O(N) Models, RG and AdS/CFTIgor Klebanov

Talk at PiTP
Institute for Advanced Study
June 16, 2014

Based mainly on• S. Giombi, IK, arXiv:1308.2337• S. Giombi, IK, B. Safdi, arXiv:1401.0825• L. Fei, S. Giombi, IK, arXiv:1404.1094

From D-Branes to AdS/CFT• Stacking D-branes and comparing the gauge theory living onthem with the curved background they create led to thegauge/gravity duality.
• In case of N D3-branes, have SU(N) gauge theory with N=4SUSY.

• Absorption of closed strings, near-extremal
entropy, the “3/4 problem” IK, Gubser, Peet, Tseytlin, …
• Zoom in on the throat:
Maldacenawith

Preoccupation• The AdS/CFT correspondence has preoccupied us
ever since.
• Many insights from weakly curved metrics aboutstrongly coupled gauge theory.
• Some non-BPS tests using integrability and SUSY
localization to get at the strongly coupled gauge
theory.
• Yet, a proof of the duality remains elusive.
• Maybe this is because both N=4 SYM and TypeIIB string are really complicated theories?

Simplify!
• Look for AdS duals of CFT’s where dynamical
fields are in the fundamental of O(N) or U(N)rather than in the adjoint. IK, Polyakov• Wilson-Fisher O(N) critical points in d=3:
• Even simpler: the O(N) singlet sector of thefree theory.
• Conserved currents of even spin

All Spins All the Time
• Similarly, can consider the U(N) singlet sector inthe d-dimensional free theory of N complex
scalars. There are conserved currents of all
integer spin.
• The dual AdSd+1 description must consist of
massless gauge fields of all integer spin, coupled
together.
• Vasiliev and others have constructed the classicalEOM for some such interacting theories.
• Complicated. No known action principle.

Matching of 3-pt functions• n-point functions of the currents do notvanish in the free CFT. This requires the bulk
theory to be interacting.
• At leading order in N, the classical EOM may
be used to calculate the 3-pt functions Giombi, Yin• The inverse Newton constant is quantized
Maldacena, Zhiboedov

Sphere Free Energy• Compare the free energy on the d-sphere at
the boundary of Euclidean AdS with the bulkcalculation• Cannot determine the leading classical piece(no known action), but focus on the one-loopcorrection.
• In the free CFT,
• For example, in d=3 U(N) singlet CFT

Conformal Scalar on Sd• In any dimension• The eigenvalues and degeneracies are• Using zeta-function regularization in d=3,• Check cancellation of theterm in Fbulk• The UV log divergence cancels using standard zetafunction regularization. This is evidence for oneloop finiteness of the Vasiliev theory in AdS4.

• The finite part for each spin Camporesi, Higuchi• Sum over spins vanishes using the Hurwitz-Lerch
function to regularize:• Perfect agreement with the CFT where theterm vanishes!
• In the minimal Vasiliev theory with only even
spins we encounter a surprise. The log divergencecancels, but the finite part does NOT vanish.

Free O(N) Model• The sum over even spins in AdS gives• This is exactly the F-value of a real massless scalarin 3 dimensions! Giombi, IK• This suggests a shift in the identification of thequantized Vasiliev coupling:• We conjecture that the classical term is• Then the sum of the classical and one loop terms
in the minimal (even spin) Vasiliev theory wouldagree with the CFT of N real scalars.

Even Boundary Dimensions• In even d, the CFT sphere free energy is UV
logarithmically divergent, the coefficient being related to
the Weyl a-anomaly.
• In the bulk, this logarithmic divergence is reflected in theIR divergence of the AdSd+1 volume for odd d+1• The coefficient of log R in the bulk free energy is dual tothe a-anomaly coefficient on the CFT side.
• There is no UV divergence in the bulk in this case (in odd
dimensional spacetime, z(0)=0 identically).

Anomaly Matching• If the CFT is free, the a-anomaly should be Nascalarwithout 1/N corrections.
• The results we find are consistent with the general
picture: Giombi, IK, Safdi

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

Example: d=4• For the Vasiliev theory in AdS5 with all integer spins, theone-loop bulk free energy• For the minimal theory with even spins only

• The +1/90 is the a-anomaly coefficient of a real scalar ind=4.
• For the even spin theory in any d,Interacting CFT’s• A scalar operator
in d-dimensional CFTis dual to a fieldin AdSd+1 which
behaves near the boundary as• There are two choices• If we insist on unitarity, then D- is allowed
only in the Breitenlohner-Freedman range
IK, Witten• Flow from a large N CFT where
hasdimension D- to another CFT with dimension D+by adding a double-trace operator. Witten; Gubser, IK• Can flow from the free d=3 scalar model in the UV
to the Wilson-Fisher interacting one in the IR. The
dimension of scalar bilinear changes from 1 to2 +O(1/N). The dual of the interacting theory is
the Vasiliev theory with D=2 boundary conditions
on the bulk scalar.
• The 1/N expansion is generated using theHubbard-Stratonovich auxiliary field.

• In 2<d<4 the quadratic term may be ignored in
the IR:• Induced dynamics for the auxiliary fieldendows it with the propagator• The 1/N corrections to operator dimensions
are calculated using this induced propagator.
For example,• For the leading correction need

• d is the regulator later sent to 0.

• When the leading correction is negative, thelarge N theory is non-unitary.
• It is positive not only for 2<d< 4,but also for 4<d<6.

• The 2-point function
coefficient
is similar

Gross-Neveu CFT• Multiple Dirac fermions with action• In 2 < d < 4 there is a UV fixed point, at least
for large N.
• In
can also be described as an IR fixed
point of the Gross-Neveu-Yukawa model
Zinn-Justin, Moshe; Hasenfratz et al• The beta functions are• IR stable fixed point Moshe, Zinn-Justin• In d=3 the U(N) singlet sector of the large Nmodel has been conjectured to be dual to type BVasiliev theory in AdS4 with the alternate
boundary conditions. Leigh, Petkou; Sezgin, Sundell

Towards Interacting 5-d O(N) Model• Scalar large N model withinteraction hasa UV fixed point for 4<d<6.
• In,• So, the UV fixed point is at a negative coupling• At large N, conjectured to be dual to Vasiliev
theory in AdS6 withboundary condition onthe bulk scalar. Giombi, IK, Safdi• Check of 5-dimensional F-theorem

Perturbative IR Fixed Points• Work inwith O(N) symmetric cubicscalar theory• The beta functions Fei, Giombi, IK• For large N, the IR stable fixed point is at realcouplings

RG Flows• Here is the flow pattern for
N=2000• The IR stable fixed points go
off to complex couplings for
N < 1039. Large N expansion
breaks down very early!

• The dimension of sigma is• At the IR fixed point this is• Agrees with the large N result for theO(N) model in d dimensions:Petkou (1995)• For N=0, the fixed point at imaginary coupling
may lead to a description of the Lee-Yang edge
singularity in the Ising model. Michael Fisher (1978)• For N=0,is below the unitarity bound• For N>1039, the fixed point at real couplings isconsistent with unitarity inCritical N• What is the critical value of N in d=5 below which the unitary
perturbative fixed point disappears?
• A two-loop calculation gives Fei, Giombi, IK, Tarnopolsky (in preparation)• If this approximation is reliable than the critical N is still verylarge in d=5, although much smaller than 1038.
• It is interesting to study the d=5 theory directly using, forexample, the conformal bootstrap.
• The first bootstrap results look encouraging. There is evidence
for a minimum of CJ and CT which for large N matches with theO(N) model results Nakayama, Ohtsuki

(Meta) Stability?
• Since the UV lagrangian is cubic, does thetheory make sense non-perturbatively?
• When the CFT is studied on or
theconformal coupling of scalar fields tocurvature renders the perturbative vacuummeta-stable.
• This suggests that the dual Vasiliev theory is
metastable, but only for the D- boundary
conditions.

Conclusions• Vasiliev theories in AdSd+1 are one-loop finite
theories of Quantum Gravity.
• Provided one-loop evidence for dualities with
U(N) and O(N) singlet sectors of scalar field
theories. In the O(N) case
• Found a new description of the UV fixed
points of the scalar O(N) model in 4<d<6 valid
for sufficiently large N.
• The (meta) stability of these theories deserves
further investigations.