Partition functions on 3d circle bundles and their gravity duals

Journal of High Energy Physics, May 2018

Abstract The partition function of a three-dimensional \( \mathcal{N}=2 \) theory on the manifold ℳg,p, an S1 bundle of degree p over a closed Riemann surface Σ g , was recently computed via supersymmetric localization. In this paper, we compute these partition functions at large N in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These configurations are solutions to \( \mathcal{N}=2 \) minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving 1/4 of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large N limit of the partition function of the dual superconformal field theory. In particular, the Σ g  × S1 = ℳg,0 partition function, which was recently shown to match the entropy of AdS4 black holes, and the S3 ≅ ℳ0,1 free energy, occur as special cases of our formalism, and we comment on relations between them.

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Partition functions on 3d circle bundles and their gravity duals

Accepted: May Partition functions on 3d circle bundles and their gravity duals Chiara Toldo 0 1 2 3 Brian Willett 0 1 2 0 Santa Barbara , CA 93106 , U.S.A 1 538 West 120 th Street, New York City, NY 10027 , U.S.A 2 Kavli Institute for Theoretical Physics, University of California , USA 3 Columbia University in the City of New York , Pupin Hall , USA The partition function of a three-dimensional N = 2 theory on the manifold via supersymmetric localization. In this paper, we compute these partition functions at large N in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These con gurations are solutions to N = 2 minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving 1=4 of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large N limit of the partition function of the dual superconformal eld theory. In particular, the g S1 = Mg;0 partition function, which was recently shown to match the entropy of AdS4 black holes, and the S3 = M0;1 free energy, occur as special cases of our formalism, and we comment on relations between them. AdS-CFT Correspondence; Supergravity Models; Supersymmetric Gauge - Theory 1 Introduction 2 Mg;p partition function at large N Supersymmetric background on Mg;p Computation of Mg;p partition function Large N computation 3 The supergravity dual Minimal N = 2 gauged supergravity NUTs and Bolts Supersymmetry properties Moduli space of solutions Uplift to 11d On-shell action via holographic renormalization 4 Holographic comparison for ABJM Truncating to minimal supergravity Holographic comparison NUTs and Bolts for S3 5 General quivers The Mg;p partition function for general quivers Fractional R-charges and the R-symmetry background Example: V 5;2 theory Minimal supergravity for general quivers and the universal twist 6 7 B C 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 5.1 5.2 5.3 5.4 { i { Comparison with S3 partition function Discussion A Explicit Killing spinors Moduli space of gravity solutions Details of partition function calculations C.1 Contributions to twisted superpotential C.2 Contributions to log ZMg;p Introduction Recently, there has been much progress in performing exact, nonperturbative computations for superconformal eld theories (SCFTs) on curved manifolds via the technique of supersymmetric localization (see the review [1] and references therein). Such methods have greatly developed in the past several years, providing a tool to study a wide variety of SCFTs in various dimensions and backgrounds, leading to non-trivial tests of holography and other known dualities. In particular, recently these techniques have been successfully applied to the computation of the partition function of three-dimensional superconformal Chern-Simons-matter theories on g S1 in presence of background magnetic ux for the R- and avor symmetries through the Riemann surface, g [2{5]. By performing a partial topological twist [6] on g, one obtains the so-called \topologically twisted Witten index" [3]. This was shown in [7] to reproduce in the large N limit1 the macroscopic entropy of supersymmetric magnetic AdS4 black holes in theories of 4d FI-gauged supergravity. These black hole con gurations, rst found in [8], consist of M2-branes wrapped around g, and thus they implement the partial topological twist for the QFT describing the low-energy dynamics of the M2-branes. This recent success led to several extensions and developments. First of all, the entropy matching was performed on more general supergravity backgrounds, including dyonic black holes [9], black hole con gurations arising from massive IIA supergravity truncations [10, 11] and solutions with hyperbolic horizon [12]. Moreover, unexpected relations were discovered between the topologically twisted index on S2 S1 and the corresponding partition function on S3 in the large N limit [13]. Speci cally, the computation of the twisted index involves, as an intermediate step, the computation of the twisted superpotential, or Bethe potential, as a function of the avor fugacities [3]. Then this quantity was shown to coincide, for a suitable mapping of parameters, with the large N limit of the S3 partition function of the same N = 2 theory [13]. Given that the partition function on S3 of three-dimensional superconformal theories, such as the ABJM theory [ 14 ], has been extensively studied, and has connections to entanglement entropy and the F -theorem [15{17], it is natural to ask if such a correspondence has a deeper meaning. In parallel with these developments, a new class of partition functions for general 3d N = 2 gauge theories was computed in [18] utilizing a three-dimensional uplift of the 2d A-model [19]. These partition functions are de ned on the manifold Mg;p, a U(1) bundle of Chern degree p 2 Z over a Riemann surface g, in particular the three-sphere S3 and the product spaces g where g 2 Z 0 denotes the genus of the Riemann surface. This set of manifolds includes M0;1 ' S3 ; Mg;0 ' g S1 : 1Upon a suitable extremization of the index with respect to the fugacities, which corresponds to the attractor mechanism in the gravity side. S g ; { 1 { (1.1) (1.2) Thus, these partition functions include both the topologically twisted index of [2{5] and the round S3 partition function of [20{22] as special cases. This then provides a natural framework to address the relation between the topologically twisted index and S3 partition function. At the same time, constructing explicit supergravity backgrounds whose boundary is such a circle bundle and computing their renormalized on-shell action provides a viable holographic check for these eld theory computations. In more detail, the partition function on Mg;p can be computed by a sum over supersymmetric \Bethe vacua," [23], X I2SBE where the index I runs over the set SBE of vacua of the theory. Here mi and si are, respectively, real masses and uxes for background avor symmetry gauge elds, and F I , HI , and iI are certain functions appearing in the 3d uplift of the A-model, described in section 2 below. We will argue that, for a class of quiver gauge theories with holographically dual M-theory descriptions, in the large N limit this sum can be approximated by a single dominant term, Idom, and we nd the result: log ZMg;p (mi) p log F Idom (mi) + (g 1) log HIdom (mi) + si log iIdom (mi) ; (1.4) leading to a very simple dependence on the geometric and ux parameters. We nd the partition function exhibits the expected N 3=2 scaling, and reproduce and generalize the results of [4, 7] in the case p = 0. However, we nd that a large N solution exists only under certain conditions on the mass and ux parameters. In the case of M0;1 = S3, these conditions di er from those under which previous large N computations of the S3 partition function were carried out, e.g., in [15], and we comment on this discrepancy in section 4 below. We reproduce this result holographically, by providing supergravity backgrounds having boundary Mg;p in the framework of minimal N = 2 U(1) gauged supergravity. Such solutions can be embedded locally in 11d on 7-dimensional Sasaki-Einstein manifolds. We construct Euclidean regular solutions which preserve 1/4 of the supersymmetries and have appropriately quantized magnetic ux. Starting from the analysis of [24{26], we nd that the boundary can be lled with multiple gravity con gurations, with di erent topology. In particular, for the boundary S3 case we can have regular \NUT" solutions, with topology R4, and for S3=Zp one nds mildly singular NUT=Zp solutions. On the other hand, for general Mg;p we nd regular \Bolt" solutions, with topology O( p) ! g. The di erent topology has non-trivial consequences for the uplift of these solutions. Indeed, while there are no requirements for the NUT solution to lift to M-theory, the Bolt uplifts to eleven dimensions only for certain values of g and p, depending on the geometrical properties of the internal Sasaki-Einstein 7-manifold. Interestingly, the same constraints are recovered in the eld theory computation by setting all uxes equal, thus reproducing the universal twist which corresponds to minimal gauged supergravity.2 2This holds provided that the reduction on Y7 does not contain Betti vector multiplets in its spectrum. the NUT con guration coincides with the free energy of the corresponding theory on S3. The renormalized on-shell action for Bolt solutions is instead of the form IBolt = p 2 N 3=2 s Vol(S7) 12 Vol(Y7) (4(1 g) p) ; with the additional constraint action in the gravity side matches with the partition function of the corresponding eld log ZMg;p = IBolt ; (1.7) as expected. In case of trivial bration, p = 0, our formulas (1.4) and (1.5) nd agreement with those of [27]. In this particular case the on-shell action of the Euclidean solution coincides with the entropy of supersymmetric 1/4 BPS black holes with constant scalars and higher genus horizon.3 Along with the matching with the Bolt solutions, we study the relation between the S3 partition function as computed by [15] and the result we obtain for the Mg;p partition function, in light of the result of [13]. In particular, we elaborate on how the interesting relation between the extremal value of the twisted superpotential and the large N partition function on S3, discovered in [13], ts in our framework by relating these both to the partition function on the lens space S3=Z2. The main text of the paper is organized as follows: in section 2 we provide the details of the computation of the large N partition function of a class of N = 2 3d quiver gauge theories on Mg;p, focusing on the example of the ABJM theory. In section 3 we describe Euclidean minimal gauged supergravity solution whose boundary is Mg;p, and examine their supersymmetry properties, along with their moduli space for regularity. Moreover, we compute the on-shell action via holographic renormalization. In section 4 we show the matching between the renormalized on-shell action of the Bolt solutions and the partition function of the dual eld theory for the ABJM theory. In section 5, we consider more general quiver gauge theories, including the V 5;2 theory, and describe the truncation to minimal supergravity for these theories, obtaining a generalization of the universal twist of [27]. In section 6, we discuss the relation between the twisted superpotential and S3 partition function observed by [13], and relate these to the lens space partition function. Finally in section 7, we discuss some open issues and future directions. Several appendices complete this paper, and they are devoted to the construction of the explicit Killing spinor 3See also the recent analysis of [28, 29] for further computations regarding the equivalence between renormalized on-shell action and BPS black hole entropy. { 3 { for the supergravity solutions, to the description of their moduli space, and to the explicit details for the computation of the partition function ZMg;p . We start in this section by discussing the computation of the Mg;p partition function for 3d N = 2 eld theories. We rst describe the supersymmetric background on Mg;p and review the computation for general, nite N theories, and then turn to the large N computation for a class of U(N ) quiver gauge theories with M-theory duals. HJEP05(218)6 2.1 Supersymmetric background on Mg;p On this space we take the following metric:4 Following [18], we consider manifolds which are U(1) bundles over a Riemann surface, g, with p 2 Z the Chern-degree of the bundle, which we take to be non-zero in this subsection. ds2 = 2(d C(z; z))2 + 2gzzdzdz ; where z; z are local coordinates on g, with gzz the metric on g, + 2 is a coordinate along the U(1) ber, which has length 2 , and C is a locally de ned 1-form on g, satisfying: (2.1) (2.2) (2.3) (2.4) We may de ne a Killing vector K = 1 @ pointing along the U(1) ber, or equivalently, a To preserve supersymmetry on this space, we must turn on additional elds in the background supergravity multiplet [30, 31]. These include the R-symmetry gauge eld, AR, a scalar, H, and a vector, V . These lead to the Killing spinor equation: 1 Z 2 g dC = p : = K dx = (d C(z; z)) : + i 2 V 1 8 { 4 { (r (2.5) In [18] the scalar parameter was set to zero, but for comparison to the supergravity background below we will take = 2p ; these choices will lead to the same Killing spinor . Here the last term in AR corresponds to a contribution from a at connection, and we will describe this in more detail below. 4Here we have rede ned C ! C relative to the background in [18] to facilitate comparison to the asymptotic supergravity solution in the next section. Although in principle we may take an arbitrary smooth metric on g, and an arbitrary connection C subject to (2.2), for concreteness, and to compare to the bulk supergravity solution, we will consider constant curvature metrics and connections: 14 (d12 (2d+2 s+ind2 2d) 2) where the connection a( ; ) is given by: a( ; ) = < 8 > 12 cos d d >: 2(g1 1) cosh d for g = 0; for g = 1 for g > 1 In all cases, + 2 is an angular coordinate. For g = 0, usual round metric on S2. For g = 1, we identify 2 [0; ] and this is the + 1, obtaining the at, rectangular metric on the torus. For g > 1, the second and third terms in (2.6) describe the metric on the hyperbolic plane, H2, and we form g by taking an appropriate quotient of the hyperbolic plane by a Fuchsian subgroup [32], with a fundamental domain Dg. In all cases we have normalized the metric on g so that vol( g) = . The connection a has curvature da proportional to the volume form on g, and satis es: HJEP05(218)6 1 Z 2 g da = 1 : AR = p + (g 1) : g p 1 (r 1 (mod p). Then the Killing spinor equation, (2.4), becomes: Background gauge elds. In addition to the elds in the background supergravity multiplet, we may include background gauge multiplets coupled to the global symmetries of the theory. If i = 1; ; rH runs over a basis of the Cartan of the avor symmetry group, H, then we may turn on background gauge multiplets Vi in con gurations labeled by: mi = i ( i + i i) 2 C; si 2 Z ; where i is the real scalar in the background gauge multiplet, and the gauge eld Ai is given by: Ai = i + (sia) ; { 5 { (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) ZMg;p (mi; si) = 1 X Z jW j ma2ZrG CJK ZMg;p6=0 (mi; si) = 1 jW j X Z ma2ZrpG CCoul (2.13) (2.14) does not change the connection Ai (modulo gauge transformations), and we will see below this is an invariance of the partition function. 2.2 Computation of Mg;p partition function In this subsection we review the computation of the Mg;p partition function for a general, nite N , 3d gauge theory, in preparation for computing the partition function at large N in the next subsection. We refer to [18] and references therein for more details. There are two equivalent methods to compute the Mg;p partition function. First, one may couple the UV action of the 3d gauge theory to the supersymmetric background on Mg;p discussed above, along the lines of [31], and use localization to reduce the path integral to a nite dimensional integral. As shown in [18], one arrives at the following integral formula for a theory with gauge group G with rank rG and Weyl group W : drG uF (ua; mi)pH(ua; mi)g 1 i(ua; mi)si a(ua; mi)ma H(ua; mi) : Here ua; ma, a = 1; ; rG are holonomies and uxes, respectively, for the gauge eld, and similarly for mi; si, i = 1; ; rH for background gauge elds coupled to the avor symmetry group, H. The functions F , H, , and H depend on the eld content of the theory, and are de ned in (2.26) below. The integral is taken over a compact contour CJK , the so-called \Je rey-Kirwan" contour [33, 34]; we refer to [18] for the precise de nition. In the case p 6= 0, this contour integral can be deformed into one over a non-compact \Coulomb" contour, CCoul, and one obtains the equivalent formula: where and a are as in (2.3) and (2.8), and is the pullback along the projection map : Mg;p ! g. Explicitly, we can write, for p 6= 0: Ai = i + si ; s i p mi ! mi + 1; si ! si + p with as in (2.9), so that the gauge eld has torsion ux si (mod p). The Mg;p partition function we compute below will be functions of the parameters mi and si, depending holomorphically on the former. Note that shifting: drG uF (ua; mi)pH(ua; mi)g 1 i(ua; mi)si a(ua; mi)ma H(ua; mi) ; where the uxes ma now takes values in Zp rather than Z. Roughly speaking,5 Ccoul = iRrG is the imaginary slice in the complex u plane, and upon making the identi cation u ! i , 5More precisely, this statement is true only when suitable conditions on the R-charges of the chiral multiplets are satis ed; more generally the contour may be deformed to pass around certain poles coming from the contributions of the chirals. See [18] for more details. { 6 { one recovers the usual integral formula for the S3 partition function [20{22] in the case g = 0; p = 1. We return to the connection to previous computations of the S3 partition function in sections 4 and 6 below. The second method starts from the observation that the Mg;p partition function is computed by a certain 2d topological quantum eld theory (TQFT) on the base space, g, of the ber bundle. Speci cally, this TQFT is the \A-twist" of the 2d N = (2; 2) theory obtained by compactifying the 3d gauge theory on a circle and studying the low energy e ective action. Then, on general grounds, we expect the partition function to be given by a sum over the supersymmetric vacua of the 3d theory on a circle. Explicitly, one nds: X involving the same functions as in (2.15). We will describe this formula in more detail below. These two methods can be shown to be equivalent for an arbitrary 3d gauge theory [18]. For the purpose of taking the large N limit, we will utilize the sum-over-vacua formula, (2.17), for the remainder of this section. However, despite these formulas being equivalent for nite N , there are some subtleties in relating the large N limit obtained by the two methods. We will return to this issue in section 4. Let us now describe the formula (2.17) in more detail, starting by reviewing the compacti cation on S1 and the vacuum structure of the resulting system. Twisted superpotential and Bethe vacua on R 2 S1. theory, we may place it on R2 S1, obtaining at low energies an e ective 2d N = (2; 2) description. Then the vacuum structure of the theory is determined by the \Bethe equaGiven a 3d N = 2 gauge tions" [23]: a = 1; ; rG ; where W(ua; mi) is the e ective twisted superpotential of this e ective 2d N = (2; 2) system, de ned in (2.20) below. Here we de ne: ua = i ( a + i(At)a); a = 1; ; rG; mi = i iBG + i AtBG i ; i = 1; (2.18) ; rH ; (2.19) where is the real scalar in the dynamical gauge multiplet, and At the component of the gauge eld along S1, and we expand them in a basis of the Cartan subalgebra of G, and similarly for the parameters, mi, i = 1; ; rH , for background gauge multiplets coupled to the avor symmetry, H. Note that ua ua + 1 and mi mi + 1 due to large gauge transformations around S1. The twisted superpotential, W(ua; mi), depends on the matter content and UV Lagrangian of the 3d theory. We consider a general theory with gauge group G of rank rG, and chiral multiplets in some representation of G. We expand the chiral multiplets in weights of G, such that they have charges Qa , a = 1; ; rG, = 1; ; M , where M is the dimension of the space of chiral multiplets. The chiral multiplets also have charges, Si , i = 1; ; rH ,under the global symmetry group, H, which may be restricted by superpotential terms in the Lagrangian. Finally, we allow Chern-Simons terms kab, kij, and kai, { 7 { for the gauge, avor, and mixed CS terms, respectively. Then the twisted superpotential is given by the following function of ua and mi: W(ua; mi) = W (Qa ua + Si mi) M X =1 + 1 rG X 1 XrH kij mi(mj + ij ) + X rG rH X kaiuami : Here W (u) is the contribution of a chiral multiplet, regulated with a level ensure gauge invariance, given by: With this de nition of the chiral multiplet contribution, the bare CS levels, kab, kij , and kai, must all be integers to ensure gauge invariance. We note that the twisted superpotential in general has branch cuts, and is only de ned modulo shifts of the form: W ! W + naua + nimi + n ; a polynomial equation in the variables xa = e2 iua and i = e2 imi . for na; ni; n 2 Z. However, one can check that (2.18) is invariant under these shifts, and is For non-abelian gauge theories, one must discard solutions which are not acted on freely by the Weyl group, W , as supersymmetry is broken at these putative vacua, and we consider the remaining solutions up to Weyl symmetry. Then we de ne the set of Bethe (2.20) (2.21) (2.22) 1 kRR 2 (2.24) where kaR; kiR; kRR 2 Z correspond to contact terms involving the R-symmetry. Here the second line is the contributions from the W -bosons of the gauge group, and the sum is over the weights of the adjoint representation of G. Note the W -bosons do not contribute to W. { 8 { vacua as: SBE = u^a ; rG; w u^a 6= u^a; 8w 2 W =W : (2.23) Ingredients in the Mg;p partition function. With this background, let us now return to the computation of the Mg;p partition function. The coupling of this theory to the curved background of Mg;p depends on the choice of a U(1)R symmetry, which is used to perform a partial topological twist along the g directions. Since we will introduce a non-trivial ux for this R-symmetry, we must pick the R-charges, r , of the chiral multiplets to be integers, so that they live in a well-de ned vector bundles over Mg;p. Given such a choice of R-symmetry, we de ne the \e ective dilaton," (ua; mi): (ua; mi) = 1) log 1 e2 i(Qa ua+Si mi) + X kaRua + X kiRmi + rG a=1 rH i=1 1 2 i + 1 2 i M X(r =1 X 2Ad(G) log(1 e2 i (u)) ; Then, as argued in [18], the Mg;p partition function is given by the following sum over Bethe vacua: X where the \ bering operator," F , \handle-gluing operator," H, and \ ux operators," , are de ned in terms of the twisted superpotential, W, and e ective dilaton, , de ned above: F (ua; mi) = exp 2 i W(ua; mi) H(ua; mi) = e2 i (ua;mi)H; H = det a a i(ua; mi) = exp 2 i (2.26) HJEP05(218)6 Such a formula arises due the topological invariance along g, which implies that the operations of gluing a handle to a unit of avor symmetry g, adding a unit of ux for the S1 bration, or adding ux, are all implemented by local operators, H, F , and i respectively, giving rise to the simple formula (2.25). To take a simple example, the Mg;p partition function of a single chiral multiplet is: Z ;Mg;p (m; s; r) = F (m)p (m)s+(r 1)(g 1) ; where m; s, and r are the its mass, avor symmetry ux, and R-charge, respectively, and: F (m) = exp 1 2 i Li2(e2 im) + m log(1 e2 im) ; (m) = 1 1 e2 im : Note this depends only on the combination: symmetry ux, s ! s + c(g symmetry. If we take: and use the di erence equation: In other words, a shift of the R-charge, r ! r + c, is equivalent to a shift of the avor 1), and re ects a mixing of the R-symmetry with this avor ` s + (r 1)(g 1) : (m; `) ! (m + 1; ` + p) ; F (m + 1) = F (m) (m) 1 ; we see the partition function, (2.27), is invariant. This is consistent with the invariance of the background gauge eld, (2.13), under this shift of parameters, and re ects the fact that, for p 6= 0, the uxes are torsion, and take values in Zp. For a general gauge theory, we may explicitly write the summand in (2.25) as:6 F (ua; mi)pH(ua; mi)g 1 i(ua; mi)si = e ikabpuaub Hg 1 Y F (Qiaua + mi)p (Qiaua + mi)si+(ri 1)(g 1) : (2.32) i 6Here for simplicity we work in a basis of the avor symmetry group where mi corresponds to the mass of the ith chiral multiplet, and we take the avor and R-symmetry CS terms to vanish. We may also treat the contribution of the vector multiplets as that of an R-charge 2 chiral multiplet in the adjoint representation of G. (2.27) (2.28) (2.29) (2.30) (2.31) { 9 { On-shell twisted superpotential. We may conveniently construct the terms in the sum above using the \on-shell" twisted superpotential and e ective dilaton, de ned by: WI (mi) = W(u^Ia; mi); I (mI ) = (u^Ia; mi) + 1 2 i log det (2.33) but this is partially xed by imposing: which is a stronger condition than (2.18), and xes the freedom to shift W by naua. Then one has: F I (mi) F (u^Ia; mi) = exp 2 i W I iI (mi) I X mi ; i I HI (mi) = H(u^Ia; mi) = exp 2 i I : (2.34) One can check that the remaining branch cut ambiguities in W expressions, and they are well-de ned. Then we may construct the partition function as: I and I drop out of these X I2SBE We will be interested in computing this partition function for a large N gauge theory. In the case p = 0, this problem was studied in [7]. There they found that, although the number of Bethe vacua, jSBEj, grows with N , in many cases there is a single vacuum, with index Idom, whose contribution is dominant compared to all other terms in (2.36).7 When this occurs, we expect that (2.36) may be approximated as: ZMg;p F Idom p Idom si HIdom g 1 i 1 ) 2 i ! log ZMg;p p WIdom 1) Idom : (2.37) i i Note, in particular, that the partition function has a very simple dependence on the geometric parameters, g and p, and the uxes, si. In cases where the theory has a holographic dual, this suggests the holographic free energy has a similar simple dependence on these parameters, which is rather non-trivial. Below we will verify this relation holds quite generally. 7More precisely, there need not be a single, strictly dominant vacuum, as other vacua which contribute at the same order will introduce extra logarithmic corrections to log Z, which will be suppressed relative to the leading N 3=2 behavior we nd below. 2.3.1 U(N ) quiver gauge theories In this section we will focus on the ABJM model [ 14 ], which we describe in more detail below. This is a special case of a more general class of U(N ) quiver gauge theories. The ingredients in the computation of the twisted superpotential and Mg;p partition function for these quivers is very similar, so we describe these general ingredients in the next few subsections, returning to a more detailed analysis of these theories in section 5. Speci cally, the class of theories we will discuss, following [13, 15], consists of N = 2 quiver gauge theories with several U(N ) gauge factors, labeled by an index = 1; ; n. We allow bifundamental chiral multiplets connecting two gauge groups, (anti-)fundamental HJEP05(218)6 chiral multiplets in a single gauge group, and Chern-Simons levels, k , for the th gauge group. However, we impose the following restrictions: The sum of all Chern-Simons levels is zero: n =1 X k = 0 : X QI = 0; I2 X(rI I2 where QI is the charge of the Ith such bifundamental chiral multiplet under any avor symmetry, and rI is its R-charge. Here adjoint chirals are counted twice in the sum. The number of bifundamental chiral multiplets entering a node is the same as the number exiting the node. are equal. The total number of fundamental and anti-fundamental chiral multiplets in the quiver For each gauge node, , there is a superpotential constraint which imposes that, for all bifundamental chiral mutiplets with a leg in this node: N 1 X N a=1 (t) = (t ta) ; These restrictions are to ensure the theory has a well behaved M-theory dual description at large N , with a characteristic N 3=2 scaling of the number of degrees of freedom. For such quivers, following [7], we will take the following large N ansatz for the eigenvalues ua : ua = va + iN 1=2ta; In the large N limit the eigenvalues become dense, and we may parameterize them by the continuous variable t, de ning: and corresponding functions, v (t). Our strategy in the rest of this section is as follows. First, we compute the twisted superpotential at large N , using the above ansatz, and nd the eigenvalue distribution (2.38) (2.39) (2.40) (2.41) which extremizes it. Then, as in (2.37), we may assume that the dominant contribution to the Bethe sum computing the Mg;p partition function is determined by this extremal distribution. Thus, we evaluate the summand in (2.37) at this extremal distribution to compute the leading behavior of the Mg;p partition function. We start by reviewing the computation of the twisted superpotential at large N , as rst computed in [7] for the ABJM theory, and studied for more general quivers of the above type in [13].8 For a given set of eigenvalues, ua , approximated by the distributions (t) and v (t) above, we may compute the value of the e ective twisted superpotential at these eigenvalues W[ (t); v (t); mi]: Let us brie y summarize the various ingredients in the functional W[ ; v ], as computed in [7, 13]. We review the derivation of these ingredients in appendix C. First, the CS terms, which satisfy P k = 0, contribute: Next, a bifundamental chiral multiplet connecting the th and th groups contributes: WCS = iN 3=2 Z dt (t) X k tv : v , m is the mass of the bifundamental, and we have introduced the following notation for the \fractional part," [u], of a complex number u: The function g(u) is given by: Here we have imposed the constraint (2.39), which implies: (2.43) (2.44) (2.45) (2.46) (2.47) (2.48) We also impose that the total number of incoming and outgoing edges at each node in the quiver are equal. Note then that the (v )3 terms contributed by Wbif will cancel, so that the functional is in general quadratic in the v . 8Let us state the relation between the notations used here and those used in [7, 13]. We have: utahem = 2 uaus; ithem = 2 mius and Vthem = (2 )2Wus : (2.42) These changes propagate into the large N ansatz, e.g., tthem = 2 tus, them = 21 us, etc.. In addition, there are contributions to W from a bifundamental chiral that are subleading in N , but whose derivatives with respect to the v get large near special points in parameter space, and so they a ect the extremization of W. Speci cally, these contributions become important when vI + mI = n^I 2 Z for some bifundamental chiral multiplet, with index I. Then if we write: for some positive function YI (t), one nds an additional \tail contribution:" HJEP05(218)6 vI (t) + mI = n^I + Ce 2 N1=2YI (t) ; Wtail = ( vI ) X mi = 0 : Finally, an (anti-)fundamental chiral multiplet contributes: with the + ( ) sign for a fundamental (anti-fundamental) chiral. Extremal value and the ABJM theory include a Lagrange multiplier term, iN 3=2 R dt As described above, we will need to nd the eigenvalue distribution which extremizes W. To do this we vary the functional W[ ; v ] with respect to (t) and the v (t). We also 1 , to impose correct normalization of . The solution is in general de ned piecewise, bounded by points where vI + mI becomes an integer, after which vI becomes locked to this value to leading order, varying at subleading order as in (2.49). Let us consider as our main example the ABJM theory [ 14 ]. This has U(N )k U(N ) k gauge group, with two bifundamentals in the (N; N ), with masses m1;2, and two in the (N ; N ) representation, with masses m3;4. We assume k > 0; the case with k < 0 can be obtained by exchanging the two gauge groups. This theory includes a quartic superpotential which imposes the following constraints on the masses: The twisted superpotential is periodic under mi ! mi + 1 (up to branch jumps), and so depends only on the fractional part of the masses, [mi]. The functional we obtain for ABJM turns out to depend only on v = v1 v2. We look for solutions with:9 [m1;2] < v < [m3;4] : 9The only other essentially di erent possibility is to have a solution with, e.g., [m1] < v < [m2], but one may check that there are no solutions of this form. (2.49) (2.50) (2.51) (2.52) (2.53) Then we simply have [ v + mi] = v + [mi] (where here and below, for \ " we take + for i = 1; 2 and for i = 3; 4), and then the functional becomes: WABJM dt BBk vt+ 2 4 X i=1 :i=1;2 +:i=3;4 g( v +[mi])CC 2 1 X[mi] i ! 1 C C A 2 i dt k vt+ 2 1 v 2 1 X( [mi]([mi] 1)) v +X g([mi]) The extremal distribution was rst derived in [7]. Note that (2.52) imposes that Pi[mi] = 1; 2 or 3. Then one nds the following solution when Pi[mi] = 1 (here we also assume [m1] < [m2] and [m3] < [m4]): (t) = < ([m4] [m3])([m1] + [m3])([m2] + [m3]) ([m1][m2] [m3][m4])kt > ([m1] + [m3])([m1] + [m4])([m2] + [m3])([m2] + [m4]) ([m1][m2] [m3][m4])+kt Pi<j<k[mi][mj][mk] t < t < t+ 8 > > > > > > > > > > > > > : 8 > > > > > > > > : > > > > [m1] + C exp 2 N 1=2 +[m2]kt Then one computes the extremal value of W as: Wext ABJM = 2iN 3=2 3 p2k[m1][m2][m3][m4] : There is a similar solution when Pi[mi] = 3, related by mi ! 1 mi and W ! W. However, for Pi[mi] = 2, we see the quadratic term in v vanishes, and we do not nd a solution. 2.3.4 Mg;p partition function at large N Next we consider the functional computing the Mg;p partition function. Speci cally, we compute the contribution to log ZMg;p from a Bethe vacuum which is, approximately at large N , given by a distribution of eigenvalues ua corresponding to the functions and v , A bifundamental chiral multiplet with mass m, avor ux s, and R-charge r contributes: log Zbif Mg;p = multiplet (which, recall, does not contribute to W) contributes as above with v + m ! 0 and ` ! (g Here we have imposed the constraints in (2.39). Once again, since we impose the number of incoming and outgoing edges at each node are equal, the cubic terms in v will cancel, and this gives an expression quadratic in the v . We also have contributions from the tail regions, where vI + mI n^I 2 Z for some I. Here we nd: as in (2.40). Once we have found the dominant such eigenvalue distribution, as above, we may plug this in to this functional to compute the leading behavior of the partition function. Here we list the various ingredients, which are derived in appendix C. The ChernSimons terms contribute: log ZCS Mg;p = where `I is as above, and the sum is over all such tail regions. Finally, for an (anti-)fundamental chiral multiplet, we have: We note that the expressions above are invariant under: log Zfun Mg;p = (m; `) ! (m + 1; ` + p) ; ! where we recall shifting m m + 1 entails shifting the corresponding integer part, n ! n + 1. This re ects the fact that the uxes ` are de ned modulo p, as in (2.30). Let us now return to the ABJM example. Then the partition function is a function of the masses, mi = [mi] + ni, avor uxes, si, and R-charges, ri, where the latter enter in log ZMtaigls;p = 2 N 3=2 X Z Ijtail v(t)I +mI n^I dt(pn^I ` I (g 1)) (t)YI (t) ; (2.63) the combination `i = si + (g 1)(ri 1). Due to the quartic superpotential, these satisfy the constraints: 4 i=1 4 i=1 X mi = X si = 0; X ri = 2 : 4 i=1 Then, one nds the functional computing the Mg;p partition function is given by: log ZMABgJ;pM = 2 N 3=2 3 log ZABJM Mg;p Pi[mi]=3 = 1 2 i i 2 N 3=2 3 2p p2k(1 X i [m1])(1 [m2])(1 [m3])(1 [m4]) (2.70) p(ni + 1) + si + (g 1 [mi] 1)ri ! : log ZMABgJ;pM = Z 2 N 3=2 dt X i p t v + 2 (g 1) v2 1 2 `i(mi ni)(mi ni 1) X i 1 6 ` i mi ni +p p(mi3 3mini(ni +1)+ni(ni +1)(2ni +1)) ; plus the contribution of the tails. Now let us plug in the extremal solution in (2.55), which, recall, required Pi[mi] = 1. Then we must impose: 0 = X mi = X[mi] + ni ) X ni = 1 : i i Plugging in the eigenvalue distribution found above and evaluating the integral, one eventually obtains the following simple result: p2k[m1][m2][m3][m4] 2p+X pni +si +(g 1)ri : (2.69) [mi] Let us make a few comments about this formula. First, in the case p = 0, where Mg;p=0 = g S1, this reproduces the results of [4, 7].10 another solution related by mi ! 1 mi, explicitly: Next, recall this formula only applies when Pi[mi] = 1; when Pi[mi] = 3, we nd (2.66) (2.67) v (2.68) (2.72) For Pi[mi] = 2, we do not nd a solution. We will return to this point in section 4 below. Also, note that the result (2.69) has the expected form (2.37): 1 2 i log ZMg;p = p WIdom 1) Idom ; (2.71) where here WIdom is given by Wext ABJM in (2.58), and: ! i i Idom = ABJM : i We will see in section 5 that this relation holds also for more general quiver gauge theories. 10To compare to their results, one makes the identi cations in footnote 8, as well as nithem = (si+ (g 1)ri)us. In this section our aim is to nd supergravity solutions whose boundary is the manifold Mg;p, a circle bundle over a closed Riemann surface g, which can be locally uplifted in M-theory.11 According to the AdS/CFT dictionary, we expect the on-shell action of these solutions, suitably renormalized, to match with the Mg;p partition function of the dual eld theory, as computed above. We will return to this comparison in the next section. Minimal N = 2 gauged supergravity Our starting point is minimal N = 2 four-dimensional gauged supergravity, whose bosonic where G4 is the four-dimensional Newton's constant and l is the AdS radius, related to the cosmological constant via = 3=l2. We work in Euclidean signature. The gravitino supersymmetry variation is = r il 1A + r = 4 F 1 2 ab + l 1 ; ; where is a Dirac spinor and are the generators of Cli (4; 0) and so they satisfy f a; bg = 2gab. We follow here closely the conventions of [25]. The Einstein's equations coming from (3.1) read R 1 2 g R = 2 g 3 l + 2 F F g F F ; 1 4 and Maxwell's ones are with with We will restrict our analysis to a set of solutions where g has constant curvature, and to con gurations with a real metric. Solutions to the system of equations of motion (3.4){(3.5) have been obtained in [37] and they have the following form12 d ? F = 0 : dr2 (r) r2 s2 (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) ds2 = (r)(d + 2sf ( ; ))2 + + (r2 s2) d 2 ; (r) = (r2 s2)2 + ( 4s2)(r2 + s2) 2M r + P 2 Q2 : 11For the solution to be globally uplifted to M-theory we need to impose further constraints on the magnetic ux for some particular classes of solutions. We will spell out this condition in section 3.5. 12In appendix A of [25] it was shown that imposing SU(2) U(1) symmetry (and a real metric) the con gurations (3.6){(3.11) with = 1 are actually the unique solutions, obtained by directly integrating the equations of motion. and f ( ; ) = < cos d d The 2d area element d 2 reads = The gauge eld has this form In these solutions, M is the mass parameter and r is the radial coordinate. parameterizes a circle bered over a 2-dimensional constant curvature surface g spanned by the coordinates and . The bration is due to the presence of the NUT parameter, which we denote by s because of its relation with the squashing of the U(1) ber relative to the base.13 Solutions of this kind for = 1 were rst discovered by Taub [39] and Newman, Unti and Tamburino [40], hence the name Taub-NUT. Their structure and thermodynamics properties were later studied in [38] and [37]. In the latter, (Lorentzian) solutions with NUT charge and planar and hyperbolic horizon were analyzed as well. In the asymptotic limit r ! 1 the metric approaches ds2 = l 2 r2 dr2 + r2 4s2 l 2 (d + f ( ; ))2 + d 2 ; (3.12) where we have de ned = =(2s). In other words, the boundary is a circle bundle over period g, and in the particular case for which f ( ; ) = cos d and is periodic with = 4 , the boundary is a squashed 3-sphere with squashing parameterized by 4s2=l2. We will see in the next subsection that, once one takes into account the appropriate compacti cations, the boundary metric in eq. (3.12) coincides with the one considered for the eld theory computation, eq. (2.6), up to a rescaling of coordinates. The bulk solutions we have described in this section can be of the type AdS-TaubNUT and AdS-Taub Bolt, depending on the value of the parameters appearing in the warp factor. These solutions are characterized by di erent topologies, since one of the Killing vectors has a zero-dimensional xed point set (\nut") or a two-dimensional one (\bolt"). We discuss the requirements for the regularity for NUTs and Bolts separately below, along with the conditions of periodicity of the coordinates. We focus rst on the spherical case, g = S2, so = 1. 13In previous literature (e.g. [37, 38]) the NUT parameter has most often been denoted by n. of the ABJM and V 5;2 models above. We conjecture these map to the M-theory uplift conditions for the corresponding supergravity backgrounds. Finally, we note that there is a similar truncation corresponding to the Bolt solution. Here we work in the background ( R = 12 ; nR = p2 + g 1), and nd: 1 2 i log ZMg;p (mi) = ( p + 4(g 1)) Wext mi = 2 i 1 SC ; 21 iSC( p + 2(g 1)) 2 Z : (5.70) In cases where both truncations exist for the same g and p, such as for the ABJM theory discussed in section 4, these typically correspond to di erent choices of masses and uxes in the eld theory, leading to two di erent truncations to minimal supergravity. In this case, one of the solutions (namely, the Bolt+ for p > 0 or the Bolt for p < 0) is dominant over the other, and this will be picked out as the preferred solution when we perform the extremization of masses outlined at the beginning of this subsection. 6 Comparison with S3 partition function In this nal section, we brie y discuss a curious relation between the extremal value of the twisted superpotential, Wext([mi]), and the large N limit of the S3 partition function, as a function of trial R-charges i, which was rst observed by [13], and which was used at the end of the previous section. Namely, they observed that:25 1 2 i log ZS3 ( i) = 4Wext([mi]) ; i = 2[mi] : where ZS3 is the S3 partition function, which was computed in the large N limit by [15], provided that we identify the trial R-charges i and the [mi] via: In particular, recall one can nd a solution to the extremization of W when one imposes the following conditions, for the superpotential constraints (5.1): X qi [mi] = 1 , i X q i i i = 2 ; namely S3 = framework. namely: which reproduces the constraint on the R-charges appearing in the S3 partition function. Since the S3 partition function appears as a special case of the Mg;p partition function, Mg=0;p=1, it is natural to ask if this relation can be understood in our More precisely, it was argued in [18] that the Mg=0;p=1 partition function in the \physical gauge", (5.27), reproduces the standard partition function on the round S3 [20{22], ZMg=0;p=1 (mi) = ZS3 ( i); where i = mi : 25Here and below we rewrite their observations in our notation. (6.1) (6.2) (6.3) (6.4) However, note that the identi cation between the mass parameters and trial R-charges here is not the one, (6.2), appearing in the relation (6.1). Moreover, we saw in section 4 that, in the case of the superconformal R-charges for the ABJM theory, i = 12 , we did not nd a solution for the Mg=0;p=1 partition function using our method, while the relation (6.1) above still holds in this case. Thus, we do not see a direct relation between our computation of the Mg=0;p=1 partition function and the observation (6.1). However, it turns out we can approach this relation from another point of view, through the lens space S3=Z2. The partition function on this space was rst computed by Benini, Nishioka, and Yamazaki [67] (see also [42]), who considered more generally the case S3=Zp. In general, their background on S3=Zp is di erent from the one considered above and in [18], as the latter includes a at R-symmetry gauge eld, while the former does not. However, we claim that one actually recovers the partition function of BNY in the case p = 2 from the Mg;p partition function if one takes the background of (5.34), i.e.: X qi ni = 0 : i i 1 2 i log ZMg=0;p=2 (mi; ni = 0) = 2Wext([mi]) : Then (5.26) implies we may take the ri 2 R. To see this, we observe that the partition function of a chiral multiplet on the S3=Z2 background of [42, 67] can be written as: ZB;NSY3=Z2 (m; s; r) = F m + r 2 2 m + r 1+s 2 ; where s 2 Z2 is the torsion ux and r 2 R is the R-charge, as one can straightforwardly show from the in nite product formula for the chiral multiplet given in [42, 67]. This agrees with (5.28) up to a rede nition m ! m=2. One may check this relation also holds for the Chern-Simons and gauge contributions, and so we nd, for a general gauge theory: ZSB3N=YZ2 (mi; si; ri) = ZMg=0;p=2 2 mi ; si; ri for ( R; nR) = ; 0 : (6.7) 1 2 Recall the background (6.5) is precisely the one which allows us to take a large N limit of the Mg;p partition function for a general quiver. Then, from (5.6) specialized to the case g = 0; p = 2, we nd: 1 2 i log ZMg=0;p=2 (mi; ni) = 2Wext([mi]) + X ni@iWext([mi]) : i Here we impose: (6.5) (6.6) (6.8) (6.9) (6.10) Then a simple choice for the ni is to take them all to be zero, in which case we nd the result: In fact, we can make a stronger statement. We claim that in the case ni = 0, and for arbitrary mi, the functionals computing 2W and 21 i log ZMg=0;p=2 for arbitrary eigenvalue distributions are identical. For example, for the contribution of a bifundamental chiral multiplet, we have, from section 2.3: 1 2 i Then one nds the di erence of the two functional gives: 2Wbif 1 2 i However, when we impose the constraint (2.39) and sum over all bifundamental chirals, we nd that this vanishes. One can similarly check the other ingredients in the two functionals agree, and so we have: 1 2 i 2Wbif [ ; v ; mi] = log ZMg=0;p=2 [ ; v ; mi; ni = 0] : (6.15) This immediately implies that their extremized values, (6.10), agree. Moreover, it implies that the two methods discussed in section 4.3 will give the same result on this space, since one is extremizing the same functional in both cases. This is indeed what happens for the supergravity solutions. For p = 2 the Bolt+ has zero ux, and its on-shell action coincides with the NUT=Z2 one. The two branches are actually continuously connected: they join at the point in phase space corresponding to For instance for ABJM we have: IBSo7lt+ p=2 = N 3=2p2 24 2p 8 = N 3=2 p ; INSU7T=Z2 = 12 INSU7T = N 3=2 p : (6.16) special role played by S3=Z2 in more detail. Given the proposed correspondence between the two methods and the free energy of the NUT and Bolt solutions, respectively, this suggests that these two supergravity solutions will agree for the case with conformal boundary S3=Z2, not just for minimal supergravity, but for general N = 2 gauged supergravity theories. It would be interesting to explore the Finally, it was argued in [42] that the large N limit of the partition function on the BNY lens space, S3=Zp, is related to that of S3 by: (6.11) (6.12) (6.13) HJEP05(218)6 log ZS3=Zp = log ZS3 : 1 p (6.17) This relation arises because the bulk llings for these boundary manifolds are simply Zp quotients of AdS4, and so their on-shell action is related by a factor of p1 to the bulk dual of the S3 partition function. Then if we combine (6.17) with (6.10), we arrive at the same relation (6.1), noted by [13]. In this paper we accomplished the task of computing the large N limit of the Mg;p partition function for three-dimensional SCFTs with holographic M-theory duals and showing that it matches with the on-shell action of the minimal supergravity Bolt solutions. A few remarks are in order here. First of all, in solving the supergravity equations of motion we have made some assumptions on the form of the solutions. For instance, for g = 0 we have imposed SU(2) U(1) symmetry in the bulk, and we looked for con gurations with a real metric. In principle we cannot exclude the existence of further M-theory solution with the same boundary data, which yield a di erent value for the on-shell action. It would be interesting to investigate further the phase space of solutions obtained when one releases one or more assumptions mentioned above. One obvious extension of this work consists of incorporating vector multiplets in the gravity side, namely working with Bolt con gurations which arise as solutions of N = 2 gauged supergravity with vector multiplets. Solutions of the U(1) Fayet-Iliopoulos STU model can be lifted up to 11 dimensions [68{70], and the analysis of Euclidean NUT or Bolt solutions in this framework is currently work in progress. Notice that Lorentzian solutions with non-trivial scalar pro les and NUT charge in theories of FI-gauged supergravity were found in [71, 72]. With the addition of vectors and scalars, we expect a more intricate phase space to arise for the NUT and Bolt con gurations. Phase transitions involving di erent solutions can arise26 and this should be re ected in the eld theory computation. It would also be interesting to study the quantum corrections to the on-shell action, and match them to the nite N result in the eld theory, as was done recently in the context of AdS4 black holes in [75]. As discussed in section 4, it would be interesting to study the large N limit of the integral formula (2.16) for the Mg;p partition function, and compare this to the large N limit we found using the sum formula (2.17). In particular, we may hope to compare these di erent large N solutions to di erent (leading or sub-leading) saddles in the supergravity partition function, such as the NUT and Bolt solutions above. Lastly, one can consider more general quiver gauge theories, including those with massive type IIA supergravity duals [76], as considered in [10, 11, 13]. It would also be desirable to obtain the partition function for the quiver theory dual to the M-theory reduction on M 32. As mentioned at the end of section 5.3, for this theory all Bolt solutions are allowed, including in particular the p = 1 Bolt+, which is a regular lling for the squashed S3 for a nite range of squashing parameters. It would be interesting to see if the eld theory computation is able to reproduce its on-shell action IBolt+ p=1 = 34 INUT, which is lower than the S3 free energy computed in [65, 66]. The question of nding the correct vacuum of the theory is of course entangled with the fact that, as already mentioned, we cannot rule out the presence of (perhaps less symmetric) branches of solutions with the same boundary data. A deeper understanding of these points would be desirable. All these questions are left for future investigation and we hope to report back in the near future. 26See for instance [73] and [74] for further studies on this topic. Acknowledgments We would like to thank D. Berenstein, C. Closset, M. Crichigno, G. Horowitz, S.M. Hosseini, K. Hristov, H. Kim, D. Klemm, J. Louko, D. Martelli, A. Passias, S. Pufu, E. Shaghoulian, J. Sparks, and A. Za aroni for discussions. We acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University for hospitality during some steps of this paper. BW was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. A Explicit Killing spinors This appendix is devoted to the explicit construction of the Killing spinor for the solutions described in section 3.3. Let's recall the Killing spinor equation: = ab + 1 2 iA + F i 4 = 0 The necessary conditions for the solutions to preserve supersymmetry, as already mentioned, can be found from the results of [44] performing the following parameter map ! i t ! it ! i Q ! iQ s ! is : The necessary conditions read and with M P Qs( 4s2) = 0 ; B+B = 0 : B = (M sQ)2 4s2)2 2P 5s2)(P 2 Q2) ; which reduce, in the case = 1 to those found by [25, 26] and are solved, for solutions preserving only 1=4 of supersymmetry, by (3.34). We are interested in constructing the Killing spinor for this latter class of solutions: the procedure we will use follows from that of [25] with minor modi cations which allow for general bases g. Spinors and vierbeins. We introduce the orthonormal frame conveniently chosen in [25]: e1 = 1pr2 s2 ; e2 = 2pr2 s2 ; e3 = 2s 3 r (r) r2 s2 ; e4 = dr s r2 s2 (r) : Depending on g, we choose the following form for 1;2;3: for S2 (g = 0) we chose the SU(3) left-invariant vielbeins 1 = cos d + sin sin d 2 = sin d + cos sin d 3 = d + cos d which in particular satis es 1 + i 2 = e i (d + i sin d ). (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) for R2 (or, upon compacti cation g = 1) we choose 1 = sin d + cos d 2 = cos d + sin d 3 = d d (A.8) for H2 (or, upon compacti cation g > 1) we have 1 = sin d + sinh cos d 2 = cos d + sin sinh d 3 = d We take the following basis of four-dimensional gamma matrices, which is the same used ) ; where where i, i = 1; 2; 3 are the Pauli matrices. The supersymmetry parameter is a Dirac spinor. It is convenient to write it in terms of its positive and negative chirality parts 0 iI2 0 iI2 ! ; 4 = 5 = 1 2 3 4 = I2 0 0 I2 1 + 5 2 = 0 2 = 0 ! : In what follows we moreover further distinguish the single parts of + and as (+); ( ) , so that the 4d Dirac spinor has this form Radial component of the Killing spinor equation. We write the warp factor (r) as (r) = (r r1)(r r4) ; where r , = 1; 2; 3; 4, are the four roots of the warp factor, and their explicit form is in (3.40). If we decompose the r component of the Killing spinor equation (A.1) into chiral parts, we obtain and 2 i 2 s r2 s r2 s2 s2 (r) (r) + + 2Q 4s2 2(r s)2 + 2Q 2(r + s)2 4s2 3 ; 3 + : In analogy to [25], we impose the following projections on the Killing spinor: (+) = (++) = 0 ( ) = i s (r s) (r (r + s) (r r1)(r r3)(r r2) ( ) r4) + : 0 (+) 1 + B (+) C = BBB@ ((+ )) CAC : C (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) 3d Killing spinor equation. Once we have inserted (A.17) the remaining components of the Killing spinor equation (A.1) reduce to the following three-dimensional equation the spinor : r(3)i iA(3)i + = 0 : is 2 i (A.17) (A.18) (A.19) (A.20) (A.21) (A.22) (A.23) boundary metric ds23 = 12 + 22 + 4s2 32 with spin connection Here r(3)i and i are respectively the covariant derivative and gamma matrices for the 3d !02 = !01 = ( s sin 2s2)f ( ) !12 = !12 = s cos sf ( ) sin !02 = s cos f ( ) !t01 = 2s2 where f ( ) is de ned in (3.8). The 3d gauge eld appearing in (A.19) is the asymptotic value of the gauge eld A, hence it has the form A(3) = P 3 = (4s2 ) 3: Choosing the vielbeins e1 = 1, e2 = 2, e3 = 2s 3 it is easy to see that the components of equation (A.19) are solved by where (0) is a constant. The full Killing spinor then reads Using the projection relation (A.16) we nd a solution to equations (A.14) and (A.15) of the form = F (r) ! G(r) ; where is a two-component spinor which does not depend on r, and the two radial functions F (r) and G(r) read: F (r) = r (r r3)(r r s r4) ; G(r) = i r (r r1)(r r + s r2) : = 0 ! (0) = BB B B q (r r3)(r r4) (0) C r s r+s 1 CC : C 1 2 0 0 The solution therefore preserves two supersymmetries out of eight, it is then 1/4 BPS. Notice that the Killing spinor only has radial dependence, hence no further supersymmetries are broken while taking the quotient to obtain a higher genus surface. Let us mention that if we had chosen the static vielbein, which for instance for = 1 reads 1 = d 2 = sin d 3 = d + cos d (A.24) the Killing spinor would have had no radial dependence, provided the asymptotic gauge eld A(3) is supplemented by at connection term A(3) = P 3 + A at where A at = d . using Morse theory.27 ditions are satis ed: We analyze here the range of existence of the Bolt+ and Bolt solutions, depending on the value of parameters p and g. Let's remind the reader that we have formally found four branches of solutions (3.44) and (3.45). However, as we will see below, not all of them are regular and in our formalism one needs to check this explicitly case by case. What happens in general is that some branches are de ned in a speci c s interval. A given solution branch, if it's not de ned everywhere, either joins other another branch (i.e. NUT solution), or ends \annihilating" another branch (as happens for the p = 1 Bolt solutions, which both end at the same point). This is consistent with an analysis of these solutions In order to obtain regular solutions one needs to ensure that the three following con3. r+(Q+) > r (Q+) for Bolt+, r+(Q ) < r (Q ) for Bolt We checked this explicitly for the cases below. The procedure used to analytically obtain the range of parameters for the g = 0 case is spelled out in section 5.2.1 of [25]. Generalizing it to higher genus case corresponds to incorporating the following modi cations in their procedure p ! p = for g 6= 1 p ! p for g = 1 j g 1 j f ! f = (16 s2 128 s2 : (B.1) (B.2) and p = 1. We summarize here the main results of our analysis. branches exist for s 2 [0; round S3 which is obtained by s = 1=2. p 4 p g = 0 Two regular Bolt+ branches exists for s 2 [0; 1+p2 ] and two regular Bolt 4 5 2 6 ]. Notice that this interval does not include the NUT solution, which has Q = 2s2, but presents conical singularities. g = 1 for this case only one regular Bolt+ solution (with Q++) exists for all s > 0. The Bolt branch with Q+ exist for s 2 [0; 14 ]. At this point s = 1=4, this joins the g > 1 the Bolt+ with Q++ exists for s 2 [0; f (g)], while the other Bolt+ with Q+ exists for s example, f (2) f (g). Therefore the two Bolt+ branches cover together the entire s axis. For 1=4, f (3) = 1=4p2 etc. Moreover, one out of the two Bolt solutions (the one corresponding to Q ) exists for all s > 0. 27We thank D. Berenstein for discussions on this topic. g = 0 We have one Bolt+ solution in the interval s 2 [0; 2p1 2 ]. At the point s = 1=2p2 the solution joins the NUT solution since r+(Q+) = s. The latter extends for all s. The two Bolt branches exist in the interval s 2 [0; 2 g = 1 the Bolt+ solution with Q++ exists for s > 0, and the Q+ Bolt solution exist in the interval s 2 [0; 2p1 2 ], merging at the point s = 1=2p2 with the NUT branch. 1=2p2, h(3) = 1=4, h(4) = 1=2p6 etc. g > 1 The Bolt solution corresponding to Q exists regular for all s > 0. The Bolt+ with Q++ exists for s 2 [0; h(g)], while the other Bolt+ with Q+ exists for s Therefore the two Bolt+ branches cover together the entire s axis. For example, h(g). p2) ]. g = 0 Bolt+ solution exists for all s > 0. The two Bolt solutions instead are regular s 2 [0; p4p ] and at the end point it merges with the planar NUT. g = 1 Bolt+ solution with Q++ exists for all s > 0. The Q+ Bolt solution exist for g > 1 One Bolt branch is once again present for all s > 0. In analogy to the For instance, for g = 3; p = 6 we have l(3; p = 6) = p3=4. previous cases, two Bolt+ solutions exist, with domains s 2 [0; lp(g)] and s lp(g). Plots of the on-shell action. Plots of the free energy of the solutions (y axis) in function of the squashing parameter s (x axis), for the sample values of g = 0; 1; 2 and for various values of p are shown in gure 1 and gure 2. The horizontal line represent the range of existence of the given solution branch. 2 The Bolt+ solution is depicted in blue and it has the same ux as the NUT=Zp with 1 units of ux (dashed, orange). The Bolt branches are depicted in green, and have 2 the same ux as the NUT=Zp with 1 ux (dashed, red). For the ABJM case Bolt+ and Bolt satisfy the quantization condition for the same values of g; p.28 The last set of plots in gure 3 nally show that the Bolt+ solution is always favored with respect to the other branches. C Details of partition function calculations In this appendix we present more details on the computation of the twisted superpotential and Mg;p partition function for large N quiver gauge theories of the type discussed in section 2.3.1. These theories may have bifundamental chiral multiplets, fundamental chiral multiplets, and Chern-Simons terms, and we refer to the main text for a description of the constraints on the matter content. Here we recall that we take the large N ansatz: ua = va + iN 1=2ta : (C.1) 28This is not true for theories like V 5;2, see discussion in section 5.3. 0.2 0.4 s s Bolt+ BoltBolt+ Bolt Bolt+ BoltBolt+ Bolt HJEP05(218)6 0 I(s) 5 8 1 2 3 8 5 4 1 1 4 0 I(s) I(s) 1 8 0 1 8 3 4 5 4 g=0,p=1 p for the values p = 1; 2; 4; 6 respectively. In the picture the Bolt+, Bolt solutions and the mildly singular NUT=Zp plus 1 unit of magnetic ux are depicted. g = 1; p = 1; 4, and the last two plots refer to g = 2; p = 1 and g = 3; p = 4 respectively. s s 0.2 s s Bolt+ NUT Bolt+ BoltNUT+flux+ NUT+flux- 5 I(s) 5 4 1 4 0 17 12 5 4 12 1 4 I(s) I(s) 1 2 0 1 2 I(s) 5 4 7 4 Bolt+ BoltBolt+ Bolt2.5 2.0 8 Bolt+ see that the Bolt+ has always lower free energy, so dominates the ensemble. In the large N limit, ta becomes a continuous variable t, and we de ne: N a (t) = ta) ; normalized by R dt (t) = 1, and va ! v (t). C.1 Contributions to twisted superpotential these works. we have: We start with the contributions to the twisted superpotential. These were derived already in [9, 13], however we review them here for completeness, and to establish the relation of our notations. We also employ a slightly di erent argument in some places than used in Our starting point is the expression for the twisted superpotential at nite N in (2.20). Let us review the various ingredients in the U(N ) quiver gauge theories. For the CS terms, WCS = X 1 ;a 2 k ua (ua 1) : If we take the ansatz (C.1), this becomes: WCS = X k ;a N 2 ta2 + iN 1=2ta v a 1 2 1 2 (va 2 va ) : Imposing the constraint P k = 0, we may write this to leading order in N as: WCS = X iN 1=2k tava ;a iN 3=2 Z dt (t) X k tv (t) : Next we compute the contribution of bifundamental and adjoint elds. Here we will take a slightly di erent approach from [7], and instead use the following argument, modi ed (C.2) (C.3) (C.4) (C.5) HJEP05(218)6 from that of [15]. We will consider the large N limit of functions of the form: f (ua u^b) ; where ua and u^b are the eigenvalues for two of the U(N ) gauge groups (or possibly the same one, in the case of adjoint elds), and f is some function, which may depend on other parameters which we suppress. Let us rewrite this as: written as: N a6=b We expect the second term to be subleading in N , but we will return to its contribution below. Focusing on the rst term for now, and plugging in the ansatz (C.1), this can be X f (va v^b +iN 1=2(ta tb)) ! N 2 Z dtdt0 (t) (t0)f (v(t) v^(t0)+iN 1=2(t t0)) : (C.8) Now, expecting most of the contribution to come from the region near t = t0, we make a change of variables t0 ! t0). Then this becomes, to leading order: = N 1=2(t N 3=2 Z 1 d f ( v(t) + i ) ; where we de ned v(t) = v(t) v^(t), and we take the principle value to exclude the contribution from t = t0. In the present case, a single bifundamental chiral multiplet contributes: Wbif = W (ua N X a;b=1 N a6=b N a=1 f (ua u^b) = u^b) + X f (ua u^a) : (C.7) (C.6) (C.9) (C.10) (C.11) (C.12) where we have de ned the contribution of a chiral multiplet in the parity-preserving regularization (i.e., without the level 12 CS term) as:29 1 (2 i)2 Li2(e2 iu) + 4 [u]([u] 1 1) + 1 24 ; where, as in the main text, we de ne, for complex u: Here we have used the freedom to change branch, taking W ! W + nu + m, n; m 2 Z, to make the twisted superpotential periodic under u ! u + 1, which will be convenient below. as u ! i1, which along Re(u) 2 Z for Im(u) < 0. 29Here and below, we work on the principle branch of the polylogarithms, de ned so that Lis(e2 iu) ! 0 xes the branch in the upper half u-plane, and extending continuously along vertical lines to the lower half plane. Then the resulting functions are periodic under u ! u + 1, with branch cuts Inserting this in (C.9), we nd: Wbif = N 3=2 Z 1 Z duW (u) = 1 (2 i)3 Li3(e2 iu) + 1 24 d W ( v + m + i ) : 1) ; Using: Next we use: and cutting o the integral over at some max, we nd: 1) HJEP05(218)6 u=i max+ v+m u= i max+ v+m 1 1 1 (2 i)3 Li3(e2 iu) (2 i)3 Li3(e 2 iu) = 1 12 [u]([u] 1) ; as well as Li3(e2 iu) ! 0 as Im u ! 1, to nd: Wbif = iN 3=2 Z = iN 3=2 Z 1)(2u 1) : However, we see this diverges as we take the cuto max ! 1. To deal with this divergence, we make two observations. First, we only expect that certain choices of matter content and superpotential terms lead to theories with appropriate supergravity duals, with N 3=2 scaling of the degrees of freedom. Speci cally, as in section 2.3.1, we impose: The number of incoming and outgoing edges (bifundamental chirals) at any node (U(N ) gauge factor) in the quiver are equal. The sum of all masses of bifundamentals charged under a node is zero.30 Then when we sum over the contributions of all chiral multiplets, we nd the coe cient of m2ax is: 2 I v I v I + mI nI 1 2 2 I nI + ; 1 2 where I runs over the bifundamentals, with the Ith fundamental connecting the I th and I th gauge groups, and nI = v I v I + mI [v I v I + mI ] 2 Z, and we have used the 30More precisely, it need only be integer for the argument below to work, and in the context of the twisted superpotential the integer part of the masses is irrelevant, so imposing the sum is precisely zero is unnecessary. However, when computing the partition function below, the integer part will be important, and we will need to impose this condition. (C.13) (C.14) : (C.15) (C.16) (C.18) (C.19) conditions above to cancel the dependence on the v and mI . Then the r.h.s. , while not necessarily zero, is 12 times an integer. Then there is one more fact we must use, which is that the twisted superpotential is only de ned modulo changes of branch. For nite N , such changes of branch take the form: W ! W + naua + nimi + n; na; ni; n 2 Z : For in nite N , we choose to focus on changes of branch which preserve Weyl symmetry of the gauge group, and consider only those terms which appear at leading order in N . Then there are two basic choices, which lead to the following terms when we plug in the n^ ub )+nimi +n0 ! N 3=2 Z d X n dt t (t) X n N a=1 Z Z u^a) ! N dt (t)f ( v(t)) : For the case of a bifundamental chiral multiplet, this gives: N Z dt (t)W ( v +m) = N dt (t) 1 1 (2 i)2 Li2(e2 i( v+m))+ 4 ([ v +m])([ v +m] 1)+ (C.20) (C.21) (C.22) (C.23) (C.24) 1 24 (C.25) : where we must take P n = P n^ so that this term is of order N 3=2. Then we note that, after summing over the contributions from all chirals, the divergent terms from (C.17) are precisely of the form of the second change of branch above, so we may remove them. Then we nd the contribution of a bifundamental chiral multiplet is: Wbif = iN 3=2 Z dt (t)2g([ v + m]) : Note that while this is cubic in v, the rst consistency condition above implies that the cubic terms for each v cancels when we sum over all chirals, and so the nal functional is always a quadratic function of the v . Finally, let us return to the subleading contribution in (C.7): i +X(n n^ )v +nimi +n0 ! 2 max i max X(n n^ )v +nimi +n0 31Note the condition that the twisted superpotential be nite xes the choice of change of branch of the second type above, while the rst type changes the twisted superpotential by a nite piece, and so is not speci ed uniquely. Naively this is subleading in N , however, note that as v + m approaches an integer, the derivative of this expression with respect to v diverges. This means the subleading term will start to compete with the leading term, and we will have to take it into account below. Speci cally, let us write, for n 2 Z: v(t) + m = n + Ce 2 N1=2YI (t) : W ( v) Then one nds the following additional contribution to the derivative of W, to leading We will have to take this term into account when nding the extremal value of W. Finally, for an (anti-)fundamental chiral, we have the following contribution to the twisted superpotential: Wfun = W ( ua + m) = W ( (va + itaN 1=2) + m) N X a=1 Z 1 X 1 24 ! N dt (t)W ( (v + itN 1=2) + m) : W (u) ! [u]([u] 1) + (C.26) (C.27) (C.28) (C.29) (C.30) (C.31) ua!( ;v ) (C.32) N Z dt (t) 1 4 tjtjN + 2 jtjN 1=2 [ v + m] : If we impose that the total number of fundamentals and anti-fundamentals are equal, the O(N 2) term cancels, and this becomes, to leading order: Wfun = iN 3=2 Z 1 : 1 2 1 2 C.2 Contributions to log ZMg;p Next we consider the contributions to ZMg;p . Here the starting point is the nite N expression for the partition function as a sum over Bethe vacua, (2.26). Speci cally, we consider the contribution from a given Bethe vacuum, and take the eigenvalues ua in this vacuum to be given by the distribution (t); v (t), as above. Then we compute the functional: log ZMg;p [ ; v (t)] = p log F (ua; mi) + (g 1) log H(ua; mi) + si log i(ua; mi) First we have the contribution of the Chern-Simons terms. These contribute only through the bering operator: p i X X k ua 2 a 2 N 3=2 Z Next consider the contribution of a bifundamental chiral. This is given by: X p log F (ua a;b u^b + m) + ` log u^b + m) ; log F (u) = (u) = e2 iu) + iu + Li2(e2 iu) + u log(1 e2 iu) : 1 2 iu2 + and we write ` = s+(g 1)(r 1), where s is the net ux felt by the chiral from background vector multiplets, and r is its R-charge. We note that this can be written in terms of the twisted superpotential as: p log F (u) + ` log (u) = 2 i pW (u) (pu = 2 i pW ([u]) (p[u] + pn [u] 2 Z, and we have used the fact that W is periodic under u ! u + 1. X sI = 0 and I2 X(rI I2 (C.34) (C.35) (C.36) (C.37) (C.40) (C.41) p log F ( v + m + i ) + ` log ( v + m + i ) : 1 2 G`( v + m + i max) + 2 G`( v + m (p( v + m) `) m2ax + G`( v + m) ; (C.38) [u]2 2 + [u] 2 1 12 (C.39) Then we have: log ZMg;p;bif d 2 N 3=2 Z 2 N 3=2 Z One can compute this directly, but we can also use (C.36) to compute: where we de ned (here n = u [u]): G`(u) = 2pg([u]) (p[u] + np l)g0([u]) = p + (` np) [u]3 [u] 12 Although the vector multiplet does not contribute to the twisted superpotential, it does contribute to the partition function, appearing in the same way as an adjoint chiral multiplet of R-charge 2. Then we can obtain its contribution from the above formula by taking v + m ! 0 and ` = (g log ZMg;p;vec = dt (t)2(g 1) 1 2 2 max 1 12 We again nd a divergence that must be dealt with. This is resolved by imposing the condition: where the sum is taken over all bifundamental chirals charged under the th gauge group, and this must hold for all gauge groups, and adjoints are counted twice. These two conditions can be summarized by: X `I = 0 ; where in the sum we include the contribution of the vector multiplet, with `I = g term quadratic in 2 max vanishes. Thus, we are left with the following nite contribution log ZMg;p;bif = In addition, there is the diagonal contribution from the bifundamental: N a=1 X p log F (ua Z u^a + m) + ` log u^a + m) = N dt (t) (p( v + m) `) log(1 e2 i( v+m)) + : Here, since the contribution is subleading, we keep only those terms which can diverge. Speci cally, the log gets large in the tail region, where: and one nds a contribution: v(t) + m = n^ + Ce 2 N1=2Y (t) ; 2 N 3=2 X Z tails v(t)+m n^ dt(pn^ `) (t)Y (t) : For an (anti-)fundamental chiral, we have the following contribution to the twisted Wfun ! N dt (t) pF ( (v + itN 1=2) + m) + ` ( (v + itN 1=2) + m) : (C.47) superpotential: Z Using the limits: 2 N 3=2 Z dt (t)jtj 1 2 p( v + m) + ` : 1 2 >><F (u) ! (u) ! iu 2 i u2 + 12 as Imu ! 1 ; N Z dt (t) 2 ptjtjN + N 1=2( pjtj( v + m) t`) : Imposing the number of fundamentals and anti-fundamentals are equal, the O(N 2) term cancels and we nd: (C.42) (C.43) (C.44) (C.45) (C.46) (C.48) (C.49) (C.50) Finally, we also have the contribution from the Hessian: W 1 A Then, naively this has order N log N if all the components of the matrix stay nite, but near the tail regions there are divergences which cause it to contribute at leading order. As argued in [7], one nds: log H Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 440301 [arXiv:1608.02952] [INSPIRE]. 01 (2015) 100 [arXiv:1405.6046] [INSPIRE]. [1] V. Pestun et al., Localization techniques in quantum eld theories, J. Phys. 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Chiara Toldo, Brian Willett. Partition functions on 3d circle bundles and their gravity duals, Journal of High Energy Physics, 2018, 116, DOI: 10.1007/JHEP05(2018)116