JHEP05(2021)209 Springer May 2, 2021 May 24, 2021 : March 8, 2021 : : December 24, 2020 : Revised Accepted Published b Received , Published for SISSA by https://doi.org/10.1007/JHEP05(2021)209 and Miguel Á. Vázquez-Mozo [email protected] a , . 3 Manuel Valle 2012.08449 a The Authors. c Anomalies in Field and String Theories, Thermal Field Theory

Using the similarity between spacetime torsion and axial gauge couplings, we , [email protected] [email protected] Departamento de Física, UniversidadApartado del 644, País 48080 Vasco UPV/EHU, Bilbao, Spain Departamento de Física Fundamental, UniversidadPlaza de de Salamanca, la Merced s/n, 37008E-mail: Salamanca, Spain b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: of linear torsionalspin magnetic energy and potential vortical do not effects,ant energy-momentum receive while tensor, corrections the on in the thetensor. axial-vector other torsion current hand, at The and linear does case order. contain the contributions of terms to The linear a the covari- in constitutive two-flavor thetorsional relations hadronic torsion electric computed. superfluid chiral Our is effect results mediated also by show analyzed, the the and charged existence pions. the of a torsional Abstract: study torsional contributions toground. the In equilibrium the partition case of function a in charged a fluid minimally stationary coupled back- to torsion, we spot the existence Juan L. Mañes, Chiral torsional effects inequilibrium anomalous fluids in thermal JHEP05(2021)209 ] – 3 9 21 10 14 ]. 22 ]. These have been shown to be 8 – 1 21 ] 20 – 1 – ]. In systems with linear dispersion relations, such 28 ]. An obvious motivation for these investigations has 19 25 – 23 ] for a review of different physical scenarios). 27 , 18 26 1 Despite its absence in standard general relativity, torsion has been the focus of attention Together with the prospects of spotting fundamental microscopic torsion in high-energy torsion ]. By placing the system on a curved background stationary metric, and performing physics, torsional geometries providecondensed a matter physics. practical way Focusing onthe of links the implementing at physics each physical node at of large effects thelattice distances, ion in lattice the irregularities. build vectors up defining an effectivelattice For dreibein dislocations whose example, geometry and models disclinations their [ curvature and torsion respectively implement in physics for almostclassical a works century, of since Élie this Cartanbeen geometrical the [ possibility notion of was ourto spacetime first new having introduced a physics small in (see albeit the [ nonvanishing torsion, giving rise dimensional reduction onto thephysical compactified effects Euclidean such time, as itbackground vorticity is Kaluza-Klein and possible (KK) acceleration, gauge to field incorporate whichmetric. and are the sourced Phenomena gradient of respectively linkedhave the by to been time the subject the component to of existence direct the of detection mixed in the gauge-gravitational laboratory anomalies [ [ relevant for the understandingmatter of to diverse cosmology. physicalfor Due phenomena, the long to ranging wavelength thefrom from modes which topological condensed the can parity-violating nature be terms20 of in computed the anomalies, using fluid differential the constitutive relations geometry effective can techniques, action be derived [ 1 Introduction The coupling of hydrodynamicgives systems rise to to external a sources variety through of anomalous chiral currents transport effects [ A Basics of spacetime torsion B A summary of expressions from ref. [ 3 The spin energy4 potential and the Torsional chiral energy-momentum effects tensor in5 a two-flavor Closing hadronic remarks superfluid Contents 1 Introduction 2 Equilibrium partition function and covariant currents with background JHEP05(2021)209 ]. The 36 the partition 1 -dimensional spacetime with torsion. ]. 49 ]. (2 + 1) 29 – 2 – ] to carry out a study of the linear effects of tor- 20 , 19 ] for a comprehensive review). The relation between the 44 ] by a first principles calculation of the effective action and 45 ]). 39 ] (see [ – 43 – 37 , 40 ]. Besides the sector associated to the Nieh-Yan anomaly, 29 , 46 ]. 28 ]. The axial anomaly receives a contribution given by the so-called Nieh-Yan 47 ], which comes from a bubble diagram with two axial-vector current insertions 33 – ], transport phenomena induced by torsion were studied, both at zero and finite 35 , 30 48 34 After analyzing the Abelian case, we focus our attention on the case of a two-flavor The effects associated with the ’t Hooft anomaly of the gauge field dual to the an- The issue of torsional transport effects in four dimensions has also received some at- In the context of chiral fluids, the interest in torsion arose in connection with the study Torsion is also known to have an interplay with chiral anomalies in quantum field It has recently been proposed, however, that there is no genuine torsional chiral dissipationless transport 1 constitutive relations from where the corresponding transport coefficients can behadronic obtained. superfluid in the presence of torsion,in in that the sector [ phase in which chiral symmetry rections linear in the torsion.which is This is written also incovariant the energy-momentum terms case tensor for of can the the be covariantvector spin covariant currents, also but energy axial-vector expressed in potential, current. in this casewritten terms the The in of coefficients terms components the depend of linearly covariant of on axial- the the the torsion, torsion they tensor. give Once rise to new torsion-induced contributions to the geometry methods employed inapply the the analysis techniques of developed anomalous insion fluids. [ in In hydrodynamics, the with presentcoupled and work, to without we an Nambu-Goldstone external bosons. electromagneticand vortical field, effects. For we a The verify axial-vector charged the current, existence fluid on of the torsional other magnetic hand, does not contain any cor- be recast in terms ofglet its dual axial-vector vector current, field. which is Thisto external affected spin source by transport couples a were to ’t the also Hooft fermionic recently sin- anomaly. studied in Torsional contributions [ tisymmetric part of the torsion can be readily computed using the standard differential function contains other contributionsated which with are the induced effective byIn axial-vector the ref. field triangle [ encoding diagramstemperature. associ- the antisymmetric The part existence offrom of fermions the a minimally torsion. coupled chiral to magnetic the and antisymmetric part electric of effects the torsion was tensor, found, which can resulting Hall viscosity and the meancurrent, orbital which spin is per sourced particlewas suggested by further a the studied connection in background withconstitutive ref. torsion. the relations [ spin of In a the fermion gas relativistic on setup, a thistention link lately [ depends of the square ofrole the of cutoff, this or term in anyfor other condensed example, matter relevant [ UV physics scale has been of explored the in theory a [ number ofof works Hall (see, transport [ background on which fermions propagate [ theory [ term [ and is quadratically divergent. As a consequence, the coefficient of the Nieh-Yan term as the case of Weyl semimetals, this geometry provides a static nondynamical effective JHEP05(2021)209 3 we (2.4) (2.1) (2.2) (2.3) B . Writing 1 AB η = 2 ψ, } B ABC . , γ κ 5 A  γ 5 , γ D , γ { ψ, γ  D γ BCA ω / ψ BCA ∇ ←− is devoted to the analysis of the A ω ABCD ψ ] 2 γ ] C i ABCD − C γ i + γ B [ B + [ γ / ψ ∇ −→ AC A ψ η γ ψγ ψ BC  B 1 4 1 4 ) η γ , we get the following expression of the left e A + − – 3 – γ − ABC BC A κ ψ A ] − (det , after a brief discussion of the linear coupling of µ κ AB B ∂ ψγ x η 4 γ µ . To make our presentation more self-contained, we 4 µ + AC A C ] A d ∂ [ η e 5 γ µ γ B A A Z BC 51 C γ γ e = , A 1 2  γ ] ] relevant to our discussion. ω = = 1 4 1 4 ], on the other hand, do not receive any contributions linear C 50 = , 20 γ The action of a massless Dirac spinor minimally coupled to = / 20 + + ∇ ←− B S [ 26 / ψ ∇ 2 −→ ψ γ BC A / ψ / ψ ∇ ∇ −→ −→ A γ some basic facts about geometric torsion, while in appendix ω = = A / ψ ∇ −→ is spontaneously broken to its vector subgroups. We compute the corrections R . U(2) A × L The present article is organized as follows. Section torsion The basics of geometric torsion, as well as the notation used in the following, are summarized in 2 where we indicate by aand bar have all used geometric the quantities gamma referred matrices to identity the Levi-Civita connection appendix covariant derivative in terms of its Levi-Civita counterpart and the Dirac matrices verify thethe Minkowskian Clifford full algebra spin connection incontorsion terms of tensor the auxiliary as torsionless Levi-Civita connection and the where the left and right covariant derivatives inside the integral are defined respectively by We begin with thea discussion of spacetime the with dynamics torsion. gravity of can massless be Dirac written fermions as propagating [ on findings are summarized inreview section in appendix list some expressions of ref. [ 2 Equilibrium partition function and covariant currents with background equilibrium partition function ofthe a computation of charged the plasmawith covariant in currents. the the calculation This of presence modelsenergy-momentum linear of is tensor. torsional further torsion, contributions elaborated In including to inNambu-Goldstone section the section bosons spin to energy torsion potentialcorrections in and to the the the Abelian constitutive case, relations we compute of the a linear two-flavor torsional hadronic superfluid. Finally, our to the covariant currents and transportexistence coefficients of linear a in torsional the chiralseparation torsion electric effects tensor effect and found mediated find in by the the [ in two the charged torsion. pions. The chiral U(2) JHEP05(2021)209 (2.8) (2.9) (2.5) (2.6) (2.7) , which ABC ABC κ and mixes for ) . ] considered an ψκ  BC  ) is in order. At . , which as shown BA 48 5 ]. ] η B γ . A BC T 2.1 52 A D BC γ  [ ABC γ T A A κ − κ .  S 5 ) gives BC  ψ γ AC 5 η ABCD γ D DA 2.2 γ B A i ABC γ ψ ( 5 + ψγ ψκ γ i  2 D , ABCD 5 BC γ η − i ψγ CD D A i 4 γ + B γ ), the authors of ref. [ / ψ T ∇ ←− + − AC ψ 2.1 η CD ABCD − AC / ψ B ∇ ←− η γ i ACB AB ψ B  κ – 4 – ), we arrive at the expression / ψ − + γ ∇ 1 8 −→ C −  γ ψ − ] 2.1 ψ AC BC  B 1 4 / ψ η η = ) ∇ −→ γ e A B ψ A A + [ γ γ S   ) − ψ ψγ (det e / ψ ∇ −→ 4 1 4 1 x ψ 4 BC  − − d )] is given by the components of the torsion tensor η (det ) A e / / x Z ∇ ∇ ←− ←− γ 4 A.9  ψ ψ 1 2 d ψ (det = = = 1 4 Z x 1 2 / S 4 ∇ ←− + d ψ = Z ). These new terms are associated with novel transport coefficients in the [see eq. ( / ψ is a “composite” expressed as the Hodge dual of the antisymmetric part of S ∇ ←− 1 2 ψ A S 3 = − S ], we call screw torsion. The minimal coupling to gravity selects only one among all 48 In addition to the couplings shown in eq. ( Before proceeding any further, some clarification on the action ( (see section constitutive relations for the corresponding energy-momentum and spin covariant currents. additional coupling of theso-called Dirac edge fermion torsion vector to field an is external proportional Abelian to the vector torsion gauge vector field. This gauge field the torsion components, whichbecause is once expressed the in “fundamental” aonly external coordinate basis on source. the the components This torsion, oftensor but is this gauge on and important, field the the depend metric not spinabsent tensor energy in as the potential well. theory of As a of the result, a theory the “fundamental” include energy-momentum gauge contributions field that axially would coupled be to a Dirac fermion ref. [ possible dimension-four operators coupling Dirac fermions to torsion [ face value, the theory itcoupled describes to seems to an be external equivalent to gauge that field. of a The Dirac crucial fermion axially difference, however, is that this external Thus, the wholefied effect through of its axial-vector background coupling torsion to on an external the effective dynamics gauge field, of which, the following fermion is codi- This form of the action suggests the introduction of the effective vector field to write contains the symmetric componentsThis of means the that contorsion in fermionsin the only appendix two couple last to indices, its cancels antisymmetric out. piece, The important point here is that the term proportional to Plugging these results into the action ( A similar calculation for the right covariant derivative in eq. ( JHEP05(2021)209 ), 2.8 (2.13) (2.14) (2.10) (2.11) (2.12) . 4 , which remains , guarantees the V = 0 2 . δ 3 T ? 3 4 = , can be written then as , ). . S T . d µν C 2.12 ? δ e αβ =  B 3 4 S e δβ, αβ ⇒ S − F A σ = e + T = σ , we also assume that the microscopic fermionic T ∇ ABC – 5 – S In the context of hydrodynamics, the coupling of , corresponds to the following transformation of T 3 8 d ? 1 3! − dα 3 4 αβ T −→ T ν = + = We have seen how the action of a Dirac fermion mini- = T T µν ) and only consider the coupling to the screw torsion, in S αβ S is factored out of the components of differential forms. F i 2.1 µν  − S → S 8 1 − = ) above, we see that the gauge variation of the screw torsion vector T µ , which can be used to define the three-form ], here no ] S 2.11 20 denotes the codifferential acting on a three-form. In components, this , ABC [ 19 T a four-form. The nihilpotency of the codifferential, ? d? ≡ ]. Here we are going to employ the differential geometry methods introduced ?α ] to build the equilibrium partition function for fluids with torsion. In the fol- 10 , ≡ − ABC 9 ∼ by an exact one-form, T δ 20 , β S Besides its coupling to torsion through Looking at eq. ( Unlike in refs. [ 19 3 presence of background torsionmore on general a case generic of static a two background flavor geometry. hadronic (super)fluid Afterdegrees will this, of be freedom the analyzed are in coupled section to an external vector Abelian gauge field the background torsion to thetion microscopic for fermionic the degrees construction of ofa freedom the gives fluid effective the [ functional prescrip- describing thein long-range [ excitations of lowing, we study the case of a fluid coupled to an external Abelian vector source in the with gauge invariance of the screw torsion fieldThe strength equilibrium ( partition function. field the torsion three-form where equation reads which can be written using the Hodge star operator as The field strength of the Abelian screw torsion, tensor Its four independent components are encoded in the screw-torsion according to eq. ( minimal coupling prescription ( the understanding that inin all the our physical results electromagnetic the field. edge torsionRemarks would on be gauge reabsorbed invariance. bymally a coupled to shift gravity only depends on the fully antisymmetric components of the torsion many practical purposes with the electromagnetic field. In what follows we stick to the JHEP05(2021)209 (2.21) (2.18) (2.19) (2.20) (2.15) (2.16) (2.17) equals 1 S ], acting according to , j 20 dx i ,
a dx 0 ) V + , x . , d i i ( 0 0 σ , V ij u, , u, − dx dx given by g 0 d 2 dx V 0 i i 2 V σ , where the length of the V S V S ue u + σ 4 − σ σ e SF 2 − SF D + ≡ ≡ − i ue e 5 e i i i 0 Z × D V + − ≡ − − 2 dx
dx dx . Vector fields are then written in terms 1 )  1
0 ) = 6 π   u S ]). Keeping only terms linear in the torsion, i V S i i x 4 S T σ ( a a da i − F 19 0 0 = dx = 0 = – 6 – = e i a , . We implement dimensional reduction onto the V 5 µ µ a 0 V E d + x − V − S CS D + F u + ] dx dx 0 i i , − µ µ 0 V S V S + S , dx (   F V h dx 0 V 5 ) B d  e x = = ω σ Γ[ ( ≡ = = e S ≡ S V ≡ V σ , S 2 V − φ V e i F ∂ − = u − = i 2 a ds → i a is the vector field strength. This gives the Bardeen form of the anomaly, V and d φ = is a five-dimensional manifold whose boundary is identified with the Euclidean + 5 V 0 D F x to be the generic static line element 5 → 0 A similar electric-magnetic decomposition can be written for the vector field strength and the KK-invariant spatial one-forms are defined by where we have introduced the four-velocity one-form the inverse of theof components equilibrium that temperature remainx invariant under KK transformations [ and take all fields tocompatified be Euclidean independent time of by setting where four-dimensional physical spacetime. Tothe compute Chern-Simons the equilibrium effective partition action,∂D function we from take the metric of the four-dimensional spacetime where which explicitly preserves vector gaugeSimons transformations. nonlocal effective The action properly encoding normalized the Chern- linear effects of torsion is then given by sions, the (nonlocal) anomalousthe part Chern-Simons of form the (see, effectivewe for have action example, can [ be computed in terms of anomaly-free and will be eventually associated to the electromagnetic field. In four dimen- JHEP05(2021)209 S . ) (2.28) (2.25) (2.26) (2.27) (2.22) (2.23) (2.24) m , dx k 2.14 . Using in i dx ) as well by E k j , whereas 0 β 0 0 and magnetic ∗ S dx T 2.10 j E i a j 2 0 ` − T i a jk 0 2 T −  σ transforms under ( ij 2 ` e , T T 0 . The electric  + S ` d  a da , σ k σ 0 B − , − 2 ` e E ) does not have any magnetic compo- . T , ue . β ∗ T j E + B E a ⊥ ) = u T + δ  2 2.14 u u j  ∗ T − σ ∗ T . 0 + − 0 , n σ − + E da 3 4 S e T e E B 0 i jk ( B ` 3 4 dx − implies the same property for E a S – 7 – d T ). Since under four-dimensional Hodge duality k T 2 u ? in eq. ( = =  + B = − ∗ T dx ` − + T 0 −→ T −→ T β j S a S S S T = ij σ , E 0 B F 2 dx  T T B  e T  T k ) ? n  0 + ≡ = 0 ` σ ` 2 k T S e 0 T j ` 2.18 F k a + T a 2 2 j `j 0 the three-dimensional Hodge dual (not to be confused with its − g ` − 2 T ∗ h jk ) and ( i` ` kn g ` mi T 2 g T  h 2.11 and we have used that  i` σ ), we can write the screw torsion field in terms of the components of the ijk g  j` − , with e g σ ∗ V )]. The equivalent expression for the field strength associated to the screw . . As a consequence, only the electric part of ijk d − d 1 1 2.24 3! 3!  ∗ e E S σ 1 8 d ≡ e = = 2.36 = uβ 1 8 − = V E B = ⊥ T T S = = δ F β 0 F S Finally, we implement the dimensional reduction on the three-form ( S components of the field strength will be later identified with the electric and magnetic torsion tensor as We see thatundergoes the the gauge standard invariance gaugeaddition of transformation eq. generated ( by the zero-form we find from eqs. ( where four-dimensional couterpart denoted by the electric and magnetic components interchange Being a four-form, thenent, gauge function In a coordinate basis, the electric and magnetic components are respectively given by with decomposing it as B fields [cf. ( torsion reads where JHEP05(2021)209 ], the (2.34) (2.31) (2.32) (2.33) (2.29) (2.30) ] . 19 i 19 , 2 0 V 11 0 S 2 ) , da . We observe that, S  + ( 2 0  = 0 V V , i As shown in [ da F S cov 0 i + 4 F 0 S . inv 0 S S ] V  V 2 S F 0 inv F + 2 hJ ] , 0 . V V + S j S V V =  F , , 2 . da , F 0 0 dx  0 0 S V 1 2 i V V V S V , 3 cov Z , 0 i  + S da dx SF SF 0 0 0 S 0 + 2 2 2 ij V V , V V + T [ 1 1 π π S 0 2 V 2 1 F 6 2 V 2 1 V da hJ π 0 W [ F 4 V 0 F S = = + ≡ – 8 – W S   δ F = h 0 V 2 S δ BZ BZ S 4 d δ 1 i i π 0 h Z F D 2 S T 2 V δ 0 = 1 anom π 0 does not pick any linear dependence on the torsion. ] T hJ = = hJ 2 S T 2 1 π F = = , da cov 4 cov i i ] S S S , = cov 19 J i J h V h V , inv ] 0 J h S , ) and coming from triangle diagrams. , da 0 V S [ , ]. We begin with the one associated with the vector current 2.25 W V , 20 , 0 S , 19 0 , V S [ 11 W In addition, the corresponding Bardeen-Zumino (BZ) currents are given by [ The covariant currents can be now computed from the invariant part of the partition Having arrived at this parametrization of the effective screw torsion, we proceed to As stated above, in this work we do not consider the sector associated with the Nieh-Yan anomaly. 4 In both cases, theunlike zero the components vector vanish, current, These covariant currents will be very relevant in the following. whereas for the one corresponding to the torsional axial-vector gauge field, the result is function [ In the second expression,ated we have with introduced the components of the field strength associ- and gauge invariance ( dimensionally-reduced effective action splits into apiece, local anomalous respectively and given a by nonlocal [ invariant is what distinguishesgauge our field theory through from theconsequences axial-vector that for current. the of constitutive As a relations already of Dirac pointed the out, fermion energy-momentum and thiscompute coupled spin has the to currents. important terms an in external the effective action induced by the ’t Hooft anomaly affecting the This dependence of the screw torsion gauge field on the metric and torsion components JHEP05(2021)209 ) ]. . i 20 , µω 2.32 (2.37) (2.38) (2.39) (2.40) (2.35) (2.36) 19 ) of the + 2 i B 2.21 → i B ) 2.22 in ( . , i  S 0 i V ω B d S S , . µµ +   k k  , a a + 2 , j j k S da i S ] and differs from the one used in [ ∂ ∂ a  0 . 0 0 j , . 11  V V S ∂ 0 i 0 µB da 0 V S + + + ijk + σ σ V µω  k V k i , − − σ 0 V + e e S + B e F j j V e i  i 1 2 – 9 – V ∂ ∂ ≡ ≡ S B 0 ∂   µ e F − S σ S µ    h − µ ijk ijk = 2 2 2 2 e   1 i π π 1 1 π π µ 5 2 2 2 2 = = ω = i i i S = = = = E B B e e BZ BZ cov cov i i i i ], covariant currents are the ones relevant for the analysis of i 0 5 V i V J em J h 20 J h , h hJ 19 , )]. 14 , 11 2.21 ). The right-hand side of the second expression, on the other hand, gives the ) to write the electromagnetic and axial-vector covariant current components the elementary electric charge. We also define the screw magnetic field as the 2.37 e 2.33 To connect with physics, we need to identify the electric and magnetic fields. These Our definition of the physical magnetic field follows [ 5 chiral magnetic and vorticaltogether effects, with both a mediated nondissipativefield by transport ( the of torsional charge chemical driven potentials, by the screw torsionExpressions in magnetic these two references can be adapted to our definition by the replacement As argued in [ transport. Thus, the conclusion to be extracted from our result is the existence of torsional With all these definitions inand hand, ( we take the Hodge duals of the covariant currents ( and the electromagnetic and torsional chemical potentials with three-dimensional Hodge dual of the magnetic component of In addition, we introduce the vorticity vector vector field strength. Thus, we have Notice that onlysions the [see magnetic eq. part ( of the vectorare field naturally strength given enters by the in electric these and expres- magnetic parts in the decomposition ( not have any correctionother at hand, linear after order dimensional in reduction the we torsion. find For the BZ vector current, on the The second equation is quadratic in the screw torsion field, so that the BZ current does JHEP05(2021)209 . i 5 J (3.3) (3.1) (3.2) (2.41) (2.42) .  0 S δ . 0 inv in terms of its dual S , δ
δW k cov i . S S + m ijk J S  h dx δ . 0 ) and isolating its local con- bulk terms  V ∧  k ` + j E S 2.31 ∂  inv j with the axial-vector current, S dx 0 ), we write this variation as 2 F S i δ S S 1 π δ δW J δ cov 2 ijk cov   S, 2.33 i 0 i inv = d i S + S , i J δ i − J h S h δW BZ  B i i 3 S 4 B Z km S + Z i µ g σ D em 0 e  1 J S j` T + 2 2 – 10 – g ] as the electromagnetic current associated with e e π π dδ S = − h 2 2 δ ijk 48 S  inv S = = inv cov 1 inv F 2! ] i δ F S δW δ BZ BZ i = em δW  F i i J , 4 3 i Z em Z V cov S D em 0 i J ≡ h F h = = J S , h 0 J h S inv cons ] 0 , i 1 S 0 ) and recasting them in term of electric and magnetic fields T i V em F [ J , ]. In doing this, we have to keep in mind that the screw torsion is a h 2.35 V δW 20 F , , 0 S 19 , , 0 11 V [ δW We also give the components of the vector BZ currents by taking the dual of the The previous expression can beone-form made current, more defined explicit as by writing Using the definition of the covariant current in eq. ( written as tribution [ composite field that depends notas only well. on Thus, the in torsion computing tensor,sider, variations but in with on addition respect the to to metricthe the the components metric implicit explicit functions dependence dependence we of in should the con- the invariant screw action torsion. on these This sources, latter also contribution can be generically 3 The spin energyThe potential knowledge and of the the energy-momentumlation covariant tensor of current associated torsional contributions tospin to the energy other screw potential covariant torsion physical andvariations allows of quantities, the the the most calcu- energy-momentum invariant notably part tensor. the of the These partition are function ( obtained by taking whose divergence vanish identically due to the presence of the Levi-Civita tensor. This BZ current wascharge identified transport in in ref. flat [ spacetime.above It by the differs however consistent from current the covariant current computed to the magnetic andorder. vortical Notice separation that effect here without we have any identified torsional the corrections current expressions at in linear eq. ( standard result for the axial-vector current of an Abelian anomalous fluid, corresponding JHEP05(2021)209 ), ,  0 (3.9) (3.7) (3.8) (3.4) (3.5) (3.6) ] k 0 2.28 26 δT − k , 0 0  cov δT k i 0  ` jk 0 cov δT . Ψ
S, j  , h i 0 j i ` k a ` δS J a ` cov h a j S, − δT cov σ i a . i 2 S, e − J i cov νσ ijk h i , µ k i  −  J k 0 σ ` h j 0 ), to give jk 0 δT a `j ` ij ` ge g δT Ψ a 3.4 δT  cov √ h g g σ j , i  2 a x ijk √ . j cov e , 3 νσ  cov a x d S, mi µ σ 3 − S, i cov g cov i − 3 i d + Ψ , i Z , S, e h S, S J `j i 3 ijk J i h g g 1 4 i Z 1 4 i  h S cov cov  cov  J ` σ J √ i i j i h = a h − + e k x a j jk – 11 – ijk jk 0 jk ` ` 3 1 4 a 0  0 0 jk ijk ijk jk a d ` 0 inv σ   σ ] Ψ Ψ Ψ δT + ijk 3 σ − σ 2 h h h S Z  S j ` e e e ` e δT δT σ jk ` cov a a a F 0 1 4 1 4 1 4 e a , − S, cov 1 4 − − − = 2 − + − i V , δT i mi S, `j The covariant spin energy potential is defined as [ i g J g F = = = = i h mi inv cov cov cov cov ,  ` J g i i i i 0 ijk h a k k  cov cov cov cov S ijk jk 0 jk 0 ) can be computed using ( δW ijk i i i i σ ,  ` ` 0 0 ijk ijk  k k e 0 σ   jk jk 0 0 σ Ψ Ψ Ψ Ψ 3.5 8 1 V − σ ` 0 ` 0 σ e [ − 1 8 Ψ Ψ Ψ Ψ ge ge ge ≡ h ≡ h ≡ h ≡ h h h h h δW √ √ = − √ x x cov cov cov cov ) takes the form x m 3 3 i i i i 3 k k d d d jk jk 0 0 3.2 δS ` 0 ` 0 3 3 3 Z Z Z S S S Ψ Ψ Ψ Ψ 1 4 1 4 1 4 h h h h − − − = inv δW Notice, however, that only onephysical of KK-invariant these combinations quantities is KK-invariant. We therefore define the plus bulk terms thatof we the omit. spin This energy allows potential the in identification terms of of the the covariant covariant screw-torsion components current as so the left-hand side of ( screw torsion field, so we onlywe have write to care about the implicit dependence. Using eq. ( The spin energy potential. All dependence of the invariant effective action on the torsion tensor is included in the This expression will bethe utilized spin energy next potential to and evaluate the the energy-momentum torsional tensor. covariant contributions to With this, eq. ( JHEP05(2021)209 6 ]. ] 53 20 , (3.14) (3.11) (3.12) (3.13) (3.10) 19 . ). In doing this, inv ] , which gives the explicitly depends σ . j , da 2.17 . inv To evaluate the com- S S σ , W appearing on the right- cov δ S, V )] σ i , cov i 0 S, J δσ, 3.4 , S i h , i  0  J k cov h k 0 V , ` [ S, 0 ij ` i T i T W cov J j` g g h  j` S, g i g √ `j S i g im x δ J F 3 ijk g two-form indicates that the components h δ  d ijk σ 0 µ depends on the metric since its computa-  ijk ijk 3  σ − Z  T ) S ]. The same can be said of the contorsion e σ − σ − S e e 1 2 − = αβ ( e 4 4 1 1 – 12 – 51 µ V , . , 1 2 + κ − − cov δ i F i + δ S 50 = 0 = = = 0 00 , 0  are evaluated next. As pointed out above, in this case m Θ σ S h cov cov cov cov 2 45 i i i i σ − V  e cov , 2 k k i ) jk jk 0 0 i − − = 26 0 ` 0 ` 0 parametrizing the static line element ( δ da = Θ Ψ Ψ m Ψ Ψ g e ( ij h h h h h δ S g √ σ  cov x δ i 0 3 T d 00 and 3 of the energy-momentum tensor as [cf. ( Θ i = Z S h a , whereas the dependence on the other two functions comes exclusively i − , cov a i σ expl cov i 00 Θ Θ h , while its symmetric part h ] ), the variation of the screw torsion with respect to αβ [ )]. Another important point to take into account is that ), we conclude that there are no linear torsional corrections to the spin energy po- µ 2.28 κ are metric-independent [ . This can be computed from the invariant part of the effective action using [ A.10 2.33 i The mixed components We begin varying the invariant effective action with respects to Other contributions to the spin current in the absence of torsion have been recently computed in [ a 6 νσ µ we have also aon contribution coming from the explicit dependence of the effective action so we arrive at the result Using ( hand side is given by through the screw torsion. components Concerning the torsion, theT definition of the tensor tion requires raising and[cf. lowering ( indices ofon the the KK metric-independent vector torsion components Torsional contributions to theponents energy-momentum of the tensor. (covariant) energy-momentum tensor,metric we functions take variations with respectit to is the very important to keep track of all dependence of the effective action on the metric. This shows that all dependencethe of the components spin of energy potentialseq. the on ( covariant the torsion current comestential. associated through Notice to that the this quantity screw is nonzero torsion. even for backgrounds Thus, with using vanishing torsion. which are found to be JHEP05(2021)209 ). ) is 2.28 (3.19) (3.20) (3.21) (3.15) (3.16) (3.17) (3.18) ). 3.21 3.15 j . δa ), and (  cov . k  ` S, 0 k ` i . . ` a δa , 3.18 T j J ij j ` cov h , ∂ i a S j 2 0 δg S, k ), ( a i i 0 S V δ + j `  k g 0 0 k J δ k T j S . 0 h 0 cov j ) 3.13 0 j  T a S, cov i  + j T i i T k i a ω i i  k + 0 ( ) J S J j S k h h + mi `k j µ 0 T 0  g ij 2 ij k ∂ i 0 ( 2 µ 0 T 0 m` ijk  g g V `k g g g T   σ  √ √ 1 4 + 4 σ + i`k − x  x − ). They are codified in the screw torsion i S k 3 e i`k 3 2 + e  V d d 1 4 B ` 1 2 j 2 − 3 3 ∂ 2.10 Z Z µ + δa S 0 S  + + 2  k V ` = + m  = 0 k 0 ` 0 k S i i S ` S – 13 – 0 ij T ), which does not depend on the torsion. After i B ij ij 2 j T δa g S g T j  δg a j a 1 2 2 − a 2.40 2 µµ ijk cov impl cov ` 2  2 − i − i ) i σ − j  ) ij 0 − g j σ − m jk i ` δ jk Θ e e Θ ( ` i jk h h T 2 2 ( k i S σ T g δ  1 1 π π ), a number of new terms appears in the constitutive relations 2 T  k − 4 4 √   ijk S x mi − −  g e  3 2.28 g = h `jk d  √ = = h = mi ijk x 3 σ cov g  Z 3 defining the three-form ( S appearing on the right-hand side of eqs. ( i e m d 8 1 8 1 ] 1 2 expl cov 1 8 ij S i 3 − − g i Z Θ cov S − 0 δ µνα h [ S, = = we discussed how, out of the 24 independent components of the torsion Θ T i = h i , whose components are given in terms of the torsion tensor in eq. ( 2 m J µ h S S a , the microscopic Dirac field only couples to four of them contained in its anti- impl cov δ i i 0 µνα Θ T h In section Finally, we evaluate the spatial components of the energy-momentum tensor tensor symmetric part gauge field The antisymmetric components ofthe the gauge torsion currents tensor and are the the spin only energy ones potential appearing tensor, in which both are fully written in terms The current the one given insubstituting the the second explicit line expressionseffective of field of given ( in the eq. ( KK-invariantof components the of energy-momentum the currents which axial-vector are linear in the torsion. Thus, we arrive at where the variation on the right-hand side of this equation takes the form This expression has to be added to the explicit contribution computed in eq. ( This leads to The implicit contribution, on the other hand, is readily computed using where the variation of the screw torsion with respect to the KK gauge field has the form nents This is a spatial two-form, whose three-dimensional Hodge dual gives the sought compo- JHEP05(2021)209 . . V αβ µ ]. T (3.22) (3.23) (3.24) at the ) plays 45 U(1) and → = 0 3.23 R µν A G and two axial- ) U(1) V , dimensions [ × 0 L ). This state of affairs V ( 2 + 1 2.28 , . Since the axial-vector and cov ) i S , , 0 αµν  j S Ψ ). This is the reason why we can ( h 00 T α k αµ 2.17 ∗ a ∇ α T − − j + in the effective action. Setting 0 cov k µ i ) T ∇ S – 14 –  µν ik Θ = + h g µ ∗ A ∇ = = , ] 0 j 0 S i cov 50 i ) shows that they contain additional components of the , T + and the screw torsion 0 26 ) µν (can) A 3.21 ( A T , h 0 A make it easy to compute the interplay of torsion with the dynamics ( 2 ), and ( enter the energy-momentum tensor through 3.18 i 00 T . In the case of the energy-momentum tensor, however, an inspection of our ), ( i S 3.13 and Finally, it should be pointed out that although our analysis includes the linear effects A relevant question here is whether the canonical energy momentum-tensor ( It is known that in the presence of torsion the Ricci tensor acquires extra terms that 0 S of Nambu-Goldstone boson forOur the theory symmetry is breaking characterized pattern byvectors, three U(1) the Abelian auxiliary gauge fields:screw a torsion vector fieldsappear couple in in the the combination same way to the fundamental fermions, they always restrict our attention to the consistentthe energy-momentum tensor gravitational without BZ having to term. consider 4 Torsional chiral effects inThe results a of two-flavor section hadronic superfluid fields, the metric energy-momentum tensor andtwo the independent spin currents, energy which potential provide can the be response regarded to as the external sources of torsion, it doesgauge-gravitational not incorporate anomaly the in contribution the to background the partition ( function of the mixed any role at all inin our condensed analysis matter of applications torsional the fluids. backgrounda spacetime In way geometry deciding of is this, nondynamical, incorporating we but certainof should microscopic the not long features forget wavelength that of modes. the As system no into field the equations effective have to theory be satisfied by the background The form of the Einsteinby equations this is new then canonical preserved, nonsymmetric provided energy-momentum its tensor, right-hand instead side of is the given metric one. force to correct theon energy-momentum the tensor spin energy by potential a [ Belinfante-Rosenfeld term depending where the modified covariant derivative is given by while they cancel out from theis expression not of a the consequence of screw dimensional torsionto reduction, in but the ( a metric result components. of taking A a variation similar with situation respect was foundrender it also nonsymmetric. in This has important consequences for the Einstein equations, which results ( torsion besides the fourcomponents included in the equilibrium partition function. In particular, the of JHEP05(2021)209 . 0 R T σ (4.6) (4.3) (4.4) (4.5) (4.1) (4.2) − in the e U(2) α = × =0 L A T

], where we  20 ψ, anom ] o 5 γ i 3 t dα, . . . 3 α, i / . We see that the Nambu- + A ∂ 0 i S . S + A B   0 + t T   ], it is enough to implement the
5 2 A 0 / . µ S π 20 , and 3 π 6 ] , 2 + dα, σ V 1 g µ  ..., π √ 1 2 19 + 2 [ , S √ 0 0 − da = W V 0 x / 0 A 3 3 ψ + 2 S π − d −  h µ 2 5 i + π 0 1 . 3 √ γ Z 0 S S − 1 A anom −   F / ] – 15 –  S σ 3  i , t 2 , transforming in the flavor group U(2) − t 0 π −→ A 1 ... , the magnetic component of the screw torsion field e 3 √ f − A ,... A S 1 2 i µ / V 5 = 0 = S 3   B Z = + S 5 A / γ + A 0 and µ 0 0 , generating in the process a triplet of Nambu-Goldstone i t t ) and we also introduced the local temperature T V 0 A 2 WZW 1 V − ] / V π = exp 6  / V 2.37 ). Expressed in components, the relevant term in the WZW i i ..., U [ − SU(2) − − α, . . . [ W 2.22 × / / ...  ∇ ∇ −→ −→ W V n  = ≡ = = U(1) WZW ] D/ψ encoded in the matrix → −→ R ± was defined in eq. ( , π α, . . . [ 0 i S U(2) π W B × L The first issue to address is how to incorporate the torsional vector fields into our After this brief discussion of the Abelian case, we turn our attention to the analysis of into account the effect ofreplacement torsion in the analysis of ref. [ while leaving all remaining fields unchanged. takes the form and similarly for the right operator. The structure of these terms shows that, to take analysis. The left Dirac operator, including the vector, axial, and screw-torsion fields, We assume the system undergoesU(2) spontaneous symmetry breaking to itsbosons vector subgroups, torsional chiral effects inrefer the the two-flavor reader hadronic for details. superfluidfields, This studied fluid respectively in couples denoted to ref. a by [ vectorThe and values an of axial-vector these external background gauge fields are restricted to the Cartan subalgebra generated by where and the chiral chemical potential where the ellipsisNambu-Goldstone in boson to the the second backgroundGoldstone fields line boson couples indicates linearly to thestrength defined torsion-independent in coupling eq.action of ( reads the local WZW action takes a particularly simple form [ end of the calculation, the linear couplings of the single Nambu-Goldstone boson JHEP05(2021)209 (4.9) (4.7) (4.8) , and (4.10) (4.11) S = a , i They represent the ,
i β 7 to include Q ]. T  ,a ν , µ 3 20 Q S  P β α − S ∂ α . ∂ and − ] QU , ν , B 1 0 ,a 20  αβ µ 1 − ,  A 3 ,
S P
U S T V β µ T β . ν µ I T h + 2  S .  H α α
β 3 , β S S β Tr t αβ
∂ , ∂ 0 ∂ µ β α ν ν α 3 + 3 α + I A αβ ∂ ∂ S I T , V α 0 0 ν µ µ ν ν ν i H t S V S u u u u u 1 3 Q ν h  + 2 ν , – 16 –  S ∂ i = , while the charge matrix is given by Q . These should be added to the expressions found i 1 µναβ µναβ µναβ µναβ µναβ Q   µ µναβ µναβ    Q −  −   cov µναβ L µ µναβ i  2 2 = = = = =  c c L 2 π π c − UU 6 7 2 µ µ 5 aA π QU c N N ,S ,S µ S S π µ N + 1 µ 1 µ 3 8 24 24 J P N i∂ − R µ P P 24 ∆ − − −  h U R = Q = = = = h h h µ L and cov cov cov cov Tr Tr Tr i i i i A A V V ≡ ≡ ≡ µ µ µ µ cov 0 3 0 3 and i µ µ J J J J H I µ U T aV µ 5 ∆ ∆ ∆ ∆ µ h h h h J P ∂ 1 ∆ h − iU ] for the corresponding currents. The results are = 20 µ R In order to compute the longitudinal and transverse components of these currents, and With all this in mind, we can compute the linear torsional corrections to the covariant For the reader’s convenience, these are summarized in appendix 7 while for the vector structuresadd we a extend new the term notation write the constitutive relations ofof the scalar hadronic and superfluid, vector we structures,linear need coupling in to of addition introduce the to a Nambu-Goldstone those bosons number used to in torsion. ref. To [ the five scalar ones, we add The first thing to beto noticed the here three-flavor is components that of the the torsional vector couplings and of axial-vector the covariant pions currents. are restricted with where we have used the tensor structures introduced in [ gauge currents at leading ordersions, in here the we derivative only expansion.we give To denote the avoid by cumbersome terms expres- inin the ref. currents [ depending linearly on the torsion, that JHEP05(2021)209 (4.14) (4.15) (4.16) (4.12) (4.13)  3 T µ  . β  , ∂ µ 5 , α P   S 3 ]. As already pointed β ν 0 µ T S u 20 µ . α . −   ∂ 3 β ν , β ,S µναβ S µ 1 ∂ u ), as well as the local temper- α α P T  ∂ = 0 S T
ν ν a 2.39 µναβ u u  − + 6 , µ 2 + i P 7 µναβ H µ µναβ  S S ω µ H T  S , + −  3 6 µ , µ 2 3 µ µ 3 + S 2 c µ π B B 2 µ 0 , ω + c N S − µ π B µ µ 0 24 µ S N ,S S – 17 – S  B

aαβ 24 − µ 3 5 5 µ µ 2 V 2  P µ µ S π ν = = π c c + 3 + 3 2 2 T u c c 6 12 N N π π H N N H + cov cov 4 4 i i = = µ 4 2 2 µναβ A A = = h  µ σ  3 3 P 2 2 1 2 S J J cov cov c c π π i i µ cov cov ∆ ∆ N N i i = A A h h µ σ 24 24 0 0 − V V µ σ µ µ σ a µ 0 0 J J u = = B J J P ∆ ∆ h h ∆ ∆ µ σ cov cov h h µ i i µ σ u µ P V V u µ σ 3 3 P J J . 0 ∆ ∆ T h h σ . µ σ µ i − u 0 e P A = or T i 0 The longitudinal and transverse components of the covariant vector and axial-vector V torsional contributions to thegiven 0-flavor by covariant the vector corresponding andthat axial-vector terms there currents are of no are the linear just currents BZ torsional and, contributions currents as to linear the a inon 0-flavor consequence, component the no of torsion. the linear consistent torsional This terms implies in the WZW action depending Let us stress once more thatcontributions all to the the expressions longitudinal given here and only transverseshould represent components the be of linear the torsional added covariant currents, toout, that the torsion respective couples to nontorsional Nambu-Goldstone terms bosons found only in through [ the 3-flavor currents. The for the 0-flavor components, whereas in the case of the 3-flavor the result is In the case of the axial-vector currents, we have The longitudinal and transversehand, components read of the 3-flavor vector current, on the other where we have introduced the magnetic field ature currents are now writtenchiral in transport terms coefficients can of be0-flavor these vector read quantities. current from the The resulting new expressions. torsional We start nondissipative with the Here we have used the screw torsion chemical potential ( JHEP05(2021)209 )  − ), all 4.20 π (4.20) (4.17) (4.18) (4.19) + matrix, π 2.37 ) for the S Q T . We write µ -odd spatial µ  2.41 3 T , k V ) ∂ 3 j π = V ( µ O ijk 0 is given in ( . V ) + i S ]) 3 T  3 2 π  B π c 20 ( k ≡ f . N 2 E 3 O 2 µ j t π e
V , 6 S + µ ) and 0  ijk − cov V  i S  3 T V µ 2.36  µ + 3 − − +  i π µ π hJ k B + 3 j e ∂ S π V ∂ 0 µ i S + − π  ), this time induced by the j B π c + ∂ 2 according to (see [ 2 π c cov N π − i f Q 3 2 N ijk 2 t 4.19 µ V e − 2 µ 18 0 π 0 5 π ]. Interstingly, the coupling showed in eq. ( T  V – 18 – j µe 12 hJ π + ∂ 20 f e 3 c
+ 2 = 3 − π − π N 3 = −  t π π 12 µ S + 3 T + cov µ π − V is set to zero, which implies i π factor is identified as the one coupling to the  k k 3 µ = k + em t ∂ E V ∂ j 0 j t hJ S S cov µ ijk i 0 ijk A ijk V i ) and the generator  T  3  2 2 π π 2 c J π c c f f 4.9 f N 2 2 ∆ 2 N 2 h π π π 2 ieN e µe 12 12 12 − − + = defined in ( cov i Q i em As for the transverse axial-vector currents, we see that the only coupling of torsion to To find the expression of the covariant currents in terms of the physical electromagnetic J ∆ h tion of a charged fluidby minimally the ’t coupled Hooft to anomaly gravity.of associated The the with terms torsion the studied tensor. screw arevortical torsion, In those chiral dual the induced torsional to Abelian the effects, case, antisymmetrictorsional whereas our part corrections. the results axial-vector show current the does existence of not magnetic have and any linear is the only torsion-induced termthis involving order. the neutral pion in the constitutive relations5 at Closing remarks In this paper we have analyzed the linear effect of background torsion in the partition func- There is therefore noand torsional vortical corrections separation to effects found the in pion-mediated [ chiral electric, magnetic, Abelian case. Weneutral see pion. that There there exits nonetheless iselectric a effect no torsional given torsion-mediated pion-dependent by contribution electromagnetic thescrew-torsion to coupling first field. the chiral term to in the eq. ( pions arises from the 3-flavor component where the electric and magneticfields fields here are defined being in KK-invariant. eq. Thecurrent ( last, of pion-independent term the is unbroken the theory, BZ electromagnetic and replicates the structure found in eq. ( whereas the field couplingnow to the torsional terms in the covariant electromagnetic current in terms of the pion fields as fields, we expand thematrix KK-invariant components of the vector field in terms of the charge As usual, the unbroken U(1) JHEP05(2021)209 ]. It is 45 dimension [ ], which explicitly depends 2 + 1 35 , 34 – 19 – ]. 20 ]. We found that no new couplings of the Nambu-Goldstone bosons to 20 , whereas no torsional vortical effect appears. Interestingly, there are no torsional ± π In this paper we have exploited the analogy between background torsion and axial- We have also studied linear torsional chiral effects in a two-flavor hadronic superfluids In this same model, the covariant spin energy potential and energy-momentum tensor A Basics of spacetimeIn torsion this appendix, we givewith a brief torsion. overview of The the focus main mathematical will features lie of spacetimes on the basic differential geometric aspects, with further Acknowledgments We thank Karlish Landsteiner for Science discussions. MinistryPGC2018-094626-B-C22 (MCIU/AEI/FEDER, grants This EU), work PGC2018-094626-B-C21grant as IT979-16. has well (MCIU/AEI/FEDER, been as EU) supported by and Basque by Government Span- on the UV cutoffin scale condensed of matter, the where theory.explore this This cutoff the Nieh-Yan arises physical term naturally. implications haspaper It of been for would this shown the be to triangle anomaly interesting be contributions. along to relevant This further the issue lines will followed be in addressed the elsewhere. present that were found in [ vector couplings to compute the linearwith effects of an torsion, which axial-vector come current fromsector triangle coupled is diagrams the to one the associated background with screw the Nieh-Yan torsion anomaly field. [ A different torsion emerge from therents. 0-flavor components The of analysis of the the covariantexistence electromagnetic vector of transverse and current, torsional axial-vector on chiral cur- the electricbosons other effect hand, mediated shows the by thecorrections two to charged the Nambu-Goldstone pion-mediated electric, magnetic, and vortical chiral separation effects vector gauge field on bothtorsional the theory metric genuinely and different thean from torsion components. external the gauge theory This field, is of as what a it makes Dirac the is fermion reflected axially instudied coupled the in to constitutive ref. relations. [ the covariant axial-vector currents, thetensor. coefficients Thus, now we do find dependactually linear include linearly components torsional on of corrections the the torsion to torsion This tensor the latter that energy-momentum do situation not tensor, is appearimportant which analogous in to the to effective stress action. the that onethe the spin already torsional energy found potential contributions in are to associated the with energy-momentum the implicit tensor dependence and of the effective axial- have been computed inonly terms depend of on the the axial-vectorto metric covariant depend current, functions. on with the Since coefficientsnot torsion the that exhibit at linear axial-vector linear torsional current order, contributions. hascovariant we energy-momentum The been conclude tensor. situation shown that is Although quite not the its different spin components in energy are the potential also case written does of in the terms of JHEP05(2021)209 . B = in ]). A ω A A = 0 e (A.5) (A.6) (A.7) (A.8) (A.1) (A.2) (A.3) (A.4) 55 , T { AB 54 , η ∇ 51 , 50 , takes the form 27 , B A 26 τ The spin connection . B 8 , where here and elsewhere C ω C = 0 A , τ and satisfying C e . AB +1 A η p B B e , , C e , . , ∇ B 1) C ω B to be metric compatible, e e A − µν BC C B ), we write the torsion two-form A A = 0 g B ω A e κ ω ω BC + ( B = − A − e A.6 + BC B ν + ω B A C B B A = A B τ e A T A ω ω µ C B 1 2 – 20 – the spacetime metric. A A de e -form tensor of the type dω e ≡ + ω B p = ) and ( = µν A B = A g + AB A A A κ η de κ B T B A.3 T A A = R dτ A T = B A τ ∇ to denote exterior products. ) ∧ ( associated with the same tetrad basis B A ω . To parametrize torsion, it is convenient to introduce an auxiliary torsionless the flat Lorentz metric and = 0 } AB ) µ η The torsion two-form can be expanded in the tetrad basis as In this work, Lorentz indices are denoted by capital Latin letters, while spacetime indices are indicated dx BC 8 ( µ A A terms of the contortion one-form as by Greek letters. Lowercasewe Latin omit indices the wedge are reserved for spatial components. To make notation lighter, which, being the difference oftransformations. two connections, Combining transforms eqs. as ( a tensor under local Lorentz This auxiliary connection isin also metric the compatible, paper weoverline. indicate The all contorsion quantities one-form associated is defined with by this Levi-Civita connection by an where the components onT the right-hand side are antisymmetricconnection in the lower two indices, Both torsion and curvatureunder are (infinitesimal) geometrical parallel transport. quantities related to the behavior of vectors Torsion is defined by the first Cartan structure equation while the second one gives the curvature defines the notion of parallel transport,operator, allowing that the construction in of the the particular covariant derivative case of In what follows, we assume the connection Let us consider a four-dimensionale curved manifold and an orthonormal tetrad basis with details being available in a number of reviews (see, for example, [ JHEP05(2021)209 ), 4.8 (B.3) (B.1) (B.2) (A.9) (A.10) (A.11) ] are expressed in . , ) 20 , 3 3) , π , , 3) are given in eq. ( ( , 3) 3) , O µ , = 0 T . This identity shows that + = 0 a = 0 ( C µ = 0 , a e 3 ( a  a ( V and (  . BC . − ] β A µ π L I κ νσ A [ + α µ . CB π ] = L κ T 2 π ν 2 4 1 2 f B BC 20 , L − A A µ  , µ a κ ), and + − T µ ω = µ a B µ  1 2 Tr ω ] µ B  A µ i + 3 T  µ − 4.13 T νσ I a BC [ π I a T µ µ , µ T = − T = µ = – 21 – = ) , ∂ = 1 2  ] 4 β ) β .  2Γ β , − I 3 β β π u aβ β β u − α ( π = aβ π ∂ α BC T I V [ ∂ ( α ∂ V α ) ∂ A O α α α α ν ∂ − = ν α O κ 3 I ∂ T ∂ ∂ ν ∂ u + ν ν ν ν − BC ν V u ( ν µ νσ + u u u u u  π µ A µ − u 0 µ µ T I κ µ µ π T ∂ π T I u . In the case of the longitudinal components of the currents, + µ + µναβ µναβ µναβ µναβ π ∂ 4     π µναβ µναβ 2 π π    2 2 µναβ µναβ µναβ f f ≡ ≡ ≡ ≡ 2 i 1 2 1 2 π    2 f − − µ 2 µ 4 ,a ,a ≡ ≡ ≡ ≡ ≡ µ 1 µ 3 P P = = = ] to write the constitutive relations for the two-flavor chiral hadronic 5 2 3 P P ,a ,a S S S µ µ 4 1 20 H I S S T Finally, the transverse components ofterms the of covariant currents the found four in tensor [ structures where the magnetic fieldwhose expansions is in defined terms in of eq. the pion ( fields are given by superfluid studied in section these are expressed in terms of the following five scalar structures B A summary ofFor expressions the from sake ref. ofintroduced [ completeness, in ref. we [ list in this appendix the scalar and tensor structures Similar expressions are obtained usingwith a the coordinate antisymmetric basis, part with of the the torsion connection being according identified to the antisymmetric part ofcomponents of the the contorsion torsion in tensor the two lower indices is determinedUsing by metric the compatibility, the symmetriccomponents piece as can be computed in terms of the torsion where in the second equality we have expanded JHEP05(2021)209 , (B.4) JHEP 103 09 , Phys. , (2012) 046 (2016) 2617 Phys. 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