arXiv:0712.4269 [hep-th] PreprinttypesetinJHEPstyle-HYPERVERSION Gravity Dual to Pure Confining Gauge 8 0 0 Theory 2 n a J 3 ] h Girma Hailu∗ t - p Newman Laboratory for Elementary Particle Physics e h Cornell University [ Ithaca, NY 14853 2 v 9 Abstract: 6 2 We find a dual gravity theory to pure confining N = 1 supersymmetric SU(N) 4 . gauge theory in four dimensions which has the correct gauge coupling running in 2 1 addition to reproducing the appropriate pattern of chiral symmetry breaking. It is 7 0 constructed in type IIB string theory on R1,3 × R1 × S2 × S3 background with N : v number of electric D5 and 2N number of magnetic D7-branes filling four dimensional i X spacetime and wrapping respectively two and four cycles. r a ∗ [email protected] Introduction.—The theory of quantum chromodynamics (QCD) of the strong nuclear interactions becomes highly nonperturbative and hard at low energies. The gauge/gravityduality[1,2,3]relatesagaugetheoryinstronglycouplednonperturba- tive region to a gravity theory in weakly coupled perturbative region and, therefore, provides the possibility for a calculable classical gravity description to low energy QCD. The first example of gauge/gravity duality in [1] involves conformal field the- ory with N = 4 supersymmetry. A gravity dual to pure N = 1 supersymmetric SU(N) gauge theory is highly desirable for several reasons. First, N = 1 super- symmetric SU(N) gauge theory exhibits phenomena such as confinement and chiral supersymmetry breaking and could serve as a laboratory to gain new insight into QCD. Second, there is a possibility that N = 1 supersymmetry may be part of nature at energies accessible in the coming generation of experiments at the Large Hadron Collider (LHC) and a gravity description is useful for calculating physical quantities such as glueball mass spectra in the N = 1 gauge theory. Third, if a suit- able supersymmetry breaking scheme which removes the gaugino in the pure N = 1 theory is found, it could be used to study the real world QCD itself. Fourth, the supergravity background has nonsingular geometry due to nonperturbative quan- tum effects which is useful for studying early universe cosmological scenarios and possibilities that a universe like ours could reside on the background. Indeed, therehasbeenextensiveefforttowardsfindingagravitydualtopureN = 1supersymmetricgaugetheory,[4,5,6]mostnotablylaidthefoundationalwork. The work in [5] produced a gravity dual to N = 1 supersymmetric SU(N +M)×SU(N) gauge theory with N number of D3 and M number of D5-branes on AdS × T1,1 5 conifold background involving novel cascading renormalization group flow towards pure N = 1 supersymmetric SU(M) gauge theory in the infrared and deformation of the conifold, but the gravity theory has constant dilaton and does not reproduce the gauge coupling running of a pure confining gauge theory. The work in [6] produced supergravity solutions with N number of NS5 or D5-branesinvolving running dilaton and appropriate pattern of chiral symmetry breaking, but it does not reproduce the gauge coupling running of pure N = 1 supersymmetric SU(N) gauge theory in four dimensions. Inthis note, we find adual gravity theoryto pure N = 1 supersymmetric SU(N) gauge theory in four dimensions which has the correct gauge coupling running and which reproduces the appropriate pattern of chiral symmetry breaking robustly. The important new ingredients which facilitate our construction are the gauge/gravity duality mapping with running dilaton and running axion obtained recently in [7] with magnetic D7 and Dirac 8-branes playing crucial role and the set of equations obtained in [8] which allows studying systematically type IIB flows with N = 1 supersymmetry. Our starting point is the running of the Yang-Mills coupling in the gauge theory and its mapping to the running of the dilaton in the gravity theory. We find a – 1 – supergravity dual in type IIB string theory on R1,3 × R1 × S2 × S3 background with N number of D5 and 2N number of D7-branes filling four dimensional (4-d) spacetime and wrapping respectively 2 and 4-cycles. The gauge theory is engineered by wrapping N electrically charged D5-branes on non-zero S2 cycle with S3 of zero- size in the base at the tip which leads to blown-down S2 and R-R 3-form F flux 3 through blown-up S3 after the familiar geometric transition which deforms the tip as in [5]. See [9] for a conifold transition on a setting in the topological A-model with S2 and S3 interchanging roles. The running of the gauge coupling leads to a running dilaton on the gravity side which is related to R-R F flux such that the 1 background follows the equations for the class of flows with imaginary self-dual 3- form flux in [8] and the supergravity solutions are read off from [7] with appropriate changes of variables. The runnings of the dilaton and the axion are due to magnetic coupling of the axion to D7-branes and Dirac 8-branes which emanate from the D7- branes. Demanding that the correct renormalization group flow of the gauge theory living on the N electrically charged D5-branes be reproduced leads to 2N magnetic D7-branes filling 4-d spacetime and wrapping 4-cycles at the ultraviolet edge of the background. The background with the F and the F fluxes induces 3-from NS-NS 1 3 H fluxandalso5-formR-RF fluxwhich canbeviewed ascomingfromthewrapped 3 5 D5-branes, which are fractional D3-branes, via backreaction NS-NS 2-form potential and the F flux. The axion potential C in the axion-dilaton coupling coefficient is 3 0 related to the Yang-Mills angle and preserves only a Z discrete symmetry in the 2N ultraviolet which matches with the anomaly-free R-symmetry in the gauge theory. The supergravity solutions preserve only a Z symmetry in the infrared and the 2 breaking of the Z symmetry down to Z gives N discrete vacua, reproducing the 2N 2 same pattern of symmetry breaking by gaugino condensation in the gauge theory. Gauge theory.—Consider N = 1supersymmetric pureSU(N) gaugetheory. The classical theory has global U(1) R-symmetry which is anomalous in the quantum theory. The anomaly-free quantum theory has a reduced Z discrete symmetry. 2N Gaugino condensation breaks the Z symmetry down to Z giving N number of 2N 2 discrete vacua. The low energy infrared dynamic of this theory at the scale Λ is described by the Veneziano-Yankielowicz superpotential [10], S 1 W = NS −NSlog( ), S = − TrWαW , (1) VY Λ3 32π2 α where S is the glueball superfield defined in terms of the gauge chiral superfield W α containing the gauge and the gaugino fields in the N = 1 vector multiplet. Ex- tremizing W with S gives the vacuum expectation value of the glueball superfield VY corresponding to the N vacua, hSi = Λ3e2πik/N, k = 1,2,··· ,N. (2) Let us define T = 8π2/g2, where g is the Yang-Mills coupling constant in the gauge theory. The quantum loop corrections to the running of the gauge coupling are – 2 – exhausted at one loop and with the exact β function we have dT = 3N, (3) dln(Λ/Λ ) c where we will take Λ to be the scale at the infrared end of the theory. c Dilaton.—The gauge coupling is related to the string coupling such that T = 2πe−Φ/g , where g is the bare string coupling and Φ is the dilaton. The scale of s s the gauge theory is mapped to the radius r such that Λ/Λ = r/r , where r is the c c c smallest value of r at the tip and ln(r/r ) ≥ 0. The appropriate running of the c dilaton then follows, d g N (e−Φ) = s , τ = 3ln(r/r ). (4) c dτ 2π Metric and fluxes.—The supergravity background we have here is similar to the one studied in [7, 11] with no regular D3-branes and the running of the dilaton now given by (4). Therefore, it is accommodated in the class of supergravity flows with imaginary self-dual 3-form flux given in [8] with running dilaton and running axion. The supergravity solutions can be read off from [7] with −s/(3+d) in [7] replaced by N/2π and the radial variable τ redefined as in (4). The difference in the signs is because the supergravity flow we have here is dual to a confining gauge theory and the magnitude of the dilaton increases towards the infrared which is opposite to the behavior of the dilaton in [7]. The metric and flux ansatz used in [7] was taken from [12, 13] with some modifications and additions to accommodate the runnings of the dilaton and the axion. We work in the string frame. For the metric, ds2 = e2A(τ)η dxµdxν +ds2(y), ds2(y) = δ GmGn, (5) µν 6 6 mn where xµ are the coordinates on R1,3 in 4-d, ym are the coordinates in the extra 6-d space, and τ is the radial variable defined in (4) and parameterizes R1 in R1,3 × R1 ×S2 ×S3. The Gm are real differential 1-forms which are expressed in terms of linear combinations of the coordinate 1-forms dyn on Y with coefficients which are functions of y, G1 = ex+2ge1, G2 = Aex+2ge2 +Bex−2gǫ˜2, G3 = ex−2gǫ˜1, G4 = Bex+2ge2 −Aex−2gǫ˜2, G5 = e−6p2−xdτ, G6 = e−6p2−xǫ˜3, (6) e = dθ , e = −sinθ dφ , ǫ˜ = ǫ −ae , ǫ˜ = ǫ −ae , 1 1 2 1 1 1 1 1 2 2 2 ǫ˜ = ǫ +cosθ dφ , ǫ = sinψsinθ dφ +cosψdθ , 3 3 1 1 1 2 2 2 ǫ = cosψsinθ dφ −sinψdθ , ǫ = dψ +cosθ dφ , (7) 2 2 2 2 3 2 2 with θ , θ ∈ [0,π], φ , φ ∈ [0,2π], and ψ ∈ [0,4π] parameterizing the base S2×S3 1 2 1 2 in R1,3×R1×S2×S3. The NS-NS H and R-R F , F , and F fluxes take the form 3 3 5 1 ′ ′ H = dτ ∧ h (ǫ ∧ǫ +e ∧e )+h (ǫ ∧e −ǫ ∧e ) +h ǫ˜ ∧(ǫ ∧e +ǫ ∧e ), (8) 3 1 1 2 1 2 2 1 2 2 1 2 3 1 1 2 2 (cid:16) (cid:17) – 3 – 1 F = − α′Ne−Φ ˜ǫ ∧(ǫ ∧ǫ +e ∧e −b(ǫ ∧e −ǫ ∧e ))+b′dτ∧(ǫ ∧e +ǫ ∧e ) , (9) 3 3 1 2 1 2 1 2 2 1 1 1 2 2 4 (cid:16) (cid:17) 1 N F = Ke ∧e ∧ǫ ∧ǫ ∧ǫ , K = − α′Ne−Φ(h +bh ), F = − ǫ˜ . (10) 5 1 2 1 2 3 1 2 1 3 2 2π The variables A, A, B, a x, g, p, h , h , b, and K in the metric and in the fluxes 1 2 above are functions of τ with A2+B2 = 1. The factor of e−Φ in the ansatz for F is 3 put in order to obtain dF = −F ∧H . 3 1 3 Solutions.—Now we want to read off the supergravity solutions for all the vari- ables in the metric and in the fluxes from [7] with the parameter s/(3+d) replaced by −N/2π. First, the running of the dilaton is given by 1 eΦ = , (11) e−Φc + gsNτ 2π where Φ is the value of the dilaton at τ = 0. The solutions involve the relations c a = −sech τ, b = −τ csch τ, eg = tanhτ, A = tanhτ, B = sech τ, (12) 1 1 h = α′g N(1−τ cothτ)csch τ, h = h coshτ, K = − e−Φα′N(h +bh ). (13) 2 s 1 2 1 2 4 2 The expressions (12)-(13) are read off from [12] and are the same as in the solutions in [5] written in the form given in [12] with nonzero Φ. The F flux is 1 N N F1 = − e6p2+xG6 = − ˜ǫ3. (14) 2π 2π Defining v = e6p+2x, u = e2x and h = e−4A, the equations for v, u and h are 3g N v′ +(2cothτ + s eΦ)v−3 = 0, (15) 2π d u 2 g N ln( ) = − s eΦ, (16) dτ h v π 1 h g N h′ + g eΦK − s eΦh = 0, (17) s 4 u 2π with Φ given by (11) and K given in (13). A consistent set of solutions requires the boundary conditions v(0) = 0, u(0) = 0, and h(0) = h . Equation (15) is easily 0 solved for v, (16) is then solved for u/h and the result is used for h/u in (17) to write integral solution for h which is then used with the expression for u/h to find u. The values of u and v increase towards the ultraviolet as τ increases. The value of h also starts increasing as τ increases from τ = 0 because of the sign of the third term in (17) which comes from the running of the dilaton in the asymptotically-free gauge theory here and is opposite to that in [7] where h decreases. – 4 – D7 and Dirac 8-branes.—As it is shown in [7], the runnings of the axion and the dilaton are due to magnetic D7-branes filling 4-d spacetime and wrapping the 4-cycle 1 ω = sinθ sinθ dθ ∧dφ ∧dθ ∧dφ = G1 ∧G2 ∧G3 ∧G4. (18) 4 1 2 1 1 2 2 u The electric dual to a D7-brane is D(-1)-brane which is a point in spacetime. The magnetic D7-branes give the F flux through ǫ˜ and the F flux is not closed because 1 3 1 of Dirac 8-branes which emanate from the D7-branes, in the same way a Dirac string emanating from a magnetic monopole in 4-d carries magnetic flux along the string. In particular, it is shown in [7] that the F flux satisfies 1 F −⋆F¯ = dC , (19) 1 9 0 ¯ where ⋆F accounts for the flux due to the Dirac 8-branes which makes dF 6= 0. 9 1 The magnetic flux F = dC = −⋆F , where C is the D7-branes potential, is closed 9 8 1 8 as a result of contributions to the flux from the magnetic charge of the D7-branes and from the Dirac 8-branes, F = −⋆dC +F¯ . The axion C and the magnitude 9 0 9 0 of the magnetic D7-branes charge Q˜ are obtained using the expression for C given 7 0 in [7] with s/(3+d) → −N/2π, N 0 C = − ψ, Q˜ = dC = 2N. (20) 0 2π 7 Z 0 4π Therefore, we have 2N number of D7-branes. Bottom of background.—Now we explore the geometry at the bottom of the background. Let us expand the variables in the metric given by (12) and solving (15)-(17) to leading order in τ near τ = 0, eg = τ +O(τ3), a = −1+O(τ3), A = τ +O(τ3), B = 1+O(τ2), h = h +O(τ), v = τ +O(τ2), u = e2Φch τ2 +O(τ3), (21) 0 0 where h = h(0). Therefore, near τ = 0, we have ex+g = u1/2eg = eΦch1/2τ2+O(τ3), 0 0 ex−g = u1/2/eg = eΦch1/2 + O(τ), e−6p−x = u1/2/v = eΦch1/2 + O(τ). The 1-forms 0 0 in the metric given by (5) near τ = 0 have the forms G1 ∼ (e2Φch )1/4τe , G2 ∼ 0 1 (e2Φch )1/4˜ǫ , G3 ∼ (e2Φch )1/4ǫ˜ , G4 ∼ −(e2Φch )1/4τǫ , G5 ∼ (e2Φch )1/4dτ, and 0 2 0 1 0 2 0 G6 ∼ (e2Φch )1/4ǫ˜ . The metric near τ = 0 is then 0 3 ds2 ∼ h−1/2η dxµdxν +eΦch1/2 dτ2 +ǫ˜2 +ǫ˜2 +ǫ˜2 +τ2(e2 +ǫ2) , (22) 0 µν 0 1 2 3 1 2 (cid:0) (cid:1) where the ǫ˜ are given by (7) with a ∼ −1. Therefore, the geometry at τ = 0 is i R1,3×S3 withtheradiusofS3 afunctionofthemagnitudeofthevacuumexpectation value of the glueball superfield S at τ = 0 which is simply the ’t Hooft coupling [14] eΦcg N and confinement via gaugino condensation is the source for the deformation s – 5 – of the tip as established in [5]. The boundary value h in solving (15)-(17) is then 0 related to theradiusofS3 atτ = 0 and, therefore, afunction ofthe’t Hooftcoupling. Large N and large ’t Hooft coupling.—Let us check the conditions on the param- eters in the theory for the supergravity description to be good. We need a large value of N so that only planar Feynman graphs survive and a large ’t Hooft coupling. The ’t Hooft coupling is related to the magnitude of the vacuum expectation value of the glueball superfield and the size of the holes in ’t Hooft’s ribbon graphs have large size and can define Riemann surface for a string worldsheet for large ’t Hooft cou- pling if filled by D-brane disks [15]. On the other hand, for small ’t Hooft coupling, the gauge theory has a good perturbative description. Let us see the implication of the large ’t Hooft coupling constraint in our case, particulary now that we have a dilaton whose magnitude decreases as τ increases. The running of the dilaton is given by (11). In order for eΦg N >> 1, we need eΦcg N(1− τ ) >> 1. Therefore, s s 2π the region in which the supergravity description is good is bounded by r/r < e2π/3 c and presents a robust region for large eΦcg N. We need the gauge/gravity flow to s be within 0 ≤ τ < 2π and we take the ultraviolet cutoff of the gauge theory at the scale Λ = Λ e2π/3 or, equivalently, the radius at the edge of the background is 0 c r = r e2π/3. The background is noncompact and can be thought of as glued to a 0 c larger background at r = r . 0 Chiral symmetry breaking.—Nextletusseehowthechiralsymmetry inthegauge theory shows up in the gravity theory. The gauge theory at the classical level has global U(1) R-symmetry which corresponds in the gravity theory to the symmetry ψ → ψ+c, where c is constant, as shown in [5, 6]. The anomaly-free quantum theory has only the Z discrete symmetry. Now we have the C potential which is nonzero 2N 0 in the ultraviolet and which depends on ψ and we want to see that it reproduces the same symmetry. The axion-dilaton coupling coefficient is given by i i N τ = e−Φ +C = e−Φ − ψ, (23) ad 0 g g 2π s s where we have used (20) for C and the subscript in τ is for axion-dilaton in order 0 ad to avoid confusion in notation with the radial variable τ. Note that the second term in (23) corresponds to a Yang-Mills angle of Θ = −Nψ and clearly shows that ψ → ψ + c, where ψ = ψ + 4π, is anomalous U(1) symmetry and Θ is left invariant under ψ → ψ + 4πn which gives Θ → Θ− 2πn. Therefore, τ preserves 2N ad a reduced Z symmetry with ψ → ψ + 4πn, where 1 ≤ n ≤ 2N. This discrete 2N 2N symmetry in τ corresponds to the symmetry in the locations on ψ, where dψ is ad related to G6 as given in (6) and (7), at which the D7-branes in the ultraviolet edge wrap the ω cycle. Both the F flux and the F flux in the ultraviolet contain dψ 4 1 3 and are single-valued. The fluxes in the supergravity solutions have non-vanishing components which explicitly contain sinψ and cosψ in the infrared as we see in the expressions (8) and (9) together with (12) and (13) and preserve only a Z symmetry 2 – 6 – corresponding to ψ → ψ +2π. Therefore, the Z symmetry is broken down to Z 2N 2 by the solutions in the gravity theory and gives the same N number of discrete vacua in the infrared as in the gauge theory. Conclusions.—The final picture we have is that the electric N = 1 supersym- metric SU(N) gauge theory lives on the electrically charged N number of D5-branes filling 4-d spacetime and wrapping S2 with vanishing S3 at the infrared end before geometric transition. The supergravity solutions involve the background with fluxes after the transition with S2 blown-down and S3 of finite size at the tip which gives the familiar gravitational description to confinement via gaugino condensation in the gauge theory. The 2N number of magnetic D7-branes fill up 4-d spacetime and wrap 4-cycles at the ultraviolet edge with invisible Dirac 8-branes filling 4-d spacetime and emanating from the D7-branes and the F flux through G6 which is related to 1 the running of the dilaton. We also have the backreaction NS-NS H flux and the 3 F flux effectively coming from the wrapped D5 fractional D3-branes. The quan- 5 tum theory has Z discrete symmetry in the ultraviolet and arises robustly from 2N our solution for the axion potential. The Z symmetry is broken down to Z by 2N 2 the supergravity solutions in the infrared giving N vacua as in the gauge theory. It is satisfying that the C potential which comes from the F flux which is a crucial 0 1 component of our construction obtained using the equations for type IIB flows with N = 1 supersymmetry we obtained in [8] and the gauge/gravity duality mapping with running dilaton and running axion we obtained in [7] has provided a consistent picture. Most importantly, the renormalization group flow of the gauge theory is reproduced in the gravity theory. What can we say about the gauge theory which lives on the 2N magnetic D7-branes? Because the number of D7-branes is the same as the order in the Z discrete symmetry in the ultraviolet, it is convenient to wrap 2N each one of the D7-branes over a 4-cycle at each one point ψ = 4πn, 1 ≤ n ≤ 2N, 2N in the ultraviolet edge of the background, and the total charge is the same. We then have U(1)2N gauge theory living on the D7-branes which is infrared-free and can be simply ignored. The supergravity solutions we have presented could be used to study additional gravitational descriptions to features in the gauge theory and the metric could be used to calculate physical quantities such as glueball mass spectra. 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