Higher Spin AdS/CFT at One Loop

Simone Giombi

Higher Spin Theories Workshop

Penn State U., Aug. 28 2015

Based mainly on:

• SG, I. Klebanov, arxiv: 1308.2337

• SG, I. Klebanov, B. Safdi, arxiv: 1401.0825

• SG, I. Klebanov, A. Tseytlin, arxiv: 1402.5396

• SG, I. Klebanov, Z. Tan, in progress

Higher spin theories in AdS

• Vasiliev wrote down a set of consistent, fully non-linear, gauge invariant equations of motion (in any dimension). They admit a vacuum solution which is AdS space (or dS if the cosmological constant is positive. Will focus on AdS case in this talk).

• In the simplest bosonic 4d theory, the linearized spectrum around the AdS vacuum consists of an infinite tower of higher spin fields plus a scalar

Vasiliev equations

• The Vasiliev equations in 4d

Linearized HS Spectrum

• Linearizing the equations around the AdS vacuum solution, one finds the standard equations for a scalar (with mass m2=-2), linearized graviton and free massless HS fields (Fronsdal).

• Going to higher order in perturbation theory, one can read off the cubic, quartic, etc. couplings. In principle, one could reconstruct a Lagrangian in terms of the physical HS fields this way (in practice very hard!).

• Free equations are ordinary wave equations. But interactions involve higher derivatives (of arbitrarily high order). Vasiliev theory is in particular a higher derivative theory of gravity.

Higher spins and AdS/CFT

• From the point of view of AdS/CFT correspondence, it is not too surprising that such theories exist.

• It is easy to see that Vasiliev higher spin theories have the correct spectrum to be dual to simple CFT’s with vector-like matter.

Higher spins and AdS/CFT

• Consider a free theory of N free complex scalar fields

• It has a U(N) global symmetry under which the scalar transforms as a vector. (This is different from familiar examples of AdS/CFT, where the CFT side is usually a gauge theory with matrix (adjoint) type fields).

Higher spins and AdS/CFT

• The free theory has an infinite tower of conserved HS currents

• If we consider U(N) invariant operators (singlet sector), these currents, together with the scalar operator

are all the “single trace” primaries of the CFT.

Higher spins and AdS/CFT

• By the usual AdS/CFT dictionary, single trace primary operators in the CFT are dual to single particle states in AdS.

• Conserved currents are dual to gauge fields (massless HS fields)

• A scalar operator is dual to a bulk scalar with

• This precisely matches the spectrum of Vasiliev’s bosonic “type A” theory in AdS4.

Higher spins and AdS/CFT

• The dual higher spin fields are necessarily interacting in order to reproduce the non-vanishing correlation functions of HS currents in the CFT

• The large N limit corresponds as usual to weak interactions in the bulk.

• So from AdS/CFT point of view we conclude that consistent theories of interacting massless higher spins in AdS indeed have to exist, as they should provide AdS duals to free vector-like CFT’s (in the singlet sector).

Higher spins and AdS/CFT

• We could also consider a theory of N free real scalars, and look at the O(N) singlet sector

• The spectrum of single trace primaries now includes a scalar plus all the even spin HS currents

• On Vasiliev’s theory side, this corresponds to a consistent truncation of the equations which retains only the even spin gauge fields (“Minimal bosonic HS theory”).

Interacting scalar vector model

• The conjecture that the (singlet sector of) the O(N)/U(N) vector model is dual to the bosonic Vasiliev theory was made by Klebanov and Polyakov (2002) (earlier closely related work: Sundborg; Witten; Sezgin, Sundell). A crucial observation is that one can also consider the Wilson-Fischer fixed point reached by a relevant “double trace” deformation of the free theory.

• The IR fixed point (“critical vector model”) is an interacting CFT whose single trace spectrum has a scalar operator of dimension and approximately conserved HS current of dimension

Fermionic vector model

• In summary, Vasiliev (type A) theory is conjectured to be dual to either free or interacting U(N)/O(N) model, depending on choice of b.c. for bulk scalar field

• A similar conjecture (Leigh-Petkou; Sezgin-Sundell) relates Vasiliev “type B” theory in AdS4 to U(N)/O(N) fermionic vector models (interacting version is the “critical Gross-Neveu” model with interaction)

• The single trace spectrum is similar to the scalar case, there is an infinite tower of conserved HS current, plus a parity odd scalar of dimension 2. This agrees with the spectrum of the type B in AdS4

• One can couple the vector models to U(N)/O(N) Chern-Simons gauge fields in order to impose singlet constraint. At finite `t Hooft coupling, this leads to a generalization of the HS/vector model conjectures that involve parity breaking HS theories in AdS4 (SG et

al; Aharony, Gur Ari, Yacoby)

• In the rest of the talk I will mainly focus on free vector model case

Tests of the duality

• These HS4d/3d-vector model dualities have passed non-trivial tests at the level of correlation functions (SG, Yin;

Maldacena-Zhiboedov; Didenko-Skvortsov; Boulanger, Kessel, Skvortsov,

Taronna; Bekaert, Erdmenger, Ponomarev, Sleight;…)

• In this talk I will focus on a different kind of test based on comparing bulk/boundary partition functions

Sphere free energies

• In particular I will be interested in the partition function, or free energy F, on a round sphere Sd

• In odd d , after regulating away power-like divergences, it is a well-defined, radius independent finite number. For RG flows in d=3, it satisfies the F-theorem (Jafferis, Klebanov,

Pufu, Safdi; Klebanov, Pufu, Safdi; Myers, Sinha; Casini. Huerta)

• For a CFT, F is also related to the entanglement entropy across a circle

Sphere free energies

• In even d, there are logarithmic UV divergences, and term in F proportional to the logarithm of the sphere radius R encodes the conformal a-anomaly of the CFT

• In d=4, it satisfies the a-theorem (Cardy; Komargodski;

Schwimmer)

• Natural generalizations of F- and a- theorems to higher dimensions have been proposed. Not proved, but non-trivial evidence from several examples

Free energy on S3

• For the free vector model on S3, we simply have to compute the Euclidean path integral

• This is just given by the one-loop determinant

• In the Wilson-Fisher interacting case, it can be computed in the 1/N expansion

(Note Fcrit < Ffree in agreement with F-theorem)

Free energy on S3 from the bulk

• The challenge is: can we reproduce these results from the bulk? In particular, can we see the vanishing of the subleading corrections in the large N expansion of the free energy from the HS dual to the free theory?

• How do we compute F from the bulk?

• We should compute the partition function of the Vasiliev’s theory on the Euclidean AdS4 vacuum (hyperbolic 4-space)

Free energy on S3 from the bulk

• In practice, we have to compute the path integral of the bulk theory, where we expand the metric around AdS4 and integrate over all quantum fluctuations

• Here GN is Newton’s constant, which scales as 1/GN ~ N

Free energy on S3 from the bulk

• The explicit bulk action is not well understood (proposals by Douroud, Smolin; Boulanger, Sundell; Sezgin; Vasiliev…). If there is an underlying Lagrangian in terms of physical fields, it should take the schematic form

• The leading term in the bulk free energy corresponds to evaluating this action on the AdS4 background metric, with all other fields set to zero.

• This is already very non-trivial, as it requires to know the form of all the higher derivative corrections in the metric sector (we know they are non-trivial from knowledge of correlation functions).

Free energy on S3 from the bulk

• One would like to show that (say for the non-minimal theory)

• Alternatively, one would need a generalization of Ryu-Takayanagi prescription to compute holographic entanglement entropy in 4d HS theory.

• This is one of the outstanding open problems in testing HS/vector model dualities.

• While we cannot show this (yet), we can start by something simpler, namely assume that this tree-level piece works, and compute the one-loop contribution to the bulk free energy.

One loop • While the classical piece is still out of reach, we can start

by something simpler, namely assume that the classical part works, and compute the one-loop contribution to the bulk free energy.

• Even in the absence of an action, the kinetic operators for all the HS fields around AdS vacuum are completely fixed

• At one-loop, we just have to compute the determinants of the corresponding differential operators

• After gauge fixing, one gets the 1-loop partition function

• In a series of papers in the ‘90’s, Camporesi and Higuchi computed the spectral zeta function (Mellin transform of the heat kernel) for operators acting on STT fields of arbitrary spin in hyperbolic space. The determinants we need can be computed using their results

AdS Spectral zeta function

• The explicit spectral zeta function in AdS is

with

• In the present case of d=3

AdS Spectral zeta function

• In terms of the spectral zeta function, the contribution to the one-loop free energy of each spin is then obtained as

• Importantly, in every even dimensional bulk spacetime (odd-dimensional boundary), there is a bulk UV logarithmic divergence proportional to the value of the spectral zeta function at z=0

UV finiteness

• For the duality to be exact and Vasiliev theory to be “UV complete”, this divergence should not be present in the full HS theory: the bulk theory should be finite.

• While each spin contributes a log divergence, can the divergence cancel in the sum over the infinite tower of fields? The explicit coefficient of the log divergence in AdS4 turns out to be

UV finiteness

• Regulating the sum with the usual Riemann zeta function regularization, and recalling that z(0)=-1/2, and z(-2)=z(-4)=0:

• So Vasiliev theory is one-loop finite (at least using this regulator)

• Regularization can be understood more systematically as analytic continuation of spectral zeta function in the spectral parameter z (summing over all spins first) (SG,

Klebanov, Safdi). The regulator should be singled out by consistency with HS symmetry.

• The same result holds for the theory with even spins only, and regardless of boundary conditions on the scalar.

The finite part

• Having shown that the log divergence cancels, one can move on to the computation of the finite piece.

• This is considerably more involved and I will omit the details. Using the known spectral zeta functions and summing up the contribution of all fields, after some work one finds that the infinite tower of spins precisely cancels the D=1 scalar field contribution:

• The one-loop bulk free energy in Vasiliev’s type A theory with D=1 boundary condition is exactly zero, in agreement with trivial large N expansion in the dual free CFT!

• In the case of the alternate D=2 boundary condition on the bulk scalar, one instead finds

which is the expected non-trivial O(N0) correction in the interacting IR fixed point.

The minimal HS theory

• We can repeat the same calculation in the minimal theory, with even spins only, which should be dual to the O(N) vector model.

• Here we find a surprise. The total one loop free energy is not zero, but it is equal to

• This is precisely equal to the value of the S3 free energy of a single real conformal scalar field…Why?

The minimal HS theory

• So far we have always assumed that Newton’s constant is given by . . But there can in principle be subleading corrections in the map between and N.

• Because the one-loop piece is precisely proportional to the expected classical piece, this suggests that the result can be consistent with the duality if we assume a shift N->N-1, i.e. , so that the classical piece is

combined with the one-loop piece would give the expected result for F.

One-loop shift

• This effect may perhaps be thought as a finite “one-loop renormalization” of the bare coupling constant in Vasiliev’s theory, somewhat similar to the one-loop shift of the level in Chern-Simons gauge theory

• The fact that the shift is simply an integer is consistent with the idea that the coupling constant in Vasiliev’s theory should be quantized (Maldacena-Zhiboedov).

• This shift will also affect loop corrections to correlation functions. For instance, it implies that the graviton 2-point function should receive non-trivial one-loop corrections (proportional to the classical result) in the minimal theory, but not in the “non-minimal” one. Would be interesting (but quite non-trivial) to check.

General dimensions

• There is a formulation of Vasiliev’s theory in arbitrary dimensions (Vasiliev ‘03). The equations of motion have a AdSd+1 vacuum solution, and the linearized spectrum around this background is

• This spectrum is in one-to-one correspondence with the single trace primaries of a free scalar vector model in dimension d (the scalar bilinear in free scalar CFT has dimension D=d-2, in agreement with D(D-d)=m2). This generalizes the 4d “type A” theory to all d.

• The spectral zeta functions for HS fields in general dimensions are also explicitly known (Camporesi-Higuchi)

Odd boundary dimensions

• It is then natural to repeat one-loop calculations for type A theory in general AdSd+1

• There are two physically distinct situations: the case of odd d (even dim. AdSd+1), or even d (odd dim. AdSd+1)

• In the odd d case, the situation is similar to the AdS4 case described earlier. The bulk free energy on Euclidean AdSd+1 is dual to the CFT free energy on Sd, which is some finite number for d odd. For free CFT, the large N expansion should be trivial, so we would expect a trivial one-loop bulk free energy.

Results: odd boundary dimensions

• For any odd d, the volume of AdSd+1 can be regulated into a finite number

• But, as in AdS4, there is in principle a UV logarithmic divergence in the bulk that should vanish for the correspondence to work

• Using the regulator defined above, we find that the UV divergence indeed vanishes in any odd d, both for minimal and non-minimal theories. Vasiliev theory is one-loop UV finite in any dimension.

Results: odd boundary dimensions

• For the non-minimal theory (s=0,1,2,..), in all d we find

• For the minimal theory (s=0,2,4,…), in all d we find again the curious (and quite non-trivial!) identity

• As in AdS4, this can be consistent with the duality if we assume a non-trivial shift in the minimal theory.

Results: even boundary dimensions

• In even d, the CFT sphere free energy is UV logarithmically divergent, and the regularized F has a logarithmic dependence on the radius of the sphere: F = a log(R)+… where a is the a-coefficient in the conformal anomaly.

• In the bulk, this logarithmic dependence is reflected in the IR divergence of the AdSd+1 volume for even d

• The coefficient of log(R) in the bulk free energy should then match the a-anomaly coefficient on the CFT side

• Note that there is no UV divergence in the bulk in this case (in odd dimensional spacetime, z(0)=0 identically)

Results: even boundary dimensions

• Again, if the CFT is free, the a-anomaly should be Nascalar, without 1/N corrections.

• The results we find are again consistent with the general picture:

All spin theory:

Minimal theory:

where ascalar is the a-anomaly coefficient of one real conformal scalar in d-dimensions (e.g. ascalar=1/90, -1/756, 23/113400… in d=4,6,8…).

This again requires in the minimal theory

Example: AdS5

• For example, for the Vasiliev theory in AdS5 with all integer spins, we find the one-loop bulk free energy

• And for the minimal theory with even spins only

The number +1/90 is precisely the a-anomaly coefficient of a real scalar in 4d, in agreement with

Type B in higher dimensions

• Let’s go back to the fermionic vector models

• In higher dimensions (d>3), the spectrum of bilinears is

more complicated than scalars because it involves mixed symmetry representations (Vasiliev; Dolan)

+ +

+ …

Example: Type B in d=4

• For example, for the d=4 Dirac fermion we have the operators

• The type B dual HS theory in AdS5 should then include two scalars with m2 = -3, two towers of totally symmetric Fronsdal fields, and one tower of mixed symmetry fields in [s,1] representation. The corresponding fully interacting theory in AdS5 (and similarly for higher dimensions) has not yet been constructed

Type B at One-loop

• Since the spectrum is known, we can still compute the one-loop corrections to the free energy. The results of Camporesi and Higuchi allow to find spectral zeta function for an arbitrary representation [m1,m2,m3,…], in any d.

• For instance in AdS5 we find

• So the one-loop free energy for the dual of Dirac fermions in the U(N) singlet sector indeed vanishes, consistently with the duality

• In d=4, we can also impose a Majorana condition and consider the O(N) singlet sector. Then the spectrum and corresponding bulk one-loop free energy in AdS5 are

• This is exactly the a-anomaly of a single Majorana fermion in d=4! Consistency of the duality then requires again the same shift encountered for scalars :

• We have checked that this pattern continues for all even d: for the dual of the Dirac fermions in U(N) singlet sector we get F (1)=0, and whenever the Majorana condition is possible, the dual minimal type B theory yields F(1) = afer log(R)

Type B puzzle in odd d

• For odd d (even dim. AdS) calculations are more complicated, but can still be carried out explicitly

• We always find that the bulk UV divergences cancel when we sum over the whole spectrum

• However, there is a puzzle with the finite part: F(1)

does not appear to vanish (not even in the non-minimal U(N) cases), and it does not correspond to a simple integer shift in GN

• This may have to do with some peculiar property of fermions in odd dimensions (parity anomaly?), but we do not understand it yet

Type C

• In even dimensions, we have another possible conformal field: the (d/2-1)-form gauge field with d/2-form field strength

• We can take N copies of complex/real such fields, and consider U(N)/O(N) singlet sector

• The spectrum of bilinears can be determined from general formulae for products of representations of the conformal group (Dolan)

• E.g., in d=4 (Beccaria, Tseytlin; SG, Klebanov, Tan)

• The spectrum can be similarly derived in higher dimensions. There should be corresponding non-linear “type C” HS theories in AdSd+1. For one-loop free energies knowledge of spectrum is sufficient, and we can be compute them with the techniques we developed

• Results are in agreement with the duality, again provided simple

shifts in GN are assumed (Beccaria, Tseytlin; SG, Klebanov, Tan)

Conclusion and summary

• Consistent interacting theories of massless higher spins can be constructed in any dimension if the cosmological constant is non-zero. They involve infinite towers of fields of all spins.

• Higher spin theories in AdS are expected to be dual to simple vector model CFT’s. Best understood case is AdS4/CFT3 , but higher dimensional generalizations are natural

• In general dimensions, we can consider vector models constructed from scalars, fermions or d/2-forms (or combinations of them) and restrict to U(N)/O(N) singlet sector. They should correspond to type “A”, “B”, “C” higher spin theories. Only type A is currently known at non-linear level in all d

Conclusion and summary

• One loop tests of higher spin/vector model dualities can be obtained by comparing sphere partition functions. (Thermal partition functions on S1xSd-1 and Casimir energies can also be computed (SG, Klebanov, Tseytlin; Beccaria, Tseytlin), though I didn’t discuss them in this talk)

• Vasiliev theory in AdSd+1 appears to be one-loop UV finite in all dimensions. A finite model of quantum gravity?

• Consistency with HS/vector model dualities requires simple shifts in the map between GN and N. It would be interesting to understand this better (quantization of HS coupling constant?), and test this from other observables (e.g. one-loop corrections to correlation functions)

EXTRA SLIDES

Partition function on S1 x Sd-1

• We can in principle compare the partition functions on other spaces. For instance, we can consider the thermal partition function of the CFT on S1 x Sd-1.

• This provides a neat way to summarize the agreement of bulk and boundary spectra.

• Unlike S3, in this case the U(N)/O(N) singlet constraint plays a crucial role.

Partition function on S1 x Sd-1

• The singlet constraint is implemented by integrating over constant holonomy of gauge field around S1.

• The result for the partition function takes the form

where Ec is the Casimir energy on Sd-1 of a single conformal scalar (it vanishes in odd d, but not in even d). This term is unaffected by the integration over holonomy.

Partition function on S1 x Sd-1 • The term has a non-trivial temperature dependence.

After averaging over holonomy, to leading order at large N, it takes the form (for U(N) case) (Shenker, Yin ; SG, Klebanov,

Tseytlin)

where

is the ``one-particle” partition function for a free conformal scalar. Similar results for O(N) and fermions.

• is of order N0. It should match the bulk one-loop term.

Higher spins in thermal AdS

• In the bulk, we compute the one-loop determinants for the HS fields on the thermal AdSd+1 with S1 x Sd-1

boundary.

• As in the CFT, we expect a structure

where Fc is a Casimir term, and the non-trivial term.

Higher spins in thermal AdS

• The non-trivial temperature dependent term for each spin takes the form

• The sum over all spins then gives

• This is indeed equal to the square of the scalar one-

particle partition function, as obtained from the singlet CFT calculation.

• Similar agreement for the minimal theory vs O(N) CFT, and for fermionic versions of the duality.

Casimir Energies

• Ec vanishes for CFT in odd d. Regularized sum over spins in global AdSd+1 for odd d indeed vanishes, both for minimal (even spins) and non-minimal (all spins).

• In even d, Ec is non-zero in CFT, and exactly proportional to N in our case.

• Regularized sum over all spins of Ec,s in even d indeed vanishes!

• The sum over even spins equals the Casimir energy of one real scalar field on Sd-1:

• This precisely agrees with the one loop shift N->N-1. Further evidence for in the minimal theories.

AdS3/WN model duality

• In AdS3, one can consider a more general higher spin theory, with a complex scalar field of mass and HS fields with s=2,3,…

• This theory was conjectured (Gaberdiel-Gopakumar) to be the large N dual of the WN minimal model CFT based on the coset . It has central charge

where

• We find that the order N0 piece is indeed reproduced by the bulk one-loop free energy in AdS3!