prof. rob leigh (university of illinois)

The Holographic Geometry of theExact Renormalization Group

Rob Leigh

University of Illinois

Wits: March 2014

with Onkar Parrikar & Alex Weiss, arXiv:1402.1430v2 [hep-th]

Rob Leigh (UIUC) ERG → HS Wits: March 2014 1 / 36

Introduction

Introduction

An appealing aspect of holography is its interpretation in terms ofthe renormalization group of quantum field theories — the ‘radialcoordinate’ is a geometrization of the renormalization scale.The usual realization of this is in the context of strongly coupledfield theories, corresponding to a bulk interpretation in terms ofsemi-classical gravity on specific backgrounds. The role of ~ in thebulk is usually played by a field theory parameter such as 1/N.This idea has largely been explored from the bulk point of view, inwhich bulk equations of motion with their boundary data isinterpreted in terms of Hamilton-Jacobi theory, with the radialcoordinate playing the role of ‘time’.e.g., [de Boer, Verlinde2 ’99, Skenderis ’02, Heemskerk & Polchinski ’10, Faulkner, Liu & Rangamani ’10 ...]

Do we understand this correspondence?


Introduction

Introduction

Consider for example the fact that (almost) everything we know indetail about RG flows in field theory is inferred from perturbationtheory (of some sort), thought of as deformations away from a freefixed point.

I path integral quantization for example is formulated in terms offunctional integration of elementary fields

So, if holographic duality is really a statement about stronglycoupled theories, how can we hope to fully understand its relationto the renormalization group?We will attack the problem directly from the field theory side.We will explore the Wilson-Polchinski exact renormalization group(ERG)

I should be thought of as the principle of regularization (cutoff)independence

I can be stated as an exact Ward identity for the partition function


Introduction

Punch Lines

We will study free field theories perturbed by arbitrary bi-local‘single-trace’ operators (→ still ‘free’, but the partition functiongenerates all correlation functions).We identify a formulation in which the operator sourcescorrespond (amongst other things) manifestly to a connection on areally big principal bundle — related to ’higher spin gaugetheories’The ‘gauge group’ can be understood directly in terms of fieldredefinitions in the path integral, and consequently there are exactWard identities that correspond to ERG equations.This can be formulated conveniently in terms of a jet bundle.The space-time structure extends in a natural way (governed byERG) to a geometric structure over a spacetime of one higherdimension, and AdS emerges as a geometry corresponding to thefree fixed point.Rob Leigh (UIUC) ERG → HS Wits: March 2014 4 / 36

Some Background Material The Exact Renormalization Group

The Exact Renormalization Group

Polchinski ’84: formulated field theory path integral by introducinga regulator given by a cutoff function accompanying the fixed pointaction (i.e., the kinetic term).extracted an exact equation describing the cutoff independence ofthe partition function by isolating (and discarding) a total derivativein the path integral.this equation describes how the couplings must depend on theRG scale in order that the partition function be independent of thecutoff.



The Exact Renormalization Group

Z =

∫[dφ]e−So[M,φ]−Sint [φ]

So[M, φ] =

∫φK−1(−�/M2)�φ

K(x)

x1

Md

dMZ = 0

implies

M∂Sint

∂M= −1

2

∫M∂K∂M

�−1[δSint

δφ

δSint

δφ+δ2Sint

δφ2

]

can extract equations for each couplingcan apply similar methods to correlation functions, and thus obtain’exact Callan-Symanzik equations’ as wellRob Leigh (UIUC) ERG → HS Wits: March 2014 6 / 36


Majorana Fermions in d = 2 + 1

To be specific, it turns out to be convenient to first consider thefree Majorana fixed point in 2 + 1. This can be described by theregulated action

S0 =

∫

xψm(x)γµPF ;µψ

m(x)

Here PF ;µ is a regulated derivative operator

PF ;µ = K−1F (−�/M2)∂(x)

µ




In 2+1, a complete basis of ‘single-trace’ operators consists of

Π(x , y) = ψm(x)ψm(y), Πµ(x , y) = ψm(x)γµψm(y)

We introduce bi-local sources for these operators in the action

Sint =12

∫

x ,yψm(x)

(A(x , y) + γµWµ(x , y)

)ψm(y)

One can think of these as collecting together infinite sets of localoperators, obtained by expanding near x → y . This quasi-localexpansion can be expressed through an expansion of the sources

A(x , y) =∞∑

s=0

Aa1···as (x)∂(x)a1· · · ∂(x)

as δ(x − y)

(similarly for Wµ). The coefficients are sources for higher spinlocal operators.Rob Leigh (UIUC) ERG → HS Wits: March 2014 8 / 36



Why do we do consider this theory?I There are no interactions, so is it interesting?

One answer is that this constitutes the unbroken phase of a higherspin theory (as we will see).Even though there are no interactions, there are still non-trivial RGequations, because given the regulator, A(x , y) and Wµ(x , y) mustbe scale dependent in order that the partition function remainindependent of the cutoff.We will see that Wµ(x , y) should be interpreted as a gauge fieldcorresponding to the linear symmetry of the fixed point.The model easily extends to other dimensions, although in d > 3there will be additional tensorial sources.Will describe how to introduce interactions later.




Define the partition function

Z [M,g(0),A,Wµ] = N∫

[dψ] ei(S0+Sint )

note that here we are asking specific questions of the O(N)fermion model, questions that only see the singlet sector — this isa ’consistent truncation’The expectation values of the single-trace operators are thengiven by

Π(x , y) = −iδ

δA(x , y)ln Z , Πµ(x , y) = −i

δ

δWµ(x , y)ln Z



Higher Spin Theories

Note that this model fits with the conjectured duality between 3dvector models and Vasiliev higher-spin theories on AdS4[Klebanov & Polyakov ’02, Sezgin & Sundell ’02, Leigh & Petkou ’03] [Vasiliev ’96, ’99, ’12] [de Mello Koch, et al ’11]

Since the conjecture involves free-field theory, such a holographicduality (if true) begs for a geometric understanding in terms of RG[Douglas, Mazzucato & Razamat ’10, Pando Zayas & Peng ’13, Sachs ’13]

The Majorana model is believed to be dual to the B-type VasilievHS theory in AdS4. (For the experts, you’ll observe that A and Wµ

are in 1–1 correspondence with fields in the Vasiliev theory).Indeed, we will arrive at a structure remarkably similar toVasiliev’s. There will be important structural differences. Should itbe obvious that we obtain precisely the Vasiliev structure?The method can be applied to other free fixed points, and higherspin theories constructed that have no analogue to Vasilievtheories.Rob Leigh (UIUC) ERG → HS Wits: March 2014 11 / 36


The O(L2) symmetry

it is conceptually important to introduce a matrix notation in whichwe re-write

PF ;µ(x , y) = K−1F (−�/M2)∂(x)

µ δ(x − y)

such that the kinetic term can be written

S0 =12

∫

x ,yψm(x)γµPF ;µ(x , y)ψm(y)

and thus the full action takes the form

S =12

∫

x ,yψm(x)

[γµ(PF ;µ + Wµ)(x , y) + A(x , y)

]ψm(y)

≡ ψm ·[γµ(PF ;µ + Wµ) + A

]· ψm



The O(L2) symmetry

This can be generalized to arbitrary products of bi-local functions

(f · g)(x , y) ≡∫

zf (x , z)g(z, y)

Now we consider the following map of elementary fields

ψm(x) 7→∫

yL(x , y)ψm(y)

The ψm are just integration variables in the path integral, and sothis is just a trivial change of integration variable. I’m using herethe same logic that might be familiar in the Fujikawa method forthe study of anomalies.So, we ask, what does this do to the partition function?



The O(L2) symmetry

We look at the action

S = ψm ·[γµ(PF ;µ + Wµ) + A

]· ψm

→ ψm · LT [γµ(PF ;µ + Wµ) + A]· L · ψm (1)

= ψm · γµLT · L · PF ;µ · ψm (2)

+ψm ·[γµ(LT ·

[PF ;µ,L

]+ LT ·Wµ · L) + LT · A · L

]· ψm

Thus, if we take L to be orthogonal,LT · L(x , y) =

∫z L(z, x)L(z, y) = δ(x , y), the kinetic term is

invariant, while the sources transform asO(L2) gauge symmetry

Wµ 7→ L−1 ·Wµ · L+ L−1 ·[PF ;µ,L

]

A 7→ L−1 · A · LRob Leigh (UIUC) ERG → HS Wits: March 2014 14 / 36


The O(L2) symmetry

But this was a trivial operation from the path integral point of view,and so we conclude that there is an exact Ward identity

Z [M,g(0),Wµ,A] = Z [M,g(0),L−1WµL+ L−1PF ;µL,L−1AL]

if we write L ∼ δ + ε, then we obtain the infinitesimal version

δWµ = [Dµ, ε]. , δA = [ε,A].

where ε is an infinitesimal parameter satisfying ε(x , y) = −ε(y , x),and the Ward identity is just the statement

[Dµ,Πµ]. + [Π,A]. = 0

where Dµ = PF ;µ + Wµ.



The O(L2) symmetry

Note what is happening here: the O(L2) symmetry leavesinvariant the (regulated) free fixed point action. Wµ is interpretedas a gauge field (connection) for this symmetry, while A transformstensorily. Dµ = PF ;µ + Wµ plays the role of covariant derivative.More precisely, the free fixed point corresponds to anyconfiguration

(A,Wµ) = (0,W (0)µ )

where W (0) is any flat connection

dW (0) + W (0) ∧W (0) = 0

It is therefore useful to split the full connection as

Wµ = W (0)µ + Wµ

This will become important when we discuss RG flow.



The CO(L2) symmetry

We can generalize the O(L2) condition to include scaletransformations ∫

zL(z, x)L(z, y) = λ2∆ψδ(x − y)

This is a symmetry (in the previous sense) provided we alsotransform the metric, the cutoff and the sources

g(0) 7→ λ2g(0), M 7→ λ−1M

A 7→ L−1 · A · LWµ 7→ L−1 ·Wµ · L+ L−1 ·

[PF ;µ,L

].

A convenient way to keep track of the scale is to introduce theconformal factor g(0) = 1

z2 η. Then z 7→ λ−1z. This z should bethought of as the renormalization scale.Rob Leigh (UIUC) ERG → HS Wits: March 2014 17 / 36


The Renormalization group

To study RG systematically, we proceed in two steps:

Step 1: Lower the cutoff M 7→ λM, by integrating out the “fast modes”

Z [M, z,A,W ] = Z [λM, z, A, W ] (Polchinski)

Step 2: Perform a CO(L2) transformation to bring the cutoff back to M,but in the process changing z 7→ λ−1z

Z [λM, z, A, W ] = Z [M, λ−1z,L−1.A.L,L−1.W .L+ L−1.[PF ,L]]

We can now compare the sources at the same cutoff, but differentz. Thus, z becomes the natural flow parameter, and we can thinkof the sources as being z-dependent. (Thus we have thePolchinski formalism with both a cutoff and an RG scale –required for a holographic interpretation).Rob Leigh (UIUC) ERG → HS Wits: March 2014 18 / 36


RG in pictures

Pictorially, we may represent the above two-step process as

Step 1 Step 2

M → λM g(0) → λ2g(0)!z → λ−1z

"

∼ 1M

∼ 1M

∼ 1λM

ℓ ℓ

λ ℓ

It is useful to think of different values of z as pertaining to differentcopies of spacetime

z

λ−1z0

z0

xµyµ

xµ+ξµ(z)yµ+ξµ(z)

z

λ−1z



Infinitesimal version: RG equations

In the infinitesimal case, we parametrize the CO(L2) gaugetransformation as

L = 1 + εzWz

The gauge transformation should be thought of as containing thez-component of the connection.The RG equations become

A(z + εz) = A(z) + εz [Wz ,A] + εzβ(A) + O(ε2)

Wµ(z + εz) = Wµ(z) + εz[PF ;µ + Wµ,Wz

]+ εzβ(W )

µ + O(ε2)

The beta functions are tensorial, and quadratic in A and W .The flat connection W (0) also satisfies a “pure-gauge” RGequation

W (0)µ (z + εz) = W (0)

µ (z) + εz[PF ;µ + W (0)

µ ,W (0)z

]+ O(ε2)



RG equations

Thus, RG extends the sources A and W to bulk fields A andW.Comparing terms linear in ε gives

∂zW(0)µ − [PF ;µ,W(0)

z ] + [W(0)z ,W(0)

µ ] = 0

∂zA+ [Wz ,A] = β(A)

∂zWµ − [PF ;µ,Wz ] + [Wz ,Wµ] = β(W)µ

These equations are naturally thought of as being part of the fullycovariant equations (e.g., the first is the zµ component of a bulk2-form equation, where d ≡ dxµPF ,µ + dz∂z .)

dW(0) +W(0) ∧W(0) = 0dA+ [W,A] = β(A)

dW +W ∧W = β(W)

The resulting equations are then diff invariant in the bulk.Rob Leigh (UIUC) ERG → HS Wits: March 2014 21 / 36


Some cross-checks

The equations that we have written are β-function equations – thebi-local sources are like couplings.Similarly, one can extract exact Callan-Symanzik equations (forexample, for the z-dependence of Π(x , y), Πµ(x , y). These extendto bulk fields P,Pµ.The full set of equations then give rise to a phase spaceformulation of a dynamical system — (A,P) and (WA,PA) arecanonically conjugate pairs from the point of view of the bulk.The form of the β-function equations suggest that the betafunctions encode 3-point functions. From the bulk point of view,they encode interactions.One can check that the solutions of the RG equations, taken backto the free fixed point do indeed give precisely the 3-pointfunctions of the free theory. [see Giombi+Yin]



The bulk extensions

Let me summarize what has happened in going from the fieldtheory spacetime to the bulk

Wµ(x , y) → W = Wµ(z; x , y)dxµ + Wz(z; x , y)dzA(x , y) → A(z; x , y) (3)

Πµ(x , y) → Pµ(z; x , y) (4)Π(x , y) → P(z; x , y) (5)

β(W )µ → β(W ) = β

(W)a ez ∧ ea + β

(W)ab ea ∧ eb (6)

β(A) → β(A) = β(A)ez + β(A)a ea

The transverse components of the beta functions (that don’tappear in the RG equations) satisfy their own flow equations(Bianchi identities)

Dβ(A) =[β(W),A

], Dβ(W) = 0



Hamilton-Jacobi Structure

Indeed, if we identify Z = eiSHJ , then a fundamental relation in H-Jtheory is

∂

∂zSHJ = −H

We can thus read off the Hamiltonian of the theory, for which theRG equations are the Hamilton equations

H = −Tr{([A,W

e(0)z

]+ β(A)

)· P}

−Tr{([

PF ;µ +Wµ,We(0)z

]+ β(W)

µ

)· Pµ

}(7)

−N2

Tr{(

∆µ · Wµ + ∆z · We(0)

z

)}

Note that this is linear in momenta — the hallmark of a free theory.



Hamilton-Jacobi Structure

The existence of this Hamiltonian structure is confirmation that(A,P) etc. are conjugate pairs (with trivial symplectic form), asrequired by holography.The full set of RG equations of the field theory (β-functions andC-S equations) are required to see this structure, and they are’integrable’ in the sense that they can be thought of as thecorresponding Hamilton equations.In terms of RG, this means that γ-functions that appear in C-Sequations are related to (derivatives of) the β-functions.Thus the (full set of) ERG equations are holographic in a standardsense.This seems to be a feature that is not shared by other attempts atinterpreting higher spin theories holographically.



The Bulk Spacetime

A natural flat background connection is given by

W(0)(x , y) = −dzz

D(x , y) +dxµ

zPµ(x , y)

where Pµ(x , y) = ∂(x)µ δ(x − y) and D(x , y) = (xµ∂(x)

µ + ∆)δ(x − y).This connection is flat because of the commutation relations ofD,Pµ.This connection is equivalent to the vielbein and spin connectionof AdS. This is appropriate, since W (0) corresponds to the freefixed point, which is conformally invariant.W and A correspond to geometric structures over AdS.

I We want to specify more precisely what this geometry is.



The Infinite Jet bundle

What sort of geometrical structure is it? We have been usinggauge theory terms — connection, gauge transformations, etc.Usually gauge ≡ local, ψ(x) 7→ eiα(x)ψ(x)

What are we to make of the non-local transformation?

ψ(x) 7→∫

yL(x , y)ψ(y)

We have been using matrix notation: space-time coordinates ≡matrix indicesIf we can regard these really as matrix indices, then we wouldhave an ordinary vector bundle, with Wµ as a connection.The proper interpretation here turns out to be in terms of amathematical construction known as the infinite jet bundle.



The Infinite Jet bundle...

The simple idea is that we can think of a differential operatorL(x , y) as a matrix by “prolongating” the field

ψm(x) 7→(ψm(x),

∂ψm

∂xµ(x),

∂2ψm

∂xµ∂xν(x) · · ·

)

The collection of such vectors at a point is called the infinite jetspace. The bundle over spacetime of such jet spaces is called theinfinite jet bundle.

Then, differential operators, such as Pµ(x , y) = ∂(x)µ δ(x − y) are

interpreted as matrices Pµ that act on these vectorsNote one effect of this organization: we have a clean demarcationbetween the vector index on Wµ and the ‘higher spin indices’ ofW ab...µ .



The Infinite Jet bundle

The bi-local transformations can be thought of as local gaugetransformations of the jet bundle.The gauge field W is a connection 1-form on the jet bundle, whileA is a section of its endomorphism bundle.RG instructs us how to extend the infinite jet-bundle of theboundary field theory into the bulk, and thenW and A are definedcorrespondingly in the bulk.



Making contact with Vasiliev theory

Recall that the flat background connection

W(0) = −dzzD +

dxµ

zδaµPa

encodes the pure-gauge RG flow of the free-fixed point.Gauge transformations ε(0) which preserveW(0) are globalsymmetries of the free fixed point. These are the isometries ofAdS.A basis for such transformations is given by D,Pa,Mab, ... (andWeyl-ordered products thereof)



Making contact with Vasiliev theory

In order to make contact with Vasiliev, a representation for thegenerators of this algebra can be given by introducing the VasilievY -oscillators, e.g.

D =12εijY d

i Y−1j , Mab =

12εijY a

i Y bj , ...

The Y Ai s satisfy the star product Y A

i ? Y Bj = Y A

i Y Bj + 1

2ηABεij .

The RG equations now become

dA+ [W,A]? = β(A)?

dW +W ∧?W = β(W)?

These are not of the form of Vasiliev’s equations, although theingredients are similar!



The Principal Jet Bundle

I have discussed Wµ in terms of a vector bundle, but this can beregarded as associated to a principal bundle (with group O(L2))A connection is a 1-form on the total space of the principal bundle.If we identify a horizontal (≡ AdS), then the connection has both’horizontal’ and ’vertical’ parts.The horizontal part is Wµ, while the vertical part is new — wesuggestively call it S. In simple terms, this amounts toincorporating Faddeev-Popov ghosts. [Thierry-Mieg ’80]

These new degrees of freedom can beintroduced subject to consistency conditions,which are just the BRST equations (QBRST ∼ dZ ).

dZW + dxS + {W,S}? = 0dZA+ [S,A]? = 0dZ S + S ∧? S = 0

Σ

∂Z∂x

PG



The full system

Putting all the equations together, we obtain

dA+ [W,A]? = β(A)?

dW +W ∧?W = β(W)?

dZW + dxS + {W,S}? = 0

dZA+ [S,A]? = 0

dZ S + S ∧? S = 0

These equations bear remarkable resemblance with Vasilievequations.But there are also significant differences – a precise mappingbetween our variables and Vasiliev is lacking.



Remarks

We have seen how the rich symmetry structure of the free-fixedpoint allows us to geometrize RG.The resulting structure has striking similarities with Vasiliev higherspin theory, and begs for a more precise matching.Although we focused on the Majorana fermion, the boson can behandled similarly, in fact in arbitrary dimension. In the bosoniccase, the action can be written in terms of PF ;µ and sources Wµ

and B.There is a unitary model based on Dirac fermions as well, whichhas interesting additional features.



Remarks, cont.

The fermion is more difficult to deal with in higher dimensions,because of additional single-trace operators like ψγµνψ, ψγµνλψ,...These will give rise to higher spin theories through the RGconstructions, but these will have no Vasiliev analogue. (TheB-model exists only in d = 4).There are many other examples of non-Vasiliev higher spintheories that we will present soon.The partition function of the interacting fixed point in d = 3 can bestudied by an integral transform. That is, multi-trace interactionscan be induced by reversing the Hubbard-Stratanovich trick. Thistransform has a large-N saddle corresponding to the (fermionic)critical O(N) model. (otherwise, N did not have to be large!)



Remarks, cont., cont.

What of standard gravitational holography?The standard higher spin lore is expected to kick in here — wheninteractions are included, the higher spin symmetry breaks (theoperators get anomalous dimensions). At strong coupling, all thatis left behind is gravity.It is an interesting challenge to show that precisely this happensgenerically.


prof. rob leigh (university of illinois)

Education

free eld theories

terms of eld redenitions

eld theory parameter

introduction introduction

gauge group

erg equations

exact equation

renormalization scale