up to integers. Wehave verified that the sum of the three contributions above exactly matches the expansion(3.12) of the integral expression, up to order O ( t ). Including the flavor fugacities wefind I { , , } SU (2) = − − χ ¯3 t − (cid:0) χ + 2 χ ¯6 (cid:1) t − (cid:0) χ ¯1¯¯0 + 2 χ + 5 (cid:1) t (3.15) − (cid:0) χ ¯1¯¯5 + 2 χ ¯1¯¯5 ′ + 2 χ + 10 χ ¯3 (cid:1) t + O ( t ) I { , , } SU (2) = − t − − χ ¯3 t − + 32 − (cid:0) χ ¯6 + 2 χ (cid:1) t + 2 χ ¯3 t − (cid:0) χ ¯1¯¯0 + 2 χ + 17 (cid:1) t + (cid:0) χ ¯6 + 3 χ (cid:1) t − (cid:0) χ ¯1¯¯5 + 2 χ ′ + 3 χ + 30 χ ¯3 (cid:1) t + 2 (cid:0) χ ¯1¯¯0 + 2 χ + 5 (cid:1) t − (cid:0) χ ¯2¯¯1 + 2 χ ¯2¯¯4 + 3 χ ′ + 31 χ ¯6 + 40 χ (cid:1) t + (cid:0) χ ¯1¯¯5 + 4 χ ¯1¯¯5 ′ + χ + 20 χ ¯3 (cid:1) t + O (cid:0) t (cid:1) I { , , } SU (2) = + 14 t − + 14 χ ¯3 t − + 32 + 14 (cid:0) χ ¯6 + 2 χ (cid:1) t + 2 χ ¯3 t + 14 (cid:0) χ ¯1¯¯0 + 2 χ + 17 (cid:1) t + (cid:0) χ ¯6 + 3 χ (cid:1) t + 14 (cid:0) χ ¯1¯¯5 + 2 χ ′ + 3 χ + 30 χ ¯3 (cid:1) t + 2 (cid:0) χ ¯1¯¯0 + 2 χ + 5 (cid:1) t + 14 (cid:0) χ ¯2¯¯1 + 2 χ ¯2¯¯4 + 3 χ ′ + 31 χ ¯6 + 40 χ (cid:1) t + (cid:0) χ ¯1¯¯5 + 4 χ ¯1¯¯5 ′ + χ + 20 χ ¯3 (cid:1) t + O (cid:0) t (cid:1) Notice that the action of T on these expressions is correctly given by (2.30). In particular, I { , , } SU (2) is invariant, while I { , , } SU (2) is obtained from I { , , } SU (2) (and viceversa). In both cases, thefact that T has finite cyclic orbits and that only integer or half-integer powers of t appear,13mplies that the triality of the SU (3) characters is correlated with the power of t . We haveverified that the sum of the three contributions above reproduces the expansion (3.13) of theintegral expression, up to order O ( t ). SU (3) We consider now the case of gauge group SU (3). We define u ≡ u and v ≡ u . TheWeyl group S of the SU (3) gauge group permutes the triplet ( u , u , u ) while it acts in thefollowing way on ( u, v ): s : ( u, v ) ( − u, v − u ) s R : ( u, v ) ( v − u, − u ) s : ( u, v ) ( u − v, − v ) s L : ( u, v ) ( − v, u − v ) s : ( u, v ) ( v, u ) . (4.1)The three reduced BAEs (2.23) are mapped one into the others by the action of the Weylgroup, therefore let us discuss how to manipulate the first of them. Introducing the function f defined in (3.2), we can recast that equation in the form Q ( u, v ; ∆ , τ ) − f ( u ; ∆ , τ ) f ( v ; ∆ , τ ) − f ( − u ; ∆ , τ ) f ( − v ; ∆ , τ ) f ( − u ; ∆ , τ ) f ( − v ; ∆ , τ ) . (4.2)Using the identity (3.4), the numerator becomesnum = (cid:20) c θ (2 u ) X k =2 , , c k θ k (2 v ) (cid:21) + (cid:20) c θ (2 v ) X k =2 , , c k θ k (2 u ) (cid:21) . (4.3)Collecting terms with the same coefficient c i and using the identities (A.6), we rewrite it asnum = c θ ( u + v ) θ (0) θ (0) θ (0) " [3] θ ( u + v ) θ ( u − v ) θ ( u − v ) (4.4) − [4] θ ( u + v ) θ ( u − v ) θ ( u − v ) − [2] θ ( u + v ) θ ( u − v ) θ ( u − v ) , where we defined [ r ] = θ r (0) θ r (∆ ) θ r (∆ ) θ r (∆ + ∆ ). They obey [3] = [4] + [2] [63], thenusing (A.6) again we obtainnum = 2 c θ ( u + v ) θ ( u ) θ ( v ) θ (0) θ (0) θ (0) × (4.5) × (cid:20) [2] θ (0) θ ( u ) θ ( v ) θ ( u − v ) − [4] θ (0) θ ( u ) θ ( v ) θ ( u − v ) (cid:21) . Notice that the first reduced BAE concerns the antisymmetric part of the product of two f ’s. Thisgeneralizes to any N : the solutions to the first reduced BAE can always be written as the zeros of theantisymmetric part of the product of N − f (while the other ones are obtained by the action ofthe Weyl group). Q − c θ ( u + v ) h ( u, v ; ∆ , τ ) θ (0) θ (0) θ (0) Q ∆ ∈{ ∆ , ∆ , − ∆ − ∆ } θ (∆ − u ) θ (∆ − v ) (4.6)where we defined the function h ( u, v ; ∆ , τ ) = θ (∆ ) θ (∆ ) θ (∆ + ∆ ) θ ( u ) θ ( v ) θ ( u − v ) − (cid:0) ↔ (cid:1) . (4.7)Crucially, this function is invariant under the gauge Weyl group. The second reduced BAEcan be obtained by acting with the gauge Weyl group on the first one. Since the denominatorof (4.6) does not have poles, the reduced BAEs can be brought to the factorized form ( θ ( u + v ) h ( u, v ; ∆ , τ )0 = θ (2 u − v ) h ( u, v ; ∆ , τ ) . (4.8)From here we clearly see the structure of the solutions.First, there are discrete solutions that follow from solving0 = θ ( u + v ) = θ (2 u − v ) . (4.9)These equations represent hyperplanes inside T , and the discrete solutions are at the in-tersections of two hyperplanes. There are four solutions on T , up to identifying those thatare related by the Weyl group action and dropping those that are fixed by some non-trivialelement of the Weyl group ( i.e. , solutions in which either u or v vanish):( u, v ) = (cid:18) , (cid:19) , (cid:18) τ , τ (cid:19) , (cid:18) τ + 13 , τ + 23 (cid:19) , (cid:18) τ + 23 , τ + 13 (cid:19) . (4.10)These are precisely the 4 HL solutions { , , } , { , , } , { , , } , { , , } , respectively,according to (2.25).Second, there is a continuous family of solutions obtained by solving the single equation0 = h ( u, v ; ∆ , τ ) . (4.11)We will analyze some properties of these solutions below, but we already see that the con-tinuous family has complex dimension 1. The solutions to (4.11), as opposed to the HLsolutions, depend on the flavor fugacities ∆ a . There are however six special points on thecurve, all related by the action of the S Weyl group, that do not depend on ∆ a :( u, v ) = (cid:18) , τ (cid:19) . (4.12)On these points, the action of P SL (2 , Z ) reduces to the action of S . These are smoothpoints, which do not seem to have any other special property. These points have been studied in [64] in the context of the N = 1 ∗ theory on S . In the context of theBethe Ansatz equations, they have been noticed in [1]. valuation of the index. The Bethe Ansatz formula (2.15) was derived in [53, 54] as-suming that all solutions to the BAEs are isolated. It is then clear that, in the presence ofcontinuous families of solutions as it happens in this case, the formula has to be modifiedsomehow in order to correctly evaluate the contribution of those families. We could hopethat, for some reason, continuous families do not contribute. Let us test this possibility, bycomparing the result of the integral formula (2.7) with that of the BA formula (2.15), inwhich the sum is restricted to isolated solutions to the BAEs (here the HL solutions), in thelow-temperature limit.The expansion of the integral formula (2.7), setting the flavor fugacities v = v = 1,gives I SU (3) = 1+6 t − t +3 t +6 t − t +27 t +18 t − t +96 t +54 t + O ( t ) , (4.13)while switching on the SU (3) fugacities it gives I SU (3) = 1 + χ t − χ t + (cid:0) χ − χ + 1 (cid:1) t + 2 χ ¯3 t + (cid:0) χ − χ ′ + χ ¯6 − χ (cid:1) t (4.14) − χ t + (cid:0) χ − χ ¯1¯¯5 ′ + χ + 5 χ ¯3 (cid:1) t + (cid:0) χ ′ − χ + 4 χ ¯6 − χ (cid:1) t + O ( t ) . The expansion of the contributions (3.9) of the HL solutions to the BA formula gives: I { , , } SU (3) = 3 + 6 χ ¯3 t + 3 (cid:0) χ ¯6 − χ (cid:1) t + 6 (cid:0) χ ¯1¯¯0 + 2 χ − (cid:1) t + 3 (cid:0) χ ¯1¯¯5 − χ ¯1¯¯5 ′ + 24 χ − χ ¯3 (cid:1) t + 6 (cid:0) χ ¯2¯¯1 + 6 χ ¯2¯¯4 + 8 χ ′ − χ ¯6 + 30 χ (cid:1) t + 3 (cid:0) χ ¯2¯¯8 − χ ¯3¯¯5 + 16 χ ¯2¯¯4 + 71( χ + χ ) − χ ¯1¯¯0 − χ − χ + 164 (cid:1) t + O ( t ) I { , , } SU (3) = χ − χ ¯3 (cid:0) χ − χ + χ ¯3 + 9 (cid:1) t − + 2 (cid:0) χ ′ − χ ¯1¯¯5 ′ + 6 χ − χ + χ ¯6 + 11 χ − χ ¯3 + 6 (cid:1) (cid:0) χ + 2 χ − χ − χ ¯3 + 27 χ − (cid:1) t − + O ( t − ) (4.15) I { , , } SU (3) = χ ¯3 + e πi/ χ (cid:0) χ + e − πi/ ( χ + χ ¯3 ) + 9 e πi/ (cid:1) t − − (cid:0) χ ′ + χ ¯6 + 11 χ − e πi/ (6 χ + 6) + e − πi/ ( χ ¯1¯¯5 ′ + 7 χ + 17 χ ¯3 ) (cid:1) (cid:0) e πi/ ( χ + 2 χ − − e − πi/ ( χ + χ ¯3 ) − χ (cid:1) t − + O ( t − )while I { , , } SU (3) is obtained from I { , , } SU (3) by taking the complex conjugate of all coefficients.The action of T on these expressions is given by (2.30). Notice that I { , ,r } SU (3) does not have awell-defined limit as v , → e − πi r , for r = 0 , , t , fractional coefficientsand characters in the denominators. This implies that the contribution from the continuousfamily of solutions is necessary in order to correctly reproduce the index. It is an open issuehow to compute the contribution from the continuous family, to which we hope to return infuture work. 16 .1 The continuous family of solutions Let us study in more detail the curve h ( u, v ; ∆ , τ ) = 0, for generic values of the flavorchemical potentials ∆ a . We rewrite the equation as θ ( u ) θ ( v ) θ ( u − v ) θ ( u ) θ ( v ) θ ( u − v ) = θ (∆ ) θ (∆ ) θ (∆ + ∆ ) θ (∆ ) θ (∆ ) θ (∆ + ∆ ) . (4.16)If we regard the left-hand side as a function of u (for fixed v ), we see that it is doubly-periodicwith periods 1 and τ , with two simple poles at u = and u = v + , and two simple zerosat u = τ and u = v + τ , i.e. , it is an elliptic function of order 2. This means that it givestwo values u ± ( v ) of u for each value of v , and they satisfy u + + u − = v (mod Z + Z τ ). Inother words we have a branched double cover of the torus.The equation can be formally solved in terms of Jacobi elliptic functions (see for instance[66]). Define the new variables z = 2 K ( m ) u , ω = 2 K ( m ) v and z i = 2 K ( m ) ∆ i , where K ( m )is the complete elliptic integral of the first kind, and m is the parameter (see Appendix A).Then the curve can be rewritten ascn( z ) cn( ω ) cn( z − ω ) = cn( z ) cn( z ) cn( z + z ) . (4.17)We use the Jacobi elliptic functions cn( z, m ), sn( z, m ), dn( z, m ) and keep the dependence onthe parameter m implicit. Then, employing addition formulas, we can isolate the dependenceon z : sn (cid:16) z − ω (cid:17) = cn( z ) cn( z ) cn( z + z ) − cn ( ω/ 2) cn( ω ) m sn ( ω/ 2) cn( z ) cn( z ) cn( z + z ) − dn ( ω/ 2) cn( ω ) . (4.18)The two values for u can finally be written in terms of the inverse function arcsn: u ± ( v ) = v ± K ( m ) arcsn s cn( z ) cn( z ) cn( z + z ) − cn ( ω/ 2) cn( ω ) m sn ( ω/ 2) cn( z ) cn( z ) cn( z + z ) − dn ( ω/ 2) cn( ω ) ! . (4.19)The two equations u = u ± ( v ) describe the curve as a double cover of the torus. The twosheets of the covering meet at all points where u + ( v ) = u − ( v ) = v/ 2. We could ask whetherthese are smooth points where the covering is branched, or rather are self-intersection pointsof the curve or intersection points of two connected components the curve is composed of.We can see that these must be smooth branch points with a simple argument. The pointslie on the hyperplane θ (2 u − v ) = 0 on which Q = 1. However, we could have equivalentlydescribed the curve as v = v ± ( u ). If those points were intersection points, and so if they hadan intrinsic geometric characterization, they would also solve v + ( u ) = v − ( u ) = u/ In the case of the SU (3) N = 1 ∗ theory on S , the continuous family of vacua has been studied in [65]. θ ( u − v ) = 0 on which Q = 1. However the intersectionsof the two hyperplanes (which intersect also the hyperplane θ ( u + v ) = 0) are preciselythe HL solutions, and one can easily check that they do not lie on the curve (4.11). Thiscontradiction implies that there are no special intersection points on the curve, only smoothbranch points. In particular this implies that the curve is composed of a single connectedcomponent.More directly, we can compute the gradient of h . Exploiting the relations (A.6) and(A.7), with some algebra we obtain that on the curve h ( u, v ; ∆ , τ ) = 0 it holds ∂ u h = − θ ′ (0) θ (2 u − v ) θ ( u ) θ ( u − v ) h (0 , v ; ∆ , τ ) , ∂ v h = θ ′ (0) θ (2 v − u ) θ ( u ) θ ( u − v ) h ( u, 0; ∆ , τ ) . (4.20)There are no points where h = ∂ u h = ∂ v h = 0, therefore all points on the curve are smooth. Acknowledgements We thank Ofer Aharony, Arash Arabi Ardehali and Jan Troost for useful discussions and cor-respondence, and in particular Paolo Milan for collaboration in the early stage of this work.We also thank the organizers and participants of the SCGP seminar series on “Supersym-metric black holes, holography and microstate counting”, where part of these results wereannounced, for interesting discussions. F.B. is partially supported by the ERC-COG grantNP-QFT No. 864583, by the MIUR-SIR grant RBSI1471GJ, by the MIUR-PRIN contract2015 MP2CX4, as well as by the INFN “Iniziativa Specifica ST&FI”. A Special functions We use fugacities and chemical potentials related by z = e πiu , p = e πiσ , q = e πiτ , (A.1)with | p | , | q | < 1. The q -Pochhammer symbol is defined as( z ; q ) ∞ = ∞ Y j =0 (1 − zq j ) . (A.2)The elliptic theta function θ is defined as θ ( u ; τ ) = ( z ; q ) ∞ ( q/z ; q ) ∞ . (A.3)18he elliptic gamma function is defined as e Γ( u ; σ, τ ) = ∞ Y m,n =0 − p m +1 q n +1 /z − p m q n z . (A.4)We use the following definitions for the Jacobi theta functions: θ ( u ; τ ) = 2 e πiτ/ sin( πu ) ∞ Y n =1 (1 − q n )(1 − q n z )(1 − q n /z ) = i e πiτ/ − πiu ( q ; q ) ∞ θ ( u ; τ ) θ ( u ; τ ) = 2 e πiτ/ cos( πu ) ∞ Y n =1 (1 − q n )(1 + q n z )(1 + q n /z ) = e πiτ/ − πiu ( q ; q ) ∞ θ (cid:0) u + ; τ (cid:1) θ ( u ; τ ) = ∞ Y n =1 (1 − q n ) (cid:0) q n − z (cid:1)(cid:0) q n − /z (cid:1) = ( q ; q ) ∞ θ (cid:0) u + + τ ; τ (cid:1) θ ( u ; τ ) = ∞ Y n =1 (1 − q n ) (cid:0) − q n − z (cid:1)(cid:0) − q n − /z (cid:1) = ( q ; q ) ∞ θ (cid:0) u + τ ; τ ) . (A.5)Notice that θ is odd while θ , , are even under u → − u . One set of identities we use is θ ( u + v ) θ ( u − v ) θ (0) θ (0) = θ ( u ) θ ( u ) θ ( v ) θ ( v ) + θ ( u ) θ ( u ) θ ( v ) θ ( v ) θ ( u + v ) θ ( u − v ) θ (0) θ (0) = θ ( u ) θ ( u ) θ ( v ) θ ( v ) + θ ( u ) θ ( u ) θ ( v ) θ ( v ) θ ( u + v ) θ ( u − v ) θ (0) θ (0) = θ ( u ) θ ( u ) θ ( v ) θ ( v ) + θ ( u ) θ ( u ) θ ( v ) θ ( v ) θ ( u + v ) θ ( u − v ) θ (0) θ (0) = θ ( u ) θ ( u ) θ ( v ) θ ( v ) − θ ( u ) θ ( u ) θ ( v ) θ ( v ) θ ( u + v ) θ ( u − v ) θ (0) θ (0) = θ ( u ) θ ( u ) θ ( v ) θ ( v ) − θ ( u ) θ ( u ) θ ( v ) θ ( v ) , (A.6)see for instance [63]. Here θ r ( u ) ≡ θ r ( u ; τ ). We also use ∂ u (cid:18) θ ( u ) θ ( u ) (cid:19) = π θ (0) θ ( u ) θ ( u ) θ ( u ) (A.7)and a similar relation with 2 and 4 exchanged, as well as θ ′ (0) = π θ (0) θ (0) θ (0).The Jacobi elliptic functions are related to the Jacobi theta functions bycn (cid:0) K ( m ) u, m (cid:1) = (cid:18) − mm (cid:19) / θ ( u ; τ ) θ ( u ; τ )sn (cid:0) K ( m ) u, m (cid:1) = (cid:18) m (cid:19) / θ ( u ; τ ) θ ( u ; τ )dn (cid:0) K ( m ) u, m (cid:1) = (1 − m ) / θ ( u ; τ ) θ ( u ; τ ) . (A.8)The parameter m is related to the modulus τ by m ( τ ) = θ (0; τ ) θ (0; τ ) . (A.9)19he function K ( m ) is the complete elliptic integral of the first kind, and is related to τ by τ = i K (1 − m ) K ( m ) or 2 K ( m ) = θ (0; τ ) π . (A.10)We use the addition formulacn( x + y, m ) cn( x − y, m ) = cn ( y, m ) − sn ( x, m ) dn ( y, m )1 − m sn ( x, m ) sn ( y, m ) . (A.11) References [1] A. Arabi Ardehali, J. Hong, and J. T. 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