Spectral sum rules for conformal field theories

Spectral sum rules for conformal ﬁeld theories in arbitrary
dimensions
Subham Dutta Chowdhury
April 7, 2017
IoP, University of Amsterdam , 2017
Subham Dutta Chowdhury Spectral sum rules for conformal ﬁeld theories in arbitrary dimensions 1/39

References
S. D. Chowdhury and J. R. David, “Spectral sum rules for conformal ﬁeld
theories in arbitrary dimensions” [arXiv:1612.00609 [hep-th]].

Introduction and motivation
The sum rule relates a weighted integral of the spectral density over frequencies
to one point functions of the theory
∞
−∞
dω
ωn
ρ(ω) ∝ One point functions ,
ρ(ω) = ImGR(ω), (1)
where GR(ω) is the retarded Green’s function at finite temperature.
The real time finite retarded correlators are difficult to obtain from lattice
calculations in QCD. However since one point functions are considerably easier
to obtain, this has led to systematic study of sum rules in QCD.
Sum rules provide important constraints on the spectral densities of any
quantum field theory.

Sum rules are a consequence of analyticity of Green’s functions in the complex
ω-plane (which in turn follows from causality).
Although the sum rules involve two point functions, it encodes information
about the three point function also.
Sum rules involve integration over the full frequency domain. The LHS of the
sum rule results from high frequency behaviour as well as low frequency
hydrodynamic behaviour of the theory.

We aim to compute sum rules for conformal ﬁeld theories in d > 2. An
operator universal to CFTs is the stress tensor Tµν . We will look at the sum
rule for the retarded correlator,
GR(t, x) = iθ(t) [Txy(t, x), Txy(0)] .
(2)
at ﬁnite temperature and zero chemical potential
We assume that no other operators apart from the stress tensor gets
expectation values.
The Fourier transform of the retarded correlator is given by,
GR(ω, 0) = dd
xeiωt
GR(t, x) (3)

General Setup
The sum rules assume the analyticity and the well behaved behaviour of the
retarded correlator as ω → i∞,
1) GR(ω) is analytic in the upper half plane including the real axis.
2) limω→i∞ GR(ω) ∼ 1
|ω|m , m > 0
By Cauchy’s theorem,
GR(ω + i ) =
1
2πi
∞
−∞
GR(z)
Z − ω − i
(4)
0 =
1
2πi
∞
−∞
GR(z)
Z − ω + i
(5)
where the contour closes in a semi circle in the upper half plane.
From these two equations, we have,
G(0) = lim
→0+
∞
−∞
dω
π
ρ(ω)
ω − i
(6)
where, ρ(ω) = ImGR(ω)

Justiﬁcations for assumptions
The physical reason why the retarded correlator is analytic in the upper half
plane is causality. It is easy to see this from the inverse Fourier transform
GR(t) =
dω
2π
e−iωt
GR(ω). (7)
For t < 0 and only when GR(ω) is holomorphic, the contour can be closed in
the upper half plane resulting in GR(t < 0) = 0 which is a requirement for the
retarded correlator.
In order to examine the second assumption, we look at CFTs at ﬁnite
temperature and retarded correlators of the stress tensor.

At high frequencies the retarded correlator exhibits a divergent behaviour.
In order to understand this behavior in a precise manner, we analytically
continue GR(ω) into the upper half plane
GR(i2πnT) = GE(2πnT). (8)
Here GE is the Euclidean time ordered correlator and 2πnT is the Matsubara
frequency.
Consider the Euclidean correlator in position space. For time intervals
δt β = 1
T
, the operator product expansion (OPE) of the stress tensor oﬀer a
good asymptotic expansion. Therefore for ω T, we can replace the
Euclidean correlator by its OPE. This allows us to obtain the asymptotic
behaviour of the GR(iω) as ω → ∞.
[Romatschke, Son, 2009]

OPE of the stress tensor in the Euclidean two point function is given by,
Txy(s)Txy(0) ∼ CT
Ixy,xy(s)
s2d
+ Âxyxyαβ(s) Tαβ(0) + · · · (9)
where expectation value has been taken over thermal states.
The stress tensor in d dimensions has scaling dimensions d. By dimensional
analysis, the tensor structure Iµνρσ is dimensionless, while the Âµνρσ scales like
1/sd
.
Their fourier transforms scale as
dd
xeiωt
CT
Ixy,xy(x)
|x2d|
≡ I ∼ ωd
log
ω
Λ
,
∞
−∞
dd
xeiωt Âxyxyαβ(x) Tαβ ≡ J ∼ ω0
âαβ
Tαβ . (10)
where λ is a cut off in the integration.
We assume that there is no chemical potential turned on and no other operator
except stress tensor gets expectation value.

We obtain
lim
ω→∞
GR(iω) ∼ ωd
f
ω
λ
+ ω0
T + O
1
ω
∼ ωd
f
ω
λ
+ J (11)
We have a divergent piece along with a ﬁnite term. The assumption about well
behaved behaviour of the retarded green’s function breaks down.

The divergent piece is identical to the Fourier transform of the retarded Greens
function at zero temperature.
To remove the divergent term we consider the following regularized Green’s
function
δGR(ω) = GR(ω)|T − GR(ω)|T =0. (12)
To remove the ﬁnite parts, we consider,
δGR(ω) = GR(ω)|T − GR(ω)|T =0 − J . (13)
We have constructed a regularized green’s function which is analytic in the
upper half plane and is well behaved as ω → i∞.

The sum rule then becomes,
lim
→0+
1
π
∞
−∞
δρ(ω)dω
ω − i
= δGR(0),
GR(t, x) = iθ(t)[Txy(t, x), Txy(0)],
(14)
where δρ(ω) is the difference of spectral densities
δρ(ω) = Im(GR(0)|T − GR(0)|T =0)
= ρ(ω) − ρ(ω)T =0,
(15)
Note that the finite term does not feature in the difference of spectral densities.
It will turn out to be real and will not contribute to the spectral densities.
The first term is obtained from the hydrodynamic behaviour of the theory while
high frequency behaviour is captured by the OPE.

Shear sum rule d ≥ 3
The OPE of the Euclidean two point function in general dimensions is well
studied.
The tensor structures in OPE are as follows,
Txy(s)Txy(0) ∼ CT
Ixy,xy(s)
s2d
+ ˆAxyxyαβ(s) Tαβ(0) + · · · (16)
[Osborn, Petkou 1993]
where CT is the normalization of the two point function of the stress tensor.
The coeﬃcients are determined in terms of the parameters of the three point
function of the stress tensor (a,b,c)

Parameters of the three point function
Let us consider the three point function of the stress tensor in d = 4. The most
general structure occurring in the three point function can be written as
TTT = ns TTT free boson + nf TTT free fermion + nv TTT free Vectors (17)
where ns, nf and nv are the number of free bosons, fermions and vectors
respectively.
[Osborn, Petkou 1993]
These tensor structures are obtained by considering free theories and
considering wick contractions.

In d = 4 the parameters a, b and c are related to the number of free scalars,
vectors and fermions.
a =
1
27π6
(ns − 54nv), b = −
1
54π6
(8ns + 27nf ), (18)
c = −
1
27π6
(ns + 27(nf + 8nv)).
Similarly CT in d = 4 is given by,
CT =
π2
3
(14a − 2b − 5c) (19)

Fourier Transform
In order to examine the high frequency behaviour, we look at the OPE in a bit
more detail.
ˆAµνρσαβCT =
(d − 2)
d + 2
(4a + 2b − c)H1
αβµνρσ(s) +
1
d
(da + b − c)H2
αβµνρσ(s)
−
d(d − 2)a − (d − 2)b − 2c
d(d + 2)
(H2
µνρσαβ(s) + H2
ρσµναβ(s))
+
2da + 2b − c
d(d − 2)
H3
αβµνρσ(s)
−
2(d − 2)a − b − c
d(d − 2)
H4
αβµνρσ(s) −
2((d − 2)a − c)
d(d − 2)
2H3
µνρσαβ(s)
+
((d − 2)(2a + b) − dc)
d(d2 − 4)
(H4
µνρσαβ + H4
ρσµναβ)(s)
+(Ch5
µνρσαβ + D(δµν h3
ρσαβ + δρσh3
µναβ))Sdδd
(s).
(20)
where,
CT =
8π
d
2
Γ(d
2
)
(d − 2)(d + 3)a − 2b − (d + 1)c
d(d + 2)
,
C =
(d − 2)(2a + b) − dc
d(d + 2)
, Sd =
2π
d
2
Γ(d
2
)

We look at the fourier transform of the following term in detail
2da + 2b − c
d(d − 2)
H3
αβµνρσ(s) (21)
The fourier transform is done by ﬁrst performing the angular integrations and
then performing the radial and time integrations.
The outcome is independent of the order of radial and time integrations.
We take ω → ∞ at the end.
Note that only α = β terms contribute.
We get,
d−1
i=1
H3
iixyxy(ω, p = pz) =
4πd/2
dω2
− p2
− ω2
dΓ d
2
− 1 (p2 + ω2)
,
H3
ttxyxy(ω, p) = −
4πd/2
dω2
− p2
− ω2
dΓ d
2
− 1 (p2 + ω2)
. (22)

This procedure can be applied to all the terms in the OPE and we obtain,
J = A + B
A =
(1 − d)P(a(d(d + 4) − 4) + d(2b − c))
2 (−a (d2 + d − 6) + 2b + cd + c)
B =
((2a + b)(−2 + d) − cd)P
−2b − c(1 + d) + a (−6 + d + d2)
,
(23)
A is the contribution from tensor structures and B is the contribution from the
contact term.
The contribution A, due the tensor structures without the contact terms, is
proportional to the Hofman-Maldacena coeﬃcient in general d dimensions.
[Meyers, Sinha 2009]

Energy ﬂux constraints
We consider the following energy functional in d dimensions.
E(n) = lim
r→∞
rd−2
∞
−∞
dtni
Tt
i (t, r¯ni
) (24)
where ni
is a unit vector over Rd−1
.
The expectation value of this quantity is examined.
E(n) =
0|O†
E(n)O|0
0|O†O|0
(25)
where expectation values are taken over states O ∼ ijTij
.
When ij
is along the tensor channel (orthogonal to t and ni
), positivity of
energy ﬂux results in the following constraint
−(a(d(d + 4) − 4) + d(2b − c))
2 (−a (d2 + d − 6) + 2b + cd + c)
≥ 0 (26)

We are in a position to state our sum rule.
δGR(ω) = GR(ω)|T − GR(ω)|T =0 − (A + B). (27)
RHS of the sum rule is given by
δGR(0) =
2c + d(c + 2bd − cd) + a 8 + d −6 + d + d2
P
2(2b + c + cd) − 2a (−6 + d + d2)
(28)
where,
GR(0)|T = P, GR(0)|T =0 = 0

Holography Check 1
The coeﬃcients a, b, c have been evaluated in AdSd+1 by evaluating the 3
point function of the stress tensor holographically.
[Arutyunov, Frolov 1999]
a = −
d4
π−d
Γ[d]
4(−1 + d)3
,
b = −
d 1 + (−3 + d)d2
π−d
Γ(1 + d)
4(−1 + d)3
,
c =
d3
(1 − 2(−1 + d)d)π−d
Γ(d)
4(−1 + d)3
.
(29)
Evaluating A and B, we have,
A = −
d
2(d + 1)
, B = P (30)

We now evaluate the sum rule,
δGR(0) =
d
2(d + 1)
lim
→0+
1
π
∞
−∞
δρ(ω)dω
ω − i
=
d
2(d + 1)
(31)
For d = 4 this becomes
lim
→0+
1
π
∞
−∞
δρ(ω)dω
ω − i
=
2
5
(32)

Holography Check 2
An alternate way of deriving the sum rule is to directly study the Green’s
functions in AdSd+1 black hole background and obtain the high frequency
behaviour.
The shear sum rule was first studied holographically in the context of AdS5.
[Son, Romatschke 2009]
The shear sum rule using the minimally coupled scalar method was first studied
by Gulotta et al. The analyticity of the Green’s functions in the upper half
plane is obtained by studying the differential equations.
[Gulotta, Herzog, Kaminski 2010]
Modifications to the shear sum rule for retarded correlators of stress tensor as
well as currents in presence of chemical potential have also been studied.
[Jain, Thakur, David 2011],
[Thakur, David 2012],

Let us evaluate the Shear correlator for N = 4 SYM from holography. We
consider the non-extremal D3 brane geometry
ds2
=
L2
dr2
r2f(r)
+
r2
L2
(−f(r)2
dt2
+ dx2
+ dy2
+ dz2
), (33)
f(r) = 1 −
r+
r
4
, T =
dr+
4πr
.
The stress-tensor is dual to the ﬂuctuation of the metric in the bulk.
δgxy =
r2
L2
φ(r)e−iωt
. (34)
This ﬂuctuation obeys the equation of a minimally coupled scalar in the AdS
back ground.
d2
φ
dr2
+ (
3
r
+
F
F
)
dφ
dr
+
ω2
F2
φ = 0, (35)
(36)
where,
F =
r2
L2
f (37)

The retarded Greens function is obtained by imposing in going boundary
conditions at the horizon r+ and obtaining the solution to φ at the boundary
r → ∞ . The Greens function is given by
GR
xy,xy(ω, T) = ˆG(ω, T) + Gcounter(ω, T) + P(T),
ˆG(ω, T) =
−1
2κ2
lim
r→∞
Fr3
φ
φ
, (38)
where κ is the gravitational coupling constant. Gcounter(ω) are the counter
terms which are necessary to remove the log(r) divergences (independent of
temperature), P(T) is the pressure (independent of frequency)
[Son, Starinets 2002]
Essentially this boils down to studying the following quantity in the ω → ∞
limit.
g(ω) = lim
r→∞
Fr3
φ
φ
(39)

We introduce the variables
iλ =
ω
r+
, y =
λr+
r
. (40)
The boundary is now at y = 0. The diﬀerential equation becomes,
d2
φ
dy2
−
1
f(y)y
3 +
y4
λ4
dφ
dy
−
φ
f2(y)
= 0,
f(y) = 1 −
yd
λd
. (41)
At λ → ∞, this is the equation of a minimally coupled scalar in AdS5.
A systematic expansion of the solution around λ → ∞ can be performed which
is consistent with the in going boundary conditions at the horizon.
The solutions obtained in this manner results in the following Green’s function
lim
λ→∞
ˆG(ω, T) =
r2
+
2κ2
lim
y→0
λ4
y3
g0 −
6
5
+ O(
1
λ4
)
g0 =
−K1(y)
K2(y)
(42)

The leading term is divergent ( 1
y2 , log y), whose divergence is taken care of by
the term Gcounter(ω, T). Note that this is independent of temperature and is a
function of ω.
The finite part and the pressure function P(T) also ensures that the Green’s
function does not satisfy the fall off property required for the derivation of the
sum rule.
We define as before ,
δGR(ω) = GR(ω)|T − GR(ω)|T =0 − (A + B) (43)
where,
GR(ω)|T =
−1
2κ2
lim
r→∞
Fr3
φ
φ
+ Gcounter(ω, T) + P(T),
GR(ω)|T =0 =
1
2r2
+κ2
lim
y→0
ω4
y3
g0 + Gcounter(ω, 0)
A = −
r2
+
2κ2
6
5
B = P(T) (44)
This satisfies the fall off property as ω → ∞.

Using Cauchy’s theorem then one gets,
1
π
∞
−∞
dω
ω
[ρ(ω) − ρT =0(ω)] =
2
5
(45)
This agrees precisely with the answer obtained from the general expression of
the sum rule, in terms of the parameters of the three point function in d = 4.
This method can be generalized to higher dimensions and we have,
1
π
∞
−∞
dω
ω
[ρ(ω) − ρT =0(ω)] =
d
2(d + 1)
(46)
This is a consistency check for shear sum rule of conformal ﬁeld theories in
arbitrary dimensions.
From this we come to the conclusion that, the parameters of the three point
function of stress tensor of any conformal ﬁeld theory with a two derivative
gravity dual must satisfy the following constraint.
2c + d(c + 2bd − cd) + a 8 + d −6 + d + d2
2(2b + c + cd) − 2a (−6 + d + d2)
=
d(d − 1)
2(d + 1)
(47)

Applications in d = 4
For free N = 4 super Yang-Mills, the parameters a, b and c take the values,
a = −
16
9π6
(N2
c − 1), b = −
17
9π6
(N2
c − 1), c = −
92
9π6
(N2
c − 1).
Substituting these values into the sum rule for d = 4 we get,
δGR(0) =
2
5
Note that this is exactly the same result from our holographic calculation. We
see that the sum rule has not been renormalized.
From the holographic consistency check we have the following relation for a
general conformal ﬁeld theory in d = 4 with a AdS5 dual.
48nb − 21nf − ns = 0, (48)
where nb, nf and ns are number of free vector bosons, fermions and scalars
respectively.

It is a well known observation that for a theory in d = 4, which admits a two
derivative gravity dual, the central charges, that appear in the trace anomaly
equation, are equal.
Tµ
µ =
ˆc
16π2
W2
−
â
16π2
E
ˆc = â (49)
Where, W is the Weyl tensor and E is the euler density
The equality of the central charges imply
â
ˆc
=
9a − 2b − 10c
2(14a − 2b − 5c)
= 1 (50)
Rewriting this in terms of the matter content in the free limit,
26nb − 7nf − 2ns = 0 (51)
The constraint from sum rule is independent of the constraint derived from the
equality of the central charges.

Positivity of energy ﬂux/ Causality constraints lead to certain bounds on the
parameters of the three point function of the stress tensor.
[Hofman, Maldacena 2008]
Let us introduce the dimensionless variables
t2 =
30(13a + 4b − 3c)
14a − 2b − 5c
,
t4 = −
15(81a + 32b − 20c)
28a − 4b − 10c
(52)
Causality constraints lead to
−
t2
3
−
2t4
15
+ 1 ≥ 0
2 −
t2
3
−
2t4
15
+ 1 + t2 ≥ 0
3
2
−
t2
3
−
2t4
15
+ 1 + t2 + t4 ≥ 0 (53)

We can rewrite the general sum rule for d = 4 in terms of Maldacena-Hofman
variables to get
δGR(0) = (
6
5
−
t2
6
−
2t4
75
)P (54)
The causality bounds then translate to the following bounds on the sum rule
P
2
≤ δGR(0) ≤ 2P (55)
The lower bound is saturated for theories with only free fermions, while the
upper bound is saturated for theories with only free bosons.

There is a similar parametrization of the parameters of the three point
functions, in d dimensions, in terms of t2, t4.
[Buchel et al 2009]
The general expression for sum rule can be expressed in terms of these
parameters.
δGR(0) =
(−1 + d)d
2(1 + d)
+
(3 − d)t2
2(−1 + d)
+
2 + 3d − d2
t4
(−1 + d)(1 + d)2
P (56)
The Causality constraints in terms of these variables is given by
[Camanho, Edelstein 2009]
1 −
1
d − 1
t2 −
2
(d + 1)(d − 1)
t4 ≥ 0, (57)
1 −
1
d − 1
t2 −
2
(d + 1)(d − 1)
t4 +
1
2
t2 ≥ 0,
1 −
1
d − 1
t2 −
2
(d + 1)(d − 1)
t4 +
d − 2
d − 1
(t2 + t4) ≥ 0.

The causality constraints imply the following bounds on shear sum rule for any
conformal ﬁeld theory in d dimensions.
1
2
P ≤ δGR(0) ≤
d
2
P. (58)
For d = 3, the causality constraints are (ignoring any parity odd structures that
might appear in the three point function)
1 −
t4
4
≥ 0
1 +
t4
4
≥ 0 (59)
The bounds on the sum rule is then,
1
2
P ≤ δGd=3
R (0) ≤ P. (60)

We evaluate the sumrule with the matter content of a single M2 brane and free
ABJM theory.
The result agrees with that calculated for AdS4.
δGR(0) =
3P
4
. (61)
Thus the sum rule for these cases are not renormalized.

Let us look at U(N) Chern Simons theory coupled to fundamental fermions at
large N.
The values of the parameters of the three point function is exactly known at
Large N.
a =
27N sin2 θ
2
sin(θ)
1024π3θ
, (62)
b =
9N sin(θ)(5 cos(θ) − 13)
2048π3θ
,
c = −
9N sin(θ)(7 cos(θ) + 9)
2048π3θ
.
(63)
The sum rule reduces to
δGR(0)CS
= −
1
4
P(cos(θ) − 3) (64)
This satisﬁes the bounds as stated before and saturates the bounds derived at
θ = 0 (free fermions) and θ = π (free bosons).
Note that at θ = π
2
,
δGR(0)CS
|θ= π
2
=
3P
4
(65)

The sumrule evaluated for a single M5 brane also matches with the sum rule
derived for AdS7

Conclusion and future directions
We have derived the sum rule corresponding to the shear channel of stress
tensor correlator for any conformal field theory in d dimensions.
lim
→0+
1
π
∞
−∞
δρ(ω)dω
ω − i
=
(−1 + d)d
2(1 + d)
+
(3 − d)t2
2(−1 + d)
+
2 + 3d − d2
t4
(−1 + d)(1 + d)2
P,
(66)
where ρ(ω is the spectral density corresponding to the retarded correlators.
Note that the other Hofman-Maldacena coefficients appear in the high
frequency behaviour of other channels. It might be worthwhile setting up the
sum rule for the other channels and explore them holographically.
Causality constrains the shear sum rule to lie within specific bounds.
1
2
P ≤ δGR(0) ≤
d
2
P. (67)
We see that for d = 4, the constraint on a, b and c from equality of the central
charges and the sum rule, can be expressed as
t2 = t4 = 0
(68)

Thank you.

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