aainstitutetext: Departamento de Física, Universidad del País Vasco UPV/EHU,
Apartado 644, 48080 Bilbao, Spainbbinstitutetext: Departamento de Física
Fundamental, Universidad de Salamanca,
Plaza de la Merced s/n, 37008 Salamanca, Spain

# Chiral Torsional Effects in Anomalous Fluids in Thermal Equilibrium

Juan L. Mañes a Manuel Valle b Miguel Á. Vázquez-Mozo<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using the similarity between spacetime torsion and axial gauge couplings, we
study torsional contributions to the equilibrium partition function in a
stationary background. In the case of a charged fluid minimally coupled to
torsion, we spot the existence of linear torsional magnetic and vortical
effects, while the axial-vector current and the spin energy potential do not
receive corrections in the torsion at linear order. The covariant energy-
momentum tensor, on the other hand, does contain terms linear in the torsion
tensor. The case of a two-flavor hadronic superfluid is also analyzed, and the
torsional contributions to the constitutive relations computed. Our results
show the existence of a torsional electric chiral effect mediated by the
charged pions.

##  1 Introduction

The coupling of hydrodynamic systems to external sources through anomalous
currents gives rise to a variety of chiral transport effects Son:2009tf ;
Fukushima:2008xe ; Sadofyev:2010pr ; Neiman:2011mj ; Kirilin:2012mw ;
Fukushima:2012vr ; Zakharov:2012vv ; Landsteiner:2016led . These have been
shown to be relevant for the understanding of diverse physical phenomena,
ranging from condensed matter to cosmology. Due to the topological nature of
anomalies, the effective action for the long wavelength modes can be computed
using differential geometry techniques, from which the parity-violating terms
in the fluid constitutive relations can be derived Banerjee:2012iz ;
Jensen:2012jh ; Jensen:2013kka ; Jensen:2012kj ; Jensen:2013rga ;
Haehl:2013hoa ; Monteiro:2014wsa ; Jain:2015jla ; Banerjee:2015hra ;
Glorioso:2017lcn ; Manes:2018llx ; Manes:2019fyw . By placing the system on a
curved background stationary metric, and performing dimensional reduction onto
the compactified Euclidean time, it is possible to incorporate physical
effects such as vorticity and acceleration, which are sourced respectively by
the background Kaluza-Klein (KK) gauge field and the gradient of the time
component of the metric. Phenomena linked to the existence of mixed gauge-
gravitational anomalies Landsteiner:2011cp  have been subject to direct
detection in the laboratory Gooth:2017mbd .

Despite its absence in standard general relativity, torsion has been the focus
of attention in physics for almost a century, since this geometrical notion
was first introduced in the classical works of Élie Cartan Cartan:1922 ;
Cartan:1923zea ; Cartan:1924yea . An obvious motivation for these
investigations has been the possibility of our spacetime having a small albeit
nonvanishing torsion, giving rise to new physics (see Hehl:1976kj ;
Shapiro:2001rz  for a review of different physical scenarios).

Together with the prospects of spotting fundamental microscopic torsion in
high-energy physics, torsional geometries provide a practical way of
implementing physical effects in condensed matter physics. Focusing on the
physics at large distances, the vectors defining the links at each node of the
ion lattice build up an effective dreibein whose geometry models lattice
irregularities. For example, their curvature and torsion respectively
implement lattice dislocations and disclinations Hughes:2012vg . In systems
with linear dispersion relations, such as the case of Weyl semimetals, this
geometry provides a static nondynamical effective background on which fermions
propagate Parrikar:2014usa .

Torsion is also known to have an interplay with chiral anomalies in quantum
field theory Zumino:1983ew ; AlvarezGaume:1985ex ; Bertlmann:1996xk ;
Fujikawa:2004cx . The axial anomaly receives a contribution given by the so-
called Nieh-Yan term Nieh:1981ww ; Nieh:1981xk , which comes from a bubble
diagram with two axial-vector current insertions and is quadratically
divergent. As a consequence, the coefficient of the Nieh-Yan term depends of
the square of the cutoff, or any other relevant UV scale of the theory
Chandia:1997hu . The role of this term in condensed matter physics has been
explored in a number of works (see, for example, Hidaka:2012rj ; Hughes:2012vg
; Parrikar:2014usa ; Nissinen:2019kld ; Nissinen:2019mkw ).

In the context of chiral fluids, the interest in torsion arose in connection
with the study of Hall transport Son:2007ny ; Haehl:2013kra ; Son:2013rqa ;
Geracie:2014mta  (see Hoyos:2014pba  for a comprehensive review). The relation
between the Hall viscosity and the mean orbital spin per particle suggested a
connection with the spin current, which is sourced by the background torsion.
In the relativistic setup, this link was further studied in Ref. Valle:2015hfa
by a first principles calculation of the effective action and constitutive
relations of a fermion gas on a $(2+1)$-dimensional spacetime with torsion.

The issue of torsional transport effects in four dimensions has also received
some attention lately Khaidukov:2018oat . Besides the sector associated to the
Nieh-Yan anomaly111It has recently been proposed, however, that there is no
genuine torsional chiral dissipationless transport in that sector
Ferreiros:2020uda ., the partition function contains other contributions which
are induced by the triangle diagrams associated with the effective axial-
vector field encoding the antisymmetric part of the torsion. In Ref.
Imaki:2020csc , transport phenomena induced by torsion were studied, both at
zero and finite temperature. The existence of a chiral magnetic and electric
effects was found, resulting from fermions minimally coupled to the
antisymmetric part of the torsion tensor, which can be recast in terms of its
dual vector field. This external source couples to the fermionic singlet
axial-vector current, which is affected by a ’t Hooft anomaly. Torsional
contributions to spin transport were also recently studied in Gallegos:2020otk
.

The effects associated with the ’t Hooft anomaly of the gauge field dual to
the antisymmetric part of the torsion can be readily computed using the
standard differential geometry methods employed in the analysis of anomalous
fluids. In the present work, we apply the techniques developed in
Manes:2018llx ; Manes:2019fyw  to carry out a study of the linear effects of
torsion in hydrodynamics, with and without Nambu-Goldstone bosons. For a
charged fluid coupled to an external electromagnetic field, we verify the
existence of torsional magnetic and vortical effects. The axial-vector
current, on the other hand, does not contain any corrections linear in the
torsion. This is also the case for the covariant spin energy potential, which
is written in terms of the covariant axial-vector current. The components of
the covariant energy-momentum tensor can be also expressed in terms of the
covariant axial-vector currents, but in this case the coefficients depend
linearly on the torsion tensor. Once written in terms of the torsion, they
give rise to new torsion-induced contributions to the constitutive relations
from where the corresponding transport coefficients can be obtained.

After analyzing the Abelian case, we focus our attention on the case of a two-
flavor hadronic superfluid in the presence of torsion, in the phase in which
chiral symmetry $\mbox{U(2)}_{L}\times\mbox{U(2)}_{R}$ is spontaneously broken
to its vector subgroups. We compute the corrections to the covariant currents
and transport coefficients linear in the torsion tensor and find the existence
of a torsional chiral electric effect mediated by the two charged pions. The
chiral separation effects found in Manes:2019fyw , on the other hand, do not
receive any contributions linear in the torsion.

The present article is organized as follows. Section 2 is devoted to the
analysis of the equilibrium partition function of a charged plasma in the
presence of torsion, including the computation of the covariant currents. This
models is further elaborated in Section 3 with the calculation of linear
torsional contributions to the spin energy potential and the energy-momentum
tensor. In Section 4, after a brief discussion of the linear coupling of
Nambu-Goldstone bosons to torsion in the Abelian case, we compute the linear
torsional corrections to the constitutive relations of a two-flavor hadronic
superfluid. Finally, our findings are summarized in Section 5. To make our
presentation more self-contained, we review in Appendix A some basic facts
about geometric torsion, while in Appendix B we list some expressions of Ref.
Manes:2019fyw  relevant to our discussion.

##  2 Equilibrium partition function and covariant currents with background
torsion

We begin with the discussion of the dynamics of massless Dirac fermions
propagating on a spacetime with torsion222The basics of geometric torsion, as
well as the notation used in the following, are summarized in Appendix A.. The
action of a massless Dirac spinor minimally coupled to gravity can be written
as Hehl:1976kj ; Ortin:2015hya ; Freedman:2012zz

$\displaystyle S={1\over 2}\int
d^{4}x\,(\det{e})\Big{(}\overline{\psi}\overrightarrow{\nabla\\!\\!\\!\\!/\,}\psi-\overline{\psi}\overleftarrow{\nabla\\!\\!\\!\\!/\,}\psi\Big{)},$
(1)

where the left and right covariant derivatives inside the integral are defined
respectively by

$\displaystyle\overrightarrow{\nabla\\!\\!\\!\\!/\,}\psi$
$\displaystyle=\gamma^{A}e_{A}^{\,\,\,\mu}\partial_{\mu}\psi+{1\over
4}\gamma^{A}\gamma^{[B}\gamma^{C]}\omega_{BCA}\psi,$
$\displaystyle\overline{\psi}\overleftarrow{\nabla\\!\\!\\!\\!/\,}$
$\displaystyle=e_{A}^{\,\,\,\mu}\partial_{\mu}\overline{\psi}\gamma^{A}-{1\over
4}\overline{\psi}\gamma^{[B}\gamma^{C]}\gamma^{A}\omega_{BCA},$ (2)

and the Dirac matrices verify the Minkowskian Clifford algebra
$\\{\gamma^{A},\gamma^{B}\\}=2\eta^{AB}\mathbb{1}$. Writing the full spin
connection in terms of the auxiliary torsionless Levi-Civita connection and
the contorsion tensor as
$\omega^{A}_{\,\,\,BC}=\overline{\omega}^{A}_{\,\,\,BC}+\kappa^{A}_{\,\,\,BC}$,
we get the following expression of the left covariant derivative in terms of
its Levi-Civita counterpart

$\displaystyle\overrightarrow{\nabla\\!\\!\\!\\!/\,}\psi$
$\displaystyle=\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi+{1\over
4}\gamma^{C}\gamma^{[A}\gamma^{B]}\kappa_{ABC}\psi$
$\displaystyle=\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi+{1\over
4}\Big{(}\gamma^{B}\eta^{AC}-\gamma^{A}\eta^{BC}+i\epsilon^{ABCD}\gamma_{D}\gamma_{5}\Big{)}\kappa_{ABC}\psi,$
(3)

where we indicate by a bar all geometric quantities referred to the Levi-
Civita connection and have used the gamma matrices identity

$\displaystyle\gamma^{A}\gamma^{[B}\gamma^{C]}=\gamma^{C}\eta^{AB}-\gamma^{B}\eta^{AC}+i\epsilon^{ABCD}\gamma_{D}\gamma_{5}.$
(4)

A similar calculation for the right covariant derivative in Eq. (2) gives

$\displaystyle\overline{\psi}\overleftarrow{\nabla\\!\\!\\!\\!/\,}$
$\displaystyle=\overline{\psi}\overleftarrow{\overline{\nabla}\\!\\!\\!\\!/\,}-{1\over
4}\overline{\psi}\gamma^{[A}\gamma^{B]}\gamma^{C}\kappa_{ACB}$
$\displaystyle=\overline{\psi}\overleftarrow{\overline{\nabla}\\!\\!\\!\\!/\,}-{1\over
4}\overline{\psi}\Big{(}\gamma^{A}\eta^{BC}-\gamma^{B}\eta^{AC}+i\epsilon^{ABCD}\gamma_{D}\gamma_{5}\Big{)}\kappa_{ABC}.$
(5)

Plugging these results into the action (1), we arrive at the expression

$\displaystyle S$ $\displaystyle={1\over 2}\int
d^{4}x\,(\det{e})\left[\overline{\psi}\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi+{1\over
4}\overline{\psi}\Big{(}\gamma^{B}\eta^{AC}-\gamma^{A}\eta^{BC}+i\epsilon^{ABCD}\gamma_{D}\gamma_{5}\Big{)}\psi\kappa_{ABC}\right.$
$\displaystyle\left.-\overline{\psi}\overleftarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi+{1\over
4}\overline{\psi}\Big{(}\gamma^{A}\eta^{BC}-\gamma^{B}\eta^{AC}+i\epsilon^{ABCD}\gamma_{D}\gamma_{5}\Big{)}\psi\kappa_{ABC}\right].$
(6)

The important point here is that the term proportional to
$(\gamma^{B}\eta^{AC}-\gamma^{A}\eta^{BC})\kappa_{ABC}$, which contains the
symmetric components of the contorsion in the two last indices, cancels out.
This means that fermions only couple to its antisymmetric piece,
$\kappa^{A}_{\,\,\,\,[BC]}$, which as shown in Appendix A [see Eq. (97)] is
given by the components of the torsion tensor

$\displaystyle S$ $\displaystyle={1\over 2}\int
d^{4}x\,(\det{e})\left(\overline{\psi}\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi-\overline{\psi}\overleftarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi+{i\over
4}\overline{\psi}\gamma^{D}\gamma_{5}\psi\epsilon_{DA}^{\,\,\,\,\,\,\,\,\,BC}T^{A}_{\,\,\,\,\,BC}\right).$
(7)

This form of the action suggests the introduction of the effective vector
field

$\displaystyle\mathcal{S}_{A}=-{1\over
8}\epsilon_{AB}^{\,\,\,\,\,\,\,\,\,\,CD}T^{B}_{\,\,\,\,\,\,CD},$ (8)

to write

$\displaystyle S$ $\displaystyle={1\over 2}\int
d^{4}x\,(\det{e})\left(\overline{\psi}\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi-\overline{\psi}\overleftarrow{\overline{\nabla}\\!\\!\\!\\!/\,}\psi-2i\overline{\psi}\gamma^{A}\gamma_{5}\psi\mathcal{S}_{A}\right).$
(9)

Thus, the whole effect of background torsion on the dynamics of the fermion is
codified through its axial-vector coupling to an external effective gauge
field, which, following Ref. Imaki:2020csc , we call screw torsion. The
minimal coupling to gravity selects only one among all possible dimension-four
operators coupling Dirac fermions to torsion Kostelecky:2007kx .

Before proceeding any further, some clarification on the action (1) is in
order. At face value, the theory it describes seems to be equivalent to that
of a Dirac fermion axially coupled to an external gauge field. The crucial
difference, however, is that this external gauge field $\mathcal{S}$ is a
“composite” expressed as the Hodge dual of the antisymmetric part of the
torsion components, which is the “fundamental” external source. This is
important, because once expressed in a coordinate basis the components of this
gauge field depend not only on the torsion, but on the metric tensor as well.
As a result, the energy-momentum tensor and the spin energy potential of the
theory include contributions that would be absent in the theory of a
“fundamental” gauge field axially coupled to a Dirac fermion (see Sec. 3).
These new terms are associated with novel transport coefficients in the
constitutive relations for the corresponding energy-momentum and spin
covariant currents.

In addition to the couplings shown in Eq. (1), the authors of Ref.
Imaki:2020csc  considered an additional coupling of the Dirac fermion to an
external Abelian vector gauge field. This so-called edge torsion vector field
is proportional to the torsion vector $T^{B}_{\,\,\,\,BA}$ and mixes for many
practical purposes with the electromagnetic field. In what follows we stick to
the minimal coupling prescription (1) and only consider the coupling to the
screw torsion, in the understanding that in all our results the edge torsion
would be reabsorbed by a shift in the physical electromagnetic field.

### Remarks on gauge invariance.

We have seen how the action of a Dirac fermion minimally coupled to gravity
only depends on the fully antisymmetric components of the torsion tensor
$\mathcal{T}_{ABC}\equiv T_{[ABC]}$, which can be used to define the three-
form333Unlike in Refs. Manes:2018llx ; Manes:2019fyw , here no $-i$ is
factored out of the components of differential forms.

$\displaystyle\mathcal{T}={1\over 3!}\mathcal{T}_{ABC}e^{A}e^{B}e^{C}.$ (10)

Its four independent components are encoded in the screw-torsion according to
Eq. (8), which can be written using the Hodge star operator as

$\displaystyle\mathcal{S}_{\mu}=-{1\over
8}\epsilon_{\mu\nu}^{\,\,\,\,\,\,\,\alpha\beta}\mathcal{T}^{\nu}_{\,\,\,\,\,\alpha\beta}\hskip
28.45274pt\Longrightarrow\hskip 28.45274pt\mathcal{S}={3\over
4}\star\mathcal{T}.$ (11)

The field strength of the Abelian screw torsion,
$\mathcal{F}_{\mathcal{S}}=d\mathcal{S}$, can be written then as

$\displaystyle\mathcal{F}_{\mathcal{S}}$ $\displaystyle={3\over
4}d\star\mathcal{T}=-{3\over 4}\star\delta\mathcal{T},$ (12)

where $\delta\equiv-\star d\star$ denotes the codifferential acting on a
three-form. In components, this equation reads

$\displaystyle\mathcal{S}_{\mu\nu}=-{3\over
8}\overline{\nabla}_{\sigma}\mathcal{T}^{\sigma}_{\,\,\,\,\,\alpha\beta}\epsilon^{\alpha\beta}_{\,\,\,\,\,\,\,\mu\nu}.$
(13)

Looking at Eq. (11) above, we see that the gauge variation of the screw
torsion vector field $\mathcal{S}$ by an exact one-form,
$\mathcal{S}\rightarrow\mathcal{S}+d\alpha$, corresponds to the following
transformation of the torsion three-form $\mathcal{T}$

$\displaystyle\mathcal{T}\longrightarrow\mathcal{T}+\delta\beta,$ (14)

with $\beta\sim\star\alpha$ a four-form. The nihilpotency of the
codifferential, $\delta^{2}=0$, guarantees the gauge invariance of the screw
torsion field strength (12).

### The equilibrium partition function.

In the context of hydrodynamics, the coupling of the background torsion to the
microscopic fermionic degrees of freedom gives the prescription for the
construction of the effective functional describing the long-range excitations
of a fluid Banerjee:2012iz ; Jensen:2012jh . Here we are going to employ the
differential geometry methods introduced in Manes:2018llx ; Manes:2019fyw  to
build the equilibrium partition function for fluids with torsion. In the
following, we study the case of a fluid coupled to an external Abelian vector
source in the presence of background torsion on a generic static background
geometry. After this, the more general case of a two flavor hadronic
(super)fluid will be analyzed in Section 4.

Besides its coupling to torsion through $\mathcal{S}$, we also assume that the
microscopic fermionic degrees of freedom are coupled to an external vector
Abelian gauge field $\mathcal{V}$, which remains anomaly-free and will be
eventually associated to the electromagnetic field. In four dimensions, the
(nonlocal) anomalous part of the effective action can be computed in terms of
the Chern-Simons form (see, for example, Manes:2018llx ). Keeping only terms
linear in the torsion, we have

$\displaystyle\widetilde{\omega}^{0}_{5}(\mathcal{S},\mathcal{F}_{V},\mathcal{F}_{\mathcal{S}})$
$\displaystyle=6\mathcal{S}\mathcal{F}_{V}^{2},$ (15)

where $\mathcal{F}_{V}=d\mathcal{V}$ is the vector field strength. This gives
the Bardeen form of the anomaly, which explicitly preserves vector gauge
transformations. The properly normalized Chern-Simons nonlocal effective
action encoding the linear effects of torsion is then given by

$\displaystyle\Gamma[\mathcal{V},\mathcal{S}]_{\rm CS}={1\over
4\pi^{2}}\int\limits_{D_{5}}\mathcal{S}\mathcal{F}_{V}^{2},$ (16)

where $D_{5}$ is a five-dimensional manifold whose boundary is identified with
the Euclidean four-dimensional physical spacetime. To compute the equilibrium
partition function from the Chern-Simons effective action, we take the metric
of the four-dimensional spacetime $\partial D_{5}$ to be the generic static
line element

$\displaystyle
ds^{2}=-e^{2\sigma(\mathbf{x})}\Big{[}dx^{0}+a_{i}(\mathbf{x})dx^{i}\Big{]}^{2}+g_{ij}(\mathbf{x})dx^{i}dx^{j},$
(17)

and take all fields to be independent of $x^{0}$. We implement dimensional
reduction onto the compatified Euclidean time by setting $D_{5}=S^{1}\times
D_{4}$, where the length of the $S^{1}$ equals the inverse of the equilibrium
temperature $T_{0}$. Vector fields are then written in terms of components
that remain invariant under KK transformations Manes:2019fyw , acting
according to $x^{0}\rightarrow x^{0}+\phi$ and $a_{i}\rightarrow
a_{i}-\partial_{i}\phi$,

$\displaystyle\mathcal{V}$
$\displaystyle\equiv\mathcal{V}_{\mu}dx^{\mu}=\bm{V}-e^{-\sigma}\mathcal{V}_{0}u,$
$\displaystyle\mathcal{S}$
$\displaystyle\equiv\mathcal{S}_{\mu}dx^{\mu}=\bm{S}-e^{-\sigma}\mathcal{S}_{0}u,$
(18)

where we have introduced the four-velocity one-form $u$ given by

$\displaystyle
u=-e^{\sigma}\big{(}dx^{0}+a_{i}dx^{i}\big{)}\equiv-e^{\sigma}\big{(}dx^{0}+a\big{)},$
(19)

and the KK-invariant spatial one-forms are defined by

$\displaystyle\bm{V}$
$\displaystyle=\Big{(}\mathcal{V}_{i}-\mathcal{V}_{0}a_{i}\Big{)}dx^{i}\equiv
V_{i}dx^{i},$ $\displaystyle\bm{S}$
$\displaystyle=\Big{(}\mathcal{S}_{i}-\mathcal{S}_{0}a_{i}\Big{)}dx^{i}\equiv
S_{i}dx^{i}.$ (20)

A similar electric-magnetic decomposition can be written for the vector field
strength

$\displaystyle\mathcal{F}_{V}$ $\displaystyle\equiv\bm{B}+u\bm{E}$
$\displaystyle=d\bm{V}-d\big{(}e^{-\sigma}u\big{)}\mathcal{V}_{0}+ue^{-\sigma}d\mathcal{V}_{0}$
(21)
$\displaystyle=\Big{(}\bm{F}_{V}+\mathcal{V}_{0}da\Big{)}+ue^{-\sigma}d\mathcal{V}_{0},$

where $\bm{F}_{V}\equiv d\bm{V}$ and we have used that $d(e^{-\sigma}u)=-da$.
The electric $\bm{E}$ and magnetic $\bm{B}$ components of the field strength
will be later identified with the electric and magnetic fields [cf. (36)]. The
equivalent expression for the field strength associated to the screw torsion
reads

$\displaystyle\mathcal{F}_{S}$ $\displaystyle\equiv\bm{B}_{S}+u\bm{E}_{S}$
$\displaystyle=\Big{(}\bm{F}_{S}+\mathcal{S}_{0}da\Big{)}+ue^{-\sigma}d\mathcal{S}_{0},$
(22)

with $\bm{F}_{S}=d\bm{S}$.

Finally, we implement the dimensional reduction on the three-form (10) as well
by decomposing it as

$\displaystyle\mathcal{T}=\mathcal{T}_{B}+u\mathcal{T}_{E}.$ (23)

In a coordinate basis, the electric and magnetic components are respectively
given by

$\displaystyle\mathcal{T}_{E}$ $\displaystyle={1\over
3!}e^{-\sigma}\Big{[}2g_{i\ell}T^{\ell}_{\,\,\,0j}+e^{2\sigma}\Big{(}T^{0}_{\,\,\,ij}-2a_{i}T^{0}_{\,\,\,0j}\Big{)}+e^{2\sigma}a_{\ell}\Big{(}T^{\ell}_{\,\,\,ij}-2a_{i}T^{\ell}_{\,\,\,0j}\Big{)}\Big{]}dx^{j}dx^{k},$
$\displaystyle\mathcal{T}_{B}$ $\displaystyle={1\over
3!}g_{j\ell}\Big{(}T^{\ell}_{\,\,\,\,kn}-2a_{k}T^{\ell}_{\,\,\,\,0n}\Big{)}dx^{j}dx^{k}dx^{n}.$
(24)

Being a four-form, the gauge function $\beta$ in Eq. (14) does not have any
magnetic component, $\beta=u\beta_{E}$. As a consequence, only the electric
part of $\mathcal{T}$ transforms under (14)

$\displaystyle\mathcal{T}_{E}$
$\displaystyle\longrightarrow\mathcal{T}_{E}+\delta_{\perp}\beta_{E},$
$\displaystyle\mathcal{T}_{B}$ $\displaystyle\longrightarrow\mathcal{T}_{B},$
(25)

where $\delta_{\perp}=*d*$, with $*$ the three-dimensional Hodge dual (not to
be confused with its four-dimensional couterpart denoted by $\star$). Since
under four-dimensional Hodge duality the electric and magnetic components
interchange

$\displaystyle\star\mathcal{T}=-*\mathcal{T}_{E}-u*\mathcal{T}_{B},$ (26)

we find from Eqs. (11) and (18)

$\displaystyle\mathcal{S}_{0}$ $\displaystyle={3\over
4}e^{\sigma}*\mathcal{T}_{B},$ $\displaystyle\bm{S}$ $\displaystyle=-{3\over
4}*\mathcal{T}_{E}.$ (27)

We see that the gauge invariance of $\mathcal{T}_{B}$ implies the same
property for $\mathcal{S}_{0}$, whereas $\bm{S}$ undergoes the standard gauge
transformation generated by the zero-form $*\beta_{E}$. Using in addition Eq.
(24), we can write the screw torsion field in terms of the components of the
torsion tensor as

$\displaystyle\mathcal{S}_{0}$ $\displaystyle={1\over
8}e^{\sigma}\epsilon^{ijk}g_{i\ell}\Big{(}T^{\ell}_{\,\,\,\,jk}-2a_{j}T^{\ell}_{\,\,\,\,0k}\Big{)},$
(28) $\displaystyle\bm{S}$ $\displaystyle=-{1\over
8}e^{-\sigma}\epsilon^{ijk}g_{mi}\Big{[}2g_{\ell
j}T^{\ell}_{\,\,\,0k}+e^{2\sigma}a_{\ell}\Big{(}T^{\ell}_{\,\,\,jk}-2a_{j}T^{\ell}_{\,\,\,0k}\Big{)}+e^{2\sigma}\Big{(}T^{0}_{\,\,\,jk}-2a_{j}T^{0}_{\,\,\,0k}\Big{)}\Big{]}dx^{m}.$

This dependence of the screw torsion gauge field on the metric and torsion
components is what distinguishes our theory from that of a Dirac fermion
coupled to an external gauge field through the axial-vector current. As
already pointed out, this has important consequences for the constitutive
relations of the energy-momentum and spin currents.

Having arrived at this parametrization of the effective screw torsion, we
proceed to compute the terms in the effective action induced by the ’t Hooft
anomaly affecting the gauge invariance (25) and coming from triangle
diagrams444As stated above, in this work we do not consider the sector
associated with the Nieh-Yan anomaly.. As shown in Jensen:2013kka ;
Manes:2018llx , the dimensionally-reduced effective action splits into a local
anomalous and a nonlocal invariant piece, respectively given by Manes:2018llx

$\displaystyle W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{V},\bm{S},da]_{\rm anom}$
$\displaystyle={1\over
4\pi^{2}T_{0}}\int\limits_{S^{3}}\Big{(}2\mathcal{V}_{0}\bm{F}_{V}+da\mathcal{V}_{0}^{2}\Big{)}\bm{S},$
(29)

and

$\displaystyle W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{V},\bm{S},da]_{\rm inv}$
$\displaystyle={1\over
4\pi^{2}T_{0}}\int\limits_{D_{4}}\Big{[}\mathcal{S}_{0}\bm{F}_{V}^{2}+2\mathcal{V}_{0}\bm{F}_{V}\bm{F}_{S}$
(30)
$\displaystyle+da\mathcal{V}_{0}\Big{(}\mathcal{V}_{0}\bm{F}_{S}+2\mathcal{S}_{0}\bm{F}_{V}\Big{)}+(da)^{2}\mathcal{S}_{0}\mathcal{V}_{0}^{2}\Big{]}.$

In the second expression, we have introduced the components of the field
strength associated with $\bm{S}$

$\displaystyle\bm{F}_{S}$ $\displaystyle=d\bm{S}\equiv{1\over
2}S_{ij}dx^{i}dx^{j}.$ (31)

The covariant currents can be now computed from the invariant part of the
partition function Jensen:2013kka ; Manes:2018llx ; Manes:2019fyw . We begin
with the one associated with the vector current

$\displaystyle\langle\bm{J}_{V}\rangle_{\rm cov}$
$\displaystyle=T_{0}{\delta\over\delta\bm{F}_{V}}W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{V},\bm{S}]_{\rm
inv}$ $\displaystyle={1\over
2\pi^{2}}\Big{[}\mathcal{S}_{0}\Big{(}\bm{F}_{V}+da\mathcal{V}_{0}\Big{)}+\mathcal{V}_{0}\bm{F}_{S}\Big{]},$
(32)

whereas for the one corresponding to the torsional axial-vector gauge field,
the result is

$\displaystyle\langle\bm{J}_{S}\rangle_{\rm cov}$
$\displaystyle=T_{0}{\delta\over\delta\bm{F}_{S}}W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{V},\bm{S}]_{\rm
inv}$ $\displaystyle={1\over 2\pi^{2}}\left(\mathcal{V}_{0}\bm{F}_{V}+{1\over
2}da\mathcal{V}_{0}^{2}\right).$ (33)

In both cases, the zero components vanish,
$\langle\mathcal{J}_{V0}\rangle_{\rm cov}=\langle\mathcal{J}_{S0}\rangle_{\rm
cov}=0$. We observe that, unlike the vector current,
$\langle\bm{J}_{S}\rangle_{\rm cov}$ does not pick any linear dependence on
the torsion. These covariant currents will be very relevant in the following.

In addition, the corresponding Bardeen-Zumino (BZ) currents are given by
Manes:2018llx

$\displaystyle\langle\mathcal{J}_{V}\rangle_{\rm BZ}$ $\displaystyle={1\over
2\pi^{2}}\mathcal{S}\mathcal{F}_{V},$
$\displaystyle\langle\mathcal{J}_{S}\rangle_{\rm BZ}$ $\displaystyle={1\over
6\pi^{2}}\mathcal{S}\mathcal{F}_{S}.$ (34)

The second equation is quadratic in the screw torsion field, so that the BZ
current does not have any correction at linear order in the torsion. For the
BZ vector current, on the other hand, after dimensional reduction we find

$\displaystyle\langle\mathcal{J}_{V0}\rangle_{\rm BZ}$ $\displaystyle={1\over
2\pi^{2}}\Big{(}\bm{F}_{V}+\mathcal{V}_{0}da\Big{)}\bm{S},$
$\displaystyle\langle\bm{J}_{V}\rangle_{\rm BZ}$ $\displaystyle={1\over
2\pi^{2}}\Big{[}\mathcal{S}_{0}\Big{(}\bm{F}_{V}+\mathcal{V}_{0}da\Big{)}+\bm{S}d\mathcal{V}_{0}\Big{]}.$
(35)

Notice that only the magnetic part of the vector field strength enters in
these expressions [see Eq. (21)].

To connect with physics, we need to identify the electric and magnetic fields.
These are naturally given by the electric and magnetic parts in the
decomposition (21) of the vector field strength. Thus, we have555Our
definition of the physical magnetic field follows Jensen:2013kka  and differs
from the one used in Manes:2018llx ; Manes:2019fyw . Expressions in these two
references can be adapted to our definition by the replacement
$\mathbb{B}^{i}\rightarrow\mathbb{B}^{i}+2\mu\omega^{i}$.

$\displaystyle e\mathbb{E}_{i}$
$\displaystyle=e^{-\sigma}\partial_{i}\mathcal{V}_{0},$ $\displaystyle
e\mathbb{B}^{i}$
$\displaystyle=\epsilon^{ijk}\Big{(}\partial_{j}V_{k}+\mathcal{V}_{0}\partial_{j}a_{k}\Big{)},$
(36)

with $e$ the elementary electric charge. We also define the screw magnetic
field as the three-dimensional Hodge dual of the magnetic component of
$\bm{B}_{S}$ in (22)

$\displaystyle
B^{i}_{S}=\epsilon^{ijk}\Big{(}\partial_{j}S_{k}+\mathcal{S}_{0}\partial_{j}a_{k}\Big{)}.$
(37)

In addition, we introduce the vorticity vector

$\displaystyle\omega^{i}$ $\displaystyle=-{1\over
2}e^{\sigma}\epsilon^{ijk}\partial_{j}a_{k},$ (38)

and the electromagnetic and torsional chemical potentials

$\displaystyle\mu$ $\displaystyle\equiv e^{-\sigma}\mathcal{V}_{0},$
$\displaystyle\mu_{S}$ $\displaystyle\equiv e^{-\sigma}\mathcal{S}_{0}.$ (39)

With all these definitions in hand, we take the Hodge duals of the covariant
currents (32) and (33) to write the electromagnetic and axial-vector covariant
current components

$\displaystyle\langle J^{i}_{\rm em}\rangle_{\rm cov}$ $\displaystyle={1\over
2\pi^{2}}\Big{(}\mu_{S}e\mathbb{B}^{i}+\mu
B_{S}^{i}+2\mu\mu_{S}\omega^{i}\Big{)},$ $\displaystyle\langle
J^{i}_{5}\rangle_{\rm cov}$ $\displaystyle={\mu\over
2\pi^{2}}\Big{(}e\mathbb{B}^{i}+\mu\omega^{i}\Big{)}.$ (40)

As argued in Jensen:2013kka ; Haehl:2013hoa ; Manes:2018llx ; Manes:2019fyw ,
covariant currents are the ones relevant for the analysis of transport. Thus,
the conclusion to be extracted from our result is the existence of torsional
chiral magnetic and vortical effects, both mediated by the torsional chemical
potentials, together with a nondissipative transport of charge driven by the
screw torsion magnetic field (37). The right-hand side of the second
expression, on the other hand, gives the standard result for the axial-vector
current of an Abelian anomalous fluid, corresponding to the magnetic and
vortical separation effect without any torsional corrections at linear order.
Notice that here we have identified the current $J^{i}_{S}$ with the axial-
vector current, $J^{i}_{5}$.

We also give the components of the vector BZ currents by taking the dual of
the expressions in Eq. (35) and recasting them in term of electric and
magnetic fields

$\displaystyle\langle J_{{\rm em}\,0}\rangle_{\rm BZ}$ $\displaystyle={e\over
2\pi^{2}}e^{\sigma}\mathbb{B}^{i}S_{i},$ $\displaystyle\langle J_{\rm
em}^{i}\rangle_{\rm BZ}$ $\displaystyle={e\over
2\pi^{2}}\Big{(}\mu_{S}\mathbb{B}^{i}+\epsilon^{ijk}S_{j}\mathbb{E}_{k}\Big{)}.$
(41)

This BZ current was identified in Ref. Imaki:2020csc  as the electromagnetic
current associated with charge transport in flat spacetime. It differs however
from the covariant current computed above by the consistent current

$\displaystyle\langle J_{\rm em}^{i}\rangle_{\rm cons}\equiv\langle J_{{\rm
em}}^{i}\rangle_{\rm cov}-\langle J_{{\rm em}}^{i}\rangle_{\rm BZ}={1\over
2\pi^{2}}\partial_{j}\big{(}\mathcal{V}_{0}\epsilon^{ijk}S_{k}\big{)},$ (42)

whose divergence vanish identically due to the presence of the Levi-Civita
tensor.

##  3 The spin energy potential and the energy-momentum tensor

The knowledge of the covariant current associated to the screw torsion allows
the calculation of torsional contributions to other covariant physical
quantities, most notably the spin energy potential and the energy-momentum
tensor. These are obtained by taking variations of the invariant part of the
partition function (2) and isolating its local contribution Jensen:2013kka ;
Manes:2018llx ; Manes:2019fyw . In doing this, we have to keep in mind that
the screw torsion is a composite field that depends not only on the torsion
tensor, but on the metric components as well. Thus, in computing variations
with respect to the metric functions we should consider, in addition to the
explicit dependence of the invariant action on these sources, also the
implicit dependence in the screw torsion. This latter contribution can be
generically written as

$\displaystyle\delta
W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{F}_{V},\bm{F}_{S}]_{\rm inv}$
$\displaystyle=\int\limits_{D_{4}}\left[{\delta W_{\rm
inv}\over\delta\bm{F}_{S}}d\delta\bm{S}+{\delta W_{\rm
inv}\over\delta\mathcal{S}_{0}}\delta\mathcal{S}_{0}\right]$
$\displaystyle=\int\limits_{S^{3}}{\delta W_{\rm
inv}\over\delta\bm{F}_{S}}\delta\bm{S}+\int\limits_{D_{4}}\left[-d\left({\delta
W_{\rm inv}\over\delta\bm{F}_{S}}\right)\delta\bm{S}+{\delta W_{\rm
inv}\over\delta\mathcal{S}_{0}}\delta\mathcal{S}_{0}\right].$ (43)

Using the definition of the covariant current in Eq. (33), we write this
variation as

$\displaystyle\delta
W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{F}_{V},\bm{F}_{S}]_{\rm inv}={1\over
T_{0}}\int\limits_{S^{3}}\langle\bm{J}_{S}\rangle_{\rm
cov}\delta\bm{S}+\mbox{bulk terms}.$ (44)

The previous expression can be made more explicit by writing
$\langle\bm{J}_{S}\rangle_{\rm cov}$ in terms of its dual one-form current,
defined as

$\displaystyle{1\over T_{0}}\langle\bm{J}_{S}\rangle_{\rm cov}$
$\displaystyle={1\over 2!}\epsilon^{ijk}g_{j\ell}g_{km}\langle
J_{i}\rangle_{S,\rm cov}dx^{\ell}\wedge dx^{m}.$ (45)

With this, Eq. (44) takes the form

$\displaystyle\delta
W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{F}_{V},\bm{F}_{S}]_{\rm inv}$
$\displaystyle=\int\limits_{S^{3}}d^{3}x\sqrt{g}\,g^{ij}\langle
J_{i}\rangle_{S,\rm cov}\delta S_{j}.$ (46)

This expression will be utilized next to evaluate the torsional covariant
contributions to the spin energy potential and the energy-momentum tensor.

### The spin energy potential.

The covariant spin energy potential is defined as Hehl:1976kj

$\displaystyle\delta W_{\rm inv}$
$\displaystyle=2\int\limits_{S^{3}}d^{3}x\sqrt{g}\langle\Psi_{\mu}^{\,\,\,\,\nu\sigma}\rangle_{\rm
cov}\delta T^{\mu}_{\,\,\,\nu\sigma}.$ (47)

All dependence of the invariant effective action on the torsion tensor is
included in the screw torsion field, so we only have to care about the
implicit dependence. Using Eq. (28), we write

$\displaystyle\delta S_{m}$ $\displaystyle=-{1\over
8}e^{\sigma}\epsilon^{ijk}g_{mi}a_{\ell}\delta T^{\ell}_{\,\,\,jk}-{1\over
4}e^{-\sigma}\epsilon^{ijk}\Big{(}g_{\ell
j}-e^{2\sigma}a_{\ell}a_{j}\Big{)}\delta T^{\ell}_{\,\,\,0k}$
$\displaystyle-{1\over 8}e^{\sigma}\epsilon^{ijk}g_{mi}\delta
T^{0}_{\,\,\,jk}+{1\over 4}e^{\sigma}\epsilon^{ijk}g_{mi}a_{j}\delta
T^{0}_{\,\,\,0k},$ (48)

so the left-hand side of (47) can be computed using (46), to give

$\displaystyle\delta W_{\rm inv}$ $\displaystyle=-{1\over
4}\int\limits_{S^{3}}d^{3}x\,\sqrt{g}e^{\sigma}\epsilon^{ijk}a_{\ell}\langle
J_{i}\rangle_{S,\rm cov}\delta T^{\ell}_{\,\,\,jk}$ $\displaystyle-{1\over
4}\int\limits_{S^{3}}d^{3}x\,\sqrt{g}e^{-\sigma}\epsilon^{ijk}\Big{(}g_{\ell
j}-e^{2\sigma}a_{\ell}a_{j}\Big{)}\langle J_{i}\rangle_{S,\rm
cov}\big{(}\delta T^{\ell}_{\,\,\,0k}-\delta T^{\ell}_{\,\,\,k0}\big{)}$ (49)
$\displaystyle-{1\over
4}\int\limits_{S^{3}}d^{3}x\,\sqrt{g}e^{\sigma}\epsilon^{ijk}\langle
J_{i}\rangle_{S,\rm cov}\delta T^{0}_{\,\,\,jk}+{1\over
4}\int\limits_{S^{3}}d^{3}x\,\sqrt{g}e^{\sigma}\epsilon^{ijk}a_{j}\langle
J_{i}\rangle_{S,\rm cov}\Big{(}\delta T^{0}_{\,\,\,0k}-\delta
T^{0}_{\,\,\,k0}\Big{)},$

plus bulk terms that we omit. This allows the identification of the covariant
components of the spin energy potential in terms of the covariant screw-
torsion current as

$\displaystyle\langle\Psi_{\ell}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle=-{1\over 4}e^{\sigma}\epsilon^{ijk}a_{\ell}\langle
J_{i}\rangle_{S,\rm cov},$
$\displaystyle\langle\Psi_{\ell}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle=-{1\over 4}e^{-\sigma}\epsilon^{ijk}\Big{(}g_{\ell
j}-e^{2\sigma}a_{\ell}a_{j}\Big{)}\langle J_{i}\rangle_{S,\rm cov},$
$\displaystyle\langle\Psi_{0}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle=-{1\over 4}e^{\sigma}\epsilon^{ijk}\langle J_{i}\rangle_{S,\rm
cov},$ (50) $\displaystyle\langle\Psi_{0}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle={1\over 4}e^{\sigma}\epsilon^{ijk}a_{j}\langle
J_{i}\rangle_{S,\rm cov}.$

Notice, however, that only one of these quantities is KK-invariant. We
therefore define the physical KK-invariant combinations

$\displaystyle\langle\bm{\Psi}_{\ell}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle\equiv\langle\Psi_{\ell}^{\,\,\,jk}\rangle_{\rm
cov}-a_{\ell}\langle\Psi_{0}^{\,\,\,jk}\rangle_{\rm cov},$
$\displaystyle\langle\bm{\Psi}_{\ell}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle\equiv\langle\Psi_{\ell}^{\,\,\,0k}\rangle_{\rm
cov}-a_{\ell}\langle\Psi_{0}^{\,\,\,0k}\rangle_{\rm
cov}+a_{j}\Big{(}\langle\Psi_{\ell}^{\,\,\,jk}\rangle_{\rm
cov}-a_{\ell}\langle\Psi_{0}^{\,\,\,jk}\rangle_{\rm cov}\Big{)},$
$\displaystyle\langle\bm{\Psi}_{0}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle\equiv\langle\Psi_{0}^{\,\,\,jk}\rangle_{\rm cov},$ (51)
$\displaystyle\langle\bm{\Psi}_{0}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle\equiv\langle\Psi_{0}^{\,\,\,0k}\rangle_{\rm
cov}+a_{j}\langle\Psi_{0}^{\,\,\,jk}\rangle_{\rm cov},$

which are found to be

$\displaystyle\langle\bm{\Psi}_{\ell}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle=0,$
$\displaystyle\langle\bm{\Psi}_{\ell}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle=-{1\over 4}e^{-\sigma}\epsilon^{ijk}g_{\ell j}\langle
J_{i}\rangle_{S,\rm cov},$
$\displaystyle\langle\bm{\Psi}_{0}^{\,\,\,jk}\rangle_{\rm cov}$
$\displaystyle=-{1\over 4}e^{\sigma}\epsilon^{ijk}\langle J_{i}\rangle_{S,\rm
cov},$ (52) $\displaystyle\langle\bm{\Psi}_{0}^{\,\,\,0k}\rangle_{\rm cov}$
$\displaystyle=0.$

This shows that all dependence of the spin energy potentials on the torsion
comes through the components of the covariant current associated to the screw
torsion. Thus, using Eq. (33), we conclude that there are no linear torsional
corrections to the spin energy potential. Notice that this quantity is nonzero
even for backgrounds with vanishing torsion666Other contributions to the spin
current in the absence of torsion have been recently computed in
Gallegos:2021bzp ..

### Torsional contributions to the energy-momentum tensor.

To evaluate the components of the (covariant) energy-momentum tensor, we take
variations with respect to the metric functions $\sigma$, $a_{i}$ and $g_{ij}$
parametrizing the static line element (17). In doing this, it is very
important to keep track of all dependence of the effective action on the
metric. Concerning the torsion, the definition of the $T^{\mu}$ two-form
indicates that the components $T^{\mu}_{\,\,\,\,\nu\sigma}$ are metric-
independent Hehl:1976kj ; Ortin:2015hya ; Freedman:2012zz ; Valle:2015hfa .
The same can be said of the contorsion tensor
$\kappa^{\mu}_{\,\,\,[\alpha\beta]}$, while its symmetric part
$\kappa^{\mu}_{\,\,\,(\alpha\beta)}$ depends on the metric since its
computation requires raising and lowering indices of the metric-independent
torsion components [cf. (98)]. Another important point to take into account is
that $W_{\rm inv}$ explicitly depends on the KK vector $a_{i}$, whereas the
dependence on the other two functions comes exclusively through the screw
torsion.

We begin varying the invariant effective action with respects to $\sigma$,
which gives the components $\langle\Theta_{00}\rangle_{\rm cov}$ of the
energy-momentum tensor as [cf. (46)]

$\displaystyle-\int\limits_{S^{3}}d^{3}x\sqrt{g}\,e^{-2\sigma}\langle\Theta_{00}\rangle_{\rm
cov}=\int\limits_{S^{3}}d^{3}x\sqrt{g}\,g^{ij}\langle J_{i}\rangle_{S,\rm
cov}\delta_{\sigma}S_{j}.$ (53)

Using (28), the variation of the screw torsion with respect to $\sigma$
appearing on the right-hand side is given by

$\displaystyle\delta_{\sigma}S_{m}$ $\displaystyle=\left(S_{m}+{1\over
2}e^{-\sigma}\epsilon^{ijk}g_{im}g_{j\ell}T^{\ell}_{\,\,\,0k}\right)\delta\sigma,$
(54)

so we arrive at the result

$\displaystyle\langle\Theta_{00}\rangle_{\rm cov}$
$\displaystyle=-e^{2\sigma}\left(S^{i}+{1\over
2}e^{-\sigma}\epsilon^{ijk}g_{j\ell}T^{\ell}_{\,\,\,0k}\right)\langle
J_{i}\rangle_{S,\rm cov}.$ (55)

The mixed components $\langle\Theta_{0}^{\,\,\,i}\rangle_{\rm cov}$ are
evaluated next. As pointed out above, in this case we have also a contribution
coming from the explicit dependence of the effective action on $a_{i}$. This
can be computed from the invariant part of the effective action using
Manes:2018llx ; Manes:2019fyw

$\displaystyle\langle\bm{\Theta}\rangle_{\rm cov}^{\rm
expl}=T_{0}\left[{\delta\over\delta(da)}-\mathcal{V}_{0}{\delta\over\delta\bm{F}_{V}}-\mathcal{S}_{0}{\delta\over\delta\bm{F}_{S}}\right]W[\mathcal{V}_{0},\mathcal{S}_{0},\bm{V},\bm{S},da]_{\rm
inv}.$ (56)

This is a spatial two-form, whose three-dimensional Hodge dual gives the
sought components

$\displaystyle\langle\Theta_{0}^{\,\,\,i}\rangle_{\rm cov}^{\rm expl}$
$\displaystyle=-{1\over
4\pi^{2}}e^{-\sigma}\epsilon^{ijk}\Big{(}2\mathcal{S}_{0}\mathcal{V}_{0}\partial_{j}V_{k}+\mathcal{V}_{0}^{2}\partial_{j}S_{k}+\mathcal{S}_{0}\mathcal{V}_{0}^{2}\partial_{j}a_{k}\Big{)}$
$\displaystyle=-{1\over
4\pi^{2}}e^{\sigma}\Big{(}2\mu\mu_{S}\mathbb{B}^{i}+\mu^{2}B^{i}_{S}+4\mu^{2}\mu_{S}\omega^{i}\Big{)}.$
(57)

The implicit contribution, on the other hand, is readily computed using

$\displaystyle\int\limits_{S^{3}}d^{3}x\sqrt{g}\,e^{-\sigma}\langle\Theta_{0}^{\,\,\,i}\rangle_{\rm
cov}^{\rm impl}\delta a_{i}$
$\displaystyle=\int\limits_{S^{3}}d^{3}x\sqrt{g}\,g^{ij}\langle
J_{i}\rangle_{S,\rm cov}\delta_{a}S_{j},$ (58)

where the variation of the screw torsion with respect to the KK gauge field
has the form

$\displaystyle\delta_{a}S_{m}$ $\displaystyle=-{1\over
8}\epsilon^{ijk}g_{mi}\Big{(}T^{\ell}_{\,\,\,jk}-2a_{j}T^{\ell}_{\,\,\,0k}\Big{)}\delta
a_{\ell}+{1\over
4}\epsilon^{ijk}g_{mi}\Big{(}T^{0}_{\,\,\,0k}+a_{\ell}T^{\ell}_{\,\,\,0k}\Big{)}\delta
a_{j}$ $\displaystyle=-{1\over
8}g_{mi}\Big{[}\epsilon^{ijk}\Big{(}T^{\ell}_{\,\,\,jk}-2a_{j}T^{\ell}_{\,\,\,0k}\Big{)}-2\epsilon^{i\ell
k}\Big{(}T^{0}_{\,\,\,0k}+a_{j}T^{j}_{\,\,\,0k}\Big{)}\Big{]}\delta a_{\ell}.$
(59)

This leads to

$\displaystyle\langle\Theta_{0}^{\,\,\,i}\rangle_{\rm cov}^{\rm impl}$
$\displaystyle=-{1\over 8}e^{\sigma}\Big{[}\epsilon^{\ell
jk}\Big{(}T^{i}_{\,\,\,jk}-2a_{j}T^{i}_{\,\,\,0k}\Big{)}+2\epsilon^{i\ell
k}\Big{(}T^{0}_{\,\,\,0k}+a_{j}T^{j}_{\,\,\,0k}\Big{)}\Big{]}\langle
J_{\ell}\rangle_{S,\rm cov}.$ (60)

This expression has to be added to the explicit contribution computed in Eq.
(57).

Finally, we evaluate the spatial components of the energy-momentum tensor

$\displaystyle{1\over
2}\int\limits_{S^{3}}d^{3}x\sqrt{g}\,\langle\Theta^{ij}\rangle_{\rm cov}\delta
g_{ij}=\int\limits_{S^{3}}d^{3}x\sqrt{g}\,g^{ij}\langle J_{i}\rangle_{\rm
cov}\delta_{g}S_{j},$ (61)

where the variation on the right-hand side of this equation takes the form

$\displaystyle\delta_{g}S_{m}$
$\displaystyle=\left(S^{k}\delta^{(i}_{k}\delta^{j)}_{m}-{1\over
2}g^{ij}S_{m}+{1\over 4}e^{-\sigma}g_{m\ell}\epsilon^{\ell
k(i}T^{j)}_{\,\,\,\,\,0k}\right)\delta g_{ij}.$ (62)

Thus, we arrive at

$\displaystyle\langle\Theta^{ij}\rangle_{\rm cov}$
$\displaystyle=\left(2S^{(i}g^{j)\ell}-g^{ij}S^{\ell}+{1\over
2}e^{-\sigma}\epsilon^{\ell k(i}T^{j)}_{\,\,\,\,\,0k}\right)\langle
J_{\ell}\rangle_{S,\rm cov}.$ (63)

The current $\langle J_{i}\rangle_{S,\rm cov}$ appearing on the right-hand
side of Eqs. (55), (60), and (63) is the one given in the second line of (40),
which does not depend on the torsion. After substituting the explicit
expressions of the KK-invariant components of the axial-vector effective field
given in Eq. (28), a number of new terms appears in the constitutive relations
of the energy-momentum currents which are linear in the torsion.

In section 2 we discussed how, out of the 24 independent components of the
torsion tensor $T_{\mu\nu\alpha}$, the microscopic Dirac field only couples to
four of them contained in its antisymmetric part $T_{[\mu\nu\alpha]}$ defining
the three-form (10). They are codified in the screw torsion gauge field
$S_{\mu}$, whose components are given in terms of the torsion tensor in Eq.
(28). The antisymmetric components of the torsion tensor are the only ones
appearing in both the gauge currents and the spin energy potential tensor,
which are fully written in terms of $\mathcal{S}_{0}$ and $S_{i}$. In the case
of the energy-momentum tensor, however, an inspection of our results (55),
(60), and (63) shows that they contain additional components of the torsion
besides the four included in the equilibrium partition function. In
particular, the components $T_{00i}$ enter the energy-momentum tensor through

$\displaystyle T^{i}_{\,\,\,0j}=g^{ik}\Big{(}T_{k0j}-a_{k}T_{00j}\Big{)},$
(64)

while they cancel out from the expression of the screw torsion in (28). This
state of affairs is not a consequence of dimensional reduction, but a result
of taking a variation with respect to the metric components. A similar
situation was found also in $2+1$ dimensions Valle:2015hfa .

It is known that in the presence of torsion the Ricci tensor acquires extra
terms that render it nonsymmetric. This has important consequences for the
Einstein equations, which force to correct the metric energy-momentum tensor
by a Belinfante-Rosenfeld term depending on the spin energy potential
Hehl:1976kj ; Ortin:2015hya

$\displaystyle\langle T^{\mu\nu}_{(\rm can)}\rangle_{\rm
cov}=\langle\Theta^{\mu\nu}\rangle_{\rm
cov}-\stackrel{{\scriptstyle*}}{{\nabla}}_{\alpha}\langle\Psi^{\alpha\mu\nu}\rangle_{\rm
cov},$ (65)

where the modified covariant derivative is given by

$\displaystyle\stackrel{{\scriptstyle*}}{{\nabla}}_{\mu}=\nabla_{\mu}+T^{\alpha}_{\,\,\,\,\alpha\mu}$
(66)

The form of the Einstein equations is then preserved, provided its right-hand
side is given by this new canonical nonsymmetric energy-momentum tensor,
instead of the metric one.

A relevant question here is whether the canonical energy momentum-tensor (65)
plays any role at all in our analysis of torsional fluids. In deciding this,
we should not forget that in condensed matter applications the background
spacetime geometry is nondynamical, but a way of incorporating certain
microscopic features of the system into the effective theory of the long
wavelength modes. As no field equations have to be satisfied by the background
fields, the metric energy-momentum tensor and the spin energy potential can be
regarded as two independent currents, which provide the response to the
external sources $G_{\mu\nu}$ and $T^{\mu}_{\,\,\,\,\alpha\beta}$.

Finally, it should be pointed out that although our analysis includes the
linear effects of torsion, it does not incorporate the contribution to the
partition function of the mixed gauge-gravitational anomaly in the background
(17). This is the reason why we can restrict our attention to the consistent
energy-momentum tensor without having to consider the gravitational BZ term.

##  4 Torsional chiral effects in a two-flavor hadronic superfluid

The results of Sec. 2 make it easy to compute the interplay of torsion with
the dynamics of Nambu-Goldstone boson for the symmetry breaking pattern
$\mbox{U(1)}_{L}\times\mbox{U(1)}_{R}\rightarrow\mbox{U(1)}_{V}$. Our theory
is characterized by three Abelian gauge fields: a vector
$(\mathcal{V}_{0},\bm{V})$ and two axial-vectors, the auxiliary
$(\mathcal{A}_{0},\bm{A})$ and the screw torsion $(\mathcal{S}_{0},\bm{S})$.
Since the axial-vector and screw torsion fields couple in the same way to the
fundamental fermions, they always appear in the combination
$(\mathcal{A}_{0}+\mathcal{S}_{0},\bm{A}+\bm{S})$ in the effective action.
Setting $\bm{A}=0$ at the end of the calculation, the linear couplings of the
single Nambu-Goldstone boson $\alpha$ in the local WZW action takes a
particularly simple form Manes:2018llx

$\displaystyle W[\alpha,\ldots]_{\rm WZW}$
$\displaystyle\equiv\Big{.}\Big{(}W[\ldots,\bm{A}+\bm{S},\ldots]_{\rm
anom}-W[\ldots,\bm{A}+\bm{S}+d\alpha,\ldots]_{\rm
anom}\Big{)}\Big{|}_{\bm{A}=0}$ $\displaystyle=\ldots-{1\over
6\pi^{2}T_{0}}\int\limits_{S^{3}}\mathcal{A}_{0}\Big{(}\bm{F}_{S}+\mathcal{S}_{0}da\Big{)}d\alpha,$
(67)

where the ellipsis in the second line indicates the torsion-independent
coupling of the Nambu-Goldstone boson to the background fields
$\mathcal{V}_{0}$, $\bm{V}$, and $\mathcal{A}_{0}$. We see that the Nambu-
Goldstone boson couples linearly to $\bm{B}_{S}$, the magnetic component of
the screw torsion field strength defined in Eq. (22). Expressed in components,
the relevant term in the WZW action reads

$\displaystyle W[\alpha,\ldots]_{\rm WZW}$
$\displaystyle=\ldots-\int\limits_{S^{3}}d^{3}x\,\sqrt{g}\,{\mu_{5}\over
6\pi^{2}T}B_{S}^{i}\partial_{i}\alpha,$ (68)

where $B_{S}^{i}$ was defined in Eq. (37) and we also introduced the local
temperature $T=e^{-\sigma}T_{0}$ and the chiral chemical potential
$\mu_{5}=e^{-\sigma}\mathcal{A}_{0}$.

After this brief discussion of the Abelian case, we turn our attention to the
analysis of torsional chiral effects in the two-flavor hadronic superfluid
studied in Ref. Manes:2019fyw , where we refer the reader for details. This
fluid couples to a vector and an axial-vector external gauge fields,
respectively denoted by $\mathcal{V}$ and $\mathcal{A}$, transforming in the
flavor group U(2)${}_{L}\times$U(2)R. The values of these background fields
are restricted to the Cartan subalgebra generated by

$\displaystyle t_{0}={1\over 2}\mathbb{1},\hskip 28.45274ptt_{3}={1\over
2}\sigma_{3}.$ (69)

We assume the system undergoes spontaneous symmetry breaking to its vector
subgroups,
$\mbox{U(2)${}_{L}\times$U(2)${}_{R}$}\rightarrow\mbox{U(1)${}_{V}$}\times\mbox{SU(2)${}_{V}$}$,
generating in the process a triplet of Nambu-Goldstone bosons
$\pi^{0},\pi^{\pm}$ encoded in the matrix

$\displaystyle U=\exp\left[{i\sqrt{2}\over
f_{\pi}}\left(\begin{array}[]{cc}\tfrac{1}{\sqrt{2}}\pi^{0}&\pi^{+}\\\
\pi^{-}&-\tfrac{1}{\sqrt{2}}\pi^{0}\end{array}\right)\right].$ (72)

The first issue to address is how to incorporate the torsional vector fields
into our analysis. The left Dirac operator, including the vector, axial, and
screw-torsion fields, takes the form

$\displaystyle\overrightarrow{D\\!\\!\\!\\!/}\,\psi$
$\displaystyle=\Big{(}\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}-i{\mathcal{V}\\!\\!\\!\\!/\,}-i{\mathcal{A}\\!\\!\\!\\!/\,}\gamma_{5}-i\mathcal{S}\\!\\!\\!/\,\mathbb{1}\gamma_{5}\Big{)}\psi$
$\displaystyle=\Big{\\{}\overrightarrow{\overline{\nabla}\\!\\!\\!\\!/\,}-i\Big{(}{\mathcal{V}\\!\\!\\!\\!/\,}_{0}t_{0}+{\mathcal{V}\\!\\!\\!\\!/\,}_{3}t_{3}\Big{)}-i\Big{[}\big{(}{\mathcal{A}\\!\\!\\!/\,}_{0}+2\,\mathcal{S}\\!\\!\\!/\,\big{)}t_{0}+{\mathcal{A}\\!\\!\\!/\,}_{3}t_{3}\Big{]}\gamma_{5}\Big{\\}}\psi,$
(73)

and similarly for the right operator. The structure of these terms shows that,
to take into account the effect of torsion in the analysis of Refs.
Manes:2019fyw , it is enough to implement the replacement

$\displaystyle\mathcal{A}_{0\mu}$
$\displaystyle\longrightarrow\mathcal{A}_{0\mu}+2\mathcal{S}_{\mu},$ (74)

while leaving all remaining fields unchanged.

With all this in mind, we can compute the linear torsional corrections to the
covariant gauge currents at leading order in the derivative expansion. To
avoid cumbersome expressions, here we only give the terms in the currents
depending linearly on the torsion, that we denote by
$\langle\Delta\mathcal{J}^{\mu}_{aV}\rangle_{\rm cov}$ and
$\langle\Delta\mathcal{J}^{\mu}_{aA}\rangle_{\rm cov}$. These should be added
to the expressions found in Ref. Manes:2019fyw  for the corresponding
currents. The results are

$\displaystyle\langle\Delta\mathcal{J}^{\mu}_{0V}\rangle_{\rm cov}$
$\displaystyle=-{N_{c}\over
8\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\mathcal{S}_{\nu}\mathcal{V}_{0\alpha\beta},$
$\displaystyle\langle\Delta\mathcal{J}_{3V}^{\mu}\rangle_{\rm cov}$
$\displaystyle=-{N_{c}\over
24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\Big{[}\big{(}\mathds{H}+3\big{)}\mathcal{S}_{\nu}\mathcal{V}_{3\alpha\beta}-\partial_{\alpha}\big{(}\mathcal{S}_{\nu}\mathds{T}_{\beta}\big{)}\Big{]},$
$\displaystyle\langle\Delta\mathcal{J}^{\mu}_{0A}\rangle_{\rm cov}$
$\displaystyle=-{N_{c}\over
24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\Big{(}\mathcal{S}_{\nu}\mathcal{A}_{0\alpha\beta}+2\mathcal{A}_{0\nu}\partial_{\alpha}\mathcal{S}_{\beta}\Big{)},$
(75) $\displaystyle\langle\Delta\mathcal{J}^{\mu}_{3A}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over
24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\partial_{\nu}\big{(}\mathcal{S}_{\alpha}\mathds{I}_{\beta}\big{)},$

where we have used the tensor structures introduced in Manes:2019fyw

$\displaystyle\mathds{H}$ $\displaystyle\equiv{\rm
Tr\,}\Big{[}\Big{(}U^{-1}QU-Q\Big{)}Q\Big{]},$ $\displaystyle\mathds{I}_{\mu}$
$\displaystyle\equiv{\rm Tr\,}\Big{[}\Big{(}R_{\mu}+L_{\mu}\Big{)}Q\Big{]},$
(76) $\displaystyle\mathds{T}_{\mu}$ $\displaystyle\equiv{\rm
Tr\,}\Big{[}Q\Big{(}R_{\mu}-L_{\mu}\Big{)}\Big{]}+2\mathcal{V}_{3\mu}{\rm
Tr\,}\Big{[}\Big{(}U^{-1}QU-Q\Big{)}Q\Big{]},$

with $R_{\mu}=iU^{-1}\partial_{\mu}U$ and $L_{\mu}=i\partial_{\mu}UU^{-1}$,
while the charge matrix is given by

$\displaystyle Q={1\over 3}t_{0}+t_{3}.$ (77)

The first thing to be noticed here is that the torsional couplings of the
pions are restricted to the three-flavor components of the vector and axial-
vector covariant currents.

In order to compute the longitudinal and transverse components of these
currents, and write the constitutive relations of the hadronic superfluid, we
need to introduce a number of scalar and vector structures, in addition to
those used in Ref. Manes:2019fyw 777For the reader’s convenience, these are
summarized in Appendix B.. They represent the linear coupling of the Nambu-
Goldstone bosons to torsion. To the five scalar ones, we add

$\displaystyle\mathds{S}_{6}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}u_{\mu}\partial_{\nu}\big{(}\mathcal{S}_{\alpha}\mathds{I}_{\beta}\big{)},$
$\displaystyle\mathds{S}_{7}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}u_{\mu}\partial_{\nu}\big{(}\mathcal{S}_{\alpha}\mathds{T}_{\beta}\big{)},$
(78)

while for the vector structures we extend the notation
$\mathds{P}^{\mu}_{1,a}$ and $\mathds{P}^{\mu}_{3,a}$ to include $a=S$, and
add a new term $\mathds{P}^{\mu}_{5}$

$\displaystyle\mathds{P}_{1,S}^{\mu}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathds{I}_{\alpha}\partial_{\beta}\left({\mu_{S}\over
T}\right),$ $\displaystyle\mathds{P}_{3,S}^{\mu}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathds{T}_{\alpha}\partial_{\beta}\left({\mu_{S}\over
T}\right),$ (79) $\displaystyle\mathds{P}_{5}^{\mu}$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathcal{S}_{\alpha}\partial_{\beta}\mathds{H}.$

Here we have used the screw torsion chemical potential (39), as well as the
local temperature $T=e^{-\sigma}T_{0}$.

The longitudinal and transverse components of the covariant vector and axial-
vector currents are now written in terms of these quantities. The new
torsional nondissipative chiral transport coefficients can be read from the
resulting expressions. We start with the 0-flavor vector current

$\displaystyle u_{\mu}\langle\Delta\mathcal{J}^{\mu}_{0V}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over 4\pi^{2}}\mathcal{S}_{\mu}\mathcal{B}_{0}^{\mu},$
$\displaystyle
P^{\mu}_{\,\,\,\sigma}\langle\Delta\mathcal{J}^{\sigma}_{0V}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over
4\pi^{2}}\left[\mu_{S}\mathcal{B}^{\mu}_{0}+T\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathcal{S}_{\alpha}\partial_{\beta}\left({\mu_{0}\over
T}\right)\right],$ (80)

where we have introduced the magnetic field

$\displaystyle\mathcal{B}_{a}^{\mu}={1\over
2}\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathcal{V}_{a\alpha\beta}\hskip
56.9055pta=0,3.$ (81)

The longitudinal and transverse components of the 3-flavor vector current, on
the other hand, read

$\displaystyle u_{\mu}\langle\Delta\mathcal{J}_{3V}^{\mu}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over
24\pi^{2}}\Big{[}2\big{(}\mathds{H}+3\big{)}\mathcal{S}_{\mu}\mathcal{B}_{3}^{\mu}-\mathds{S}_{7}\Big{]},$
$\displaystyle
P^{\mu}_{\,\,\,\sigma}\langle\Delta\mathcal{J}_{3V}^{\sigma}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over
24\pi^{2}}\bigg{[}2\big{(}\mathds{H}+3\big{)}\mu_{S}\mathcal{B}^{\mu}_{3}+\big{(}\mathds{H}+6\big{)}T\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathcal{S}_{\alpha}\partial_{\beta}\left({\mu_{3}\over
T}\right)$
$\displaystyle-\mu_{S}\mathds{P}_{4}^{\mu}+T\mathds{P}_{3,S}^{\mu}+\mu_{3}\mathds{H}\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}\mathcal{S}_{\beta}-\mu_{3}\mathds{P}_{5}^{\mu}\bigg{]}.$
(82)

In the case of the axial-vector currents, we have

$\displaystyle u_{\mu}\langle\Delta\mathcal{J}^{\mu}_{0A}\rangle_{\rm cov}$
$\displaystyle={N_{c}\mu_{5}\over 6\pi^{2}}\mathcal{S}_{\mu}\omega^{\mu},$
$\displaystyle
P^{\mu}_{\,\,\,\sigma}\langle\Delta\mathcal{J}^{\sigma}_{0A}\rangle_{\rm cov}$
$\displaystyle={N_{c}\mu_{5}\over
12\pi^{2}}\bigg{(}-2\mu_{S}\omega^{\mu}+\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}\mathcal{S}_{\beta}\bigg{)},$
(83)

for the 0-flavor components, whereas in the case of the 3-flavor the result is

$\displaystyle u_{\mu}\langle\Delta\mathcal{J}^{\mu}_{3A}\rangle_{\rm cov}$
$\displaystyle={N_{c}\over 24\pi^{2}}\mathds{S}_{6},$ $\displaystyle
P^{\mu}_{\,\,\,\sigma}\langle\Delta\mathcal{J}^{\sigma}_{3A}\rangle_{\rm cov}$
$\displaystyle=-{N_{c}\over
24\pi^{2}}\Big{(}\mu_{S}\mathds{P}_{2}^{\mu}-T\mathds{P}_{1,S}^{\mu}\Big{)}.$
(84)

Let us stress once more that all the expressions given here only represent the
linear torsional contributions to the longitudinal and transverse components
of the covariant currents, that should be added to the respective nontorsional
terms found in Manes:2019fyw . As already pointed out, torsion couples to
Nambu-Goldstone bosons only through the 3-flavor currents. The torsional
contributions to the 0-flavor covariant vector and axial-vector currents are
just given by the corresponding terms of the BZ currents linear in the
torsion. This implies that there are no linear torsional contributions to the
0-flavor component of the consistent currents and, as a consequence, no linear
torsional terms in the WZW action depending on $V_{0i}$ or $A_{0i}$.

To find the expression of the covariant currents in terms of the physical
electromagnetic fields, we expand the KK-invariant components of the vector
field in terms of the charge matrix $Q$ defined in (77) and the generator
$t_{3}$ according to (see Manes:2019fyw )

$\displaystyle
V_{0\mu}t_{0}+V_{3\mu}t_{3}=3V_{0\mu}Q+\big{(}V_{3\mu}-3V_{0\mu}\big{)}t_{3}.$
(85)

As usual, the unbroken U(1)V factor is identified as the one coupling to the
$Q$ matrix, whereas the field coupling to $t_{3}$ is set to zero, which
implies $V_{\mu}\equiv 3V_{0\mu}=V_{3\mu}$. We write now the torsional terms
in the covariant electromagnetic current

$\displaystyle\langle\mathcal{J}^{\mu}_{\rm em}\rangle_{\rm cov}={e\over
3}\langle\mathcal{J}_{0V}^{\mu}\rangle_{\rm
cov}+e\langle\mathcal{J}^{\mu}_{3V}\rangle_{\rm cov},$ (86)

in terms of the pion fields as

$\displaystyle\langle\Delta\mathcal{J}^{i}_{\rm em}\rangle_{\rm cov}$
$\displaystyle=-{e^{2}N_{c}\over
12\pi^{2}f_{\pi}^{2}}\epsilon^{ijk}S_{j}\mathbb{E}_{k}\pi^{+}\pi^{-}-{\mu
e^{2}N_{c}\over 12\pi^{2}f_{\pi}^{2}}B_{S}^{i}\pi^{+}\pi^{-}$
$\displaystyle-{ieN_{c}\over
12\pi^{2}f_{\pi}^{2}}T\epsilon^{ijk}\partial_{k}\left[{\mu_{S}\over
T}\Big{(}\pi^{+}\partial_{j}\pi^{-}-\pi^{-}\partial_{j}\pi^{+}\Big{)}\right]-{e^{2}N_{c}\over
6\pi^{2}f_{\pi}^{2}}T\epsilon^{ijk}V_{j}\partial_{k}\Big{(}{\mu_{S}\over
T}\pi^{+}\pi^{-}\Big{)}$ $\displaystyle+{\mu e^{2}N_{c}\over
12\pi^{2}f_{\pi}^{2}}\epsilon^{ijk}S_{j}\partial_{k}\big{(}\pi^{+}\pi^{-}\big{)}+{5e^{2}N_{c}\over
18\pi^{2}}\Big{(}\mu_{S}\mathbb{B}^{i}+\epsilon^{ijk}S_{j}\mathbb{E}_{k}\Big{)}+\mathcal{O}(\pi^{3}),$
(87)

where the electric and magnetic fields are defined in Eq. (36) and $B_{S}^{i}$
is given in (37), all fields here being KK-invariant. The last, pion-
independent term is the BZ electromagnetic current of the unbroken theory, and
replicates the structure found in Eq. (41) for the Abelian case. We see that
there is no torsion-mediated electromagnetic coupling to the neutral pion.
There exits nonetheless a torsional pion-dependent contribution to the chiral
electric effect given by the first term in Eq. (87), this time induced by the
$\mathsf{T}$-odd spatial screw-torsion field.

As for the transverse axial-vector currents, we see that the only coupling of
torsion to pions arises from the 3-flavor component

$\displaystyle\langle\Delta\mathcal{J}^{i}_{3A}\rangle_{\rm cov}$
$\displaystyle=-{N_{c}\over
12\pi^{2}f_{\pi}}T\epsilon^{ijk}\partial_{j}\pi^{0}\partial_{k}\left({\mu_{S}\over
T}\right)+\mathcal{O}(\pi^{3}).$ (88)

There is therefore no torsional corrections to the pion-mediated chiral
electric, magnetic, and vortical separation effects found in Manes:2019fyw .
Interestingly, the coupling showed in Eq. (88) is the only torsion-induced
term involving the neutral pion in the constitutive relations at this order.

##  5 Closing remarks

In this paper we have analyzed the linear effect of background torsion in the
partition function of a charged fluid minimally coupled to gravity. The terms
studied are those induced by the ’t Hooft anomaly associated with the screw
torsion, dual to the antisymmetric part of the torsion tensor. In the Abelian
case, our results show the existence of magnetic and vortical chiral torsional
effects, whereas the axial-vector current does not have any linear torsional
corrections.

In this same model, the covariant spin energy potential and energy-momentum
tensor have been computed in terms of the axial-vector covariant current, with
coefficients that only depend on the metric functions. Since the axial-vector
current has been shown not to depend on the torsion at linear order, we
conclude that the spin energy potential does not exhibit linear torsional
contributions. The situation is quite different in the case of the covariant
energy-momentum tensor. Although its components are also written in terms of
the covariant axial-vector currents, the coefficients now do depend linearly
on the torsion tensor. Thus, we find linear torsional corrections to the
energy-momentum tensor, which actually include components of the torsion
tensor that do not appear in the effective action. This latter situation is
analogous to the one already found in $2+1$ dimension Valle:2015hfa . It is
important to stress that the torsional contributions to the energy-momentum
tensor and the spin energy potential are associated with the implicit
dependence of the effective axial-vector gauge field on both the metric and
the torsion components. This is what makes the torsional theory genuinely
different from the theory of a Dirac fermion axially coupled to an external
gauge field, as it is reflected in the constitutive relations.

We have also studied linear torsional chiral effects in a two-flavor hadronic
superfluids studied in Ref. Manes:2019fyw . We found that no new couplings of
the Nambu-Goldstone bosons to torsion emerge from the 0-flavor components of
the covariant vector and axial-vector currents. The analysis of the
electromagnetic transverse current, on the other hand, shows the existence of
torsional chiral electric effect mediated by the two charged Nambu-Goldstone
bosons $\pi^{\pm}$, whereas no torsional vortical effect appears.
Interstingly, there are no torsional corrections to the pion-mediated
electric, magnetic, and vortical chiral separation effects that were found in
Manes:2019fyw .

In this paper we have exploited the analogy between background torsion and
axial-vector couplings to compute the linear effects of torsion, which come
from triangle diagrams with an axial-vector current coupled to the background
screw torsion field. A different sector is the one associated with the Nieh-
Yan anomaly Nieh:1981ww ; Nieh:1981xk , which explicitly depends on the UV
cutoff scale of the theory. This Nieh-Yan term has been shown to be relevant
in condensed matter, where this cutoff arises naturally. It would be
interesting to further explore the physical implications of this anomaly along
the lines followed in the present paper for the triangle contributions. This
issue will be addressed elsewhere.

###### Acknowledgements.

We thank Karl Landsteiner for discussions. This work has been supported by
Spanish Science Ministry grants PGC2018-094626-B-C21 (MCIU/AEI/FEDER, EU) and
PGC2018-094626-B-C22 (MCIU/AEI/FEDER, EU), as well as by Basque Government
grant IT979-16.

##  Appendix A Basics of spacetime torsion

In this appendix, we give a brief overview of the main mathematical features
of spacetimes with torsion. The focus will lie on the basic differential
geometric aspects, with further details being available in a number of reviews
(see, for example, Hehl:1976kj ; Shapiro:2001rz ; Ortin:2015hya ; Trautman2006
; Hehl:2007bn ; Freedman:2012zz ). Let us consider a four-dimensional curved
manifold and an orthonormal tetrad basis
$\\{e^{A}=e^{A}_{\,\,\,\mu}dx^{\mu}\\}$

$\displaystyle\eta_{AB}e^{A}_{\,\,\,\mu}e^{B}_{\,\,\,\nu}=g_{\mu\nu},$ (89)

with $\eta_{AB}$ the flat Lorentz metric and $g_{\mu\nu}$ the spacetime
metric888In this work, Lorentz indices are denoted by capital Latin letters,
while spacetime indices are indicated by Greek letters. Lowercase Latin
indices are reserved for spatial components. To make notation lighter, we omit
the wedge $(\wedge)$ to denote exterior products.. The spin connection
$\omega^{A}_{\,\,\,B}$ defines the notion of parallel transport, allowing the
construction of the covariant derivative operator, that in the particular case
of $p$-form tensor of the type $\tau^{A}_{\,\,\,B}$ takes the form

$\displaystyle\nabla\tau^{A}_{\,\,\,B}=d\tau^{A}_{\,\,\,B}+\omega^{A}_{\,\,\,C}\tau^{C}_{\,\,\,B}+(-1)^{p+1}\tau^{A}_{\,\,\,C}\omega^{C}_{\,\,\,B}.$
(90)

In what follows, we assume the connection $\omega^{A}_{\,\,\,BC}$ to be metric
compatible, $\nabla\eta_{AB}=0$. Torsion is defined by the first Cartan
structure equation

$\displaystyle T^{A}=de^{A}+\omega^{A}_{\,\,\,B}e^{B},$ (91)

while the second one gives the curvature

$\displaystyle R^{A}_{\,\,\,B}$
$\displaystyle=d\omega^{A}_{\,\,\,B}+\omega^{A}_{\,\,\,C}\omega^{C}_{\,\,\,B}.$
(92)

Both torsion and curvature are geometrical quantities related to the behavior
of vectors under (infinitesimal) parallel transport.

The torsion two-form can be expanded in the tetrad basis as

$\displaystyle T^{A}={1\over 2}T^{A}_{\,\,\,BC}e^{B}e^{C},$ (93)

where the components on the right-hand side are antisymmetric in the lower two
indices, $T^{A}_{\,\,\,(BC)}=0$. To parametrize torsion, it is convenient to
introduce an auxiliary torsionless connection
$\overline{\omega}^{A}_{\,\,\,B}$ associated with the same tetrad basis
$e^{A}$ and satisfying

$\displaystyle de^{A}+\overline{\omega}^{A}_{\,\,\,B}e^{B}=0.$ (94)

This auxiliary connection is also metric compatible,
$\overline{\nabla}\eta_{AB}=0$, where here and elsewhere in the paper we
indicate all quantities associated with this Levi-Civita connection by an
overline. The contorsion one-form is defined by

$\displaystyle\kappa^{A}_{\,\,\,B}\equiv\omega^{A}_{\,\,\,B}-\overline{\omega}^{A}_{\,\,\,B},$
(95)

which, being the difference of two connections, transforms as a tensor under
local Lorentz transformations. Combining Eqs. (91) and (94), we write the
torsion two-form $T^{A}$ in terms of the contortion one-form as

$\displaystyle
T^{A}=\kappa^{A}_{\,\,\,\,\,B}e^{B}=-\kappa^{A}_{\,\,\,BC}e^{B}e^{C},$ (96)

where in the second equality we have expanded
$\kappa^{A}_{\,\,\,B}=\kappa^{A}_{\,\,\,BC}e^{C}$. This identity shows that
the antisymmetric part of the contorsion in the two lower indices is
determined by the components of the torsion tensor

$\displaystyle\kappa^{A}_{\,\,\,[BC]}=-{1\over 2}T^{A}_{\,\,\,\,\,BC}.$ (97)

Using metric compatibility, the symmetric piece can be computed in terms of
the torsion components as

$\displaystyle\kappa^{A}_{\,\,\,(BC)}={1\over
2}T^{\,\,\,\,A}_{B\,\,\,\,C}+{1\over 2}T^{\,\,\,\,A}_{C\,\,\,\,B}.$ (98)

Similar expressions are obtained using a coordinate basis, with the torsion
being identified with the antisymmetric part of the connection according to

$\displaystyle
T^{\mu}_{\,\,\,\,\nu\sigma}=-2\Gamma^{\mu}_{\,\,\,[\nu\sigma]}=-2\kappa^{\mu}_{\,\,\,[\nu\sigma]}.$
(99)

##  Appendix B A summary of expressions from Ref. Manes:2019fyw

For the sake of completeness, we list in this Appendix the scalar and tensor
structures introduced in Ref. Manes:2019fyw  to write the constitutive
relations for the two-flavor chiral hadronic superfluid studied in Section 4.
In the case of the longitudinal components of the currents, these are
expressed in terms of the following five scalar structures

$\displaystyle\mathds{S}_{1,a}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}\mathds{I}_{\mu}u_{\nu}\partial_{\alpha}{\mathcal{V}}_{a\beta}=\mathds{I}_{\mu}\mathcal{B}^{\mu}_{a}\hskip
56.9055pt(a=0,3),$ $\displaystyle\mathds{S}_{2}$ $\displaystyle\equiv{1\over
2}\epsilon^{\mu\nu\alpha\beta}\mathds{I}_{\mu}u_{\nu}\partial_{\alpha}u_{\beta}=\mathds{I}_{\mu}\omega^{\mu},$
$\displaystyle\mathds{S}_{3}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}u_{\mu}\left[{\mathcal{V}}_{3\nu}\partial_{\alpha}\mathds{I}_{\beta}-{i\over
3}{\rm Tr\,}\Big{(}L_{\nu}L_{\alpha}L_{\beta}\Big{)}\right],$ (100)
$\displaystyle\mathds{S}_{4,a}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}\mathds{T}_{\mu}u_{\nu}\partial_{\alpha}{\mathcal{V}}_{a\beta}=\mathds{T}_{\mu}\mathcal{B}_{a}^{\mu}\hskip
56.9055pt(a=0,3),$ $\displaystyle\mathds{S}_{5}$ $\displaystyle\equiv{1\over
2}\epsilon^{\mu\nu\alpha\beta}\mathds{T}_{\mu}u_{\nu}\partial_{\alpha}u_{\beta}=\mathds{T}_{\mu}\omega^{\mu},$

where the magnetic field is defined in Eq. (81), and $\mathds{I}_{\mu}$ and
$\mathds{T}_{\mu}$ are given in Eq. (76), whose expansions in terms of the
pion fields are given by

$\displaystyle\mathds{H}$ $\displaystyle=-{2\over
f_{\pi}^{2}}\pi^{+}\pi^{-}+\mathcal{O}(\pi^{4}),$
$\displaystyle\mathds{I}_{\mu}$ $\displaystyle=-{2\over
f_{\pi}}\partial_{\mu}\pi^{0}+\mathcal{O}(\pi^{3}),$ (101)
$\displaystyle\mathds{T}_{\mu}$ $\displaystyle={2i\over
f_{\pi}^{2}}\Big{(}\pi^{+}\partial_{\mu}\pi^{-}-\pi^{-}\partial_{\mu}\pi^{+}\Big{)}-{4\over
f_{\pi}^{2}}\pi^{+}\pi^{-}\mathcal{V}_{3\mu}+\mathcal{O}(\pi^{3}).$

Finally, the transverse components of the covariant currents found in
Manes:2019fyw  are expressed in terms of the four tensor structures

$\displaystyle\mathds{P}^{\mu}_{1,a}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathds{I}_{\alpha}\partial_{\beta}\left({\mu_{a}\over
T}\right)\hskip 56.9055pt(a=0,3),$ $\displaystyle\mathds{P}^{\mu}_{2}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}\mathds{I}_{\beta},$
$\displaystyle\mathds{P}^{\mu}_{3,a}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}u_{\nu}\mathds{T}_{\alpha}\partial_{\beta}\left({\mu_{a}\over
T}\right)\hskip 56.9055pt(a=0,3),$ (102) $\displaystyle\mathds{P}^{\mu}_{4}$
$\displaystyle\equiv\epsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}\mathds{T}_{\beta}.$

These expressions have been written in terms of the chemical potentials

$\displaystyle\mu_{a}$ $\displaystyle=e^{-\sigma}\mathcal{V}_{a0}.$ (103)

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