Remarks on SYK Holography

Sumit R. Das(S.R. Das, A. Jevicki and K. Suzuki : 1704.07208)

(S.R. Das, A. Ghosh, A. Jevicki and K. Suzuki : 1711.09839; 1712.02725; in progress)

Large N and Space-time• Examples of holography usually involve large-N field theories.• Generally the large N degrees of freedom get metamorphosed into additional

dimensions. In most cases the details of this metamorphism are not known.• Exceptions : Duality of Matrix Quantum Mechanics and 2 dim string theory

Vector Model – Higher Spin Theory duality• This talk deals with a toy model of holography where this could be possible.

Vector Models and Bilocal Fields

• Klebanov and Polyakov (2002) : conformal 3d O(N) vector model is dual to Higher Spin Gauge theories of Vasiliev type in one higher dimensional AdS space-time.

• S.R.D. and A. Jevicki (2003) : this may be understood by recognizing that all invariants can be expressed in terms of Yukawa type bilocal collective fields.

• Now introduce center of mass and relative coordinates and express the relative coordinate in terms of the magnitude and angles, and perform an expansion

• Our first instinct was to identify with the Poincare coordinate in an , and then the fields would be the infinite tower of massless higher spin fields.

• If this is true, all the conformal transformations of and , or equivalently , should reproduce the Killing symmetries of acting on higher spin fields.

• This works for dilatations, translations and rotations

• Here is the scaling dimension of

are angular momentum generators in the space of relative coordinates which become interpreted as spin in the d+1 dimensional theory.• These are now isometries of with r being identified as the Poincare

coordinate.

• However for arbitrary d this does not work for the special conformal transformations

• The last term is not present for the corresponding isometry. In fact this term mixes up fields of different spins.

• However for arbitrary d this does not work for the special conformal transformations

• The last term is not present for the corresponding isometry. In fact this term mixes up fields of different spins.

• Vasiliev theory is supposed to be dual to a gauged version of the model e.g. with a Chern Simons. On the plane this can be done by considering bi-locals at equal time.

• In d=3 de Mello Koch, Rodrigues, Jevicki and Jin (2011) invented a non-local transformation of equal time bi-locals to fields in light front quantization such that these symmetries are realized correctly. Also some progress in canonical quantization

• For recent progress in understanding this, see Jevicki’s talk.

HOWEVER SOMETHING INTERESTING HAPPENS WHEN

Bilocals in d=1 • When we have one dimension, the SL(2,R) transformations are

• Defining• We get the following transformations

• These are exactly the Killing isometries of Lorentzian

• This is why these have become interesting for the SYK model

SYK Model

• This is a model of N real fermions which are all connected to each other by a random coupling. The Hamiltonian is

• The couplings are random with a Gaussian distribution with width

• This model is of interest since this displays quantum chaos and thermalization.• When N is large, Sachdev and Kitaev showed that one can replace the quenched

average by an annealed average.

• Averaging over the couplings gives rise to the action

• There is now a O(N) global symmetry.• We can now express the path integral in terms of the bilocal collective field. This was done by

(Jevicki, Suzuki and Yoon)

• The path integral is now

• Where the collective action includes the jacobian for transformation from the original variables to the new bilocal fields

• The equations of motion are the large N Dyson-Schwinger equations

• At strong coupling – which is the IR of the theory – the first term can be neglected, and there is an emergent reparametrization symmetry

• In this limit, the saddle point solution is (at zero temperature)

The Strong Coupling Spectrum• Expand the bilocal action around the large N saddle point

• Where denotes a complete orthonormal set of combination of Bessel functions (Kitaev, Polchinski and Rosenhaus)

• The order is real discrete or purely imaginary continuous

• These appear as simultaneous eigenfunctions for the kernel for quadratic fluctuations and the SL(2,R) Casimir which is the wave operator of a scalar field in with mass at the BF bound

• The orthonormality and completeness relations are

• The integral here is a shorthand for a sum over discrete modes and an integral over imaginary values.

• This combination of Bessel functions is forced on us by the requirement that the SL(2,R) generators commute with the kernel. (Polchinski and Rosenhaus)

• This leads to the quadratic action

where

• The spectrum is therefore given by the solutions of the equation

• There are always an infinite number of solutions.• For any q is always a solution. • This is a zero mode at strong coupling coming from reparametrization symmetry

broken by the saddle point solution.

The Bilocal Propagator• The 4 point function of the fermions is the two point function of the bilocal

fluctuations. • The expression for this is

• Performing the integral over the propagator can be expressed as a sum over poles

• denotes greater (smaller) of z and z’.• is the residue of the pole at• Actually there is a double pole at 3/2. In the above expression we have implicitly used

a regulator to shift this pole – this anticipates a proper treatment of this mode.

• Explicit expression for the residue

• Here

• This expression has a divergent contribution from the mode since

• At finite coupling, however, this pole is shifted since the reparametrization invariance is explicitly broken (Maldacena and Stanford)

• Using this shift the contribution from this mode can be calculated

• The answer is proportional to - hence called an enhanced propagator• This is the quantity which is reproduced from the Schwarzian action.

A Tower of Fields • The bilocal propagator can be thought of a sum of two point functions of composite

operators. The primary operators are

• Then the dimensions of these operators can be read off from the form of the bilocalpropagator,

• In standard AdS/CFT, each primary operator of the theory should correspond to a field – so one would expect that there are an infinite number of fields in

• This is happening because even though looks like a 1+1 dimensional field its action is non-polynomial in derivatives

AdS Interpretation

• Maldacena, Stanford and Yang; Engelsoy, Mertens and Verlinde argued that the gravity sector should be Jackiw-Teitelboim type of dilaton-gravity in 1+1 dimensions coupled to some matter (Alihemri and Polchinski)

• Classical solutions have a metric, e.g.• And a dilaton• A nonzero a breaks the conformal symmetry. Agrees with SYK where this breaking is

in the UV ( finite z ). Naturally,

• There is also a black hole solution

• This is supposed to be dual to the SYK model at finite temperature

• The gravity sector of the theory is thought to come from the mode of the SYK model which is a zero mode at infinite coupling.

• In fact, the action – which comes entirely from boundary terms – is a Schwarzianaction.

• Mandal, Nayak and Wadia : Polyakov’s action for 2d gravity.• In fact the Schwarzian action is quite universal and describes the physics near the

horizon of black holes in 2+1 and 3+1 dimensions(Nayak, Shukla, Soni, Trivedi & Vishal; Gaikwad, Joshi, Mandal & Wadia)

Bilocal space and Dual space • The coordinates of the bilocal space act as Poincare coordinates in• In fact the bi-local propagator is a sum of scalar propagators with masses

• However these are not any of the standard propagators.• One possibility : maybe the bilocal space should be considered as• More significantly – all our calculations were in a Euclidean theory (which is why

there is no in front of the propagator). How could its dual live in a Lorentzian space-time ?

• The bi-local space is not the dual space in the standard AdS/CFT sense.• Before addressing these issues, let me discuss an intruiging aspect of the model

viewed as a theory in the bilocal space.

The Matter as a KK tower : q=4• It turns out that the spectrum and the bilocal propagator can be exactly reproduced

by a three dimensional model. • For q=4 we have a single scalar field with a standard kinetic term living in the metric

• The third direction is and interval with size 2L – with Dirichlet boundary conditions at the ends.

• To leading order in the scalar equation of motion is

• Pretty much like Horava-Witten

• Decompose the field

• Where the are eigenfunctions of the Schrodinger operator

• This is a standard problem in quantum mechanicsThe even parity modes obeyChoosing

This becomes exactly the equation

which determines the SYK spectrum for q=4

• Substituting the expansion of the field in the action we get

• where

• And are the solutions of the equation

• If we now choose the poles of the propagator in space are exactly the poles of SYK

• Note that is the BF bound of

The Green’s function

• The position space propagator is now given by

• Once again the integral over stands for a sum over the discrete set and an integral over the imaginary axis.

• This integral can be performed with the result

wave function in 3rd direction

• If we evaluate this at we get

• This nontrivial factor comes from the normalized wavefunctions evaluated at y = 0. and from other normalization factors.

• This is exactly the SYK bilocal propagator at q=4 apart from an overall number. There factor was the residue at the pole.

• Once again the contribution from the m=0 mode is infinite since

Eigenvalue shift• The metric along the 3rd direction contains the parameter

• We now calculate the change of the spectrum perturbatively in this parameter.

• where

• We can now calculate the modification to the eigenvalues perturbatively.• The “zero mode” gets a correction which is of the form

• Since this is exactly of the form of the correction obtained by Maldacena and Stanford in the SYK model.

• If we now substitute this modified eigenvalue in the contribution to the propagator from this mode we reproduce the “enhanced propagator”.

• The agreement of the propagator coefficients with wavefunctions should haveimplications for three point functions of primary operators in the model.

Three dimensional model for all q

• The story continues to hold for arbitrary q.• The background is now conformal to

• While the background for q=4 can be obtained from the near-horizon limit of extremal black hole, we do not know of a natural origin of this background.

• The scalar field action now involves a non-trivial potential as well

• The boundary conditions are just as in q=4

• Where we have restricted to even parity modes and imposed the suitable boundary condition – since we are only interested in correlators whose both ends are at x=0.

• The number V is dependent on q

• The eigenvalue problem can be now solved in terms of hypergeometric functions, and the eigenvalues are determined by the equation

• Which exactly agrees with the SYK spectrum.• Likewise (an unconventional) propagator between two points with x=0 also agree

with the SYK propagator – we could verify this numerically to a very high accuracy.

Large q limit• In the large q limit remains an exact solution. The other poles are (Gross

and Rosenhaus; Maldacena and Stanford)

• The residue at remains finite

• The residues at the other poles vanish for large q - these do not contribute to the propagator.

• Unlike in string theory, the infinite tower modes have masses of the same order even at large q. They decouple in the propagator due to vanishing residues.

• For reasons discussed earlier this 3d space-time cannot be directly considered as the dual space in the sense of standard AdS/CFT

• We could regard this as a convenient packaging of the infinite tower of modes of the model.

• In fact, there is no reason to expect that the theory has local interactions in these three dimensions.

Towards a Dual Space• How did the bilocal space end up as a Lorentzian space ?• This is of course because the underlying symmetry is SL(2,R), which is the isometry

group of Lorentzian• However Euclidean also has the same isometry – we would usually write this

as SO(1,2). • In standard AdS/CFT we expect to get a • We now ask how can we get from the bilocal space to a• Note : the following discussion is independent of the 3d interpretation..in the latter

the interval goes along for a ride.

• Let us write the metric on the bilocal space as

• Whose isometries are

• Let us denote the metric on as

• Whose isometries are

• Is there a canonical transformation which takes one to the other ?• A priori, this is an overdetermined problem. However the answer is YES.

• This can be implemented as transformations on momentum space fields.• It turns out that one of these – written in position space – is the X-ray (or radon)

transformation acting on a function on

• The Radon transform of a function on is the integral of the function evaluated on a geodesic with end-points on the boundary.

• The radon transform intertwines between generators of isometries and those of which implies

• This kind of transform appeared in the literature of c=1 Matrix model. It is similar in spirit to the construction of the Vasiliev higher spin fields by de Mello et.al.

• In the context of AdS/CFT, Radon transforms have been used to relate OPE blocks of 2d CFT to bulk fields on a single time slice of (Czech et.al.; Bhowmick et.al.)

• Our use of the radon transform is quite different : we of relate a pair of operators at different times to a bulk field.

• In fact the situation in higher dimensional O(N) theories is quite different (see Jevicki’s talk).

• In any case if our guess is correct, the Radon transform should map the correct eigenfunctions of to the eigenfunctions of the SYK kernel

• Remarkably this is precisely what happens.• The normalizable eigenfunctions of are

• For the discrete series however

• One might have thought that this would mean that the inverse Radon will transform the SYK propagator into a standard Euclidean propagator.

• This is almost correct, but not quite.• Recall the SYK propagator is

• We need to perform the inverse transform on the combinations of the Bessel functions and then perform the integration over

• Inverse radon transform on the SYK propagator

• Inverse radon transform on the SYK propagator

Residues of the poles – wavefunctions in the 3d picture

Usual Euclidean propagator for a field

• Inverse radon transform on the SYK propagator

Residues of the poles – wavefunctions in the 3d picture

Usual Euclidean propagator for a field

Additional “Leg pole” factors. In 3d interpretation another transformation in the 3rd direction- Similar factors appear in the c=1 matrix model

• Inverse radon transform on the SYK propagator

?? ????? Don’t understand : reminiscent of Discrete states in c=1

Finite Temperature• So far we have dealt with the zero temperature theory.• At strong coupling the finite temperature theory is described by compactifying the

Euclidean time – this may be done by performing a reparamterization

• One would have thought that this would transform the metric on the bilocal space to the metric in Rindler – or a black hole. This is NOT what happens. Rather

• This is actually global with identifications.• On the other hand Rindler (black hole) has the metric

• The Euclidean continuation of this is

• This is diffeomorphic to the whole upper half plane (just as in flat space).• In these coordinates a geodesic which joins two points on the boundary is

• The radon transform of a function is

• This radon transform correctly intertwines the laplacians on Euclidean BH and on the lorentzian FTSYK metric

• The solution of the eigenvalue problem for the finite temperature kernel is complicated

• However they are known for (Maldacena and Stanford). The operator which needs to be diagonalized is precisely

• The solutions which satisfy the periodicity conditions and regularity are (even n)

• In the zero temperature limit these reduce to the

Maldacena and Stanford

• On the other hand the eigenfunctions of which are regular everywhere are, for given by

• On the other hand the eigenfunctions of which are regular everywhere are, for given by

• We have calculated the radon transform of this numerically for several values of and and compared them with the eigenfunctions of with the same values

Radon Transform of Euclidean BH eigenfunction forr=3 and n=5

eigenfunction of for r=3 and n=5

• The fact that the Euclidean propagator had these additional terms is disturbing.• The radon transform – or its analog for the O(N) model – is known to reproduce the

standard bulk-boundary map. From this point of view its use in reconstructing the bulk space is quite natural.

• This is what led to these additional terms – we do not know how to interpret those.• We clearly do not understand the story fully – but hopefully we assembled some of

the correct ingredients.

THANK YOU