DSF-3-2003 PreprinttypesetinJHEPstyle. -HYPERVERSION hep-th/0301171 Perturbative and non-perturbative aspects of = 1 pure super Yang-Mills theory from N 3 wrapped branes 0 0 2 n a J 2 W. M¨uck 2 Dipartimento di Scienze Fisiche, Universita` di Napoli “Federico II”, 1 and INFN, Sezione di Napoli, Via Cintia, 80125 Napoli, Italy v 1 E-mail: [email protected] 7 1 1 0 Abstract: The Maldacena-Nun˜ez solution is generalized to include a number of 3 0 integration constants, one of which controls the resolution of the singularity of / h the wrapped D5-brane background. Some features of the dual pure = 1 su- t N - per Yang-Mills (SYM) theory are calculated, amongst which the gluino condensate, p e the beta function of the gauge coupling and a brane probe potential, which is re- h : lated to the Veneziano-Yankielowicz effective potential. Each integration constant v i has a precise meaning in the dual SYM theory, e.g., the amount of non-perturbative X SYM physics captured by the gravity configuration is described by the singularity r a resolution parameter. Keywords: Brane Dynamics in Gauge Theories, D-branes, Nonperturbative Effects. Contents 1. Introduction and Summary 1 2. Review of the MN solution 4 3. Gluino condensate and effective superpotential 7 4. Brane probe analysis 8 5. Beta function 10 6. Probe potential 12 1. Introduction and Summary The possibility of studying super Yang-Mills(SYM) theories using their gravityduals has been a surprising manifestation of ’t Hooft’s old idea that gauge theories have a string theoretical microscopic origin [1]. After the success of the AdS/CFT corre- spondence [2, 3, 4] (see also the recent lecture notes [5, 6]) in describing the large N limit of (super) conformal SYM theories (e.g., = 4 SYM theory) by asymp- N totically AdS super gravity (SUGRA) backgrounds, a systematic formulation of a more general “gauge/gravity” duality describing also non-conformal SYM theories (or gauge theories with fewer supersymmetries) is still an outstanding problem. A huge amount of work has recently been devoted to the study of specific cases of this duality, and much relevant information of the SYM theories has been extracted from their SUGRA duals. One way to reduce supersymmetry is to consider SUGRA backgrounds generated by branes wrapping supersymmetric cycles [7, 8]. Thus, the identification of such a SUGRA dual of pure = 1 SYM theory by Maldacena and N Nun˜ez (MN) [9] was a major achievement and has spurred a lot of activity.1 For a list of references to other known cases of the gauge/gravity correspondence, we refer the reader to [11]. TheMNsolutionwasoriginallyfoundasaBPSmagneticmonopolebyChamsed- dine and Volkov [12, 13], and since MN’s work it has been the subject of a number of articles. A qualitative analysis of its implications for = 1 SYM theory has been N 1Another dual of pure = 1 SYM theory is the Klebanov-Strassler solution [10] describing N fractional 3-branes at the apex of a deformed conifold. 1 performed by Loewy and Sonnenschein [14]. The importance of the gaugino con- densate for de-singularizing the SUGRA solution was discussed by Apreda, Bigazzi, Cotrone, Petrini and Zaffaroni [15]. Building on this observation, Di Vecchia, Lerda and Merlatti [11, 16, 17] used a D5-brane embedded in the MN background and wrapping a certain two-cycle in order extract the running of the gauge coupling and the θ angle. In particular, they found agreement to leading order with the YM Novikov-Shifman-Vainshtain-Zakharov (NSVZ) beta function [18, 19] and evidence for fractional instantons. However, their specific way of wrapping the D5-branes led to a number of problems, most notably, their full beta function contains terms, which are logarithmic in the coupling and cannot be interpreted in field theory. Although the two-loop coefficient of the beta function can be adjusted by chang- ing the radial/energy relation [20], the problem of the logarithmic terms remained. Its resolution involves a suitable change of renormalization scheme, which could be translated to a change of the two-cycle around which the D5-brane is wrapped. In fact, Bertolini and Merlatti [21] solved this problem by wrapping the D5-brane on a different cycle, which had already been indicated in the paper by MN. Other aspects of the MN solution, which have been studied in the literature, include the resolution of the conifold singularity using black holes [22], a different approach to the radial/energy relation [23], non-supersymmetric deformations [24, 25, 26], its non-commutative extension [27] and its Penrose (pp-wave) limit [28, 29, 30]. In this paper, we shall again consider the MN solution and add a number of new ingredients to the discussion. Thus, we hope not only to contribute to a better understanding of this specific case of the gauge/gravity duality, but also to provide a guideline for analyzing other cases. Let us now summarize our paper and interpret the main results. One of the main ideas of the gauge/gravity duality is that there exists a dictionary relating the SUGRA fields to certain SYM operators. Taking the simplest approach, we generalize the MN solution by allowing for three integration constants, which have a precise physical meaning in the dual SYM theory. These are the following. First, we introduce a constant angle, ψ , by performing in the MN solution a global rotation 0 of the frame of the twisted three-sphere. In the SYM theory, ψ is identified with 0 the phase of the gluino condensate. Moreover, it also determines the vacuum angle, θ , which is the value of θ at the vacuum. For each θ , there are N inequivalent 0 YM 0 values of ψ giving rise to the N physically inequivalent vacua. Second, the dilaton 0 constant Φ relates the value of the dynamically generated SYM scale Λ to the string 0 parameters by the relation e2Φ0 = 2(2π)4(Λ/L)3/(N3g2), where L is the SUGRA s scale given by L−2 = Ng α′. This result stems from a direct calculation of the gluino s condensate. Finally, we include an integration constant c, which can be found in the solution byGubser, Tseytlin andVolkov [31], andwhich controlstheresolution ofthe singularity. In fact, for c > 0, the bulk geometries possess a bad naked singularity, 2 whereas the case c = 0 corresponds to the regular MN solution. The most natural interpretation of this constant is that it measures the amount of non-perturbative SYM physics captured by the dual SUGRA geometry. For c = , the SUGRA ∞ solution contains only perturbative effects, whereas the MN solution describes also the non-perturbative physics. This interpretation is supported by the running of the gauge coupling, the value of the gluino condensate and the behaviour of the probe brane potential. The (generalized) MN solution is reviewed in Sec. 2. The MN solution is of the form R1,3 M , where M is a (non-compact) Calabi- 6 6 × Yau manifold. Its geometry encodes a variety of SYM quantities [32, 33, 34, 35]. Of these, we calculate in Sec. 3 the gluino condensate and the effective superpotenial. The gluino condensate is a constant, which, for the regular MN solution, we identify with Λ3, where Λ is the dynamically generated scale of the SYM theory [36]. The remaining sections deal with the application of the brane probe technique, which is laid out in Sec. 4.2 We obtain the gauge coupling and θ angle and, by YM considering the terms of the probe action that are independent of the gauge fields, the probe potential. Our analysis generalizes the calculation of [21], in that we do not fix the angular coordinate of the embedding. Thus, we obtain the correct (and in general non-vanishing) θ . In contrast, the wrapping of [11] yields a θ differing YM YM from our result by a factor 1/2 (with a somewhat difficult interpretation of the chiral symmetry), while the result of [21] corresponds to the special case θ = 0. YM Sec. 5 focuses on the analysis of the gauge coupling obtained by the brane probe, but the breaking of the chiral symmetry by perturbative and non-perturbative effects (U(1) Z Z ) shall also be discussed. The main result will be the calculation 2N 2 → → of the beta function. In contrast to [21] we shall average the probe gauge coupling over all inequivalent vacua in order to confront its running with a perturbative field theory analysis. This will not only remove a spurious energy dependence from θ , YM but we will also be able to exactly re-write the beta function in terms of gauge theory quantities. It will turn out that the (singular) solution with c = correctly ∞ yields the complete perturbative running in terms the NSVZ beta function, and non- perturbativeeffectsappearintermsofa(c-dependent)valueofthegluinocondensate, which might be re-written in terms of fractional instanton contributions in the far UV. Thus, our beta function predicts new terms, which have not been obtained in the field theory. In fact, the pole of the NSVZ beta function seems to disappear in the complete theory, which is dual to the (regular) MN solution. For the singular solutions, the locationof the pole of the beta function coincides with the minimum of the effective superpotential from the Calabi-Yau geometry, which is thus interpreted as a vacuum-averaged result. 2The embedded object carrying the D5-brane action will be called a brane probe, although the usualzero-forceconditioncannotbe imposed. Instead,weshalltrytointerprettheprobepotential as an effective potential of the dual SYM theory. 3 Whilethegaugecoupling isrelevant forthemicroscopic(UV)degreesoffreedom, the probe potential provides a measure of the effective degrees of freedom around the vacuum. We shall analyze it in detail in Sec. 6. For the singular solutions, the probe brane will fall into the singularity, and the state of lowest energy appears to be chirally invariant. Fortunately, the perturbative analysis of the coupling breaks down before the brane reaches the singularity. In contrast, for the regular case, the minimum of the probe potential is not chirally invariant, and we obtain a good description of the behaviour of the composite operator λ2, where λ is the gluino field, around the vacuum. In the same region the probe potential will turn out to be closely related to the Veneziano-Yankielowicz effective potential [37]. 2. Review of the MN solution We consider the SUGRA solution corresponding to a system of N D5-branes. One way of finding it is by using d = 7 gauged SUGRA, which is obtained as a con- sistent truncation of the d = 10 SUGRA by compactification on an S3 [38, 39]. This is a natural setting to incorporate the twist condition necessary to retain some supersymmetry [7, 8]. The metric in the string frame is [9, 11] e2h 1 1 3 ds2 = eΦ dx2 + (dθ˜2 +sin2θ˜dφ˜2)+ dρ2 + (σa LAa)2 . (2.1) 10 1,3 L2 L2 L2 − " # a=1 X Here, θ˜ [0,π) and φ˜ [0,2π) parameterize a two-sphere, S2, which is part of the ∈ ∈ gauged SUGRA solution. The compactification three-sphere, S3, is parameterized by the left-invariant one-forms σa (a = 1,2,3) and is twisted by the SU(2) gauge field A = τaAa, where τa denote the Pauli matrices. For completeness, we give here the expressions for the σa, 1 σ1 = [cos(ψ ψ )dθ+sin(ψ ψ )sinθdφ] , (2.2) 0 0 2 − − 1 σ2 = [ sin(ψ ψ )dθ+cos(ψ ψ )sinθdφ] , (2.3) 0 0 2 − − − 1 σ3 = [dψ +cosθdφ] , (2.4) 2 which satisfy dσa = εabcσb σc . (2.5) − ∧ The angles are defined in the intervals φ [0,2π), θ [0,π) and ψ [0,4π). In ∈ ∈ ∈ addition to the solution in the literature, we have included the constant angle ψ , 0 which arises from a global U(1) gauge transformation. More precisely, one could consider the transformed gauge field A′ = g−1Ag +ig−1dg, where g SU(2). Then, ∈ 4 the part of the metric that belongs to the twisted 3-sphere can be written in the form (apart from the warp factor) 3 1 1 (σa LA′a)2 = Tr(σ LA′)2 = Tr(gσg−1 iLdgg−1 LA)2 . (2.6) − 2 − 2 − − a=1 X Hence, a global gauge transformation corresponds to a pure rotation of the frame on S3, while local transformations will, in addition to the rotation, contribute to the twisting. Our frame is obtained from the MN frame by the transformation g = exp( iψ τ3/2). 0 − The dilaton Φ and the prefactor e2h in the metric are functions of the radial variable ρ and are given by sinh(2ρ+c) e2Φ = e2Φ0f(c) , (2.7) 2eh 1 e2h = ρcoth(2ρ+c) [a(ρ)2 +1] , (2.8) − 4 2ρ a(ρ) = . (2.9) sinh(2ρ+c) In eqn. (2.7), f(c) is part of the overall constant, but we choose not to absorb it into Φ . The reason for this is that we want to consider c and Φ as independent 0 0 integration constants related to distict features of the dual gauge theory. We impose that for the MN solution (c = 0), f(0) = 1. Moreover, if the solution with c = ∞ and finite Φ is to make sense, we also need f(c) e−c for c . f(c) shall be 0 ∼ → ∞ determined in Sec. 5. The SU(2) gauge fields, Aa, are given by a(ρ) a(ρ) 1 A1 = dθ˜, A2 = sinθ˜dφ˜ , A3 = cosθ˜dφ˜ , (2.10) 2L 2L 2L with the field strengths Fa = dAa +LǫabcAb Ac, ∧ a˙(ρ) a˙(ρ) 1 F1 = dρ dθ˜, F2 = dρ sinθ˜dφ˜ , F3 = a(ρ)2 1 dθ˜ sinθ˜dφ˜ . 2L ∧ 2L ∧ 2L − ∧ (2.11) (cid:2) (cid:3) The dot denotes a derivative with respect to ρ. Furthermore, the solution contains a 2-form potential 1 C(2) = ψ sinθdθ dφ)+sinθ˜dθ˜ dφ˜ +cosθcosθ˜dφ dφ˜ 4L2 ∧ ∧ ∧ (2.12) a(ρh) (cid:16) (cid:17) i dθ˜ σ1 +sinθ˜dφ˜ σ2 , − 2L2 ∧ ∧ (cid:16) (cid:17) whose 3-form field strength F(3) = dC(2) is 3 2 1 F(3) = (σ1 LA1) (σ2 LA2) (σ3 LA3) Fa (σa LAa) . (2.13) L2 − ∧ − ∧ − − L ∧ − a=1 X 5 The metric (2.1) is real for ρ ρˆ, where ρˆ is defined by e2h(ρˆ) = 0. It is not ≥ difficult to show that ρˆis implicitly determined by the transcedental equation 2ρˆ[coth(2ρˆ+c)+1] = 1 , (2.14) which has a unique solution 0 ρˆ 1/4. The limiting cases are ρˆ= 0 for c = 0 and ≤ ≤ ρˆ= 1/4 for c = . ∞ The constant L is related to the number N of wrapped D5-branes by the usual charge quantization condition 1 F(3) = Nτ . (2.15) 2κ2 5 10 ZS3 Using κ = 8π7/2g α′2 and τ = (2π)−5g−1α′−3 one obtains [11] 10 s 5 s L−2 = Ng α′ . (2.16) s In addition to the fields listed so far, there is a non-zero 6-form potential, C(6), defined by dC(6) = ⋆F(3), where the Hodge dual is taken with respect to the string frame metric (2.1). Using eqns. (2.1), (2.10), (2.11) and (2.13), it is straightforward to obtain 1 1 dC(6) = v(4) 2e2Φ e2h + e−2h(1 a4) dρ dθ˜ sinθ˜dφ˜ −L2 ∧ 16 − ∧ ∧ (cid:26) (cid:20) (cid:21) 1 + d e2Φa˙ dθ˜ σ2 sinθ˜dφ˜ σ1 (a2 1)e2Φ−2hdρ (σ3 LA3) , 4 ∧ − ∧ − − ∧ − (cid:27) h (cid:16) (cid:17) i (2.17) where we have abbreviated v(4) = dx1 dx2 dx3 dx4. Thus, the 6-form potential ∧ ∧ ∧ is 1 C(6) = v(4) Ψ(ρ)dθ˜ sinθ˜dφ˜ −L2 ∧ ∧ 1 n + e2Φa˙ dθ˜ σ2 sinθ˜dφ˜ σ1 (a2 1)e2Φ−2hdρ (σ3 LA3) , 4 ∧ − ∧ − − ∧ − (cid:27) h (cid:16) (cid:17) i (2.18) where the function Ψ(ρ) satisfies 1 Ψ˙(ρ) = 2e2Φ e2h + e−2h(1 a4) . (2.19) 16 − (cid:20) (cid:21) We were not able to integrate this equation, except for the case c = , where ∞ Ψ = e2Φ(ρ 1/2)+ Ψ , with Ψ being an integration constant. We shall comment 0 0 − further on the function Ψ in Sec. 6. 6 3. Gluino condensate and effective superpotential Let us start our analysis by considering the geometry of the “internal” manifold. The bulk solution is—apart from the warp factor—of the form R1,3 M , where M is a 6 6 × K¨ahler manifold and geometrically encodes various aspects of the dual gauge theory. It encodes, first, the effective superpotential W F(3) Ω , (3.1) eff ∼ ∧ ZM6 where Ω is the holomorphic 3-form of the complex manifold M . Since M is not 6 6 compact, W explicitly depends on a cut-off. The holomorphic 3-form Ω is given by eff [40, 41]3 e2Φ Ω = 1 2 3 , (3.2) L3 E ∧E ∧E where the complex 1-forms i are defined by E 1 = (σ3 LA3) idρ , (3.3) E − − 2 = (σ1 LA1)+iXehsinθ˜dφ˜ iP(σ2 LA2) , (3.4) E − − − 3 = ehdθ˜+iX(σ2 LA2)+iPehsinθ˜dφ˜ , (3.5) E − and sinh(4ρ+2c) 4ρ 2eh P = − , X = (1 P2)1/2 = . (3.6) 2sinh2(2ρ+c) − sinh(2ρ+c) A straightforward calculation yields ρ0 16π3 W (ρ ) e2Φ0f(c) dρ[2ρcoth(2ρ+c) 1] . (3.7) eff 0 ∼ L5 − Z ρˆ The effective superpotential can be recast in terms of a pre-potential after intro- ducing a canonical basis of homology 3-cycles of M [33, 34], 6 W F(3) Ω Ω F(3) . (3.8) eff ∼ − ZA ZB ZA ZB In our case the compact 3-cycle is A = S3, and the non-compact 3-cycle B has a complicated form. Since we have already found W , we shall consider only the eff compact S3. The integral of F(3) over S3 is proportional to the number of D-branes, N, see eqn. (2.15), while the integral of Ω over S3 encodes the gluino condensate, 2π2 2π2 2πi λ2 = τ Ω = iτ e2ΦX = iτ e2Φ0f(c) . (3.9) |h ic| 5 5 L3 5 L3 ZS3 3We have chosen the phase so that Weff is real. 7 The constant τ is needed for dimensional reasons. Following our interpretation of 5 the integration constants, we re-write eqn. (3.9) as λ2 = λ2 f(c) = Λ3f(c) , (3.10) c 0 |h i | |h i | where we have used the convention f(0) = 1 and the fact that the regular solution is the true dual of = 1 SYM theory, i.e. λ2 = Λ3, where Λ is the dynamically 0 N h i generated mass scale. Thus, we identify the precise role of Φ relating Λ to the 0 SUGRA parameters by the relation 2(2π)4 Λ 3 e2Φ0 = π−1τ−1Λ3L3 = . (3.11) 5 N3g2 L s (cid:18) (cid:19) Obviously, for c = we have λ2 ∞ = 0, in agreement with the fact that a purely ∞ |h i | perturbative calculation fails to exhibit the gluino condensate. 4. Brane probe analysis Another way of obtaining information about the dual field theory is by using the probe technique. Let us consider a D5-brane embedded in the background (2.1). Its action is given by S = τ d6ξe−Φ det(G+2πα′F)+τ C(n) e2πα′F . (4.1) 5 5 − − ∧ ! Z p Z Xn 6-form ˆ We consider a D5-brane wrapping a two-sphere parameterized by two angles θ and ˆ φ. Expanding the Born-Infeld part of the action (4.1) and demanding that the non- abelian gauge fields F live only in the 4d part of the D5-branes, one finds 1 θ +2πn S = d4x V + FAFµν YM FA(⋆F )µν , (4.2) − 4g2 µν A − 32π2 µν A Z (cid:20) YM (cid:21) where the raising of the indices and the dual of the gauge fields are taken using the 4d Minkowski metric, and we have used the convention tr(TATB) = 1δAB for the 2 colour trace over the non-abelian generators. The potential V is given by V = τ dθˆdφˆ e−Φ√ G C(6) . (4.3) 5 − − 1234θˆφˆ Z (cid:16) (cid:17) For the gauge coupling, g , and the theta angle, θ , one obtains [11] YM YM 1 = 2π2α′2τ dθˆdφˆe−3Φ√ detG , (4.4) g2 5 − YM Z θ = (2π)4α′2τ dθˆdφˆC(2) mod 2π . (4.5) YM 5 θˆφˆ Z 8 In eqns. (4.2) and (4.5) we have used the fact that the physics of Yang-Mills theory is periodicinthethetaanglewithperiod2π,andweadopttheconventionθ [0,2π). YM ∈ The metric, the 2-form and the 6-form are induced from the respective bulk fields. In order to proceed we have to specify how the world volume coordinates of the D5-brane are related to the bulk coordinates of the MN solution. The flat 4d part is obvious, but the wrapped S2 needs some care. In order to use the coordinates ρ and ψ as parameters, we have to ensure that both of them are trivially fibred over the world volume [9]. This is done by imposing the four embedding conditions4 ˜ ˆ ˜ ˆ θ = θ = θ , φ = φ = φ , ψ = const , ρ = const . (4.6) Thus, we have dρ = 0 and σ3 LA3 = 0 on the world volume. Notice that the first − two conditions differ from the ones used in [11], where θ and φ are kept constant. Hence, the induced metric on the world volume of the D5-branes becomes 1 ds2 = eΦ dx2 + 4e2h +a(ρ)2 +1 2a(ρ)cos(ψ ψ ) dθˆ2 +sin2θˆdφˆ2 , 6 1,3 4L2 − − 0 (cid:26) (cid:27) (cid:2) (cid:3)(cid:16) (cid:17)(4.7) so that 1 1 √ G = e3Φsinθˆ ρcoth(2ρ+c) a(ρ)cos(ψ ψ ) . (4.8) − L2 − 2 − 0 (cid:20) (cid:21) Moreover, the induced 2- and 6-forms are 1 C(2) = sinθˆdθˆ dφˆ[ψ a(ρ)sin(ψ ψ )] , (4.9) 2L2 ∧ − − 0 1 1 C(6) = sinθˆ Ψ(ρ)+ e2Φa˙ cos(ψ ψ ) , (4.10) 1234θˆφˆ −L2 4 − 0 (cid:20) (cid:21) respectively. Inserting these equations into the general expressions for V, g and θ , we YM YM find 4πτ 1 V = 5 e2Φρcoth(2ρ+c)+Ψ(ρ) e2Φ(2a a˙)cos(ψ ψ ) , (4.11) L2 − 4 − − 0 (cid:20) (cid:21) as well as 1 N 1 = ρcoth(2ρ+c) a(ρ)cos(ψ ψ ) , (4.12) g2 4π2 − 2 − 0 YM (cid:20) (cid:21) θ = N [ψ a(ρ)sin(ψ ψ )] mod 2π . (4.13) YM 0 − − 4The alternative θ = θ˜is physically equivalent. − 9