EExact results for 5d SCFTs of long quiver type
Christoph F. Uhlemann ∗ Mani L. Bhaumik Institute for Theoretical PhysicsDepartment of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA
Exact results are derived for 5d SCFTs with holographic duals in Type IIB supergravity.These theories have relevant deformations that flow to linear quiver gauge theories, withthe number of nodes large in the large- N limits described by supergravity. Starting from asuitable formulation of the matrix models resulting from supersymmetric localization of thesquashed S partition functions, the saddle point equations are solved for generic quiverswith N f = 2 N at all interior nodes, which includes the T N theories, and for a sample oftheories with N f (cid:54) = 2 N nodes including theories with Chern-Simons terms. The resultingexact expressions for the free energies and conformal central charges are consistent withsupergravity predictions and, where available, with previous numerical field theory analyses. CONTENTS
I. Introduction 2II. 5d Partition Functions for long quivers 3A. 5d partition functions 4B. Long linear quiver gauge theories 5C. Saddle point equation 8D. Boundary conditions 9E. Junction conditions 10F. Conformal central charge C T N f = 2 N quivers 13IV. 5d SCFTs with gravity duals in Type IIB 15A. + N,M theory 15B. T N theory 16C. Y N theory 17D. (cid:30) + N theory 21E. T K,K, theory 23F. T N,K,j theory 25G. +
N,M,j theory 26V. Discussion 27Acknowledgments 29A. Free energies for N f = 2 N theories 29References 31 ∗ [email protected] a r X i v : . [ h e p - t h ] S e p I. INTRODUCTION
Five-dimensional superconformal field theories (SCFTs) play an interesting role in the gen-eral understanding of quantum field theory (QFT). On the one hand, they are hard to constructdirectly using field theory methods, and are typically defined through constructions in string the-ory and M-theory [1–12]. On the other hand, many of them admit relevant deformations thatflow to perturbatively non-renormalizable Lagrangian gauge theories, for which the SCFTs providestrongly-coupled UV fixed points. In between the maximal six-dimensional theories and theories inlower dimensions, five-dimensional theories allow for an intriguing interplay between string theoryconstructions, AdS/CFT and field theory methods, and upon compactification they provide manyinteresting insights into lower-dimensional theories.This interplay has been very successful for the 5d
U Sp ( N ) theories, realized by D4/D8/O8configurations in Type IIA, which for large N have holographic duals in massive Type IIA super-gravity [13, 14]. These theories have been studied extensively e.g. in [15–28]. However, many more5d gauge theories are believed to have strongly-coupled UV fixed points, and indeed many moretheories can be engineered in Type IIB string theory using 5-brane webs [3–5], 5-brane webs with7-branes [29] and further generalizations [30–33]. Type IIB supergravity solutions describing thenear-horizon limits of 5-brane webs and 5-brane webs with 7-branes have been constructed in [34–36] and [37], and result in large classes of explicit holographic dualities for the corresponding 5dSCFTs. Various aspects of the dualities have been studied in [43–56]. In particular, the free energywas found to scale quartically with the parameter controlling the large- N limit in supergravity [43],and this scaling was confirmed numerically using field theory methods [48, 56].In this paper we study 5d SCFTs that have holographic duals in Type IIB supergravity usingfield theory methods. The theories we consider have relevant deformations that are describedby linear quiver gauge theories with SU ( N ) gauge nodes, with bifundamental hypermultipletsconnecting adjacent nodes. In addition there may be fundamental hypermultiplets attached toindividual nodes and Chern-Simons terms. The general quiver diagram is shown in eq. (2.1). Weuse the gauge theory deformations and supersymmetric localization to analytically compute thefree energies of the 5d SCFTs on the round S and on squashed spheres, in the large- N limitsdescribed by supergravity.Localization of the (squashed) 5-sphere partition function has been worked out for generic 5dgauge theories in [57–61]. The path integral is reduced to a matrix integral over the Cartan algebraof the gauge group. At large N , instanton contributions are expected to be suppressed, and thezero-instanton part of the partition function is expected to be captured exactly by a saddle pointcomputation. However, for the gauge theories of interest here, the number of nodes in the quiverdiagrams is large in the large- N limits of interest. Each gauge node contributes a matrix integral,so that the number of matrix integrals itself is large, and for each gauge node one needs to find an a-priori independent saddle point eigenvalue distribution. We set up a suitable language to describethe localized partition functions for long quiver gauge theories, and cast the saddle point equationsin a form akin to a 2d electrostatics problem. The saddle point equations are solved analyticallyfor generic quivers where all interior nodes have an effective number of flavors equal to twice the Earlier studies of the BPS equations can be found in [40–42], and T-duals of the Type IIA solution were discussedin [38, 39]. For the free energy of quivers arising from orbifolds of the
USp ( N ) theory discussed in [15], the saddle pointeigenvalue distributions are independent of the gauge node, effectively reducing the problem to a single node. Forthe theories considered here the saddle point configurations depend non-trivially on the quiver node. number of colors, N f = 2 N , and for a sample of theories which also have N f (cid:54) = 2 N nodes, leadingto exact expressions for the squashed sphere free energies. This includes the theories for which the S free energies were computed numerically in [48, 56], which provided valuable intuition for theanalytic computations shown here. Following [62, 63], the conformal central charge characterizingthe energy-momentum tensor two-point function is extracted, and we show that a universal relationbetween conformal central charge and S free energy, found numerically for two example theoriesin [48], holds for all theories of the long quiver type to be discussed below.The remaining parts are organized as follows. In sec. II we discuss the squashed S free energiesfor long quiver gauge theories at a general level, derive the saddle point conditions and establish arelation between the conformal central charge and the S free energy. In sec. III the saddle pointconditions are summarized and solved explicitly for theories with N f = 2 N at all interior nodes.In sec. IV we discuss the free energies of a sample of 5d gauge SCFTs that have supergravity dualsin Type IIB. We conclude with a discussion in sec. V. II. 5D PARTITION FUNCTIONS FOR LONG QUIVERS
This section contains a discussion of partition functions on squashed spheres for 5d long quivergauge theories with holographic duals in Type IIB. The theories of interest are linear quiver gaugetheories with SU ( N ) gauge nodes, denoted by ( N ), with bifundamental hypermultiplets betweenadjacent gauge nodes and optionally additional fundamental hypermultiplets denoted by [ k ] for k fundamentals. 5d SU ( N ) gauge theories with N > c , if non-zero, by a subscript, e.g. ( N ) c . The form ofthe quiver gauge theories then is( N ) c − ( N ) c − . . . − ( N L − ) c L − − ( N L ) c L | | | | (2.1)[ k ] [ k ] [ k L − ] [ k L ]In the limits of interest here, the length of the quiver in (2.1) is large, L (cid:29)
1. Moreover, with t = 1 , . . . , L labeling the quiver node, at least some N t are large, of order L , but not necessarily allof them. The term “large N ” will be used to refer to this limit with L (cid:29) L limit, it will be convenient to introduce an effectively continuousparameter z ∈ [0 ,
1] labeling the gauge node, z = tL . (2.2)The left end of the quiver (2.1) corresponds to z = 0, the right end to z = 1. The data { N t , k t , c t } characterizing the quiver is then encoded in functions N ( z ), k ( z ), c ( z ) of the effectively continuousvariable z , defined by N ( z ) = N zL , k ( z ) = k zL , c ( z ) = c zL . (2.3)In the examples discussed below N ( z ) will be a continuous, piece-wise linear, concave function.Fundamental hypermultiplets and Chern-Simons terms only appear at the isolated nodes where N ( z ) has kinks, but for a uniform treatment we nevertheless introduce k ( z ) and c ( z ), which willbe sums of δ -functions, to encode them.The effective number of flavors at an interior gauge node, including fundamental hypermultipletsand bifundamental hypermultiplets with adjacent gauge nodes, is N f,t = N t +1 + N t − + k t . Thecontinuous version is N f ( z ) = 2 N ( z ) + 1 L ∂ z N ( z ) + k ( z ) . (2.4)At nodes where N ( z ) is smooth and k ( z ) zero, this effective number of flavors is equal to twicethe number of colors, N f = 2 N . At nodes where N ( z ) has a kink, N f may be smaller or equal totwice the number of colors, depending on the number of fundamental hypermultiplets attached tothat node.The general form of the matrix models resulting from supersymmetric localization of thesquashed sphere partition function of 5d gauge theories is reviewed in sec. II A. A suitable for-malism for long quiver gauge theories is set up in sec. II B. The saddle point conditions will bediscussed in three parts: Along parts of the quiver with no Chern-Simons terms and N f = 2 N ateach node, they take the form of a partial differential equation, which is derived in sec. II C. It issupplemented by boundary conditions at z = 0 and z = 1 which are discussed in sec. II D. Finally,junction conditions for nodes with N f (cid:54) = 2 N are derived in sec. II E. A universal large- N relationbetween the conformal central charge and the S free energy is derived in sec. II F. A. 5d partition functions
Supersymmetric 5d gauge theories can be formulated on squashed 5-spheres [60, 61], for whichan explicit metric can be written as ds = (cid:88) i =1 (cid:0) dρ i + ρ i dθ i (cid:1) −
11 + (cid:80) i =1 φ i ρ i (cid:32) (cid:88) i =1 φ i ρ i dθ i (cid:33) , (2.5)with real coordinates θ i ∈ (0 , π ) and ρ i ≥
0, constrained by (cid:80) i =1 ρ i = 1. The φ i are thesquashing parameters, and for φ = φ = φ = 0 the metric reduces to the round S . Forgeneric φ i the isometry is reduced from SO (6) to U (1) . The perturbative part of the squashed S partition function has been derived in [61]. For a 5d gauge theory with gauge group G and N f hypermultiplets in a real representation R f ⊗ ¯ R f of G , it is given by Z (cid:126)ω = S (cid:48) (0 | (cid:126)ω ) rk G |W| (2 π ) rk G (cid:34) rk G (cid:89) i =1 (cid:90) ∞−∞ dλ i (cid:35) e − (2 π )3 ω ω ω F ( λ ) × (cid:81) α S ( − iα ( λ ) | (cid:126)ω ) (cid:81) N f f =1 (cid:81) ρ f S (cid:0) iρ f ( λ ) + ω tot | (cid:126)ω (cid:1) . (2.6)It depends on the squashing parameters through the periods ω i = 1 + iφ i , which are collected in (cid:126)ω = ( ω , ω , ω ), with ω tot ≡ ω + ω + ω . For the round S with (cid:126)ω = (1 , ,
1) we will also use S as a subscript. The roots of G are denoted by α , the flavor hypermultiplets are labeled by f = 1 , . . . N f , and ρ f are the weights of the corresponding representation R f ⊗ ¯ R f . S ( z | (cid:126)ω ) is thetriple sine function, W the Weyl group of G , and F ( λ ) is the classical flat space prepotential. Atthe UV fixed point, where g YM → ∞ , only Chern-Simons terms remain, F ( λ ) = c π ) tr( λ ) . (2.7)The triple sine function can be represented as S ( z | (cid:126)ω ) = exp (cid:18) − πi B , ( z | (cid:126)ω ) − I ( z | (cid:126)ω ) (cid:19) , I ( z | (cid:126)ω ) = (cid:90) R + i + dxx e zx ( e ω x −
1) ( e ω x −
1) ( e ω x − ,B , ( z | (cid:126)ω ) = 1 ω ω ω (cid:20) (cid:16) z − ω tot (cid:17) −
14 ( ω + ω + ω ) (cid:16) z − ω tot (cid:17) (cid:21) , (2.8)where B , is a generalized Bernoulli polynomial. The contour in I runs over the real axis, avoidingthe origin via a semi-circle around x = 0 going into the positive half-plane. B. Long linear quiver gauge theories
The zero-instanton partition function (2.6) for generic quiver gauge theories of the form (2.1)can be written conveniently as Z (cid:126)ω = S (cid:48) (0 | (cid:126)ω ) rk G |W| (2 π ) rk G (cid:90) ∞−∞ (cid:34) L (cid:89) t =1 N t − (cid:89) i =1 dλ ( t ) i (cid:35) exp (cid:18) − ω ω ω F (cid:126)ω (cid:19) , (2.9)with F (cid:126)ω = L (cid:88) t =1 N t (cid:88) (cid:96),m =1 (cid:96) (cid:54) = m F V (cid:0) λ ( t ) (cid:96) − λ ( t ) m (cid:1) + L − (cid:88) t =1 N t (cid:88) (cid:96) =1 N t +1 (cid:88) m =1 F H (cid:0) λ ( t ) (cid:96) − λ ( t +1) m (cid:1) + L (cid:88) t =1 k t N t (cid:88) i =1 F H (cid:0) λ ( t ) i (cid:1) + L (cid:88) t =1 π c t (cid:16) λ ( t ) (cid:96) (cid:17) , (2.10)where F V ( x ) ≡ − ω ω ω [ln S ( ix | (cid:126)ω ) + ln S ( − ix | (cid:126)ω )] ,F H ( x ) ≡ ω ω ω ln S (cid:16) ix + ω tot | (cid:126)ω (cid:17) . (2.11)We will need the asymptotic behavior of F H and F V for large arguments. The relevant asymptoticsof the triple sine function were collected in appendix A of [63]. With large real | x | , F V ( x ) ≈ + π | x | − ω + ω ω + ω ω + ω ω π | x | ,F H ( x ) ≈ − π | x | − ω + ω + ω π | x | . (2.12)In the large- N limit instanton constributions are expected to be suppressed, such that the pertur-bative part captures the large- N behavior, and the saddle point approximation is expected to beexact. To evaluate the partition functions we introduce normalized eigenvalue densities ρ t for the t th gauge node, such that 1 N t N t (cid:88) (cid:96) =1 f ( λ ( t ) (cid:96) ) −→ (cid:90) dλ ρ t ( λ ) f ( λ ) . (2.13)In the examples to be discussed below the rank may not be large for all gauge groups, but wecan nevertheless introduce the densities for gauge groups of small rank and approximate them bysmooth functions. F (cid:126)ω in eq. (2.10) then becomes F (cid:126)ω = (cid:90) dλ d ˜ λ (cid:34) L (cid:88) t =1 N t ρ t ( λ ) ρ t (˜ λ ) F V (cid:0) λ − ˜ λ (cid:1) + L − (cid:88) t =1 N t N t +1 ρ t ( λ ) ρ t +1 (˜ λ ) F H (cid:0) λ − ˜ λ (cid:1)(cid:35) + (cid:90) dλ L (cid:88) t =1 N t ρ t ( λ ) (cid:104) k t F H ( λ ) + π c t λ (cid:105) . (2.14)At leading order for large arguments, F H and F V are opposite-equal. This suggests to combinethe two terms in the first line to form differences of the eigenvalue distributions at the t and t + 1nodes. To this end we introduce F ( x ) ≡ F V ( x ) + F H ( x ) ω , ∆ t ( λ ) ≡ N t +1 ρ t +1 ( λ ) − N t ρ t ( λ ) , (2.15)where a factor of ω − has been included in the definition of F for later convenience. Then (2.14)can be written as F (cid:126)ω = (cid:90) dλ d ˜ λ (cid:34) L (cid:88) t =1 N t ρ t ( λ ) ρ t (˜ λ ) ω F (cid:0) λ − ˜ λ (cid:1) − L − (cid:88) t =1 ∆ t ( λ )∆ t (˜ λ ) F H (cid:0) λ − ˜ λ (cid:1)(cid:35) − (cid:88) t ∈{ ,L } (cid:90) dλ d ˜ λ N t ρ t ( λ ) ρ t (˜ λ ) F H (cid:0) λ − ˜ λ (cid:1) + (cid:90) dλ L (cid:88) t =1 N t ρ t ( λ ) (cid:104) k t F H ( λ ) + π c t λ (cid:105) . (2.16)Note the remainder terms at the first and last node in the second line. To implement large L , acontinuous parameter z ∈ [0 ,
1] as defined in (2.2) is introduced to label the gauge nodes, suchthat the quiver data is encoded in the functions N ( z ), k ( z ) and c ( z ) defined in (2.3). Likewise, thefamily of eigenvalue densities { ρ t } is replaced by one function of two continuous parameters, ρ ( z, λ ) = ρ zL ( λ ) , z = tL . (2.17)With the definitions in (2.3), (2.17) the sums over quiver nodes are replaced by integrals, L (cid:88) t =1 f t → L (cid:90) dzf ( zL ) , (2.18)and the finite difference terms in (2.16) turn into derivatives with respect to z ,∆ zL ( λ ) → L ∂ z [ N ( z ) ρ ( z, λ )] . (2.19)The expression for F (cid:126)ω in (2.16) becomes F (cid:126)ω = L (cid:90) dz (cid:90) dλ d ˜ λ (cid:104) N ( z ) ρ ( z, λ ) ρ ( z, ˜ λ ) ω F ( λ − ˜ λ ) − L ∂ z (cid:0) N ( z ) ρ ( z, λ ) (cid:1) ∂ z (cid:0) N ( z ) ρ ( z, ˜ λ ) (cid:1) F H ( λ − ˜ λ ) (cid:105) − (cid:88) z ∈{ , } (cid:90) dλ d ˜ λ N ( z ) ρ ( z, λ ) ρ ( z, ˜ λ ) F H ( λ − ˜ λ )+ L (cid:90) dz (cid:90) dλ N ( z ) ρ ( z, λ ) (cid:104) k ( z ) F H ( λ ) + π c ( z ) λ (cid:105) . (2.20)To determine the scaling of the eigenvalues in the large- N limits described below (2.1), we focuson the first integral in (2.20), assuming that both terms in the square brackets are non-vanishing.The first term in the square brackets is linear in the eigenvalues and the second term is cubic:as consequence of the definition in (2.15), the cubic terms in F ( λ − ˜ λ ) cancel and the explicitexpressions for the leading-order behavior for large argument are given by F ( x ) = − π | x | , F H ( x ) = − π | x | . (2.21)Both terms in the square brackets in the first integral in (2.20) are O ( N ( z ) ) without scaling theeigenvalues, so the eigenvalues are not expected to scale non-trivially with N ( z ). This leaves thescaling with L . Assume that the eigenvalues scale as λ = L α x , with x of order 1. Then thefirst term in the square brackets scales as L α , and the second term as L α − . They can combinenon-trivially if α = 1, such that the eigenvalues scale linearly with L . We thus introduce λ = Lω tot x , ˆ ρ ( z, x ) = Lω tot ρ ( z, Lω tot x ) , (2.22)with x of order one and ˆ ρ a normalized density with ˆ ρ ( z, x ) dx = ρ ( z, λ ) dλ . The factor ω tot , whichis order one, does not affect the scaling but has been included to isolate the dependence on thesquashing parameters. Keeping only the leading terms, we find F (cid:126)ω = ω F , (2.23)where F is independent of the squashing parameters and given by F = L (cid:90) dz (cid:90) dx dy L − L (cid:88) z ∈{ , } (cid:90) dx dy N ( z ) ˆ ρ ( z, x ) ˆ ρ ( z, y ) F H (cid:0) x − y (cid:1) + L (cid:90) dz (cid:90) dx N ( z ) ˆ ρ ( z, x ) (cid:104) k ( z ) F H ( x ) + π c ( z ) x (cid:105) , (2.24)with L = N ( z ) ˆ ρ ( z, x ) ˆ ρ ( z, y ) F ( x − y (cid:1) − ∂ z (cid:0) N ( z ) ˆ ρ ( z, x ) (cid:1) ∂ z (cid:0) N ( z ) ˆ ρ ( z, y ) (cid:1) F H (cid:0) x − y (cid:1) , (2.25)and F and F H as given in (2.21). The requirements for proper normalization of the eigenvaluedistributions and the constraint that the eigenvalues sum to zero amount to (cid:90) dx ˆ ρ ( z, x ) = 1 , (cid:90) dx x ˆ ρ ( z, x ) = 0 . (2.26)This calls for a constrained extremization of F to determine the saddle points. For some of thetheories considered below an unconstrained extremization leads to solutions which already satisfythe constraints, such that working with (2.24) is sufficient, but not for all of them. The constraintscan be implemented by Lagrange multiplier terms. For the two sets of constraints, each labeled bythe continuous parameter z , two Lagrange multiplier functions µ ( z ) and τ ( z ) are introduced, withfactors of L and N ( z ) included in their definition for later convenience. This leads to F = L (cid:90) dz (cid:90) dx dy L − L (cid:88) z ∈{ , } (cid:90) dx dy N ( z ) ˆ ρ ( z, x ) ˆ ρ ( z, y ) F H (cid:0) x − y (cid:1) + L (cid:90) dz (cid:90) dx N ( z ) ˆ ρ ( z, x ) (cid:104) k ( z ) F H ( x ) + π c ( z ) x (cid:105) + L (cid:90) dz N ( z ) (cid:20) µ ( z ) (cid:18)(cid:90) dx ˆ ρ ( z, x ) − (cid:19) + τ ( z ) (cid:90) dx x ˆ ρ ( z, x ) (cid:21) . (2.27)In summary, the leading large- N version of the partition function Z (cid:126)ω in (2.9) becomes Z (cid:126)ω = (cid:90) D ˆ ρ exp (cid:18) − ω ω ω ω F (cid:19) , (2.28)with F given in (2.27) and L in (2.25), and ˆ ρ a distribution on [0 , × R . The overall constantsin (2.9) are subleading in the large N limits of interest and have been dropped, along with othersubleading terms. The free energy in the saddle point approximation is given by F (cid:126)ω ≡ − ln Z (cid:126)ω ≈ ω ω ω ω F (cid:12)(cid:12) ˆ ρ =ˆ ρ s , (2.29)with ˆ ρ s denoting the saddle point configuration. For the round S with (cid:126)ω = (1 , , C. Saddle point equation
To derive the saddle point conditions, the expression for F in eq. (2.27) is first simplified byintroducing a rescaled eigenvalue distribution (cid:37) , (cid:37) ( z, x ) ≡ N ( z ) ˆ ρ ( z, x ) , (cid:90) dx (cid:37) ( z, x ) = N ( z ) . (2.30)Then F = L (cid:90) dz (cid:90) dx dy L − L (cid:88) z ∈{ , } (cid:90) dx dy (cid:37) ( z, x ) (cid:37) ( z, y ) F H (cid:0) x − y (cid:1) + L (cid:90) dz (cid:90) dx (cid:37) ( z, x ) (cid:104) k ( z ) F H ( x ) + π c ( z ) x (cid:105) + L (cid:90) dz (cid:20) µ ( z ) (cid:18)(cid:90) dx (cid:37) ( z, x ) − N ( z ) (cid:19) + τ ( z ) (cid:90) dx x (cid:37) ( z, x ) (cid:21) , (2.31)with L = (cid:37) ( z, x ) (cid:37) ( z, y ) F ( x − y (cid:1) − ∂ z (cid:37) ( z, x ) ∂ z (cid:37) ( z, y ) F H (cid:0) x − y (cid:1) . (2.32)The saddle point equation derived in this section is obtained from (2.31) by varying (cid:37) in a partof the interior of the interval z ∈ [0 ,
1] along which (cid:37) is assumed to be smooth. The resultingcondition is local in z and reads (cid:90) dy (cid:2) (cid:37) ( z, y ) F ( x − y ) + ∂ z (cid:37) ( z, y ) F H ( x − y ) (cid:3) + L (cid:104) k ( z ) F H ( x ) + π c ( z ) x + µ ( z ) + xτ ( z ) (cid:105) = 0 . (2.33)Using the explicit expressions for the leading large-argument behavior of F and F H in (2.21) andintegration by parts in the first term of the integral, along with the required fall-off behavior of (cid:37) ( z, x ) for large x , this may be rewritten as (cid:90) dy (cid:104) ∂ y (cid:37) ( z, y ) + ∂ z (cid:37) ( z, y ) + L k ( z ) δ ( y ) (cid:105) F H ( x − y ) + L (cid:104) π c ( z ) x + µ ( z ) + xτ ( z ) (cid:105) = 0 . (2.34)The saddle point equation (2.34) is consistent for large | x | along parts of the quiver withoutChern-Simons terms and where the effective number of flavors N f ( z ) defined in (2.4) at each nodeis equal to twice the number of colors: The leading terms at large x in (2.34) are cubic, with F H ( x ) ≈ − π | x | independent of y . The first term in the integral thus drops out at cubic order,due to the required fall-off behavior of (cid:37) . In the second term one may exchange the integrationand the derivative ∂ z , and use the normalization of (cid:37) to arrive at − π | x | (cid:2) ∂ z N ( z ) + L k ( z ) (cid:3) + π L c ( z ) x = 0 . (2.35)This is consistent for large positive and negative x in regions where c ( z ) = 0 and ∂ z N ( z ) = − L k ( z ),and the latter is precisely the condition that the effective number of flavors is twice the number ofcolors, also at nodes where N ( z ) may have a kink. In these regions (2.34) can be imposed for all x ∈ R . A local condition can then be derived by acting on (2.34) with ( ∂ x ) , leading to14 ∂ x (cid:37) ( z, x ) + ∂ z (cid:37) ( z, x ) + L k ( z ) δ ( x ) = 0 . (2.36)Using this condition in (2.34) shows that, in these regions, the functions µ ( z ) and τ ( z ) vanish. D. Boundary conditions
To derive the boundary conditions at z = 0 and z = 1, let z b ∈ { , } . We assume that N ( z ) issmooth in a neighborhood of z b , such that (2.36) holds, and that there are k b fundamental flavors atthe boundary node with Chern-Simons level c b . Since the continuous version of δ t,t is L − δ ( z − t L ), k ( z ) and c ( z ) become k ( z ) = k b L δ ( z − z b ) + . . . , c ( z ) = c b L δ ( z − z b ) + . . . , (2.37)where the integral over z in (2.27) is understood to include the end points, in the sense that δ ( z )and δ ( z −
1) contribute. Likewise, the Lagrange multiplier functions take the form µ ( z ) = µ b L δ ( z − z b ) + . . . , τ ( z ) = τ b L δ ( z − z b ) + . . . . (2.38)The variation of F in (2.27), with δ ˆ ρ ( z, x ) non-vanishing only in the aforementioned neighborhoodof z = z b , reads δ F = L N ( z b ) (cid:90) dx δ ˆ ρ ( z b , x ) (cid:34) − (cid:90) dy (cid:104) n b ∂ z ( N ( z ) ˆ ρ ( z, y )) (cid:12)(cid:12) z = z b + LN ( z b ) ˆ ρ ( z b , y ) (cid:105) F H ( x − y )+ L (cid:16) k b F H ( x ) + π c b x + µ b + τ b x (cid:17) (cid:35) . (2.39)The first term in the inner square brackets in (2.39) results from integration by parts in L , where n b is the outward-pointing unit vector normal to the boundary, n b = 1 for z b = 1 and n b = − z b = 0. The second term is due to the explicit boundary terms in (2.27).The argument for the boundary conditions depends on whether N ( z ) is non-zero or zero as z → z b : If N ( z b ) >
0, the first term in the inner square bracket is subleading with respect to the0second one, and the boundary conditions are determined from the vanishing of the leading terms.The case that will be encountered below is N ( z b ) > , k b = N ( z b ) + O (1) , c b = 0 . (2.40)That is, the rank of the boundary gauge node is large, with no Chern-Simons term, but with anumber of fundamental hypermultiplets which is large as well and differs from the rank of thegauge group N ( z b ) only by an order one number. That means the effective number of flavors atthe boundary node, including the bifundamentals with the adjacent gauge node, to leading orderis equal to twice the number of colors. The leading-order part of the condition δ F = 0 with (2.39)and only N ( z b ) and k b non-vanishing then reduces to − (cid:90) dy LN ( z b ) ˆ ρ ( z b , y ) F H ( x − y ) + Lk b F H ( x ) = 0 . (2.41)For k b = N ( z b ) this is solved by ˆ ρ ( z b , x ) = δ ( x ) . (2.42)If N ( z ) vanishes as z → z b , the boundary terms in δ F in (2.39) vanish, and there are noconstraints from extremality of F . This case corresponds to a quiver tail along which the rank ofthe gauge groups decreases from order L to order one. In this case, the relation between ˆ ρ and (cid:37) in (2.30) becomes singular. The requirement that the eigenvalue distributions ˆ ρ be well behavedat z = z b is therefore non-trivial, and regular eigenvalue distributions at z = z b can be obtained iflim z → z b N ( z ) ˆ ρ ( z, x ) = (cid:37) ( z b , x ) = 0 . (2.43)This condition can also be deduced from the normalization condition in (2.30). Both scenarios,(2.42) and (2.43), are summarized by (cid:37) ( z b , x ) = N ( z b ) δ ( x ) . (2.44)With these boundary conditions the explicit boundary terms in the first line of (2.27) and contri-butions from fundamental flavors at the boundary nodes drop out. E. Junction conditions
As discussed in sec. II C, the saddle point equation (2.34) is consistent for large positive andlarge negative x only along nodes with no Chern-Simons terms and effective number of flavorsequal to twice the number of colors. At (isolated) nodes which are not of this type, two solutionsto the local condition (2.36) are joined, and we now derive the junction conditions.Let z t ∈ (0 ,
1) label an interior node where N ( z ) has a kink, with k t fundamental hypermultipletsand a Chern-Simons term with level c t , k ( z ) = k t L δ ( z − z t ) + . . . , c ( z ) = c t L δ ( z − z t ) + . . . ,µ ( z ) = µ t L δ ( z − z t ) + . . . , τ ( z ) = τ t L δ ( z − z t ) + . . . . (2.45)1The junction condition for the local solutions in the regions z < z t and z > z t are derived fromextremality of F in (2.27), by using eq. (2.34) in the regions z < z t and z > z t separately. For avariation δ ˆ ρ ( z, x ) which is non-vanishing only in a region around z t in which z t is the only kink,the variation of F reads δ F = L (cid:90) dz (cid:90) dx dy δ L + L N ( z t ) (cid:90) dx δ ˆ ρ ( z t , x ) (cid:104) k t F H ( x ) + π c t x + µ t + τ t x (cid:105) . (2.46)Using (2.36) with integration by parts and the required fall-off behavior of ˆ ρ ( z, y ) for large y in thefirst term, the condition δ F = 0 leads to (cid:90) dy [ ∂ z (cid:37) ( z, y )] z = z t + (cid:15)z = z t − (cid:15) F H ( x − y ) + Lk t F H ( x ) + π Lc t x + Lµ t + Lτ t x = 0 , (2.47)which has to be imposed for all x for which ˆ ρ and thus δ ˆ ρ are allowed to be non-vanishing. Forlarge | x | , (2.47) leads to the condition found in (2.35) before. Depending on the value of theChern-Simons level, the support of (cid:37) thus has to be bounded from below, from above, or both, byDirichlet boundary conditions as follows (cid:37) ( z t , x ) = 0 ∀ x ≶ x c t = ± (cid:18) L [ ∂ z N ( z )] z = z t + (cid:15)z = z t − (cid:15) + k t (cid:19) ,(cid:37) ( z t , x ) = 0 ∀ x / ∈ ( x , x ) | c t | < (cid:12)(cid:12)(cid:12)(cid:12) L [ ∂ z N ( z )] z = z t + (cid:15)z = z t − (cid:15) + k t (cid:12)(cid:12)(cid:12)(cid:12) . (2.48)Acting with ( ∂ x ) on (2.47) shows that ∂ z (cid:37) ( z, x ) is continuous for x (cid:54) = 0 in the intervals where (cid:37) isnot constrained by (2.48), with a source at x = 0 if fundamental flavors are present. The bounds x and x in (2.48) will be determined by the constraints (2.26) and the junction condition (2.47). F. Conformal central charge C T We now derive a general relation between the S free energy and the conformal central charge C T for the type of theories discussed in the previous section. It follows from the general form of thefree energy in (2.29) and does not need an explicit solution to the saddle point conditions. Following[62, 63] (see also [64]), the conformal central charge C T can be computed from the squashed spherefree energy. Namely, with ω i = 1 + a i it can be obtained from an expansion for small a i via F (1+ a , a , a ) = F S − π C T (cid:32) (cid:88) i =1 a i − (cid:88) i The combination of the local saddle point equation (2.36) with the boundary and junctionconditions derived in the previous section poses a problem akin to 2d electrostatics. The generalproblem is summarized in the following. In sec. III A we solve it explicitly for quiver gauge theorieswith N f = 2 N at all interior gauge nodes. A number of the theories that were considered explicitlyso far, e.g. in [47, 48], are of this type, including the T N and + N,M theories. Theories with N f (cid:54) = 2 N interior nodes are also included in sec. IV, in the form of the Y N and (cid:30) + N theories, andthe corresponding saddle points will be discussed there.As discussed in sec. II C, along parts of the quiver where N f = 2 N with no Chern-Simons terms,the support of (cid:37) ( z, x ) is unrestricted and it satisfies14 ∂ x (cid:37) ( z, x ) + ∂ z (cid:37) ( z, x ) + L k ( z ) δ ( x ) = 0 . (3.1)This is a Poisson equation and (cid:37) can thus be interpreted as an electrostatics potential on the strip[0 , × R , which has to be non-negative. Fundamental flavor fields account for the source termsgiven by k ( z ) δ ( x ). For the theories considered below, there are fundamental flavors only at a finitenumber of nodes, such that k ( z ) takes the form k ( z ) = L (cid:88) t =1 k t L δ ( z − z t ) , z t ≡ tL , (3.2)with only a finite number of terms in the sum non-vanishing. The boundary conditions at z = 0and z = 1 as derived in sec. II D are (cid:37) (0 , x ) = N (0) δ ( x ) , (cid:37) (1 , x ) = N (1) δ ( x ) . (3.3)3The number of fundamental flavors at the boundary nodes was assumed to be of the same orderas the number of colors, which will be the case in all examples considered below. We thus need anelectrostatics potential between two infinite plates with prescribed Dirichlet boundary conditions.For normalizability of the eigenvalue distributions, (cid:37) also needs to vanish for x → ±∞ . At interiornodes with N f < N there may be Chern-Simons terms, and as discussed in sec. II E the followingconstraints have to be imposed, depending on the value of the Chern-Simons level c t , (cid:37) ( z t , x ) = 0 ∀ x ≶ x c t = ± (cid:18) L [ ∂ z N ( z )] z = z t + (cid:15)z = z t − (cid:15) + k t (cid:19) ,(cid:37) ( z t , x ) = 0 ∀ x / ∈ ( x , x ) | c t | < (cid:12)(cid:12)(cid:12)(cid:12) L [ ∂ z N ( z )] z = z t + (cid:15)z = z t − (cid:15) + k t (cid:12)(cid:12)(cid:12)(cid:12) . (3.4)The end points x and x are determined from the junction condition (cid:90) dy [ ∂ z (cid:37) ( z, y )] z = z t + (cid:15)z = z t − (cid:15) F H ( x − y ) + Lk t F H ( x ) + π Lc t x + Lµ t + Lτ t x = 0 , (3.5)and from the normalization and SU ( N ) constraints in (2.26). Both cases in (3.4) are compatiblewith the bound 2 | c | ≤ N − N f of [2]. In the electrostatics analogy the conditions in (3.4) amountto the insertion of semi-infinite, perfectly conducting plates parallel to the plates at z = 0 and z = 1. The charge at generic z = z t is determined from Gauß’s law Q t = (cid:90) dx [ ∂ z (cid:37) ( z, x )] z = z t + (cid:15)z = z t − (cid:15) = [ ∂ z N ( z )] z t + (cid:15)z t − (cid:15) , (3.6)where the second equality follows from the normalization of (cid:37) . At nodes where N ( z ) has a kink,the charge may be provided entirely by fundamental fields, leading to the case N f = 2 N , or at leastin part by the conducting plates restricting the support of (cid:37) at z = z t . A schematic representationof the electrostatics problem is shown in fig. 1. A. General solution for N f = 2 N quivers If all interior nodes have N f = 2 N , (cid:37) satisfies the saddle point equation (3.1) on the entire strip,with boundary conditions given by (3.3). With k ( z ) in (3.2) the equation satisfied in the interiorof the strip becomes 14 ∂ x (cid:37) ( z, x ) + ∂ z (cid:37) ( z, x ) + L − (cid:88) t =2 Lk t δ ( z − z t ) δ ( x ) = 0 . (3.7)The flavors at the first and last node of the quiver are crucial in the discussion of boundaryconditions and are reflected in (3.3); they do not play a role for the equation in the interior of thestrip. We solve this equation by mapping the strip to the upper half plane via u = e πx + iπz . (3.8)The boundaries at z = 0 and z = 1 are mapped to the positive and negative real line, respec-tively. The points at the origin and at infinity correspond to large negative and large positive x ,respectively. The saddle point equation (3.7) becomes ∂ u ∂ ¯ u (cid:37) + 12 L L − (cid:88) t =2 k t δ ( u, u t ) = 0 , u t = e iπz t . (3.9)4That is, the flavors contribute as point charges on the unit circle. The boundary conditions at z = 0 and z = 1 become Dirichlet boundary conditions on the real line, (cid:37) (cid:12)(cid:12) u ∈ R = 2 πN (0) δ ( u, − 1) + 2 πN (1) δ ( u, . (3.10)Since the problem is linear, a solution can be found by first constructing the solution to (3.9) withvanishing Dirichlet boundary condition, and then superimposing a harmonic function to implement(3.10). The Green’s function on the upper half plane with vanishing Dirichlet boundary conditionand ∂ u ∂ ¯ u G ( u, v ) = δ ( u, v ) is given by G ( u, v ) = 1 π ln (cid:12)(cid:12)(cid:12)(cid:12) u − vu − ¯ v (cid:12)(cid:12)(cid:12)(cid:12) . (3.11)The solution to (3.9) with vanishing Dirichlet boundary condition then is (cid:37) ( u ) = (cid:90) d v G ( u, v ) (cid:34) − L L − (cid:88) t =2 k t δ ( u, u t ) (cid:35) = − L L − (cid:88) t =2 k t G (cid:0) u, e iπz t (cid:1) . (3.12)The complete solution for (cid:37) , with an added harmonic function to satisfy (3.10), reads (cid:37) s ( u ) = iN (0) (cid:20) u + 1 − u + 1 (cid:21) + iN (1) (cid:20) u − − u − (cid:21) − L π L − (cid:88) t =2 k t ln (cid:12)(cid:12)(cid:12)(cid:12) u − u t u − ¯ u t (cid:12)(cid:12)(cid:12)(cid:12) . (3.13)Transforming back to the strip leads to (cid:37) s ( z, x ) = N (0) sin( πz )cosh(2 πx ) − cos( πz ) + N (1) sin( πz )cosh(2 πx ) + cos( πz ) − L π L − (cid:88) t =2 k t ln (cid:18) cosh(2 πx ) − cos ( π ( z − z t ))cosh(2 πx ) − cos ( π ( z + z t )) (cid:19) , z t = tL . (3.14)The actual eigenvalue distributions are obtained via (2.30),ˆ ρ s ( z, x ) = (cid:37) s ( z, x ) N ( z ) . (3.15)Since (3.14) is symmetric under x → − x , the SU ( N ) constraint, requiring that the eigenvaluessum to zero, is satisfied. The norm evaluates to (cid:90) dx (cid:37) s ( z, x ) = (1 − z ) N (0) + zN (1) − L L − (cid:88) t =2 k t ( | z − z t | + 2 z t z − z − z t ) . (3.16)This is precisely N ( z ), as can be seen from the fact that both are piece-wise linear, agree on theboundary values and have identical second derivatives, such that ˆ ρ s is properly normalized.To obtain the free energies, F in (2.27) is evaluated on the saddle point configuration (3.15)with (3.14). The details are given in app. A, the result is F (cid:12)(cid:12) ˆ ρ =ˆ ρ s = − L π (cid:2) N (0) + 2 N (1) + 3 N (0) N (1) (cid:3) ζ (3) − L π L − (cid:88) t =2 k t (cid:104) N (0) D (cid:0) e iπz t (cid:1) + N (1) D (cid:0) e iπ (1 − z t ) (cid:1)(cid:105) + L π L − (cid:88) t =2 L − (cid:88) s =2 k t k s (cid:104) D (cid:0) e iπ ( z s + z t ) (cid:1) − D (cid:0) e iπ ( z s − z t ) (cid:1)(cid:105) , (3.17)5 M NMN (a) NN N (b) N NN (c) (a) N (d) FIG. 2. 5-brane junctions for the + N,M , T N , Y N and (cid:30) + N theories from left to right. ( p, q ) 5-branesare represented by straight lines at angles determined by the p, q charges, the filled black dots representcorresponding [ p, q ] 7-branes. The 5d SCFTs are realized by intersections at a point; the external 5-braneshave been resolved slightly to visually represent the involved branes. with the Riemann ζ -function ζ ( s ) = Li s (1) and the single-valued polylogarithms D n ( e iα ) = (cid:40) Im (cid:0) Li n ( e iα ) (cid:1) for n evenRe (cid:0) Li n ( e iα ) (cid:1) for n odd . (3.18)The free energy is given by (2.29) with F | ˆ ρ =ˆ ρ s in (3.17). IV. 5D SCFTS WITH GRAVITY DUALS IN TYPE IIB In this section the general results of the previous sections are used to compute squashed S freeenergies for a sample of theories with gravity duals in Type IIB. The 5d SCFTs are engineeredin Type IIB string theory by ( p, q ) 5-brane junctions. We will consider two classes of theories,distinguished from the string theory perspective by whether or not s -rule constraints for multiple5-branes ending on one 7-brane play a crucial role. The theories of the first class are shown in fig. 2,those of the second class in fig. 4. They all have relevant deformations that flow to long quivergauge theories of the form (2.1) in the IR. From the field theory perspective they are distinguishedby whether or not there are fundamental hypermultiplets at interior gauge nodes. The T N , + N,M , T K,K, , T N,K,j and + N,M,k theories have N f = 2 N at all interior gauge nodes, such that theresults follow straightforwardly from sec. III A. The Y N and (cid:30) + N theories have interior nodes with N f (cid:54) = 2 N , and the Y N theory also has a non-trivial Chern-Simons term. A. + N,M theory In Type IIB string theory the + N,M theory is defined on the intersection of N D5 and M NS5branes, fig. 2(a), and was discussed already in [5]. In field theory it can be defined as the UV fixedpoint of the linear quiver gauge theory[ N ] − ( N ) − . . . − ( N ) − [ N ] , (4.1) The functions D n agree, for example, with Zagier’s single-valued polylogarithms [65] evaluated on a phase. M − SU ( N ) nodes and Chern-Simons levels zero for all gauge nodes. For N = M = 2 this is the E theory of [1]. The S-dual gauge theory deformation of the + N,M theory leadsto a quiver of the same form, but with N and M exchanged. The limit described by supergravitycorresponds to N, M (cid:29) N/M fixed. For this gauge theory all gauge groups have largerank in the supergravity limit. The sphere partition function has been obtained numerically fromlocalization in [48] and matched to an analytic supergravity prediction obtained in [43]. The matrixmodel for the quiver (4.1) is defined by (2.16) with N t = N for all t and L = M − 1. The onlynon-vanishing k t are k = k M − = N . The continuous versions appearing in (2.27) are N ( z ) = N , k ( z ) = NL ( δ ( z ) + δ ( z − , c ( z ) = 0 . (4.2)With a slight abuse of notation we denote by N ( z ) the continuous function describing the quiverand by N the integer number of D5 branes. Since N f = 2 N at all nodes, the results of sec. III Aapply. The Lagrange multiplier functions µ ( z ) and τ ( z ) vanish and do not need to be included inorder to find consistent results. With this data F in eq. (2.27) for the + N,M theory becomes F + N,M = M (cid:90) dz dx dy L + N M (cid:88) z ∈{ , } (cid:90) dx ˆ ρ ( z, x ) (cid:20) F H ( x ) − (cid:90) dy ˆ ρ ( z, y ) F H ( x − y ) (cid:21) . (4.3)The saddle point eigenvalue distribution is given by (3.14) with the data in (4.2), which isˆ ρ s ( z, x ) = 4 sin( πz ) cosh (2 πx )cosh (4 πx ) − cos(2 πz ) . (4.4)At the center node, z = , the eigenvalues are the largest; as the boundaries z = 0 and z = 1 areapproached, the eigenvalues become concentrated at zero. The squashed S free energy is obtainedfrom (2.29) with the general form of F | ˆ ρ =ˆ ρ s in (3.17) and the quiver data in (4.2), which yields F M,N (cid:126)ω = − π ω ω ω ω ζ (3) N M . (4.5)For the round sphere with (cid:126)ω = (1 , , N free energy was computed for the5d U Sp ( N ) theories and their orbifolds introduced in [14]. For example, the quiver obtainedfrom a Z k orbifold shown in fig. 1(c) of [15] involves the same gauge nodes and bifundamentalhypermultiplets as the quiver for the + N,M theory in (4.1). However, the flavor hypermultiplets atthe boundary nodes of the respective quivers are different, N fundamental hypermultiplets for the+ N,M theory compared to one antisymmetric hypermultiplet at each end for the Z k orbifold of the U Sp ( N ) theory, and the length k of the quiver in [15] is order one. For the quivers in [15], saddlepoints were found with equal eigenvalue distributions for all gauge nodes. For the + N,M theories,on the other hand, the saddle point configuration (4.4) depends non-trivially on the gauge nodelabel z . For the free energy this leads to different scalings: N M for the + N,M theory comparedto N / k / for the orbifolds of the U Sp ( N ) theory. B. T N theory The (unconstrained) T N theory is defined by a junction of N D5, N NS5 and N (1 , 1) 5-branes[66], as shown in fig. 2(b). It is the strongly-coupled UV fixed point of the linear quiver gauge7theory [67, 68] [2] − (2) − (3) − . . . − ( N − − ( N − − [ N ] , (4.6)with all Chern-Simons levels zero. The S-dual deformation leads to the same quiver. For N = 3this is the rank-1 E theory. The S free energy at large N has been obtained numerically fromlocalization in [48], and matched to an analytic supergravity prediction from [43]. For the large- N limit of the matrix model, one can strictly speaking not expect ˆ ρ t to be a smooth distribution forsmall t where SU ( t +1) has small rank. But one can nevertheless use it as an approximation, whichwill lead to consistent results. In (2.16) the T N quiver corresponds to L = N − N t = t + 1,and the only non-vanishing k t are k = 2 and k N − = N . In the continuous version (2.27), N ( z ) = N z , k ( z ) = 2 L δ ( z ) + NL δ ( z − , c ( z ) = 0 . (4.7)This theory has N f = 2 N at all interior gauge nodes, the only exception is the boundary node atthe left end of the quiver tail, where N ( z ) vanishes. Thus, the Lagrange multiplier functions µ ( z )and τ ( z ) in (2.27) can again be set to zero. The expression for F in eq. (2.27) becomes F T N = N (cid:90) dz dx dy L + N (cid:90) dx ˆ ρ (1 , x ) (cid:20) F H ( x ) − (cid:90) dy ˆ ρ ( z, y ) F H ( x − y ) (cid:21) . (4.8)The two flavors at the SU (2) gauge node only produce subleading contributions; they drop out inthe large N limit due to N (0) = 0. The saddle point eigenvalue distribution is given by (3.14) withthe quiver data (4.7), ˆ ρ s ( z, x ) = sin( πz ) z πx ) + cos( πz ) . (4.9)The free energy is obtained from (2.29) with (3.17) and (4.7), which yields F T N (cid:126)ω = − π ω ω ω ω ζ (3) N . (4.10)For the round S with (cid:126)ω = (1 , , 1) this provides an analytic result matching the field theorynumerics of [48] and the supergravity computations of [43]. The two flavors at the left end ofthe quiver (4.6) did not explicitly play a role in the derivation, but regularity of the eigenvaluedistribution at the quiver tail did. C. Y N theory The Y N theories were defined in [47] on junctions of N (1 , 1) 5-branes, N ( − , 1) 5-branesand 2 N NS5-branes, as shown in fig. 2(c). The theory admits two quiver deformations that werediscussed in [47], and we compute the free energy from both of them.The quiver gauge theory obtained directly from the Y -shaped 5-brane junction reads[2] − (2) − (3) − · · · − ( N − − ( N ) ± − ( N − − · · · − (3) − (2) − [2] . (4.11)Along the two quiver tails, N f = 2 N for each node, and the Chern-Simons levels are zero. At thecentral node N f = 2( N − c cl + N f ∈ Z [2] requires an integer8 (1 , − , 1) ( N − , − N + 1 , N − 1) ( N − , N − Y N junction with the central node of the quiver deformation (4.11) partly resolved. The solid linesshow the subweb correspobding to the central node; it can be obtained from a + N, web by integrating outtwo flavors. The quiver tails correspond to the dashed lines. Chern-Simons level. The brane web realization of the central node is shown in fig. 3. It can beobtained from a + ,N web, which has Chern-Simons level zero, by integrating out two flavors. TheChern-Simons level at the central node therefore is ± 1. The quiver is also related to the quiverfor the T K,K, theory in (4.52) below by replacing the two flavors at the central node by theChern-Simons term.The data describing the quiver (4.11) in (2.1) is L = 2 N − N t = t + 1 for 1 ≤ t ≤ N − N t = 2 N − t − N ≤ t ≤ L . The non-vanishing k t are k = k L = 2. The only non-vanishingChern-Simons level is c N − = ± 1. The continuous version is N ( z ) = N (cid:40) z z ≤ − z z ≥ , k ( z ) = 2 L δ ( z ) + 2 L δ ( z − ,c ( z ) = c N − L δ ( z − ) . (4.12)Since µ ( z ) and τ ( z ) vanish along parts of the quiver where N f = 2 N , they take the form µ ( z ) = µ L δ (cid:0) z − (cid:1) , τ ( z ) = τ L δ (cid:0) z − (cid:1) . (4.13)In the expression for F in (2.27) the flavors at the boundary nodes drop out, and the boundaryterms vanish, such that F Y N = L (cid:90) dz (cid:90) dx dy L + π c N − L N (cid:90) dx ˆ ρ ( , x ) x + L N (cid:20) µ (cid:90) dx x ˆ ρ ( , x ) + τ (cid:18)(cid:90) dx ˆ ρ ( , x ) − (cid:19)(cid:21) . (4.14) 1. Saddle point The quiver (4.11) is symmetric under z → − z , as reflected in N ( z ) = N (1 − z ), and the sameis expected for the saddle point configuration. One therefore has to find a non-negative harmonicfunction (cid:37) ( z, x ) on the strip ( z, x ) ∈ [0 , ] × R , with, since N ( z ) vanishes at z = 0, (cid:37) (0 , x ) = 0 . (4.15)The boundary condition at z = follows from the junction condition in (2.47), which with thesymmetry under z → − z becomes − (cid:90) dy ∂ z (cid:37) ( z, y ) (cid:12)(cid:12) z = z − (cid:15) F H ( x − y ) + π Lc N − x + Lµ + Lτ x = 0 . (4.16)9Since 2 L | c N − | equals the discontinuity in ∂ z N ( z ) at z = , following (2.48) the support at z = has to be restricted to x < x for c N − = +1 and to x > x for c N − = − 1. Consequently, (4.16)has to hold for c N − ( x − x ) < 0. Acting on (4.16) with ( ∂ x ) shows that ∂ z (cid:37) ( z, x ) | z = = 0 for c N − ( x − x ) < 0. Thus, the boundary conditions at z = are (cid:37) (cid:0) , x (cid:1) = 0 for c N − ( x − x ) > ,∂ z (cid:37) (cid:0) z, x (cid:1) | z = 12 = 0 for c N − ( x − x ) < . (4.17)To construct (cid:37) , the half of the strip with z ∈ [0 , ] is mapped to the upper half plane withcomplex coordinate u via u = e − πc N − ( x − x )+2 πiz . (4.18)The range c N − ( x − x ) < z = is mapped to ( −∞ , − 1) on the real line, while c N − ( x − x ) > z = is mapped to ( − , 0) on the real line. The boundary at z = 0 is mapped to the positivereal line. The boundary conditions (4.15), (4.17) thus require (cid:37) ( u ) to satisfy Neumann boundarycondition on ( −∞ , − 1) and vanishing Dirichlet on ( − , ∞ ). Moreover, (cid:37) should have at most anintegrable divergence at u = − u → ∞ . The entire strip with z ∈ [0 , 1] maps to the entire complex plane, withthe same conditions imposed on the real line. Since c N − ( x − x ) < z = corresponds to( −∞ , − ⊂ R , (cid:37) should be smooth across that part of the real line, but may have a branch cutfrom − (cid:37) ( u ) = 2 N √− u − . c . (4.19)with the branch cut of the square root √· along the negative real axis, such that the branch cutof (cid:37) extends from u = − x implicit in the definition of u in (4.18) is determined by the SU ( N ) constraint, which yields x = c N − π ln 2 . (4.20)The final result for the saddle point configuration for z ∈ [0 , 1] therefore isˆ ρ s ( z, x ) = 2 NN ( z ) √− − e − πc N − x +2 πiz + c . c . (4.21)It remains to verify that the condition (4.16) is satisfied, and determine µ and τ which willbe needed for computing the free energy. The first term in (4.16), T ( x ) ≡ (cid:90) dy | x − y | ∂ z (cid:37) s ( z, y ) (cid:12)(cid:12)(cid:12) z = 12 − (cid:15) , (4.22)is a polynomial of degree 4 in x : Acting with ( ∂ x ) n with n ≥ δ -functions inthe integral. Using integration by parts and the fall-off behavior of (cid:37) shows that these derivativesvanish. Thus, T ( x ) = (cid:88) n =0 a n x n , a n = 1 n ! ( ∂ x ) n T ( x ) (cid:12)(cid:12)(cid:12) x =0 . (4.23)0The coefficients can be evaluated using the support properties of ∂ z (cid:37) at z = . This shows that(4.16) is indeed satisfied with µ = ζ (3)16 π , τ = − π c N − . (4.24) 2. Free energy To derive the free energy, the expression for F Y N in (4.14) is evaluated on the saddle pointconfiguration (4.21). The Lagrange multiplier terms do not contribute since they multiply theconstraints, leaving F Y N (cid:12)(cid:12) ˆ ρ =ˆ ρ s = L (cid:90) dz (cid:90) dx dy L (cid:12)(cid:12) ˆ ρ =ˆ ρ s + π c N − L N (cid:90) dx ˆ ρ s ( , x ) x . (4.25)Using integration by parts and that (cid:37) s is harmonic, the first term reduces to boundary terms.Using also the symmetry of (cid:37) s under z → − z , we find F Y N (cid:12)(cid:12) ˆ ρ =ˆ ρ s = L (cid:90) dx (cid:37) s ( , x ) (cid:20) π c N − Lx − (cid:90) dy ∂ z (cid:37) s ( z, y ) (cid:12)(cid:12) z = − (cid:15) F H ( x − y ) (cid:21) . (4.26)With the junction condition (4.16) this becomes F Y N (cid:12)(cid:12) ˆ ρ =ˆ ρ s = L (cid:90) dx (cid:37) s ( , x ) (cid:104) π c N − x − µ − τ x (cid:105) . (4.27)The term proportional to τ vanishes by virtue of the SU ( N ) constraint, the one with µ can beevaluated using the normalization of (cid:37) s . With (4.24) and (2.29) one finds F Y N (cid:126)ω = − π ω ω ω ω ζ (3) N . (4.28)This free energy is related to that of the T N theory by a factor 4. This relation becomes moretransparent in the S-dual quiver deformation, which will be discussed next. 3. S-dual quiver The quiver deformation arising after performing an S-duality on the brane web is given by(2) − (4) − (6) − . . . − (2 N − − [2 N ] , (4.29)with all Chern-Simons levels zero and N f = 2 N at all nodes. For N = 2 this is the rank-1 E theory. In the matrix model (2.16) this quiver corresponds to L = N − N t = 2 t and k N − = 2 N .The continuous version (2.27) is specified by N ( z ) = 2 N z , k ( z ) = 2 NL δ ( z − , c ( z ) = 0 . (4.30)Since N f = 2 N at all nodes, µ ( z ) = τ ( z ) = 0. Consequently, F in (2.27) becomes F Y N = N (cid:90) dz (cid:90) dx dy L + 2 N (cid:90) dx ˆ ρ (1 , x ) (cid:20) F H ( x ) − (cid:90) dy ˆ ρ (1 , y ) F H ( x − y ) (cid:21) . (4.31)1Up to an overall factor of four this is equivalent to F T N for the T N theory in (4.8), where the [2]fundamentals in the quiver (4.6) only produce subleading corrections. The saddle point conditionsare insensitive to this overall factor. Consequently, the free energy for the Y N theory is related tothat of the T N theory in (4.10) by a factor 4, F Y N (cid:126)ω = 4 F T N (cid:126)ω , leading to (4.28). From the supergravityperspective this relation between the T N and Y N theories at large N follows from the discussion ofcombined SL (2 , R ) transformations and overall rescaling of the 5-brane charges in sec. 4 of [47]. D. (cid:30) + N theory The (cid:30) + N theory was defined in [47] on a sextic intersection of N D5-branes, N NS5-branes and N (1 , 1) 5-branes, as shown in fig. 2(d). It describes the strongly-coupled UV fixed point of thequiver gauge theory[ N ] − ( N + 1) − . . . − (2 N − − (2 N ) − (2 N − − . . . − ( N + 1) − [ N ] , (4.32)with all Chern-Simons levels zero (the subweb corresponding to the central node is symmetricunder rotation by π , corresponding to charge conjugation). For N = 1 this is the rank-1 E theory.The data characterizing this theory in (2.16) is L = 2 N − N t = N + t for t ≤ N while N t = 3 N − t for t ≥ N . The non-zero k t are k = k N − = N . The continuous version in (2.27) isdefined by N ( z ) = N (cid:40) z , z ≤ − z , z ≥ , k ( z ) = NL ( δ ( z ) + δ ( z − , c ( z ) = 0 . (4.33)Since the Lagrange multipliers vanish along parts of the quiver where N f = 2 N , their form is µ ( z ) = µ L δ (cid:0) z − (cid:1) , τ ( z ) = τ L δ (cid:0) z − (cid:1) . (4.34)With this data F in eq. (2.27) for the (cid:30) + N theory becomes F (cid:30) + N = 4 N (cid:90) dz (cid:90) dx dy L + 4 N (cid:88) z ∈{ , } (cid:90) dx ˆ ρ ( z, x ) (cid:20) F H ( x ) − (cid:90) dy ˆ ρ ( z, y ) F H ( x − y ) (cid:21) + 2 L N (cid:20) µ (cid:90) dx x ˆ ρ ( , x ) + τ (cid:18)(cid:90) dx ˆ ρ ( , x ) − (cid:19)(cid:21) . (4.35)The quiver is symmetric under z → − z and the saddle point eigenvalue distributions areexpected to be symmetric as well. We therefore have to construct a harmonic function on ( z, x ) ∈ [0 , ] × R . The boundary condition at z = 0 reads (cid:37) (0 , x ) = N δ ( x ) . (4.36)The boundary condition at z = follows from the junction condition (2.47). With the symmetryunder z → − z it becomes2 (cid:90) dy ∂ z (cid:37) ( z, y ) (cid:12)(cid:12) z = − (cid:15) F H ( x − y ) = L ( µ + τ x ) . (4.37)2Since the Chern-Simons level is zero, the support of the eigenvalue distribution at z = hasto be bounded from below and from above. Since, with no Chern-Simons terms, the problem issymmetric under x → − x , the boundary conditions at z = are (cid:37) (cid:0) , x (cid:1) = 0 for | x | > x ,∂ z (cid:37) ( z, x ) (cid:12)(cid:12) z = = 0 for | x | < x . (4.38) 1. Saddle point To construct (cid:37) , the problem is mapped to the upper half plane with a following SL (2 , R )transformation, u = e πx +2 πiz , v = ue πx + 1 u + e πx . (4.39)In the v coordinate we need a non-negative function satisfying Neumann boundary conditions for v in ( −∞ , ⊂ R and Dirichlet boundary conditions for v ∈ R + , (cid:37) ( v ) (cid:12)(cid:12) v ∈ R + = N δ ( v, . (4.40)With this condition the eigenvalue distributions vanish for x → ±∞ , as required for normalizability.The additional requirements are the following: There should be at most integrable divergences at v = 0 and v = ∞ , for normalizable eigenvalue distributions. Aside from the δ -function pole at v = 1 these should be the only divergences. The entire strip is mapped to the entire complex plane,and (cid:37) should be smooth across the negative real axis. Moreover, the eigenvalue distributions shouldbe symmetric under x → − x , i.e. v → / ¯ v .The function (cid:37) is constructed in two steps. A function satisfying the specified boundary condi-tions, symmetry under v → / ¯ v and the remaining requirements is given by (cid:37) = a √− v − v + c . c . (4.41)with the branch cut of the square root along the negative real axis, such that (cid:37) has the branchcut along the positive real axis. We may add an arbitrary function satisfying vanishing Dirichletboundary conditions on all of R + and all other requirements. Such a function is given by (cid:37) = b (1 − v ) √− v + c . c . (4.42)It has square root divergences at the origin and at the point at infinity, and the relative coefficientsof the terms in the numerator are fixed by the requirement for invariance under v → / ¯ v .The SU ( N ) constraint is satisfied automatically due to the symmetry under x → − x , and theparameters a and b are fixed by the normalization conditions, a = 2 N tanh(2 πx ) , b = 12 N coth( πx ) sech(2 πx ) . (4.43)Finally, x is determined from the junction condition (4.37). The left hand side is a polynomial ofdegree 4 in x , by the same argument as for the Y N theory, T ( x ) ≡ (cid:90) dy | x − y | ∂ z (cid:37) s ( z, y ) (cid:12)(cid:12)(cid:12) z = 12 − (cid:15) = (cid:88) n =0 a n x n , a n = 1 n ! ( ∂ x ) n T ( x ) (cid:12)(cid:12)(cid:12) x =0 . (4.44)3The quartic term vanishes due to the Neumann boundary condition on ( − x , x ) at z = . Thelinear and cubic terms vanish by symmetry of (cid:37) s under x → − x . The condition that the quadraticterm be zero leads to cosh(2 πx ) = 2 . (4.45)The resulting saddle point configuration for z ∈ (0 , 1) is given byˆ ρ s = NN ( z ) 1 − (2 πx + iπz ) √ πx + iπz ) (cid:115) √ πx + iπz ) + 2 √ πx + iπz ) − . c . (4.46)The junction condition (4.37) is satisfied with µ = 7 ζ (3) N π L , τ = 0 . (4.47) 2. Free energy The free energy is obtained by evaluating F in (4.35) on the saddle point configuration (4.46).With the boundary conditions at z = 0 and z = 1, the local saddle point equation and the symmetryunder z → − z this leads to F (cid:30) + N (cid:12)(cid:12) (cid:37) = (cid:37) s = − N (cid:90) dx dy [ (cid:37) s ( z, x ) ∂ z (cid:37) s ( z, y )] z = − (cid:15)z =0 F H ( x − y ) . (4.48)Using the boundary condition at z = 0 this further evaluates to F (cid:30) + N (cid:12)(cid:12) (cid:37) = (cid:37) s = 4 N (cid:90) dx (cid:104) N ∂ z (cid:37) s ( z, x ) (cid:12)(cid:12) z =0 F H ( x ) − (cid:37) s ( , x ) (cid:90) dy ∂ z (cid:37) s ( z, y ) (cid:12)(cid:12) z = − (cid:15) F H ( x − y ) (cid:105) . (4.49)The remaining integral in the second term can be evaluated using (4.37) with (4.47) and thenormalization of (cid:37) s , which yields F (cid:30) + N (cid:12)(cid:12) (cid:37) = (cid:37) s = 4 N (cid:20)(cid:90) dx ∂ z (cid:37) s ( z, x ) (cid:12)(cid:12) z =0 F H ( x ) − Lµ (cid:21) = − π ζ (3) N . (4.50)The resulting free energy is F (cid:30) + N = − π ω ω ω ω ζ (3) N . (4.51)For the round sphere this agrees with a supergravity computation of the same quantity along thelines of [43]. As in the previous examples the result involves an overall ζ (3) and has a simpledependence on the parameters characterizing the field theory. Examples where the free energy hasmore complicated dependence on the parameters are discussed in the following sections. E. T K,K, theory The T K,K, theories, as defined in [53], are realized in Type IIB string theory by junctionsinvolving the same 5-branes as the unconstrained T N theories of sec. IV B, but with the D5 branespartitioned into two groups of K D5 branes, with each group ending on one D7-brane, fig. 4(a).4 [ K, K ] 2 K K (a) [ K j , N − jK ] NN (b) jN MM (c) FIG. 4. Constrained 5-brane junctions with multiple 5-branes ending on the same 7-branes. From left toright for the T K,K, , T N,K,j , and + N,M,j theories. These theories are obtained from the T N theories with N = 2 K by renormalization group flows.The supergravity duals and aspects of the field theories were discussed in [53]. They may also bedefined as UV fixed points of the linear quiver gauge theories[2] − (2) − (3) − . . . − ( K − − ( K ) − ( K − − . . . − (3) − (2) − [2] , | (4.52)[2]with all Chern-Simons levels zero. For K = 2 this is the rank-1 E theory. The S free energywas obtained numerically from localization and in supergravity in [56], and matched to very goodaccuracy between the two descriptions.In (2.16), the T K,K, quiver corresponds to L = 2 K − N t = t + 1 for t ≤ K − N t = 2 K − t − t ≥ K . The non-vanishing k t are k = k K − = k K − = 2. The continuousversion in (2.27) is defined by N ( z ) = (cid:40) Kz z < K (1 − z ) z > , k ( z ) = 2 L (cid:0) δ ( z ) + δ (cid:0) z − (cid:1) + δ ( z − (cid:1) , (4.53)with c ( z ) = 0. Since N f = 2 N at all interior nodes, µ ( z ) = τ ( z ) = 0. In (2.27) the flavors at the SU (2) nodes produce subleading contributions only and drop out. Thus, F T K,K, = 4 K (cid:90) dz (cid:90) dx dy L + 16 K (cid:90) dx ˆ ρ (cid:16) , x (cid:17) F H ( x ) . (4.54)The quiver is symmetric under reflection across the central node, and the saddle point eigenvaluedistribution is expected to be symmetric as well. Since N f = 2 N at all interior nodes, the discussionof sec. III A applies, and the saddle point is given by (3.14) with the data in (4.53),ˆ ρ s ( z, x ) = 2 KπN ( z ) ln (cid:20) cosh (2 πx ) + sin ( πz )cosh (2 πx ) − sin ( πz ) (cid:21) . (4.55)The free energy is obtained via (2.29) with (3.17) and (4.53), which yields F T K,K, (cid:126)ω = − π ω ω ω ω ζ (5) K . (4.56)5For the round sphere with (cid:126)ω = (1 , , ζ (5) instead of ζ (3) may be understoodas follows. The general AdS solutions providing the holographic duals for 5-brane junctionsare defined by a pair of locally holomorphic functions A ± . These functions have meromorphicdifferentials for solutions corresponding to unconstrained junctions [36], while the differentialsinvolve logarithms for solutions corresponding to constrained junctions [37]. In the convention of[69], A ± as functions of the poles have transcendentality degree one for solutions without 7-branesand two for solutions with 7-branes. The free energies are computed from certain integrals of thefunctions A ± , which are thus of higher transcendentality degree for solutions corresponding toconstrained junctions. From the field theory perspective the appearance of ζ (5) is an effect of thecharges due to fundamental flavors at internal nodes, as seen explicitly from (3.17). F. T N,K,j theory The 5-brane realization of the T N,K,j theories is obtained from the one for the T N theories bytaking the N D5-branes and separating out j groups of K D5-branes that each end on a single D7brane. This leaves N − jK unconstrained D5-branes, as shown in fig. 4(b). The holographic dualswere discussed in [53]. A quiver deformation for N > jK is given by[ N − jK ] − ( N − jK + j − x − . . . x K − − ( N − K ) x K − . . . x N − − (2) − [2] , | (4.57)[ j ]with all Chern-Simons levels zero. Between the links labeled by x K and x N − the rank of thegauge groups decreases in steps of one. There are K − x and x K − , with rank increasing in steps of j − 1. For j = 1 there is a total of K SU ( N − K )gauge nodes. For j = 2 and N = 2 K the would-be (1) gauge node on the left end is replacedby two fundamental hypermultiplets; this case was discussed in sec. IV E. The T N,N − , theoriescorrespond to the R ,N theories of [67, 70], and the χ kN theories of [67] correspond to T N,N − k − , .The supergravity limit corresponds to N, K (cid:29) j of order one.In (2.16) the quiver (4.57) corresponds to L = N − N t = N − jK + t ( j − 1) for t ≤ K and N t = N − t for t > K , as well as k = N − jK , k K = j and k N − = 2. The continuous version in(2.27) is defined by N ( z ) = N (cid:40) − j k + ( j − z , z ≤ k − z , z ≥ k , k ≡ KN ,k ( z ) = N − jKL δ ( z ) + jL δ ( z − k ) + 2 L δ ( z − , c ( z ) = 0 . (4.58)The quiver deformation has N f = 2 N at all interior nodes, such that µ ( z ) = τ ( z ) = 0. Explicitly,(2.27) becomes F T N,K,j = N (cid:90) dz (cid:90) dx dy L + N j ( N − K ) (cid:90) dx ˆ ρ ( k , x ) F H ( x )+ N ( N − jK ) (cid:90) dx ˆ ρ (0 , x ) (cid:20) F H ( x ) − (cid:90) dy ˆ ρ (0 , y ) F H ( x − y ) (cid:21) . (4.59)6The flavors at the SU (2) node on the right end only produce contributions that are subleadingat large N , but the j flavors at z = k are important. Since N f = 2 N at all internal nodes, thediscussion of sec. III A applies. The saddle point eigenvalue distribution is given by (3.14) with(4.58),ˆ ρ s ( z, x ) = 1 N ( z ) (cid:18) ( N − jK ) sin( πz )cosh (2 πx ) − cos( πz ) − N j π ln (cid:18) cosh (2 πx ) − cos( π ( k − z ))cosh (2 πx ) − cos( π ( k + z )) (cid:19)(cid:19) . (4.60)The free energy is given by (2.29) with (3.17) and (4.58), F T N,K,j (cid:126)ω = − N π ω ω ω ω (cid:34) (1 − j k ) ζ (3) + 2 jπ (1 − j k ) D (cid:0) e i k π (cid:1) + j π (cid:16) ζ (5) − D (cid:0) e i k π (cid:17) (cid:35) , (4.61)with D n ( z ) defined in (3.18). The limit K → 1, leads back to the unconstrained T N theory, and(4.61) reduces to (4.10). The limit j k → T jK,K,j theories, and for j = 2 the resultagrees with (4.56). The result in (4.61) shows that the parameters of the field theory can in generalappear as arguments of polylogarithms, and that the form of the free energy is not limited to thesimple dependence on the field theory parameters found in the examples of the previous sections.For the special cases studied numerically in [56], F T N,K,j (cid:126)ω in (4.61) reduces to F T N,N/ , S = − N π (cid:18) ζ (3) + 4 π D ( i ) + 318 π ζ (5) (cid:19) ,F T N,N/ , S = − N π (cid:18) ζ (3)4 + 6 π D (cid:0) e iπ (cid:1) + 4743256 π ζ (5) (cid:19) , (4.62)and matches the numerical field theory and supergravity results. For general j , k one can compareto the supergravity results of [56], by rearranging (4.61) to match the parametrization of the freeenergy in (3.26) of [56] as follows, F T N,K,j S = − π ζ (3) N (cid:104) − jF (1) T N,K,j + j F (2) T N,K,j (cid:105) ,F (2) T N,K,j = 2 k F (1) T N,K,j − k − D (cid:0) e i k π (cid:1) − ζ (5)2 π ζ (3) , F (1) T N,K,j = k − D (cid:0) e i k π (cid:1) πζ (3) . (4.63)With this parametrization F (1) T N,K,j matches the plot of S (1) in [56] and F (2) T N,K,j matches S (2) . G. + N,M,j theory The brane realization of the + N,M,j theories is obtained from that of the + N,M theories bypartitioning one group of N D5-branes into N/j subgroups and terminating each subgroup on asingle D7-brane, as shown in fig. 4(c). These theories describe the UV fixed points of the quivergauge theories( j ) − (2 j ) − . . . − ( N − j ) − ( N − j ) − ( N ) − ( N ) N − Nj − − [ N ] . | (4.64)[ j ]7For j = 1 the would-be (1) gauge node on the left end is replaced by two fundamental hypermul-tiplets, [2]. The Chern-Simons levels are zero for all nodes. The supergravity limit corresponds to N, M (cid:29) j of order one. Aspects of the spectrum were studied in [47].In (2.16) the quiver for the + N,M,j theories with j > L = M − N t = tj for t ≤ N/j and N t = N for t ≥ N/j , with k N/j = j and k L = N . For j = 1 the first gauge node isreplaced by 2 fundamental flavors. In the large- N limit the continuous version of F + N,M,j is givenby (2.27) with N ( z ) = (cid:40) M jz , z ≤ k N , z ≥ k , k ≡ NjM , (4.65)and k ( z ) = 2 L δ j, δ ( z ) + jL δ ( z − k ) + NL δ ( z − , c ( z ) = 0 . (4.66)Since N f = 2 N at all interior nodes, µ ( z ) = τ ( z ) = 0. Explicitly, (2.27) becomes F + N,M,j = M (cid:90) dz (cid:90) dx dy L − M N (cid:90) dx dy ˆ ρ (1 , x ) ˆ ρ (1 , y ) F H (cid:0) x − y (cid:1) + M N j (cid:90) dx ˆ ρ ( k , x ) F H ( x ) + M N (cid:90) dx ˆ ρ (1 , x ) F H ( x ) . (4.67)As in previous examples, the flavors at the left end of the quiver that appear for j = 1 produceonly subleading contributions and drop out.Since N f = 2 N at all interior nodes, the saddle point eigenvalue distributions are given by(3.14) with (4.65),ˆ ρ s ( z, x ) = 1 N ( z ) (cid:18) N sin( πz )cosh (2 πx ) + cos( πz ) + jM π ln (cid:18) cosh (2 πx ) − cos( π ( k + z ))cosh (2 πx ) − cos( π ( k − z )) (cid:19)(cid:19) . (4.68)The free energy is given by (2.29) with (3.17), F + N,M,j (cid:126)ω = − M π ω ω ω ω (cid:34) N ζ (3) + 2 jM Nπ D (cid:0) e iπ (1 − k ) (cid:1) + j M π (cid:16) ζ (5) − D (cid:0) e i k π (cid:1)(cid:17) (cid:35) , (4.69)with D n defined in (3.18). For j = 1 and M > N , the brane web for the + N,M,j theory is equivalent(after moving 7-branes) to the one for the T M,M − N, theory. The free energy indeed agrees with(4.61) after identifying the parameters appropriately. Strictly speaking, the case j = N , leadingback to the unconstrained + N,M theory, is outside the range of validity for this result, since j wasassumed to be of order one for the derivation. By an appropriate expansion for small k , however,one can still recover (4.5). V. DISCUSSION We have obtained exact results for the free energies of 5d SCFTs with holographic duals in TypeIIB on squashed spheres. The SCFTs have relevant deformations that flow to quiver gauge theoriesof the form (2.1), and in the large- N limits described by supergravity the number of quiver nodesis large. The ranks of at least some gauge groups are large as well, but not necessarily all of them.8The saddle point conditions for the matrix models resulting from supersymmetric localization ofthe squashed sphere partition function were formulated as problems akin to 2d electrostatics, whichcan be solved using standard methods. They were solved explicitly for theories with N f = 2 N atall interior gauge nodes, and for a sample of theories with N f (cid:54) = 2 N nodes including theories withChern-Simons terms. The conformal central charge at large N was shown to generally be relatedto the round sphere free energy by C T = − π − F S .For a number of theories the saddle point solutions and free energies were discussed explicitly.This includes the + N,M , T N , Y N and (cid:30) + N theories, which are engineered by the unconstrained5-brane junctions in fig. 2. They admit quiver gauge theory deformations given in (4.1), (4.6),(4.11) and (4.32), respectively. These theories do not have fundamental flavors at internal nodes.The free energies on squashed spheres with metric (2.5), where φ i are the squashing parameters,are F + N,M (cid:126)ω = − π ω ω ω ω ζ (3) N M , F T N (cid:126)ω = − π ω ω ω ω ζ (3) N ,F Y N (cid:126)ω = − π ω ω ω ω ζ (3) N , F (cid:30) + N (cid:126)ω = − π ω ω ω ω ζ (3) N . (5.1)The squashing parameters are encoded in ω i = 1 + iφ i , with ω tot ≡ ω + ω + ω , and the round S is recovered for ω = ω = ω = 1. We also discussed the T K,K, , T N,K,j and + N,M,k theories,which are engineered in Type IIB by the constrained 5-brane junctions shown in fig. 4. Theirquiver gauge theory deformations are given in (4.52), (4.57) and (4.64), respectively, and thesehave fundamental flavors at interior nodes. The free energies are F T N,K,j (cid:126)ω = − N π ω ω ω ω (cid:34) (1 − j k ) ζ (3) + 2 jπ (1 − j k ) D (cid:0) e iπ k (cid:1) + j π (cid:16) ζ (5) − D (cid:0) e iπ k (cid:17) (cid:35) ,F + N,M,j (cid:126)ω = − M π ω ω ω ω (cid:34) N ζ (3) + 2 jM Nπ D (cid:0) e iπ (1 − k ) (cid:1) + j M π (cid:16) ζ (5) − D (cid:0) e iπ k (cid:1)(cid:17) (cid:35) , (5.2)where D ( e iα ) = Im(Li ( e iα )) and D ( e iα ) = Re(Li ( e iα )). For the T N,K,j theory k = K/N ,and for the + N,M,j theory k = N/ ( jM ). The T K,K, theory was discussed separately since thegauge theory deformation involves extra flavors, but the free energy is given by setting N = 2 K , j = 2 in the result for the generic T N,K,j theories. The results for theories with flavors at interiornodes involve polylogarithms up to degree five, while the maximal degree for theories withoutflavors is three. Moreover, while the results in (5.1) show a simple dependence on the field theoryparameters, in (5.2) the field theory parameters appear as arguments of polylogarithms.The analytic results for the S free energies of the + N,M and T N theories match the analyticsupergravity results of [43] and the numerical field theory computations of [48]. For the T N,K,j theories the S free energy matches the numerical supergravity and field theory results of [56]. Forthe Y N and (cid:30) + N theories the supergravity results are included in [43], and match the analytic fieldtheory results (5.1). Finally, for the + N,M,k theories a supergravity computation for a sample ofparameter choices matches the analytic result in (5.2). The results presented here therefore supportthe AdS /CFT dualities proposed in [35, 36] for 5d SCFTs engineered by 5-brane junctions, andthe dualities proposed in [45, 47, 53] for AdS solutions with 7-branes [37]. However, taking inspiration from [71], where transcendentality weight n is assigned to ζ ( n ) with n ≥ 2, such that π has weight one, and extending it so as to assign weight n to Li n ( z ) for an arbitrary phase z , gives a homogeneoustranscendentality weight for the free energies of all theories considered. − φ = φ = φ = i √ − s can be obtained with the consistent truncations from theType IIB solutions of [35–37] to 6d gauged supergravity worked out in [50, 51, 54]. Namely, byuplifting the 6d solutions of [72, 73]. The relation between the free energies on these squashedspheres and on the round S found in [72, 73], F s = (3 − √ − s ) / (1 − √ − s ) F S , thereforealso holds when computing the free energies in Type IIB supergravity. This relation is a specialcase of the general relation (2.50), providing a further match between field theory and supergravity.In a similar vein, the universal relation between conformal central charge and sphere free energyderived in sec. II F follows on the supergravity side from the existence of a consistent truncationto 6d supergravity, as spelled out in [48].The analytic results obtained here provide a stepping stone for many more field theory studies.With the saddle point eigenvalue distributions in hand, one may compute other BPS quantities suchas Wilson loops or supersymmetric correlators, or the flavor central charges from mass deformationsfollowing [62]. One may also expect similar methods to allow for an analytic computation of thetopologically twisted indices studied numerically in [56], and matched to the supergravity predictionbased on the solutions of [74–76]. More generally, it would be interesting to study long quiver gaugetheories in other dimensions, such as the 4d SCFTs discussed in [77]. ACKNOWLEDGMENTS It is a pleasure to thank Martin Fluder and Morteza Hosseini for fruitful collaborations anddiscussions, and Diego Rodriguez-Gomez for interesting discussions. I am also grateful to theorganizers and participants of the workshop “Holography, Generalized Geometry and Duality” atthe Mainz Institute for Theoretical Physics for the inspiring workshop, and to MITP for hospitality.This work is generously supported by the Mani L. Bhaumik Institute for Theoretical Physics. Appendix A: Free energies for N f = 2 N theories In this appendix, F in (2.27) is evaluated on the saddle point configuration for quivers with N f = 2 N at all interior nodes as given in (3.15) with (3.14). This is done by directly evaluating F in (2.31) on (3.14). With the Chern-Simons terms and the Lagrange multiplier terms vanishing,(2.31) reduces to F (cid:12)(cid:12) (cid:37) = (cid:37) s = L (cid:90) dz (cid:90) dx dy L (cid:12)(cid:12) (cid:37) = (cid:37) s − L (cid:88) z ∈{ , } (cid:90) dx dy (cid:37) s ( z, x ) (cid:37) s ( z, y ) F H (cid:0) x − y (cid:1) + L (cid:90) dz (cid:90) dx (cid:37) s ( z, x ) k ( z ) F H ( x ) . (A.1)With the boundary condition (3.3), the explicit boundary terms vanish, as do possible flavorcontributions from the first and last node, leaving F = L (cid:90) dz (cid:90) dx dy L (cid:12)(cid:12) (cid:37) = (cid:37) s + L (cid:90) dz (cid:90) dx (cid:37) s ( z, x ) k ( z ) F H ( x ) . (A.2)0With integration by parts, this becomes F (cid:12)(cid:12) (cid:37) = (cid:37) s = − L (cid:90) dx dy [ (cid:37) s ( z, x ) ∂ z (cid:37) s ( z, y )] z =1 z =0 F H ( x − y )+ L (cid:90) dx dy (cid:37) s ( z, x ) (cid:18) ∂ x (cid:37) s ( z, x ) + 12 ∂ z (cid:37) s ( z, x ) (cid:19) F H ( x − y )+ L (cid:90) dz (cid:90) dx (cid:37) s ( z, x ) k ( z ) F H ( x ) . (A.3)The first term can be simplified using the boundary condition (3.3). For the second term one usesthe saddle point equation (3.7). With k ( z ) in (3.2) this leads to F (cid:12)(cid:12) (cid:37) = (cid:37) s = − L (cid:90) dx [ N ( z ) ∂ z (cid:37) s ( z, x )] z =1 z =0 F H ( x ) + 12 L L − (cid:88) t =2 k t (cid:90) dx (cid:37) s ( z t , x ) F H ( x ) . (A.4)The saddle point configurations (3.14) are symmetric under x → − x , so the integral can be reducedto non-negative x at the expense of a factor 2, thus eliminating the absolute values in F H . Explicitly, F (cid:12)(cid:12) (cid:37) = (cid:37) s = π L (cid:90) ∞ dx x (cid:34) [ N ( z ) ∂ z (cid:37) s ( z, x )] z =1 z =0 − L L − (cid:88) t =2 k t (cid:37) s ( z t , x ) (cid:35) . (A.5)This can be further evaluated using the explicit solution (3.14), leading to integrals that can befound e.g. in [78] or performed using Mathematica.To explicitly evaluate the integral in (A.5), (cid:37) s is decomposed as follows, (cid:37) s ( z, x ) = (cid:37) ( z, x ) − L π L − (cid:88) t =1 k t ˜ (cid:37) t ( z, x ) ,(cid:37) ( z, x ) = N (0) sin( πz )cosh(2 πx ) − cos( πz ) + N (1) sin( πz )cosh(2 πx ) + cos( πz ) , ˜ (cid:37) t ( z, x ) = ln (cid:18) cosh(2 πx ) − cos ( π ( z − z t ))cosh(2 πx ) − cos ( π ( z + z t )) (cid:19) . (A.6)Correspondingly splitting the terms in (A.5) according to their power in L leads to F (cid:12)(cid:12) (cid:37) = (cid:37) s = L F + L L − (cid:88) t =2 k t F t + L L − (cid:88) t =2 L − (cid:88) s =2 k s k t F st , (A.7)with F , F t and F st independent of L . F can be evaluated using integration by parts, F = π (cid:90) ∞ dx x [ N ( z ) ∂ z (cid:37) ( z, x )] z =1 z =0 = π (cid:90) ∞ dx x (cid:20) ( N (0) + N (1)) ∂ x csch(2 πx ) + ( N (0) − N (1)) ∂ x (coth(2 πx ) − (cid:21) = − N (0) + 3 N (0) N (1) + 2 N (1) π ζ (3) . (A.8)Evaluating F t leads to F t = − π (cid:90) ∞ dx x (cid:20) (cid:37) ( z t , x ) + 12 π [ N ( z ) ∂ z ˜ (cid:37) t ( z, x )] z =1 z =0 (cid:21) = − π (cid:90) ∞ dx x (cid:20) πz t ) N (0)cosh(2 πx ) − cos( πz t ) + 2 sin( πz t ) N (1)cosh(2 πx ) + cos( πz t ) (cid:21) = iN (0)8 π (cid:0) Li (cid:0) e iπz t (cid:1) − Li (cid:0) e − iπz t (cid:1)(cid:1) + iN (1)8 π (cid:16) Li (cid:0) e iπ (1 − z t ) (cid:1) − Li (cid:0) e − iπ (1 − z t ) (cid:1)(cid:17) . 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