Prof. Rob Leigh (University of Illinois)

The Holographic Geometry of the
Exact 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 of
the renormalization group of quantum field theories — the ‘radial
coordinate’ is a geometrization of the renormalization scale.
The usual realization of this is in the context of strongly coupled
field theories, corresponding to a bulk interpretation in terms of
semi-classical gravity on specific backgrounds. The role of in the
bulk is usually played by a field theory parameter such as 1/N.
This idea has largely been explored from the bulk point of view, in
which bulk equations of motion with their boundary data is
interpreted in terms of Hamilton-Jacobi theory, with the radial
coordinate 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 in
detail about RG flows in field theory is inferred from perturbation
theory (of some sort), thought of as deformations away from a free
fixed point.
path integral quantization for example is formulated in terms of
functional integration of elementary fields
So, if holographic duality is really a statement about strongly
coupled theories, how can we hope to fully understand its relation
to the renormalization group?
We will attack the problem directly from the field theory side.
We will explore the Wilson-Polchinski exact renormalization group
(ERG)
should be thought of as the principle of regularization (cutoff)
independence
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 function
generates all correlation functions).
We identify a formulation in which the operator sources
correspond (amongst other things) manifestly to a connection on a
really big principal bundle — related to ’higher spin gauge
theories’
The ‘gauge group’ can be understood directly in terms of field
redefinitions in the path integral, and consequently there are exact
Ward 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 by
ERG) to a geometric structure over a spacetime of one higher
dimension, and AdS emerges as a geometry corresponding to the
free fixed point.

Some Background Material The Exact Renormalization Group
The Exact Renormalization Group
Polchinski ’84: formulated ﬁeld theory path integral by introducing
a regulator given by a cutoff function accompanying the ﬁxed point
action (i.e., the kinetic term).
extracted an exact equation describing the cutoff independence of
the partition function by isolating (and discarding) a total derivative
in the path integral.
this equation describes how the couplings must depend on the
RG scale in order that the partition function be independent of the
cutoff.

The Exact Renormalization Group
Z = [dφ]e−So[M,φ]−Sint [φ]
So[M, φ] = φK−1
(− /M2
) φ
K(x)
x1
M
d
dM
Z = 0
implies
M
∂Sint
∂M
= −
1
2
M
∂K
∂M
−1 δSint
δφ
δSint
δφ
+
δ2Sint
δφ2
can extract equations for each coupling
can apply similar methods to correlation functions, and thus obtain
’exact Callan-Symanzik equations’ as well

Majorana Fermions in d = 2 + 1
To be specific, it turns out to be convenient to first consider the
free Majorana fixed point in 2 + 1. This can be described by the
regulated action
S0 =
x
ψm
(x)γµ
PF;µψm
(x)
Here PF;µ is a regulated derivative operator
PF;µ = K−1
F (− /M2
)∂(x)
µ

In 2+1, a complete basis of ‘single-trace’ operators consists of
ˆΠ(x, y) = ψm
(x)ψm
(y), ˆΠµ
(x, y) = ψm
(x)γµ
ψm
(y)
We introduce bi-local sources for these operators in the action
Sint =
1
2 x,y
ψm
(x) A(x, y) + γµ
Wµ(x, y) ψm
(y)
One can think of these as collecting together inﬁnite sets of local
operators, obtained by expanding near x → y. This quasi-local
expansion 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 spin
local operators.

Why do we do consider this theory?
There are no interactions, so is it interesting?
One answer is that this constitutes the unbroken phase of a higher
spin theory (as we will see).
Even though there are no interactions, there are still non-trivial RG
equations, because given the regulator, A(x, y) and Wµ(x, y) must
be scale dependent in order that the partition function remain
independent of the cutoff.
We will see that Wµ(x, y) should be interpreted as a gauge ﬁeld
corresponding to the linear symmetry of the ﬁxed point.
The model easily extends to other dimensions, although in d > 3
there 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 is
a ’consistent truncation’
The expectation values of the single-trace operators are then
given by
Π(x, y) = −i
δ
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 3d
vector 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 holographic
duality (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 Vasiliev
HS 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 to
Vasiliev’s. There will be important structural differences. Should it
be obvious that we obtain precisely the Vasiliev structure?
The method can be applied to other free fixed points, and higher
spin theories constructed that have no analogue to Vasiliev
theories.

The O(L2) symmetry
it is conceptually important to introduce a matrix notation in which
we re-write
PF;µ(x, y) = K−1
F (− /M2
)∂(x)
µ δ(x − y)
such that the kinetic term can be written
S0 =
1
2 x,y
ψm
(x)γµ
PF;µ(x, y)ψm
(y)
and thus the full action takes the form
S =
1
2 x,y
ψm
(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) ≡
z
f(x, z)g(z, y)
Now we consider the following map of elementary ﬁelds
ψm
(x) →
y
L(x, y)ψm
(y)
The ψm are just integration variables in the path integral, and so
this is just a trivial change of integration variable. I’m using here
the same logic that might be familiar in the Fujikawa method for
the 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
· γµ
LT
· 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 as
O(L2) gauge symmetry
Wµ → L−1
· Wµ · L + L−1
· PF;µ, L
A → L−1
· A · L

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−1
WµL + L−1
PF;µL, L−1
AL]
if we write L ∼ δ + , then we obtain the inﬁnitesimal version
δWµ = [Dµ, ]. , δA = [ , A].
where is an inﬁnitesimal parameter satisfying (x, y) = − (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 leaves
invariant the (regulated) free fixed point action. Wµ is interpreted
as a gauge field (connection) for this symmetry, while A transforms
tensorily. Dµ = PF;µ + Wµ plays the role of covariant derivative.
More precisely, the free fixed point corresponds to any
configuration
(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 scale
transformations
z
L(z, x)L(z, y) = λ2∆ψ
δ(x − y)
This is a symmetry (in the previous sense) provided we also
transform the metric, the cutoff and the sources
g(0) → λ2
g(0), M → λ−1
M
A → L−1
· A · L
Wµ → L−1
· Wµ · L + L−1
· PF;µ, L .
A convenient way to keep track of the scale is to introduce the
conformal factor g(0) = 1
z2 η. Then z → λ−1z. This z should be
thought of as the renormalization scale.

The Renormalization group
To study RG systematically, we proceed in two steps:
Step 1: Lower the cutoff M → 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 → λ−1z
Z[λM, z, A, W] = Z[M, λ−1
z, L−1
.A.L, L−1
.W.L + L−1
.[PF , L]]
We can now compare the sources at the same cutoff, but different
z. Thus, z becomes the natural ﬂow parameter, and we can think
of the sources as being z-dependent. (Thus we have the
Polchinski formalism with both a cutoff and an RG scale –
required for a holographic interpretation).

RG in pictures
Pictorially, we may represent the above two-step process as
Step 1 Step 2
M → λM g(0)
→ λ2
g(0)
z → λ−1
z
∼ 1
M
∼ 1
M
∼ 1
λM
ℓ ℓ
λ ℓ
It is useful to think of different values of z as pertaining to different
copies of spacetime
z
λ
−1z0
z0
xµ
yµ
xµ
+ξµ
(z)
yµ
+ξµ
(z)
z
λ−1
z

Infinitesimal version: RG equations
In the infinitesimal case, we parametrize the CO(L2) gauge
transformation as
L = 1 + εzWz
The gauge transformation should be thought of as containing the
z-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” RG
equation
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 and W.
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 fully
covariant equations (e.g., the ﬁrst is the zµ component of a bulk
2-form equation, where d ≡ dxµPF,µ + dz∂z.)
dW(0)
+ W(0)
∧ W(0)
= 0
dA + [W, A] = β(A)
dW + W ∧ W = β(W)
The resulting equations are then diff invariant in the bulk.

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

The bulk extensions
Let me summarize what has happened in going from the ﬁeld
theory spacetime to the bulk
Wµ(x, y) → W = Wµ(z; x, y)dxµ
+ Wz(z; x, y)dz
A(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’t
appear 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-J
theory is
∂
∂z
SHJ = −H
We can thus read off the Hamiltonian of the theory, for which the
RG equations are the Hamilton equations
H = −Tr A, We
(0)
z
+ β(A)
· P
−Tr PF;µ + Wµ, We
(0)
z
+ β(W)
µ · Pµ
(7)
−
N
2
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), as
required by holography.
The full set of RG equations of the ﬁeld theory (β-functions and
C-S equations) are required to see this structure, and they are
’integrable’ in the sense that they can be thought of as the
corresponding Hamilton equations.
In terms of RG, this means that γ-functions that appear in C-S
equations are related to (derivatives of) the β-functions.
Thus the (full set of) ERG equations are holographic in a standard
sense.
This seems to be a feature that is not shared by other attempts at
interpreting higher spin theories holographically.

The Bulk Spacetime
A natural flat background connection is given by
W(0)
(x, y) = −
dz
z
D(x, y) +
dxµ
z
Pµ(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 of
D, Pµ.
This connection is equivalent to the vielbein and spin connection
of AdS. This is appropriate, since W(0) corresponds to the free
fixed point, which is conformally invariant.
W and A correspond to geometric structures over AdS.
We want to specify more precisely what this geometry is.

The Inﬁnite Jet bundle
What sort of geometrical structure is it? We have been using
gauge theory terms — connection, gauge transformations, etc.
Usually gauge ≡ local, ψ(x) → eiα(x)ψ(x)
What are we to make of the non-local transformation?
ψ(x) →
y
L(x, y)ψ(y)
We have been using matrix notation: space-time coordinates ≡
matrix indices
If we can regard these really as matrix indices, then we would
have an ordinary vector bundle, with Wµ as a connection.
The proper interpretation here turns out to be in terms of a
mathematical construction known as the inﬁnite jet bundle.

The Infinite Jet bundle...
The simple idea is that we can think of a differential operator
L(x, y) as a matrix by “prolongating” the field
ψm
(x) → ψm
(x),
∂ψm
∂xµ
(x),
∂2ψm
∂xµ∂xν
(x) · · ·
The collection of such vectors at a point is called the infinite jet
space. The bundle over spacetime of such jet spaces is called the
infinite jet bundle.
Then, differential operators, such as Pµ(x, y) = ∂
(x)
µ δ(x − y) are
interpreted as matrices Pµ that act on these vectors
Note one effect of this organization: we have a clean demarcation
between the vector index on Wµ and the ‘higher spin indices’ of
Wab...
µ .

The Infinite Jet bundle
The bi-local transformations can be thought of as local gauge
transformations of the jet bundle.
The gauge field W is a connection 1-form on the jet bundle, while
A is a section of its endomorphism bundle.
RG instructs us how to extend the infinite jet-bundle of the
boundary field theory into the bulk, and then W and A are defined
correspondingly in the bulk.

Making contact with Vasiliev theory
Recall that the flat background connection
W(0)
= −
dz
z
D +
dxµ
z
δa
µPa
encodes the pure-gauge RG flow of the free-fixed point.
Gauge transformations (0) which preserve W(0) are global
symmetries of the free fixed point. These are the isometries of
AdS.
A basis for such transformations is given by D, Pa, Mab, ... (and
Weyl-ordered products thereof)

Making contact with Vasiliev theory
In order to make contact with Vasiliev, a representation for the
generators of this algebra can be given by introducing the Vasiliev
Y-oscillators, e.g.
D =
1
2
ij
Yd
i Y−1
j , Mab
=
1
2
ij
Ya
i Yb
j , ...
The YA
i s satisfy the star product YA
i YB
j = YA
i YB
j + 1
2 ηAB
ij.
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 the
ingredients are similar!

The Principal Jet Bundle
I have discussed Wµ in terms of a vector bundle, but this can be
regarded 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 — we
suggestively call it S. In simple terms, this amounts to
incorporating Faddeev-Popov ghosts. [Thierry-Mieg ’80]
These new degrees of freedom can be
introduced subject to consistency conditions,
which are just the BRST equations (QBRST ∼ dZ ).
dZ W + dx S + {W, S} = 0
dZ A + [S, A] = 0
dZ 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)
dZ W + dx S + {W, S} = 0
dZ A + [S, A] = 0
dZ S + S ∧ S = 0
These equations bear remarkable resemblance with Vasiliev
equations.
But there are also signiﬁcant differences – a precise mapping
between our variables and Vasiliev is lacking.

Remarks
We have seen how the rich symmetry structure of the free-ﬁxed
point allows us to geometrize RG.
The resulting structure has striking similarities with Vasiliev higher
spin theory, and begs for a more precise matching.
Although we focused on the Majorana fermion, the boson can be
handled similarly, in fact in arbitrary dimension. In the bosonic
case, 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, which
has 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 RG
constructions, but these will have no Vasiliev analogue. (The
B-model exists only in d = 4).
There are many other examples of non-Vasiliev higher spin
theories that we will present soon.
The partition function of the interacting ﬁxed point in d = 3 can be
studied by an integral transform. That is, multi-trace interactions
can be induced by reversing the Hubbard-Stratanovich trick. This
transform 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 — when
interactions are included, the higher spin symmetry breaks (the
operators get anomalous dimensions). At strong coupling, all that
is left behind is gravity.
It is an interesting challenge to show that precisely this happens
generically.

Prof. Rob Leigh (University of Illinois)

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