aa r X i v : . [ h e p - t h ] J un Prepared for submission to JHEP
Topologically twisted index of T [ SU ( N )] at large N Lorenzo Coccia a,b a Dipartimento di Fisica, Università di Milano-Bicocca, I-20126 Milano, Italy b INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italy
E-mail: [email protected]
Abstract:
We compute, in the large N limit, the topologically twisted index of the 3d T [ SU ( N )] theory, namely the partition function on Σ g × S , with a topological twist onthe Riemann surface Σ g . To provide an expression for this quantity, we take advantageof some recent results obtained for five dimensional quiver gauge theories. In case of auniversal twist, we correctly reproduce the entropy of the universal black hole that can beembedded in the holographically dual solution. ontents T [ SU ( N )]
33 Large N free energy on S N matrix model 53.2 Saddle point and boundary conditions 93.3 Evaluation of the free energy 10 T [ SU ( N )] twisted superpotential 146 Topologically twisted index for T [ SU ( N )] S
23C Formulas for the computation of the twisted index 26 – 1 –
Introduction
The topologically twisted index Z Σ g × S for three dimensional N ≥ g × S with a topological twist on the Riemann surfaceΣ g . Thanks to the insight of [3], the index for the ABJM theory in the large N limithas been used to provide the first holographic microscopic counting of the entropy of anasymptotically AdS black hole. Afterwards, this result has been extended (see [4] for areview) and the large N limit of the index has been studied for other quiver gauge theorieswith an AdS dual [5–8]. To compute the topologically twisted index, one can resort tothe localization results in [1, 2] which allow to write the index as a contour integral of ameromorphic form, summed over a lattice of magnetic fluxes. It is convenient, as in [3], tofirst perform the sum and to apply the residue theorem to evaluate the expression left. Inthis procedure, one can introduce an auxiliary quantity, the twisted superpotential , whosecritical points give the position of the poles of the integrand obtained after performing thesum.As we will better discuss in Section 4, in the large N limit of all the studied cases,the twisted superpotential turns out to be related to the free energy on the three-sphereby a simple relation, pointed out in [5] (see also [7]). Moreover, for a particular choice oftopological twist, called universal topological twist in [9], the free energy F S is also relatedto the twisted index of the same theory bylog Z Σ g × S = ( g − F S . (1.1)This simple relation has an equally simple explanation, in the holographic perspective,in terms of an universal black hole [9]. This black hole is a solution in minimal fourdimensional gauged supergravity and in the large N limit the topologically twisted indexreproduces its entropy. The fact that the black hole solution can be embedded in infinitelymany ways in eleven-dimensional and massive type IIA supergravity justifies the fact that(1.1) holds for a large class of theories [9].However, despite all these remarkable progresses, the index has been computed only forfew theories in the large N limit. First of all, the theories considered so far are non chiral,namely for each bi-fundamental connecting two nodes there is another bi-fundamental inthe opposite direction and the number of fundamental and (anti-)fundamental fields isequal. It is not yet clear how to compute the large N limit of the twisted index for chiralquivers. Moreover, even among non chiral theories, many cases are not are not covered inthe literature. An example is the T [ SU ( N )] theory. T [ SU ( N )] is a three dimensional N = 4 gauge theory, originally introduced in thestudy of S-duality of boundary conditions in N = 4 four dimensional SYM theory [12]. Itcan be represented in terms of a linear quiver with N − S of T [ SU ( N )] has been computed exactly This quantity is referred to as
Bethe potential in [3] and part of the literature. This situation is similar to the one found in [10] for the large N limit of the free energy on S . We note, however, that the topologically twisted index for T [ SU ( N )] has been considered in [11] fromthe point of view of the factorisation into holomorphic blocks. or each N [13, 14] and the leading behaviour of the free energy F S at large N is F S = 12 N log N + O ( N ) . (1.2)This result has also been reproduced considering the gravity dual solution of T [ SU ( N )][15, 16]. For our discussion, the important point is that the universal black hole can beembedded in this dual solution. This means that, if we compute the twisted index for T [ SU ( N )] in case of the universal twist, we expect to verify relation (1.1), reproducing theentropy of the universal black hole. This will be the main aim of this paper.In [17] a new approach has been proposed to compute the free energy of some theorieson a five dimensional sphere, in the planar limit. Theories considered in [17] are describedby linear quivers with a large number of nodes. In three dimensions, this is exactly whathappens for T [ SU ( N )] in the large N limit. Indeed, in this work we will show that theapproach of [17] can be also applied in three dimensions. First of all, we will be able toreproduce the known results for the free energy on the three-sphere, testing in this way themethod. Using the same framework, we will then move to the computation of the twistedsuperpotential, recovering the known relation with the free energy, previously mentioned.Finally we will also be able to compute the topologically twisted index for T [ SU ( N )],verifying the relation (1.1).The plan of the rest of the paper is the following. In the next section we will introducethe T [ SU ( N )] theory, recalling some useful aspects and describing the notation used inthe paper. In Section 3, we will explicitly compute the free energy in the large N limit,following the procedure of [17]. In this computation we will turn on an arbitrary R-charge,in prevision of a comparison with the expressions for the twisted superpotential and theindex, which are functions of chemical potentials. Section 4 is devoted to the review ofthe general aspects of the topologically twisted index, together with some known results.In Section 5 we compute the twisted superpotential for T [ SU ( N )] and finally in Section6 we will provide the expression for the index, in particular in the case of the universaltopological twist. We will conclude the paper with few comments and three appendices,containing useful formulas and computations. T [ SU ( N )] T [ SU ( N )] is a three dimensional N = 4 gauge theory at the IR superconformal fixedpoint. It was originally introduced in [12] and admits a nice description in terms of thelinear quiver (in N = 2 notation):1 2 3 ... N − N (2.1)– 3 –ach round node in the quiver is labelled by a number t = 1 , . . . , N − U ( t ) in the gauge group of the theory, with an associated N = 2 vector multiplet. On top ofeach node, an arc indicates the presence of a chiral field Φ t in the adjoint representation ofthe group U ( t ). Bi-fundamentals fields are identified with lines connecting adjacent nodes:we denote Q ( t ) the bi-fundamental which goes from the node t to the node t +1 and ˜ Q ( t ) thebi-fundamental in the opposite direction. For the last gauge node, the lines and the relatedsymbols ˜ Q ( N − , Q ( N − actually denotes (anti-)fundamental fields transforming under the SU ( N ) global symmetry, represented by the square box at the end of the quiver. Withthese conventions, the superpotential for T [ SU ( N )] can be schematically written as W = N − X t =1 Tr h ˜ Q ( t ) Φ ( t ) Q ( t ) − Q ( t − Φ ( t ) ˜ Q ( t − i (2.2)where Q (0) = 0 (we refer to [18, 19] for a more careful notation). Possible deformations ofthe theory are obtained considering mass terms for the N = 4 hypermultiplet and Fayet-Iliopoulos (FI) parameters for the U (1) gauge factors. In this work, however, we will setthese deformations to zero.Finally, we recall that T [ SU ( N )] is invariant under mirror symmetry [20]. This sym-metry acts on the vacuum moduli space and exchanges Higgs and Coulomb branches, massand FI parameters, and the two factors of the SU (2) R × SU (2) L R-symmetry of the theory.
Notation
In the next sections, following the treatment of [17], it will be convenient torewrite the quiver (2.1) in a slightly more general way. We use the symbol L to indicatethe length of the quiver and introduce a coordinate t to label the gauge nodes of the quiver,with t = 1 , . . . , L . We also denote the rank of the gauge group in the t th node with N t andnumber of flavours in the last node with k L . The resulting quiver is N N N N L ... k L t (2.3)At the end of our computations, we will substitute N t = t , L = N − k L = N ,obtaining results for the original quiver (2.1). N free energy on S We start considering the T [ SU ( N )] partition function on the three dimensional sphere S .Resorting to the results obtained in [21] by a localization procedure, this quantity has beencomputed exactly for each N [13, 14]. Moreover, the related free energy F S = − log | Z S | (3.1)as been studied, in the large N limit, both from the field theory and the gravity dual side [15, 16, 22]. In case of vanishing mass deformations and topological charges, the leadingbehaviour is F S = 12 N log N + O ( N ) . (3.2)Our first aim in this work is to take advantage of the recent approach of [17] in studying longlinear quivers, to reproduce the behaviour (3.2). Indeed, we will see that the techniquesapplied in [17] to compute partition functions on the five-sphere can be re-proposed inthree dimensions, providing consistent results. N matrix model As shown in [21], the partition function on the three-sphere of a superconformal theorywith N = 2 or more supersymmetry localizes to a matrix model. In these cases the R-charges of the theory are fixed to their canonical values. This result has been extended in[23, 24] where more general N = 2 theories with an arbitrary R-charge assignment havebeen considered. For these theories, the partition function on S still reduces to a matrixmodel and it is a function of a set of trial R-charges. Extremizing the partition functionreturns the exact values for the R-charges [23].In this section, we write the partition function for T [ SU ( N )] in language N = 2 andturn on an arbitrary R-charge r , equal for all the bi-fundamental and (anti-)fundamentalfields in the theory. The R-charge ˜ r for all the adjoint fields is then fixed by the constraint˜ r = 2(1 − r ) (3.3)in such a way that the superpotential (2.2) has R-charge 2. Using the results in [23], wecan write the partition function as a matrix model and we will then apply the saddle pointapproximation to compute the free energy (3.1) as a function of the charge r . At the end,maximizing the expression with respect to r , we expect to find (3.2) when r = 1 /
2. Thechoice of turning an R-charge on is motivated by a future comparison of F S with thetopological twisted index, where chemical potentials will play the role of R-charges (seeSection 4).Therefore, the partition function we are interested in localizes to a finite matrix inte-gral, which we schematically write [23] Z S = 1 | W | Z Y
Cartan dλ e −F ( λ ) (3.4)where | W | is the order of the Weyl group of the gauge group. The exponential in theintegrand is the sum of the various contributions in the theory F = F vec + F adj + F bif + F (a)f . (3.5)More explicitly, and using the notation introduced in the quiver (2.3), the contributionfrom all the gauge nodes in the theory is F vec = − L X t =1 N t X i = j =1
12 log (cid:16) ( π ( λ ( t ) i − λ ( t ) j )) (cid:17) ; (3.6) In [15, 16] a larger class of T ρσ [ SU ( N )] has been considered. – 5 –rom the bi-fundamental pairs we have F bif = − L − X t =1 N t X i =1 N t +1 X j =1 (cid:16) ℓ (1 − r + i ( λ ( t ) i − λ ( t +1) j )) + ℓ (1 − r − i ( λ ( t ) i − λ ( t +1) j )) (cid:17) (3.7)with ℓ ( z ) = − z log (cid:16) − e πiz (cid:17) + i (cid:18) πz + 1 π Li (cid:16) e πiz (cid:17)(cid:19) − iπ
12 (3.8)and from the (anti-)fundamentals in the last node with k L flavours F (a)f = − k L N L X i =1 (cid:16) ℓ (1 − r + iλ ( L ) i ) + ℓ (1 − r − iλ ( L ) i ) (cid:17) . (3.9)Adjoint fields are identified with a pair of bi-fundamentals connecting the same gaugegroup, with an overall factor 1 /
2, and this gives the expression for F adj .Our plan is to evaluate (3.4) in the large N limit, using the saddle point approximation.The idea is then to find the configuration of eigenvalues λ extremizing (3.5). Note thatthe prefactor 1 / | W | in front of (3.4) compensates for the fact that there are | W | distinctcritical points in which the integral takes the same value. Hence, once we evaluate F ( λ ) inone of such configurations λ , we can approximate the integral (3.4) as Z ∼ e −F ( λ ) . It isimportant to observe that, by this method, we do not expect to be able to reproduce sub-leading terms of order N in the asymptotic expansion. Indeed, the number of integrationvariables in the matrix model is of order N as well, meaning that all the orders in theexpansion of F ( λ ) around the saddle point could in principle contribute to the order N of the free energy.We now follow the treatment of [17], adapting it to our case. The extremizationproblem can be tackled introducing a density for the eigenvalues (i.e. the integrationvariables) of each node ρ t ( λ ) = 1 N t N t X i =1 δ ( λ − λ ( t ) i ) . (3.10)We assume that, for large N t , this density becomes a continuous function, with the correctnormalization Z dλ ρ t ( λ ) = 1 . (3.11)Obviously, a continuous distribution is not a good approximation for the eigenvalues asso-ciated with a group of small rank, i.e. for the first nodes of T [ SU ( N )]. However, as thequiver becomes longer and longer (i.e. for large N ), we expect the contribution from thelarge groups to be more and more important and the approximation to be reliable. Hence,in the expression (3.5) we simply substitute N t X i =1 → N t Z dλ ρ t ( λ ) (3.12)or each node t obtaining F = Z dλdλ ′ L X t =1 N t ρ t ( λ ) ρ t ( λ ′ ) F V ( λ − λ ′ ) + L − X t =1 N t N t +1 ρ t ( λ ) ρ t +1 ( λ ′ ) F H ( λ − λ ′ ) ! + k L Z dλN L ρ L ( λ ) F H ( λ ) (3.13)where we introduced F V ( λ ) = −
12 log (cid:16) ( πλ ) (cid:17) −
12 [ ℓ (1 − ˜ r + iλ ) + ℓ (1 − ˜ r − iλ )] ,F H ( λ ) = − ℓ (1 − r + iλ ) − ℓ (1 − r − iλ ) . (3.14)Here the first term in F V is the vector contribution and the second the adjoint one. Theremaining bi-fundamental contributions are in F H . After a simple manipulation, (3.13)becomes F = Z dλdλ ′ " L X t =1 N t ρ t ( λ ) ρ t ( λ ′ ) F ( λ − λ ′ ) − L − X t =1 η t ( λ ) η t ( λ ′ ) F H ( λ − λ ′ ) − X t ∈{ ,L } Z dλdλ ′ N t ρ t ( λ ) ρ t ( λ ′ ) F H ( λ − λ ′ ) + k L Z dλN L ρ L ( λ ) F H ( λ ) (3.15)with η t ( λ ) ≡ N t +1 ρ t +1 ( λ ) − N t ρ t ( λ ) and F ( λ ) = F V ( λ ) + F H ( λ ) . (3.16)At this point, the procedure of [17] consists in replacing the variable t , which labelsnodes of the quiver, with the variable z = t/L which can be considered continuous in thelarge L limit. The boundaries of the quiver are given by z = 0 and z = 1. At the sametime, we replace the family of densities in (3.12), parametrized by t , with a single functionof two continuous parameters ρ t ( λ ) = ρ zL ( λ ) ≡ ρ ( z, λ ) . (3.17)Promoting the rank of the gauge groups N t to a continuous function of z , we also makethe substitution N t +1 ρ t +1 ( λ ) − N t ρ t ( λ ) → L ∂ z ( N ( z ) ρ ( z, λ )) . (3.18)We can then rewrite (3.15) as F = L Z dz Z dλdλ ′ " N ( z ) ρ ( z, λ ) ρ ( z, λ ′ ) F (cid:0) λ − λ ′ (cid:1) − L ∂ z ( N ( z ) ρ ( z, λ )) ∂ z (cid:0) N ( z ) ρ ( z, λ ′ ) (cid:1) F H (cid:0) λ − λ ′ (cid:1) − X z ∈{ , } Z dλdλ ′ N ( z ) ρ ( z, λ ) ρ ( z, λ ′ ) F H (cid:0) λ − λ ′ (cid:1) + k L N (1) Z dλ ρ (1 , λ ) F H ( λ ) . (3.19)– 7 –or T [ SU ( N )], N ( z ) = zL ∼ zN . Expression (3.19) is completely analogous to the expres-sions found in [17] for free energies on the five-sphere. However, integrands in (3.19) arerather complicated expressions, which need to be simplified. For this purpose, note thatthe terms inside the square brackets are balanced in N ( z ) but not in L . Therefore, weassume the eigenvalues to scale as λ = L α x , (3.20)with α >
0. Under this assumption and after some computations described in AppendixB, we obtain the expressions F ( L α x ) ∼ πL α (1 − r ) r δ ( x ) + . . . ,F H ( L α x ) ∼ πL α (1 − r ) | x | + . . . . (3.21)Before proceeding with the computation, let us pause a bit on these functions. We note thatwhen we plug the expression just written for F in (3.19), it produces a local contributionin the eigenvalues. Looking at the computation in Appendix B, we see that this is truebecause the leading term in the expansion of F = F V + F H π (2 − r − ˜ r ) N ( z ) Z dλdλ ′ ρ ( z, λ ) ρ ( z, λ ′ ) (cid:12)(cid:12) λ − λ ′ (cid:12)(cid:12) (3.22)vanishes under condition (3.3) imposed by the superpotential. This feature is often called long-range force cancellation , meaning that the free energy, which is schematically a func-tion of the entire sum P i,j ( λ i − λ j ), only gets contributions from i ∼ j . However, this isnot completely our case, because of the presence of integrals with F H , which is non local.Let us now go back to the computations. In order to have a non trivial combinationbetween the terms in the square brackets of (3.19), we require them to have the samescaling with respect to L ; this leads us to α = 1. With this choice, the leading order of thefree energy (3.19) becomes F = Z dz Z dx " ̺ ( z, x ) ˜ F ( r ) −
12 ˜ F H ( r ) ∂ z ̺ ( z, x ) Z dx ′ ∂ z ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) − L X z ∈{ , } Z dxdx ′ ̺ ( z, x ) ̺ ( z, x ′ ) ˜ F H ( r ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + Lk L Z dx̺ (1 , x ) ˜ F H ( r ) | x | + L Z dz µ ( z ) (cid:18)Z dx̺ ( z, x ) − N ( z ) (cid:19) (3.23)where, performing the change of variable λ = Lx , we introduced the rescaled density dx̺ ( z, x ) = dλN ( z ) ρ ( z, λ ) and defined˜ F ( r ) = 2 π (1 − r ) r , ˜ F H ( r ) = 2 π (1 − r ) . (3.24)We also inserted a Lagrange multiplier µ ( z ) to impose the normalization condition Z dx ̺ ( z, x ) = N ( z ) . (3.25) .2 Saddle point and boundary conditions Having the expression (3.23), we can proceed with the saddle point approximation andevaluate it around the critical point. Hence, we need to take the variation of (3.23) w.r.t. ̺ ( z, x ). In this procedure, we get contributions both from the bulk, namely the interior ofthe interval z ∈ [0 , made of z = { , } . The functional variation inthe interior of [0 ,
1] gives2 ˜ F ( r ) ̺ ( z, x ) + ˜ F H ( r ) Z dx ′ ∂ z ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + Lµ ( z ) = 0 . (3.26)This equation has to be satisfied for each x . In particular, for large x , the first term issubleading, since we assume the density to decay at infinity, and (3.26) gives | x | ˜ F H ( r ) Z dx ′ ∂ z ̺ ( z, x ′ ) = 0 ⇒ ∂ z N ( z ) = 0 (3.27)where we used the normalization condition (3.25). Eq. (3.27) is the continuous version ofthe condition for balanced nodes 2 N t = N t − + N t +1 , which is certainly satisfied, for each t , by T [ SU ( N )]. Following [17], in order to solve (3.26) we consider its second derivativewith respect to x : ˜ F ˜ F H ∂ x ̺ ( z, x ) + ∂ z ̺ ( z, x ) = 0 (3.28)with ˜ F ˜ F H = r . (3.29)Remarkably, if (3.28) is solved, then also (3.26) is automatically solved, with vanishingLagrange multipliers. Boundary conditions
The boundary contribution in the variation comes from the ex-plicit terms in the second line of (3.23) and from the derivatives in the first line. However,due to the normalization condition (3.25), for T [ SU ( N )] when z = 0 we need to have ̺ (0 , x ) = 0 (3.30)since N ( z ) = zL . So we only consider the boundary at z = 1. Assuming vanishingmultipliers even on the boundary, the variation gives − ˜ F H ( r ) Z dx ′ (cid:0) ∂ z ̺ ( z, x ′ ) + L̺ ( z, x ′ ) − Lk L δ ( x ′ ) (cid:1) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =1 = 0 . (3.31) Recall that, in general, when one has a functional F [ φ ] F [ φ ] = Z M g ( φ, ∇ φ ) dV + Z ∂ M h ( φ ) d Σover a volume M with boundary ∂ M the variation is given by δF [ φ ] = Z M (cid:18) ∂g∂φ − ∇ · ∂g∂ ( ∇ φ ) (cid:19) δφdV + Z ∂ M (cid:18) ∂g∂ ( ∇ φ ) · n + ∂h∂φ (cid:19) δφd Σwith n outward-pointing unit vector, normal to the boundary. The two terms have to independently vanish. – 9 –or large L , the first term in the brackets is subleading and the equation can be satisfied if ̺ (1 , x ) = k L δ ( x ) . (3.32)The saddle point equation (3.28), together with the boundary conditions (3.30) and (3.32),defines a two-dimensional "electrostatic" problem, equal to the one found in [17] for the5d T N theory [25]. Up to an appropriate rescaling, then, we can directly read the solutionobtained in that work ̺ s ( z, x ) = k L sin( πz )2 r (cosh (cid:0) πr x (cid:1) + cos( πz )) . (3.33)Note that this (rescaled) density is defined on the entire x axis and it is properly normal-ized (see [17] for more details), consistently with the assumption of vanishing Lagrangemultipliers. We can now evaluate F in the saddle point, i.e. we have to substitute in (3.23) the ̺ s ( z, x )just found. The result gives the required expression for the free energy F S . Thanks tothe conditions (3.30) and (3.32), the explicit boundary terms disappear leaving (after anintegration by parts) F S = Z dz Z dx ̺ s ( z, x ) (cid:20) ̺ s ( z, x ) ˜ F ( r ) + 12 ˜ F H ( r ) Z dx ′ ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:21) − Z dz Z dxdx ′ ˜ F H ( r ) ∂ z (cid:2) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:3) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) . (3.34)Using the saddle point equation (3.26) with no Lagrange multipliers, the terms inside thesquare brackets vanish and the expression becomes F S = −
12 ˜ F H ( r ) Z dz Z dxdx ′ ∂ z (cid:2) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:3) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) == −
12 ˜ F H ( r ) Z dxdx ′ (cid:2) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:3) z =1 z =0 (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) . (3.35)Finally, using again the boundary conditions (3.30) and (3.32), we can reduce this expres-sion to a one dimensional integral F S = − k L F H ( r ) Z dx [ ∂ z ̺ s ( z, x )] z =1 | x | . (3.36)The domain of integration should be the entire real axis but this integral turns out to bedivergent in 0. The origin of this divergence is the fact that we substituted the expressionsof F ( Lx ) and F H ( Lx ) with their asymptotic expansions (3.21), motivated by the fact that L is large. However, this substitution holds until x becomes of the order β/L with an The useful equations in [17] are (3.1), (3.14) and (4.9). Indeed, at least in case of canonical R-charge, it is possible to repeat the whole discussion with thecomplete expressions of F and F H and check that there are no divergences. rbitrary finite β . Hence, using the fact that ̺ ( z, x ) is even in x , we introduce a cut-off inthe integral F S = π r ˜ F H ( r ) k L Z ∞ βL dx x cosh (cid:0) πr x (cid:1) − π r ˜ F H ( r ) k L Z ∞ βL dx x sinh ( πx r ) == (cid:16) ˜ F ( r ) ˜ F H ( r ) (cid:17) / π k L Z ∞ β ′ L dx x sinh ( x ) (3.37)where we decided, for future convenience, to express everything in terms of ˜ F and ˜ F H ,using (3.29). Solving the last integral at the leading order in L we have F S = (cid:16) ˜ F ( r ) ˜ F H ( r ) (cid:17) / π k L log L + . . . . (3.38)Finally, substituting L = N and k L = N we find the free energy F S = 2 r (1 − r ) N log N + O ( N ) . (3.39)As expected, this quantity has its maximum when r = 1 /
2, where its value is exactly(3.2). Moreover, note that, under mirror symmetry, the R-charge r is sent into r → − r and expression (3.39) is invariant under this substitution, consistently with the self-mirrorproperties of T [ SU ( N )]. As a final remark, we note that if we consider slightly differentquiver theories with N t = at , t = 1 , . . . , L/a and a much smaller of the length of the quiver,the previous discussion can be repeated leading to the same result (3.39). This observationis in agreement with [16]. The topologically twisted index [1, 2] for a 3 d N ≥ g × S , with a topological twist on the Riemann surface Σ g of genus g . Theindex is expressed in terms of complex fugacities y for the global symmetries and in terms ofa set of integer magnetic fluxes n on Σ g , parametrizing inequivalent twists. It is convenient,as in [5], to assign a magnetic flux n I and a fugacity y I to each of the chiral fields in thetheory, with the constraint that, for each term W a in the superpotential of the theory, X I ∈ W a n I = 2(1 − g ) , Y I ∈ W a y I = 1 . (4.1)It will also be useful to introduce chemical potentials ∆ I , such that y I = e i ∆ I and X I ∈ W a ∆ I ∈ π Z (4.2)as a consequence of (4.1). In the following, we will take the chemical potentials to be real.Using localizations techniques [1, 2, 4], it is possible to reduce the topologically twistedindex to a matrix model. Explicitly, for a theory with gauge group G , the index is givenby Z Σ g × S ( y, n ) = 1 | W | X m ∈ Γ I C Z pert ( λ, y, m , n ) det ∂ log Z pert ( λ, y, m , n ) ∂iu∂ m ! g . (4.3)– 11 –ere, | W | denotes the order of the Weyl group of G and the sum is over magnetic fluxes m living in the co-root lattice Γ of G . The integration is over the zero-mode gauge variable λ i ( A t + iβσ ) where A t is a Wilson line along S running over the maximal torus of the gaugegroup G and σ is the real scalar in the vector multiplet running over the correspondingsubalgebra. β is the radius of S . In (4.3) we also introduced a Cartan-complex valuedquantity u = A t + iβσ , such that λ = e iu .Consider now the integrand in (4.3). For a theory with no Chern-Simons terms and aset of chiral multiplets transforming in representations R I of G , the function in the integral(4.3) is given by Z pert = Y α ∈ G (1 − λ α ) − g ( idu ) rank G Y I Y ρ I ∈ R I λ ρ I / y / I − λ ρ I y I ! ρ I ( m ) − n I +1 − g (4.4)where α are the roots of G , ρ I are the weights of the representation R I and we used thenotation λ ρ ≡ λ iρ ( u ) . We also included the measure of the integrand in this expression.Lastly, supersymmetry selects a particular contour of integration in (4.3), which can beformulated in terms of the Jeffrey-Kirwan residue. We refer to [1, 2] for more details.To compute the index, we follow the procedure described in [2, 3]. After interchangingsum and integral in (4.3), we obtain a geometric series. Resumming this series and followingthe appropriate prescription for the poles, the index can be written as a sum over residues[2] Z Σ g × S = ( − rank G | W | X residues Z pert | m =0 det ∂ log Z pert ∂ m ∂iu ! g − (4.5)where, defining iB ( a ) i = ∂ log Z pert ∂ m ( a ) i , (4.6)the residues are those satisfying the Bethe ansatz equations (BAEs) e iB ( a ) i = 1 (4.7)and we will briefly give a more explicit expression for the left hand side of this equation.First, however, note that rewriting (4.7) as iB ( a ) i − πin ( a ) i = 0 we can conveniently seethe solutions of (4.7) as critical points of an appropriate twisted superpotential W . Thispotential has some ambiguity in its definition and we will stick with the conventions of [5].The expression of the topologically twisted index as sum over the critical points ofthe twisted superpotential has been first derived in the contest of the Bethe/gauge corre-spondence, see [26–28] and the general discussions in [29, 30]. From this perspective, W isinterpreted as the twisted superpotential of the two dimensional theory obtained after thecompactification of the 3d theory on S [29, 30]. See also [4, 31–35]. As for the free energy on the three-sphere, we introduce two different indices: a superscript runningover the different nodes in the quiver theory and a subscript running over the Cartan of the single node. In fact, we should only keep solutions for which the Vandermonde determinant Q α ∈ G (1 − λ α ) doesn’tvanish. This quantity is sometimes referred to as
Bethe potential . .1 Twisted superpotential As said, the twisted superpotential is such that its critical points satisfy equation (4.7).The explicit expression for B ( a ) i is [5] e iB ( a ) i = Y bi-fundamentals( a,b ) and ( b,a ) N b Y j =1 s λ ( a ) i λ ( b ) j y ( a,b ) − λ ( a ) i λ ( b ) j y ( a,b ) − λ ( b ) j λ ( a ) i y ( b,a ) s λ ( b ) j λ ( a ) i y ( b,a ) Y fund. a q λ ( a ) i y a − λ ( a ) i y a Y anti. a − λ ( a ) i ˆ y a r λ ( a ) i ˆ y a (4.8)where the different terms coming from bi-fundamental and (anti-)fundamental fields canbe identified. Adjoints can be thought as bi-fundamentals connecting the same group.After few manipulations of (4.8), one can find the different contributions to the twistedsuperpotential. Explicitly, a pair of bi-fundamentals, one with chemical potential ∆ ( a,b ) transforming in the ( N a , ¯ N b ) of U ( N a ) × U ( N b ) and the other with chemical potential∆ ( b,a ) and transforming in the ( ¯ N a , N b ) of the same group, gives a contribution [5, 7] W bi-fund = X bi-fundamentals( a,b ) and ( b,a ) N a X i =1 N b X j =1 (cid:20) Li (cid:18) e i ( u ( b ) j − u ( a ) i +∆ ( b,a ) ) (cid:19) − Li (cid:18) e i ( u ( b ) j − u ( a ) i − ∆ ( a,b ) (cid:19)(cid:21) − X bi-fundamentals( a,b ) and ( b,a ) N a X i =1 N b X j =1 " (∆ ( b,a ) − π ) + (∆ ( a,b ) − π )2 ( u ( b ) j − u ( a ) i ) (4.9)where we used λ = e iu . Similarly, the (anti)-fundamentals contribution is W (anti-)fund = N a X i =1 " X anti. a Li (cid:18) e i ( − u ( a ) i + ˆ∆ a ) (cid:19) − X fund. a Li (cid:18) e i ( − u ( a ) i − ∆ a ) (cid:19) + 12 N a X i =1 " X anti. a ( ˆ∆ a − π ) u ( a ) i + X fund. a (∆ a − π ) u ( a ) i − N a X i =1 " X anti. a (cid:16) u ( a ) i (cid:17) − X fund. a (cid:16) u ( a ) i (cid:17) . (4.10)Note that, for non chiral quivers like T [ SU ( N )], the last line vanishes. As mentioned in the Introduction, in the large N limit the twisted index has been computedfor many N ≥ X I ∈ W a ∆ I = 2 π , (4.11)the twisted superpotential and the free energy on S of the same theory are related by therelation, pointed out in [5], − iπ f W (∆ I ) = F S (cid:18) ∆ I π (cid:19) (4.12)– 13 –here with f W we denote the extremal value of twisted superpotential with respect to theeigenvalues u . It is interesting to observe that this equation relates the free energy toan apparently auxiliary quantity. Moreover, the chemical potentials, which are angularvariables, play the role of the R-charges. However, recall that the chemical potentialsare constrained by the superpotential and, under the condition (4.11), they can be safelyidentified with a set of R-charges.Another remarkable result is the index theorem introduced in [5] (see also [7]), whichrelates twisted superpotential and topologically twisted index of the same theory throughlog Z Σ g × S (∆ I , n I ) = (1 − g ) iπ f W (∆ I ) + i X I "(cid:18) n I − g − ∆ I π (cid:19) ∂ f W (∆ I ) ∂ ∆ I . (4.13)As said, for a fixed Riemann surface Σ g , different choices of fluxes parametrize differenttopological twists. If, in particular, one chooses the fluxes to be proportional to the exactR-charge ¯∆ I of the theory ¯ n I = ¯∆ I π (1 − g ) (4.14)one obtains log Z Σ g × S ( ¯∆ I , ¯ n I ) = ( g − F S ¯∆ I π ! . (4.15)The choice (4.14) is referred to as universal twist in [9] where the authors provide a niceholographic interpretation of (4.15) in terms of the magnetically charged black hole of[36, 37]. Being a solution of minimal gauged supergravity, this black hole can be embeddedin eleven dimension and massive type IIA supergravity in infinitely ways, providing a simpleand unique explanation of (4.15).We can now go back to the evaluation of the topological twisted index for T [ SU ( N )].As mentioned in the Introduction, the universal black hole can be also embedded in theholographically dual solution of T [ SU ( N )]. Indeed, we will show the relation (4.15) to holdeven in our case. Moreover, as an intermediate step in the computation, we will obtain anexplicit expression for the twisted superpotential, also verifying (4.12) and (4.13). T [ SU ( N )] twisted superpotential By analogy with the choice of R-charges for the free energy on S , we assume that every(anti-)fundamental and bi-fundamental field has chemical potential ∆ and every adjointfield a chemical potential ˜∆. We fix the angular ambiguity requiring that 0 < ∆ , ˜∆ < π and we choose 2∆ + ˜∆ = 2 π (5.1)to satisfy the constraint (4.2). Applying the rules shown in the previous section to the T [ SU ( N )] case (and using again the notation of the quiver (2.3)), we write the twistedsuperpotential as W = L X t =1 N t X i,j =1 V A ( u ( t ) j − u ( t ) i ) + L − X t =1 N t +1 X j =1 N t X i =1 V H ( u ( t +1) j − u ( t ) i ) + k L N L X i =1 V H ( − u ( L ) i ) (5.2)here V H ( x ) =Li (cid:16) e i ( x +∆) (cid:17) − Li (cid:16) e i ( x − ∆) (cid:17) − (∆ − π ) x ,V A ( x ) = 12 " Li (cid:16) e i ( x + ˜∆) (cid:17) − Li (cid:16) e i ( x − ˜∆) (cid:17) − ( ˜∆ − π ) x . (5.3)The first term in (5.2) represents the contribution from the adjoints, the second from thebi-fundamentals and the third from the (anti-)fundamentals in the last node. Note thatthere is no contribution to the twisted superpotential from the N = 2 vector multiplet. Large N limit Our first task is to manipulate the expression (5.2) and put it, after thelong quiver limit (namely the large N limit), in a form analogous to (3.23). First of all, werewrite (5.2) as W = L X t =1 N t X i,j =1 (cid:16) V A ( u ( t ) j − u ( t ) i ) + V H ( u ( t ) j − u ( t ) i ) (cid:17) − L − X t =1 N t +1 X i,j =1 V H ( u ( t +1) j − u ( t +1) i ) − N t +1 X j =1 N t X i =1 V H ( u ( t +1) j − u ( t ) i ) + N t X i,j =1 V H ( u ( t ) j − u ( t ) i ) − X t ∈{ ,L } N t X i,j =1 V H ( u ( t ) j − u ( t ) i ) + k L N L X i =1 V H ( − u ( L ) i ) . (5.4)We suppose the eigenvalues u to be pure imaginary and to scale with the length L of thequiver according u ( t ) j = iL α x ( t ) j (5.5)with x ( t ) j real. We will fix α later. As for the free energy, we introduce a density for eachnode t N t X i =1 f ( x ( t ) i ) → Z dx ̺ t ( x ) f ( x ) , Z dx ̺ t ( x ) = N t (5.6)and, in the limit of large L , we can simplify the various contributions in (5.4). Considerfor example N q X i =1 N p X j =1 V H ( u ( p ) j − u ( q ) i ) . (5.7)This term is the contribution to the twisted superpotential from a pair of bi-fundamentalfields. In case of variables scaling with a large parameter, in our case L α , its expressioncan be obtained from the results in [5, 7]. Following the conventions of [5], we have a localterm iL − α Z dx̺ p ( x ) ̺ q ( x ) g + (∆) (5.8) The equations of [5] one has to look at are (A.24) and (A.28), with δv ( t ) = 0 and with a straightforwardgeneralization to the case of gauge groups with different densities. In [5] the eigenvalues are supposed toscale with N α and the local bi-fundamental contribution scales as N − α ; in our (5.8) a factor N ( z ) ishidden in the densities. See also [7] where gauge groups with different ranks are considered. – 15 –lus a non local term(∆ − π ) X i = j ( u ( p ) j − u ( q ) i )sign( i − j ) → − iL α (∆ − π ) Z dxdx ′ ̺ p ( x ) ̺ q ( x ′ ) (cid:12)(cid:12) x ′ − x (cid:12)(cid:12) . (5.9)In (5.8) we introduced the function g + ( u ) = u − π u + π u . (5.10)The contribution from adjoint fields, which in the expression (5.4) corresponds to theterms with V A , is simply obtained considering p = q , with an overall factor 1 /
2. For the(anti-)fundamentals, instead N p X j =1 V H ( − u ( p ) i ) (5.11)the leading order is the contribution − iL α (∆ − π ) Z dx̺ p ( x ) | x | . (5.12)For more details on similar computations we refer to [5, 7]. However, we note that similarcomputations are shown in Appendix C, since we will use them to evaluate the topologicallytwisted index. Manipulations
We can now use the results reviewed in the previous paragraph to write(5.4) in a more convenient form. Consider the first line for fixed t , namely N t X i,j =1 (cid:16) V A ( u ( t ) j − u ( t ) i ) + V H ( u ( t ) j − u ( t ) i ) (cid:17) . (5.13)This term is the sum of the contribution of an adjoint and a pair of bi-fundamentalsconnecting the same node. Looking at (5.8), then, we see that it gives2 iL − α Z dx̺ t ( x ) g + (∆) + iL − α g + ( ˜∆) Z dx̺ t ( x ) = iL − α ∆ ( π − ∆) Z dx ̺ t ( x ) , (5.14)where we used the definition (5.10) of g + ( u ) and (5.1). Together with this term, local inthe density, the first line also produces a long-range contribution (see eq. (5.9))(2∆ + ˜∆ − π )2 X i = j (cid:16) u ( t ) j − u ( t ) i (cid:17) sign( i − j ) . (5.15)However, in the definition of the twisted superpotential there is an angular ambiguity and,for each node, we can add a term − π N t X i =1 n ( t ) i u ( t ) i (5.16)ith n ( t ) i integer. Following [5], we can use this term to absorb the non local contribution(5.15), once one imposes the superpotential constraint 2∆ + ˜∆ = 2 π Z . Note that, in thefree energy computation, something similar happens for the term F = F V + F H . Therethe long-range forces from vector, adjoint and bi-fundamental fields are perfectly balancedwhen the R-charges satisfy 2 r + ˜ r = 2. For the twisted superpotential case, the vectorcontribution is absent but we can use the ambiguity (5.16). To summarize, from the firstline of (5.4) we only have the contribution given by (5.14), summed over the nodes.Consider now the second line − L − X t =1 N t +1 X j =1 N t +1 X i =1 V H ( u ( t +1) j − u ( t +1) i ) − N t +1 X j =1 N t X i =1 V H ( u ( t +1) j − u ( t ) i ) + N t X j =1 N t X i =1 V H ( u ( t ) j − u ( t ) i ) . In the planar limit, the leading order is given by the term iL α (∆ − π )2 L − X t =1 Z dxdx ′ (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) ( ̺ t +1 ( x ) − ̺ t ( x )) (cid:0) ̺ t +1 ( x ′ ) − ̺ t ( x ′ ) (cid:1) . (5.17)Note that in this case we remain with a long force contribution, as for the computation forthe free energy on the three sphere.In the end, in the third line of (5.4), we have the boundary terms i L α (∆ − π ) X t ∈{ ,L } Z dxdx ′ (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) ̺ t ( x ) ̺ t ( x ′ ) − iL α k L Z dx̺ L ( x ) | x | (∆ − π ) . (5.18) Evaluation of f W All in all, the twisted superpotential in (5.4), at the leading order in L , becomes W i = L − α ˜ V (∆) L X t =1 Z dx̺ t ( x ) − L α V H (∆) L − X t =1 Z dxdx ′ (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) η t ( x ) η t +1 ( x ′ ) − L α X t ∈{ ,L } ˜ V H (∆) Z dxdx ′ (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) ̺ t ( x ) ̺ t ( x ′ ) + L α k L ˜ V H (∆) Z dx̺ L ( x ) | x | (5.19)with η t ( x ) = ̺ t +1 ( x ) − ̺ t ( x ) and ˜ V (∆) = ∆ ( π − ∆) , ˜ V H (∆) = π − ∆ . (5.20)In order to balance the first two terms in (5.19) and find the value of α , we consider thecontinuum limit for the variable t , as we did for the free energy z = tL , ̺ ( z, λ ) ≡ ̺ zL ( λ ) . (5.21) In fact, this is actually true only when the dimension of the rank of the gauge group is odd. Whenwe are considering a node with even dimension, instead, we have to include an extra ( − m in the twistedpartition function which can be reabsorbed in the definition of the topological fugacity. See [5, 7] andespecially [33] for a discussion on sign ambiguities. – 17 –e obtain W i = Z dz Z dx " L − α ̺ ( z, x ) ˜ V (∆) − L α − V H (∆) Z dx ′ ∂ z ̺ ( z, x ) ∂ z ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) − L α X z ∈{ , } ˜ V H (∆) Z dxdx ′ ̺ ( z, x ) ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + L α k L ˜ V H (∆) Z dx ̺ (1 , x ) | x | . (5.22)We want to balance the terms in the square brackets and this leads to α = 1. The expressionwe find with this choice is equal to the expression (3.23) found for the free energy, up tothe substitutions ˜ V (∆) → ˜ F ( r ) , ˜ V H (∆) → ˜ F H ( r ) (5.23)and the adding of Lagrange multipliers. Therefore, we can avoid to repeat the wholediscussion of Section 3 and immediately write the results. In particular, the saddle pointequation for the twisted superpotential is2 ˜ V (∆) ̺ ( z, x ) + ˜ V H (∆) Z dx ′ ∂ z ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + Lµ ( z ) = 0 (5.24)with boundary conditions ̺ (0 , x ) = 0 , ̺ (1 , x ) = k L δ ( x ) , (5.25)satisfied, for vanishing Lagrange multipliers, by the (rescaled) density ̺ s ( z, x ) = k L sin( πz )2∆(cosh (cid:0) π ∆ x (cid:1) + cos( πz )) . (5.26)Moreover, starting from (3.38) and using the substitution (5.23), we can immediately writedown the expression for the twisted superpotential in the saddle point f W (∆) = i ∆( π − ∆) π N log N + O ( N ) . (5.27)Finally, comparing this expression with (3.39) we find the relation − iπ f W (∆) = F S (cid:18) ∆ π (cid:19) (5.28)as expected. T [ SU ( N )] We now go back to the evaluation of the topologically twisted index. In Section 4, we notedthat the index can be written as Z Σ g × S = ( − rank G | W | X BEAs Z pert | m =0 det ∂ log Z pert ∂iu∂ m ! g − (6.1)nd we are now going to consider the logarithm of this expression. Note that the physicallyunambiguous quantity in the planar limit is the real part of the logarithm of the index; sowe ignore the overall phase in (6.1).First of all, let us evaluate the determinant. Recalling the definition (4.6) and how weintroduced the twisted superpotential, we can also writedet ∂ log Z pert ∂iu∂ m ! = det ∂ W ∂u∂u ′ ! ≡ det B . (6.2)Suppose all the entries of B to be bounded by some constant c . In general, we havethat the t th node’s contribution to the matrix is made of N t lines with N t − + N t +1 + N t non-vanishing entries, coming from bi-fundamentals and adjoint terms. Solog det ∂ W ∂u∂u ′ ! ≤ log L Y t =1 c N t N t ( N t − + N t + N t +1 ) ! . (6.3)Since for T [ SU ( N )] we have N t = t and L = N −
1, we can also writelog det ∂ W ∂u∂u ′ ! ≤ log c N N Y t =1 N ! ∼ N . (6.4)Hence, the contribution from the determinant in (6.1) is at most of order N and, we shallsee, is subleading in the computation of the index.Consider now the remaining part of the index, namely log Z pert ; to simplify the nota-tion, we will omit the subscript. In the end, we will be interested in the universal twist(4.14) so, by analogy with our choice of R-charges for the free energy, we associate thesame flux n h to each bi-fundamental field (and to the (anti-)fundamentals) and a flux n v to each adjoint field, with the relation2 n h + n v = 2(1 − g ) . (6.5)After some manipulations shown in Appendix C, we can writelog Z = L X t =1 N t X i = j =1 ( g − (cid:18) e i ( u ( t ) j − u ( t ) i ) (cid:19) + L X t =1 N t X i,j =1 Z A (cid:16) u ( t ) j − u ( t ) i (cid:17) + L X t =1 N t X i =1 N t +1 X j =1 Z H (cid:16) u ( t +1) j − u ( t ) i (cid:17) + k L N L X i =1 Z H (cid:16) u ( L ) i (cid:17) (6.6)where the first term is the vector contribution, and the adjoint and bi-fundamentals con-tributions are respectively expressed in terms of the functions Z A ( x ) = 12 ( − n v + 1 − g ) h Li (cid:16) e i ( x + ˜∆) (cid:17) + Li (cid:16) e i ( x − ˜∆) (cid:17) + i ( x − π ) i ,Z H ( x ) = ( − n h + 1 − g ) h Li (cid:16) e i ( x − ∆) (cid:17) + Li (cid:16) e i ( x +∆) (cid:17) + i ( x − π ) i . (6.7) This assumption can be explicitly verified taking the derivatives of the building blocks (4.9) and (4.10):the outcome is that divergences can only occur in regions where u is equal or opposite to the chemicalpotential ∆. This however is not possible since, with our assumptions, u is pure imaginary and the chemicalpotential real and non vanishing. A similar discussion on the determinant has been previously proposedin [3]. In that work u also has an imaginary part and it is necessary to consider some "tail contributions"related to divergent entries in the matrix. – 19 –y analogy with the computation of the twisted superpotential, we writelog Z = L X t =1 N t X i = j =1 ( g − (cid:18) e i ( u ( t ) j − u ( t ) i ) (cid:19) + L X t =1 N t X i,j =1 (cid:16) Z A ( u ( t ) j − u ( t ) i ) + Z H ( u ( t ) j − u ( t ) i ) (cid:17) − L − X t =1 N t +1 X i,j =1 Z H ( u ( t +1) j − u ( t +1) i ) − N t +1 X j =1 N t X i =1 Z H ( u ( t +1) j − u ( t ) i ) + N t X i,j =1 Z H ( u ( t ) j − u ( t ) i ) − X t ∈{ ,L } N t X j =1 N t X i =1 Z H ( u ( t ) j − u ( t ) i ) + k L N L X i =1 Z H ( u ( L ) i ) . (6.8)Substituting u = iLx , we now consider the large L limit of this expression, using the resultsin Appendix C. The procedure retraces the one used for the twisted superpotential and wewill not explicitly repeat it here but we report all the details in Appendix C. We only notethat the first line of (6.8) produces a non local term i X j>i (cid:18) u ( t ) i − u ( t ) j + 12 π (cid:19) (2 n h + n v − − g )) (6.9)which vanishes imposing the condition (6.5). This is completely analogous to what foundfor the free energy and the twisted superpotential when we used the conditions on R-chargesand chemical potential to make the non local terms vanish. From the other lines we getlong-range contributions and, all in all, we can recast log | Z | aslog | Z | = Z dz Z dx " ̺ ( z, x ) ˜ Z (∆ , n ) −
12 ˜ Z H ( n ) ∂ z ̺ ( z, x ) Z dx ′ ∂ z ̺ ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) − L X z ∈{ , } Z dxdx ′ ̺ ( z, x ) ̺ ( z, x ′ ) ˜ Z H ( n ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + Lk L Z dx̺ (1 , x ) ˜ Z H ( n ) | x | (6.10)with ˜ Z (∆ , n ) = ∆( π −
32 ∆) n v + 2∆(∆ − π )(1 − g ) , (6.11)˜ Z H ( n ) = − ( − n h + 1 − g ) = − n v . (6.12) The last step is to evaluate (6.10) in the saddle point configuration (5.26). For futureconvenience, we recall that this configuration satisfies Z dx ′ ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) = − ̺ s ( z, x )2∆ . (6.13)Using the boundary conditions (5.25), the explicit terms form the boundary z ∈ { , } andthe contributions from the flavours in the last node disappear. Integrating by parts theremaining expression we findlog | Z | = Z dz Z dx̺ s ( z, x ) (cid:18) ̺ s ( z, x ) ˜ Z (∆ , n ) + ˜ Z H ( n ) 12 Z dx ′ ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:19) − ˜ Z H ( n ) 12 Z dxdx ′ (cid:0) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) z =1 z =0 (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) . (6.14)ubstituting ˜ Z and ˜ Z H log | Z | = Z dz " Z dx̺ s ( z, x ) (cid:18) ∆( π −
32 ∆) n v + 2∆(∆ − π )(1 − g ) (cid:19) − n v Z dxdx ′ ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) + n v Z dxdx ′ (cid:0) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) z =1 z =0 (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) (6.15)and using the saddle point condition (6.13) together with the relation 2 n h + n v = 2(1 − g ),we obtainlog | Z | = 2∆(∆ − π ) n h Z dz Z dx̺ s ( z, x ) + n v Z dxdx ′ (cid:0) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) z =1 z =0 (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) . (6.16)The second integral is the one found in the computation of the free energy, equation (3.35).After introducing an appropriate cut-off, it gives n v Z dxdx ′ (cid:0) ̺ s ( z, x ) ∂ z ̺ s ( z, x ′ ) (cid:12)(cid:12) z =1 z =0 (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) = − ∆2 π n v k L log L + . . . (6.17)Consider now the first integral in (6.16). Using that the density is even in x and introducinga cut-off β/L with β arbitrary, we write it as4∆(∆ − π ) n h Z dz Z βL dx k L sin( πz )2∆(cosh (cid:0) π ∆ x (cid:1) + cos( πz )) ! ==(∆ − π ) n h Z dz Z β ′ L dx ′ (cid:18) k L sin( πz )(cosh( πx ′ ) + cos( πz )) (cid:19) ==(∆ − π ) n h k L Z β ′ L dx ′ (cid:0) coth (cid:0) πx ′ (cid:1) − (cid:1) (6.18)where we used the change of variable x ′ = x/ ∆ and performed the z integral (using e.g.Mathematica). Considering the leading order in L , we find(∆ − π ) n h k L Z β ′ L dx ′ (cid:0) coth (cid:0) πx ′ (cid:1) − (cid:1) = (∆ − π ) π n h k L log L + . . . . (6.19)As promised, the determinant contribution in (6.4) is actually subleading if comparedwith (6.17) and (6.19), which represent the leading contribution to the index. Putting alltogether, we finally find at the leading order ( L = N and k L = N )log Z Σ g × S (∆ , n ) = N log N " (∆ − π ) π n h − ∆2 π n v = N log N (cid:20) (2∆ − π ) π n h − ∆ π (1 − g ) (cid:21) . (6.20)Comparing (5.27) and (6.20), it is easy to check that the index theorem (4.13) is actuallyverified. Finally, evaluating (6.20) on the universal twist¯ n h = ¯∆ π (1 − g ) , ¯∆ = π , (6.21)– 21 –e obtain log Z Σ g × S ( ¯∆ , ¯ n ) = ( g −
1) 12 N log N (6.22)and, hence log Z Σ g × S ( ¯∆ , ¯ n ) = ( g − F S ¯∆ π ! (6.23)as we wanted. In this work, we have computed the topologically twisted index for the T [ SU ( N )] theory,in the large N limit. In particular, we focused on the case of the universal topologicaltwist, obtained with the choice of magnetic fluxes and chemical potentials given in (6.21).The expression we found correctly reproduces the entropy of the universal black hole [9],satisfying (6.22) and (6.23).As an intermediate step, we also provided the computation of the free energy on thethree-sphere at large N , turning on an arbitrary R-charge. A natural idea is then to applythe same procedure to compute the free energy of other three dimensional theories describedby long linear quivers. However, our discussion relies on a saddle point approximation. Aswe briefly mentioned in Section 3, we were able to use this approximation because theleading order of the free energy of T [ SU ( N )], when masses and FI parameters are turnedoff, scales as N log N and not as N . Indeed, the localization procedure gives an integralin N variables and, in principle, all the terms in the expansion around the saddle pointcould contribute to this order in the free energy. This should be kept in mind before tryingto apply the procedure to other theories described by long linear quivers. We also stressthat, as in [17], the equation we found for the saddle point configuration requires to have abalanced quiver. Finally, it is interesting to observe that the saddle point configuration forthe three dimensional T [ SU ( N )] theory is the same found in [17] for the five dimensional T N theory, up to a rescaling.With our discussion, we applied the method of [17], proposed for the computation offree energy on five-spheres, to the three dimensional case. As future directions of research,it would be then natural to study what happens in other dimensions. Moreover, as recentlydone in [38] for Wilson loops, one could also try to exploit the expressions for the saddlepoint configurations to compute other quantities in the field theory side, comparing theresults with holographic predictions. Acknowledgments
I am deeply grateful to Alberto Zaffaroni for suggesting me this project, for his continuedhelp and for many comments on the draft. I would also like to thank Ivan Garozzo, AndreaGrigoletto, Gabriele Lo Monaco and Matteo Sacchi for many useful discussions and clarifi-cations. I am supported by the INFN and by the MIUR-PRIN contract 2017CC72MK003.
Polylogarithms
For ease of reading, we here recall the definition of polylogarithmsLi s ( z ) = ∞ X k =1 z k k s (A.1)together with some useful propertiesLi ( e iu ) + Li ( e − iu ) = − , Li ( e iu ) − Li ( e − iu ) = − iu + iπ , Li ( e iu ) + Li ( e − iu ) = u − πu + π , Li ( e iu ) − Li ( e − iu ) = i u − i π u + i π u , (A.2)where we assumed 0 < R e( u ) < π . Relations in the region − π < R e( u ) < u → − u . We also define the functions: g + ( u ) = u − π u + π u , g ′ + ( u ) = u − πu + π . (A.3) B Formulas for the computation of the free energy on S As argued in Section 3, the free energy on the three-sphere for T [ SU ( N )] can be writtenin the form F = L Z dz Z dλdλ ′ N ( z ) ρ ( z, λ ) ρ ( z, λ ′ ) (cid:0) F V (cid:0) λ − λ ′ (cid:1) + F H (cid:0) λ − λ ′ (cid:1)(cid:1) − L Z dz Z dλdλ ′ L ∂ z ( N ( z ) ρ ( z, λ )) ∂ z (cid:0) N ( z ) ρ ( z, λ ′ ) (cid:1) F H (cid:0) λ − λ ′ (cid:1) − X z ∈{ , } Z dλdλ ′ N ( z ) ρ ( z, λ ) ρ ( z, λ ′ ) F H (cid:0) λ − λ ′ (cid:1) + k L N (1) Z dλρ (1 , λ ) F H ( λ ) . (B.1)However, we now show how it is possible to simplify this expression assuming the scaling λ = L α x (B.2)with α > L limit. Bi-fundamentals contribution
We start considering integrals in (B.1) containing F H (see [7, 10, 39–41] for related com-putations). Consider for example Z dxdy̺ ( z, x ) ̺ ( z, y ) F H ( L α ( x − y )) == − Z dxdy̺ ( z, x ) ̺ ( z, y ) [ ℓ (1 − r + iL α ( x − y )) + ℓ (1 − r − iL α ( x − y ))] (B.3)– 23 –here we used the rescaled density dx̺ ( z, x ) = dλN ( z ) ρ ( z, λ ) and the definition (3.14) of F H . We also recall that ℓ ( z ) = − z log (cid:16) − e πiz (cid:17) + i (cid:18) πz + 1 π Li (cid:16) e πiz (cid:17)(cid:19) − iπ . (B.4)It is convenient to separately consider the different contributions inside ℓ ( z ), starting fromthe terms involving − z log(1 − exp[2 πiz ]), i.e. − ∞ X k =1 k Z dxdy̺ ( z, x ) ̺ ( z, y ) " (1 − r + iL α ( x − y )) (cid:16) e πk ( − ir − L α ( x − y )) (cid:17) + (1 − r − iL α ( x − y )) (cid:16) e πk ( − ir + L α ( x − y )) (cid:17) (B.5)where we noted that − log(1 − z ) = Li ( z ) = ∞ X k =1 z k k . (B.6)Integrating by parts, we can obtain the first terms of the large L expansion for the previousintegral. Explicitly, consider, in the region x > y , the integral − Z x −∞ dy ̺ ( z, y ) " (1 − r + iL α ( x − y )) (cid:16) e πk ( − ir − L α ( x − y )) (cid:17) (B.7)which, after an integration by parts, becomes ̺ ( z, x ) L α (cid:18) ( r − πk − i π k (cid:19) e − πikr + 1 L α Z x −∞ ∂ y ̺ ( z, y ) e πk ( − ir − L α ( x − y ) ( i + 2 kπ (1 − r + iL α ( x − y )))4 k π . (B.8)If one keep integrating by parts the second line, obtains an expansion in 1 /L α . Saving onlythe leading order, then, we write the first line of (B.5) in the region x > y as − ∞ X k =1 k Z dx ̺ ( z, x ) Z x −∞ dy ̺ ( z, y )(1 − r + iL α ( x − y )) (cid:16) e πk ( − ir − L α ( x − y )) (cid:17) == ∞ X k =1 k " Z dx ̺ ( z, x ) (cid:18) ( r − πkL α e − πikr − i π k L α e − πikr (cid:19) == Z dx ̺ ( z, x ) (cid:18) ( r − πL α Li ( e − πir ) − i π L α Li ( e − πir ) (cid:19) . (B.9)If we try to apply the same procedure in the region x < y , we encounter divergences in theintegration by parts. Fortunately, we can use (A.2) to invert the sign in the exponentialand avoid divergences. With this procedure, however, we also obtain a non local term: Z dx Z ∞ x dy ̺ ( z, x ) ̺ ( z, y )(1 − r + iL α ( x − y )) ( − πir − πL α ( x − y ) + iπ )+ Z dx ̺ ( z, x ) (cid:18) ( r − πL α Li ( e πir ) + i π L α Li ( e πir ) (cid:19) . (B.10)umming (B.9) and (B.10), we find for the first line of (B.5) Z dx Z ∞ x dy̺ ( z, x ) ̺ ( z, y ) (1 − r + iL α ( x − y )) ( − πir − πL α ( x − y ) + iπ )+ Z dx ̺ ( z, x ) (cid:20) ( r − πL α g ′ + (2 πr ) − g + (2 πr )4 π L α (cid:21) (B.11)with g + ( u ) defined in (A.3). The second line in (B.5) can be computed in the same wayand gives Z dx Z x −∞ dy̺ ( z, x ) ̺ ( z, y ) (1 − r − iL α ( x − y )) ( − πir + 2 πL α ( x − y ) + iπ )+ Z dx ̺ ( z, x ) (cid:20) ( r − πL α g ′ + (2 πr ) − g + (2 πr )4 π L α (cid:21) . (B.12)All together, the leading contribution in L from (B.5) is Z dx Z dy̺ ( z, x ) ̺ ( z, y ) (1 − r − iL α | x − y | ) ( − πir + 2 πL α | x − y | + iπ )+ Z dx ̺ ( z, x ) (cid:20) ( r − πL α g ′ + (2 πr ) − g + (2 πr )2 π L α (cid:21) . (B.13)The computation of dilogarithms contributions in (B.3) is very similar in the procedure.The result is − i π Z dxdy ̺ ( z, x ) ̺ ( z, y ) " ( − πiL α | x − y | − πr ) π ( − πiL α | x − y | − πr ) + π − π L α Z dx̺ ( z, x ) g + (2 πr ) . (B.14)Expressions (B.13) and (B.14), together with the remaining quadratic and constant termsin ℓ ( z ), finally give the leading order contribution Z dxdy̺ ( z, x ) ̺ ( z, y ) F H ( L α ( x − y )) == π L α ( r − r ( r − Z dx ̺ ( z, x ) + 2 πL α (1 − r ) Z dxdy ̺ ( z, x ) ̺ ( z, y ) | x − y | . (B.15)A completely analogous argument can be used for the other integrals involving F H . Adjoint and vector contribution
Next, we need to consider the term in (B.1) involving F V , which represents the adjointand the vector contributions. The former can be thought as bi-fundamental connecting thesame gauge group and we can then use the result in (B.15), with an overall factor 1 / − Z dxdy̺ ( z, x ) ̺ ( z, y ) [ ℓ (1 − ˜ r + iL α ( x − y )) + ℓ (1 − ˜ r − iL α ( x − y ))] == π L α (˜ r − r (˜ r − Z dx ̺ ( z, x ) + πL α (1 − ˜ r ) Z dxdy ̺ ( z, x ) ̺ ( z, y ) | x − y | . (B.16)– 25 –he vector contribution is, instead, given by the integral of (see again (3.14)) −
12 log (cid:16) ( L α π ( x − y )) (cid:17) = − πL α | x − y | − log (cid:16) − e − πL α | x − y | (cid:17) (B.17)which in the planar limit becomes − Z dxdy̺ ( z, x ) ̺ ( z, y ) log (cid:16) ( L α π ( x − y )) (cid:17) == − L α π Z dxdy ̺ ( z, x ) ̺ ( z, y ) | x − y | + π L α Z dx ̺ ( z, x ) . (B.18)Here, we integrated by parts the term with the logarithm as we did before, throwing awaysubleading orders in L α . So, all together Z dxdy̺ ( z, x ) ̺ ( z, y ) F V ( L α ( x − y )) == − π ˜ rL α Z dxdy ̺ ( z, x ) ̺ ( z, y ) | x − y | + π L α [(˜ r − r (˜ r − Z dx ̺ ( z, x ) . (B.19)We conclude this appendix with a couple of important observations. In the expression(B.1), the combination F = F V + F H appears. Summing together (B.15) and (B.19) we seethat the non local term is zero when 2 r + ˜ r = 2, condition required by the superpotential.The term in the integral involving F is then local in the density and we can write Z dxdy̺ ( z, x ) ̺ ( z, y ) F ( L α ( x − y )) = 2 π (1 − r ) r L α Z dx̺ ( z, x ) . (B.20)Conversely, all the other terms in (B.1) in which only F H appears are non local and wewill only keep the leading long-range force contribution from (B.15). C Formulas for the computation of the twisted index
In the computation of the index, we need to evaluatelog Z pert (cid:12)(cid:12)(cid:12) m =0 = log Y α ∈ G (1 − λ α ) − g Y I Y ρ I ∈ R I λ ρ I / y / I − λ ρ I y I ! − n I +1 − g (C.1)where the product over the roots α of G is the contribution from the N = 2 vectormultiplet and the other products denote the contribution of the chiral multiplets in thetheory. Computations of this appendix can be compared with those of [5, 7]. We also recallthat, in the upcoming computations, overall phases in the index can be neglected, since wewill be interested in log | Z | . Bi-fundamentals contribution
Let us start considering the logarithm of bi-fundamentals contribution, made of two fieldsconnecting two adjacent nodes labelled by t and t + 1. Associated with them we have aagnetic flux n h and a fugacity y h = e i ∆ , equal for both the fields. Hence, we have N t X i =1 N t +1 X j =1 Z H ( u ( t +1) j − u ( t ) i ) ≡ log N t Y i =1 N t +1 Y j =1 − λ ( t +1) j λ ( t ) i − λ ( t +1) j λ ( t ) i y ( − h ! − λ ( t +1) j λ ( t ) i y h ! − n h +1 − g (C.2)which we rewrite N t X i =1 N t +1 X j =1 Z H ( u ( t +1) j − u ( t ) i ) == ( − n h + 1 − g ) N t X i =1 N t +1 X j =1 " Li e i (cid:16) u ( t +1) j − u ( t ) i +∆ (cid:17)! + Li e i (cid:16) u ( t +1) j − u ( t ) i − ∆ (cid:17)! + i (cid:16) u ( t +1) j − u ( t ) i (cid:17) − iπ (C.3)using λ = e iu and Li ( z ) = − log(1 − z ). Now, consider the first logarithm in (C.3). Withthe assumption u = iLx , in the large L limit the region j > i gives Z dx̺ t ( x ) Z ∞ x dx ′ ̺ t +1 ( x ′ ) Li (cid:16) e − L ( x ′ − x )+ i ∆ (cid:17) . (C.4)This integral is analogous to those found in Appendix B. As we did in Appendix B for thefree energy, we apply the definition (A.1) and integrate by parts to obtain the expansionin 1 /L . We find Z dx̺ t ( x ) Z x dx ′ ̺ t +1 ( x ′ ) Li (cid:16) e − L ( x ′ − x )+ i ∆ (cid:17) = 1 L Z dx̺ t ( x ) ̺ t +1 ( x )Li (cid:16) e i ∆ (cid:17) + . . . (C.5)When j < i instead, we need to invert the integrand to avoid divergences in the procedureof integration by parts. This can be done using the properties (A.2). All together, fromthe first logarithm in (C.3), at the leading order in L we have1 L Z dx̺ t ( x ) ̺ t +1 ( x ) (cid:16) Li (cid:16) e i ∆ (cid:17) + Li (cid:16) e − i ∆ (cid:17)(cid:17) = 1 L Z dx̺ t ( x ) ̺ t +1 ( x ) g ′ + (∆) (C.6)plus non local terms from the inversion formula − i X i>j (cid:16) u ( t +1) j − u ( t ) i + ∆ − π (cid:17) . (C.7)The second logarithm in (C.3) can be treated analogously and gives the same local term(C.6), plus − i X i>j (cid:16) u ( t +1) j − u ( t ) i − ∆ + π (cid:17) . (C.8)– 27 –otice that we obtained this contribution applying relations (A.2) in the region − π < R e( u ) <
0. In the end, considering all the terms in (C.3), we have2 L ( − n h + 1 − g ) Z dx̺ t ( x ) ̺ t +1 ( x ) g ′ + (∆) (C.9)and the long-range term − i ( − n h + 1 − g ) X i = j h(cid:16) u ( t +1) j − u ( t ) i (cid:17) sign( i − j ) + π i . (C.10)For the moment, it is convenient not to keep the continuous limit of this sum. Adjoint contribution
Adjoint terms can be identified with bi-fundamentals connecting the same node. Calling n v the magnetic flux and ˜∆ the chemical potential N t X i,j =1 Z V ( u ( t ) j − u ( t ) i ) ≡
12 ( − n v + 1 − g ) N t X i,j =1 " Li e i (cid:16) u ( t ) j − u ( t ) i + ˜∆ (cid:17)! + Li e i (cid:16) u ( t ) j − u ( t ) i − ˜∆ (cid:17)! + i (cid:16) u ( t ) j − u ( t ) i (cid:17) − iπ . (C.11)Hence, we can use the results (C.9), (C.10) and conclude that, in the continuous limit, thecontribution from each adjoint field is1 L ( − n v + 1 − g ) Z dx̺ t ( x ) g ′ + ( ˜∆) (C.12)plus the non local term − i − n v + 1 − g ) N t X i = j h(cid:16) u ( t ) j − u ( t ) i (cid:17) sign( i − j ) + π i . (C.13) (Anti-)fundamental contribution For the (anti-)fundamental contribution we have( − n h + 1 − g ) N t X i =1 log − x ( t ) i (cid:16) − x ( t ) i y h (cid:17) (cid:16) − x ( t ) i y ( − h (cid:17) ==( − n h + 1 − g ) N t X i =1 (cid:20) Li (cid:18) e i ( u ( t ) i +∆) (cid:19) + Li (cid:18) e i ( u ( t ) k − ∆) (cid:19) + iu ( t ) i − iπ (cid:21) (C.14)which, in the continuous limit, has the leading order − ( − n h + 1 − g ) Z dx̺ t ( x ) | x | . (C.15) auge vector contribution To conclude, we consider the term coming from the vector multiplet in the t th nodelog N t Y i = j − x ( t ) i x ( t ) j − g (C.16)which we rewrite(1 − g ) X i>j log (cid:18) − e i ( u ( t ) i − u ( t ) j ) (cid:19) + X j>i log (cid:18) − e i ( u ( t ) i − u ( t ) j ) (cid:19) == (1 − g ) X i>j log (cid:18) − e i ( u ( t ) i − u ( t ) j ) (cid:19) + X j>i (cid:18) log (cid:18) − e i ( u ( t ) j − u ( t ) i ) (cid:19) + i ( u ( t ) i − u ( t ) j ) + iπ (cid:19) == ( g − X i>j Li (cid:18) e i ( u ( t ) i − u ( t ) j ) (cid:19) − i X j>i (cid:16) u ( t ) i − u ( t ) j + π (cid:17) (C.17)and in the large L limit gives the local term( g − π L Z dx̺ t ( x ) (C.18)plus i (1 − g ) X j>i (cid:16) u ( t ) i − u ( t ) j + π (cid:17) . (C.19) Other manipulations
Let us now use the previous results to compute the continuous limit of Eq. (6.8). The firstterm we need to consider is the combination L X t =1 N t X i = j =1 ( g − (cid:18) e i ( u ( t ) j − u ( t ) i ) (cid:19) + L X t =1 N t X i,j =1 (cid:16) Z A ( u ( t ) j − u ( t ) i ) + Z H ( u ( t ) j − u ( t ) i ) (cid:17) . (C.20)Using the results just found, we see that the long range contribution from this combination i X j>i (cid:18) u ( t ) i − u ( t ) j + 12 π (cid:19) (2 n h + n v − − g )) (C.21)vanishes when 2 n h + n v = 2(1 − g ). Hence, (C.20) only produces a local term in thecontinuous limit, namely (cid:18) n v ∆( π −
32 ∆) + (1 − g )2∆(∆ − π ) (cid:19) Z dz Z dx̺ ( z, x ) . (C.22)The second line of (6.8), instead, is − L − X t =1 N t +1 X i,j =1 Z H ( u ( t +1) j − u ( t +1) i ) − N t +1 X j =1 N t X i =1 Z H ( u ( t +1) j − u ( t ) i ) + N t X i,j =1 Z H ( u ( t ) j − u ( t ) i ) (C.23)– 29 –nd in this case the long-range term doesn’t disappear. In fact, this is the leading contri-bution which, in the large L limit, becomes12 ( − n h + 1 − g ) L Z dxdx ′ (cid:12)(cid:12) x − x ′ (cid:12)(cid:12) ( ̺ t +1 ( x ) − ̺ t ( x )) (cid:0) ̺ t +1 ( x ′ ) − ̺ t ( x ′ ) (cid:1) (C.24)and introducing the variable z = t/L
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