week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
PRL 113, 141601 (2014)
Gravity Dual of Supersymmetric Gauge Theories on a Squashed Five-Sphere Luis F. Alday,* Martin Fluder,† Paul Richmond,‡ and James Sparks,§ Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom (Received 8 July 2014; published 29 September 2014) We present the gravity dual of large N supersymmetric gauge theories on a squashed five-sphere. The one-parameter family of solutions is constructed in Euclidean Romans Fð4Þ gauged supergravity in six dimensions, and uplifts to massive type IIA supergravity. By renormalizing the theory with appropriate counterterms we evaluate the renormalized on-shell action for the solutions. We also evaluate the large N limit of the gauge theory partition function, and find precise agreement. DOI: 10.1103/PhysRevLett.113.141601
PACS numbers: 11.25.Tq, 04.65.+e, 11.30.Pb, 11.10.Kk
Supersymmetric gauge theories on a squashed fivesphere.—In Ref. [1] supersymmetric gauge theories with general matter content were defined on the SUð3Þ × Uð1Þ symmetric squashed five-sphere. The background metric is 1 1 ds25 ¼ 2 ðdτ þ CÞ2 þ dσ 2 þ sin2 σðdθ2 þ sin2 θdφ2 Þ 4 s 1 2 ð1Þ þ cos σ sin2 σðdψ þ cos θdφÞ2 ; 4 where C ¼ − 12 sin2 σðdψ þ cos θdφÞ and s is the squashing parameter. The round sphere corresponds to s ¼ 1. The theory preserves 3=4 of the supersymmetry of the round sphere, provided one turns on a background SUð2ÞR gauge field AR ¼
1þ
pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 1 − s2 s2
ðdτ þ CÞ;
ð2Þ
where we have embedded Uð1ÞR ⊂ SUð2ÞR . The background then admits a Killing spinor that solves the Killing spinor equation in Ref. [1] and transforms in the 3 of SUð3Þ. The perturbative partition function of the gauge theories was computed in Ref. [2] (see also Ref. [3]) and the final formula involves triple sine functions, generalizing the double sine functions that appear for squashed threespheres [4]. A particular class of five-dimensional gauge theories, with gauge group USpð2NÞ and arising from a D4-D8 system, is expected to have a large N description in terms of massive type IIA supergravity [5,6]. In Ref. [7] the large N limit of the partition function of these theories on the round sphere was computed and successfully compared to the entanglement entropy of the dual warped AdS6 × S4 supergravity solution. One can compute the large N limit of the USpð2NÞ gauge theory partition function Zs for the squashed background (1), (2). The corresponding free energy F s ¼ − log Zs is given by 0031-9007=14=113(14)=141601(5)
Fs ¼
1 27s2
pffiffiffiffiffiffiffiffiffiffiffiffi3 1 − s2 pffiffiffiffiffiffiffiffiffiffiffiffi F 1 : 1 − 1 − s2
3−
ð3Þ
Here F 1 is the free energy on the round sphere, which scales as N 5=2 [6,7]. The computation of Eq. (3) involves asymptotic expansions of the triple sine function and standard large N matrix model techniques, and details will appear in Ref. [8]. Similarly, we have computed the large N limit of the vacuum expectation value (VEV) of a Wilson loop wrapping the τ circle at σ ¼ 0, finding pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 pffiffiffiffiffiffiffiffiffiffiffiffi loghWi1 ; loghWis ¼ 3ð1 þ 1 − s2 Þ 3−
ð4Þ
where loghWi1 scales as N 1=2 [9]. In the remainder of this Letter we will reproduce Eqs. (3) and (4) from a dual supergravity computation. Euclidean Romans supergravity.—In order to find supergravity duals of the above theories put on general background five-manifolds it is natural to work in the sixdimensional Romans Fð4Þ supergravity theory [10]. The key here is that, as shown in Ref. [11], the Romans theory is a consistent truncation of massive type IIA supergravity on S4 . In particular, the AdS6 vacuum uplifts to the warped AdS6 × S4 solution mentioned above, relevant for the round five-sphere. The bosonic fields consist of the metric, a dilaton ϕ, a two-form potential B, a one-form potential A, together with an SOð3Þ ∼ SUð2Þ gauge field Ai , i ¼ 1, 2, 3. It is convenient to introduce the scalar field X ≡ pffiffiffi expð−ϕ=2 2Þ, and define the field strengths H ¼ dB, F ¼ dA þ 23 B, Fi ¼ dAi − 12 ϵijk Aj ∧ Ak , where without loss of generality we have set the gauge coupling to 1. The equations of motion for the Romans theory in Lorentz signature appear in Refs. [10,11]. However, in order to compute the holographic free energy we will work in Euclidean signature. This Wick rotation is not entirely straightforward due to Chern-Simons-type couplings. The Euclidean equations of motion are [8]
141601-1
© 2014 American Physical Society
PRL 113, 141601 (2014)
PHYSICAL REVIEW LETTERS
i i 2 dðX4 HÞ ¼ F ∧ F þ Fi ∧ Fi þ X−2 F; 2 2 3 dðX−2 FÞ ¼ −iF ∧ H; DðX−2 Fi Þ ¼ −iFi ∧ H; 1 −6 2 −2 1 2 −1 dðX dXÞ ¼ − X − X þ X 1 6 3 2 1 − X−2 ðF ∧ F þ Fi ∧ Fi Þ 8 1 ð5Þ þ X4 H ∧ H: 4 Here Dωi ¼ dωi − ϵijk Aj ∧ ωk is the SOð3Þ covariant derivative. Finally, the Einstein equation is 1 −6 2 −2 1 2 −2 X − X − X gμν Rμν ¼ 4X ∂ μ X∂ ν X þ 18 3 2 1 4 1 2 1 −2 2 1 2 2 Fμν − F gμν þ X H μν − H gμν þ X 4 6 2 8 1 1 þ X−2 ðFi Þ2μν − ðFi Þ2 gμν ; ð6Þ 2 8 where F2μν ¼ Fμρ Fν ρ , H2μν ¼ Hμρσ Hρσ ν . A solution to the above equations of motion is supersymmetric provided the Killing spinor equation and dilatino equation hold [8]: i 1 −3 1 Dμ ϵI ¼ pffiffiffi X þ X γ μ γ 7 ϵI − X2 Hνρσ γ νρσ γ μ γ 7 ϵI 3 48 4 2 i − pffiffiffi X−1 Fνρ ðγ μ νρ − 6δμ ν γ ρ ÞϵI 16 2 1 þ pffiffiffi X−1 Fiνρ ðγ μ νρ − 6δμ ν γ ρ Þγ 7 ðσ i ÞI J ϵJ ; ð7Þ 16 2 1 0 ¼ −iX−1 ∂ μ Xγ μ ϵI þ pffiffiffi ðX − X−3 Þγ 7 ϵI 2 2 i 1 þ X2 H μνρ γ μνρ γ 7 ϵI − pffiffiffi X−1 Fμν γ μν ϵI 24 8 2 i − pffiffiffi X−1 Fiμν γ μν γ 7 ðσ i ÞI J ϵJ : 8 2
ð8Þ
Here ϵI , I ¼ 1, 2, are two Dirac spinors, γ μ generate the Clifford algebra Cliff(6,0) in an orthonormal frame, and we
week ending 3 OCTOBER 2014
have defined the chirality operator γ 7 ¼ iγ 123456, which satisfies γ 27 ¼ 1. The SOð3Þ ∼ SUð2Þ gauge field Ai is an Rsymmetry gauge field, with the spinor ϵI transforming in the two-dimensional representation via the Pauli matrices ðσ i ÞI J . Thus the covariant derivative acting on the spinor is Dμ ϵI ¼ ∇μ ϵI þ 2i Aiμ ðσ i ÞI J ϵJ . The theory possesses a gauge invariance A → A þ 23 λ, B → B − dλ, where λ is any one-form. Using this freedom we fix the gauge A ¼ 0, leaving F ¼ 23 B: the B field “eats” the Uð1Þ gauge field A in a Higgs-like mechanism. The solution.—The squashed five-sphere background in Eq. (1) has SUð3Þ × Uð1Þ symmetry. One expects this symmetry to be preserved by the bulk solution. This leads to the following ansatz for the supergravity fields: ds26 ¼ α2 ðrÞdr2 þ γ 2 ðrÞðdτ þ CÞ2 1 þ β2 ðrÞ dσ 2 þ sin2 σðdθ2 þ sin2 θdφ2 Þ 4 1 2 2 2 þ cos σsin σðdψ þ cos θdφÞ ; 4 1 B ¼ pðrÞdr ∧ ðdτ þ CÞ þ qðrÞdC; 2 i i A ¼ f ðrÞðdτ þ CÞ;
ð9Þ
where also X ¼ XðrÞ. We have constructed a smooth, supersymmetric, asymptotically locally Euclidean AdS solution to Eqs. (5,6,7,8), which has as conformal boundary the squashed five-sphere background of Eq. (1). The function βðrÞ can be set to its AdS6 value by ffiusing pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi reparametrization invariance, βðrÞ ¼ 3 6r2 − 1= 2. Furthermore, we have performed an SOð3Þ ∼ SUð2Þ rotation so as to set f 1 ðrÞ ¼ f 2 ðrÞ ¼ 0, and renamed f 3 ðrÞ ≡ fðrÞ. Even though we are not able to give a closed expression for the solution, it is possible to give it as an expansion around different limits. Expansion around the conformal boundary: Finding the gravity dual to a theory on a prescribed conformal boundary may be regarded as a filling problem in supergravity. As such, it is natural to solve the supergravity equations order by order in an expansion around the boundary at r ¼ ∞. We have computed this expansion up to order Oð1=r9 Þ. The first terms are given by
3 1 8 þ s2 1 αðrÞ ¼ pffiffiffi þ pffiffiffi 2 3 þ ; 2 r 36 2s r pffiffiffi 3 3 −16 þ 7s2 1 −1280 þ 1120s2 þ 241s4 1 pffiffiffi pffiffiffi rþ − þ ; γðrÞ ¼ s r3 12 3s3 r 2592 3s5 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 − 3 1 − s2 1 s2 1 − s2 κ 1 þ XðrÞ ¼ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffi 3 þ ; 2 2 r 54s 12 1 − s2 þ 1 − s2 r
141601-2
PRL 113, 141601 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
qffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i 23 s2 þ 3 1 − s2 − 1 1 pðrÞ ¼ − þ ; 2 s3 qffiffi prffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 3i 6 1 − s2 þ 9 1 − s 5s − 5 i 1 − s 3 1 rþ þ ; qðrÞ ¼ − 3 s r 3s pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 − s2 þ 1 − s2 2 −2 þ 2s − ð2 þ s Þ 1 − s 1 κ þ þ þ : fðrÞ ¼ 9s4 r2 r3 s2
Notice that the squashing parameter s arises as the pffiffiffi boundary value limr→∞ γðrÞ=3 3r ¼ s−1 . In the limit s ¼ 1 the solution collapses to Euclidean AdS6 . The whole solution depends on the single parameter s. The extra parameter κ is fixed pffiffiffi by requiring a nonsingular solution at the origin r ¼ 1= 6. Alternatively, this can be computed as an expansion, as is done at the end of next subsection. Expansion around Euclidean AdS6 : The solution presented in this Letter is continuously connected to Euclidean
ð10Þ
AdS6 . Hence it can be given as a perturbation around this background. It is convenient to use the real expansion parameter δ, related to the squashing parameter by 1 ¼ 1 þ δ2 : s
ð11Þ
We have explicitly computed the solution up to sixth order in δ. At leading order we find
pffiffiffi pffiffiffi 4 pffiffiffi 6 pffiffiffi 8 pffiffiffi 2 pffiffiffi 3 5 − 3744r þ 1620 6 r þ 8640r − 7560 6 r þ 5184 6r −5 6 þ 330 6 r 3 3 pffiffiffi 2 2 αðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ δ2 þ ; 9 2r ð6r − 1Þ9=2 6r2 − 1 pffiffiffi pffiffiffi 2 pffiffiffi 4 pffiffiffi 6 pffiffiffi 8 pffiffiffi pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55 2 − 384 3 r þ 1080 2 r 2 þ 768 3 r − 5400 2 r þ 11232 2 r − 11664 2r 3 6r − 1 pffiffiffi − δ2 þ ; γðrÞ ¼ 6ð6r2 − 1Þ7=2 2 hpffiffiffi i pffiffiffi 2 1 − 2 6r þ 6r2 XðrÞ ¼ 1 − δ þ ; 3ð6r2 − 1Þ2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3i 2 −4 þ 9 6r − 24r2 − 12 6r3 þ 36 6r5 δ qðrÞ ¼ − þ ; ð6r2 − 1Þ2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 18i 2 6 − 16r þ 12 6r2 − 12 6r4 pðrÞ ¼ δ þ ; ð6r2 − 1Þ3 pffiffiffi pffiffiffi 2 −3 þ 8 6r − 36r2 þ 36r4 fðrÞ ¼ δ þ . ð12Þ ð6r2 − 1Þ2 One can explicitly check that each term ofpthe ffiffiffi solution above is nonsingular at the origin r ¼ 1= 6, giving a regular solution on a manifold M 6 with the topology of a six-ball. By comparing the two expansions we find pffiffiffi pffiffiffi 3 3 25 1127 5 2 2 113 3 κ ¼δþ δ þ pffiffiffi δ4 þ δ δ þ 4 36 288 3 9 2 35 þ pffiffiffi δ6 þ ; 9 2
ð13Þ
which is used in evaluating the on-shell action below.
Comparison.—The bulk supergravity action of the Romans theory, in Euclidean signature in the gauge A ¼ 0, is Z 1 Sbulk ¼ − R 1 − 4X−2 dX ∧ dX 16πGN M6 2 −6 8 −2 2 X − X − 2X 1 − 9 3 1 −2 4 1 i i − X B ∧ B þ F ∧ F − X 4 H ∧ H 2 9 2 2 1 − iB ∧ ð14Þ B ∧ B þ Fi ∧ Fi : 27 2
141601-3
PRL 113, 141601 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
Here GN is the six-dimensional Newton constant. More precisely, we should cut off the manifold M 6 at some large constant radius r ¼ ρ, and include the Gibbons-Hawking boundary term Z pffiffiffiffiffiffiffiffiffiffi 1 SGH ¼ − K det hd5 x; ð15Þ 8πGN ∂M6
where hij is the induced metric on the boundary ∂M6 ¼ fr ¼ ρg ≅ S5 , and K denotes the trace of the second fundamental form. The total action is divergent as one sends ρ → ∞, but may be regularized using holographic renormalization techniques. This leads to the following boundary counterterms [8]:
pffiffiffi 4 2 1 1 3 15 3 þ pffiffiffi RðhÞ − pffiffiffi ∥B∥2h þ pffiffiffi RðhÞij RðhÞij − pffiffiffi RðhÞ2 − pffiffiffi ∥Fi ∥2h 3 2 2 6 2 4 2 64 2 4 2 ∂M 6 pffiffiffi pffiffiffi
1 5 i 2 1 i 2 1 B ∧ B∥2h − pffiffiffi B; dδh B þ d h B ∧ B − pffiffiffi ∥dB∥2h þ pffiffiffi Trh B4 þ pffiffiffi ∥d h B þ 3 3 12 2 8 2 4 2 2 h pffiffiffi p ffiffiffiffiffiffiffiffiffi ffi 4 2 1 9 13 pffiffiffi ∥B∥4h det hd5 x ð1 − XÞ2 − pffiffiffi hRicðhÞ∘B; Bih þ pffiffiffi RðhÞ∥B∥2h − þ 3 2 32 2 192 2 pffiffiffi 1 2 2i B ∧ δh B − B ∧ h ðB ∧ BÞ : − pffiffiffi B ∧ ½d h dB þ 9 3 4 2
1 Sct ¼ 8πGN
Z
Here RicðhÞij ¼ RðhÞij denotes the Ricci tensor of the metric hij , with RðhÞ the Ricci scalar. The inner product of pffiffiffiffiffiffiffiffiffiffi two p-forms ν1 , ν2 is defined by hν1 ; ν2 ih det hd5 x ¼ ν1 ∧ h ν2 , which then also defines the square norm via ∥ν∥2h ¼ hν; νih . The adjoint δh of d with respect to hij acting on the two-form B is δh B ¼ h d h B, and we have also defined Trh B4 ≡ Bji Bkj Blk Bil . Finally, we have defined the p-form ðS∘νÞi1 ip ≡ S½i1 j νjjji2 ip , where Sij is any symmetric 2-tensor, and ν is any p-form. Adding the contributions and taking the cutoff to infinity we obtain pffiffiffi 27π 2 8 2 16 2 3 68 4 δ þ δ 1þ δ þ Sbulk þ SGH þ Sct ¼ − 3 27 27 4GN pffiffiffi 28 2 5 32 6 δ þ δ þ : þ ð17Þ 27 27 This should be identified with the holographic free energy. Recalling that s−1 ¼ 1 þ δ2 , this precisely agrees with Eq. (3) to sixth order in δ. It should be straightforward to extend this agreement to higher orders. The BPS Wilson loop (4) maps to a fundamental string in type IIA, at the “pole” of the internal S4 [9]. The string wraps the surface Σ spanned by the τ and r directions at σ ¼ 0. The renormalized string action is Z pffiffiffiffiffiffiffiffiffi 3 Sstring ¼ ½X−2 detγ d2 x þ iB − pffiffiffi lengthð∂ΣÞ; ð18Þ 2 Σ
Sstring
pffiffiffi pffiffiffi 4 2δ 8δ2 5 2δ3 4δ4 þ ¼ 1− − þ 3 3 3 3 7δ5 − pffiffiffi þ 0δ6 þ Sstring ∣δ¼0 ; 12 2
ð16Þ
ð19Þ
which precisely matches Eq. (4). A conjecture.—A supersymmetric solution admits an SUð2Þ doublet of Killing spinors ϵI . Provided the Killing spinor satisfies a symplectic Majorana condition CϵI ¼ εI J ϵJ , where C is the charge conjugation matrix (C−1 γ μ C ¼ γ μ ), it can be shown [8] that K μ ¼ εIJ ϵTI Cγ μ ϵJ
ð20Þ
is a real Killing vector. For our solution the Killing spinor has three integration constants, corresponding to the fact that it is 3=4 BPS, and for an appropriate choice of the Killing spinor in Eq. (20) we obtain K ¼ b1 ∂ φ1 þ b2 ∂ φ2 þ b3 ∂ φ3 ;
ð21Þ
where b1 ¼ 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 ;
b2 ¼ b3 ¼ 1 −
pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 ; ð22Þ
and φ1 , φ2 , φ3 are the standard 2π periodic azimuthal variables φ1 ¼ −τ, φ2 ¼ τ − 12 ðψ þ φÞ, φ3 ¼ τ − 12 ðψ − φÞ, embedding S5 ⊂ R2 ⊕ R2 ⊕ R2 . Note that the large N free energy Eq. (3) can then be written as
where γ ab is the induced metric and the second term is a boundary counterterm. We may evaluate this up to sixth order in δ for our solution to obtain 141601-4
F¼
ðb1 þ b2 þ b3 Þ3 F round : 27b1 b2 b3
ð23Þ
PRL 113, 141601 (2014)
PHYSICAL REVIEW LETTERS
It is then natural to make the following conjectures. (1) For any supersymmetric supergravity solution with the topology of the six-ball, with at least Uð1Þ3 isometry, and for which the Killing vector Eq. (20) takes the form of Eq. (21), the holographic free energy is equal to Eq. (23). (2) If we define a supersymmetric gauge theory on the conformal boundary of the background in point 1, the finite N partition function depends only on b1 , b2 , b3 . These conjectures extend to 5d=6d the results proven for the analogous 3d=4d context in Refs. [12,13]. Conjecture 1 also extends to the BPS Wilson loop wrapping φi , at the origin of the perpendicular R4. In this case loghWis ¼ (ðb1 þ b2 þ b3 Þ=3bi ) loghWi1 . In Ref. [8] we construct further families of supersymmetric backgrounds satisfying the conditions of point 1 and verify the conjecture for these cases. These include a supersymmetric solution with the SUð2Þ gauge field turned off, with a squashing parameter but for which bi ¼ 1. For this case the free energy does not depend on the squashing parameter, in full agreement with conjecture 1. The work of L. F. A., M. F., and P. R. is supported by ERC STG Grant No. 306260. J. S. is supported by a Royal Society University Research Fellowship.
*
week ending 3 OCTOBER 2014
[email protected] [email protected] ‡ [email protected] § [email protected] [1] Y. Imamura, Prog. Theor. Exp. Phys. 2013, 013B04 (2013). [2] Y. Imamura, arXiv:1210.6308. [3] G. Lockhart and C. Vafa, arXiv:1210.5909. [4] N. Hama, K. Hosomichi, and S. Lee, J. High Energy Phys. 05 (2011) 014. [5] S. Ferrara, A. Kehagias, H. Partouche, and A. Zaffaroni, Phys. Lett. B 431, 57 (1998). [6] A. Brandhuber and Y. Oz, Phys. Lett. B 460, 307 (1999). [7] D. L. Jafferis and S. S. Pufu, J. High Energy Phys. 05 (2014) 032. [8] L. F. Alday, M. Fluder, C. M. Gregory, P. Richmond, and J. Sparks, J. High Energy Phys. 09 (2014) 067.. [9] B. Assel, J. Estes, and M. Yamazaki, Ann. Inst. Henri Poincaré, A 15, 589 (2014). [10] L. J. Romans, Nucl. Phys. B269, 691 (1986). [11] M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. Lett. 83, 5226 (1999). [12] L. F. Alday, D. Martelli, P. Richmond, and J. Sparks, J. High Energy Phys. 10 (2013) 095. [13] D. Farquet, J. Lorenzen, D. Martelli, and J. Sparks, arXiv:1404.0268.
141601-5
†
PHYSICAL REVIEW LETTERS
PRL 113, 141601 (2014)
Gravity Dual of Supersymmetric Gauge Theories on a Squashed Five-Sphere Luis F. Alday,* Martin Fluder,† Paul Richmond,‡ and James Sparks,§ Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom (Received 8 July 2014; published 29 September 2014) We present the gravity dual of large N supersymmetric gauge theories on a squashed five-sphere. The one-parameter family of solutions is constructed in Euclidean Romans Fð4Þ gauged supergravity in six dimensions, and uplifts to massive type IIA supergravity. By renormalizing the theory with appropriate counterterms we evaluate the renormalized on-shell action for the solutions. We also evaluate the large N limit of the gauge theory partition function, and find precise agreement. DOI: 10.1103/PhysRevLett.113.141601
PACS numbers: 11.25.Tq, 04.65.+e, 11.30.Pb, 11.10.Kk
Supersymmetric gauge theories on a squashed fivesphere.—In Ref. [1] supersymmetric gauge theories with general matter content were defined on the SUð3Þ × Uð1Þ symmetric squashed five-sphere. The background metric is 1 1 ds25 ¼ 2 ðdτ þ CÞ2 þ dσ 2 þ sin2 σðdθ2 þ sin2 θdφ2 Þ 4 s 1 2 ð1Þ þ cos σ sin2 σðdψ þ cos θdφÞ2 ; 4 where C ¼ − 12 sin2 σðdψ þ cos θdφÞ and s is the squashing parameter. The round sphere corresponds to s ¼ 1. The theory preserves 3=4 of the supersymmetry of the round sphere, provided one turns on a background SUð2ÞR gauge field AR ¼
1þ
pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 1 − s2 s2
ðdτ þ CÞ;
ð2Þ
where we have embedded Uð1ÞR ⊂ SUð2ÞR . The background then admits a Killing spinor that solves the Killing spinor equation in Ref. [1] and transforms in the 3 of SUð3Þ. The perturbative partition function of the gauge theories was computed in Ref. [2] (see also Ref. [3]) and the final formula involves triple sine functions, generalizing the double sine functions that appear for squashed threespheres [4]. A particular class of five-dimensional gauge theories, with gauge group USpð2NÞ and arising from a D4-D8 system, is expected to have a large N description in terms of massive type IIA supergravity [5,6]. In Ref. [7] the large N limit of the partition function of these theories on the round sphere was computed and successfully compared to the entanglement entropy of the dual warped AdS6 × S4 supergravity solution. One can compute the large N limit of the USpð2NÞ gauge theory partition function Zs for the squashed background (1), (2). The corresponding free energy F s ¼ − log Zs is given by 0031-9007=14=113(14)=141601(5)
Fs ¼
1 27s2
pffiffiffiffiffiffiffiffiffiffiffiffi3 1 − s2 pffiffiffiffiffiffiffiffiffiffiffiffi F 1 : 1 − 1 − s2
3−
ð3Þ
Here F 1 is the free energy on the round sphere, which scales as N 5=2 [6,7]. The computation of Eq. (3) involves asymptotic expansions of the triple sine function and standard large N matrix model techniques, and details will appear in Ref. [8]. Similarly, we have computed the large N limit of the vacuum expectation value (VEV) of a Wilson loop wrapping the τ circle at σ ¼ 0, finding pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 pffiffiffiffiffiffiffiffiffiffiffiffi loghWi1 ; loghWis ¼ 3ð1 þ 1 − s2 Þ 3−
ð4Þ
where loghWi1 scales as N 1=2 [9]. In the remainder of this Letter we will reproduce Eqs. (3) and (4) from a dual supergravity computation. Euclidean Romans supergravity.—In order to find supergravity duals of the above theories put on general background five-manifolds it is natural to work in the sixdimensional Romans Fð4Þ supergravity theory [10]. The key here is that, as shown in Ref. [11], the Romans theory is a consistent truncation of massive type IIA supergravity on S4 . In particular, the AdS6 vacuum uplifts to the warped AdS6 × S4 solution mentioned above, relevant for the round five-sphere. The bosonic fields consist of the metric, a dilaton ϕ, a two-form potential B, a one-form potential A, together with an SOð3Þ ∼ SUð2Þ gauge field Ai , i ¼ 1, 2, 3. It is convenient to introduce the scalar field X ≡ pffiffiffi expð−ϕ=2 2Þ, and define the field strengths H ¼ dB, F ¼ dA þ 23 B, Fi ¼ dAi − 12 ϵijk Aj ∧ Ak , where without loss of generality we have set the gauge coupling to 1. The equations of motion for the Romans theory in Lorentz signature appear in Refs. [10,11]. However, in order to compute the holographic free energy we will work in Euclidean signature. This Wick rotation is not entirely straightforward due to Chern-Simons-type couplings. The Euclidean equations of motion are [8]
141601-1
© 2014 American Physical Society
PRL 113, 141601 (2014)
PHYSICAL REVIEW LETTERS
i i 2 dðX4 HÞ ¼ F ∧ F þ Fi ∧ Fi þ X−2 F; 2 2 3 dðX−2 FÞ ¼ −iF ∧ H; DðX−2 Fi Þ ¼ −iFi ∧ H; 1 −6 2 −2 1 2 −1 dðX dXÞ ¼ − X − X þ X 1 6 3 2 1 − X−2 ðF ∧ F þ Fi ∧ Fi Þ 8 1 ð5Þ þ X4 H ∧ H: 4 Here Dωi ¼ dωi − ϵijk Aj ∧ ωk is the SOð3Þ covariant derivative. Finally, the Einstein equation is 1 −6 2 −2 1 2 −2 X − X − X gμν Rμν ¼ 4X ∂ μ X∂ ν X þ 18 3 2 1 4 1 2 1 −2 2 1 2 2 Fμν − F gμν þ X H μν − H gμν þ X 4 6 2 8 1 1 þ X−2 ðFi Þ2μν − ðFi Þ2 gμν ; ð6Þ 2 8 where F2μν ¼ Fμρ Fν ρ , H2μν ¼ Hμρσ Hρσ ν . A solution to the above equations of motion is supersymmetric provided the Killing spinor equation and dilatino equation hold [8]: i 1 −3 1 Dμ ϵI ¼ pffiffiffi X þ X γ μ γ 7 ϵI − X2 Hνρσ γ νρσ γ μ γ 7 ϵI 3 48 4 2 i − pffiffiffi X−1 Fνρ ðγ μ νρ − 6δμ ν γ ρ ÞϵI 16 2 1 þ pffiffiffi X−1 Fiνρ ðγ μ νρ − 6δμ ν γ ρ Þγ 7 ðσ i ÞI J ϵJ ; ð7Þ 16 2 1 0 ¼ −iX−1 ∂ μ Xγ μ ϵI þ pffiffiffi ðX − X−3 Þγ 7 ϵI 2 2 i 1 þ X2 H μνρ γ μνρ γ 7 ϵI − pffiffiffi X−1 Fμν γ μν ϵI 24 8 2 i − pffiffiffi X−1 Fiμν γ μν γ 7 ðσ i ÞI J ϵJ : 8 2
ð8Þ
Here ϵI , I ¼ 1, 2, are two Dirac spinors, γ μ generate the Clifford algebra Cliff(6,0) in an orthonormal frame, and we
week ending 3 OCTOBER 2014
have defined the chirality operator γ 7 ¼ iγ 123456, which satisfies γ 27 ¼ 1. The SOð3Þ ∼ SUð2Þ gauge field Ai is an Rsymmetry gauge field, with the spinor ϵI transforming in the two-dimensional representation via the Pauli matrices ðσ i ÞI J . Thus the covariant derivative acting on the spinor is Dμ ϵI ¼ ∇μ ϵI þ 2i Aiμ ðσ i ÞI J ϵJ . The theory possesses a gauge invariance A → A þ 23 λ, B → B − dλ, where λ is any one-form. Using this freedom we fix the gauge A ¼ 0, leaving F ¼ 23 B: the B field “eats” the Uð1Þ gauge field A in a Higgs-like mechanism. The solution.—The squashed five-sphere background in Eq. (1) has SUð3Þ × Uð1Þ symmetry. One expects this symmetry to be preserved by the bulk solution. This leads to the following ansatz for the supergravity fields: ds26 ¼ α2 ðrÞdr2 þ γ 2 ðrÞðdτ þ CÞ2 1 þ β2 ðrÞ dσ 2 þ sin2 σðdθ2 þ sin2 θdφ2 Þ 4 1 2 2 2 þ cos σsin σðdψ þ cos θdφÞ ; 4 1 B ¼ pðrÞdr ∧ ðdτ þ CÞ þ qðrÞdC; 2 i i A ¼ f ðrÞðdτ þ CÞ;
ð9Þ
where also X ¼ XðrÞ. We have constructed a smooth, supersymmetric, asymptotically locally Euclidean AdS solution to Eqs. (5,6,7,8), which has as conformal boundary the squashed five-sphere background of Eq. (1). The function βðrÞ can be set to its AdS6 value by ffiusing pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi reparametrization invariance, βðrÞ ¼ 3 6r2 − 1= 2. Furthermore, we have performed an SOð3Þ ∼ SUð2Þ rotation so as to set f 1 ðrÞ ¼ f 2 ðrÞ ¼ 0, and renamed f 3 ðrÞ ≡ fðrÞ. Even though we are not able to give a closed expression for the solution, it is possible to give it as an expansion around different limits. Expansion around the conformal boundary: Finding the gravity dual to a theory on a prescribed conformal boundary may be regarded as a filling problem in supergravity. As such, it is natural to solve the supergravity equations order by order in an expansion around the boundary at r ¼ ∞. We have computed this expansion up to order Oð1=r9 Þ. The first terms are given by
3 1 8 þ s2 1 αðrÞ ¼ pffiffiffi þ pffiffiffi 2 3 þ ; 2 r 36 2s r pffiffiffi 3 3 −16 þ 7s2 1 −1280 þ 1120s2 þ 241s4 1 pffiffiffi pffiffiffi rþ − þ ; γðrÞ ¼ s r3 12 3s3 r 2592 3s5 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 − 3 1 − s2 1 s2 1 − s2 κ 1 þ XðrÞ ¼ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffi 3 þ ; 2 2 r 54s 12 1 − s2 þ 1 − s2 r
141601-2
PRL 113, 141601 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
qffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i 23 s2 þ 3 1 − s2 − 1 1 pðrÞ ¼ − þ ; 2 s3 qffiffi prffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 3i 6 1 − s2 þ 9 1 − s 5s − 5 i 1 − s 3 1 rþ þ ; qðrÞ ¼ − 3 s r 3s pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 − s2 þ 1 − s2 2 −2 þ 2s − ð2 þ s Þ 1 − s 1 κ þ þ þ : fðrÞ ¼ 9s4 r2 r3 s2
Notice that the squashing parameter s arises as the pffiffiffi boundary value limr→∞ γðrÞ=3 3r ¼ s−1 . In the limit s ¼ 1 the solution collapses to Euclidean AdS6 . The whole solution depends on the single parameter s. The extra parameter κ is fixed pffiffiffi by requiring a nonsingular solution at the origin r ¼ 1= 6. Alternatively, this can be computed as an expansion, as is done at the end of next subsection. Expansion around Euclidean AdS6 : The solution presented in this Letter is continuously connected to Euclidean
ð10Þ
AdS6 . Hence it can be given as a perturbation around this background. It is convenient to use the real expansion parameter δ, related to the squashing parameter by 1 ¼ 1 þ δ2 : s
ð11Þ
We have explicitly computed the solution up to sixth order in δ. At leading order we find
pffiffiffi pffiffiffi 4 pffiffiffi 6 pffiffiffi 8 pffiffiffi 2 pffiffiffi 3 5 − 3744r þ 1620 6 r þ 8640r − 7560 6 r þ 5184 6r −5 6 þ 330 6 r 3 3 pffiffiffi 2 2 αðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ δ2 þ ; 9 2r ð6r − 1Þ9=2 6r2 − 1 pffiffiffi pffiffiffi 2 pffiffiffi 4 pffiffiffi 6 pffiffiffi 8 pffiffiffi pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 55 2 − 384 3 r þ 1080 2 r 2 þ 768 3 r − 5400 2 r þ 11232 2 r − 11664 2r 3 6r − 1 pffiffiffi − δ2 þ ; γðrÞ ¼ 6ð6r2 − 1Þ7=2 2 hpffiffiffi i pffiffiffi 2 1 − 2 6r þ 6r2 XðrÞ ¼ 1 − δ þ ; 3ð6r2 − 1Þ2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3i 2 −4 þ 9 6r − 24r2 − 12 6r3 þ 36 6r5 δ qðrÞ ¼ − þ ; ð6r2 − 1Þ2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 18i 2 6 − 16r þ 12 6r2 − 12 6r4 pðrÞ ¼ δ þ ; ð6r2 − 1Þ3 pffiffiffi pffiffiffi 2 −3 þ 8 6r − 36r2 þ 36r4 fðrÞ ¼ δ þ . ð12Þ ð6r2 − 1Þ2 One can explicitly check that each term ofpthe ffiffiffi solution above is nonsingular at the origin r ¼ 1= 6, giving a regular solution on a manifold M 6 with the topology of a six-ball. By comparing the two expansions we find pffiffiffi pffiffiffi 3 3 25 1127 5 2 2 113 3 κ ¼δþ δ þ pffiffiffi δ4 þ δ δ þ 4 36 288 3 9 2 35 þ pffiffiffi δ6 þ ; 9 2
ð13Þ
which is used in evaluating the on-shell action below.
Comparison.—The bulk supergravity action of the Romans theory, in Euclidean signature in the gauge A ¼ 0, is Z 1 Sbulk ¼ − R 1 − 4X−2 dX ∧ dX 16πGN M6 2 −6 8 −2 2 X − X − 2X 1 − 9 3 1 −2 4 1 i i − X B ∧ B þ F ∧ F − X 4 H ∧ H 2 9 2 2 1 − iB ∧ ð14Þ B ∧ B þ Fi ∧ Fi : 27 2
141601-3
PRL 113, 141601 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
Here GN is the six-dimensional Newton constant. More precisely, we should cut off the manifold M 6 at some large constant radius r ¼ ρ, and include the Gibbons-Hawking boundary term Z pffiffiffiffiffiffiffiffiffiffi 1 SGH ¼ − K det hd5 x; ð15Þ 8πGN ∂M6
where hij is the induced metric on the boundary ∂M6 ¼ fr ¼ ρg ≅ S5 , and K denotes the trace of the second fundamental form. The total action is divergent as one sends ρ → ∞, but may be regularized using holographic renormalization techniques. This leads to the following boundary counterterms [8]:
pffiffiffi 4 2 1 1 3 15 3 þ pffiffiffi RðhÞ − pffiffiffi ∥B∥2h þ pffiffiffi RðhÞij RðhÞij − pffiffiffi RðhÞ2 − pffiffiffi ∥Fi ∥2h 3 2 2 6 2 4 2 64 2 4 2 ∂M 6 pffiffiffi pffiffiffi
1 5 i 2 1 i 2 1 B ∧ B∥2h − pffiffiffi B; dδh B þ d h B ∧ B − pffiffiffi ∥dB∥2h þ pffiffiffi Trh B4 þ pffiffiffi ∥d h B þ 3 3 12 2 8 2 4 2 2 h pffiffiffi p ffiffiffiffiffiffiffiffiffi ffi 4 2 1 9 13 pffiffiffi ∥B∥4h det hd5 x ð1 − XÞ2 − pffiffiffi hRicðhÞ∘B; Bih þ pffiffiffi RðhÞ∥B∥2h − þ 3 2 32 2 192 2 pffiffiffi 1 2 2i B ∧ δh B − B ∧ h ðB ∧ BÞ : − pffiffiffi B ∧ ½d h dB þ 9 3 4 2
1 Sct ¼ 8πGN
Z
Here RicðhÞij ¼ RðhÞij denotes the Ricci tensor of the metric hij , with RðhÞ the Ricci scalar. The inner product of pffiffiffiffiffiffiffiffiffiffi two p-forms ν1 , ν2 is defined by hν1 ; ν2 ih det hd5 x ¼ ν1 ∧ h ν2 , which then also defines the square norm via ∥ν∥2h ¼ hν; νih . The adjoint δh of d with respect to hij acting on the two-form B is δh B ¼ h d h B, and we have also defined Trh B4 ≡ Bji Bkj Blk Bil . Finally, we have defined the p-form ðS∘νÞi1 ip ≡ S½i1 j νjjji2 ip , where Sij is any symmetric 2-tensor, and ν is any p-form. Adding the contributions and taking the cutoff to infinity we obtain pffiffiffi 27π 2 8 2 16 2 3 68 4 δ þ δ 1þ δ þ Sbulk þ SGH þ Sct ¼ − 3 27 27 4GN pffiffiffi 28 2 5 32 6 δ þ δ þ : þ ð17Þ 27 27 This should be identified with the holographic free energy. Recalling that s−1 ¼ 1 þ δ2 , this precisely agrees with Eq. (3) to sixth order in δ. It should be straightforward to extend this agreement to higher orders. The BPS Wilson loop (4) maps to a fundamental string in type IIA, at the “pole” of the internal S4 [9]. The string wraps the surface Σ spanned by the τ and r directions at σ ¼ 0. The renormalized string action is Z pffiffiffiffiffiffiffiffiffi 3 Sstring ¼ ½X−2 detγ d2 x þ iB − pffiffiffi lengthð∂ΣÞ; ð18Þ 2 Σ
Sstring
pffiffiffi pffiffiffi 4 2δ 8δ2 5 2δ3 4δ4 þ ¼ 1− − þ 3 3 3 3 7δ5 − pffiffiffi þ 0δ6 þ Sstring ∣δ¼0 ; 12 2
ð16Þ
ð19Þ
which precisely matches Eq. (4). A conjecture.—A supersymmetric solution admits an SUð2Þ doublet of Killing spinors ϵI . Provided the Killing spinor satisfies a symplectic Majorana condition CϵI ¼ εI J ϵJ , where C is the charge conjugation matrix (C−1 γ μ C ¼ γ μ ), it can be shown [8] that K μ ¼ εIJ ϵTI Cγ μ ϵJ
ð20Þ
is a real Killing vector. For our solution the Killing spinor has three integration constants, corresponding to the fact that it is 3=4 BPS, and for an appropriate choice of the Killing spinor in Eq. (20) we obtain K ¼ b1 ∂ φ1 þ b2 ∂ φ2 þ b3 ∂ φ3 ;
ð21Þ
where b1 ¼ 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 ;
b2 ¼ b3 ¼ 1 −
pffiffiffiffiffiffiffiffiffiffiffiffi 1 − s2 ; ð22Þ
and φ1 , φ2 , φ3 are the standard 2π periodic azimuthal variables φ1 ¼ −τ, φ2 ¼ τ − 12 ðψ þ φÞ, φ3 ¼ τ − 12 ðψ − φÞ, embedding S5 ⊂ R2 ⊕ R2 ⊕ R2 . Note that the large N free energy Eq. (3) can then be written as
where γ ab is the induced metric and the second term is a boundary counterterm. We may evaluate this up to sixth order in δ for our solution to obtain 141601-4
F¼
ðb1 þ b2 þ b3 Þ3 F round : 27b1 b2 b3
ð23Þ
PRL 113, 141601 (2014)
PHYSICAL REVIEW LETTERS
It is then natural to make the following conjectures. (1) For any supersymmetric supergravity solution with the topology of the six-ball, with at least Uð1Þ3 isometry, and for which the Killing vector Eq. (20) takes the form of Eq. (21), the holographic free energy is equal to Eq. (23). (2) If we define a supersymmetric gauge theory on the conformal boundary of the background in point 1, the finite N partition function depends only on b1 , b2 , b3 . These conjectures extend to 5d=6d the results proven for the analogous 3d=4d context in Refs. [12,13]. Conjecture 1 also extends to the BPS Wilson loop wrapping φi , at the origin of the perpendicular R4. In this case loghWis ¼ (ðb1 þ b2 þ b3 Þ=3bi ) loghWi1 . In Ref. [8] we construct further families of supersymmetric backgrounds satisfying the conditions of point 1 and verify the conjecture for these cases. These include a supersymmetric solution with the SUð2Þ gauge field turned off, with a squashing parameter but for which bi ¼ 1. For this case the free energy does not depend on the squashing parameter, in full agreement with conjecture 1. The work of L. F. A., M. F., and P. R. is supported by ERC STG Grant No. 306260. J. S. is supported by a Royal Society University Research Fellowship.
*
week ending 3 OCTOBER 2014
[email protected] [email protected] ‡ [email protected] § [email protected] [1] Y. Imamura, Prog. Theor. Exp. Phys. 2013, 013B04 (2013). [2] Y. Imamura, arXiv:1210.6308. [3] G. Lockhart and C. Vafa, arXiv:1210.5909. [4] N. Hama, K. Hosomichi, and S. Lee, J. High Energy Phys. 05 (2011) 014. [5] S. Ferrara, A. Kehagias, H. Partouche, and A. Zaffaroni, Phys. Lett. B 431, 57 (1998). [6] A. Brandhuber and Y. Oz, Phys. Lett. B 460, 307 (1999). [7] D. L. Jafferis and S. S. Pufu, J. High Energy Phys. 05 (2014) 032. [8] L. F. Alday, M. Fluder, C. M. Gregory, P. Richmond, and J. Sparks, J. High Energy Phys. 09 (2014) 067.. [9] B. Assel, J. Estes, and M. Yamazaki, Ann. Inst. Henri Poincaré, A 15, 589 (2014). [10] L. J. Romans, Nucl. Phys. B269, 691 (1986). [11] M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. Lett. 83, 5226 (1999). [12] L. F. Alday, D. Martelli, P. Richmond, and J. Sparks, J. High Energy Phys. 10 (2013) 095. [13] D. Farquet, J. Lorenzen, D. Martelli, and J. Sparks, arXiv:1404.0268.
141601-5
†
