Global gravitational anomalies and transport

Global gravitational anomalies and transport
Subham Dutta Chowdhury
March 28, 2017
Instituto de Fisica Teorica, 2017
Subham Dutta Chowdhury Global gravitational anomalies and transport 1/48

References
S. D. Chowdhury and J. R. David, “Anomalous transport at weak coupling,”
JHEP 1511, 048 (2015). [arXiv:1508.01608 [hep-th]].
S. D. Chowdhury and J. R. David, “Global gravitational anomalies and
transport,”JHEP 1612, 116 (2016) [arXiv:1604.05003 [hep-th]].

Introduction and motivation
A symmetry of classical physics is not necessarily a symmetry of quantum
theory.
A chiral transformation leaves the free fermion action invariant but not the
path integral measure. This results in the non-conservation of the current and
the stress tensor,
µjµ
= 0
µTµν
= Fµν
jµ (1)
The modiﬁcations of the macroscopic equations of hydrodynamics due to the
presence of quantum anomalies of the underlying theory has been the focus of
recent interest.

As an example let us recall the anomalous conservation laws for free fermions in
d = 2, the anomalous conservation law for a U(1) current and stress tensor is
µjµ
= c(1/2)
s
µν
Fµν ,
ν Tµν
= Fµ
ν jν
+ c(1/2)
g
µν
ν R. (2)
The macroscopic theory admits the following anomalous constitutive relations,
Tµν
= ( + P) uµ
uν
− Pηµν
+ λ(2)
(uµ νρ
uρ + uν µρ
uρ),
jµ
= nuµ
+ ζ(2) µν
uν . (3)
where, λ(2)
, ζ(2)
are the two anomalous transport coeﬃcients.

The following relation hold true,
ζ(2)
= −2c(1/2)
s µ−, (4)
The parity odd transport coefficients are non dissipative. This fact was utilized
to determine this relationship using equilibrium partition function method by
Banerjee et al.
[Banerjee, Datta, Jain, loganayagam, Sharma ’13]
For the four dimensional case, the relation between microscopic gauge
anomalies and the macroscopic parity odd transport coefficient was first shown
by son et al. from general considerations involving equations of hydrodynamics,
anomalous conservation laws and second law of thermodynamics.
[D. T. Son and P. Surowka ’09]
The anomalous conservation laws and the constitutive relations tell us that the
order of derivative at which gravitational anomaly occurs in the conservation
laws is two orders higher than that of the constitutive relations. Thus it is not
possible to constrain λ(2)
using this method since it relies on matching
derivative expansion at every order.

Constraints on the gravitational anomalous transport coefficient was first
imposed by Loganayagam et al using consistency conditions of euclidean
vacuum.
[Jensen,Loganayagam,Yarom, ’12]
˜c
(1/2)
2d = −8π2
c(1/2)
g , (5)
where,
λ(2)
= ˜c
(1/2)
2d T2
+ · · · (6)
This has been confirmed by explicit perturbative calculations in finite
temperature field theory.
[David, Dutta Chowdhury ’15]
The main focus of the talk will be on the transport coefficients which capture
the gravitational anomalies.
It is more useful to formulate the anomaly coefficients in terms of an anomaly
polynomial rather than explicit conservation equations.

For example we consider fermions in d = 2. The anomaly polynomial is given
by
Pd=1+1( ˆF, ˆR) = c(1/2)
s F ∧ F + c(1/2)
g tr( ˆR ∧ ˆR),
ˆRab =
1
2
Rabcddxc
∧ dxd
. (7)
It was also observed by loganayagam et al that the second or the higher
pontryangin classes in the anomaly polynomial of a chiral field does not
contribute to the transport coefficient.
For example let us consider the anomaly polynomial in d = 6
Pd=6 = cγ(Tr( ˆR)2
)2
+ cδ(
1
4
Tr( ˆR4
) −
1
8
Tr( ˆR2
)2
), (8)
Where cγ is the coefficient which occurs with the square of the first Pontryagin
class while the cδ occurs with the second Pontryagin class.

If we parametrize the transport coeﬃcient in d = 6, which is constrained by the
gravitational anomaly as
λ(6)
= 9˜c6d
g T4
, (9)
Then ‘replacement rule’ predicts the relation
˜c6d
g = −(8π2
)2
cγ. (10)
Note that the second Pontryagin class does not contribute to the transport
coeﬃcient according to this rule.
It would be nice to have an explanation for this observation using symmetry
arguments rather than explicit calculation.

Let us consider a system of free chiral gravitinos. For d = 2 there are not
propagating degrees of freedom, but one can, in principle, study gravitino like
theories.
Loganayagam et al had predicted thet the replacement rule is violated if there
are chiral gravitinos in the theory.
The contribution from gravitinos seem to behave like that of a single weyl
fermion instead of a spin3
2
particle.
˜c
(3/2)
2d = −8π2
c(1/2)
g , (11)
instead of
˜c
(3/2)
2d = −8π2
c(3/2)
g . (12)
This has been explicitly seen using ﬁnite temperature ﬁeld theory calculations.

More generally the problem of chiral gravitinos can be stated as follows.
Replacement Rule predicts that, in general 4k + 2 dimensions, if we
parametrize the coefficient capturing the gravitational anomaly of gravitinos as
λ
(4k+2)
g = α(4k+2)˜c
3/2
g T2k+2
and λ
(4k+2)
f = α(4k+2)˜c
1/2
g T2k+2
, for that of
fermions, we have according to the Replacement rule,
˜c3/2
g = (d − 1)˜c1/2
g (13)
To summarize it all, the euclidean partition function methods relate only the
gauge anomalies and the mixed gauge-gravitational anomalies to the transport.
The relationship between the pure gravitational anomaly and transport seems
to breakdown for gravitinos.
Sethi et al have recently shown that it is possible to write down a low energy
effective action for fermions constrained by global diffeomorphisms in d = 2.
The effective action correctly reproduces the anomalous transport coefficient in
d = 2.
[Sethi, Golkar ’15]

What are global anomalies?
Global anomalies are phase shifts that appear in the euclidean partition
function under large gauge transformations and large diffeomorphisms. For
example on a compact manifold endowed with the metric gµν , if we have a
symmetry transformation (large diffeomorphism)
π : gµν → gπ
µν (14)
the partition function changes by
Z(gµν ) → Z(gπ
µν ) = e−iπη
Z(gµν ) (15)
[Witten ’85]
Existence of global anomalies require specific couplings to be present in the low
energy effective action (hydrodynamic limit). This low energy effective action
must correctly reproduce the global anomaly under the symmetry
transformation.
e−se(gπ
µν )
→ e−iπη
e−se(gµν )
(16)
We wish to study this approach in more detail and address the issues discussed
before from the effective action perspective rather than perturbative
calculations.

Results
We have used the method of global anomaly matching to determine the
thermal effective actions for Weyl fermion, chiral gravitinos and Self dual
tensors in d = 2, 6. For the fermions and self dual tensors, the results obtained
for transport coefficients match with perturbative results.
For the gravitino the results are consistent with perturbative calculations and
hence the Replacement rule upto mod 2.
The calculation for topological η invariant clearly shows that second order
pontryangin terms do not contribute to the transport in d = 6. This provides a
topological explanation to the observation of Loganayagam et al in their
Replacement rule
As a check of our calculations for self dual tensor, we calculate the relevant
transport coefficient for self dual tensors in d = 6 using finite temperature field
theory. In order to do so we promote the Feynman rules proposed by Witten et
al to finite temperature. The results match with the replacement rule and
global anomaly analysis.
Finally we propose an alternate way of computing the η invariant. We claim
that the η invariant computed using the index theorem can be verified by weak
coupling analysis of various correlators. We extrapolate our formalism and
propose the η invariants for various fields in d = 10

Free fermions in d = 2
Let us consider the partition function, Z(ψ, ¯ψ), for a system of free weyl
fermions on a 2- torus with the modular parameter iβ
L
( also called τ). Note
that the modular parameter τ is the ratio of lengths along the two directions.
Let the fermions obey the following anti periodic boundary conditions on the
torus.
ψ(z + 1) = −ψ(z), ψ(z + τ) = −ψ(z) (17)
Explicit evaluation of the partition function, under these boundary conditions,
gives us
ZAA(τ) =
θ3(τ)
η(τ)
(18)
where θ3 and η are the theta and dedekind eta functions respectively

The symmetry of the 2-torus is given by
T : τ → τ + 1 S : τ → −
1
τ
(19)
The properties of the thermal partition function, under such transformations,
are given by
T2
: ZAA(τ + 2) = e−i π
6 ZAA(τ), S : ZAA(−
1
τ
) = ZAA(τ), (20)
We see that under the symmetry transformations of a torus, the thermal
partition function of fermions pick up a phase under deﬁnite boundary
conditions.
A natural outcome of this calculation is the question, whether we can see this
phase shift, under torus transformations, without computing the explicit
partition function? Is it possible to write down an eﬀective action which
captures this?

Anomaly matching and effective action
Consider a system of weyl fermions (this can be extended to gravitinos and self
dual tensors) on a compact even dimensional manifold M of d dimensions,
endowed with the metric gµν . We are interested in the large diffeomorphism
π : gµν → gπ
µν ds2
= gµν dxµ
dxν
(21)
If the theory has a global anomaly, the partition function picks up a phase
Z(ψ, ¯ψ, gπ
µν ) = e
−iπη 1
2 Z(ψ, ¯ψ, gµν ) (22)
[Witten ’85]
The quantity η1
2
is calculated as follows. We construct an interpolating metric.
gµν (y) = (1 − y)gµν + ygπ
µν (23)
where the parameter y interpolates between the original metric and the
diffeomorphed one.
A higher dimensional manifold is then constructed by promoting the
interpolating parameter to a coordinate of the metric.
ds2
= dy2
+ gµν dxµ
dxν
(24)
The manifold at y = 0 is then identified with the manifold at y = 1 resulting in
a compact space called the mapping torus (Σ)

Anti periodic boundary conditions are chosen for the fermions on the time
circle t which will eventually be the thermal circle. The quantity η1
2
is then
obtained by solving the dirac equation in this d + 1 dimensional manifold with
the boundary condition that y = 0 and y = 1 are glued together.
Let λ denote the eigen value of the Dirac operator. The η invariant is then
defined by
η1
2
=
λ 1
2
sign(λ) (25)
Solving the dirac equation on this complicated manifold is not easy. If there
exists a manifold B such that ∂B = Σ, we can appeal to the
Atiyah-Patodi-Singer index theorem to obtain η1
2
Ind( /D 1
2
) =
B
Â(R) + Iboundary −
η1
2
2
. (26)
where Â(R) is a curvature form on B and Iboundary, defined on the boundary
Σ, are corrections to the APS theorem . As we shall see, η1
2
correctly
reproduces the phase shift for Weyl fermions in d = 2.

Now that we have the phase change without actually calculating the partition
function, we want to write down a low energy effective action which reproduces
the change in the partition function under the global diffeomorphism.
e−se
(gπ
µν ) → e
−iπη 1
2 e−se
(gµν ) (27)
In high temperature or low energy limit, the thermal circle cannot be resolved
and the theory is gapped from a effective field theory perspective. Hence the
theory is a local functional of back ground fields and depends on the rest d − 1
dimensions.
However in order to calculate transport coefficients, we need the theory on
S1
× Rd−1
. We decompactify the thermal action we have written down along
the spatial directions. We assume that the coefficient does not change as we
smoothly decompactify as this must reproduce the global anomaly.
We calculate the anomalous transport coefficients, that appear in the
hydrodynamic description, from our constructed effective action and match it
with perturbative calculations.

Fermions in d = 2 (revisited)
We consider a theory of free Weyl fermions in d = 2 on a torus ˆT2
. The
coordinates on the torus are given by (t, x) with periodicities given below.
(t, x) ∼ (t + 2πn, x + 2πm) n, m ∈ Z (28)
The metric on the torus is given by
g : ds2
= (dt + a(x)dx)2
+ dx2
. (29)
We consider the following large gauge transformation of the torus (labelled as
T2
)
t
x
→
1 2
0 1
t
x
. (30)
Under this change the metric under goes the following transformation
gT 2
: ds2
= (dt + (a + 2)dx)2
+ dx2
. (31)

The mapping torus is constructed, which maps the metric g to gT 2
, through a
coordinate y. Let us call this three dimensional manifold Σ.
ds2
Σ = dy2
+ [dt + (a + 2y)dx]2
+ dx2
. (32)
We also have the identification that the torus at y = 0 is identified with the
torus at y = 1, resulting in a compact space.
(t, x, y) ∼ (t − 2x, x, y + 1). (33)
In order to use the results of the APS index theorem, we need to construct a
higher dimensional manifold B such that ∂B = Σ. This is done by filling up
the t circle of the manifold Σ
ds2
B = dr2
+ dy2
+ f(r)2
[dt + (a + 2y)dx]2
+ dx2
. (34)
where r takes values from 0 to 1. The metric at the boundary is given by,
ds2
P = dr2
+ dy2
+ f(1)2
[dt + (a + 2y)dx]2
+ dx2
(35)
The function f(r) is a filling function which has the property,
lim
r→0
f(r) = r (36)
This ensures there is no conical singularity at r = 0.

Note that at r = 1, this metric is strictly not that of Σ. It reduces to Σ for
f(1) = 1. However as we will see, the salient features of the calculation are
better captured by a general function f(r).
The APS index theorem relates the η invariant to the following geometric
quantity on the manifold B.
Ind( /D 1
2
) =
1
24 × 8π2
B
Tr(R ∧ R) +
Σ
IΣ=∂B −
η1
2
2
.
where IΣ=∂B is a Chern Simons term defined on the boundary which satisfies
dIΣ=∂B = −
1
24 × 8π2
Tr(R ∧ R) (37)
[Witten, ’80]
Recall that we had defined a quantity called Iboundary which were corrections to
the general APS index theorem. For fermions in d = 2 on a torus, Σ
IΣ=∂B is
the term that removes non topological contributions of the curvature integrals.

Explicit calculations give,
1
24 × 8π2
B
Tr(R ∧ R) = −
2f(1)4
+ −1 + f (1)2
12
(38)
We notice that this is not a topological invariant since it depends on the ﬁlling
function f and its derivatives.
The deﬁning relation of the correction term can be explicitly solved to give
IΣ=∂B =
−1
24π2 × 8
(ω ∧ dω +
2
3
ω ∧ ω ∧ ω) (39)
We evaluate this term explicitly on the boundary to give,
Σ
IΣ=∂B =
2f(1)4
+ f (1)2
12
Putting all this together, we have the η1
2
as,
η1
2
=
1
6
+ 2Ind( /D 1
2
). (40)

Since the manifold B has the topology of a solid torus (homogenous), the index
of the Dirac operator is an integer. Thus this term contributes to a trivial phase
change of the partition function under the T2
diffeomorphisms of the torus.
η1
2
=
1
6
(41)
Thus the phase picked up by the T2
transformation is given by
Z[gT 2
] = e−iπη1/2
Z[g] = e−i π
6 Z[g]. (42)
This is precisely the phase picked up the T2
transformation for fermions with
the (A, A) boundary conditions.
We want to write the low energy effective action that reproduces this anomaly.
In order to do so we first restore the dimensions to the metric we have been
working with.
˜x =
Lx
2π
, ˜t =
βt
2π
, ˜ds2
=
β2
(2π)2
ds2
. (43)

Under the T2
transformation we have,
T : (˜t, ˜x) → (˜t +
2β˜x
L
, ˜x), ã → ã +
2β
L
. (44)
We decompactify along the x direction by taking the periodicity L to be large.
However we expect the coefficient of the effective action to not change as we
smoothly change since it must reproduce the same anomaly.
The only parameter on which the effective action can depend on is the metric
component ã(˜x) , since the background is flat and there is no curvature. An
action which satisfies all this is
Sf
eff =
iπ
12β
ã(˜x)d˜x, Z[g] = e−S
f
eff . (45)
Writing this effective action in momentum space, we can obtain the one point
function of the stress tensor T ˜τ ˜x
T
˜t˜x
(p) =
1
√
g
δ ln Z
δg˜t˜x
=
δ ln Z
δã(p)
= −
iπ
β212
2πβδ(p). (46)

We go over to minkowski space by analytic continuation ˜t = −i˜t
T
˜t ˜x
(p) = −
π
12β2
(47)
where 2πβδ(0) has been stripped off which occurs in the overall momentum
conservation.
For a theory of Weyl fermions in d = 2, the anomalous transport coefficient
capturing the gravitational anomaly is computed using finite temperature field
theory and is given by,
λ
(2)
f = − T
˜t ˜x
=
π
12β2
(48)
The correlator constrained by the global anomaly and that computed using
perturbation theory are in agreement with each other.

Chiral bosons in d = 2
The chiral boson or the self dual tensor in d = 2 (euclidean signature) are
deﬁned as bosons obeying the following constraints
∂µ
φ = i
µν
√
g
∂ν φ. (49)
A real self dual tensor in d = 2 is dual to a single complex Weyl fermion.
Hence we expect the results to be the same.
The general APS index theorem for the self dual tensors is given by
σ(B)
8
=
1
8
L(R) − Iboundary +
ηS
2
. (50)
Here L is the Hirzebruch polynomial constructed out of the curvature tensor, σ
the Hirzebruch signature of B and IΣ(R) is the term added to make the
curvature term a topological invariant. In d = 2 it takes the form
σ(B)
8
= −
1
24 × 8π2
B
Tr(R ∧ R) +
Σ
IΣ=∂B +
ηS
2
The calculation proceeds similar to that of the fermions.
ηS =
1
6
+
σ(B)
4
. (51)

The non topological contribution to ηS has been removed by adding a Chern
Simons term IΣ=∂B as for the fermions. For a solid torus (homogenous), the
signature is a multiple of 8, we are left with
ηS =
1
6
mod2 (52)
Alternatively one can start with the refined APS index theorem due to Monnier
et al. which states that on the right hand side of the index theorem, the
signature of manifold σ(B)
4
, is replaced by λ ∧ λ where λ is a two form.
Topological arguments show that this can be chosen to vanish.
The effective action is thus the same as that of fermions and the resulting one
point function, there fore agrees with perturbative computation of transport
coefficient.
Sb
eff =
iπ
12β
ã(˜x)d˜x λ
(2)
b = − Ttx
=
π
12β2
(53)

Chiral gravitinos in d = 2
There are no gravitinos in d = 2 with propagating degrees of freedom. We
study a ’gravitino like theory’.
The APS index theorem takes the general form
Index( /D3/2(B)) =
B
ˆA(R) TreiR/2π
− 1 + Iboundary −
η3
2
2
, (54)
which for d = 2 takes the form
Ind( /D 3
2
(B)) = −
23
24 × 8π2
B
Tr(R ∧ R) + 23
Σ
IΣ=∂B −
η3
2
2
. (55)
Explicit calculation results in
η3/2 =
−23
6
+ 2Ind( /D 3
2
(B)). (56)
As before the non-topological contribution is removed by adding a Chern
Simons term.
The index of the spin 3
2
operator is taken to be integer since the topology is
that of a solid torus.
η3/2 =
−23
6
mod2
=
1
6
mod2
(57)

We note that upto mod 2, the contribution to the global anomaly of a chiral
gravitino is same as that of a Weyl fermion. Hence the effective action must
also be the same.
Sg
eff =
iπ
12β
ã(˜x)d˜x (58)
The resulting one point function of the stress tensor matches with that of a
single Weyl fermion upto mod 2.
Recall that the perturbative calculations imply that in d dimensions, the
transport coefficient for gravitinos is d − 1 times that of a single Weyl fermion.
˜c3/2
g = (d − 1)˜c1/2
g (59)
We conclude that in d = 2 the transport coefficient for the gravitinos obtained
from matching global anomalies is consistent with perturbative calculations
upto mod 2.

Fermions in d = 6
We now study the theory of Weyl fermions in d = 6. We start with the
following metric on ˆT6
.
ds2
= (dt + a1(a)da + a2(b)dz + a3(y)dx)2
+ dx2
+ dz2
+ da2
+ db2
+ dy2
. (60)
The coordinates are periodic with period 2π. Anti-periodic boundary conditions
have been imposed along all the directions, for the fermions.
t ∼ t + 2π, a ∼ a + 2π, b ∼ b + 2π, (61)
x ∼ x + 2π, y ∼ y + 2π, z ∼ z + 2π.
As we will see later, this conﬁguration constrains the functional form of the
metric components ai in a non-trivial way.

Similar to the d = 2 case, we construct the seven dimensional mapping torus
which extrapolates between the original metric and the T2
transformed metric.
ds2
Σ = du2
+ (dt + [a1(a) + 2u] da + a2(b)dz + a3(y)dx)2
(62)
+dx2
+ dz2
+ da2
+ db2
+ dy2
.
Here the coordinate u interpolates between the original torus and the one
related to it, by T2
diﬀeomorphism described above, as u runs from 0 to 1.
We also have the identiﬁcation,
(t, a, u, b, z, x, y) ∼ (t − 2a, a, u + 1, b, z, x, y). (63)
under the T2
transformation.
Filling up the metric along the radial direction we have,
ds2
B = dr2
+ du2
+ f(r)2
(dt + [a1(a) + 2u] da + a2(b)dz + a3(y)dx)2
+dx2
+ dz2
+ da2
+ db2
+ dy2
. (64)
with the condition f(r)r→0 = r.

The product metric is then given by,
ds2
P = dr2
+ du2
+ f(1)2
(dt + [a1(a) + 2u] da + a2(b)dz + a3(y)dx)2
+dx2
+ dz2
+ da2
+ db2
+ dy2
.
(65)
In order to compute η1
2
for this manifold, we need to invoke the APS index
theorem as discussed for d = 2 case.
Ind( /D 1
2
(B)) =
−1
6! B
p2(R)
2
−
7
8
p1(R)2
+
Σ
IΣ=∂B,p1 +
Σ
IΣ=∂B,p2
−
η1/2
2
= Ip1 + Ip2 −
η1/2
2
(66)
where
p2(R) =
−1
(2π)4
1
4
Tr(R ∧ R ∧ R ∧ R) −
1
8
Tr(R ∧ R)Tr(R ∧ R) ,
p1(R) =
−1
(2π)2
Tr(R ∧ R)
2
, (67)
and IΣ=∂B,p1 , IΣ=∂B,p2 are the corrections to the APS index theorem
corresponding to p1(R)2
and p2(R) respectively.

The expression for IΣ=∂B is given by
dIΣ=∂B,p2 =
1
2 × 6!
p2(R)
dIΣ=∂B,p1 = −
7
8 × 6!
p1(R)2
(68)
Recall that Replacement rule by Loganayagam et al predicted that the second
and higher Pontryagin classes do not contribute to the transport coeﬃcient.
Let us try to evaluate Ip2 integral ﬁrst.
Ip2 =
−1
6! B
p2(R)
2
+
Σ
IΣ=∂B,p2 .

Explicit evaluation on the manifold B gives,
Ip2 = 0
Indeed we have no contribution of Ip2 to η1
2
!!
We proceed to evaluate the second topological contribution to the APS index
theorem,
Ip1 =
7
8 × 6! B
p1(R)2
+
Σ
IΣ=∂B,p1 ,
(69)
After explicit computation, we have a purely topological contribution from the
ﬁrst Pontryagin class. The APS index theorem then becomes,
η1/2
2
= −
7
480(2π)2
dbdydadza2(b)a3(y) − Ind( /D1/2(B)).
(70)

The right hand side of APS index theorem is composed of functions of metric
components and its derivatives. As it stands it is not a pure number.
In order to see that indeed this is a number, let us consider two torus directions
(x, y) for example. (A, A) boundary conditions along the torus implies the
following allowed conﬁguration for the metric component a3
a3(y) = 2n
y
2π
, n ∈ Z. (71)
This ensures that the a3 → a3 + 2n under y → y + 2π. Thus we have a T2
transformation along (x, y) torus and hence the boundary conditions remain
unchanged.
This leads to the metric components having non-trivial windings along the
compact direction.
Index theorem now becomes,
η1/2
2
= −
7nm
60
− Ind( /D1/2(B)), n, m ∈ Z
(72)
Note that the right hand side is now a pure number.

Utilizing this information and the fact that the Dirac index on a homogenous
space is an integer, we have
η1/2 = −
7
240(2π)2
dbdydxdza2(b)a3(y) mod 2. (73)
Note that we have restored back the derivatives of the metric components.
Given the phase shift that occurs due to the large diffeomorphism, we can write
the effective action as,
Seff = −
i7π
480(2π)3
dadbdxdydza1(a)a2(b)a3(y). (74)
We decompactify along the spatial directions and to introduce temperature, we
rescale the coordinates as before . In fourier space the effective action
becomes,
Seff =
−i7π
β3480
d5
pd5
k
((2π)5)2
(ikb
ipy
)ã1(−p − k)ã2(k)ã3(p) (75)

We can compute transport coefficient using the rescaled, decompactified
effective action. The relevant transport coefficient which captures the
gravitational anomaly is as follows
λ6
= −
3 Tta
(−p − k)Ttx
(p)Ttz
(k)
2(ipy)(ikb)
,
= −
3i Tτa
(−p − k)Tτx
(p)Tτz
(k)
2(ipy)(ikb)
,
= −
3i
(2ipy)(ikb)
δ3
ln Z
δgτaδgτxδgτz
. (76)
From our effective action, the transport coefficient is computed to be,
λ6
(1/2) =
7π
320β4
(77)
This matches with what was computed using perturbative methods.

Gravitinos in d = 6
We now consider a system of free Chiral gravitinos on a torus ˆT6
. The APS
index theorem for the gravitinos is given by,
Ind( /D 3
2
) =
−1
6! B
245p2(R)
2
−
275
8
p1(R)2
+245
Σ
IΣ=∂B,p1 +
275
7 Σ
IΣ=∂B,p2 −
η3
2
2
.
(78)
The calculation proceeds exactly similar to that of the fermions. The Index
theorem becomes,
η3/2 = −
1
(2π)2
275
240
dbdydxdza2(b)a3(y) (79)
where we have dropped the index of the spin-3
2
operator, since it is an integer
for torus.

Recall that the metric components ai have non trivial windings on the torus.
η3
2
then becomes
η3/2 = −
275
60
nm, (80)
= −
35
60
nm − 4nm = −
35
60
nm mod 2.
The eﬀective action which reproduces this phase shift under a1 → a1 + 2 is
given by
Seff = −
i35π
480(2π)3
dbdydxdadza1(a)a2(b)a3(y). (81)
Note that this is 5 times the result obtained for the Weyl fermion.
Therefore on decompactifying the spatial directions and extracting out the
transport coeﬃcient for the gravitinos we obtain
λ
(6)
(3/2) =
35π
320β4
. (82)
which matches with perturbative calculations and the replacement rule.

Self Dual Tensors in d = 6
The self dual tensor in d = 6, is defined as the field strength of the gauge field
Aµ1µ2 , which itself transforms as a two index anti symmetric tensor.
Fµ1µ2µ3
= ∂µ1
Aµ2µ3
+ permutations. (83)
The self dual condition is given by
Fµ1µ2µ3
=
i
√
g
µ1µ2µ3ν1ν2ν3
Fν1ν2ν3 . (84)
The stress tensor for such a theory is defined as
Tµν (F) = −
1
2
FµαβF αβ
ν +
1
12
gµν FαβγFαβγ
, (85)
whereas the self dual condition is imposed by considering
Tµν (F+
) = Tµν (
1
2
(F + i ˜F)) (86)
We wish to find a low energy effective action for such a theory in order to
compute the transport coefficient that captures the gravitational anomaly.

The APS index theorem reads
σS(B)
8
=
−1
6! B
28p2(R)
2
−
16
8
p1(R)2
+28
Σ
IΣ=∂B,p1 +
16
7 Σ
IΣ=∂B,p2 −
ηSD
2
.
(87)
Going through the same steps of evaluating the curvature polynomial and using
the fact that the Hirzebruch index for a solid torus vanishes (Monnier et al), we
obtain
ηSD =
−16
240(2π)2
dbdydadza2(b)a3(y) (88)
The thermal eﬀective action which reproduces this phase shift under
a1 → a1 + 2 is given by
Seﬀ =
−i16π
480(2π)3
dbdydxdadza1(a)a2(b)a3(y). (89)

We decompactify the spatial directions and extract out the transport coeﬃcient
to obtain the following result for self-dual tensors
λ
(6)
SD =
16π
320β4
=
π
20β4
. (90)
We proceed to verify this result from explicit perturbative thermal ﬁeld theory
calculation .

The transport coeﬃcient of interest is given by λ(6)
,
λ
(6)
SD = −
3
2
lim
pb,ky→0
Tta
(k + p)Ttx
(−k) ˜Ttz
(−p)
ipbiky
. (91)
where ˜T indicates that the self dual condition has been imposed on one stress
tensor only.
We have used the propagator derived by Witten and Alvarez-Gaume and
promoted it to thermal propagator with matsubara frequencies.
SB(ωn, p) = Fµ1µ2µ3
(ωn, p)Fν1ν2ν3
(−ωn , −p3)
= −
pµ1
pν1
gµ2ν2
gµ3ν3
(iωn)2 − p2
+ Permutations 2πβδn,n δ(p − p3),
(92)
where
ωn = 2nπT. (93)
All the wick contractions have been evaluated using a mathematica code.
λ
(6)
SD =
3
64π3
(
16π4
T4
15
). (94)
We ﬁnd that this is in agreement with the replacement rule and the global
anomaly calculation.

Details of the calculation
We explore the intricacies of the perturbative calculation in some more detail.
The hyrdrodynamic correlation function that captures the gravitational
anomaly is formally defined as
Tµα
Tνβ
Tρσ
E = Tµα
fl Tνβ
fl Tρσ
fl E (95)
−2
δTµα
√
gδgνβ
Tρσ
E − 2
δTµα
√
gδgρβ
Tνσ
E
−2 Tµα δTνβ
√
gδgρσ
E + 4
δ2
Tµα
√
gδgναδgρσ
E.
The first term is three point function of tensors evaluated on flat background.
To be explicit we write down the (τx) component of the stress tensor
Tτx
fl (−k) = −
1
β
Σωm
d5
p2
(2π)5
Fτab
(−p2 − k)Fxab
(p2) + Fτay
(−p2 − k)
×Fxay
(p2) + Fτaz
(−p2 − k)Fxaz
(p2) + Fτby
(−p2 − k)Fxby
(p2)
+Fτbz
(−p2 − k)Fxbz
(p2) + Fτyz
(−p2 − k)Fxyz
(p2) .
It is sufficient to impose self dual condition on just one stress tensor (similar to
fermions).

Figure: Contributions to λ(6) from Wick contractions of the stress tensor

The terms in second and third lines are the contact terms, which needs to be
treated carefully. We expand the stress tensor by considering metric fluctuations
hτa, hτx and hτz. For example we demonstrate the action of hτz on Tτx
,
δTτx
(−k)
δhτz(p)
= −
ωm
d5
p3
(2π)5
Fzab
(−p3 − p − k)Fxab
(p3)
+Fzay
(−p3 − p − k)Fxay
(p3) + Fzby
(−p3 − p − k)Fxby
(p3) .
Explicit thermal field theory calculations show that the contact terms do not
contribute to the transport coefficient.
This is remarkably different from the perturbative calculation of this correlator
for fermions , where the contact terms had explicit contribution to the three
point functions.
The only contribution comes from the wick contractions of the correlators in
flat space (term in the first line).

η invariants in higher dimensions
We have seen that given the η invariant corresponding to the T2
transformation of a torus, we can determine the contribution of the chiral
matter to the parity odd transport coefficient.
But the calculation of η invariants themselves can be very tedious. Hence it is
convenient to turn the problem around and evaluate η invariant using transport
coefficients.
We compute η invariants for fermions, gravitinos, self dual tensors in
d = 2, 6, 10 based on the values of parity odd transport coefficients in these
dimensions.
[Loganayagam ’11]

Dimension Species η invariant (upto mod 2)
d = 2 Fermions 1
6
Gravitinos 1
6
Chiral Bosons 1
6
d = 6 Fermions − 7
60
nm
Gravitinos −35
60
nm
Self Dual Tensors −16
60
nm
d = 10 Fermions 31
126
mnop
Gravitinos 279
126
mnop
Self Dual Tensors 256
126
mnop
Table: η invariants in various dimensions
where m, n, o, p ∈ Z

Thank you.

Global gravitational anomalies and transport

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