Torsional Constitutive Relations at Finite Temperature

: The general form of the linear torsional constitutive relations at finite temperature of the gauge chiral current, energy-momentum tensor, and spin energy potential are computed for a chiral fermion fluid minimally coupled to geometric torsion and with nonzero chiral chemical potential. The corresponding transport coefficients are explicitly calculated in terms of the energy and number densities evaluated at vanishing torsion. A microscopic calculation of these constitutive relations in some particular backgrounds is also presented, confirming the general structure found.

In ref. [24] we examined the linear torsional constitutive relations at zero temperature for a fluid of chiral fermions minimally coupled to geometric torsion.A descent analysis was applied to construct the Chern-Simons equilibrium partition function from the sixform torsional anomaly polynomial.This in turn was used to find the general form of the torsional constitutive relations, all of them expressed in terms of the model-and cutoffdependent global normalization of the anomaly polynomial.
The aim of the present work is to go beyond vacuum contributions and study thermal linear torsional corrections to the constitutive relations of a chiral fermion fluid minimally coupled to torsion and in the presence of a nonzero chiral chemical potential.Exploiting the equivalence with the coupling to an effective axial gauge field [42,43], we determine the form of the equilibrium partition function linear in the torsion to first order in the derivative expansion.From this, the finite temperature constitutive relations for the chiral current, energy-momentum tensor, and spin energy potential are obtained, finding closed expressions for all of them.The corresponding transport coefficients are expressed in terms of thermodynamical functions evaluated at vanishing torsion, in particular the number and energy densities, as well as the derivative of the latter with respect to the chiral chemical potential.
We also present a first-principles computation of the finite temperature constitutive relations for three particular torsional backgrounds by solving the corresponding thermal one-particle Green function equations at linear order in the background data.An important feature of all the three cases considered is that they have nonzero spin connection and curvature.Besides serving as a crosscheck of the general results obtained, these analyses also shed some light on the interplay between torsion and other background data, most particularly vorticity.Another worth-mentioning result about the examples studied is that the torsional contributions to the vacuum, zero-temperature parts of the chiral current can be set to zero by an appropriate regularization prescription.The same happens with the Nieh-Yan anomaly, in the case when it receives contributions at linear order in torsion.
The paper is organized as follows.In section 2 we present the calculation of the general structure of the finite temperature linear torsional corrections to the chiral current, energy-momentum tensor, and spin energy potential of a chiral fermion fluid.Section 3 is devoted to the analysis of the finite temperature constitutive relations of a chiral fluid in a background with purely magnetic torsion and nonvanishing curvature that generalizes the one used in refs.[13,30] to describe torsional Landau levels.In section 4 a second geometry is investigated, this time with magnetic torsion and vorticity, extending the background studied in ref. [11], where all constitutive relations are free from torsional corrections at linear order.As a last instance, in sec.5 we consider a geometry describing a chiral fermion fluid with electric torsion and nonzero spin chemical potential for which the chiral current, unlike the other particular geometries considered earlier, is affected by a nonzero Nieh-Yan anomaly.Finally, our results are summarized and discussed in sec.6.

Linear torsional constitutive relations at finite temperature
In order to study the general structure of the linear torsional thermal corrections for a fluid of chiral fermions, we start from the action of a left-handed Weyl fermion minimally coupled to torsion and to an external gauge field A µ [44] where the covariant derivatives are defined in terms of the spin connection by In writing these expressions, we have introduced the matrices σ µ ± ≡ e µ a σ a ± , where e µ a is the vierbein and σ a ± are given in terms of the three Pauli matrices by σ a ± ≡ (±1, σ x , σ y , σ z ).
We also assume nonvanishing geometric torsion 1 and write the spin connection as ω a b = ω a b + κ a b , with ω a b the Levi-Civita part satisfying de a + ω a b e b = 0 and κ a b ≡ κ a bµ dx µ the contortion tensor [1,44].Using a number of Pauli matrices identities, the action (2.1) can be recast as The barred covariant derivatives are obtained by replacing the full spin connection ω a b in (2.2) with its Levi-Civita part ω a b , while the effective gauge field A µ is defined by The field encodes all torsion dependence, where the spacetime components of the torsion tensor are defined by and similarly for the ones of the contortion tensor, κ µ αβ = e µ a e b α κ a bβ .The effect of background torsion can be thus recast as a coupling of the chiral fermions to an effective external gauge field [42,43].
It should be kept in mind that in all our expressions the four-dimensional Levi-Civita tensor is normalized according to ϵ 0123 = √ −G and ϵ 0123 = −1/ √ −G, with G the determinant of the spacetime metric.Notice that the expression of the torsional gauge field one-form involves a Hodge dual, S ≡ S µ dx µ = − 1  4 ⋆ (e a T a ), and therefore its components depend on the background metric.This fact will play an important role later on in the analysis of the constitutive relations of the energy-momentum tensor components.
Using the identity σ + = G µν 1, the Dirac-Weyl equation derived from (2.4) can be recast as From this, we read the one-particle Hamiltonian operator 1 Greek letters are used throughout the paper for spacetime indices.Spatial coordinates are indicated by i, j, . .., while a, b, . . .denote four-dimensional Lorentz indices.For simplicity, we also omit the wedge symbol ∧ to indicate the exterior product.
which is self-adjoint with respect to the inner product with Σ t a constant-time section and G the metric determinant.
The equilibrium partition function.To study a fluid of negative chirality fermions in the presence of geometric torsion, we follow the effective action approach [45,46], assuming that chiral fermions propagate on a generic static background spacetime metric The fluid's equilibrium four-velocity, acceleration, and vorticity are related to the metric functions by the standard expressions ) while the chiral chemical potential and the local equilibrium temperature are respectively defined by with T −1 0 ≡ β 0 the length of the compatified thermal Euclidean circle in the imaginarytime formalism to be used in our computations of the following sections.In the absence of torsion, the equilibrium partition function can be written as2 where P(T, µ L ) is the fluid's pressure, µ L = u µ A µ the chiral chemical potential, and g the determinant of the transverse metric in (2.11).It should be taken into account that gauge invariance prevents the pressure from depending on the spatial components of the gauge field.Although a dependence on the static electric and magnetic fields is allowed in principle, it would induce linear torsional corrections only at second order in derivatives.This is the reason why we do not consider this possibility here.
From the form of the microscopic action (2.4), we reach the key observation that the coupling to background torsion is implemented in the equilibrium partition function by a shift of the chiral chemical potential where µ S ≡ u µ S µ contains all dependence on the torsion tensor [20] µ To write the expression in the second line, we used the explicit form of the fluid four-velocity in (2.12), taking into account the antisymmetry of T µ να in its two lower indices, as well as the definition of the three-dimensional Levi-Civita tensor ϵ ijk = −e σ ϵ 0ijk , normalized as ϵ 123 = 1/ √ g.Equation (2.16) explicitly shows that µ S is independent of σ and also invariant under the Kaluza-Klein (KK) transformations induced by space-dependent time reparametrizations, t → t + ϕ(x).Following ref. [37], the contortion tensor is taken to be first order in the derivative expansion 3 .This means that, unlike the standard bona fide microscopic gauge field A µ , the effective gauge field (2.6) is first order in derivatives, an important fact to bear in mind.Thus, implementing the shift µ L → µ L + µ S in eq.(2.14) and expanding to linear order in µ S , we obtain the linear torsional correction to the equilibrium partition function at first order in derivatives where ⟨n⟩ is the thermal-averaged number density and the subscript on the right-hand side indicates that the derivative is evaluated at vanishing torsion.
Chiral current.The linear torsional corrections to the chiral current are found applying the expression given ref.[45] to the partition function (2.17), taking into account the relation between A 0 and µ L shown in eq.(2.13).For the (covariant) zero component, we obtain while the (contravariant) spatial components are equal to zero This last result follows from the independence of the pressure with respect to A i , as explained after eq.(2.14).
The energy-momentum tensor.Besides parametrizing velocity, vorticity, and acceleration in the fluid, the metric functions in (2.11) are the classical sources coupling to the components of the fluid energy-momentum tensor, which are computed by taking functional derivatives of the partition function with respect to them [45].Let us recall that, at the level of the microscopic action (2.1), the minimal coupling to torsion is equivalent to a coupling of the chiral fermions to the external effective gauge field (2.6), while for the thermodynamics all dependence on the background torsion comes from the shift (2.15) in the equilibrium partition function.There is however a crucial physical difference between this and the coupling to a standard external gauge field: the torsional gauge field, and therefore also the torsional chemical potential µ S , depends not only on the torsion tensor but on the metric functions as well [see eq.(2.16)].This results in additional terms in the constitutive relations of various energy-momentum tensor components, that can be explicitly computed by taking the appropriate functional derivatives of eq.(2.17).
Taking this basic fact into account, we compute the torsional corrections to the energymomentum tensor, beginning with the 00 component where we have applied that, unlike T and ⟨n⟩, the torsional chemical potential µ S is independent of σ, as we pointed out after eq. ( 2.16).To recast this expression in terms of the fluid's energy density, we make use the thermodynamic relation Using eq.(2.13), we see that the two partial derivatives can be combined into a single total one with respect to σ and differentiating this expression with respect to µ L , we arrive at The total and partial derivatives on the right-hand side, however, do not commute with one another, but we have instead Applying this identity in eq. ( 2.24), we find and arrive at the result after substituting the definition (2.16) in the second line.
The mixed components of the energy-momentum tensor, on the other hand, are obtained by taking variations with respect to the KK gauge field a i (2.28) Using the definition ⟨Θ i 0 ⟩ = q i u 0 , we find the linear torsional contribution to the heat current with q 0 = 0 (or q 0 = −a i q i ) due to transversality.Interestingly, this result is obtained from the covariant expression particularized to the static line element (2.6) and with the equilibrium four-velocity taking the form given in eq.(2.12).We finally compute the spatial components, taking into account that √ gϵ ijk is independent of the transverse metric g ij The derivative in the last expression can be explicitly computed from eq. (2.16), with the result 0ℓ . (2.32) The right-hand side is the equilibrium form of the covariant expression written in terms of the projector ∆ µ ν = δ µ ν + u µ u ν .The tensor T µν can actually be split into a trace and a traceless parts To obtain the first identity we have applied again eq.(2.16), while in the second one we introduced the symmetric tensor which is both transverse and traceless.Collecting all previous results together, we find the covariant form of the linear torsional contributions to the energy-momentum tensor, at first order in the derivative expansion We notice the existence of nondissipative torsional terms to the heat current, dynamical pressure, and anisotropic stress, with their corresponding transport coefficients given in eqs.(2.30) and (2.35).In particular, the term π µν can be interpreted as a four-dimensional counterpart of the Hall viscosity term in 2+1 dimensions [47].Let us recall that in this case the torsional contribution ⟨Θ ij ⟩ at equilibrium is obtained from the covariant expression [31] where σ µν is the shear tensor linear in the torsion components.There is however an important difference with respect to the four-dimensional result (2.35), since T µν 2+1 is traceless and includes no torsional corrections to the dynamic pressure.
Another relevant feature of the linear torsional corrections (2.37) is that they preserve the trace of the energy-momentum tensor.Indeed, using the properties of q µ and π µν , as well as eq.(2.26), we find which vanishes as a consequence of the homogeneity property of the number density imposed by classical scale invariance.Here we find a further difference with respect to the (2 + 1)-dimensional case, where the linear torsional corrections to the trace of the energy-momentum tensor are given by [31] ⟨Θ µ µ ⟩ 2+1 = e 2σ d⟨ΨΨ⟩ dσ This expression does not vanish after imposing the corresponding homogeneity relation for the fermion condensate To understand this physically, we recall that in 2 + 1 dimensions the coupling to torsion amounts to a shift in the fermion mass [31], which necessarily breaks classical scale invariance.In the (3 + 1)-dimensional case, by contrast, the effect of torsion is equivalent to a coupling to an external gauge field, which does not conflict with classical scale invariance.As a consequence, ⟨Θ µ µ ⟩ 3+1 does not receive corrections linear in the torsion.
The spin energy potential.Being determined by the variation of the equilibrium partition function with respect to the torsion tensor [1,20], the leading order contribution to the spin energy potential comes from the part of the partition function linear in the torsion and computed in eq (2.17) and is of order zero in the derivative expansion.Applying eq. ( 2.16), we compute the nonvanishing components of the spin energy potential a result that can be obtained from the covariant expression evaluated at equilibrium.The nonvanishing KK-invariant components can be found using the explicit expressions given in [20], with the result The linear torsional contribution to the spin energy potential, on the other hand, come from the second-order correction to the partition function (2.48) Applying then the definition (2.44), we find This is linear in the torsion and therefore first order in the derivative expansion.

Thermal Weyl fermions in magnetic background torsion
After having analyzed the general structure of linear torsional terms in the finite temperature constitutive relations for the chiral current, energy-momentum tensor, and spin energy potential, we focus our attention on some particular geometrical backgrounds with different types of torsion.The first to be considered is defined by the tetrad and the spin connection one-form ) depending on two real background data B and T .A short calculation shows that the latter parametrizes the background torsion while the curvature takes the form This geometry is a generalization of the one applied in refs.[13,30] to the study of torsional Landau levels, that is recovered when B = T and both the spin connection and curvature vanish.By including a nonzero spin connection, however, here we are able to disentangle the "torsional magnetic field" parameter B from the bona fide torsion T .The background (3.1)-(3.2) describes a fluid in equilibrium with four-velocity u ≡ −e 0 = −dt and vanishing vorticity and acceleration, ω = a = 0. Implementing the electricmagnetic decomposition of the torsion two-form with respect to u we see that (3.3) is purely magnetic with B a = T δ a 3 dxdy, while the corresponding decomposition for the vierbein and the spin connection e a = e e e a − χ a u, shows that χ a = δ a 0 and the spin chemical potential is zero, µ a b = 0.Moreover, the equilibrium constraint [24] E a − D + a χ a + µ a b e e e b = 0, ( with D the covariant derivative defined from the magnetic component of the spin connection, can be seen to be trivially satisfied.This shows the independence of the two background data B and T .Finally, the tetrad (3.1) defines a static line element of the form (2.11) with σ = a i = 0 In addition to its minimal coupling to the background geometry just described, we assume that the fundamental chiral fermion field is also coupled to an external, purely electrical gauge field of the form A = −µ L u, with µ L the chiral chemical potential.The fermion Hamiltonian (2.9) now takes the form We see that, as in the background analyzed in ref. [13], B parametrizes a sort of external magnetic field coupling to fermions through their momentum p z , whereas the torsion T enters the Hamiltonian only through a shift in the chiral chemical potential.For later convenience, we write where the shifted chemical potential µ is defined by The thermal one-particle Green function.To compute the constitutive relations at finite temperature, we use the imaginary time formalism where the Euclidean time is compactified to a circle of length β 0 = T −1 0 and fermions are antiperiodic along the thermal circle.More specifically, we work with the fermion Green function where T τ [. ..] represents (Euclidean) time ordering.Since we are interested in computing thermal averages in a system of microscopic chiral fermions interacting only with external sources, it suffices to consider the one-particle Green function, defined by the matrix equation where the Hamiltonian H is acting on the r coordinate.It is convenient to recast the Green function in terms of its Euclidean time Fourier transform with ω n the Matsubara frequencies The new matrix function G(ω n ; r, r ′ ) then satisfies the equation whose solution gives the thermal averages of fermion bilinears via the identity [48-50] with A an arbitrary linear operator.
In fact, the thermal one-particle Green function can be expressed in terms of the heat kernel associated to the square of the operator H defined in (3.10) where K(r, r ′ ; s) is a 2 × 2 matrix satisfying the boundary condition Using (3.18), it is straightforward to check that the function solves indeed eq.(3.16).
Being interested in corrections linear in the torsion, we would only need to solve eq.(3.16) in perturbation theory at first order in T , as we will actually do in forthcoming sections.In the case of the background (3.1)-(3.3),however, both the heat kernel K(r, r ′ ; s) and the one-particle Green function G(r, r ′ ; ω n ) can be computed exactly.Using the explicit expression of H extracted from eq. (3.9), we find the following solution to the heat kernel equation (3.18) (see also ref. [51]) with K 12 (r, r ′ ; s) = K 21 (r, r ′ ; s) = 0. Substituting these expressions into eq.(3.20), the explicit form of the Green-function to be used in the computation of thermal torsional corrections of the various currents is obtained.Let us proceed with the evaluation of the thermal average of the chiral current J µ ≡ −ψ † σ µ − ψ.Since it offers a template for future analyses, here we present the calculations in some detail.We start with the time component that at zeroth order in the background data gives the number density ⟨n⟩, the main ingredient of the general expressions computed in section 2. Applying eq.(3.17) and the expression of the one-particle Green function in terms of the heat kernel (3.20), we find after some algebra , where the sums over Matsubara frequencies have been packed into Jacobi theta functions [52] and the prime, as usual, indicates differentiation with respect to the first argument.In addition, the last integral in eq.(3.23) has been regularized with a proper time UV cutoff, while the one in the second line is finite due to the absence of a zero mode in θ ′ 4 (z|τ ).At this point, we carry out the expansion in B and T , keeping only terms linear in both quantities Applying then the identity (see Appendix A) valid for n > 1, we find that all dependence on B cancels from the cutoff-independent part The remaining integrals are computed using the expansion and expressed in terms of polylogarithms give Finally, using the identity [53] Li with B n (x) the n-th Bernoulli polynomial, we arrive at the result As already pointed out, the only dependence on the parameter B appears on the vacuum contribution.The terms independent of either B or T give the number density of a chiral fermion at finite temperature with nonzero chiral chemical potential [54] ⟨n⟩ ≡ ⟨J 0 ⟩ (0) = 1 6 which is the quantity to be used as one of the main ingredients of the general expressions obtained in section 2.
The computation of the current spatial components is carried out along similar lines.An evaluation of the relevant traces leads to the conclusion that all three components are equal to zero This result, together with the torsional terms in eq.(3.31), show total agreement with the general expression for the torsional corrections to the current components given in (2.19) and (2.20), taking into account that in our background as it can be checked from eqs. (2.16) and (3.3).Next, we apply the thermal one-particle Green function computed above to evaluate the thermal linear torsional corrections to the energy-momentum tensor beginning with where in writing this expression we took into account that ω a b0 = 0 [cf.(3.2)].The calculation is similar to the one of the chiral current, leading also to a cancellation of all terms proportional to B From the finite, T -independent terms we read the perfect fluid energy density [54] Using (3.36), as well as the expression of ⟨n⟩ obtained in eq.(3.32), we find that the linear torsional terms in (3.39) reproduce the result derived from eq. (2.27).
The remaining diagonal components are computed similarly, now including the nonzero contributions from the spin connection (3.2) in the covariant derivatives.These terms, proportional to T − B, cancel those scaling with B in the one-particle thermal Green function, rendering again B-independent expressions We see that only ⟨Θ 33 ⟩ picks up corrections linear in the torsion, which are proportional to ⟨n⟩ as given in eq.(3.32).This is totally consistent with (2.32), taking into account that in our background T i jk = 0 for i = 1, 2 or j, k = 3.Moreover, from the finite T -independent terms we read the perfect fluid pressure [54] Notice that, unlike the case of the chiral current, the vacuum contributions to the diagonal components of the energy-momentum tensor do not depend on the background data.
As for the off-diagonal terms, omitting the vacuum part, we find that the only nonzero components are all independent of the torsion parameter T .In fact, the components just computed are part of the perfect-fluid energy-momentum tensor Indeed, using eq.(3.8) we find which, together with (3.43), reproduces (3.44).In addition, since the B-dependent terms in G 33 are of second order in this parameter, PG 33 does not contribute any linear correction to ⟨Θ 33 ⟩ computed in the second identity of eq.(3.42).To compare with the expressions derived in sec.2, we should keep in mind that T i 0j = 0 in the background of interest, so the absence of torsional corrections found in the microscopic calculation of the off-diagonal components matches with the result of applying eqs.(2.32) and (2.28).
To close this section, let us compute the thermal-averaged spin energy potential.Taking the variation of the fermionic action (2.4) with respect to the torsion tensor T µ αβ , we find that the spin energy potential is expressed in term of the chiral current as Inserting eqs.(3.32) and (3.35), the only nonzero components are found to be in accord with the result shown in eq.(2.47).The contribution linear in T is obtained from the corresponding terms in (3.31) and also agrees with eq.(2.49) evaluated in the present background4 .

Disentangling torsion from vorticity
The second torsional background to be studied also has magnetic torsion and is closely related to the geometry studied in ref. [11].Its vierbein takes the form Bxdy, dx, dy, dz , ( with B a real constant, while the spin connection has the following nonzero components The curvature is computed to be nonzero and of second order in B and T B , while the purely magnetic torsion is linear in T B .This geometry describes a fermion fluid with velocity and the following values for the fluid's vorticity and acceleration These expressions provide the interpretation of the two background data B and T B as respectively parametrizing vorticity and torsion.Carrying out the electric-magnetic decompositions shown in eq.(3.6), we find that χ a = δ a 0 and the fluid has a nonzero spin chemical potential, µ 12 = −µ 21 = − 1 2 (T B − B).The equilibrium constraint (3.7) is also identically satisfied, implying also here the independence of B and T B .
Unlike in the case of the geometry studied in the previous section, now the Green function equation cannot be solved exactly, so we rely on perturbation theory to compute thermal corrections linear in B and T B .We thus split the Hamiltonian (2.9) particularized to this background as H = H (0) + H (1) , ( where H (0) is its free part and describes the interaction of the microscopic Weyl fermions with the background geometry linear in the background data.Equation (3.16) then takes the form and is solved by splitting the Green function into a free part and a perturbation where G (0) (ω n ; r, r ′ ) satisfies the equation whose solution is given by The linear correction G (1) (ω n ; r, r ′ ), on the other hand, takes the form where the perturbation Hamiltonian H (1) acts on the r ′′ argument of the free Green function to its right, and the last term originates in the piece linear in B of the right-hand side of eq.(4.10).The terms linear in the background data in the finite temperature constitutive relation of the chiral current are given by Applying the same techniques as in the previous section, we find where the vanishing of ⟨J 1 ⟩ (1) and ⟨J 2 ⟩ (2) results from a cancellation between the contributions of the two terms on the right-hand side of eq.(4.14).As for ⟨J 3 ⟩ (1) , we find a nontrivial cancellation of all finite temperature contributions proportional to This absence of thermal linear torsional corrections agrees with the general expressions (2.19) and (2.20), given that µ S = 0 for the torsion tensor shown in eq.(4.4).
Let us focus for a moment in the cutoff-dependent vacuum terms in (4.17).In our perturbative computation, the coincidence limit r ′ → r in (4.14) has been taken before carrying out the integration over proper time coming from the zeroth-order Green function (4.13).However, would we have performed this integration prior to the coincidence limit, the torsional term in the vacuum contribution would be regularized to zero, with only the part scaling as BΛ 2 remaining.The finite temperature terms, on the other hand, are not affected by the order ambiguity.It should also be mentioned that, taking the coincidence limit first, the presence of a term proportional to T B Λ 2 in (4.17) agrees with the general form of the vacuum chiral current obtained in ref. [24] using the method of descent.
To be more precise, the origin of the ambiguity traces back to the integral which is convergent and vanishes when evaluated for r ′ ̸ = r.Taking the coincidence limit first, on the other hand, makes the proper time integration ill defined and forces the introduction of a UV cutoff.After inserting the sums over Matsubara frequencies, the proper time integration is carried out to give with no power-like contributions in either the temperature or the chemical potential.Recasting the sums in terms of Jacobi theta functions, it can be seen that (4.19) follows again from applying the identity (3.26) with n = 2.An explicit evaluation of the energy momentum tensor components shows the absence of linear torsional corrections, confirming the results found in sec. 2 for a background with T i jk = T i k0 = 0.Only the heat current picks up a linear dependence on the vortical parameter B. The finite temperature contributions to the spin energy potential linear in the background data show no dependence on T B either.Indeed, the only nonzero components are together with the ones obtained by permutation of the three antisymmetric indices.
To close this section, let us point out that the analysis presented above can be easily generalized to the geometry defined by the tetrad with spin connection for constant ω ij and T 0 ij , both antisymmetric in their lower indices.This gives nonzero values for the vorticity and torsion two-forms where from the latter identity we conclude that µ S = 0.A calculation of the thermal constitutive relations renders the results Here we have introduced the vorticity vector ω i = 1 of the axial current are found5 .To avoid this state of affairs, let us consider the background defined by the trivial tetrad e a = dt, dx, dy, dz , ( together with a spin connection one-form describing a nonzero spin chemical potential with and vanishing magnetic components, ω a b = 0 [see the second identity in (3.6)].The associated torsion two-form is found to be purely electric while the nonzero components of the curvature are linear in the single background datum T E .Since u = −dt, this geometry can be used to describes a chiral fermion fluid coupled to torsion with no vorticity or acceleration, for which the equilibrium constraint (3.7) is trivially satisfied.The one-particle Green thermal equation (3.16) for the Weyl Hamiltonian can be solved in perturbation theory with respect to T E , as we did in the case discussed in sec.4. For the chiral current, we find the following results for the linear torsional terms where Λ is the proper time cutoff.Unlike the background studied in the previous section, here there are no finite-temperature linear torsional corrections, again as a consequence of the identity (3.26) with n = 2.We now take the divergence of the computed current to obtain a nonzero value for the Nieh-Yan anomaly (cf.[24]) where the Nieh-Yan invariant in the second line is computed to linear order in T E and the star denotes the four-dimensional Hodge dual.These results are totally consistent with the general structure of the vacuum chiral current found in [24].It should be stressed as well that this absence of linear thermal corrections in both the chiral current and the Nieh-Yan anomaly provide a counterexample to the conjecture stated in [14], according to which the coefficient of the Nieh-Yan anomaly should receive corrections proportional to T 2 0 .As in the case of the background studied in the previous section, here as well the presence of torsional vacuum terms in (5.6) depends on the order in which the coincidence limit and the proper time integration are carried out.Integrating first in proper time when evaluating the Green function in perturbation theory, we find that the vacuum contribution vanishes after the coincidence limit is properly taken, and as a consequence the Nieh-Yan anomaly is also regularized to zero.The ambiguity once again stems from an integral of the type shown in eq.(4.18).This suggests that, at least in this case, the Nieh-Yan anomaly could be regarded as an artifact of the regularization 6 .
Concerning the energy-momentum tensor, only the heat current acquires a term proportional to T E with q 0 = q 1 = q 2 = 0. Once more, the results for both the chiral current and the energymomentum tensor obtained from the microscopic calculation match with the expressions derived in sec. 2 applied to a background with σ = 0 = a i , g ij = δ ij , and a torsion tensor whose only nonzero components are

Closing remarks
In this paper we have studied finite temperature effects in chiral fluids at equilibrium coupled to background torsion.At the level of the equilibrium partition function, all dependence on the torsion at first order in the derivative expansion enters through a shift in the chiral chemical potential.Our main result is a full determination of the general structure of the linear torsional corrections to the constitutive relations of the chiral current, energy-momentum tensor, and spin energy potential.One interesting conclusion to be extracted from these results is the existence of nonzero torsion configurations which do not leave imprints in the constitutive relations of these three conserved currents.Another important upshot of our analysis is that the constitutive relation of the energy-momentum tensor contains nondissipative torsional terms analog to the (2 + 1)-dimensional Hall viscosity.
We also carried out a first-principle, microscopic computation of these constitutive relations for some particular geometries, confirming in all cases the general results obtained in sec.2. Unlike most of the torsional backgrounds with potential applications to condensed matter physics studied in the literature (see, for example, [9-11, 13, 15, 17, 30]), here we have focused our attention on geometries with nonzero spin connection and curvature.This not only opens a way of adding curvature and thus implementing the effects of disclinations as well as dislocations, but also allows the disentanglement of torsion from the background data parametrizing other fluid properties.This feature might in fact shed light on the interplay between torsion and vorticity, a subject of some recent controversy (see, for example, [19,22,23]).
The geometrical background studied in sec.4 offers a good test bench for this task.As explained there, the model includes both vorticity and torsion, parametrized by the two independent background data B and T B (or ω ij and T 0 ij , in the generalization discussed at the section's end), their difference (resp., T 0 ij + 2ω ij ) determining the spin connection and the curvature.At linear order, however, thermal corrections to the constitutive relations only depend on vorticity, thus indicating the absence of genuine torsional transport effects.The particular case B = T B (resp., ω ij = − 1 2 T 0 ij ) deserves closer attention.The torsional background in this instance reduces to the one used in ref. [11], whose results we generalized to allow for a nonzero chiral chemical potential.Since ω a b = 0 = a, torsion can now be interpreted as a particular way of implementing vorticity, based on the identification Let us point out however that this identity is not Lorentz covariant, the left-hand side being a scalar while the right-hand side is the time component of a Lorentz vector, so such a direct link between torsion and vorticity is problematic in a generic geometry.This indicates that the identification of vorticity with torsion only works in very particular cases.
Our microscopic analyses also bring forward an interesting feature of the vacuum contributions to the chiral current, with some bearing on the results of ref. [24].We have seen how the torsional vacuum terms in the backgrounds studied in secs.4 and 5, including the Nieh-Yan anomaly in the second case, can be removed by an appropriate prescription in the order in which the integration over proper time and the coincidence limit are taken.It is important to stress that this ambiguity only affects the torsional part of the vacuum contribution, whereas both finite temperature contributions and the vacuum terms proportional to other background data remain unchanged.
This last trait is also manifest in the background of secs.3, where it can be shown that the vacuum term proportional to BΛ 2 is free from order ambiguities.The distinct feature of this case is that all torsion dependence in the Green function exclusively enters through a nonzero value of the torsional chemical potential µ S , whereas for the other two backgrounds studied here this quantity is zero and torsion dependence comes from the spatial components of the torsional gauge field.This is why the corresponding finite temperature constitutive relations of the chiral current display no torsional linear corrections, as implied also by the results of sec.2. Notice, however, that choosing the prescription preserving the torsional terms in the current cutoff-dependent piece leads to results agreeing with the general form of the zero temperature current derived in [24].
Finally, although this work is not directly concerned with the nature of the Nieh-Yan anomaly, it is worth-mentioning that the ambiguity we have found related to the order in which the integration over proper time and the coincidence limit are carried out seems to indicate that this anomaly could be avoided, at least in some cases.Anyhow, on the broad issue of the physical role of the Nieh-Yan anomaly the jury is still out (see, for example, refs.[12-18, 21-24, 55-57]).