Chiral torsional effects in anomalous fluids in thermal equilibrium

JHEP05(2021)209 Published for SISSA by Springer Received:December 24, 2020 Revised:March 8, 2021 Accepted:May 2, 2021 Published:May 24, 2021 Chiral torsional effects in anomalous fluids in thermal equilibrium Juan L. Mañes,aManuel Valleaand Miguel Á. Vázquez-Mozob aDepartamento de Física, Universidad del País Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain bDepartamento de Física Fundamental, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain E-mail: [email protected],[email protected], [email protected] 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. Keywords: Anomalies in Field and String Theories, Thermal Field Theory ArXiv ePrint: 2012.08449 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2021)209 JHEP05(2021)209 Contents 1 Introduction 1 2 Equilibrium partition function and covariant currents with background torsion 3 3 The spin energy potential and the energy-momentum tensor 10 4 Torsional chiral effects in a two-flavor hadronic superfluid 14 5 Closing remarks 18 A Basics of spacetime torsion 19 B A summary of expressions from ref. [20]21 1 Introduction The coupling of hydrodynamic systems to external sources through anomalous currents gives rise to a variety of chiral transport effects [1–8]. 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 [9– 20]. 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 [21] have been subject to direct detection in the laboratory [22]. 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 [23–25]. 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 [26,27] 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 [28]. In systems with linear dispersion relations, such – 1 – JHEP05(2021)209 as the case of Weyl semimetals, this geometry provides a static nondynamical effective background on which fermions propagate [29]. Torsion is also known to have an interplay with chiral anomalies in quantum field theory [30–33]. The axial anomaly receives a contribution given by the so-called Nieh-Yan term [34,35], 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 [36]. The role of this term in condensed matter physics has been explored in a number of works (see, for example, [28,29,37–39]). In the context of chiral fluids, the interest in torsion arose in connection with the study of Hall transport [40–43] (see [44] 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. [45] 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 [46]. Besides the sector associated to the Nieh-Yan anomaly,1the 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. [48], 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 [49]. 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 [19,20] 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 axialvector 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 1It has recently been proposed, however, that there is no genuine torsional chiral dissipationless transport in that sector [47]. – 2 – JHEP05(2021)209 U(2)L×U(2)Ris 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 [20], on the other hand, do not receive any contributions linear in the torsion. The present article is organized as follows. Section 2is 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 Asome basic facts about geometric torsion, while in appendix Bwe list some expressions of ref. [20] 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 torsion.2The action of a massless Dirac spinor minimally coupled to gravity can be written as [26,50,51] S=1 2Zd4x(det e)ψ−→ ∇/ ψ −ψ←− ∇/ ψ,(2.1) where the left and right covariant derivatives inside the integral are defined respectively by −→ ∇/ ψ =γAeµ A∂µψ+1 4γAγ[BγC]ωBCAψ, ψ←− ∇/=eµ A∂µψγA−1 4ψγ[BγC]γAωBCA,(2.2) and the Dirac matrices verify the Minkowskian Clifford algebra {γA, γB}= 2ηAB1. Writing the full spin connection in terms of the auxiliary torsionless Levi-Civita connection and the contorsion tensor as ωA BC =ωA BC +κA BC, we get the following expression of the left covariant derivative in terms of its Levi-Civita counterpart −→ ∇/ ψ =−→ ∇/ ψ +1 4γCγ[AγB]κABCψ =−→ ∇/ ψ +1 4γBηAC −γAηBC +iABCDγDγ5κABC ψ, (2.3) where we indicate by a bar all geometric quantities referred to the Levi-Civita connection and have used the gamma matrices identity γAγ[BγC]=γCηAB −γBηAC +iABCDγDγ5.(2.4) 2The basics of geometric torsion, as well as the notation used in the following, are summarized in appendix A. – 3 – JHEP05(2021)209 A similar calculation for the right covariant derivative in eq. (2.2) gives ψ←− ∇/=ψ←− ∇/−1 4ψγ[AγB]γCκACB =ψ←− ∇/−1 4ψγAηBC −γBηAC +iABCDγDγ5κABC .(2.5) Plugging these results into the action (2.1), we arrive at the expression S=1 2Zd4x(det e)ψ−→ ∇/ ψ +1 4ψγBηAC −γAηBC +iABCDγDγ5ψκABC −ψ←− ∇/ ψ +1 4ψγAηBC −γBηAC +iABCDγDγ5ψκABC .(2.6) The important point here is that the term proportional to (γBηAC −γAηBC )κABC , 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, κA[BC], which as shown in appendix A[see eq. (A.9)] is given by the components of the torsion tensor S=1 2Zd4x(det e)ψ−→ ∇/ ψ −ψ←− ∇/ ψ +i 4ψγDγ5ψ BC DA TABC .(2.7) This form of the action suggests the introduction of the effective vector field SA=−1 8CD AB TBCD,(2.8) to write S=1 2Zd4x(det e)ψ−→ ∇/ ψ −ψ←− ∇/ ψ −2iψγAγ5ψSA.(2.9) 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. [48], we call screw torsion. The minimal coupling to gravity selects only one among all possible dimension-four operators coupling Dirac fermions to torsion [52]. Before proceeding any further, some clarification on the action (2.1) 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 Sis 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 section 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. (2.1), the authors of ref. [48] 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 TB BA and mixes for – 4 – JHEP05(2021)209 many practical purposes with the electromagnetic field. In what follows we stick to the minimal coupling prescription (2.1) 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 TABC ≡T[ABC], which can be used to define the three-form3 T=1 3!TABCeAeBeC.(2.10) Its four independent components are encoded in the screw-torsion according to eq. (2.8), which can be written using the Hodge star operator as Sµ=−1 8αβ µν Tναβ =⇒ S =3 4?T.(2.11) The field strength of the Abelian screw torsion, FS=dS, can be written then as FS=3 4d ? T=−3 4? δT,(2.12) where δ≡ − ? d? denotes the codifferential acting on a three-form. In components, this equation reads Sµν =−3 8∇σTσαβαβµν.(2.13) Looking at eq. (2.11) above, we see that the gauge variation of the screw torsion vector field Sby an exact one-form, S → S +dα, corresponds to the following transformation of the torsion three-form T T −→ T +δβ, (2.14) with β∼?α a four-form. The nihilpotency of the codifferential, δ2= 0, guarantees the gauge invariance of the screw torsion field strength (2.12). 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 [9,10]. Here we are going to employ the differential geometry methods introduced in [19,20] 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 S, we also assume that the microscopic fermionic degrees of freedom are coupled to an external vector Abelian gauge field V, which remains 3Unlike in refs. [19,20], here no −iis factored out of the components of differential forms. – 5 – JHEP05(2021)209 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, [19]). Keeping only terms linear in the torsion, we have e ω0 5(S,FV,FS)=6SF2 V,(2.15) where FV=dVis the vector field strength. This gives the Bardeen form of the anomaly, which explicitly preserves vector gauge transformations. The properly normalized ChernSimons nonlocal effective action encoding the linear effects of torsion is then given by Γ[V,S]CS =1 4π2Z D5 SF2 V,(2.16) where D5is 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 ∂D5to be the generic static line element ds2=−e2σ(x)hdx0+ai(x)dxii2+gij(x)dxidxj,(2.17) and take all fields to be independent of x0. We implement dimensional reduction onto the compatified Euclidean time by setting D5=S1×D4, where the length of the S1equals the inverse of the equilibrium temperature T0. Vector fields are then written in terms of components that remain invariant under KK transformations [20], acting according to x0→x0+φand ai→ai−∂iφ, V ≡ Vµdxµ=V−e−σV0u, S ≡ Sµdxµ=S−e−σS0u, (2.18) where we have introduced the four-velocity one-form ugiven by u=−eσdx0+aidxi≡ −eσdx0+a,(2.19) and the KK-invariant spatial one-forms are defined by V=Vi−V0aidxi≡Vidxi, S=Si−S0aidxi≡Sidxi.(2.20) A similar electric-magnetic decomposition can be written for the vector field strength FV≡B+uE =dV−de−σuV0+ue−σdV0(2.21) =FV+V0da+ue−σdV0, – 6 – JHEP05(2021)209 where FV≡dVand we have used that d(e−σu) = −da. The electric Eand magnetic Bcomponents of the field strength will be later identified with the electric and magnetic fields [cf. (2.36)]. The equivalent expression for the field strength associated to the screw torsion reads FS≡BS+uES =FS+S0da+ue−σdS0,(2.22) with FS=dS. Finally, we implement the dimensional reduction on the three-form (2.10) as well by decomposing it as T=TB+uTE.(2.23) In a coordinate basis, the electric and magnetic components are respectively given by TE=1 3!e−σh2gi`T`0j+e2σT0 ij −2aiT0 0j+e2σa`T`ij −2aiT`0jidxjdxk, TB=1 3!gj`T`kn −2akT`0ndxjdxkdxn.(2.24) Being a four-form, the gauge function βin eq. (2.14) does not have any magnetic component, β=uβE. As a consequence, only the electric part of Ttransforms under (2.14) TE−→ TE+δ⊥βE, TB−→ TB,(2.25) where δ⊥=∗d∗, with ∗the three-dimensional Hodge dual (not to be confused with its four-dimensional couterpart denoted by ?). Since under four-dimensional Hodge duality the electric and magnetic components interchange ?T=−∗TE−u∗TB,(2.26) we find from eqs. (2.11) and (2.18) S0=3 4eσ∗TB, S=−3 4∗TE.(2.27) We see that the gauge invariance of TBimplies the same property for S0, whereas S undergoes the standard gauge transformation generated by the zero-form ∗βE. Using in addition eq. (2.24), we can write the screw torsion field in terms of the components of the torsion tensor as S0=1 8eσijkgi`T`jk −2ajT`0k,(2.28) S=−1 8e−σijkgmih2g`jT`0k+e2σa`T`jk −2ajT`0k+e2σT0 jk −2ajT0 0kidxm. – 7 – JHEP05(2021)209 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 (2.25) and coming from triangle diagrams.4As shown in [11,19], the dimensionally-reduced effective action splits into a local anomalous and a nonlocal invariant piece, respectively given by [19] W[V0,S0,V,S, da]anom =1 4π2T0Z S32V0FV+daV2 0S,(2.29) and W[V0,S0,V,S, da]inv =1 4π2T0Z D4 hS0F2 V+ 2V0FVFS(2.30) +daV0V0FS+ 2S0FV+ (da)2S0V2 0i. In the second expression, we have introduced the components of the field strength associated with S FS=dS≡1 2Sijdxidxj.(2.31) The covariant currents can be now computed from the invariant part of the partition function [11,19,20]. We begin with the one associated with the vector current hJVicov =T0 δ δFV W[V0,S0,V,S]inv =1 2π2hS0FV+daV0+V0FSi,(2.32) whereas for the one corresponding to the torsional axial-vector gauge field, the result is hJSicov =T0 δ δFS W[V0,S0,V,S]inv =1 2π2V0FV+1 2daV2 0.(2.33) In both cases, the zero components vanish, hJV0icov =hJS0icov = 0. We observe that, unlike the vector current, hJSicov 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 [19] hJViBZ =1 2π2SFV, hJSiBZ =1 6π2SFS.(2.34) 4As stated above, in this work we do not consider the sector associated with the Nieh-Yan anomaly. – 8 – JHEP05(2021)209 end of the calculation, the linear couplings of the single Nambu-Goldstone boson αin the local WZW action takes a particularly simple form [19] W[α, . . .]WZW ≡W[. . . , A+S, . . .]anom −W[. . . , A+S+dα, . . .]anomA=0 =. . . −1 6π2T0Z S3A0FS+S0dadα, (4.1) where the ellipsis in the second line indicates the torsion-independent coupling of the Nambu-Goldstone boson to the background fields V0,V, and A0. We see that the NambuGoldstone boson couples linearly to BS, the magnetic component of the screw torsion field strength defined in eq. (2.22). Expressed in components, the relevant term in the WZW action reads W[α, . . .]WZW =. . . −Z S3 d3x√gµ5 6π2TBi S∂iα, (4.2) where Bi Swas defined in eq. (2.37) and we also introduced the local temperature T=e−σT0 and the chiral chemical potential µ5=e−σA0. 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. [20], where we refer the reader for details. This fluid couples to a vector and an axial-vector external gauge fields, respectively denoted by Vand A, transforming in the flavor group U(2)L×U(2)R. The values of these background fields are restricted to the Cartan subalgebra generated by t0=1 21, t3=1 2σ3.(4.3) We assume the system undergoes spontaneous symmetry breaking to its vector subgroups, U(2)L×U(2)R→U(1)V×SU(2)V, generating in the process a triplet of Nambu-Goldstone bosons π0, π±encoded in the matrix U= exp  i√2 fπ   1 √2π0π+ π−−1 √2π0  .(4.4) 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 −→ D/ ψ =−→ ∇/−iV/−iA/ γ5−iS/1γ5ψ =n−→ ∇/−iV/0t0+V/3t3−ihA/0+ 2 S/t0+A/3t3iγ5oψ, (4.5) 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 ref. [20], it is enough to implement the replacement A0µ−→ A0µ+ 2Sµ,(4.6) while leaving all remaining fields unchanged. – 15 – JHEP05(2021)209 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 h∆Jµ aV icov and h∆Jµ aAicov. These should be added to the expressions found in ref. [20] for the corresponding currents. The results are h∆Jµ 0Vicov =−Nc 8π2µναβSνV0αβ, h∆Jµ 3Vicov =−Nc 24π2µναβhH+ 3SνV3αβ −∂αSνTβi, h∆Jµ 0Aicov =−Nc 24π2µναβSνA0αβ + 2A0ν∂αSβ,(4.7) h∆Jµ 3Aicov =Nc 24π2µναβ∂νSαIβ, where we have used the tensor structures introduced in [20] H≡Tr hU−1QU −QQi, Iµ≡Tr hRµ+LµQi,(4.8) Tµ≡Tr hQRµ−Lµi+ 2V3µTr hU−1QU −QQi, with Rµ=iU−1∂µUand Lµ=i∂µUU−1, while the charge matrix is given by Q=1 3t0+t3.(4.9) 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. [20].7They represent the linear coupling of the Nambu-Goldstone bosons to torsion. To the five scalar ones, we add S6=µναβuµ∂νSαIβ, S7=µναβuµ∂νSαTβ,(4.10) while for the vector structures we extend the notation Pµ 1,a and Pµ 3,a to include a=S, and add a new term Pµ 5 Pµ 1,S =µναβuνIα∂βµS T, Pµ 3,S =µναβuνTα∂βµS T,(4.11) Pµ 5=µναβuνSα∂βH. 7For the reader’s convenience, these are summarized in appendix B. – 16 – JHEP05(2021)209 Here we have used the screw torsion chemical potential (2.39), as well as the local temperature T=e−σT0. 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 uµh∆Jµ 0Vicov =Nc 4π2SµBµ 0, Pµ σh∆Jσ 0Vicov =Nc 4π2µSBµ 0+TµναβuνSα∂βµ0 T,(4.12) where we have introduced the magnetic field Bµ a=1 2µναβuνVaαβ a= 0,3.(4.13) The longitudinal and transverse components of the 3-flavor vector current, on the other hand, read uµh∆Jµ 3Vicov =Nc 24π2h2H+ 3SµBµ 3−S7i, Pµ σh∆Jσ 3Vicov =Nc 24π22H+ 3µSBµ 3+H+ 6TµναβuνSα∂βµ3 T −µSPµ 4+TPµ 3,S +µ3Hµναβuν∂αSβ−µ3Pµ 5.(4.14) In the case of the axial-vector currents, we have uµh∆Jµ 0Aicov =Ncµ5 6π2Sµωµ, Pµ σh∆Jσ 0Aicov =Ncµ5 12π2−2µSωµ+µναβuν∂αSβ,(4.15) for the 0-flavor components, whereas in the case of the 3-flavor the result is uµh∆Jµ 3Aicov =Nc 24π2S6, Pµ σh∆Jσ 3Aicov =−Nc 24π2µSPµ 2−TPµ 1,S.(4.16) 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 [20]. 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 V0ior A0i. – 17 – JHEP05(2021)209 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 Qdefined in (4.9) and the generator t3according to (see [20]) V0µt0+V3µt3= 3V0µQ+V3µ−3V0µt3.(4.17) As usual, the unbroken U(1)Vfactor is identified as the one coupling to the Qmatrix, whereas the field coupling to t3is set to zero, which implies Vµ≡3V0µ=V3µ. We write now the torsional terms in the covariant electromagnetic current hJµ emicov =e 3hJµ 0Vicov +ehJµ 3Vicov,(4.18) in terms of the pion fields as h∆Ji emicov =−e2Nc 12π2f2 π ijkSjEkπ+π−−µe2Nc 12π2f2 π Bi Sπ+π− −ieNc 12π2f2 π Tijk∂kµS Tπ+∂jπ−−π−∂jπ+−e2Nc 6π2f2 π TijkVj∂kµS Tπ+π− +µe2Nc 12π2f2 π ijkSj∂kπ+π−+5e2Nc 18π2µSBi+ijkSjEk+O(π3),(4.19) where the electric and magnetic fields are defined in eq. (2.36) and Bi Sis given in (2.37), 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. (2.41) 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. (4.19), this time induced by the 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 h∆Ji 3Aicov =−Nc 12π2fπ Tijk∂jπ0∂kµS T+O(π3).(4.20) There is therefore no torsional corrections to the pion-mediated chiral electric, magnetic, and vortical separation effects found in [20]. Interstingly, the coupling showed in eq. (4.20) 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. – 18 – JHEP05(2021)209 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 [45]. 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 axialvector 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. [20]. 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 π±, whereas no torsional vortical effect appears. Interestingly, there are no torsional corrections to the pion-mediated electric, magnetic, and vortical chiral separation effects that were found in [20]. In this paper we have exploited the analogy between background torsion and axialvector 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 [34,35], 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. Acknowledgments 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. 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 – 19 – JHEP05(2021)209 details being available in a number of reviews (see, for example, [26,27,50,51,54,55]). Let us consider a four-dimensional curved manifold and an orthonormal tetrad basis {eA= eA µdxµ} ηABeA µeB ν=gµν,(A.1) with ηAB the flat Lorentz metric and gµν the spacetime metric.8The spin connection ωA 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 τA Btakes the form ∇τA B=dτA B+ωA CτC B+ (−1)p+1τA CωC B.(A.2) In what follows, we assume the connection ωA BC to be metric compatible, ∇ηAB = 0. Torsion is defined by the first Cartan structure equation TA=deA+ωA BeB,(A.3) while the second one gives the curvature RA B=dωA B+ωA CωC B.(A.4) 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 TA=1 2TA BCeBeC,(A.5) where the components on the right-hand side are antisymmetric in the lower two indices, TA (BC)= 0. To parametrize torsion, it is convenient to introduce an auxiliary torsionless connection ωA Bassociated with the same tetrad basis eAand satisfying deA+ωA BeB= 0.(A.6) This auxiliary connection is also metric compatible, ∇ηAB = 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 κA B≡ωA B−ωA B,(A.7) which, being the difference of two connections, transforms as a tensor under local Lorentz transformations. Combining eqs. (A.3) and (A.6), we write the torsion two-form TAin terms of the contortion one-form as TA=κABeB=−κA BCeBeC,(A.8) 8In 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 (∧)to denote exterior products. – 20 – JHEP05(2021)209 where in the second equality we have expanded κA B=κA BCeC. This identity shows that the antisymmetric part of the contorsion in the two lower indices is determined by the components of the torsion tensor κA [BC]=−1 2TABC .(A.9) Using metric compatibility, the symmetric piece can be computed in terms of the torsion components as κA (BC)=1 2TA B C +1 2TA C B.(A.10) Similar expressions are obtained using a coordinate basis, with the torsion being identified with the antisymmetric part of the connection according to Tµνσ =−2Γµ [νσ]=−2κµ [νσ].(A.11) B A summary of expressions from ref. [20] For the sake of completeness, we list in this appendix the scalar and tensor structures introduced in ref. [20] 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 S1,a ≡µναβIµuν∂αVaβ =IµBµ a(a= 0,3), S2≡1 2µναβIµuν∂αuβ=Iµωµ, S3≡µναβuµV3ν∂αIβ−i 3Tr LνLαLβ,(B.1) S4,a ≡µναβTµuν∂αVaβ =TµBµ a(a= 0,3), S5≡1 2µναβTµuν∂αuβ=Tµωµ, where the magnetic field is defined in eq. (4.13), and Iµand Tµare given in eq. (4.8), whose expansions in terms of the pion fields are given by H=−2 f2 π π+π−+O(π4), Iµ=−2 fπ ∂µπ0+O(π3),(B.2) Tµ=2i f2 ππ+∂µπ−−π−∂µπ+−4 f2 π π+π−V3µ+O(π3). Finally, the transverse components of the covariant currents found in [20] are expressed in terms of the four tensor structures Pµ 1,a ≡µναβuνIα∂βµa T(a= 0,3), Pµ 2≡µναβuν∂αIβ, Pµ 3,a ≡µναβuνTα∂βµa T(a= 0,3),(B.3) Pµ 4≡µναβuν∂αTβ. – 21 – JHEP05(2021)209 These expressions have been written in terms of the chemical potentials µa=e−σVa0.(B.4) Open Access. 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