Holographic RG flows in N = 3 Chern-Simons-Matter theory from N = 3 4D gauged supergravity Parinya Karndumri∗ String Theory and Supergravity Group, Department of Physics, Faculty of Science, 6 Chulalongkorn University, 254 Phayathai Road, 1 0 Pathumwan, Bangkok 10330, Thailand 2 l (Dated: August 2, 2016) u J Abstract 0 3 We study various supersymmetric RG flows of N = 3 Chern-Simons-Matter theory in three ] h dimensionsbyusingfour-dimensionalN = 3gaugedsupergravitycoupledtoeightvectormultiplets t - p with SO(3) SU(3) gauge group. The AdS critical point preserving the full SO(3) SU(3) 4 e × × h provides a gravity dual of N = 3 superconformal field theory with flavor symmetry SU(3). We [ 4 study the scalar potential and identify a new supersymmetric AdS critical point preserving the 4 v 3 full N = 3 supersymmetry and unbroken SO(3) U(1) symmetry. An analytic RG flow solution 0 × 7 interpolating between SO(3) SU(3) and SO(3) U(1) critical points is explicitly given. We then 5 × × 0 investigate possible RG flows from these AdS critical points to non-conformal field theories in the . 4 1 0 IR. All of the singularities appearing in the IR turn out to be physically acceptable. Furthermore, 6 1 we look for supersymmetric solutions of the form AdS Σ with Σ being a two-sphere or a two- 2 2 2 : × v Xi dimensional hyperbolic space and find a number of AdS2 geometries preserving four supercharges r with SO(2) SO(2) SO(2) and SO(2) SO(2) symmetries. a × × × ∗ REVTeX Support: [email protected] 1 I. INTRODUCTION AdS /CFT correspondence is interesting in many aspects such as its applications in 4 3 the study of M2-brane dynamics and in the holographic dual of condensed matter physics systems. There are a few examples of supersymmetric AdS backgrounds with known M- 4 theoryorigins. Apartfromthemaximallysupersymmetic N = 8AdS S7 compactification, 4 × there is an AdS background with N = 3 supersymmetry arising from a compactification 4 of M-theory on a tri-sasakian manifold N010 [1]. This is a unique solution for 2 < N < 8 supersymmetry. The spectrum of the former example has been studied in [2] and the mass- less modes can be described by the maximally SO(8) gauged supergravity constructed in [3]. The lowest modes of the latter are on the other hand encompassed in the gauged N = 3 supergravity coupled to eight vector multiplets constructed in [4], see also [5, 6]. The holo- graphic study of this background within the framework of N = 8 gauged supergravity and eleven-dimensional supergravity has appeared in many previous works, see for example [7– 9]. The analysis of the complete spectrum of the Kaluza-Klein reduction of M-theory on AdS N010 has been carried out in [10], see also [11]. It has been argued that the com- 4 × pactification can be described by a four-dimensional effective theory in the form of N = 3 supergravity coupled to eight vector multiplets with SO(3) SU(3) gauge group. From × the AdS/CFT point of view, the SO(3) and SU(3) factors correspond respectively to the SO(3) R-symmetry and SU(3) flavor symmetry of the dual N = 3 superconformal field theory (SCFT) in three dimensions with the superconformal group OSp(3 4) SU(3). The | × structure of N = 3 multiplets and some properties of the dual SCFT have been studied in [12–17]. Furthermore, a generalization to quiver gauge theories has been considered more recently in [18–23]. In the present work, we are interested in exploring possible supersymmetric solutions within four-dimensional N = 3 gauged supergravity. The N = 3 gauged supergravity cou- pled to n vector multiplets has been constructed in [4]. The theory contains 6n scalar fields parametrizing the SU(3,n)/SU(3) SU(n) U(1) coset manifold. We will focus on the × × case of n = 8 which, together with the other three vectors from the supergravity multiplet, gives rise to eleven vector fields corresponding to a gauging of the SO(3) SU(3) subgroup × of the global symmetry group SU(3,8). The maximally supersymmetric AdS critical point 4 of the resulting gauged supergravity with all scalars vanishing is expected to describe the 2 AdS N010 background of eleven-dimensional supergravity. 4 × We will look for other possible supersymmetric AdS critical points. According to the 4 standard dictionary of the AdS/CFT correspondence, these should be dual to other confor- mal fixed points in the IR of the UV N = 3 SCFT with the SU(3) flavor symmetry. We find that indeed there exists a non-trivial supersymmetric AdS critical point with SO(3) U(1) 4 × symmetry and unbroken N = 3 supersymmetry. We will also investigate holographic RG flows from the UV N = 3 SCFT to non-conformal field theories by looking for domain wall solutions interpolating between the AdS critical points and some singular domain wall ge- 4 ometries in the IR. Finally, wewilllookforsupersymmetric AdS Σ solutionswithΣ beingaRiemannsur- 2 2 2 × face. Like the higher-dimensional solutions, these solutions should be interpreted as twisted compactifications of the N = 3 SCFTs in three dimensions to one dimensional space-time. These results could be interesting both in the holography of three-dimensional SCFTs and in the context of AdS /CFT correspondence which plays an important role in black hole 2 1 physics, see for example [24] and [25]. Along this line, the topologically twisted indices for these theories on S2 have been computed in [26, 27]. These results can be used to find the microscopic entropy of AdS black holes by following the approach of [28]. 4 The paper is organized as follow. In section II, we review N = 3 gauged supergravity in four dimensions coupled to eight vector multiplets. In section III, we will give an explicit parametrization of SU(3,8)/SU(3) SU(8) U(1) coset and study the scalar potential × × for the SO(3) singlet scalars and identify possible supersymmetric vacua. An analytic diag RG flow from the UV SO(3) SU(3) SCFT to a new IR fixed point with residual sym- × metry SO(3) U(1) is also given. We then move to possible supersymmetric RG flows diag × to non-conformal field theories in section IV. Supersymetric AdS backgrounds obtained 2 from twisted compactifications of AdS on a Riemann surface are given in section V. Some 4 conclusions and comments on the results reported in this paper are presented in section VI. II. N = 3 GAUGED SUPERGRAVITY COUPLED TO VECTOR MULTIPLETS Inorder to fix the notationand describe the relevant framework fromwhich all the results are obtained, we will give a brief description of N = 3 gauged supergravity coupled to n vector multiplets and finally restrict ourselves to the case of n = 8. The theory has been constructed in[4]byusing thegeometricgroupmanifoldapproach. Forthepresent work, the 3 space-time bosonic Lagrangian and supersymmetry transformations of fermionic component fields are sufficient. Therefore, we will focus only on these parts. The interested reader can find a more detailed construction in [4]. In four dimensions, the matter fields allowed in N = 3 supersymmetry are given by the fields in a vector multiplet with the following field content (A ,λ ,λ,z ). µ A A Indices A,B,... = 1,2,3 denote the fundamental representation of the SU(3) part of R the full SU(3) U(1) R-symmetry. Each vector multiplet contains a vector field A , R R µ × four spinor fields λ and λ which are respectively singlet and triplet of SU(3) , and three A R complex scalars z in the fundamental of SU(3) . For n vector multiplets, we use indices A R i,j,... = 1,...,n to label each of them. Space-time and tangent space indices will be denoted by µ,ν,... and a,b,..., respectively. In contrast to the construction in [4], we will use the metric signature ( +++) throughout this paper. − The N = 3 supergravity multiplet consists of the following fields (ea,ψ ,A ,χ). µ µA µA ea is the usual graviton, and ψ are three gravitini. The gravity multiplet also contains µ µA three vector fields A and an SU(3) singlet spinor field χ. It should be noted that the µA R fermions are subject to the chirality projection conditions ψ = γ ψ , χ = γ χ, λ = γ λ , λ = γ λ. (1) µA 5 µA 5 A 5 A 5 − These also imply ψA = γ ψA and λA = γ λA. µ − 5 µ − 5 With n vector multiplets, there are 3n complex scalar fields z i living in the coset space A SU(3,n)/SU(3) SU(n) U(1). These scalars are conveniently parametrized by the coset × × representative L(z) Σ. From now on, indices Λ,Σ,... will take the values 1,...,n + 3. Λ The coset representative transforms under the global G = SU(3,n) and the local H = SU(3) SU(n) U(1) symmetries by a left and right multiplications, respectively. It is × × convenient to split the index corresponding to H transformation as Σ = (A,i), so we can write L Σ = (L A,L i). Λ Λ Λ Together with three vector fields fromthe gravity multiplet, there are n+3 vectors which, accompanying by their magnetic dual, transform as the fundamental representation n+3 of the global symmetry SU(3,n). These vector fields will be grouped together by a single 4 notation A = (A ,A ). From the result of [4], after gauging, a particular subgroup of Λ A i SO(3,n) SU(3,n) becomes a local symmetry. The corresponding non-abelian gauge field ⊂ strengths are given by F = dA +f ΣΓA A (2) Λ Λ Λ Σ ∧ Γ where f Γ denote the structure constants of the gauge group. The gauge generators T ΛΣ Λ satisfy [T ,T ] = f ΓT . (3) Λ Σ ΛΣ Γ Indices on f Γ are raised and lowered by the SU(3,n) invariant tensor ΛΣ J = JΛΣ = (δ , δ ) (4) ΛΣ AB ij − which will become the Killing form of the gauge group in the presence of gauging. As pointed out in [4], one of the possible gauge groups takes the form of SO(3) H with n × SO(3) SU(3) and H being an n-dimensional subgroup of SO(n) SU(n). In this case, n ⊂ ⊂ only electric vector fields participate in the gauging. As a general requirement, gaugings consistent with supersymmetry impose the condition that f obtained from the gauge ΛΣΓ structure constants via fΛΣΓ = fΛΣΓ′JΓ′Γ are totally antisymmetric. In the present paper, we are interested only in this compact gauge group with a particular choice of H = SU(3) 8 with f Γ = (g ǫ ,g f ). This choice clearly satisfies the consistency condition. f ΛΣ 1 ABC 2 ijk ijk denote the SU(3) structure constants while g and g are SO(3) SU(3) gauge couplings. 1 2 × The independent, non-vanishing, components of f can be explicitly written as ijk 1 f = 1, f = f = f = f = , 123 147 246 257 345 2 1 √3 f = f = , f = f = . (5) 156 367 458 678 −2 2 Other possible gauge groups will be explored in the forthcoming paper [29]. The bosonic Lagrangian of the resulting gauged supergravity can be written as 1 1 e−1 = R PiAPµ aΛΣF+ F+µν a¯ΛΣF− F−µν L 4 − 2 µ Ai − Λµν Σ − Λµν Σ i e−1ǫµνρσ(aΛΣF+ a¯ΛΣF− )F V . (6) −2 Λµν − Λµν Σρσ − We have translated the first order Lagrangian in the differential form language given in [4] to the usual space-time Lagrangian. In addition, we have multiplied the whole Lagrangian by a factor of 3. This results in a factor of 3 in the scalar potential compared to that given in [4]. 5 Before giving the definitions of all quantities appearing in the above Lagrangian, we will present the fermionic supersymmetry transformations read off from the rheonomic parametrization of the fermionic curvatures as follow δψ = D ǫ 2ǫ GB γνǫC +S γ ǫB, (7) µA µ A − ABC µν AB µ 1 δχ = GA γµνǫ + Aǫ , (8) −2 µν A U A δλ = P Aγµǫ + ǫA, (9) i − iµ A NiA δλ = P Bγµǫ ǫC G γµνǫ + Bǫ . (10) iA − iµ ABC − iµν A MiA B From the coset representative, we can define the Mourer-Cartan one-form Ω Π = (L−1) ΣdL Π +(L−1) Σf ΩΓA L Π. (11) Λ Λ Σ Λ Σ Ω Γ The inverse of L Σ is related to the coset representative via the following relation Λ (L−1) Σ = J JΣ∆(L Π)∗. (12) Λ ΛΠ ∆ The component ΩA = (Ω i)∗ gives the vielbein P A of the SU(3,n)/SU(3) SU(n) U(1) i A i × × coset. Other components give the composite connections (Q B,Qj,Q) for SU(3) SU(n) A i × × U(1) symmetry Ω B = Q B nδBQ, Ωj = Qj +3δjQ. (13) A A − A i i i It should be noted that Q A = Qi = 0. A i The covariant derivative for ǫ is defined by A 1 1 Dǫ = dǫ + ωabγ ǫ +Q Bǫ + nQ. (14) A A 4 ab A A B 2 The scalar matrices S , A, and B are given in terms of the “boosted structure AB U NiA MiA constants” CΛ as follow ΠΓ 1 S = ǫ C PQ +ǫ C MC AB 4 BPQ A ABC M 1 (cid:16) (cid:17) = C PQǫ +C PQǫ , 8 A BPQ B APQ (cid:16)1 1 (cid:17) A = C MA, = ǫ C PQ, U −4 M NiA −2 APQ i 1 B = (δBC M 2C B) (15) MiA 2 A iM − iA where CΛΠΓ = LΛ′Λ(L−1)ΠΠ′(L−1)ΓΓ′fΠ′Γ′Λ′ and CΛΠΓ = JΛΛ′JΠΠ′JΓΓ′(CΛΠ′ ′Γ′)∗ (16) 6 With all these definitions, the scalar potential can be written as 2 1 1 V = 2S SCM + A + iA + iB A − AC 3UAU 6NiAN 6M AMiB 1 1 1 = C B 2 + C PQ 2 C PQ 2 C 2 (17) 8| iA | 8| i | − 4 | A | −| P| (cid:16) (cid:17) with C = C M. P − PM We now come to the gauge fields. The self-dual and antiself-dual field strengths are defined by i F± = F ǫ Fcd (18) Λab Λab ∓ 2 abcd Λ with 1ǫ F±cd = iF± and F± = (F∓ )∗. The explicit form of the symmetric matrix 2 abcd Λ ± Λab Λab Λab a in term of the coset representative is quite involved. We will not repeat it here, but the ΛΣ interested reader can find a detailed discussion in the appendix of [4]. Finally, the field strengths appearing in the supersymmetry transformations are given in terms of F± by Λµν 1 1 Gi = Mij(L−1) ΛF− , GA = MAB(L−1) ΛF+ (19) µν −2 j Λµν µν 2 B Λµν where Mij and MAB are respectively inverse matrices of M = (L−1) Λ(L−1) ΠJ and M = (L−1) Λ(L−1) ΠJ (20) ij i j ΛΠ AB A B ΛΠ In subsequent sections, we will study supersymmetric solutions to this gauged supergrav- ity with SO(3) SU(3) gauge group. × III. FLOWS TO SO(3) U(1) IR FIXED POINT WITH N = 3 SUPERSYMME- diag × TRY We now consider the case of n = 8 vector multiplets and SO(3) SU(3) gauge group. × There are 48 scalars transforming in (3,8¯) + (3¯,8) representation of the local symmetry SU(3) SU(8). It is efficient and more convenient to study the scalar potential on a partic- × ular submanifold of the full SU(3,8)/SU(3) SU(8) U(1) coset space. This submanifold × × consists of all scalars which are singlets under a particular subgroup of the full gauge group SO(3) SU(3). All vacua found on this submanifold are guaranteed to be vacua on the full × scalar manifold by a simple group theory argument [30]. 7 A. Supersymmetric AdS critical points 4 In term of the dual N = 3 SCFT, the SO(3) part of the full gauge group corresponds to the R-symmetry of N = 3 supersymmetry in three dimensions while the SU(3) part plays the role of the global symmetry. There are no singlet scalars under the SO(3) R-symmetry. In order to have SO(3)symmetry, we then consider scalars invariant under a diagonal SO(3) subgroup of SO(3) SO(3) SO(3) SU(3). × ⊂ × Before going to the detail of an explicit parametrization, we first introduce an element of 11 11 matrices × (e ) = δ δ . (21) ΛΣ ΠΓ ΛΠ ΣΓ The SO(3) SU(3) gauge generators can be obtained from the structure constant (T ) Γ = × Λ Π f Γ. Accordingly, the SO(3) part is generated by (T(1)) Γ = f Γ, A = 1,2,3, and the ΛΠ A Π AΠ SU(3) generators are given by (T(2)) Γ = f Γ, i = 1,...,8. The SO(3) is then i Π i+3,Π diag generated by (T(1)) Γ +(T(2)) Γ. A Π A Π Under SU(3) SO(3) U(1), we have the branching → × 8 = 3 +1 +2 +2 . (22) 0 0 3 −3 This implies that the 48 scalars transform under SO(3) U(1) as diag × 2 [3 (3 +1 +2 +2 )] = 2 (1 +3 +5 +2 +4 +2 +4 ). (23) 0 0 0 3 −3 0 0 0 3 3 −3 −3 × × × A factor of 2 comes from the fact that both (3,8¯) and (3¯,8) of SU(3) SU(8) become (3,8) × under SO(3) SU(3). We see that there are two SO(3) singlets. These correspond to diag × the SU(3,8) non-compact generators ˆ Y = e +e +e +e +e +e , 1 14 41 25 52 36 63 Yˆ = ie +ie ie +ie ie +ie . (24) 2 14 41 25 52 36 63 − − − These two generators are non-compact generators of SL(2,R) SU(3,8) commuting with ⊂ SO(3) . The SO(2) compact generator of this SL(2,R) is given by diag J = diag(2iδAB, 2iδi+3,j+3,0,0,0,0,0), i,j = 1,2,3. (25) − From (23), it should be noted that the two singlets are uncharged under the U(1) factor from SU(3). Therefore, the full symmetry of Yˆ is in fact SO(3) U(1). 1,2 diag × 8 By using an Euler angle parametrization of SL(2,R)/SO(2) SO(2,1)/SO(2) ∼ ∼ SU(1,1)/U(1), we parametrize the coset representative by L = eϕJeλYˆ1e−ϕJ . (26) The resulting scalar potential can be written as 3 V = e−6λ (1+e4λ) (1+e2λ)4g2 +(e2λ 1)4g2 −64 1 − 2 +2(e4λ (cid:2)1)3cos(4ϕ(cid:2))g g . (cid:3) (27) 1 2 − (cid:3) The above potential admits two supersymmetric AdS critical points. The first one is a 4 trivial critical point, preserving the full SO(3) SU(3) symmetry, with all scalars vanishing × 3 λ = ϕ = 0, V = g2 (28) 0 −2 1 where V is the value of the potential at the critical point, the cosmological constant. This 0 AdS criticalpointshouldbeidentifiedwithacompactificationofM-theoryonN010 manifold 4 anddual to anN = 3 SCFT in three dimensions with SU(3) flavor symmetry. In the present convention, the AdS radius L is related to the value of the cosmological constant by 4 3 1 L2 = = . (29) −2V g2 0 1 At this critical point, all of the 48 scalars have m2L2 = 2 in agreement with the spectrum − of M-theory on AdS N010. These scalars are dual to operators of dimension ∆ = 1,2 in 4 × the dual SCFT. Another supersymmetric critical point is given by 1 g g 3g2g2 ϕ = 0, λ = ln 2 − 1 , V = 1 2 . (30) 2 g +g 0 −2(g2 g2) (cid:20) 2 1(cid:21) 2 − 1 This critical point is an AdS critical point for g2 > g2 as required by the reality of λ. 4 2 1 That this critical point preserves supersymmetry can be checked from the supersymmetry transformations given in the next subsection. The AdS radius can be found to be 4 g2 g2 L2 = 2 − 1 . (31) g2g2 1 2 More precisely, there are many critical points, equivalent to the one given above, with sin(4ϕ ) = 0 or ϕ = nπ, n Z. At this critical point, we can determine the full scalar 0 4 ∈ masses as shown in table I. From the table, we see seven massless scalars corresponding to Goldstone bosons of the 9 SO(3) U(1) representations m2L2 ∆ diag × 1 4, 2 4, (1,2) 0 − 2 0 , 2 3, (1,2) 3 (×2) (×2) − 2 0 , 2 3, (1,2) −3 (×2) (×2) − 3 0 , 2 3, (1,2) 0 (×3) (×3) − 4 9 , 2 3, (1,2) 3 −4(×4) − (×4) 2 4 9 , 2 3, (1,2) −3 −4(×4) − (×4) 2 5 2 (1,2) 0 (×10) − TABLE I. Scalar masses at the N = 3 supersymmetric AdS critical point with SO(3) U(1) 4 diag × symmetry and the corresponding dimensions of the dual operators symmetry breaking of SO(3) SU(3) to SO(3) U(1). The singlet scalar λ is dual to diag × × an irrelevant operator of dimension 4 at this critical point while ϕ is still dual to a relevant operator of dimension ∆ = 1,2. It should also be noted that all the masses satisfy the BF bound as expected for a supersymmetric critical point. There is also a non-supersymmetric critical point, but we will not give its location and value of the cosmological constant here due to its complexity. B. A supersymmetric RG flow In this subsection, we will find a supersymmetric domain wall solution interpolating between two AdS critical points identified previously. In order to do this, we will set up the 4 corresponding BPS equations by setting the supersymmetry transformations of fermions to zero. The non-vanishing bosonic fields are the metric and SO(3) singlet scalars. diag We adopt the standard domain wall ansatz for the four-dimensional metric ds2 = e2A(r)dx2 +dr2 (32) 1,2 with dx2 being the flat Minkowski metric in three dimensions. We will use the same 1,2 convention as in [31]. All spinors will be written as chiral projected Majorana spinors. For example, we have 1 1 ǫ = (1+γ )ǫ˜A, ǫA = (1 γ )ǫ˜A (33) A 5 5 2 2 − where ǫ˜A is a Majorana spinor. In this Majorana representation, all of the gamma matrices γa are real while γ = iγ γ γ γ is purely imaginary. As a consequence, ǫA and ǫ are simply 5 0 1 2 3 A 10