One Loop Tests of Supersymmetric Higher Spin Ad S 4 /CF T 3
PPrepared for submission to JHEP
One Loop Tests of Supersymmetric Higher Spin
AdS / CFT Yi Pang, Ergin Sezgin and Yaodong Zhu a Max-Planck-Insitut f¨ur Gravitationsphysik (Albert-Einstein-Institut) Am M¨uhlenberg 1, DE-14476Potsdam, Germany b George and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We compute one loop free energy for D = 4 Vasiliev higher spin gravitiesbased on Konstein-Vasiliev algebras hu ( m ; n | ho ( m ; n |
4) or husp ( m ; n |
4) and subject tohigher spin preserving boundary conditions, which are conjectured to be dual to the U ( N ), O ( N ) or U Sp ( N ) singlet sectors, respectively, of free CFTs on the boundary of AdS .Ordinary supersymmetric higher spin theories appear as special cases of Konstein-Vasilievtheories, when the corresponding higher spin algebra contains OSp ( N |
4) as subalgebra. In
AdS with S boundary, we use a modified spectral zeta function method, which avoidsthe ambiguity arising from summing over infinite number of spins. We find that thecontribution of the infinite tower of bulk fermions vanishes. As a result, the free energy isthe sum of those which arise in type A and type B models with internal symmetries, theknown mismatch between the bulk and boundary free energies for type B model persists,and ordinary supersymmetric higher spin theories exhibit the mismatch as well. The onlymodels that have a match are type A models with internal symmetries, corresponding to n = 0. The matching requires identification of the inverse Newton’s constant G − N with N plus a proper integer as was found previously for special cases. In AdS with S × S boundary, the bulk one loop free energies match those of the dual free CFTs for arbitrary m and n . We also show that a supersymmetric double-trace deformation of free CFT basedon OSp (1 |
4) does not contribute to the O ( N ) free energy, as expected from the bulk. a r X i v : . [ h e p - t h ] A ug ontents AdS with S boundary 6 S and comparison 125 One loop free energies of supersymmetric higher spin theories in AdS with S β × S boundary 15 N = 1 SCFT 207 Conclusions 23A Comparison of ζ (∆ ,s ) ( z ) with (cid:101) ζ (∆ ,s ) ( z ) A.1 Bosonic case 27A.2 Fermionic case 28
It has been known for sometime that the conjectured holographic duals of higher spin(HS) gravities [1] can be as simple as free CFTs living on the boundary of anti-de Sit-ter spacetime. Moreover, it has also been noted that the duality is expected to arise inweakly coupled regimes of both bulk and boundary field theories. Therefore, one expectsthat higher spin AdS/CFT correspondence should be amenable to test order by order inperturbation theory. – 1 –ree CFTs arise in conjectured dualities in the context of parity invariant HS gravitiesin 4D subject to HS symmetry preserving boundary conditions. There are two typesof parity invariant Vasiliev HS gravities, known as type A and B [2]. In their simplestforms, they both contain an infinite tower of massless even spin fields, each occurring once.They differ from each other in the parity of the spin-0 field, which is parity even (odd)in type A (B) theory. It has been conjectured that type A theory with ∆ = 1 boundarycondition imposed on the scalar is dual to the O ( N ) singlet sector of N free real scalars [3],while type B theory with ∆ = 2 boundary condition imposed on the pseudoscalar is dualto the O ( N ) singlet sector of N free Majorana fermions [2] (for earlier work in which HSholography involving CFTs with matrix valued free fields, see [4]). These are HS symmetrypreserving boundary conditions, with standard boundary conditions imposed on all otherfields understood. The dual CFT can be altered by changing the boundary conditionsimposed on the spin-0 field in such a way that they break HS symmetry. For instance,type A model with ∆ = 2 boundary condition on the scalar is conjectured to be dual tothe critical O ( N ) vector model [3], while type B model with ∆ = 1 boundary conditionimposed on the pseudoscalar is conjectured to be dual to O ( N ) Gross-Neveu model [2].An important test of the holography is to match the free energy of the bulk theorywith that of the CFT defined on the conformal boundary of the bulk geometry. Assumingthe bulk HS theory possesses an action formulation, the partition function evaluated onEuclidean AdS can be expanded in terms of G N as F bulk = 1 G N F (0)bulk + F (1)bulk + G N F (2)bulk + · · · . (1.1)When the bulk Euclidean AdS is the hyperbolic space H whose conformal boundary is around S , the free energy of the bulk HS theory should match with that of a free CFT ona round S . The free energy of a free CFT on S takes the simple form [5] F CFT = N F (0)CFT , (1.2)where F (0) CF T is the free energy of a single component in U ( N ) or O ( N ) vector model. Thezeroth-order contribution F (0)bulk has not been computed so far due to the lack of an actionfor Vasiliev theory with all the required properties. We will return to this point in theconclusions. Matching F bulk with F CFT necessarily requires that F bulk is proportional to F (0)CFT at each order in the small G N expansion and that G N is identified in terms of N as G − N → γ ( N + ∆ N ) , (1.3)with γ and ∆ N being constants, and ∆ N should be a fixed integer for a given bulk/boundarydual pair. Therefore, the higher order quantum affects simply the relation between G N and N . Assuming Fronsdal type quadratic action for the massless HS fields, one loop compu-tations have shown that these requirements are fulfilled in the conjectured duality betweentype A theory and the bosonic O ( N ) vector model [6]. However, for the conjectured du-ality between type B theory and the fermionic O ( N ) vector model [2], these requirementsare not satisfied since F (1)bulk and F (0)CFT are not proportional to each other. Matching of– 2 –ree energy was also found in the type A/critical O ( N ) vector duality, but not in the typeB/ O ( N ) Gross-Neveu duality. In critical O ( N ) vector model, the conformal dimensions ofHS currents receive quantum corrections. The leading 1 /N corrections are summarized in[7]. These anomalous dimensions of HS currents at O (1 /N ) should be compared with theone loop corrections to the AdS energies of HS fields computed directly from the bulk HStheory. It would be interesting to check whether they match precisely.The principal aim of this paper is to extend the one loop tests by computing thefree energies in a wider class of HS theories in 4D that are expected to be dual to freeCFTs on the boundary of
AdS . In particular, we wish to study the consequences ofsupersymmetry which combine type A and type B spectra of fields with an infinite towerof massless fermions. The underlying HS algebras, denoted by hu ( m ; n | ho ( m ; n | husp ( m ; n | SO (3 ,
2) which also carry fundamentalrepresentations of classical Lie groups. Vasiliev equations for these theories are describedin detail in [9]. Their spectral properties will be summarized in the next section. Sufficesto mention here that generically their underlying HS algebras serve as infinite dimensionalsupersymmetry algebras, and only in special cases, namely when m = n = 2 k for some k corresponds to the fundamental spinor representation of O ( N ), they contain the AdS superalgebra OSp ( N | R -symmetry group SO ( N ) . We shall also consider extension ofthese models by introduction of internal symmetry [9].When the boundary of AdS is S , we compute the one loop free energy by using themodified spectral zeta function method, which avoids the ambiguity arising from summingover infinite number of spins. As a side result, we obtain the contributions of the even andodd spin towers of HS fields separately. Furthermore we find that the contribution of theinfinite tower of fermionic fields to the free energy vanishes. Putting all results together,we find that the bulk free energy may match that of the dual free CFT only for type Amodels. Their spectrum consist of bosonic fields arising from the tensor product of twobosonic singletons in fundamental representation of classical Lie groups. The matchingrequires identification of the inverse Newton’s constant G − N with N plus a proper integeras was found previously for special cases. Note that mismatch in the free energy at one loopoccurs in particular for type B models whose spectrum consists of bosonic fields arisingfrom the tensor product of two spinor singletons in fundamental representation of classicalLie groups.When AdS is written in the thermal AdS coordinates, with the boundary being S × S , we find that the bulk one loop free energies match those of the dual free CFTsfor generic Konstein-Vasiliev models. In order to distinguish the notion of supersymmetry in generic Konstein-Vasiliev models versus thespecial cases where
OSp ( N |
4) arises as a subalgebra, we shall sometimes refer to the latter ones as “ordinarysupersymmetric HS theories”. – 3 –he N = 1 higher spin theory admits N = 1 mixed boundary condition which corre-sponds to adding a supersymmetric double-trace deformation in the free CFT. We showthat such a double-trace deformation does not contribute to the O ( N ) free energy, com-patible with the fact that imposing mixed boundary condition does not change the bulkspectrum and therefore the bulk one loop free energy remains the same.The rest of the paper is organized as follows. In Section 2 we review the spectra of HSgravities based on HS algebras hu ( m ; n | ho ( m ; n |
4) and husp ( m ; n | AdS with S boundary, where wealso consider the ordinary supersymmetric HS theories with internal symmetry. We adoptan alternate regularization scheme introduced in [10] in the bosonic sector, then generalizethe method also to the fermionic sector. In Section 4, we compare the results obtained inthe bulk with the corresponding ones in the boundary CFTs. In Section 5, we implementthe one loop test to HS theories in thermal AdS with the dual CFTs on boundary S × S .In Section 6 we study a possible mixed boundary condition for N = 1 higher spin theoryand the effect on the free energy on the CFT side where a supersymmetric double-tracedeformation is turned on. We summarize and comment on our results in Section 7, andcomment on possible ways to approach the problem of mismatch of free energies in type Band ordinary supersymmetric HS theories and their conjectured duals. We also commenton the action formulation proposed in [11] in the context of classical free energy in thebulk. The validity and detailed calculation of the alternate regularization method adoptedin this paper are shown in Appendix A. The group theoretical building blocks for the construction of the physical spectra of HStheories in
AdS are the singleton representations of SO (3 , D ( E , s ) for the discrete unitaryrepresentations of sp (4; R ) ∼ SO (3 , E is the lowest energy and s is the spin ofthe lowest weight state, Di refers to the D (1 , /
2) and Rac refers to the D (1 / ,
0) repre-sentations. An important property these representations have is given by Flato-Fronsdaltheorem which states thatRac ⊗ Rac = ∞ (cid:88) s =0 D (1 + s, s ) , Di ⊗ Di = D (2 ,
0) + ∞ (cid:88) s =1 D (1 + s, s ) , Di ⊗ Rac = ∞ (cid:88) s =0 D (3 / s, / s ) , (2.1)where s = 0 , , , ... . The representations D (1 + s, s ) are massless spin s fields, and D (2 , S + := (Rac , m ) ⊕ (Di , n ) , S − := (Di , m ) ⊕ (Rac , n ) . (2.2)– 4 –here m labels the fundamental representations of u ( m ) or usp ( m ) or a vector represen-tation of so ( m ). It has been shown that the physical spectra of three types of HS theories,based on HS algebras denoted by hu ( m ; n | , ho ( m ; n | , husp ( m ; n | hu ( m ; n |
4) : S + ⊗ ¯ S + , hu ( n ; m |
4) : S − ⊗ ¯ S − , (2.3) ho ( m ; n |
4) : ( S + ⊗ S + ) S , ho ( n ; m |
4) : ( S − ⊗ S − ) S , (2.4) husp ( m ; n |
4) : ( S + ⊗ S + ) A , husp ( m ; n |
4) : ( S − ⊗ S − ) A , (2.5)where ( · ) S and ( · ) A stand for symmetric and antisymmetric tensor products, respectively.These algebras contain u ( m ) ⊗ u ( n ), o ( m ) ⊗ o ( n ) and usp ( m ) ⊗ usp ( n ) as maximal bosonicsubalgebras. The resulting spectra are as follows [8] hu ( m ; n |
4) : ( m − , ⊕ (1 , n − ⊕ (1 , ⊕ (1 , s = 0 , , , , . . . ( m, ¯ n ) ⊕ ( ¯ m, n ) s = , , , . . .ho ( m ; n |
4) : ( m ( m − , ⊕ (1 , n ( n − s = 1 , , . . . ( m ( m + 1) − , ⊕ (1 , n ( n + 1) − ⊕ (1 , ⊕ (1 , s = 0 , , , . . . ( m, n ) s = , , , . . .husp ( m ; n |
4) : ( m ( m + 1) , ⊕ (1 , n ( n + 1)) s = 1 , , . . . ( m ( m − − , ⊕ (1 , n ( n − − ⊕ (1 , ⊕ (1 , s = 0 , , , . . . ( m, n ) s = , , , . . . , (2.6)where the dimensions of the representations are shown. While there are the isomorphisms hu ( m ; n | ∼ hu ( n ; m | ho ( m ; n | ∼ ho ( n ; m |
4) and husp ( m ; n | ∼ husp ( n ; m | { m , m ( m +1) / , m ( m − / } scalars in D (1 ,
0) representations, and { n , n ( n +1) / , n ( n − / } scalars in D (2 ,
0) representations of SO (3 , hu ( m ; n | ho ( m ; n | husp ( m ; n |
4) respectively. The models with mn > m = n = 2 N/ − or m = n = 2 ( N − / ,these algebras do not contain a finite dimensional superalgebra and as such they are infinitedimensional algebras. In the case of m = n = 2 N/ − , the Rac and Di belong to left andright handed fundamental spinor representations of SO ( N ) and we have the isomorphisms shs E ( N | ∼ = hu (cid:16) N − ; 2 N − (cid:12)(cid:12)(cid:12) (cid:17) N = 2 mod 4 ,husp (cid:16) N − ; 2 N − (cid:12)(cid:12)(cid:12) (cid:17) N = 4 mod 8 ,ho (cid:16) N − ; 2 N − (cid:12)(cid:12)(cid:12) (cid:17) N = 8 mod 8 . (2.7)– 5 –he HS superalgebra shs E ( N |
4) contains the N extended AdS superalgebra OSp ( N | m = n = 2 ( N − / , the Di and Rac belong to the 2 ( N − / dimensional fundamental spinor representations of SO ( N ) and we have the isomorphisms shs E ( N | ∼ = ho (cid:16) ( N − / ; 2 ( N − / (cid:12)(cid:12)(cid:12) (cid:17) N = 1 mod 8 ,husp (cid:16) ( N − / ; 2 ( N − / (cid:12)(cid:12)(cid:12) (cid:17) N = 5 mod 8 . (2.8)As for the case of N =3 mod 4, it has been shown in [9] that it is equivalent to the case of N =4 mod 4. The OSp ( N |
4) supermultiplet content of the spectra described above can bedetermined in a straightforward way but this information is not needed for the purposesof this paper.The supersymmetric HS models described above can be extended by introduction ofinternal symmetry. In this case, the Di and Rac representations not only carry the spinorrepresentation of SO ( N ) but also a fundamental representation of a classical Lie algebra.Working out their tensor products yields the spectrum of the expected dual HS theory,which can be found in Table 5 of [9]. AdS with S boundary In this section we shall compute the free energy of Konstein-Vasiliev HS theories in
AdS with S boundary, imposing the HS symmetry preserving boundary conditions. Free energyof bosonic HS fields in AdS has been studied in [6, 12–14]. The regularization scheme thathas been used in summing over infinite tower of HS fields, however, is very complicated.Here, we employ an alternate method which is much simpler, utilizing the character ofirreducible representation of SO (2 , F (1) = − log Z (1) where Z (1) is the one loop partition function. For HS theory with n S real scalars, n P pseudoscalars, n copies of fields with s = 1 , , ..., ∞ , n copies of fields with s = 2 , , ..., ∞ fields and n F – 6 –opies of spin 1 / , / , ..., ∞ fields, we have F (1) ( n S , n P , n , n , n F ) = n S log det D B (1 ,
0) + n P log det D B (2 , n ∞ (cid:88) k =0 (cid:104) log det D B (2 k + 2 , k + 1) − log det D B (2 k + 3 , k ) (cid:105) (3.1)+ n ∞ (cid:88) k =1 (cid:104) log det D B (2 k + 1 , k ) − log det D B (2 k + 2 , k − (cid:105) − n F log det D F ( , ) − n F ∞ (cid:88) k =1 (cid:104) log det D F ( k + , k + ) − log det D F ( k + , k − ) (cid:105) , where we have defined D B (∆ , s ) = (cid:2) −∇ + ∆(∆ − − s (cid:3) , D F (∆ , s ) = (cid:104) − / ∇ + ∆(∆ −
3) + (cid:105) . (3.2)The negative contributions in the bosonic sector and the positive contributions in thefermionic sector are due to ghosts. In computing det and det , the irregular (∆ − = 1)and regular (∆ + = 2) boundary conditions are to be used.For a differential operator of the form D = −∇ + X , or D = − / ∇ + Y , writing − log det D = (cid:90) ∞ dtt K D ( t ) , K D ( t ) := Tr (cid:2) e − t D (cid:3) , (3.3)and defining the spectral zeta function ζ D ( z ) := 1Γ( z ) (cid:90) ∞ dt t z − K D ( t ) , (3.4)one finds the standard result [15] − log det D = ζ D (0) log( (cid:96) Λ ) + ζ (cid:48)D (0) , (3.5)where (cid:96) is the AdS radius and Λ is the renormalization scale. For fields of aribrary spinsin hyperbolic space H , the spectral zeta function technique has been developed in [16, 17]to compute their one loop effective potentials.– 7 – .1 Bosons Upon Euclideanization of
AdS to H , the boundary is S and in this setting various freeenergies of the bosonic HS theory are given by F (1)even 1 = − (cid:104) ζ B (1 , (0) + ∞ (cid:88) s =2 , , ··· (cid:16) ζ B ( s +1 ,s ) (0) − ζ B ( s +2 ,s − (0) (cid:17)(cid:105) log( (cid:96) Λ ) − (cid:104) ζ B (cid:48) (1 , (0) + ∞ (cid:88) s =2 , , ··· (cid:16) ζ B (cid:48) ( s +1 ,s ) (0) − ζ B (cid:48) ( s +2 ,s − (0) (cid:17)(cid:105) ,F (1)even 2 = − (cid:104) ζ B (2 , (0) + ∞ (cid:88) s =2 , , ··· (cid:16) ζ B ( s +1 ,s ) (0) − ζ B ( s +2 ,s − (0) (cid:17)(cid:105) log( (cid:96) Λ ) − (cid:104) ζ B (cid:48) (2 , (0) + ∞ (cid:88) s =2 , , ··· (cid:16) ζ B (cid:48) ( s +1 ,s ) (0) − ζ B (cid:48) ( s +2 ,s − (0) (cid:17)(cid:105) ,F (1)odd = − ∞ (cid:88) s =1 , , ··· (cid:16) ζ B ( s +1 ,s ) (0) − ζ B ( s +2 ,s − (0) (cid:17) log( (cid:96) Λ ) − ∞ (cid:88) s =1 , , ··· (cid:16) ζ B (cid:48) ( s +1 ,s ) (0) − ζ B (cid:48) ( s +2 ,s − (0) (cid:17) , (3.6)where F (1)even 1 and F (1)even 2 denote the total free energy of all even spin fields s = 0 , , · · · ,in which the scalar satisfies ∆ = 1 and ∆ = 2 boundary conditions, respectively, and F (1)odd denotes the total free energy of all odd spin fields s = 1 , , · · · .As stated earlier, we now employ a simpler method than those used previously, uti-lizing the character of irreducible representation of SO (2 , s field can be recast in theform ζ B (∆ ,s ) ( z ) = 1Γ( z ) (cid:90) ∞ dβ (cid:104) µ ( z, β ) + ν ( z, β ) ∂ ∂α (cid:105) χ ∆ ,s ( β, α ) (cid:12)(cid:12)(cid:12) α =0 , (3.7)in which χ ∆ ,s ( β, α ) = e − β (∆ −
32 ) sin[( s + ) α ]4 sinh β sin α (cosh β − cos α ) ,µ ( z, β ) = sinh β (cid:104) f ( z, β ) (cid:16) − β (cid:17) + 4 f ( z, β ) sinh β (cid:105) ,ν ( z, β ) = − f ( z, β ) sinh β ,f n ( z, β ) = √ π (cid:90) ∞ duu n tanh( πu )( β u ) z − J z − / ( uβ ) , (3.8)where χ ∆ ,s ( β, α ) is the character of a representation of SO (3 ,
2) labeled by D (∆ , s ). Owingto the e − β (∆ −
32 ) factor in the character, (cid:80) s ζ (∆ ,s ) ( z ) is convergent. Therefore, no regu-larization is needed in performing the sum over infinitely many spins. This is the desired– 8 –eature for computing the one loop free energy of HS theory where the summation overinfinitely many spins is encountered. It was also noticed by [10] that since the one loopfree energy depends only on ζ (0) and ζ (cid:48) (0), an alternate zeta function (cid:101) ζ ( z ) is physicallyequivalent to the original ζ ( z ), provided that (cid:101) ζ (0) = ζ (0), and (cid:101) ζ (cid:48) (0) = ζ (cid:48) (0). Thus, forthe convenience of calculation, one can in fact utilize an alternate zeta function which isphysically equivalent to the original zeta function. For bosonic HS fields, one choice of thealternate zeta function takes the form [10] (cid:101) ζ B (∆ ,s ) ( z ) = 1Γ(2 z ) (cid:90) ∞ dβ β z − coth β (cid:104) (cid:16) sinh β (cid:17) ∂ α (cid:105) χ ∆ ,s ( β, α ) (cid:12)(cid:12)(cid:12) α =0 . (3.9)The physical equivalence between the alternate spectral zeta function and the original one(3.7) is shown in the appendix. The total character of all even spin fields and that of allodd spin fields are computed as χ even 1 ( β, α ) = χ , ( β, α ) + (cid:88) s =2 , , ··· ( χ s +1 ,s ( β, α ) − χ s +2 ,s − ( β, α ))= 1 + cos α + cosh β + cosh 2 β α − cosh β ) (cos α + cosh β ) , (3.10) χ even 2 ( β, α ) = χ , ( β, α ) + (cid:88) s =2 , , ··· ( χ s +1 ,s ( β, α ) − χ s +2 ,s − ( β, α ))= 1 + cos α + cos 2 α + cosh β α − cosh β ) (cos α + cosh β ) , (3.11) χ odd ( β, α ) = (cid:88) s =1 , , ··· ( χ s +1 ,s ( β, α ) − χ s +2 ,s − ( β, α ))= cos α + cosh β + 2 cos α cosh β α − cosh β ) (cos α + cosh β ) . (3.12)Substituting the results above into (3.9), we find (cid:101) ζ B even,1 ( z ) = 1Γ(2 z ) (cid:90) ∞ dββ z − cosh β β , (cid:101) ζ B even,2 ( z ) = − z ) (cid:90) ∞ dββ z − β β , (cid:101) ζ B odd ( z ) = − (cid:101) ζ B even 1 ( z ) . (3.13)With the help of the following identities1sinh β = 2 β ∂ ∂x βx | x =1 −
12 sinh β , − z ζ (2 z, a z ) (cid:90) ∞ dββ z − e − aβ − e − β , (3.14)– 9 –here ζ ( a, b ) is the Hurwitz zeta function, we finally obtain (cid:101) ζ B even 1 ( z ) = 4 − (2+ z ) (cid:104) ζ (2 z, − ) + 4 ζ (2 z − , − ) + 8 ζ (2 z − , − )+(4 z − ζ (2 z ) + 3(4 z − ζ (2 z − − z − ζ (2 z − (cid:105) , (cid:101) ζ B even 2 ( z ) = 4 − (1+ z ) (cid:104) − ζ (2 z − , − ζ (2 z − ,
0) + (4 z − ζ (2 z ) − z ζ (2 z −
2) + 4 ζ (2 z − (cid:105) . (3.15)By using the relation between F (1) and spectral zeta function, one arrives at the results F (1)even 1 = 116 (cid:18) − ζ (3) π (cid:19) , F (1)even 2 = 116 (cid:18) − ζ (3) π (cid:19) ,F (1)odd = − F (1)even 1 . (3.16)Note that the potential logarithmic divergences in F (1)even 1 and F (1)even 2 have canceled out,and the above finite results are from (cid:101) ζ B (cid:48) (0) terms, in agreement with [6]. Furthermore,these results can be used as building blocks for the computation of the free energies of theKonstein-Vasiliev models we are interested in, thanks to the observation that for all thosemodels discussed in Section 2, it is always the case that n = n S + n P , (3.17)where we recall that n is number of copies of even fields with s = 2 , , . . . ∞ , n S is thenumber of scalars and n P is the number of pseudoscalars. We now compute the one loop free energy of all fermionic HS fields. The spectral zetafunction of a spin- s fermionic fields is given by ζ F (∆ ,s ) ( z ) = 1Γ( z ) (cid:90) ∞ dβ (cid:104) µ ( z, β ) + ν ( z, β ) ∂ ∂α (cid:105) χ ∆ ,s ( β, α ) (cid:12)(cid:12)(cid:12) α =0 , (3.18)where χ ∆ ,s ( β, α ) = e − β (∆ −
32 ) sin[( s + ) α ]4 sinh β sin α (cosh β − cos α ) ,µ ( z, β ) = sinh β (cid:104) f ( z, β ) (cid:16) − β (cid:17) + 4 f ( z, β ) sinh β (cid:105) ,ν ( z, β ) = − f ( z, β ) sinh β ,f n ( z, β ) = √ π (cid:90) ∞ duu n coth( πu )( β u ) z − J z − / ( uβ ) . (3.19)To compute the one loop free energy of all fermionic HS fields, we propose the followingalternate spectral zeta function, which is much easier to use. The physical equivalence– 10 –etween the alternate spectral zeta function (3.20) and the original one (3.18) is shown inthe appendix. (cid:101) ζ F (∆ ,s ) ( z ) = 1Γ(2 z ) (cid:90) ∞ dββ z − (cid:104) sinh β + 1sinh β + sinh β ∂ α (cid:105) χ ∆ ,s ( β, α ) (cid:12)(cid:12)(cid:12) α =0 . (3.20)The sum of characters of all fermionic HS fields is computed as χ , ( β, α ) + ∞ (cid:88) s =3 / (cid:104) χ s +1 ,s ( β, α ) − χ s +2 ,s − ( β, α ) (cid:105) = cos α cosh β (cos α − cosh β ) . (3.21)It is straightforward to check that (cid:104) sinh β + 1sinh β + (cid:16) sinh β (cid:17) ∂ α (cid:105) × (cid:16) χ , ( β, α ) + ∞ (cid:88) s =3 / (cid:104) χ s +1 ,s ( β, α ) − χ s +2 ,s − ( β, α ) (cid:105)(cid:17)(cid:12)(cid:12)(cid:12) α =0 = 0 , (3.22)which indicates that the total one loop free energy of fermionic HS fields in fact vanishes. For a Konstein-Vasiliev higher theory consisting of n S real scalars, n P pseudoscalars, n copies of fields with s = 1 , , ..., ∞ , n = n S + n P copies of fields with s = 2 , , ..., ∞ fieldsand n F copies of spin 1 / , / , ..., ∞ fields, we have F (1) ( n S , n P , n , n , n F ) = log 28 ( n S + n P − n ) − ζ (3)16 π (3 n S + 5 n P − n ) , (3.23)where we have used the relation n = n S + n P . The values of n S , n P and n can be readoff from (2.6) for various Konstein-Vasiliev models. Substituting them into the equationabove, we obtain hu ( m ; n |
4) : F (1) hu = − ζ (3)8 π n , (3.24) ho ( m ; n |
4) : F (1) ho = log 28 ( m + n ) − ζ (3)16 π (3 m + 4 n + n ) , (3.25) husp ( m ; n |
4) : F (1) husp = − log 28 ( m + n ) + ζ (3)16 π (3 m + 4 n − n ) . (3.26)The one loop free energy of husp ( m ; n |
4) model is related to the one of ho ( m ; n |
4) modelvia m → − m, n → − n . The ordinary supersymmetric HS models correspond to the cases m = n = 2 N − for even N and m = n = 2 ( N − / for odd N .As for the ordinary supersymmetric HS models with internal symmetries, we recallthat their spectra can be obtained by assigning fundamental representations of the internalsymmetry group to the OSp ( N |
4) singletons, and working out the their two-fold tensor– 11 –roducts. The resulting spectra are provided in Table 5 of [9]. In particular, the numberof fermions with s = mod 2 and s = mod 2 are the same. As a consequence, thecontributions of the fermions to the one loop free energy will continue to vanish since in(3.20) we found that fermions with each half integer spin occurring once give vanishingcontribution. Consequently, the bulk free energy becomes the sum of free energies of typeA and type B models with the desired internal symmetries. In this case both log 2 and ζ (3) terms will show up in the one loop free energy, and their coefficients involve n dependence, where n fund is the dimension of fundamental representation of the internalsymmetry group. This information is sufficient to perform the one loop test by means ofcomparing the bulk and boundary free energies, as we shall see at the end of next section. S and comparison The free energies of free scalars and free fermions which are conformally coupled to S have been studied in [5]. A conformally coupled free scalar and a free fermion on S aredescribed by the following two actions respectively S S = 12 (cid:90) d x √ g (cid:104) ( ∇ φ ) + 34 L φ (cid:105) , S D = 12 (cid:90) d x √ gψ † (i /Dψ ) , (4.1)where L is the radius of the round S . Free energies of the above two theories are definedas usual F S = − log Z S = 12 log det[Λ − O S ] , O = −∇ + 34 L ,F D = − log Z D = − log det[Λ − O D ] , O = i /D . (4.2)Using zeta function, F S and F D can be computed straightforwardly and the results are [5] F S = 116 (cid:16) − ζ (3) π (cid:17) , F D = 18 (cid:16) ζ (3) π (cid:17) . (4.3)Notice that the free energy of a Majorana fermion on S is F D .A bulk HS theory is conjectured to be dual to a free vector model when the boundaryconditions of the bulk fields preserve the HS symmetry [3, 4], which is the case here.Assuming the bulk HS theory possesses an action, its free energy associated with AdS should have the form displayed in (1.1) where G N is the Newton’s constant. In caseswhere the boundary of AdS is S , the bulk free energy should be compared with thatof a free vector model on S order by order in 1 /N expansion. Hence the comparisonrequires an identification between G N and N . It was suggested by [6] that in general therelation between G N and N is of the form given in (1.3) where γ and ∆ N are constantsand especially ∆ N should be an integer. The basic fields in the vector model constitutea vector in the fundamental representation of a classical Lie group, which can be U ( N ), O ( N ) or U Sp ( N ) in our cases. The free energy of a free vector model can be computed– 12 –xactly and be put in the form F CFT = N F (0)CFT , (4.4)where we use F (0)CFT to denote the contribution of a single component in the vector. For F bulk to match with F CFT , it is clear that the bulk free energy at each order in G N expansionshould all be proportional to F (0)CFT .Various one loop tests of HS holography have been carried out in the literature [6, 12].For instance, the non-minimal type A model is conjectured to be dual to the U ( N ) singletsector of N complex scalars. When HS symmetry is preserved by the boundary condition, F (1)bulk was found to be 0, indicating that G − N is identified with N at one loop order. Forminimal A model, the conjectured dual CFT is the O ( N ) singlet sector of N real scalars.In this case, F (1)bulk is equal to F S , the free energy of a real free scalar (4.3). Thus, matchingthe bulk and boundary free energies at one loop order requires G − N being identified with N −
1. The husp (2; 0 |
4) Vasiliev theory is conjectured to be dual to the
U Sp ( N ) singletsector of N complex scalars and F (1)bulk is equal to − F S . Therefore, for husp (2; 0 |
4) higherspin theory, G − N is identified with N + 1 at one loop order.In this section, we consider the cases in which the bulk HS symmetry is preservedby the boundary condition, thus the CFT duals are certain singlet sectors of free CFTscomposed by free scalars and free fermions. For the hu ( m ; n |
4) theory, the dual CFTconsists of
N m complex free scalars φ ia , i = 1 , , ...N , a = 1 , , ...m and N n
Dirac fermions ψ ir , r = 1 , , ...n . The m ∆ = 1 scalars and n ∆ = 2 pseudoscalars correspond to theoperators ¯ φ ia φ ib , ¯ ψ ia ψ ib . (4.5)Free energy of this theory is given by F CFT = N F (0)CFT , F (0)CFT = 2 mF S + nF D , (4.6)where F S and F D are given in (4.3).For the ho ( m ; n |
4) theory, the dual CFT consists of
N m real free scalars φ ia , i =1 , , ...N , a = 1 , , ...m and N n majorana fermions ψ ir , r = 1 , , ...n . The m ∆ = 1 scalarfields and n ∆ = 2 pseudoscalars correspond to the operators φ ia φ jb δ ij , ¯ ψ ia ψ jb δ ij . (4.7)The free energy is given by F CFT = N F (0)CFT , F (0)CFT = mF S + nF D . (4.8) Strictly speaking, the bulk HS theory is dual to the U ( N ), O ( N ) or USp ( N ) singlet sector of a freeCFT. The partition function of a free CFT on S is evaluated in the vacuum which is already a singletstate under the corresponding symmetry group in each case. Thus, imposing the singlet constraint shouldnot affect the free energy. – 13 –or the husp ( m ; n |
4) theory, the dual CFT consists of
N m complex free scalars φ ia , i =1 , , ...N , a = 1 , , ...m and N n
Dirac fermions ψ ir , r = 1 , , ...n , subject to the symplecticreality condition. The m ∆ = 1 scalar fields and n ∆ = 2 pseudoscalars correspond tothe operators φ ia φ jb Ω ij , ¯ ψ ia ψ jb Ω ij , (4.9)where Ω ij is the U Sp ( N ) invariant tensor. Free energy of this theory is given by F CFT = N F (0)CFT , F (0)CFT = mF S + nF D . (4.10)Since supersymmetric HS theories can be mapped to special cases of Konstein-Vasilievmodels, we will not give separate discussions on them.As discussed before, duality between the bulk HS theory and boundary free CFT maybe achieved only if F (1)bulk is proportional to F (0)CFT . Using (3.23), (4.3), (4.6), (4.8) and(4.10), we find that this requirement amounts to( m + n )(3 n S + 5 n P − n ) = 3( m − n )( n S + n P − n ) , (4.11)obtained by setting the ratios of log 2 and ξ (3) dependent terms equal to each other. Takingthe values of n S , n P and n from (2.6), these ratios for the bulk sides can be read off from(3.24), (3.25) and (3.26) in terms of m and n . One can show that for all three Konstein-Vasiliev models, the only solution to the equation above is given by n = 0, which impliesbosonic type A models. In this case the log 2 and ζ (3) dependent terms arise in the sameratio as of a single real scalar field, and we have the result F (1) hu ( m ;0 | = 0 , F (1) ho ( m ;0 | = mF S , F (1) ho ( m ;0 | = − mF S . (4.12)Therefore, assuming that F (0)bulk = F (0)CFT , the bulk and boundary free energies match witheach other provided that hu ( m ; 0 |
4) : G − N → N ,ho ( m ; 0 |
4) : G − N → N − ,husp ( m ; 0 |
4) : G − N → N + 1 . (4.13)The holographic dictionaries relating G N to N in various HS models have been put forwardin [6] via testing the holography of hu (1; 0 | ho (1; 0 |
4) and husp (2; 0 |
4) models at one looplevel. Here, we have extended the validity of these holographic mappings to hu ( m ; 0 | ho ( m ; 0 |
4) and husp ( m ; 0 |
4) Konstein-Vasiliev models. We see that the inclusion of infinitetower of bulk fermions does not cure the problem with the mismatch of the free energiesin the type B model, which corresponds to the case in which m = 0 and n (cid:54) = 0, and itsconjectured dual.Finally, we consider the ordinary supersymmetric models with internal symmetry dis-cussed earlier, whose spectra are given in Table 5 of [9]. In Section 3 we found that the– 14 –ontributions of the bulk fermions give vanishing contributions to free energy and conse-quently the bulk free energy becomes the sum of free energies of type A and type B modelswith the desired internal symmetries. Furthermore, we noted that the coefficient of the ζ (3)dependent contribution to the free energy will have n dependence, where n fund is thedimension of the fundamental representation of the internal symmetry group. On the otherhand it is easy to show that the ζ (3) dependent terms on the CFT side vanish. Therefore,we conclude the problem of free energy mismatch will persist in ordinary supersymmetricHS theories with internal symmetry. AdS with S β × S boundary In thermal
AdS , the one loop free energy of the bulk theory takes the form [13] F (1)bulk = F ( β ) bulk + βE c bulk + a bulk log Λ , (5.1)where β is the period of the imaginary time, F ( β ) bulk is the thermal free energy which canbe computed by taking the log of the thermal partition function as F ( β ) bulk ≡ β − log Z bulk with Z bulk ≡ tr e − βH bulk , and a bulk is the anomaly coefficient related to the Seeley coeffi-cient. The trace denotes the sum over all HS particle states. a bulk is proportional to theintegral of local curvature invariants, and should be the same for AdS with S boundaryand for the thermal AdS . Thus, after summing over spins the total a bulk should vanishas shown in previous sections. E c bulk is the one loop contribution to the Casimir energywhich can be extracted from the thermal free energy in a standard way (cf. (5.5), (5.6)).The free energy of the U ( N ) , O ( N ) or U Sp ( N ) singlet sector of a free vectorial CFTon S β × S takes similar form F CFT = F singlet ( β ) CFT + βE c CFT + a CFT log Λ , (5.2)in which F ( β ) CFT is the free energy of the subsector in Hilbert space consisting of only thestates that are invariant under the required symmetry group. The Casimir energy E c CFT isgiven by
N E , where E is the Casimir energy of a single conformally invariant free field on S β × S . The anomaly coefficient a CFT vanishes on S β × S , which is conformally flat andhas vanishing Euler number. Therefore, there are no logarithmic divergent terms on boththe bulk and the boundary sides. There remains comparison of the thermal part of the freeenergies and the Casimir energies on both sides. The thermal part of the free energies areexpected to match since, by definition, the bulk and boundary thermal partition functionswhich give rise to the corresponding thermal free energies are both equal to the characterof the HS algebra associated with the spectrum of the HS theory. The comparison betweenthe bulk and boundary Casimir energies, however, is not straightforward, since differentfrom E c bulk , the Casimir energy on the CFT side is not directly related to the thermal freeenergy of the singlet sector through (5.5). Holographic matching of the free energies at O ( N ) demands that E c bulk is an integer times the Casimir energy of a single conformallyinvariant free field on S β × S . – 15 –n this section, we first study the one loop free energy of Konstein-Vasiliev theory inthermal AdS with S β × S boundary. We then compare the bulk result with the freeenergy of the corresponding dual CFT at O ( N ). Recall that there exist generalizations of d > U ( N ) or O ( N ) singlet sector of free scalars orfermions [18]. Free energy of this type of HS theory in thermal AdS d has been calculatedin [13] and compared with O ( N ) term in the free energy of the large N U ( N ) or O ( N )vectorial free CFT. It was found that the matching of free energy implies shifts in therelation between G − N and N at leading order by an integer.Different from [13] where the bulk theories are purely bosonic, in our case the bulktheory includes also fermionic HS fields. Accordingly, the dual CFT consists of both scalarsand fermions. In particular, the fermionic HS fields are dual to the bilinear conservedcurrents built out of both scalars and fermions. State operator correspondence then impliesthe existence of scalar-fermion mixed states in the Hilbert space that are singlet under therequired symmetry group. These scalar-fermion mixed states contribute to the thermalfree energy of the singlet sector nontrivially, which means that the F singlet ( β ) for a CFTinvolving both scalars and fermions cannot be obtained by a simple sum of the F singlet ( β )’sof a pure-scalar CFT and of a pure-fermion CFT.Below we start with the computation of the free energies in Konstein-Vasiliev models,which include supersymmetric HS theories as special cases. The story is far more elaboratein higher dimensions. In particular, we refer the reader to [19, 20] and [21] for the case of5D, and [22] for the case of 7D. As stated earlier, the one loop free energy of a massless field in thermal
AdS has thestructure displayed in (5.1) with the vanishing log divergence. F ( β ) can be obtained fromthe grand canonical partition function asFor bosons: F ( β ) bulk = − ∞ (cid:88) m =1 m Z ( mβ ) , (5.3)For fermions: F ( β ) bulk = ∞ (cid:88) m =1 ( − m m Z ( mβ ) . (5.4)Here Z ( β ) is the one-particle canonical partition function. The Casimir energy E c bulk canbe obtained from the energy ζ -function as E c bulk = ± ζ E ( − , (5.5)where ± correspond to bosonic and fermionic cases respectively. The energy ζ -function isrelated to the one-particle partition function by a Mellin transform ζ E ( z ) = 1Γ( z ) (cid:90) ∞ dββ z − Z ( β ) . (5.6)– 16 –n d = 4, the thermal one-particle partition function for a scalar field is given by Z (∆)0 = q ∆ (1 − q ) ∆ > , (5.7)where ∆ is the AdS energy and q = e − β [23]. Thermal one-particle partition function for s ≥ massless field takes the form Z s ( β ) = q s +1 (1 − q ) (cid:2) s + 1 − (2 s − q (cid:3) . (5.8)From the results derived in [13], we deduce the useful formulae F (1)even 1 = F ( β ) even 1 = − ∞ (cid:88) m =1 m Z even 1 ( mβ ) , Z even 1 ( β ) = q (1 + q ) (1 − q ) + q (1 + q )(1 − q ) = [ (cid:101) Z ( β )] + (cid:101) Z (2 β ) ,F (1)even 2 = F ( β ) even 2 = − ∞ (cid:88) m =1 m Z even 2 ( mβ ) , Z even 2 ( β ) = 2 q (1 − q ) − q (1 − q ) = [ (cid:101) Z ( β )] − (cid:101) Z (2 β ) ,F (1)odd 1 = F ( β ) odd = − ∞ (cid:88) m =1 m Z odd ( mβ ) , Z odd ( β ) = q (1 + q ) (1 − q ) − q (1 + q )(1 − q ) = [ (cid:101) Z ( β )] − (cid:101) Z (2 β ) , (5.9)where for later convenience we express the results in terms of the characters (cid:101) Z ( β ) and (cid:101) Z ( β ) of the conformally coupled free scalar and the free real fermion which realize thespin-0 and spin- singleton representations of the SO (3 , (cid:101) Z ( β ) = q (1 + q )(1 − q ) , (cid:101) Z ( β ) = 2 q (1 − q ) . (5.10)By using (5.5) and (5.6), one can show that Z even 1 ( β ), Z even 2 ( β ) and Z odd ( β ) all lead tovanishing Casimir energy [13]. Therefore we simply dropped E c term in (5.9). Also oneshould note that [ (cid:101) Z ( β )] + (cid:101) Z (2 β ) = [ (cid:101) Z ( β )] − (cid:101) Z (2 β ) . (5.11)For all the fermionic fields, we find that the total one-particle canonical partition functionis given by Z F ( β ) = ∞ (cid:88) s = q s +1 (1 − q ) (cid:104) s + 1 − (2 s − q (cid:105) = 2 q (1 + q )(1 − q ) = (cid:101) Z ( β ) (cid:101) Z ( β ) . (5.12) In the rest of this subsection the thermal free energies and partition functions refer to those of the bulktheory. – 17 –sing the total one-particle canonical partition function, we can construct the energy ζ -function for fermions ζ FE ( z ) = 1Γ( z ) (cid:90) ∞ dββ z − e − β (1 + e − β )(1 − e − β ) = 2 ∞ (cid:88) n =1 (cid:18) n + 23 (cid:19) [( n + ) − z + ( n + ) − z ]= ζ ( z, ) − ζ ( z − , ) − ζ ( z − , ) + ζ ( z − , ) − ζ ( z, ) − ζ ( z − , ) + ζ ( z − , ) + ζ ( z − , ) . (5.13)This vanishes at z = −
1. Therefore, the total Casimir energy for fermionic HS fieldsvanishes in thermal
AdS as well, and the correspoding one loop free energy is simply F (1) F = F ( β ) F bulk = ∞ (cid:88) m =1 ( − m m Z F ( mβ ) . (5.14)Summarizing the results above and using the spectra given in (2.6), we find that the oneloop free energies for generic Konstein-Vasiliev HS theories are given by hu ( m ; n |
4) : F (1) hu = − ∞ (cid:88) k =1 k (cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) , (5.15) ho ( m ; n |
4) : F (1) ho = − ∞ (cid:88) k =1 k (cid:16)(cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) + m (cid:101) Z (2 kβ ) − n (cid:101) Z (2 kβ ) (cid:17) , (5.16) husp ( m ; n |
4) : F (1) husp = − ∞ (cid:88) k =1 k (cid:16)(cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) − m (cid:101) Z (2 kβ ) + n (cid:101) Z (2 kβ ) (cid:17) . (5.17)The free energy of husp ( m ; n |
4) theory can be obtained from that of the ho ( m ; n |
4) theoryby m → − m , n → − n . In this section, we calculate the partition function of the singlet sector of free CFTs on S β × S . We closely follow the technique developed in [24, 25]. The partition function of aCFT on S β × S is equal to the thermal partition function due to the vanishing of Casimirenergy and logarithmic divergence. Therefore, we have Z ( β ) = (cid:88) i ∈ physical states q E i , q = e − β , (5.18)where the physical states are restricted to be the singlet states of U ( N ), O ( N ) or U Sp ( N )for our purpose. We have also used the fact that there is no non-trivial chemical potential– 18 –n the system. The thermal partition functions of the U ( N ) and O ( N ) singlet sectors offree scalar and free fermion theories have been studied in [13, 26]. We generalize theirresults to the cases with both scalars and fermions. We first consider the U ( N ) singletsector of a free CFT with N m complex free scalars and
N n
Dirac fermions. As shown in[13, 26], the thermal partition function can be expressed as a path integral localized on theeigenvalues of U ( N ) matrix Z U ( N ) ( β ) = e − F ( β ) U ( N ) = (cid:90) N (cid:89) i =1 dα i e − S ( α ,...α N ) ,S ( α , ...α N ) = − N (cid:88) i (cid:54) = j =1 log sin α i − α j N (cid:88) i =1 f β ( α i ) ,f β ( α ) = N (cid:88) k =1 c k ( β ) cos( kα ) , c k ( β ) = − k (cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) , (5.19)where the matter contents affect the effective action through c k ( β ). In the large N limit,the integral over α i can be replaced by the path integral over the eigenvalue density ρ ( α ), α ∈ ( − π, π ). ρ ( α ) satisfies the standard normalization (cid:90) π − π dαρ ( α ) = 1 . (5.20)The effective action in terms of ρ ( α ) takes the form S ( ρ ) = N (cid:90) dαdα (cid:48) K ( α − α (cid:48) ) ρ ( α ) ρ ( α (cid:48) ) + 2 N (cid:90) dαρ ( α ) f β ( α ) ,K ( α − α (cid:48) ) = − log(2 − α ) , f β ( α ) = N (cid:88) k =1 c k ( β ) cos( kα ) . (5.21)Integrating out ρ , one obtains F ( β ) U ( N ) = − ∞ (cid:88) k =1 k [ c k ( β )] = − ∞ (cid:88) k =1 k (cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) , (5.22)which coincides with one loop free energy for hu ( m ; n |
4) higher spin theory (5.15). Next,we study the O ( N ) singlet sector of a free CFT with N m real free scalars and
N n
Majoranafermions. This is a generalization of the results in [13], where the free CFT consists of onlyscalars or fermions. It is suggested in [13] that, one can choose N to be even, namely N =2N for simplicity in the large N . The difference between even N and odd N casesis at the next order in 1 /N expansion. Free energy of the O (2N) singlet sector of a freeCFT with N m real free scalars and
N n
Majorana fermions can again be written as a pathintegral over the eigenvalues of O ( N ) matrix. The effective potential of the O ( N ) singletsector is given by [13] S ( α , ...α N ) = −
12 N (cid:88) i (cid:54) = j =1 log sin α i − α j −
12 N (cid:88) i (cid:54) = j =1 log sin α i + α j N (cid:88) i =1 f β ( α i ) , (5.23)– 19 –here f β is the same as the one in (5.19). The effective potential for the O ( N ) singletsector differs from that of the U ( N ) by the log sin α terms which come from the Van derMonde determinant or the Haar measure. In the large N limit, the path integral over α i can again be recast into an integral over the eigenvalue density ρ ( α ). After integrating out ρ , one obtains F ( β ) O ( N ) = − ∞ (cid:88) k =1 k (cid:16) [ c k ( β )] − k c k ( β ) (cid:17) (5.24)= − ∞ (cid:88) k =1 k (cid:16)(cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) + m (cid:101) Z (2 kβ ) − n (cid:101) Z (2 kβ ) (cid:17) , which matches the one loop free energy of ho ( m ; n |
4) HS theory in (5.16). In the last case,we consider the
U Sp ( N ) singlet sector of a free CFT with N m complex free scalars φ ia , i = 1 , , ...N , a = 1 , , ...m and N n
Dirac fermions subject to the symplectic real condition.Since N is even in this case, we denote N by 2N. The effective potential of the U Sp ( N )singlet sector takes the form S ( α , ...α N ) = −
12 N (cid:88) i (cid:54) = j =1 log sin α i − α j −
12 N (cid:88) i,j =1 log sin α i + α j −
12 N (cid:88) i =1 log sin α i + 2 N (cid:88) i =1 f β ( α i ) . (5.25)In the large N limit, the path integral over α i can be evaluated by using the same techniqueas before. The free energy of the U Sp ( N ) singlet sector of a free CFT is obtained as F ( β ) USp ( N ) = − ∞ (cid:88) k =1 k (cid:16) [ c k ( β )] + k c k ( β ) (cid:17) (5.26)= − ∞ (cid:88) k =1 k (cid:16)(cid:104) m (cid:101) Z ( kβ ) + n ( − ) k +1 (cid:101) Z ( kβ ) (cid:105) − m (cid:101) Z (2 kβ ) + n (cid:101) Z (2 kβ ) (cid:17) , which matches one loop free energy of husp ( m ; n |
4) HS theory in (5.17). N = 1 SCFT In N = 1 HS theory, the OSp (1 |
4) invariant boundary conditions are given in [2] . Todescribe this, we write the boundary behavior ( ρ →
0) of the complex scalar φ = A + i B as A = ρα + + ρ β + , B = ρα − + ρ β − , (6.1)and define the 3d, N = 1 superfieldsΦ − = α − + i¯ θη − − ¯ θθ β + , Φ + = α + + i¯ θη + + ¯ θθ β − . (6.2) Here we correct a sign error in the result given by [2]. – 20 –he boundary conditions preserving
OSp (1 |
4) take the formΦ − = λ Φ + , (6.3)where λ is an arbitrary real number. In terms of the new scalar fields we have A (cid:48) = sin ϑA − cos ϑB , B (cid:48) = cos ϑA + sin ϑB , (6.4)where tan ϑ = λ , and the boundary condition (6.3) is equivalent to α (cid:48) + = 0 , β (cid:48)− = 0 . (6.5)The linearized bulk scalar field equations would remain the same form under the SO (2)rotation, thus the newly defined scalar fields A (cid:48) and B (cid:48) possess the same Feffer-Grahamexpansion as the original scalar fields A and B . The boundary condition (6.5) implies thatnear the boundary A (cid:48) = ρ β (cid:48) + , B (cid:48) = ρα (cid:48)− . (6.6)Therefore, in computing the one loop free energy, A (cid:48) should have ∆ = 2, while B (cid:48) shouldhave ∆ = 1, which does not affect the N = 1 HS spectrum and the corresponding one loopcalculation. On the CFT side, the boundary condition (6.3) implies the N = 1 free CFTbeing deformed by a supersymmetric double-trace term∆ S = λ (cid:90) d xd θ O , (6.7)where O is given by O = 1 √ N W , W = ϕ + i¯ θψ + ¯ θθ f . (6.8)We compute the difference between the free energy of the deformed CFT and that of thefree CFT, following the procedure adopted in [5, 27]. Denoting the partition function ofthe free CFT by Z , we calculate ∆ F = − log ZZ . (6.9)Using the Hubbard-Stratonovich transformation, we have ZZ = 1 (cid:82) D Σexp( λ (cid:82) dz (cid:48) Σ ) (cid:90) D Σ (cid:68) exp (cid:104) (cid:90) dz (cid:16) λ Σ + Σ O (cid:17)(cid:105)(cid:69) , (6.10)where Σ is an auxiliary superfield and z denotes the supercoordinate. In the large N limit,the higher point functions of O are suppressed. This allows us to write (cid:68) exp (cid:104) (cid:90) dz Σ O (cid:105)(cid:69) = exp (cid:104) (cid:68)(cid:16) (cid:90) dz Σ O (cid:17) (cid:69) + o (1 /N ) (cid:105) . (6.11)Note that Σ and O are single-trace operators of N = 1 superfields, say M and W respec-tively, each with component fields A i , λ i , B i and φ i , ψ i , f i , where B and f are auxiliary– 21 –elds, and the index i stands for the representation of O ( N ). The component fields obeythe following superconformal transformations δA = 14 ξλ δφ = 14 ξψ (6.12) δλ = /∂Aξ − Bξ + Aη δψ = /∂φξ − f ξ + φη (6.13) δB = − ξ / ∇ λ δf = − ξ / ∇ ψ (6.14)where ξ and η are spinors satisfying the conformal Killing spinor equation ∇ µ ξ = γ µ η .Integrating out the spinor coordinates θ and ¯ θ , we obtain (cid:90) dz λ Σ = 1 λ (cid:90) dx √ g ( B i A i A j A j + 12 λ i λ i A j A j + λ i λ j A i A j )= 1 λ (cid:90) dx √ g (Σ Σ + Σ / Σ / ) , (6.15) (cid:90) dz Σ O = (cid:90) dx √ g ( f i φ i A j A j + 12 ψ i ψ i A j A j + B i A i φ j φ j + 12 λ i λ i φ j φ j + 2 ψ i λ j φ i A j )= (cid:90) dx √ g ( O Σ + Σ O + 2 O / Σ / ) , (6.16)where we defined Σ = A i A i , O = φ i φ i , Σ / = A i λ i , O / = φ i ψ i , Σ = B i A i + 12 λ i λ i , O = f i φ i + 12 ψ i ψ i , (6.17)with the lower indices labeling the dimension of the single-trace operators.With the above preparation the second factor of (6.10) at large N is (cid:90) D Σ exp (cid:104) λ (cid:90) dz Σ + 12 (cid:68)(cid:16) (cid:90) dz Σ O (cid:17) (cid:69) (cid:105) = (cid:90) D Σ exp (cid:104) λ (cid:90) dx √ g (Σ Σ + Σ / Σ / )+ 12 (cid:68)(cid:16) (cid:90) dx √ g ( O Σ + Σ O + 2 O / Σ / ) (cid:17) (cid:69) (cid:105) = (cid:90) D Σ exp (cid:104) λ (cid:90) dV (Σ Σ + Σ / Σ / )+ 12 (cid:90) (cid:90) dV dV (cid:48) (cid:16) Σ ( x )Σ ( x (cid:48) ) (cid:68) O ( x ) O ( x (cid:48) ) (cid:69) + Σ ( x )Σ ( x (cid:48) ) (cid:68) O ( x ) O ( x (cid:48) ) (cid:69) + 4Σ / ( x )Σ / ( x (cid:48) ) (cid:68) O / ( x ) O / ( x (cid:48) ) (cid:69) (cid:17)(cid:105) , (6.18)where dV ≡ dx √ g , and we dropped vanishing terms in the two-point function to reachthe last line. – 22 –he integral in (6.10) then becomes gaussian, which integrates to give ZZ = det (cid:16) + 2 λ (cid:104)O / O / (cid:105) (cid:17)(cid:110) det (cid:16) λ (cid:104)O O (cid:105) (cid:17) det (cid:16) λ (cid:104)O O (cid:105) (cid:17) det (cid:16) − ( λ (cid:104)O O (cid:105) ) − ( λ (cid:104)O O (cid:105) ) − (cid:17)(cid:111) . (6.19)At λ → ∞ , the change of the free energy compared to the free theory is∆ F = − log ZZ = − tr log (cid:16) (cid:104)O / O / (cid:105) (cid:17) + 12 tr log (cid:16) (cid:104)O O (cid:105) (cid:17) + 12 tr log (cid:16) (cid:104)O O (cid:105) (cid:17) . (6.20)The two-point functions (cid:104)O O (cid:105) and (cid:104)O O (cid:105) can be expanded in terms of scalar har-monics on S [27] (cid:104)O ∆ ( x ) O ∆ ( x (cid:48) ) (cid:105) = (cid:88) (cid:96)m g ∆ (cid:96) Y ∗ (cid:96)m ( x ) Y (cid:96)m ( x (cid:48) ) , (6.21)where g ∆ (cid:96) is given by g ∆ (cid:96) = R − π − ∆ Γ( − ∆)Γ(∆) Γ( (cid:96) + ∆)(3 + (cid:96) − ∆) . (6.22)Since the harmonics satisfy orthonormal relations, we have (cid:90) √ gd y (cid:104)O ( x ) O ( y ) (cid:105) (cid:104)O ( y ) O ( x (cid:48) ) (cid:105) = (cid:88) (cid:96)m g ∆=2 (cid:96) g ∆=1 (cid:96) Y ∗ (cid:96)m ( x ) Y (cid:96)m ( x (cid:48) ) . (6.23)It is straightforward to see that g ∆=2 (cid:96) g ∆=1 is independent of (cid:96) , and therefore according to[27], tr log (cid:104)O O (cid:105) + tr log (cid:104)O O (cid:105) does not contribute to ∆ F .Similarly, for fermionic two-point function, it is shown in [5] that tr log (cid:104)O / O / (cid:105) isalso zero. Therefore, in the IR there is no modification to the free energy given by thedouble-trace deformation.When λ is small, one can apply perturbation theory to compute ∆ F induced by thedeformation. As shown in [5] the change of free energy caused by the deformation isproportional to the beta function of the deformation coupling. The deformation appearinghere is exactly marginal in the N → ∞ limit, which implies that the beta function of thecoupling constant is suppressed by 1 /N . Thus, at small coupling it can also be seen thatthe deformation does not affect the O ( N ) free energy. In summary, although we have notcomputed the free energy of the deformed theory for arbitrary λ , the vanishing of ∆ F at O ( N ) in both the strong and weak coupling limits provides strong evidence that ∆ F doesnot receive O ( N ) contribution from the supersymmetric double-trace deformation, whichis exactly marginal in the N → ∞ limit. We have carried out a one loop test of the conjectured dualities between Konstein-VasilievHS theories in
AdS with S and S β × S boundaries. These theories are based on the– 23 –S algebras hu ( m ; n | ho ( m ; n |
4) and husp ( m ; n |
4) which contain u ( m ) ⊕ u ( n ), o ( m ) ⊕ o ( n ) and usp ( m ) ⊕ usp ( n ) as bosonic subalgebras. Generically these HS algebras can beinterpreted as infinite dimensional supersymmetry algebras and they do not contain theextended AdS superalgebra OSp ( N |
4) as a subalgebra. They do so only in the specialcase of m = n = 2 N − for even N or 2 ( N − / for odd N . Our results for the free energiesextend previous ones [6, 12, 13] by inclusion of fermionic bulk degrees of freedom. Incomputing the one loop free energies of bosonic and fermionic HS fields in AdS with S boundary, we have adopted the modified spectral zeta function method suggested by [10],thereby reproducing the one loop free energy for bosonic HS fields in a much simpler waywithout the ambiguities encountered in [6, 12]. We also find that the total one loop freeenergy of an infinite tower of bulk fermionic fields vanishes.Matching the bulk fields with boundary operators suggests that the possible CFT dualsof Konstein-Vasiliev theories based on hu ( m ; n | ho ( m ; n |
4) and husp ( m ; n | U ( N ), O ( N ) and U Sp ( N ) singlet sectors of free scalars and free fermions vector representations of the bosonicsubalgebras conformally coupled to S . We find that the free energy of the HS theory maymatch with that of the free CFT only when the bulk theories are hu ( m ; 0 | ho ( m ; 0 | husp ( m ; 0 |
4) Konstein-Vasiliev theories, and with identifications G − N = γ ( N + ∆ N ) withsuitable integers ∆ N . These are generalized type A theories with bosonic scalars on theboundary and bosonic bulk HS fields containing even parity scalars. Thus, in particular,the free energies for generalized type B models with fermions on the S boundary andbosonic HS fields including odd parity scalar fields do not match. The mismatch in thecase of m = 0 , n = 1 corresponding to the simplest type B model has already been notedin [6] where the one loop free energy F (1) = − ζ (3) / (8 π ) obtained in the bulk does notagree with the free energy of Dirac fermions on S boundary. We have also calculated thefree energies of Konstein-Vasiliev theories in AdS with S β × S boundary. In this case, wefind that the free energies of all three families of Konstein-Vasiliev theories match those ofthe conjectured dual free CFTs.Turning to the problem of mismatch in free energies of type B model and its conjectureddual, one may have to take into account the issue of how to impose the O ( N ) invariancecondition on the CFT side. A natural way of implementing it is to gauge the O ( N )symmetry by means of vector gauge field with level k Chern-Simons kinetic term. Thisterm breaks parity but the result for the free energy of the parity invariant model can beobtained in a limit in which the CS gauge field decouples. It has been suggested in [6] thatas the fermions coupled to CS on the boundary give rise to a shift in the level k , it maynot be justified to obtain the result for parity-preserving case by a naive subtraction of CScontribution from the free energy on the CFT side. However, one expects that this effectbecomes irrelevant in the decoupling limit in which k → ∞ . In fact, we have examined theprocedure of decoupling CS in the large k limit by evaluating the S free energies for ABJmodel based on U ( N ) k × U (1) − k [28, 29] and a few N = 3 CS matter theories in whichthe matter sector consists of fundamental hypermultiplets [30–32]. After subtracting thecontribution from pure CS term, we indeed obtain the free energies of free vector models.– 24 –herefore, the puzzle of free energy mismatch in type B remains unresolved and its solutionrequires deeper understanding of HS/vector model holography. In this context, it has beensuggested by [33] and explored further in [34] that the vector-like limit of ABJ model basedon U ( N ) k × U ( M ) − k is given by N, k → ∞ with λ ≡ Nk and M finite . (7.1)In this limit, the ABJ theory effectively behaves like a N = 6 CS gauged vector modelwith U ( M ) flavor symmetry [33]. Its bulk dual is conjectured to be the parity violating N = 6 U ( M ) gauged Vasiliev theory [33]. The parity violating angle θ is conjectured tobe related to the CFT t’Hooft coupling by θ = πλ/ .Turning to the question of free energy in the parity invariant HS theory, we may firstkeep λ finite, and consider the limit λ → Different from the parity preserving HS theories, in the N = 6 parity violatingHS theory a mixed boundary condition needs to be imposed on the bulk spin-1 gaugedfield in order to preserve the N = 6 supersymmetry [33]. The effect of the mixed spin-1boundary condition was mimicked by introducing an N -dependent anomalous dimensionfor the bulk spin-1 gauge field, which is responsible for the log N term in the one loop freeenergy of the bulk theory. The bulk one loop free energy is then compared with the freeenergy of ABJ theory in the vector-like limit (7.1), and with the free energy of pure U ( M )CS subtracted. Matching of the log N terms present in the free energies on both sides leadsto the identification [34] G N = γN πλ sin( πλ ) . (7.2)On the other hand, an exact expression for G N has been obtained from correlation functionfor two stress tensors on the CFT side in [35]. Comparing the relevant terms in theseexpressions for G N one deduces that γ = 2 /π . Assuming the stated value of γ , in the limit λ →
0, required for obtaining the parity invariant HS theory, one finds the relation G N =2 / ( N π ) which differs from the one that appears in the HS/free vector model holographyby a factor of π . This is due to the fact that, while we assume that F (0)bulk = F (0)CFT in theHS/free vector model holography, the example of HS/ABJ holography seems to suggestthat F (0)bulk = F (0)CFT /γ . The above approach may seem to resolve the free energy problemin type B model, however what is missing in this picture is that a bulk computation of theAdS energy for the vector field to justify this value of γ . Furthermore, the beyond log N Besides the Newton constant which is small in the limit described above, there is also a bulk t’Hooftcoupling g M ∼ M/N (cid:28)
1. String theory emerges when
M/N ∼
1. Due to strong interactions, theHS particles form U ( M ) singlet states which are described by the color neutral string states. Since the Mtheory circle R ∼ ( M/k ) / shrinks and √ α (cid:48) /R AdS ∼ ( k/M ) / → ∞ , this is type IIA string in the highenergy limit. The N = 6 parity violating U ( M ) gauged Vasiliev theory can be perceived as a deconfinementphase of type IIA string when M/N (cid:28)
1, in which the string states fragment into HS particles coloredunder U ( M ) [33]. There are subtleties regarding the λ → – 25 –ependence, the terms of higher order in 1 /N have not been compared in the matching ofthe free energies. These issues clearly deserves further study.Another interesting future direction is to consider HS/free matrix model holography.In this case, the corresponding bulk HS theory contains infinitely many massive HS fieldsin addition to the usual massless ones. Recently, a preliminary one loop test of HS/freematrix model holography was carried out in [10]. A dual pair considered in [10] consists ofa free scalar field, namely the bosonic singleton, namely Rac, in the adjoint representationof SU ( N ) and a HS theory in AdS whose spectrum can be constructed from the two,three and four-fold tensor products of the Rac. The bulk fields are dual to the single-traceof product of multiple Rac’s. The one loop free energies of the bulk fields belonging to thefirst few Regge trajectories were computed in [10]. The one loop free energy of the firsttrajectory comprised of massless HS fields is equal to that of a real conformally coupledscalar, however, such feature ceases to exist for higher trajectories. It is possible that aftersumming over all trajectories the total bulk free energy may possess a nice property. Butsuch a difficult task has not been completed. It is also possible that supersymmetry mayprovide simplifications, as we recall that in AdS , the long multiplet of SU (2 , |
4) givesrise to vanishing one loop free energy [19]. It should be noted that the matrix phase ofABJ model based on U ( N ) k × U ( M ) − k with M ∼ N has conserved HS currents emergingin the limit λ →
0, which implies the presence of massless HS particles in the spectrum oftype IIA string. Thus, in the regime M ∼ N , λ = N/k → , (7.3)the duality between IIA string on AdS × CP and ABJ theory may provide an exampleof HS/free matrix model duality [33] if the contribution from CS term in the CFT can besimply subtracted. For the string theory interpretation of this limit, we refer the readerto [36]. The point we wish to stress here is that there are two regimes of type IIA stringtheory on AdS × CP which remarkably give two different supersymmetric HS theoriesone of which is expected to be dual to a vector model, and the other to a matrix modelon the boundary of AdS , and that the puzzle we have encountered in the one loop testof holography by computing the free energies in the case of vector model remains to beinvestigated thoroughly in the case of matrix model.A complete matching of the free energies on both sides requires the knowledge of F (0)bulk which can only be computed from the full action for HS theory. There exists an action thattakes the form of a Chern-Simons action in a generalized spacetime of form M = X × Z where Z is a twistor space with no boundary, and the spacetime M resides on an openregion of the boundary of X [11]. The action contains Lagrange multiplier master fieldsbut they do not propagate to produce unwanted degrees of freedom. What remains to bedone is to add suitable HS invariant deformations that reside on the boundary of M , whichare highly restricted and for which candidates have been proposed [11], and to constructa boundary action that resides on the boundary of asymptotically AdS spacetime M which has not been constructed so far. These are needed for obtaining the field equationsthrough an appropriate variational principle, and once they are constructed, the full action– 26 –an be quantized in a path integral approach and the Feynman rules can be derived, eventhough the action does not have the traditional form consisting an infinite sum of Einstein-Hilbert term and powers of curvature tensors and their derivatives. It remains to be seenwhether the result for the one loop free energy computed in this fashion agrees with thatobtained under the assumption that the quadratic action for the HS fluctuations around AdS has the standard Fronsdal form with two derivatives. In particular, it would beinteresting to determine if the mismatch in the free energies encountered in the type B andordinary supersymmetric HS theories and their conjectured duals may find a resolution ina computation based on the action discussed above. Acknowledgements
We thank S. Giombi, I. Klebanov, E. Skvortsov, P. Sundell, A. Tseytlin, M. Vasiliev and X.Yin for discussions. Y.P. would like to thank Nordita Institute, Beijing Normal Universityand Sun Yet-sen University for hospitality during various stages of this work. Y.P andE.S. would also like to thank Munich Institute for Astro-and Particle Physics (MIAPP) forproviding a wonderful work environment. The work of E.S. is supported in part by NSFgrant PHY-1214344. Y.P. is supported by Alexander von Humboldt fellowship.
A Comparison of ζ (∆ ,s ) ( z ) with (cid:101) ζ (∆ ,s ) ( z ) In this section, we will show that the alternate spectral zeta function is physically equivalentto the original spectral zeta in computing the one loop free energy of HS fields.
A.1 Bosonic case
For bosonic HS fields, the physical equivalence of alternate spectral zeta function and theoriginal spectral zeta function has been studied in [10] in the case of summing over allinteger spins. The crucial point is that for a given HS field labeled by (∆ , s ), the differencebetween the alternate spectral zeta function and the original zeta function can be expressedas a contour integral encircling β = 0 [10] (cid:101) ζ B (∆ ,s ) ( z ) − ζ B (∆ ,s ) ( z ) = 13 (cid:18) s + 12 (cid:19) ν (cid:34) ν − (cid:18) s + 12 (cid:19) (cid:35) (A.1)= z πi (cid:73) dβ β β (cid:32) β + 2sinh β −
13 + 4 ∂ α (cid:33) χ ∆ ,s ( β, α ) (cid:12)(cid:12)(cid:12) α =0 + O ( z ) . It has been shown in [10] that upon summing over all integer spins, the contour integralvanishes. We have also checked that this is also true for summing over all even spins orodd spins separately. – 27 – .2 Fermionic case
For fermionic HS fields, we will elaborate on the physical equivalence of alternate spectralzeta function and the original spectral zeta function which has not been studied elsewhere.For a fermionic HS field labeled by (∆ , s ), the alternate spectral zeta function can becomputed exactly (cid:101) ζ F (∆ ,s ) ( z ) =(2 s + 1) (cid:18) − s ( s + 1)24 (cid:19) z ) (cid:90) ∞ dββ z − e − νβ β + 2 s + 116 1Γ(2 z ) (cid:90) ∞ dββ z − e − νβ β = 2 s + 124 (cid:2) ν (cid:0) (2 s + 1) − ν (cid:1) ζ (2 z, ν ) + 4 ζ (2 z − , ν ) − νζ (2 z − , ν )+ (cid:0) ν − s ( s + 1) − (cid:1) ζ (2 z − , ν ) (cid:3) , (A.2)from which we see that (cid:101) ζ F (∆ ,s ) (0) matches ζ F (∆ ,s ) (0). The latter takes the form ζ B (∆ ,s ) (0) = s + (cid:20) ν − ( s + 12 ) ν (cid:21) −
13 (2 s + 1) (cid:34) s + ) (cid:35) . (A.3)Next, we compute the first derivative of (cid:101) ζ F (∆ ,s ) ( z ) at z = 0, which is given by (cid:101) ζ F (cid:48) (∆ ,s ) (0) = 2 s + 112 (cid:2) ν (cid:0) (2 s + 1) − ν (cid:1) ζ (cid:48) (0 , ν ) + 4 ζ (cid:48) ( − , ν ) − νζ (cid:48) ( − , ν )+(12 ν − s ( s + 1) − ζ (cid:48) ( − , ν ) (cid:3) . (A.4)After some algebra, we obtain the difference between (cid:101) ζ F (cid:48) (∆ ,s ) (0) and ζ F (cid:48) (∆ ,s ) (0) (cid:101) ζ F (cid:48) (∆ ,s ) (0) − ζ F (cid:48) (∆ ,s ) (0) = −
124 (2 s + 1) ν + 2 s + 19 ν . (A.5)This can again be converted to a contour integral of β circling β = 0 (cid:101) ζ F (cid:48) (∆ ,s ) (0) − ζ F (cid:48) (∆ ,s ) (0) = 2 πi (cid:73) dβ β β (cid:16) β + 2sinh β −
13 + 4 ∂ α (cid:17) χ ∆ ,s ( β, α ) . (A.6)From (3.21), one can see that the total character of fermionic sector including the contri-butions of all physical fermionic higher fields and their ghosts gives rise to an even functionof β which has vanishing contour integral. Therefore, we have shown that in computingthe one loop free energy of the whole fermionic sector, the alternate spectral zeta functionis physically equivalent to the original one.– 28 – eferences [1] M. A. Vasiliev, More on equations of motion for interacting massless fields of all spins in(3+1)-dimensions , Phys. Lett. B (1992) 225.[2] E. Sezgin and P. Sundell,
Holography in 4D (super) higher spin theories and a test via cubicscalar couplings , JHEP (2005) 044 [hep-th/0305040].[3] I. R. Klebanov and A. M. Polyakov,
AdS dual of the critical O(N) vector model , Phys. Lett. B (2002) 213 [hep-th/0210114].[4] E. Sezgin and P. Sundell,
Massless higher spins and holography , Nucl. Phys. B (2002) 303[hep-th/0205131].[5] I. R. Klebanov, S. S. Pufu and B. R. Safdi,
F-Theorem without Supersymmetry , JHEP (2011) 038 arXiv:1105.4598 [hep-th].[6] S. Giombi and I. R. Klebanov,
One Loop Tests of Higher Spin AdS/CFT , JHEP (2013)068 arXiv:1308.2337 [hep-th].[7] E. D. Skvortsov,
On (Un)Broken Higher-Spin Symmetry in Vector Models , arXiv:1512.05994[hep-th].[8] S. E. Konstein and M. A. Vasiliev,
Extended Higher Spin Superalgebras and Their MasslessRepresentations , Nucl. Phys. B (1990) 475.[9] E. Sezgin and P. Sundell,
Supersymmetric Higher Spin Theories , J. Phys. A (2013) 214022,arXiv:1208.6019 [hep-th].[10] J. B. Bae, E. Joung and S. Lal, One-loop test of free SU(N) adjoint model holography , JHEP (2016) 061, arXiv:1603.05387 [hep-th].[11] N. Boulanger, E. Sezgin and P. Sundell,
4D Higher Spin Gravity with Dynamical Two-Formas a Frobenius-Chern-Simons Gauge Theory , arXiv:1505.04957 [hep-th].[12] S. Giombi, I. R. Klebanov and B. R. Safdi,
Higher Spin AdS d +1 /CFT d at One Loop , Phys.Rev. D (2014) no.8, 084004 arXiv:1401.0825 [hep-th][13] S. Giombi, I. R. Klebanov and A. A. Tseytlin, Partition Functions and Casimir Energies inHigher Spin AdS d +1 / CFT d , Phys. Rev. D (2014) no.2, 024048 arXiv:1402.5396 [hep-th].[14] S. Giombi, TASI Lectures on the Higher Spin - CFT duality , arXiv:1607.02967 [hep-th].[15] S. W. Hawking,
Zeta Function Regularization of Path Integrals in Curved Space-Time ,Commun. Math. Phys. (1977) 133. BF01626516[16] R. Camporesi, zeta function regularization of one loop effective potentials in anti-de Sitterspace-time , Phys. Rev. D (1991) 3958.[17] R. Camporesi and A. Higuchi, Arbitrary spin effective potentials in anti-de Sitter space-time ,Phys. Rev. D (1993) 3339.[18] M. A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d) ,Phys. Lett. B (2003) 139, hep-th/0304049.[19] M. Beccaria and A. A. Tseytlin,
Higher spins in AdS at one loop: vacuum energy, boundaryconformal anomalies and AdS/CFT , JHEP (2014) 114, arXiv:1410.3273 [hep-th].[20] M. Beccaria and A. A. Tseytlin, Vectorial AdS /CFT duality for spin-one boundary theory ,J. Phys. A (2014) no.49, 492001, arXiv:1410.4457 [hep-th]. – 29 –
21] J. B. Bae, E. Joung and S. Lal,
On the Holography of Free Yang-Mills , arXiv:1607.07651[hep-th].[22] M. Beccaria, G. Macorini and A. A. Tseytlin,
Supergravity one-loop corrections on AdS andAdS , higher spins and AdS/CFT , Nucl. Phys. B (2015) 211, arXiv:1412.0489 [hep-th].[23] G. W. Gibbons, M. J. Perry and C. N. Pope, Partition functions, the Bekenstein bound andtemperature inversion in anti-de Sitter space and its conformal boundary , Phys. Rev. D (2006) 084009 [hep-th/0606186].[24] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, TheHagedorn - deconfinement phase transition in weakly coupled large N gauge theories , Adv.Theor. Math. Phys. (2004) 603, hep-th/0310285.[25] H. J. Schnitzer, Confinement/deconfinement transition of large N gauge theories with N(f )fundamentals: N(f )/N finite , Nucl. Phys. B (2004) 267, hep-th/0402219.[26] S. H. Shenker and X. Yin,
Vector Models in the Singlet Sector at Finite Temperature ,arXiv:1109.3519 [hep-th].[27] S. S. Gubser and I. R. Klebanov,
A Universal result on central charges in the presence ofdouble trace deformations , Nucl. Phys. B (2003) 23, hep-th/0212138.[28] A. Kapustin, B. Willett and I. Yaakov,
Exact Results for Wilson Loops in SuperconformalChern-Simons Theories with Matter , JHEP (2010) 089 arXiv:0909.4559 [hep-th].[29] H. Awata, S. Hirano and M. Shigemori,
The Partition Function of ABJ Theory , PTEP (2013) 053B04 [arXiv:1212.2966].[30] M. Marino,
Lectures on localization and matrix models in supersymmetricChern-Simons-matter theories , J. Phys. A (2011) 463001 arXiv:1104.0783 [hep-th].[31] D. R. Gulotta, C. P. Herzog and T. Nishioka, The ABCDEF’s of Matrix Models forSupersymmetric Chern-Simons Theories , JHEP (2012) 138 arXiv:1201.6360 [hep-th].[32] M. Mezei and S. S. Pufu,
Three-sphere free energy for classical gauge groups , JHEP (2014) 037 arXiv:1312.0920 [hep-th].[33] C. M. Chang, S. Minwalla, T. Sharma and X. Yin,
ABJ Triality: from Higher Spin Fields toStrings , J. Phys. A (2013) 214009 arXiv:1207.4485 [hep-th].[34] S. Hirano, M. Honda, K. Okuyama and M. Shigemori, ABJ Theory in the Higher Spin Limit ,arXiv:1504.00365 [hep-th].[35] M. Honda,
Identification of Bulk coupling constant in Higher Spin/ABJ correspondence ,JHEP (2015) 110 arXiv:1506.00781 [hep-th].[36] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena,
N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,
JHEP (2008) 091,arXiv:0806.1218 [hep-th].(2008) 091,arXiv:0806.1218 [hep-th].