The holographic supersymmetric Casimir energy
Pietro Benetti Genolini, Davide Cassani, Dario Martelli, James Sparks
aa r X i v : . [ h e p - t h ] D ec The holographic supersymmetric Casimir energy
Pietro Benetti Genolini, Davide Cassani, Dario Martelli, and James Sparks Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, U.K. LPTHE, Sorbonne Universit´es UPMC Paris 06 and CNRS, UMR 7589, F-75005, Paris, France Department of Mathematics, King’s College London, The Strand, London, WC2R 2LS, U.K.
We consider a general class of asymptotically locally AdS solutions of minimal gauged supergravity,which are dual to superconformal field theories on curved backgrounds S × M preserving twosupercharges. We demonstrate that standard holographic renormalization corresponds to a schemethat breaks supersymmetry. We propose new boundary terms that restore supersymmetry, andshow that for smooth solutions with topology S × R the improved on-shell action reproduces boththe supersymmetric Casimir energy and the field theory BPS relation between charges. I. THE SUPERSYMMETRIC CASIMIR ENERGY
In [1, 2] a new observable of d = 4 superconformalfield theories has been introduced: the supersymmetricCasimir energy. This is defined by putting the theoryon certain curved backgrounds M = S β × M , where S β is a circle of length β and M is a compact three-manifold. These are rigid supersymmetric backgrounds,and the supersymmetric Casimir energy is defined as E susy = − lim β →∞ dd β log Z susy S β × M . (1)Here the partition function Z susy is computed with pe-riodic boundary conditions for the fermions around S β .A key point is that, unlike the vacuum energy of general d = 4 conformal field theories (CFTs), E susy is scheme-independent and thus an intrinsic observable.The rigid supersymmetric backgrounds of interest com-prise a metric on M of the form g = d τ + g = d τ + (d ψ + a ) + 4e w d z d¯ z , (2)where τ ∼ τ + β is a coordinate on S β . The vector ∂ ψ is Killing, and generates a transversely holomorphic foli-ation of M , with local transverse complex coordinate z . The local one-form a satisfies d a = i u e w d z ∧ d¯ z ,where w = w ( z, ¯ z ), u = u ( z, ¯ z ). In addition there is anon-dynamical Abelian gauge field, which couples to theR-symmetry current and arises when the field theory iscoupled to background conformal supergravity, given by A = i8 u d τ + u (d ψ + a ) + i4 ( ∂ ¯ z w d¯ z − ∂ z w d z )+ γ d ψ + d λ ( z, ¯ z ) . (3)Notice that the second line is locally pure gauge; however,the constant γ will play an important role.The background geometry thus depends on the choiceof the two functions w ( z, ¯ z ), u ( z, ¯ z ), and via (1) the su-persymmetric Casimir energy also a priori depends onthis choice. These backgrounds admit two superchargesof opposite R-charge, and associated to each of theseis an integrable complex structure ( i.e. they are ambi-Hermitian). In [3] it is argued that the supersymmetricpartition function depends on the background only via the choice of complex structure(s). In the present set-up,this implies that Z susy depends only on the transverselyholomorphic foliation generated by ∂ ψ . In particular, de-formations of w ( z, ¯ z ) and u ( z, ¯ z ) that leave this foliationfixed should not change E susy .Later in this paper we will focus on the case that topo-logically M ∼ = S . Here we may embed S ⊂ R = R ⊕ R , and write ∂ ψ = b ∂ ϕ + b ∂ ϕ , where ϕ , ϕ are standard 2 π periodic azimuthal angles. In this casethe above statements imply that E susy should dependonly on b , b , and the explicit calculation in [1] gives E susy = 2( b + b ) b b (3 c − a ) + 23 ( b + b )( a − c ) . (4)Here a and c are the usual trace anomaly coefficientsfor a d = 4 CFT. For field theories admitting a large N gravity dual in type IIB supergravity, to leading orderin the N → ∞ limit one has a = c = π /κ , where κ is the five-dimensional effective gravity constant and wehave set the AdS radius to 1. In this limit (4) reduces to E susy = ( b + b ) b b π κ . (5)In particular the conformally flat S β × S , where M ∼ = S is equipped with the standard round metric of radius r ,has b = b = 1 /r , leading to E susy = 16 π / r κ . Wewill reproduce (5) from a dual supergravity calculation. II. DUAL SUPERGRAVITY SOLUTIONS
The gravity duals are constructed in d = 5 minimalgauged supergravity, whose solutions uplift to type IIBsupergravity. In Euclidean signature, the bosonic part ofthe action reads S bulk = − κ Z M h d x √ det G ( R G − F µν F µν + 12) − √ A ∧ F ∧ F i . (6) In this paper our conventions are such that b , b > Here G = ( G µν ) denotes the five-dimensional metric, R G is its Ricci scalar, A is the graviphoton and F = d A .We are interested in supersymmetric solutions that areasymptotically locally Anti-de Sitter (AlAdS), with met-ric and graviphoton on the conformal boundary givenby (2) and (3). Employing a coordinate system definedcanonically by supersymmetry, we have solved the super-symmetry conditions and equations of motion in a seriesexpansion near the boundary. We have then cast the so-lution in Fefferman-Graham coordinates [4], where themetric is G = d ρ /ρ + h ij ( x, ρ )d x i d x j , and h = 1 ρ h h (0) + h (2) ρ + (cid:16) h (4) + ˜ h (4) log ρ (cid:17) ρ + O ( ρ ) i , A = A (0) + (cid:16) A (2) + ˜ A (2) log ρ (cid:17) ρ + O ( ρ ) . (7) Here the conformal boundary is at ρ = 0. The terms atleading order in the expansions, h (0) ≡ g and A (0) ≡− A/ √
3, coincide with the metric (2) and gauge field (3),respectively. These depend only on the functions w ( z, ¯ z )and u ( z, ¯ z ), which we therefore refer to as boundary func-tions . h (2) , ˜ h (4) , and ˜ A (2) are uniquely fixed in terms ofthese, whereas h (4) and A (2) are not determined by theconformal boundary, and parametrize the one-point func-tions of the dual field theories. These depend on four newfunctions k ( z, ¯ z ), k ( z, ¯ z ), k ( z, ¯ z ) and k ( z, ¯ z ), that werefer to as non-boundary functions . The first three ofthese appear in the expansion of the gauge field: A (2) = √ h (cid:0) − k − uk + 4 u (cid:3) w + u (cid:1) id τ + 1 u (cid:16) k − uk − k + 16 (cid:3) k − k (cid:3) w − u k + 3 (cid:3) ( (cid:3) w + u ) − (cid:3) w ) − u (cid:3) w − − w ∂ z u∂ ¯ z u − u (cid:17) ( − id τ + d ψ + a ) − ∗ d (cid:0) k + u (cid:1)i , (8)˜ A (2) = √ (cid:2) (cid:3) u id τ + (cid:0) (cid:3) u − u (cid:3) w − u (cid:1) (d ψ + a ) + ∗ d (cid:0) (cid:3) w + u (cid:1)(cid:3) , where (cid:3) ≡ e − w ∂ z ∂ ¯ z and ∗ d ≡ i(d¯ z ∂ ¯ z − d z ∂ z ). A moreexhaustive discussion will be presented in [5].The bulk action evaluated on a solution is divergentand must be renormalized by the addition of countert-erms. As usual, we include the Gibbons-Hawking term S GH = − κ Z ∂M ǫ d x √ det h K , (9)to have a well-defined variational principle. Here h is themetric (7) induced on a four-dimensional hypersurface ∂M ǫ = { ρ = ǫ = constant } , and K the trace of its secondfundamental form. The counterterms S ct = 1 κ Z ∂M ǫ d x √ det h (cid:0) R h (cid:1) , (10)cancel all divergences as ǫ →
0. In general there is also alog ǫ divergence in the action, related to the field theoryWeyl anomaly; but in the limit ǫ → E ≡ C ijkl C ijkl ≡ F ij F ij , andthis term vanishes identically [6]. Here R h , C ijkl and E are the Ricci scalar, Weyl tensor and Euler densityof the metric h , respectively. We also include a linearcombination of the standard finite counterterms∆ S st = − κ Z ∂M ǫ d x √ det h (cid:0) ς R h − ς ′ F ij F ij (cid:1) , (11)where ς and ς ′ are arbitrary constants. These affect theholographic one-point functions, as well as the on-shellaction. The ordinary renormalized action is obtained as S ≡ lim ǫ → ( S bulk + S GH + S ct + ∆ S st ) . (12) A variation of the total on-shell action with respect toboundary data takes the form δS = Z M d x p det g (cid:16) − T ij δg ij + j i δA i (cid:17) , (13)where g is the ρ -independent metric (2) on the confor-mal boundary. The holographic energy-momentum ten-sor T ij and R-symmetry current j i may be computedwith standard formulas (see e.g. [7]). The former is par-ticularly unwieldy, but we have verified that these sat-isfy the expected Ward identities. In particular, the R-symmetry current is conserved, ∇ i j i = 0, and the energy-momentum tensor obeys the correct conservation equa-tion ∇ i T ij = j i F ji , (14)with F = d A . However, we will show next that imposingsupersymmetric Ward identities requires a non-standardmodification of the holographic renormalization scheme. III. SUPERSYMMETRIC HOLOGRAPHICRENORMALIZATION
According to the gauge/gravity duality, the renormal-ized on-shell gravitational action is identified with minusthe logarithm of the partition function of the dual fieldtheory, in the large N limit. Namely Z susy S β × M = e − S [ M ] , (15)where S [ M ] is evaluated on an appropriate supergrav-ity solution, as described in the previous section. As-suming (15), the field theory results summarized in thefirst section imply that S should be invariant under de-formations of the boundary geometry that leave fixed the transversely holomorphic foliation generated by ∂ ψ .Concretely, this implies that S should be invariant under w → w + δw , u → u + δu , where δw ( z, ¯ z ), δu ( z, ¯ z ) are ar-bitrary smooth global functions on M , invariant under ∂ ψ . The corresponding variation of S may be computedexplicitly using the general formula (13). We find δ w S = Z M d x √ det g κ δw (cid:2) (1 − ς + 16 ς ′ ) u R d + (1 − ς + 28 ς ′ ) (cid:3) u + 12 ς ′ u (cid:3) u − (19 − ς + 192 ς ′ ) u − − ς + ς ′ )( R d + 2 (cid:3) R d ) + γ (2 uR d + 2 (cid:3) u − u ) (cid:3) , (16) δ u S = Z M d x √ det g κ δu (cid:2) − − ς + 16 ς ′ ) uR d − ς ′ (cid:3) u + (19 − ς + 192 ς ′ ) u + γ (3 u − R d ) (cid:3) . Here R d ≡ − (cid:3) w is the Ricci scalar of the transverse two-dimensional metric 4e w d z d¯ z . We emphasize that thisis locally, but not globally, a total derivative. Noticethe dependence on the constant γ , which appears in theboundary gauge field A in (3). In the first variation in(16) we hold d a fixed, meaning that δ ( u e w ) = 0 andhence δu = − u δw ; while the second variation in (16)is the change in S under an arbitrary variation δu . Inobtaining these expressions we have used Stokes’ theoremto discard total derivative terms. In particular, we findthat all dependence on the non-boundary functions dropsout of these integrals, as does d λ ( z, ¯ z ) in (3).Crucially we see that there is no choice of ς , ς ′ forwhich these variations are zero for an arbitrary back-ground. The standard holographic renormalization of theprevious section hence does not correspond to the super-symmetric renormalization scheme used in field theory.This result explains why previous attempts to obtain theholographic supersymmetric Casimir energy have failed.Remarkably, we have found that if we define the new“finite counterterms”∆ S new = − κ Z M (i A ∧ Φ + Ψ) , (17)whereΦ ≡ (cid:0) u − uR d (cid:1) i e w d z ∧ d¯ z ∧ (2 d ψ + i d τ ) , Ψ ≡ (cid:0) u − u R d (cid:1) d x p det g , (18)then (16) implies that S susy ≡ lim ǫ → ( S bulk + S GH + S ct ) + ∆ S new (19)is invariant under w → w + δw , u → u + δu . We claim that(19) is the correct renormalized supergravity action forthe class of backgrounds introduced in the first section,in the sense that this corresponds to the unique super-symmetric renormalization scheme used in field theory.In particular, this result should be valid for arbitrarytopology of M . Specializing to the case M ∼ = S , in the next section we shall not only show that (19) correctlyreproduces (5), but moreover we are able to determinethe holographic charges in this scheme, and prove thatthese satisfy the correct BPS relation in field theory. IV. ON-SHELL ACTION AND HOLOGRAPHICCHARGES
In general to evaluate the bulk action one needs toknow the full solution. However, with some additionaltopological assumptions, and assuming that a bulk fillingexists, one can compute S susy in (19) explicitly.We henceforth take M ∼ = S . In this case the bound-ary supercharges are sections of a trivial bundle, and cor-respondingly A in (3) is a global one-form. As shown in[1] this fixes the constant γ = ( b + b ) /
2, which phys-ically is the charge of the spinors under ∂ ψ . As in thesolution of [7], we assume the bulk filling is smooth withtopology S × R , with the bulk graviphoton A smoothlyextending A on the boundary. These assumptions, to-gether with supersymmetry, allow one to write the bulkaction as a total derivative, and hence express S susy as thelimit of a term evaluated near the conformal boundary.However, this expression still depends on non-boundaryfunctions, which are only determined by regularity in thedeep interior of the solution. Fortunately, we may bypassthis problem using another idea from [7]. If C ∼ = R isa regular hypersurface at τ = constant, with boundary M ∼ = S at infinity, then combining the Maxwell equa-tion and Stokes’ theorem on C one can show that Z M (cid:16) ∗ F + √ A ∧ F (cid:17) = 0 . (20)Substituting (7) and (8) in, this identity may be used toeliminate all dependence of the on-shell action on non-boundary functions. Also discarding terms which aretotal derivatives on M , and noting that (17) leads toextensive cancellations, (19) evaluates to the remarkableformula S susy = γ κ Z M d x p det g R d . (21)We reiterate that this has been derived here for M ∼ = S β × S , although as we shall explain in [5] this formulahas larger validity. As remarked earlier, R d is locally butnot globally a total derivative. Its integral is a topologicalinvariant of the foliation, proportional to the transversefirst Chern class. Using the explicit formulas in [1] forthe metric functions and coordinate ranges for M ∼ = S with ∂ ψ = b ∂ ϕ + b ∂ ϕ , we find Z M d x p det g R d = 2(2 π ) b + b b b . (22)Substituting this into (21), using γ = ( b + b ) / τ has period β , we find that S susy = βE susy , where E susy is the field theory result (5)!The above argument applies to any solution with topol-ogy S × R , but it is worth emphasizing that thereare explicit examples. The new counterterms (17) arenon-zero even for AdS in global coordinates, whoseboundary is the conformally flat S β × S geometry with b = b = 1 /r mentioned at the end of the first section.The solution of [7] has a squashed S β × S v boundary, withthe bulk solution depending non-trivially on the squash-ing parameter v . However, b = b = 1 /vr , and we findthat S susy is a simple rescaling of the action of AdS .Finally, we turn to the holographic charges. Let usstart from the standard charges, which may be obtainedfrom T ij and j i (defined through (12), (13)). Due tothe Ward identity (14) the canonical Hamiltonian H andangular momentum J associated to translations along ∂ τ and − ∂ ψ are defined as H ≡ Z M d x p det g ( T ττ + j τ A τ ) ,J ≡ i Z M d x p det g ( T τψ + j τ A ψ ) , (23)respectively. On the other hand, the holographic R-charge is defined as Q ≡ − i Z M d x p det g j τ . (24)In the dual field theory, these are identified with the vevof the corresponding operators h H i , h J i , and h Q i .Utilizing a trick introduced in [7] and elaborated in [5],one can then show that βH = S and J = 0 . (25)Recall that the supersymmetry algebra implies that inthe field theory vacuum the BPS relation h H i + h J i + γ h Q i = 0 , (26) should hold, with h H i = E susy [2]. However, for theEuclidean AdS solution, which is expected to correspondto the vacuum of theories in conformally flat space, onefinds that J | EAdS = Q | EAdS = 0, implying that (26) isviolated.Assuming that the identity (13) holds replacing S with S susy , and correspondingly T ij → T susy ij , j i → j susy i , wecan define “supersymmetric” versions of the holographiccharges, via formulas analogous to (23) and (24). In par-ticular, the improved electric charge may be defined as Q susy ≡ − i Z M δS susy δA τ = Q − κ Z M Φ , (27)and by direct computation we find γQ susy = − β S susy . (28)Moreover, using the relations (25) applied to the im-proved Hamiltonian and angular momentum, we deducethat βH susy = S susy and J susy = 0, thus showing thatthese obey the BPS relation (26). V. CONCLUDING REMARKS
We have constructed new boundary terms of five-dimensional minimal gauged supergravity that we arguedare necessary to restore supersymmetry of the gravita-tional action in a large class of AlAdS solutions. In-cluding these counterterms, we have reproduced the su-persymmetric Casimir energy and the field theory BPSrelation between charges [1, 2]. More details, as wellas a number of generalizations, will be presented in [5].For example, we will perform an analogous computationin four-dimensional gauged supergravity, finding that nonew counterterms are needed. In five dimensions we willconsider M with more general topology, making contactwith [8], as well as a twisting of S β over M .P.B.G. is supported by EPSRC and a Scatcherd Schol-arship. D.C. is supported by the European CommissionMarie Curie Fellowship PIEF-GA-2013-627243. D.M. ac-knowledges support from ERC Starting Grant N. 304806. [1] B. Assel, D. Cassani and D. Martelli, “Localiza-tion on Hopf surfaces,” JHEP , 123 (2014),[arXiv:1405.5144].[2] B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski,J. Lorenzen and D. Martelli, “The Casimir Energy inCurved Space and its Supersymmetric Counterpart,”JHEP , 043 (2015), [arXiv:1503.05537].[3] C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Ko-margodski, “The Geometry of Supersymmetric PartitionFunctions,” JHEP , 124 (2014), [arXiv:1309.5876]. [4] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holo-graphic reconstruction of space-time and renormalizationin the AdS / CFT correspondence,” Commun. Math.Phys. (2001) 595, [hep-th/0002230].[5] P. B. Genolini, D. Cassani, D. Martelli and J. Sparks,“Holographic renormalization and supersymmetry,”arXiv:1612.06761 [hep-th].[6] D. Cassani and D. Martelli, “Supersymmetry on curved spaces and superconformal anomalies,” JHEP (2013) 025, [arXiv:1307.6567].[7] D. Cassani and D. Martelli, “The gravity dual of super-symmetric gauge theories on a squashed S × S ,” JHEP , 044 (2014), [arXiv:1402.2278].[8] D. Martelli and J. Sparks, “The character of the su-persymmetric Casimir energy,” JHEP1608