Torsional Newton-Cartan Geometry

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Having in mind applications to Condensed Matter Physics, we perform a null-reduction of General Relativity in d + 1 spacetime dimensions thereby obtaining an extension to arbitrary torsion of the twistless-torsional Newton-Cartan geometry. We shortly discuss the implementation of the equations of motion.

Torsional Newton-Cartan Geometry


application/pdf Torsional Newton-Cartan Geometry Eric Bergshoe, Athanasios Chatzistavrakidis, Luca Romano, Jan Rosseel
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Torsional Newton-Cartan Geometry


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	    <date dateType="Created">Sat 3 Mar 2018</date>
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            <description descriptionType="Abstract">Having in mind applications to Condensed Matter Physics, we perform a null-reduction of General Relativity in d + 1 spacetime dimensions thereby obtaining an extension to arbitrary torsion of the twistless-torsional Newton-Cartan geometry. We shortly discuss the implementation of the equations of motion.

Torsional Newton-Cartan Geometry Eric Bergshoeff1 , Athanasios Chatzistavrakidis1,2 , Luca Romano, and Jan Rosseel3 1 Van Swinderen Institute for Particle Physics and Gravity, Nijenborgh 4, 9747 AG Groningen, The Netherlands, 2 Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia 3 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria, Abstract. Having in mind applications to Condensed Matter Physics, we perform a null-reduction of General Relativity in d + 1 spacetime dimensions thereby obtaining an extension to arbitrary torsion of the twistless-torsional Newton-Cartan geometry. We shortly discuss the im- plementation of the equations of motion. 1 Introduction Usually, when discussing Newton-Cartan gravity, one defines an absolute time by imposing that the curl of the time-like Vierbein τµ vanishes.4 This condition, sometimes called the zero torsion condition, allows one to solve for τµ in terms of a single scalar field τ(x): ∂µτν − ∂ντµ = 0 ⇒ τµ = ∂µτ . (1) Choosing for this function the time t, i.e. τ(x) = t, defines the absolute time direction: τ(x) = t ⇒ τµ = δ0 µ . (2) The zero-torsion condition (1) is sufficient but not required to obtain a causal behaviour of the theory. A more general condition, that guarantees a time flow orthogonal to Riemannian spacelike leaves, is the so-called hypersurface orthog- onality condition: τab ≡ ea µ eb ν τµν = 0 , τµν = ∂[µτν] . (3) 4 Most of this presentation applies to any spacetime dimension. We will therefore from now on use the word Vielbein instead of Vierbein and take µ = 0, 1, · · · d − 1 . Here ea µ , together with τµ , are the projective inverses of the spatial and timelike Vielbeine eµ a and τµ, respectively, with a = 1, 2, · · · d − 1. They are defined by the following projective invertibility relations: eµ aeν a = δµ ν − τµ τν , eµ aeµ b = δa b , τµ τµ = 1 , eµ aτµ = 0 , τµ eµ a = 0 . (4) The condition (3), also called the twistless torsional condition, was encoun- tered in the context of Lifshitz holography when studying the coupling of Newton- Cartan gravity to the Conformal Field Theory (CFT) at the boundary of space- time [1]. Twistless-torsional Newton-Cartan geometry has also been applied in a study of the Quantum Hall Effect [2]. It not surprising that the twistless-torsional condition (3) was found in the context of a CFT. The stronger condition (1) sim- ply does not fit within a CFT since it is not invariant under spacetime-dependent dilations δτµ ∼ ΛD(x)τµ. Instead, the condition (3) is invariant under spacetime- dependent dilatations due to the relation ea µ τµ = 0, see eq. (4). One can define a dilatation-covariant torsion as τC µν ≡ ∂[µτν] − 2b[µτν] = 0 , (5) where bµ transforms as the gauge field of dilatations, i.e. δbµ = ∂µΛD. Since τµ τµ = 1 one can use the space-time projection of the equation τC µν = 0 to solve for the spatial components of bµ: τC 0a ≡ τµ ea ν τC µν = 0 ⇒ ba ≡ ea µ bµ = −τ0a . (6) This implies that in a conformal theory only the spatial components of the conformal torsion can be non-zero: τC ab ≡ ea µ eb ν τC µν = τab 6= 0 . (7) At first sight, it seems strange to consider the case of arbitrary torsion since causality is lost in that case. However, in condensed matter applications, one often considers gravity not as a dynamical theory but as background fields that couple to the energy and momentum flux. It was pointed out a long time ago in the seminal paper by Luttinger [3] that to describe thermal transport one needs to consider an auxiliary gravitational field ψ(x) that couples to the energy and is defined by τµ = eψ(x) δµ,0 (8) corresponding to the case of twistless torsion. Later, it was pointed out that, for describing other properties as well, one also needs to introduce the other compo- nents of τµ that couple to the energy current. This leads to a full un-restricted τµ describing arbitrary torsion [4]. For an earlier discussion on torsional Newton- Cartan geometry, see [5]. For applications of torsion in condensed matter, see [6, 7]. To avoid confusion we will reserve the word ‘geometry’ if we only consider the background fields and their symmetries whereas we will talk about ‘gravity’ if also dynamical equations of motion are valid. In this presentation, we will construct by applying a null reduction of Gen- eral Relativity, the extension of NC geometry to the case of arbitrary torsion, i.e. τµν 6= 0, see Table 1. Null-reductions and Newton-Cartan geometry with or without torsion have been discussed before in [8–11]. The outline of this presentation is as follows. In the next section we will derive NC geometry with arbitrary torsion from an off-shell null reduction, meaning we do not perform a null reduction of the equations of motion, of General Relativity in d + 1 spacetime dimensions. We point out that performing a null reduction of the equations of motion as well we obtain the equations of motion of NC gravity with zero torsion thereby reproducing the result of [8, 9]. We comment in the Conclusions on how one could go on-shell keeping arbitrary torsion. Table 1. Newton-Cartan geometry with torsion. NC geometry Geometric Constraint arbitrary torsion τ0a 6= 0 , τab 6= 0 twistless torsional τ0a 6= 0 , τab = 0 zero torsion τ0a = 0 , τab = 0 2 The null reduction of General Relativity One way to obtain NC geometry with arbitrary torsion is by performing a dimen- sional reduction of General Relativity (GR) from d+1 to d spacetime dimensions along a null direction [9]. We show in detail how to perform such a null reduction off-shell, i.e. at the level of the transformation rules only. At the end of this sec- tion, we point out that, after going on-shell, we obtain the equations of motion of NC gravity with zero torsion [8, 9]. Our starting point is General Relativity in d + 1 dimensions in the second order formalism, where the single independent field is the Vielbein êM A . Here and in the following, hatted fields are (d+1)-dimensional and unhatted ones will denote d-dimensional fields after dimensional reduction. Furthermore, capital indices take d+1 values, with M being a curved and A a flat index. The Einstein- Hilbert action in d + 1 spacetime dimensions is given by S (d+1) GR = − 1 2κ Z dd+1 x ê êM AêN BR̂MN AB (ω̂(ê)) , (9) where κ is the gravitational constant and ê is the determinant of the Vielbein. The inverse Vielbein satisfies the usual relations êM AêM B = δB A , êM AêN A = δM N . (10) The spin connection is a dependent field, given in terms of the Vielbein as ω̂M BA (ê) = 2êN[A ∂[M êN] B] − êN[A êB]P êMC∂N êP C , (11) while the curvature tensor is given by R̂MN AB (ω̂(ê)) = 2∂[M ω̂N] AB − 2ω̂[M AC ω̂N]C B . (12) Under infinitesimal general coordinate transformations and local Lorentz trans- formations, the Vielbein transforms as δêM A = ξN ∂N êM A + ∂M ξN êN A + λA BêM B . (13) In order to dimensionally reduce the transformation rules along a null direc- tion, we assume the existence of a null Killing vector ξ = ξM ∂M for the metric ĝMN ≡ êM A êN B ηAB, i.e. LξĝMN = 0 and ξ2 = 0 . (14) Without loss of generality, we may choose adapted coordinates xM = {xµ , v}, with µ taking d values, and take the Killing vector to be ξ = ξv ∂v. Then the Killing equation implies that the metric is v-independent, i.e. ∂vĝMN = 0, while the null condition implies the following constraint on the metric: 5 ĝvv = 0 . (15) A suitable reduction Ansatz for the Vielbein should be consistent with this constraint on the metric. Such an Ansatz was discussed in [9], and we repeat it below in a formalism suited to our purposes. First, we split the (d+1)-dimensional tangent space indices as A = {a, +, −}, where the index a is purely spatial and takes d − 1 values, while ± denote null directions. Then the Minkowski metric components are ηab = δab and η+− = 1. The reduction Ansatz is specified upon choosing the inverse Vielbein êM + to be proportional to the null Killing vector ξ = ξv ∂v. A consistent parametrization is êM A =     µ v a eµ a eµ amµ − Sτµ Sτµ mµ + 0 S−1     . (16) The scalar S is a compensating one and will be gauge-fixed shortly. Given the expression (16) for the inverse Vielbein, the Vielbein itself is given by êM A = a − + µ eµ a S−1 τµ −Smµ v 0 0 S ! . (17) 5 Due to this constraint, we are not allowed to perform the null reduction in the action but only in the transformation rules and equations of motion [9]. To avoid confusion, recall that the index a takes one value less than the index µ; thus the above matrices are both square although in block form this is not manifest. Note that the Ansatz (17) has two zeros. The zero in the first column, êv a = 0, is due to the fact that we gauge-fixed the Lorentz transformations with param- eters λa +. On the other hand, the zero in the second column, êv − = 0, is due to the existence of the null Killing vector ξ = ξv ∂v: ξ2 = ξv ξv ĝvv = 0 ⇒ ĝvv = êv A êv B ηAB = 0 ⇒ êv − = 0 . (18) We will call λa ≡ λa − and λ ≡ λ+ +. A simple computation reveals that the invertibility relations (10), after sub- stitution of the reduction Ansatz, precisely reproduce the projected invertibility relations (4) provided we identify {τµ , eµ a } as the timelike and spatial Viel- bein of NC gravity, respectively. Starting from the transformation rule (13) of the (d + 1)-dimensional Vielbein, we derive the following transformations of the lower-dimensional fields: δτµ = 0 , (19) δeµ a = λa beµ b + S−1 λa τµ , (20) δmµ = −∂µξv − S−1 λaeµ a , (21) δS = λS . (22) Next, fixing the Lorentz transformations with parameter λ by setting S = 1 and defining σ := −ξv we obtain, for arbitrary torsion, the transformation rules δτµ = 0 , δeµ a = λa beµ b + λa τµ , (23) δmµ = ∂µσ + λa eµa of Newton-Cartan geometry in d dimensions provided we identify mµ as the central charge gauge field associated to the central charge generator of the Bargmann algebra. Note that we have not imposed any constraint on the torsion, i.e. τµν = ∂[µτν] 6= 0. We next consider the null-reduction of the spin-connection given in eq. (11). Inserting the Vielbein Ansatz (17) with S = 1 into eq. (11) we obtain the following expressions for the torsionful spin-connections: ω̂µ ab (ê) ≡ ωµ ab (τ, e, m) = ω̊µ ab (e, τ, m) − mµτab , ω̂µ a+ (ê) ≡ ωµ a (τ, e, m) = ω̊µ a (e, τ, m) + mµτ0 a , (24) where ω̊µ ab (e, τ, m) and ω̊µ a (e, τ, m) are the torsion-free Newton-Cartan spin connections given by ω̊µ ab (τ, e, m) = eµceρa eσb ∂[ρeσ] c − eνa ∂[µeν] b + eνb ∂[µeν] a − τµeρa eσb ∂[ρmσ],(25) ω̊µ a (τ, e, m) = τν ∂[µeν] a + eµ c eρa τσ ∂[ρeσ]c + eνa ∂[µmν] + τµτρ eσa ∂[ρmσ] . (26) Furthermore, we find that the remaining components of the spin-connections are given by ω̂v ab (ê) = τab , ω̂v a+ (ê) = −τ0 a , ω̂µ a− (ê) = −τµτ0 a − eµ b τb a , ω̂v a− (ê) = 0 , ω̂µ −+ (ê) = −eµ b τ0b , ω̂v −+ (ê) = 0 . (27) At this point, we have obtained the transformation rules of the independent NC gravitational fields describing NC geometry with arbitrary torsion. Furthermore, we obtained the expressions for the dependent (torsional) spin-connections. The same method cannot be used to obtain the equations of motion of NC gravity with arbitrary torsion [8, 9]. A simple argument for this will be given in the next section. Instead, it has been shown [8, 9] that reducing the Einstein equations of motion leads to the torsion-less NC equations of motion. More precisely, using flat indices, the equations of motion in the −−, −a and ab directions yield the NC equations of motion while those in the ++, +− and +a direction constrain the torsion to be zero [8, 9]. Since the two sets of equations of motion transform to each other under Galilean boosts, it is not consistent to leave out the second set of equations of motion in the hope of obtaining NC equations of motion with arbitrary torsion. As a final result, we find the following zero torsion NC equations of motion: R0a(Ga ) = 0 , Rcā(Jc b) = 0 , (28) where R(G) and R(J) are the covariant curvatures of Galilean boosts and spatial rotations and where in the last equation we collected two field equations into one by using an index ā = (a, 0). 3 Comments In this presentation we applied the null-reduction technique to construct the transformation rules corresponding to Newton-Cartan geometry with arbitrary torsion. The null reduction technique has the advantage that the construction is algorithmic and can easily be generalized to other geometries, such as the Schrödinger geometry, as well. To explain why the on-shell null reduction leads to zero torsion equations of motion, it is convenient to consider the Schrödinger field theory (SFT) 6 that can be associated to the first NC equation of motion of eq. (28) by adding compensating scalars. To this end, we introduce a complex scalar Ψ = ϕ eiχ that transforms under general coordinate transformations, with parameter ξµ (x), as a scalar and under local dilatations and central charge transformations, with parameters λD(x) and σ(x), with weight w and M, respectively: δΨ = ξµ ∂µΨ + wλD + iMσ  Ψ . (29) 6 SFTs are explained in [12]. The discussion below is partly based upon [12]. We first consider the case of NC geometry with zero torsion. Since the zero torsion NC equations of motion (28) are already invariant under central charge transformations, we consider a real scalar ϕ, i.e. χ = 0. Due to the time/space components of the zero torsion condition, i.e. τ0a = 0, this real scalar must satisfy the constraint ∂aϕ = 0 . 7 At the same time, the first NC equation of motion in eq. (28) leads to the scalar equation of motion ∂0∂0ϕ = 0 . Given these constraints the second NC equation of motion in eq. (28) does not lead to additional restrictions. Summarizing, one can show that the two equations just derived form a SFT with the correct Schrödinger symmetries: SFT1 : ∂0∂0φ = 0 , ∂aϕ = 0 . (30) We next consider the case of NC geometry with arbitrary torsion. In that case one lacks the second equation of (30) that followed from the torsion con- straint τ0a = 0. The real scalar ϕ now satisfies only the first constraint of (30) which does not constitute a SFT. Instead, one is forced to introduce the second compensating scalar χ, together with a mass parameter M, such that φ and χ together form the following SFT: SFT2 : ∂0∂0ϕ − 2 M (∂0∂aϕ)∂aχ + 1 M2 (∂a∂bϕ)∂aχ∂bχ = 0 . (31) Since χ is the compensating scalar for central charge transformations, this im- plies that the extension to arbitrary torsion of the first NC equation of motion in (28) cannot be invariant under central charge transformations. Because null- reductions by construction lead to equations of motion that are invariant under central charge transformations, this explains why we did find zero torsion equa- tions of motion in the previous section. The easiest way to obtain the Newton-Cartan equations of motion with ar- bitrary torsion would be to couple the SFT2 given in eq. (31) to Schrödinger gravity with arbitrary torsion that by itself can be obtained by a null-reduction of conformal gravity. Indeed, such a null reduction has been performed and the coupling of SFT2 to Schrödinger gravity can be constructed in three spacetime dimensions [13]. The extension to higher dimensions remains an open question. As a closing remark, it would be interesting to apply the null reduction tech- nique to supergravity theories. The case of d = 3 should lead to a generalization of the off-shell 3d NC supergravity constructed in [14, 15] to the case of arbi- trary torsion. More interestingly, one can also take d = 4 and construct on-shell 4D NC supergravity with zero torsion thereby obtaining, after gauge-fixing, the very first supersymmetric generalization of 4D Newtonian gravity. An intriguing feature of the 3D case is that the Newtonian supergravity theory contains both a Newton potential as well as a dual Newton potential [16] . In analogy to the 3d case, it would be interesting to see which representations of the Newton po- tential would occur in the 4d case and investigate whether this could have any physical effect. 7 Note that the space/space components of the zero torsion constraint are already invariant under dilatations. Acknowledgements. This presentation is based upon work that at the time of writing was in progress [13]. E.A.B. and J.R. gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research described in this presentation was performed during the workshop Applied Newton-Cartan Geometry. E.A.B wishes to thank the University of Vienna for its hospitality. References 1. M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, “Torsional Newton- Cartan Geometry and Lifshitz Holography,” Phys. Rev. D 89 (2014) 061901 doi:10.1103/PhysRevD.89.061901 [arXiv:1311.4794 [hep-th]]. 2. M. Geracie, D. T. Son, C. Wu and S. F. Wu, “Spacetime Symmetries of the Quan- tum Hall Effect,” Phys. Rev. D 91 (2015) 045030 doi:10.1103/PhysRevD.91.045030 [arXiv:1407.1252 [cond-mat.mes-hall]]. 3. J. M. Luttinger, “Theory of Thermal Transport Coefficients,” Phys. Rev. 135 (1964) A1505. doi:10.1103/PhysRev.135.A1505 4. A. Gromov and A. G. Abanov, “Thermal Hall Effect and Geometry with Tor- sion,” Phys. Rev. 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