# Relativistic hydrodynamics with the parity anomaly

Napat Poovuttikul  Department of Mathematical Sciences, Durham University,
Durham, DH1 3LE, United Kingdom

###### Abstract

We consider the hydrodynamic regime of a 2+1 dimensions QFT with the parity
anomaly. Beyond the known constraints from positivity of entropy production,
we show that the anomaly inflow mechanism, from a corresponding bulk SPT
phase, together with thermodynamic consistency of equilibrium partition
functions, restricts the form of non-dissipative transport coefficients. This
included the known form of quantised Hall conductivity, which is fixed to be
$\sigma_{xy}=e^{2}/2h$, along with new constraints on other three non-
dissipative parity-odd transport coefficients.

###### Contents

  1. I Introduction and motivations
  2. II Parity-odd fluid, thermal equilibrium and results
  3. III Derivation of constraints from parity anomaly
  4. IV Discussions
  5. A A very short interlude on APS index theorem and anomaly inflow
  6. B Euler density contribution to equilibrium partition function?

##  I Introduction and motivations

The intertwining of hydrodynamic and quantum regimes of quantum field theory
provides an interesting playground of interesting mathematical ideas with
observable consequences. This regime can be found in both (very clean)
condensed matter system Bandurin2016 ; Crossno1058 ; moll2016evidence  and in
quark-gluon plasma Romatschke:2007mq  (see also Romatschke:2017ejr  for recent
extensive review). Symmetry structures of quantum systems thereby enrich the
ordinary hydrodynamics to be much more interesting than, arguably,
conservation of energy-mentum with extra steps.

Perhaps one of the most interesting developments in this direction is the
interplay between quantum anomaly and hydrodynamics. Take a hydrodynamic limit
of a QFT with $U(1)^{3}$ ’t Hooft anomaly111For standard reviews on anomaly,
see e.g. Harvey:2005it ; Bilal:2008qx  or §13 of nakahara2018geometry  in
$3+1$ dimensions whose Ward identity, in the presence of background gauge
field $a_{\mu}$, can be written as

$\partial_{\mu}j^{\mu}=\kappa\epsilon^{\mu\nu\rho\sigma}(da)_{\mu\nu}(da)_{\rho\sigma}\,.$
(1)

This anomaly will remain fixed all the way to the low energy description of
this theory tHooft:1979rat . According the hydrodynamic principle (see e.g.
LLfluid ; forster1995 ; Kovtun:2012rj ), one may attempt to write down a
parity-odd (due to anomaly) current $j^{\mu}$ in terms of hydrodynamic
variables – the temperature $T$, the chemical potential $\mu$ and fluid
velocity $u^{\mu}$ order by order in the gradient expansions. The most general
terms one can write down, up to the first derivative level is

$j^{\mu}=\rho
u^{\mu}+\xi\,\epsilon^{\mu\nu\rho\sigma}u_{\nu}\nabla_{\rho}u_{\sigma}+\frac{1}{2}\xi_{B}\,\epsilon^{\mu\nu\rho\sigma}u_{\nu}(da)_{\rho\sigma}+...$
(2)

where we omitted the usual dissipative first derivative terms (interested
readers can find the explicit relations in the above textbook references).
Intriguingly, the parity odd transport coefficients $\\{\xi,\xi_{B}\\}$ are
completely fixed in terms of anomaly coefficients $\kappa$ and thermodynamic
variables. This result can be derived from purely macroscopic arguments such
as positivity of entropy production Son2009 , properties of thermal
correlation function Jensen:2012jy , consistency of equilibrium partition
function Banerjee:2012iz  and hydrodynamic effective action Dubovsky:2011sk ;
Haehl2014 ; Glorioso:2017lcn . This has a vast impact on the studies of
transport phenomena that we cannot do justice here222We recommend the reader
to a recent review Landsteiner:2016led  for further informations and key
references on this aspect.. Many generalisation of the anomaly constraints on
macroscopic transport have been made over the years such as generalisation of
$U(1)$ to arbitrary group Lie group $G$, inclusion of gravitational anomaly
and different kind of ’t Hooft anomaly across arbitrary (even)dimensions, see
e.g. Neiman:2010zi ; Banerjee:2012cr ; Jensen:2013kka .

The origin of all anomalies of continuous group $G$, types discussed above,
can be traced back to existence of chirality that is exclusive to even
spacetime dimensions (such as chiral fermion in the seminal ABJ anomaly).
These are not only anomalies in nature. By unrestricted ourselves from
continuous group, one finds that there can be anomaly in both odd and even
spacetime dimensions and the matter may or may not involves massless fermion
Wen:2013oza ; Kapustin:2014lwa ; Kapustin:2014zva 333 We also find an
entertaining and insightful lecture series by Y. Tachikawa to be a good start
for familiarising the concept of discrete anomaly.. The oldest and most well-
known (albeit a lot less than its continuous symmetry counter part) is the
parity anomaly in $2+1$ dimensions Redlich:1983kn ; Niemi:1983rq ;
AlvarezGaume:1984nf . Some of its notable properties, in comparison to the
usual anomalies, should be mentioned here

  * •

While anomalies of continuous group can be detected through perturbative
computation (such as triangle diagram in $3+1$ dimensions), the information of
parity anomaly and the kind involving discrete anomaly cannot be obtained in
such a way. Hence there is a nomenclature, perturbative and non-perturbative
anomaly for the former and latter case respectively.

  * •

In the langauge of partition function, one can define anomaly by how it
transformed under continuous background gauge transformation e.g.

$Z[a+d\lambda]=Z[a]\exp\left(i\Phi[\lambda,da]\right)$ (3)

where $\Phi[\lambda,da]$ is a phase ambiguity of the partition function and
$\lambda$ is the transformation parameter of the background gauge field. In
contrast, the partition function of a theory with parity anomaly in such a way
that is invariant under $a\to a+d\lambda$. However, it does transformed under
time-reversal $Z\to Z^{\mathcal{T}}$, namely

$Z^{\mathcal{T}}=Z\exp(-i\pi\boldsymbol{\eta})$ (4)

where the phase ambiguity is the Atiyah-Patodi-Singer (APS)
$\boldsymbol{\eta}-$invariant atiyah1975spectral 444In a simple term, it is a
(regulated) difference between positive and negative eigenvalues of the Dirac
operator $i\not{D}$ on a given manifold.. As a result, this anomaly does not
show up in the anomalous Ward identity, unlike the example in (1) and its
generalisation.

  * •

The partition function of anomalous theory can be made invariant when treated
as a boundary of a topological field theory in one higher dimensions (through
anomaly inflow mechanism Callan1985 ). For continuous symmetry in $d+1\in
2\mathbb{Z}$ dimensions, the bulk is described by Chern-Simons term in $d+2$
dimensions. There is no Chern-Simons term in even $d+2$ dimensions, however.
As it turns out, the bulk theory that cancel the parity anomaly of the
boundary QFT is555We use the convention where $\oint F=2\pi\mathbb{Z}/e$. The
prefactor $e^{2}/32\pi^{2}$ is chosen such that, on a closed manifold
$S_{top}=\theta n$ with $n\in\mathbb{Z}$. This kind of bulk theory is often
referred to as SPT phase. Unlike Chern-Simons theory, it has a unique ground
state. Its partition function can be written as a complex number
$Z_{bulk}=\exp(i\Phi)$ of dimension 1 and its inverse is simply a complex
conjugate $Z_{bulk}^{-1}=\exp(-i\Phi)$, see Witten:2015aba  for similar
discussion.

$S_{bulk}=\frac{\theta e^{2}}{32\pi^{2}\hbar}\int
d^{4}X\,\epsilon^{abcd}F_{ab}F_{cd}\qquad\text{at}\qquad\theta=\pm\pi$ (5)

where $F$ is the field strength of the (boundary) background gauge field $a$
extended to the bulk. As it turns out, this bulk action is an effective
description of a topological insulator Qi:2008ew . Note that, for a fixed
$\theta$, this bulk action (5) is odd under time-reversal but the combined
bulk + boundary system is invariant through the discrete version of inflow
mechanism. In this particular case, the cancellation is a manifestation of the
APS index theorem. The reader may find a (very)brief demonstration of how APS
index works in this particular case in Appendix A.

All these information, particularly the part concerning anomaly inflow were
nicely explained in Witten:2015aba ; Witten:2019bou . We highly recommend to
readers for more detailed explanation. Studies on effects of perturbative
anomaly in the effective theory description in a setup closely related to ours
can be found in Chan:2015zhy ; Golkar:2015oxw .

The goal this work is to understand whether or not the parity anomaly can fix,
without resorting to microscopic details, the value of parity-odd
hydrodynamics in $2+1$ d in the same manner as its perturbative anomaly
counter parts in even dimensions. It should be note that, since the parity
anomaly does not altered the Ward identity, such constraint cannot be derived
using positivity of entropy production argument of Son2009 . To see the effect
of parity anomaly, we restrict ourselves to the hydrodynamic equilibrium
partition function approach of Jensen2012 . The technology to study anomaly
inflow in such configuration was developed in Jensen:2013kka  and we simply
apply it to the current context of parity anomaly. With this method, we show
that there are constraints on parity-odd fluid transport coefficients as a
consequence of thermal equilibrium and anomaly inflow. There is no microscopic
details required other than that parity anomaly persist all the way from the
UV to the hydrodynamic regime in the deep IR.

Next in Section II, we outlined what is the parity-odd fluid in 2+1 dimensions
(its transport coefficients and previously known constraints) then stating our
result on which of these transport coefficients are fixed by parity anomaly.
One of the constraint resulting in the fixed value of Hall conductivity

$\sigma_{xy}=\frac{e^{2}}{2h}\,.$ (6)

where the factor $e^{2}$ appears as a result of quantisation condition of flux
$\oint F=2\pi\text{ (integers)}/e$ in (5). This is a well-known result
obtained form the free theory computation at the boundary of topological
insulator (see e.g. Section III of Qi:2011zya  and Section V.A. of
Hasan:2010xy ). Others transport coefficients that are fixed by anomaly will
be elaborate in the same section. The derivation of these results are provided
in Section III. We will discuss the result, possible applications and future
directions in Section IV.

##  II Parity-odd fluid, thermal equilibrium and results

By fluid or hydrodynamics, we do mean the conventional definition classic
textbooks such as LLfluid ; forster1995 . It is an IR effective description a
gapless theory at finite temperature and possibly chemical potential/density.
A possible justification for hydrodynamic limit in a QFT is that, at finite
temperature, generic operators that decays away at late-time and long-
distance, except those that are commuted with the Hamiltonian. In a
relativistic system with Poincaré and $U(1)$ global symmetry, it means that
all conserved currents $T^{\mu\nu},j^{\mu}$ can be written in terms of
$\\{T,u^{\mu},\mu\\}$ which are conjugate to energy, momentum and $U(1)$
charge666 This should be contrasted with other effective theory, also dubbed
hydrodynamics, but do not include temperature, fluid velocity as well as
dissipative effects in the description, such as in Chan:2015zhy . The
equilibrium sector of such EFT can be obtained from what described here upon
turning off the temperature and fixing the fluid velocity at rest. .
Expressions for $T^{\mu\nu},j^{\mu}$ in terms of these variables are called
constitutive relations. For a parity-invariant fluid, the terms up to first
order in the derivative expansions and their constraint from positivity of
entropy production have been classified and can be found in LLfluid . The
profiles of these hydrodynamic variables can be obtained by solving the Ward
identities

$\nabla_{\mu}T^{\mu\nu}=(da)^{\nu\mu}j_{\mu}\,,\qquad\nabla_{\mu}j^{\mu}=0\,,$
(7)

where the theory is defined on a manifold with metric $g_{\mu\nu}$ and
background $U(1)$ gauge field $a_{\mu}$. Below, we briefly review the
notations and machinery that used to arrived at the result.

When parity is broken, by whatever means, the constitutive relations need to
be modified to included terms constructed from pseudoscalars, vectors and
tensors. It was notice many decades ago that these new terms have consequences
in transport phenomena avron1998odd . A complete classification of
thermodynamically consistent parity odd in 2+1 dimensions was done relatively
recently in Jensen:2011xb ; Banerjee:2012iz  (see also Kaminski:2013gca  for
its non-relativistic version and Lucas:2014sia  for recent discussion on
applications). One finds the modified constitutive relations to be777This
constitutive relation chose the form of off-equilibrium fluid variables in
such a way that $u_{\mu}j^{\mu}=\rho$, $u_{\mu}u_{\nu}T^{\mu\nu}=\varepsilon$
and $u_{\mu}T^{\mu\nu}=(\varepsilon+p)u^{\nu}$ without any derivative
corrections. This choice of variables is known as Landau frame, see §XV of
Landau&Lifshitz’s book LLfluid . This is not a unique choice of variables and
another popular “frame” known as the Eckart frame can be found in the rest of
the same book.

$\displaystyle T^{\mu\nu}$ $\displaystyle=\varepsilon
u^{\mu}u^{\nu}+\left(p-\zeta\nabla_{\lambda}u^{\lambda}-\tilde{\chi}_{B}B-\tilde{\chi}_{\Omega}\Omega\right)-\eta\sigma^{\mu\nu}-\tilde{\eta}\sigma^{\mu\nu}\,,$
(8) $\displaystyle j^{\mu}$ $\displaystyle=\rho u^{\mu}+\sigma
V^{\mu}+\tilde{\sigma}\epsilon^{\mu\nu\lambda}u_{\nu}V_{\lambda}+\tilde{\chi}_{E}\epsilon^{\mu\nu\lambda}u_{\nu}E_{\lambda}+\tilde{\chi}_{T}\epsilon^{\mu\nu\lambda}u_{\nu}\nabla_{\lambda}T\,.$

Here, the $\varepsilon,p,\rho$ are the usual energy density, pressure and
$U(1)$ density whereas $\eta,\zeta$ are the shear and bulk viscosities present
in the ordinary fluid. The scalar, vector and tensor structures found above
are defined as

$\displaystyle\sigma^{\mu\nu}$
$\displaystyle=2\nabla_{(\mu}u_{\nu)}-\Delta^{\mu\nu}\nabla_{\lambda}u^{\lambda}\,,\qquad\quad$
$\displaystyle\tilde{\sigma}^{\mu\nu}$
$\displaystyle=\frac{1}{2}\left(\epsilon^{\mu\alpha\beta}u_{\alpha}\sigma_{\beta}^{\;\;\nu}+\epsilon^{\nu\alpha\beta}u_{\alpha}\sigma_{\beta}^{\;\;\mu}\right)\,,$
(9) $\displaystyle V^{\mu}$
$\displaystyle=E^{\mu}-T\Delta^{\mu\nu}\nabla_{\nu}\frac{\mu}{T}\,,\qquad\quad$
$\displaystyle\tilde{V}^{\mu}$
$\displaystyle=\epsilon^{\mu\alpha\beta}u_{\alpha}V_{\beta}\,,$ $\displaystyle
E^{\mu}$ $\displaystyle=(da)^{\mu\nu}u_{\nu}\,,\qquad\quad$
$\displaystyle\tilde{E}^{\mu}$
$\displaystyle=\epsilon^{\mu\alpha\beta}u_{\alpha}E_{\beta}\,,$
$\displaystyle\Omega$
$\displaystyle=-\epsilon^{\mu\alpha\beta}u_{\mu}\omega_{\mu\nu}\,,\qquad\quad$
$\displaystyle\mathcal{B}$
$\displaystyle=-\frac{1}{2}\epsilon^{\mu\alpha\beta}u_{\mu}(da)_{\alpha\beta}$

with $\Delta=g^{\mu\nu}+u^{\mu}u^{\nu}$ be a projector along the plane
perpendicular to $u^{\mu}$ and
$\omega_{\mu\nu}=\Delta^{\mu\alpha}\Delta^{\nu\beta}\nabla_{[\alpha}u_{\beta]}$
is the vorticity. The tilded vectors/tensors are pseudovectors/tensors and
$\tilde{\eta},\tilde{\chi}_{B},\tilde{\chi}_{\Omega},\tilde{\sigma},\tilde{\chi}_{E},\tilde{\chi}_{T}$
are additional transport coefficients in the parity-odd fluid. Requiring that
the entropy production is positive will implies positivity of terms governing
dissipative effect i.e. $\eta\geq 0,\zeta\geq 0$ and $\sigma\geq 0$. There is
no constraint on non-dissipative transport coefficients
$\tilde{\eta},\tilde{\sigma}$ and but there are relations among
$\tilde{\chi}_{B},\tilde{\chi}_{\Omega},\tilde{\chi}_{E},\tilde{\chi}_{T}$
Jensen:2011xb .

Constraints on the last set of transport coefficient is best understood as a
consistency of equilibrium partition function which we will briefly elaborate.
It means that there is a time-like Killing vector $K^{\mu}=u^{\mu}/T$ and
consistent set of sources $g_{\mu\nu},a_{\mu}$. This resulting in the
vanishing of entropy production, the structures
$\sigma^{\mu\nu},\tilde{\sigma}^{\mu\nu}$ and $V^{\mu}$, along with the
following relations

$\nabla_{\mu}T=-Tu^{\lambda}\nabla_{\lambda}u_{\mu}\,,\qquad\partial_{\mu}\mu=-\mu
u^{\lambda}\nabla_{\lambda}u_{\mu}+E_{\mu}\,,\qquad\nabla_{\mu}u_{\nu}=-u_{\mu}u^{\lambda}\nabla_{\lambda}u_{\nu}+\omega_{\mu\nu}$
(10)

Since this configuration describes non-dissipative sector of the theory, it
can be captured by the ordinary generating function888This should be contrast
with Closed-Time-Path formalism where the degrees of freedom are doubled in
the complex time contour of Schwinger:1960qe ; Keldysh:1964ud  to take into
account the dissipative effect and fluctuation-dissipation theorem. See
e.g.kamenev2011field  for a more modern notations. . There is a nice geometric
interpretation to the hydrodynamic variables when analytically continued the
direction along $K^{\mu}$ to the thermal $S^{1}$ i.e. the temperature and
chemical potential is the (proper)size of and the $U(1)$ holonomy around the
thermal cycle and $u^{\mu}$ is the normalised Killing vector along $S^{1}$
direction. One can then proceed to write down every possible independent
(pseudo)scalars out of these geometric quantities up to the first order in the
derivative expansions, as done in Jensen2012 ; Banerjee:2012iz , and find the
following expression for $W=-i\log Z$

$W=\int
d^{3}x\left(\sqrt{-g}\,p(T,\mu)+\tilde{\alpha}_{1}\mathcal{B}+\tilde{\alpha}_{2}\Omega\right)+\mathcal{O}(\partial^{2})\,.$
(11)

where $p,\tilde{\alpha}_{1},\tilde{\alpha}_{2}$ are, a priori, arbitrary
function of $T,\mu$ to be determined by microscopic computations. It turns out
that, all the non-dissipative transport coefficients in (8), except
$\tilde{\eta},\tilde{\sigma}$, are related to
$\tilde{\alpha}_{1},\tilde{\alpha}_{2}$ in the following way

$\displaystyle\tilde{\chi}_{B}$ $\displaystyle=-\frac{\partial
p}{\partial\varepsilon}\left(T\frac{\partial\tilde{\alpha}_{1}}{\partial
T}+\mu\frac{\partial\tilde{\alpha}_{1}}{\partial\mu}-\alpha_{1}\right)-\frac{\partial
p}{\partial\rho}\frac{\partial\tilde{\alpha}_{1}}{\partial\mu}\,,$ (12)
$\displaystyle\tilde{\chi}_{E}$
$\displaystyle=\frac{\partial\tilde{\alpha}_{1}}{\partial\mu}-\frac{\rho}{\varepsilon+p}\left(\frac{\partial\tilde{\alpha}_{2}}{\partial\mu}-\tilde{\alpha}_{1}\right)\,,$
$\displaystyle\tilde{\chi}_{\Omega}$ $\displaystyle=-\frac{\partial
p}{\partial\varepsilon}\left(T\frac{\partial\tilde{\alpha}_{2}}{\partial
T}+\mu\frac{\partial\tilde{\alpha}_{2}}{\partial\mu}-2\tilde{\alpha}_{2}\right)-\frac{\partial
p}{\partial\rho}\left(\frac{\partial\tilde{\alpha}_{2}}{\partial\mu}-\tilde{\alpha}_{1}\right)\,,$
$\displaystyle\tilde{\chi}_{T}$
$\displaystyle=\frac{1}{T}\left(T\frac{\partial\tilde{\alpha}_{1}}{\partial
T}+\mu\frac{\partial\tilde{\alpha}_{1}}{\partial\mu}-\tilde{\alpha}_{1}\right)-\frac{\rho}{T(\varepsilon+p)}\left(T\frac{\partial\tilde{\alpha}_{2}}{\partial
T}+\mu\frac{\partial\tilde{\alpha}_{2}}{\partial\mu}-2\tilde{\alpha}_{2}\right)\,.$

To the best of our knowledge, these are the most exhausted macroscopic
constraints of a consistent parity-odd fluid so far.

Now that we are done with the technicality and terminology, let us state the
new result in the work. If the parity-violation can be removed by coupled the
fluid to the bulk SPT phase in Eq.(5), we will show, in the next section, that

$\tilde{\alpha}_{1}=\mu\left(\frac{\theta e^{2}}{2\pi
h}\right)\,,\qquad\tilde{\alpha}_{2}=\frac{1}{2}\mu^{2}\left(\frac{\theta
e^{2}}{2\pi h}\right)\,,\qquad\text{with}\qquad\theta=\pm\pi\,.$ (13)

and consequently

$\tilde{\chi}_{\Omega}=\tilde{\chi}_{T}=0\,,\qquad\tilde{\chi}_{B}=-\frac{\partial
p}{\partial\rho}\theta\,,\qquad\tilde{\chi}_{E}=\theta\,.$ (14)

in the unit of $e^{2}/(2\pi h)$. One can immediately see what this implies for
a Hall conductivity. We can considering the setup where we have a fixed
$u^{\mu}=(1,0,0)$, which can be obtained in a system with disorder that break
translation symmetry, keeping the temperature constant and set
$E^{i}=\partial_{i}\mu$ so that the system remains in equilibrium. As a
result, we find that there is an equilibrium Hall current:

$j^{x}=\tilde{\chi}_{E}E_{y}\,,\qquad j^{y}=-\tilde{\chi}_{E}E_{x}\,.$ (15)

Compared with the definition of Hall conductivity
$j^{i}=\sigma_{xy}\epsilon^{ij}E_{j}$, we find the known value of Hall
conductivity on the edge of topological insulator advertised in (6).

##  III Derivation of constraints from parity anomaly

By writing the non-dissipative sector of parity-odd fluid in the equilibrium
partition function, it is not very surprising that parity anomaly will further
restrict the form of coefficients $\tilde{\alpha}_{1}$ and
$\tilde{\alpha}_{2}$ in (11). After all, the anomalous can be canceled by
anomaly inflow mechanism which is to say that the combined bulk+boundary
system is time-reversal invariant

$Z_{inv}=Z_{hydro}[a]\exp(iS_{top})\,.$ (16)

where $Z_{hydro}=\exp(iW)$ in (11) and $S_{top}$ is the action for SPT phase
in (5). The challenge is then to write the topological term (in one dimension
higher) in terms of hydrodynamic variables. Luckily, such machinery developed
in Jensen:2013kka  for perturbative anomalies can be immediately applied to
this case. We will use this approach to derived the results in (13)-(14).

Let’s consider the topological term (5) on a manifold $\mathcal{N}$ whose
boundary $\partial\mathcal{N}=\mathcal{M}$ support the parity-odd fluid. Let
$u=u_{a}dX^{a}$ be a 1-form dual of timelike Killing vector on $\mathcal{N}$
whose boundary is the fluid velocity $u_{\mu}dx^{\mu}$ such that
$u_{\mu}u^{\mu}=u_{a}u^{a}=-1$. Similarly
$\mu,\omega=\frac{1}{2}\omega_{ab}dX^{a}\wedge
dX^{b},A=A_{a}dX^{a},F=\frac{1}{2}F_{ab}dX^{a}\wedge dX^{b}$ are the local
chemical potential, vorticity, background gauge field $a$ and field strength
$da$ extended from $\mathcal{M}$ to $\mathcal{N}$. The bulk is also required
to be in equilibrium, that is there is a timelike Killing vector
$K^{a}=u^{a}/T$ and the chemical potential is the (analytically
continued)holonomy $\mu=u^{a}A_{a}$ that become $K^{\mu}=u^{\mu}/T$ and
$\mu=u^{\mu}a_{\mu}$ at $\mathcal{M}=\partial\mathcal{N}$. One can define
another $U(1)$ connection to be

$A^{\perp}=A+\mu\,u\,,\qquad\text{such that}\qquad u^{a}A^{\perp}_{a}=0\,.$
(17)

We can define the field strength out of $A^{\perp}$ via
$F^{\perp}=dA^{\perp}$. This field strength has vanishing $\int
F^{\perp}\wedge F^{\perp}$. An easy way to see this is to imagine $u^{a}$ to
be along the time direction and for it to be in equilibrium, we need $A_{a}$
to be independent of time. Thus $A^{\perp}$ and $F^{\perp}$ has no component
in time direction and therefore $\epsilon^{abcd}(F^{\perp}\wedge
F^{\perp})_{abcd}$ vanished. This means that

$S_{top}\sim\int_{\mathcal{N}}F\wedge F=\int\left(F\wedge F-F^{\perp}\wedge
F^{\perp}\right)$ (18)

The r.h.s. is particularly useful since we can invoke the Chern-Weil theorem
to which states that the difference between anomaly polynomial corresponds to
two different connections (in this case $A$ and $A^{\perp}$) can be written as
an exact form. This means that

$\mathcal{P}[F]-\mathcal{P}[F^{\perp}]=dV_{\mathcal{P}}\,,\qquad\text{where}\qquad\mathcal{P}[F]=F\wedge
F\,,\quad\mathcal{P}[F^{\perp}]=F^{\perp}\wedge F^{\perp}\,,$ (19)

and that $\int_{\mathcal{N}}F\wedge F=\int_{\mathcal{M}}V_{\mathcal{P}}$. The
3-form $V_{\mathcal{P}}$ is known as transgression. More details on this can
be found in appendix D of Jensen:2013kka  or more formal discussion in §11 of
nakahara2018geometry 999Had we chose $\mathcal{P}[F^{\perp}=0]$, one will have
$\int_{\mathcal{N}}F\wedge F=\int_{\mathcal{M}}S_{CS}$ which is the usual
Chern-Simons form. This is but one representation of the transgression and
only a globally well-defined when $F$ can be written as $da$ everywhere at the
boundary. .

To determined the transgression $V_{\mathcal{P}}$, let consider a parameter
$\tau\in[0,1]$ connecting $A$ and $A^{\perp}$ as

$A(\tau)=A+(1-\tau)\mu\,u$ (20)

and that the field strength

$F(\tau)=-u\wedge(E+(\tau-1)d\mu)+(B-2(1-\tau)\mu\,\omega)\,.$ (21)

where $B=F+u\wedge E$ is a 2-form magnetic field. Using the commutativity
between differential $d$ and derivative along $\tau$ direction
$\partial_{\tau}$, one finds that

$\displaystyle V_{\mathcal{P}}$
$\displaystyle=\int^{\tau=1}_{\tau=0}d\tau\left(\partial_{\tau}A_{\tau}\wedge\frac{\partial\mathcal{P}[F(\tau)]}{\partial
F(\tau)}\right)\,,$ (22) $\displaystyle=-2\mu\,u\wedge
F-2\mu^{2}\,u\wedge\omega\,,\qquad\qquad\text{for}\qquad\mathcal{P}[F(\tau)]=F(\tau)\wedge
F(\tau)\,.$

Evaluated the transgression at the boundary and convert it to the index
notation, we find that

$S_{top}=-\frac{\theta
e^{2}}{4\pi^{2}\hbar}\int_{\mathcal{M}}d^{3}x\left(\mu\,\mathcal{B}+\frac{1}{2}\mu^{2}\,\Omega\right)$
(23)

Combining $S_{top}$ with hydrodynamic equilibrium partition function, we find
that, for

$-i\log Z_{inv}=W+S_{top}$ (24)

to be invariant, one need cancel the terms odd under time-reversal. That is
$\tilde{\alpha}_{1}$ and $\tilde{\alpha}_{2}$ must follows (13). Upon
obtaining $T^{\mu\nu},j^{\mu}$ from $W$ and convert it to the Landau frame, as
in Jensen2012  using (12), one finds the transport coefficients in (14). This
conclude our derivation.

Readers who familiar with the full form of APS index theorem (or notice the
appendix A) may worry that we have been ignoring the contribution from Euler
density. We would like to reassured that it does not contribute to the
hydrodynamic transport at this order in the derivative expansions. A
verification of this statement can be found in appendix B.

An important remark also has to be made concerning the periodicity of
$\theta-$angle and whether or not presented effect is simply a pure contact
term. It can be easily seen that the topological partition function
$Z=\exp(iS_{top})$ on $\mathcal{N}$, if $\mathcal{N}$ is closed, is invariant
under $\theta\to\theta+2\pi\mathbb{Z}$. Similarly, one may wonder if one
simply add a Chern-Simons counter term to $\mathcal{M}$ and altered or make
the transport coefficient vanishes. As it turns out, one can indeed using
thermal equilibrium identities (10) to write the Chern-Simons action in the
form similar to (23). However, the allowed counter term has to be quantised
property (see e.g. Closset:2012vp ) and can only produce a term in (23) with
$\theta=2\pi\mathbb{Z}$. One can therefore conclude that the ambiguity of
$\theta-$angle from the bulk can be removed by appropriately adjusted counter
term and vice-versa. Only the fractionaly part i.e. $\theta/2\pi=\pm 1/2$ is
physical and is therefore procedure independent.

##  IV Discussions

Let us reiterate new results in the work. Perhaps the most striking features
of this analysis is the existence of equilibrium Hall conductivity which is
fixed to, $\sigma_{xy}=\tilde{\chi}_{E}=\theta e^{2}/2h$ for $\theta=\pm\pi$,
which is a hallmark of topological insulator. Any naive EFT that claims to be
in the same universality class as boundary of topological insulator would have
to put in this condition, either by hand or certain microscopic inputs. The
consistency of anomaly inflow and equilibrium partition function not only
obtained this result purely from macroscopic point of view but also put
restrictions the other non-disspative transport coefficients. From (14), we
saw that, if the parity anomaly is the only source of broken time-reversal
symmetry, then

  * (i)

The pressure response to magnetic field $\tilde{\chi}_{B}=-(\partial
p/\partial\rho)\theta$ is fixed by thermodynamic functions and
$\theta=\pm\pi$.

  * (ii)

The pressure response to vorticity $\tilde{\chi}_{\Omega}$ and current
responds to perpendicular temperature gradient $\tilde{\chi}_{T}$ are forced
to vanish.

Measurements of these quantities could served as additional indication whether
or not a system of interest could have parity anomaly. It should be emphasised
here that our analysis only relies on the (anomalous)global symmetries in the
IR, thermodynamic consistencies and assumption of gradient expansions. Thus,
we expect this result to be robust even when the microscopic descriptions
involved strong interactions.

Another way the system can be realised without being at the boundary of $3+1$
bulk material is when one spontaneously break the $\mathbb{Z}_{2}$ time-
reversal symmetry. When this happens, one end up with system with domain wall
separating the two sectors with $\theta=\pi$ and $\theta=-\pi$. The
differences between two sectors is
$j^{i}[\theta=\pi]-j^{i}[\theta=-\pi]=(e^{2}/h)\epsilon^{ij}E_{j}$. This
situation is ‘realised’ in e.g. Haldane model Haldane:1988zza  (see also
section II.B. of Hasan:2010xy ).

We find no constraint on two transport coefficients $\tilde{\eta}$ and
$\tilde{\sigma}$ as tensor and vector structure multiplying them, see Eq.(8),
vanished in equilibrium. The first coefficient $\tilde{\eta}$, known as the
Hall viscosity, is particularly interesting: first because of its difficulty
in measuring (see e.g. Hoyos:2014pba ) and recent success in hydrodynamic
regime of graphene berdyugin2019measuring ; scaffidi2017hydrodynamic .
Admittedly, our computation cannot say anything about this transport
coefficient. Nevertheless, it is tempting to conclude that the parity anomaly
does not contribute to the Hall viscosity. A strong indication being that
effective theories which produce Hall viscosity requires additional
topological term of different nature Read:2010epa ; Hoyos:2011ez ;
Gromov:2014gta  known as the Wen-Zee term (see also Golkar:2014wwa  for its
relativistic generalisation). It would be interesting to understand whether or
not the Wen-Zee term is a result of discrete anomaly of different kind.

Last but not least, we hope that this paper demonstrate that, despite not
affecting the Ward identity, the discrete anomaly can affect hydrodynamic
transport in a nontrivial way. We only consider only one, arguably the
simplest, of such examples but there are plethora of SPT phases out there.
Given a rich physics that came out of the study of perturbative anomaly
induced transport, this would be a very interesting direction to explore.

## Acknowledgements

I would like to thanks Nabil Iqbal and Tin Sulejmanpasic for collaborating on
related topics and various discussions. I am also very grateful to Iñaki
García-Etxerbarria, Nakarin Lohitsiri for illuminating discussions as well as
Paolo Glorioso, Nabil Iqbal and Kristan Jensen for commenting on the
manuscript. I would like to thank Chulalongkorn University for hospitality
during the pandemic refuge period and my work is supported by STFC grant
number ST/T000708/1.

##  Appendix A A very short interlude on APS index theorem and anomaly inflow

For self-containment sake, we will briefly outlined reasons why free massless
pseudoreal fermion is anomalous as well as how APS index theorem provide a
derivation of anomaly inflow mechanism for parity anomaly. All of these
information are explained in more details in AlvarezGaume:1984nf ;
Witten:2015aba . More examples on how APS index theorem provide anomaly inflow
mechanism and computation’s subtleties can be found in Witten:2019bou  and for
a more familiar chiral anomaly in Kobayashi:2021jbn . This will also help
justify the reason for $\theta=\pi$ beyond the argument of Qi:2008ew .

Start by considering a prototype of parity anomaly: the (massless) Dirac
fermion in $2+1$d whose eigenvalue obtained via

$i\not{D}\psi_{k}=\lambda_{k}\psi_{k}$ (25)

From this, we can obtain the partition function of free massless Dirac fermion

$Z_{\psi}=\det(i\not{D})=\prod_{k}\lambda_{k}$ (26)

which, just like any QFT partition function, is infinite. To make sense of it,
we add a regulator by adding

$S_{reg}=\int d^{3}x\chi\left(i\not{D}+iM\right)\chi$ (27)

for $\chi$ be a 2-components scalars that satisfies $(i\not{D}+iM)\chi=0$. The
resulting regulated action is

$Z_{reg}=\prod_{k}\frac{\lambda_{k}}{\lambda_{k}+iM}=\exp\left(\sum_{k}\log\left(\frac{\lambda_{k}}{\lambda_{k}+iM}\right)\right)$
(28)

Even when $M\to+\infty$, this theory regulated theory turns out to not be
invariant under time-reversal as the partition function is not real. To check
the possible imaginary part, we can consider the imaginary part of the
exponent

$\displaystyle\text{Im
}\sum_{k}\log\left(\frac{\lambda_{k}}{\lambda_{k}+iM}\right)$
$\displaystyle=\sum_{k}\left[-\log(\lambda_{k}+iM)+\log(\lambda_{k}-iM)\right]\,,$
(29) $\displaystyle=\sum_{k}\tan^{-1}\left(\frac{M}{\lambda_{k}}\right)\,,$
$\displaystyle=\frac{\pi}{2}\left[\sum_{\lambda_{k}>0}1-\sum_{\lambda_{k}<0}1\right]\qquad\text{as
}M\to+\infty$

The last term in the $[..]$ is the APS $\boldsymbol{\eta}-$invariant.
Computing $\boldsymbol{\eta}-$invariant requires another kind of regulator but
we will not talk about that for now. What is interesting is that we can then
write the regulated partition function as

$Z_{reg}=|Z_{\psi}|\exp\left(i\pi\boldsymbol{\eta}/2\right)$ (30)

where $|Z_{\psi}|$ is the modulus of the (regulated)partition function. We can
see from there that the time reversal, $TiT^{-1}=-i$, transformed the
regulated partition function as

$T:\qquad Z_{reg}\to Z_{reg}\exp(-i\pi\boldsymbol{\eta})$ (31)

The APS index theorem state that for $\boldsymbol{\eta}-$invariant on
$\mathcal{M}=\partial\mathcal{N}$, we have

$-\frac{\boldsymbol{\eta}}{2}-\int_{\mathcal{N}}(\mathcal{P}-\hat{A}(R))=\mathfrak{J}$
(32)

where $\mathfrak{J}\in\mathbb{Z}$ is the Dirac index on $\mathcal{N}$ computed
with the APS boundary condition, see e.g. Witten:2015aba ; Witten:2019bou .
The terms $\mathcal{P}\sim F\wedge F$ is the anomaly polynomial in Section III
(with proper normalisation to $\int\mathcal{P}=1$ on a closed manifold) and
$\hat{A}(R)$ is the Euler density. Taking the background metric to be flat, we
find that

$\displaystyle Z_{reg}\exp\left(i\pi\int_{\mathcal{N}}\mathcal{P}\right)$
$\displaystyle=|Z_{\psi}|\exp\left(i\frac{\boldsymbol{\eta}\pi}{2}+i\pi\int_{\mathcal{N}}\mathcal{P}\right)\,,$
(33) $\displaystyle=|Z_{\psi}|(-1)^{\mathfrak{J}}$

the combined bulk+boundary partition function is therefore real and invariant
under time-reversal. Notice that the topological term $i\pi\int\mathcal{P}$ is
nothing but the SPT phase in (5) with $\theta=\pi$.

##  Appendix B Euler density contribution to equilibrium partition function?

As one may see else where or from (32) that ${\boldsymbol{\eta}}/2$ and
$\mathcal{P}=F\wedge F$ does not only give integer $\mathfrak{J}$ but also the
Euler density $\hat{A}(R)$. We have been conveniently ignored the contribution
from $\hat{A}(R)$ so far. Here, we will justify the reason for that. In simple
words, even if we include the Euler density, it will only contribute to
transport coefficients at third order in the derivative expansions.

To verify this claim, we want to find an analgue of $A^{\perp}$, which for the
geometric quantity turns out to involve the connection 1-form
${\bf\Gamma}=\Gamma^{\mu}_{\;\;\mu\lambda}dx^{\lambda}$ on $\mathcal{N}$

${\bf\Gamma}^{\perp}=\Gamma+\boldsymbol{\mu}_{R}u\,,\qquad(\boldsymbol{\mu}_{R})^{a}_{\;\;b}:=T\nabla_{b}\left(\frac{u^{a}}{T}\right)$
(34)

The curvature 2-form
$\boldsymbol{R}=d{\bf\Gamma}+{\bf\Gamma}\wedge{\bf\Gamma}$ can also be written
in a form analogous to the electric and magnetic field

${\bf R}=-u\wedge{\bf E}_{R}+{\bf
B}_{R}\,,\qquad(E_{R})^{\mu}_{\;\;\nu}=R^{\mu}_{\;\;\nu\alpha\beta}u^{\beta}dx^{\alpha}$
(35)

and the perpendicular field strength can be written (after imposing
equilibrium conditions) as

${\bf R}^{\perp}={\bf B}_{R}+2\boldsymbol{\mu}_{R}\omega$ (36)

We can now play to game of deriving the transgression form for
${\mathrm{tr}}({\bf R}\wedge{\bf R})$. Again, with Chern-Weil theorem, we can
write

${\mathrm{tr}}({\bf R}\wedge{\bf R})-{\mathrm{tr}}({\bf R}^{\perp}\wedge{\bf
R}^{\perp})=dV_{\mathcal{P}}^{grav}$ (37)

The flow from $\boldsymbol{\Gamma}^{\perp}$ to $\boldsymbol{\Gamma}$ can be
written as

$\boldsymbol{\Gamma}(\tau)=\Gamma+(1-\tau)\boldsymbol{\mu}_{R}u\,,\qquad{\bf
R}(\tau)=u\wedge(...)+({\bf B}_{R}+2(1-\tau)\boldsymbol{\mu}_{R}\omega)$ (38)

The transgression form for this case is therefore

$\displaystyle V_{\mathcal{P}}^{grav}$
$\displaystyle={\mathrm{tr}}\left[\int^{\tau=1}_{\tau=0}d\tau\left\\{\partial_{\tau}{\bf\Gamma}\wedge\frac{\partial\mathcal{P}[{\bf
R}(\tau)]}{\partial{\bf R(\tau)}}\right\\}\right]\,,$ (39)
$\displaystyle=-{\mathrm{tr}}\left[2\boldsymbol{\mu}_{R}u\wedge{\bf
B}_{R}+2[\boldsymbol{\mu}_{R}]^{2}u\wedge\omega\right]$

All of these terms are third order in derivatives of hydrodynamic variables
$(u^{\mu},T)$ and metric.

There is no known classification of parity-odd hydrodynamic with $U(1)$
symmetry at this order. The closest attempt in this direction is a
classification transport coefficients for relativistic hydrodynamic without
$U(1)$ charge upto thrid order in the derivative expansions, which found 68
terms in total Grozdanov:2015kqa . No inclusions of $U(1)$ symmetry has been
done, let alone the parity-violation terms. So we hope, given the scope of
this work, that this will justify leaving the consequences of Euler density
part to more powerful future generations.

It should be mentioned that similar situation occur in the gravitational
anomaly in the $2n$ dimensions e.g. take $n=2$ where $d\star
j\propto{\mathrm{tr}}[{\bf R}\wedge{\bf R}]$. In a naive derivative counting,
one would have thought that the gravitational, which only appears in $4^{th}$
order in the derivative expansions cannot affect the $\xi,\xi_{B}$ in (2)
which enters the Ward identity at $2^{nd}$ order. However, by requiring that
the theory is consistent in a singular geometry e.g. a cone with the metric

$ds^{2}=r^{2}d\tau^{2}+dr^{2}+ds_{\perp}^{2}$ (40)

with transverse flat metric $ds_{\perp}^{2}$ and $\tau\sim\tau+2\pi\delta$
with $\delta<1$ be a deficit angle signifying conical singularity. The
presence of a cone inhibit a ‘derivative jump’ that bring $4^{th}$ derivative
down to $2^{nd}$ derivative level. By demanding that momentum of the combined
bulk+boundary system toward the tip of the cone, $T^{\tau r}$, must vanishes,
Ref. Jensen:2012kj ; Jensen:2013rga  show that this condition add additional
fixed remaining unknowns in Son2009 ; Neiman:2010zi  in terms of the mixed
$U(1)$-gravitational anomaly coefficient101010We emphasise that this condition
is not a universal low energy relations and should only be applied to a theory
that is continuously connected to free theories Glorioso:2017lcn . The
situation here is amount to say that the process of taking the geometry to be
a cone in the UV and taking the RG flow to low energy limit does not commuted.
. In $2+1$ dimensions, on the other hand, one can check that putting the
theory on a cone does not produce any extra constraints on transport. To see
it explicitly, consider perturbing the metric of a cone in the form of

$\delta(ds^{2})=h_{\tau r}(r,z)d\tau dr\,.$ (41)

One can check that there is no term in the “inflowed” action
$\int_{\mathcal{M}}V_{\mathcal{P}}^{grav}$ contains no term linearly in
$h_{\tau r}$ except the total derivative terms. Thus the condition $T^{\tau
r}=0$ of Jensen:2012kj  is trivially satisfied in $2+1$ dimension and does not
producing any nontrivial constraints to the hydrodynamic transport.

That being said, while the anomaly inflow of Euler density from $\mathcal{N}$
to $\mathcal{M}=\partial\mathcal{N}$ does not constrain any transport on
$\mathcal{M}$ (where our fluid lives), it can have non-trivial effect if
$\mathcal{M}$ has a boundary. This can be understood as the Euler density on
$\mathcal{N}$ can be written as a gravitational term
$S_{CS}[\Gamma]\sim{\mathrm{tr}}\int_{\mathcal{M}}({\bf\Gamma}\wedge
d{\bf\Gamma}+\frac{2}{3}{\bf\Gamma}\wedge{\bf\Gamma}\wedge{\bf\Gamma})$ whose
gravitational Chern-Simons coupling is fixed by $\theta=\pm\pi$ via (32)-(33).
While having no effect on the hydrodynamic modes on $\mathcal{M}$,
$S_{CS}[\Gamma]$ can induce anomaly inflow mechanism to $\partial\mathcal{M}$
Stone:2012ud . The method of Jensen:2012kj  can then be used to determine the
transport coefficients in terms of the Chern-Simons coupling, resulting in the
quantised transport coefficient responsible for thermal hall conductivity
Kane:1997fda .

## References

  * (1) D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A. Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, and M. Polini, “Negative local resistance caused by viscous electron backflow in graphene,” Science (2016) , arXiv:1509.04165.
  * (2) J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, and K. C. Fong, “Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene,” Science 351 no. 6277, (2016) 1058–1061.
  * (3) P. J. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie, “Evidence for hydrodynamic electron flow in pdcoo2,” Science 351 no. 6277, (2016) 1061–1064.
  * (4) P. Romatschke and U. Romatschke, “Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?,” Phys. Rev. Lett. 99 (2007) 172301, arXiv:0706.1522 [nucl-th].
  * (5) P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 5, 2019.  arXiv:1712.05815 [nucl-th].
  * (6) J. A. Harvey, “TASI 2003 lectures on anomalies,”  9, 2005.  arXiv:hep-th/0509097.
  * (7) A. Bilal, “Lectures on Anomalies,” arXiv:0802.0634 [hep-th].
  * (8) M. Nakahara, Geometry, Topology and Physics.  CRC Press, 2018.
  * (9) G. ’t Hooft, “Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,” NATO Sci. Ser. B 59 (1980) 135–157.
  * (10) L. D. Landau and E. M. Lifshitz, Fluid Mechanics.  Butterworth-Heinemann, 2nd ed., 1987.
  * (11) D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions.  Perseus Books, 1995.
  * (12) P. Kovtun, “Lectures on hydrodynamic fluctuations in relativistic theories,” J. Phys. A45 (2012) 473001, arXiv:1205.5040 [hep-th].
  * (13) D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev. Lett. 103 (2009) 191601, arXiv:0906.5044 [hep-th].
  * (14) K. Jensen, “Triangle Anomalies, Thermodynamics, and Hydrodynamics,” Phys. Rev. D 85 (2012) 125017, arXiv:1203.3599 [hep-th].
  * (15) N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, and T. Sharma, “Constraints on Fluid Dynamics from Equilibrium Partition Functions,” JHEP 09 (2012) 046, arXiv:1203.3544 [hep-th].
  * (16) S. Dubovsky, L. Hui, and A. Nicolis, “Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions,” Phys. Rev. D 89 no. 4, (2014) 045016, arXiv:1107.0732 [hep-th].
  * (17) F. M. Haehl, R. Loganayagam, and M. Rangamani, “Effective actions for anomalous hydrodynamics,” JHEP 03 (2014) 034, arXiv:1312.0610 [hep-th].
  * (18) P. Glorioso, H. Liu, and S. Rajagopal, “Global Anomalies, Discrete Symmetries, and Hydrodynamic Effective Actions,” JHEP 01 (2019) 043, arXiv:1710.03768 [hep-th].
  * (19) K. Landsteiner, “Notes on Anomaly Induced Transport,” Acta Phys. Polon. B 47 (2016) 2617, arXiv:1610.04413 [hep-th].
  * (20) Y. Neiman and Y. Oz, “Relativistic Hydrodynamics with General Anomalous Charges,” JHEP 03 (2011) 023, arXiv:1011.5107 [hep-th].
  * (21) N. Banerjee, S. Dutta, S. Jain, R. Loganayagam, and T. Sharma, “Constraints on Anomalous Fluid in Arbitrary Dimensions,” JHEP 03 (2013) 048, arXiv:1206.6499 [hep-th].
  * (22) K. Jensen, R. Loganayagam, and A. Yarom, “Anomaly inflow and thermal equilibrium,” JHEP 05 (2014) 134, arXiv:1310.7024 [hep-th].
  * (23) X.-G. Wen, “Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders,” Phys. Rev. D 88 no. 4, (2013) 045013, arXiv:1303.1803 [hep-th].
  * (24) A. Kapustin and R. Thorngren, “Anomalies of discrete symmetries in three dimensions and group cohomology,” Phys. Rev. Lett. 112 no. 23, (2014) 231602, arXiv:1403.0617 [hep-th].
  * (25) A. Kapustin and R. Thorngren, “Anomalies of discrete symmetries in various dimensions and group cohomology,” arXiv:1404.3230 [hep-th].
  * (26) A. N. Redlich, “Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions,” Phys. Rev. Lett. 52 (1984) 18.
  * (27) A. J. Niemi and G. W. Semenoff, “Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times,” Phys. Rev. Lett. 51 (1983) 2077.
  * (28) L. Alvarez-Gaume, S. Della Pietra, and G. W. Moore, “Anomalies and Odd Dimensions,” Annals Phys. 163 (1985) 288.
  * (29) M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and riemannian geometry. i,” in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 77, pp. 43–69, Cambridge University Press.  1975\.
  * (30) C. G. Callan, Jr. and J. A. Harvey, “Anomalies and Fermion Zero Modes on Strings and Domain Walls,” Nucl. Phys. B250 (1985) 427–436.
  * (31) E. Witten, “Fermion Path Integrals And Topological Phases,” Rev. Mod. Phys. 88 no. 3, (2016) 035001, arXiv:1508.04715 [cond-mat.mes-hall].
  * (32) X.-L. Qi, T. Hughes, and S.-C. Zhang, “Topological Field Theory of Time-Reversal Invariant Insulators,” Phys. Rev. B 78 (2008) 195424, arXiv:0802.3537 [cond-mat.mes-hall].
  * (33) E. Witten and K. Yonekura, “Anomaly Inflow and the $\eta$-Invariant,” in The Shoucheng Zhang Memorial Workshop.  9, 2019.  arXiv:1909.08775 [hep-th].
  * (34) A. P. O. Chan, T. Kvorning, S. Ryu, and E. Fradkin, “Effective hydrodynamic field theory and condensation picture of topological insulators,” Phys. Rev. B 93 no. 15, (2016) 155122, arXiv:1510.08975 [cond-mat.str-el].
  * (35) S. Golkar and S. Sethi, “Global Anomalies and Effective Field Theory,” JHEP 05 (2016) 105, arXiv:1512.02607 [hep-th].
  * (36) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, “Towards hydrodynamics without an entropy current,” http://arxiv.org/abs/1203.3556v2.
  * (37) X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83 no. 4, (2011) 1057–1110, arXiv:1008.2026 [cond-mat.mes-hall].
  * (38) M. Z. Hasan and C. L. Kane, “Topological Insulators,” Rev. Mod. Phys. 82 (2010) 3045, arXiv:1002.3895 [cond-mat.mes-hall].
  * (39) J. Avron, “Odd viscosity,” Journal of statistical physics 92 no. 3, (1998) 543–557.
  * (40) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, “Parity-Violating Hydrodynamics in 2+1 Dimensions,” JHEP 05 (2012) 102, arXiv:1112.4498 [hep-th].
  * (41) M. Kaminski and S. Moroz, “Nonrelativistic parity-violating hydrodynamics in two spatial dimensions,” Phys. Rev. B 89 no. 11, (2014) 115418, arXiv:1310.8305 [cond-mat.mes-hall].
  * (42) A. Lucas and P. Surówka, “Phenomenology of nonrelativistic parity-violating hydrodynamics in 2+1 dimensions,” Phys. Rev. E 90 no. 6, (2014) 063005, arXiv:1403.5239 [cond-mat.mes-hall].
  * (43) J. S. Schwinger, “Brownian motion of a quantum oscillator,” J. Math. Phys. 2 (1961) 407–432.
  * (44) L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Zh. Eksp. Teor. Fiz. 47 (1964) 1515–1527.
  * (45) A. Kamenev, Field Theory of Non-Equilibrium Systems.  Cambridge University Press, 2011.
  * (46) C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski, and N. Seiberg, “Comments on Chern-Simons Contact Terms in Three Dimensions,” JHEP 09 (2012) 091, arXiv:1206.5218 [hep-th].
  * (47) F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ’Parity Anomaly’,” Phys. Rev. Lett. 61 (1988) 2015–2018.
  * (48) C. Hoyos, “Hall viscosity, topological states and effective theories,” Int. J. Mod. Phys. B 28 (2014) 1430007, arXiv:1403.4739 [cond-mat.mes-hall].
  * (49) A. I. Berdyugin, S. Xu, F. Pellegrino, R. K. Kumar, A. Principi, I. Torre, M. B. Shalom, T. Taniguchi, K. Watanabe, I. Grigorieva, et al., “Measuring hall viscosity of graphene’s electron fluid,” Science 364 no. 6436, (2019) 162–165.
  * (50) T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, and J. E. Moore, “Hydrodynamic electron flow and hall viscosity,” Physical review letters 118 no. 22, (2017) 226601.
  * (51) N. Read and E. H. Rezayi, “Hall viscosity, orbital spin, and geometry: paired superfluids and quantum Hall systems,” Phys. Rev. B 84 (2011) 085316, arXiv:1008.0210 [cond-mat.mes-hall].
  * (52) C. Hoyos and D. T. Son, “Hall Viscosity and Electromagnetic Response,” Phys. Rev. Lett. 108 (2012) 066805, arXiv:1109.2651 [cond-mat.mes-hall].
  * (53) A. Gromov and A. G. Abanov, “Density-curvature response and gravitational anomaly,” Phys. Rev. Lett. 113 (2014) 266802, arXiv:1403.5809 [cond-mat.str-el].
  * (54) S. Golkar, M. M. Roberts, and D. T. Son, “Effective Field Theory of Relativistic Quantum Hall Systems,” JHEP 12 (2014) 138, arXiv:1403.4279 [cond-mat.mes-hall].
  * (55) S. K. Kobayashi and K. Yonekura, “Atiyah-Patodi-Singer index theorem from axial anomaly,” arXiv:2103.10654 [hep-th].
  * (56) S. Grozdanov and N. Kaplis, “Constructing higher-order hydrodynamics: The third order,” Phys. Rev. D 93 no. 6, (2016) 066012, arXiv:1507.02461 [hep-th].
  * (57) K. Jensen, R. Loganayagam, and A. Yarom, “Thermodynamics, gravitational anomalies and cones,” JHEP 02 (2013) 088, arXiv:1207.5824 [hep-th].
  * (58) K. Jensen, R. Loganayagam, and A. Yarom, “Chern-Simons terms from thermal circles and anomalies,” JHEP 05 (2014) 110, arXiv:1311.2935 [hep-th].
  * (59) M. Stone, “Gravitational Anomalies and Thermal Hall effect in Topological Insulators,” Phys. Rev. B 85 (2012) 184503, arXiv:1201.4095 [cond-mat.mes-hall].
  * (60) C. L. Kane and M. P. A. Fisher, “Quantized thermal transport in the fractional quantum Hall effect,” Phys. Rev. B 55 no. 23, (1997) 15832–15837, arXiv:cond-mat/9603118.