Gravitational Landau levels and the chiral anomaly

A popular physical picture of the mechanism behind the four-dimensional chiral anomaly is provided by the massless Dirac equation in the presence of constant electric and magnetic background fields. The magnetic field creates highly degenerate Landau levels, the lowest of which is gapless. Any parallel component of the electric field drives a spectral flow in the gapless mode that causes particles to emerge from, or disappear into, the Dirac sea. Seeking a similar picture for the gravitational contribution to the chiral anomaly, we consider the massless Dirac equation in a background spacetime that creates gravitational Landau levels. We find that in this case the resulting spectral flow, with its explicit particle production, accounts for only a small part of the anomalous creation of chiral charge. The balance is provided by the vacuum expectation charge arising from the spectral asymmetry.


Introduction
When charge-e right-handed Weyl field evolves in both a background electromagnetic field F µν and a gravitational field described by Riemann curvature tensor R α βµν the associated chiral particle-number current J µ R is anomalous.Although classically we expect the current to be conserved, the quantum operator has a non-zero divergence [1,2,3] On the right-hand side of (1) both the electromagnetic Chern-character density ∝ ǫ µνρσ F µν F ρσ and the gravitational Pontryagin density ∝ ǫ µνρσ R α βµν R β αρσ serve as sources for chiral charge that apparently appears from nowhere.The physical mechanism behind the term involving the electromagnetic field is well understood [4,5].In flat space with (−, +, +, +) metric and ǫ 0123 = +1 we have In the case of a uniform B the quantized cyclotron motion in the plane perpendicular to B leads to Landau levels with energy Here k is the momentum component parallel to B, and each energy level has degeneracy e|B|/2π per unit area perpendicular to B. Considered as functions of k these m > 0 levels are gapped, but there is an additional m = 0 Landau level with the same degeneracy that is distinguished from the m > 0 levels in that it is both gapless and only one sign is allowed: the one with ǫ = +k .A component of the electric field parallel to B causes k to increase at a rate k = eE so the gapless modes rise through the ε = 0 Fermi energy at a rate eE /2π per unit length.The net result is that righthanded particles are seen to emerge out of the Dirac-sea vacuum at a rate of e 2 E • B/4π 2 per unit volume per unit time.Left-handed Weyl particles disappear into the sea at the same rate.For Dirac particles the axial current is therefore also anomalous with coefficients that are twice those in (1).The vector current J µ = J µ R + J µ L is conserved.There appears to be no comparably simple physical picture of the gravitational contribution to the anomaly arising from the Pontryagin term.The formal mathematics is understood via the index theorem and Euclideansignature zero modes [6] but the necessity of tidal forces encoded in the curvature tensor obscures the physical mechanism.In particular there is no exact gravitational analogue of the constant E, B field configuration that we can use to aid our understanding.
In the present paper we consider a family of space-times that come close to the E, B case in that they support a gravitational version of Landau levels and for which the rate of level crossing can be calculated and compared with that predicted by the anomaly equation.We will find that despite the similarity of the Landau-level picture there are some surprising differences between the gauge-field and gravitational cases.
In section 2 we introduce the space-time geometry and its analogue Landau levels.We then allow the metric to become time dependent and compare the resulting chiral particle creation with the total chiral charge creation predicted by the anomaly.We find a substantial difference.In section 3 we account for the discrepancy by showing that the vacuum charge arising from the spectral asymmetry is much larger than the explicit particle creation.
The detailed calculations are somewhat intricate so we have relegated most of them to appendices.Appendix A explains the group theory behind the geometric and spectral properties of the spatial part of our spacetime.Appendix B provides a review of the Dirac equation in curved space-time, and constructs and solves the Weyl equation for our family of spacetimes.Appendix C computes the spectral asymmetry directly from the Weyl Hamiltonian eigenspectrum thus demonstrating the consistency of the anomaly equation with the total chiral charge and also identifying which eigenstates contribute the most to the vacuum charge.

Gravitational Landau levels
We seek a space-time geometry that has similar physics to the constant E, B, gauge field configuration that gave us the physical picture of spectral flow creating particles out of the Dirac sea.Our example is based on the metric which describes the geometry of a Bianchi type-II spacetime associated with the Heisenberg group.Consider first the static case in which a and b (both assumed positive) are independent of time.The associated scalar field Hamiltonian is H = −∇ 2 where ∇ 2 is the Laplacian To obtain a discrete set of eigenfunctions of H we impose periodic boundary conditions on the wave-functions.The reason for this seeming odd choice is that each of the t = constant space slices described by ( 5) can be identified with the group manifold of the Heisenberg Lie group.As such, it possesses three nonobvious translation symmetries arising from the group action.The periodicity (7) is the simplest that is compatible with these symmetries, and is the gravitational analogue of the twisted periodic boundary conditions required when we solve the Schrödinger equation in the presence of a Landau-gauge uniform magnetic field.The construction of mode expansions that automatically satisfy these conditions is described in section A.2.Note that while the x, y, z coordinates are now restricted to the unit cube, the volume of the compactified space is a 2 b, and this can be as large as desired.
In section A.3 we show that these boundary conditions lead to a complete set of eigenfunctions of −∇ 2 that are parameterized by three quantum numbers.The first, n, labels the momentum in the z direction and can take any integer value; the second, k, labels a Harmonic-oscillator eigenstate and can be any non-negative integer; the third, m, is an integer in the range 0, 1, . . ., |n| − 1.The corresponding eigenvalues are [7] The |n|-fold degeneracy arises because the eigenvalues are independent of m.These eigenstates arise from essentially the same mathematics as the usual Landau-gauge Landau levels, so it is natural to refer to them as Gravitational Landau levels.In the magnetic case, though, the degeneracy is the same for all levels and equal to the number N = eB z L x L y /2π of magnetic flux units passing through the L x -by-L y torus in the x-y plane.In the gravitational version the analogue of "BL x L y " is unity, but the effective charge of a particle interacting with gravity depends on its energy-momentum, and, as a result, the particle's charge e is replaced by the z-momentum 2πn.The number of "gravitational flux units" passing through the unit square in the x-y plane becomes n, and this determines the degeneracy.

Weyl Hamiltonian
The Hamiltonian for a two-component right-handed Weyl fermion moving in the metric ( 5) is constructed in section B.2, and is The diagonal term −(b/4a 2 )I 2 arises from the spin connection and can be interpreted as a geometry-dependent chemical potential.It will be the principal source of spectral flow.The eigenstates of (9) are obtained in section B.3.Here we simply summarize the results.
The spectrum of H R falls into three broad classes with two subclasses: On states of the form we have eigenvalues with one-state for each pair (n 1 , n 2 ).On 2-spinors of the form there are two different cases depending on whether the z-momentum n is positive or negative.Case i) n > 0: The eigenvalues are These do not depend on m so each level has degeneracy |n| and the corresponding eigenstates are the Weyl-equation version of the gravitational Landau levels.
The special case k = 0 has where ϕ 0 is a harmonic-oscillator ground state wavefunction, and eigenvalues From our experience with the magnetic field case, we might anticipate that these will be "topological" gapless modes that provide the spectral flow.
Case ii) n < 0: We again have eigenvalues that appear similar to the n > 0 case, but the k = 0 eigenstates are now with eigenvalues Observe that "topological" k = 0 states have strictly negative energy eigenvalues The reason for the |n| in (19) is that changing the sign of the z-momentum changes the direction of the gravity-analogue of the magnetic flux.Being always negative, these energy levels never cross the ε = 0 Fermi level -so our anticipation that the "topological" states provide the spectral flow was premature.This is our first surprise.

Counting level-crossings
We have found a gravitational analogue the B in the e 2 E • B/4π 2 anomaly term.To find an analogue of E we allow a and b to vary slowly with time and seek the anomaly by counting the number of states that cross zero energy and emerge from the Dirac sea.There are two sources of Fermi-energy level crossing: i) the "chemical potential" −b/4a 2 will shift up or down, ii) the individual eigenstate energies will shift because of the changing parameters in the ± √ . . .expressions.
The symmetry point at which the ± √ . . .terms become zero always lies below the ε = 0 Fermi energy.No levels cross the symmetry point when a, b change, so the number of states that cross from the ε < 0 Dirac sea to positive energy is equal to the decrease in the number of states lying between the ± symmetry point and the ε = 0 Fermi level.Counting the exact number states in this energy-range is not easy, but if we assume that b ≫ a we can use approximate density-of-states methods.
For the n = 0, states with So and the approximate total is The Landau-level states with energy have ε(±|n| max , k) = 0 at So, taking into account the |n|-fold degeneracy, we have A numerical evaluation of the sum over k gives C ≈ 635.For b ≫ a the number of level crossings from the n = 0 states is negligible compared to those from the Landau-levels because b 4 /a 4 is much larger than b 2 /a 2 .Suppose now we start at time −∞ with a stationary metric, have a period of time in which a and b vary slowly, then return to a stationary metric by t = ∞, then the estimated number of Landau-level level crossings is The minus sign appears because an increase in the number between ε = 0 and the symmetry point means a decrease in the number of levels above ε = 0. We can compare this result with that expected from the anomaly.The gravitational source term in the anomaly is the four-form part, A 1 , of the total A-roof genus that that appears in the general Dirac index theorem [8,9].
For general time-dependent a, b we can use a symbolic tensor analysis package such as ccgrg [10] to compute The coefficient of dtdxdydz is an exact derivative as expected of a topological quantity.The coefficient in the anomaly equation is minus the A-roof expression because, as we will see in the next section, the topological index that counts zero modes has a plus sign for right-handed energy levels that sink into the Dirac sea and a minus sign for levels that emerge from the sea.If we again start at time −∞ with a stationary metric, have a period of time in which a and b vary slowly, then return to a stationary metric by t = ∞, then the anomaly equation tells us change in the total charge for right-handed particles should be This is a much larger number than the estimated number of particles created by the spectral flow.This particle-creation shortfall is our second surprise.
3 Vacuum charge and the eta-invariant.
We need to account for the discrepancy between the chiral charge arising from the anomaly equation and the one arising from explicit particle creation via spectral flow.We will see that the discrepancy is explained by the vacuumstate contribution to the charge.
We start with a rapid review of the connection between the anomaly and the Dirac index theorem.When the metric is of the form the Euclidean-signature Dirac operator appropriate for a pair of left and right Weyl fermions is (see B.1) where, in flat space, is the Hermitian Hamiltonian of a right-handed Weyl fermion.This remains true in curved space.Similarly −H is the Hamiltonian of a left-handed Weyl fermion.
It is this Euclidean-signature Dirac operator that is related, via the Atiyah-Singer index-theorem, to the spectral-flow interpretation of the anomaly.When the geometry changes slowly1 , an instantaneous Hamiltonian can still be defined but it now depends on τ .When a family of eigenfunctions u(τ ) of H(τ ) have eigenvalues ε(τ ) that change sign from positive to negative as τ goes from −∞ to ∞ the operator D has a normalizable γ 5 → +1 zero mode2 Conversely, when a family of eigenfunctions v(τ ) of H(τ ) have eigenvalues ε(τ ) that change sign from negative to positive as τ goes from −∞ to ∞ we find a normalizable γ 5 → −1 zero mode The Dirac index is the difference n + − n − in the number of normalizable right-and left-handed zero modes of D.
For a closed Euclidean-signature manifold M the index is given by where the integral on the RHS is necessarily a whole number.Our manifold is not closed, but is of the form M = N × R, where R is Euclidean time parameterized by τ .
In such an open manifold M Â is no longer guaranteed to be integral but the index on the LHS of (37) must still be an integer.A generally non-integral eta-invariant was introduced by Atiyah, Patodi and Singer [12] to supply the difference.The eta invariant is obtained from where the sum is over the spectrum of H (counted with multiplicity).The sum is convergent for large Re(s) and has a meromorphic continuation to all of s ∈ C. The eta-invariant itself is defined by η H = lim s→0 η H (s) and is always finite.The APS theorem [12] for Dirac operators is now the statement that (The dim Ker(H) term is only present when an energy level is precisely zero, so we will ignore it.)After multiplication by two we can rewrite the APS formula as in which the LHS is the integral over space-time of the RHS of anomaly equation for the chiral current of a Dirac fermion which comprises both a left-and a right-handed Weyl fermion.If the anomaly was solely due to spectral flow the change in the total chiral charge would be because an eigenvalue of H crossing from negative to positive means one extra right-handed particle and one fewer left-handed particle, and at the same time adds unity to n − .An eigenvalue of H crossing from positive to negative means one fewer right-handed particle and one more left handed particle, and adds unity to n + .The η-invariant correction can therefore be interpreted as the vacuum contributions that must be added to the count of explicitly-created particle pairs in order to compute the total chiral charge.The interpretation of −η(0)/2 as a nonintegral vacuum charge is of course familiar from the theory of charge fractionalization [13].The surprise here is that −η(0)/2 is not a small fractional correction, but actually supplies the bulk of the anomaly-induced charge.We are tempted to conjecture that this charge is due to the large spectral asymmetry from the "topological" modes that lie solely below the Fermi level.
We can investigate the conjecture by observing that, when b/4a 2 is sufficiently small that the chemical potential lies so close to the ± √ . . .symmetry point that no level crossing has occurred, we should have In appendix C we compute η H (0) in the small-b/4a 2 region directly from the Weyl-Hamiltonian eigenvalue spectrum and find that3 which is consistent with (29) and the induced charge from (30).We also make the decomposition of the eta invariant into parts due to the different branches of the spectrum.The −1/64 is the part due to the always-negative k = 0 modes, so, although not dominant, the do play a significant part in creating the anomalous chiral charge.

Discussion
We have constructed a family of space-times that possess gravitational analogues of Landau levels, and have used them to investigate the gravity contribution to the chiral anomaly.Although there are similarities to the magnetic field case there are also some surprises.The first is that although there are analogues of the topological modes that play a key role in the E•B electromagnetic case these modes do not contribute to any particle creation through level crossing.The second surprise is that when we let the metric vary (slowly) with time the number of explicitly created particles is small compared to the vacuum charge arising from the spectral asymmetry.The "topological" modes do, however, contribute a significant amount to this vacuum charge.
It must be noted, however, that in curved space, the decomposition of the charge into explicit particle number and a vacuum contribution is frame dependent.For some theories this can be understood by initially defining operators such as the current or energy-momentum tensor to be normal ordered with respect to the positive and negative frequencies as seen by an observer moving along a world line of a time-like Killing vector.Different Killing vectors lead to different normal-ordered operators, and so the intitially-defined operators do not transform as tensors.To obtain physically meaningful tensor quantities we must add c-number terms to the normal-ordered operators.
A classic example of this occurs in Hawking radiation where a Schwarzschild time-coordinate observer sees the asymptotic energy being carried by actual particles while in Kruskal coordinates the operator part of the energy momentum tensor has vanishing expectation value everywhere and the asymptotic energy flux comes entirely from the c-number vacuum term -see section III-A in [15] for recent discussion of this.The −η(0)/2 vacuum contribution to the chiral currents is an example of such a c-number.

Appendices A Heisenberg manifolds
In this appendix we will review the group theory that allows us to set-up and solve Dirac equation in the metric (5).This subject has been extensively explored by mathematicians: see [7], and in particular in [22].

A.1 The Heisenberg Group
The three-dimensional Heisenberg Lie Group He[R] is the set of matrices of the form The entries x, y, z can take any real values, so the group manifold is the whole of R 3 .The group product and inverse are given by The associated Lie algebra he has elements where the X, Ŷ , Ẑ matrices have commutators The physicists' Heisenberg algebra identifies X → q, Ŷ → p and Ẑ → i so [q, p] = i .
The exponential map Exp : he → He given by exp is a bijection and the exponential coordinates and the original defining coordinates are globally related by As with any Lie group, multiplication of a point in the group manifold by infinitesimal group elements from the right gives left-invariant vector fields, which in this case are Their vector-field Lie brackets obey [X, Y ] = Z etc.
The corresponding left-invariant Maurer-Cartan forms are extracted from as They satisfy ω X (X) = 1 and ω X (Y ) = 0 etc., so these are indeed the 1-forms dual to the left-invariant vector fields.Any constant-coefficient quadratic expression in the left-invariant Maurer-Cartan forms can serve as a left-invariant metric, meaning that the Lie derivative L X g = 0 etc., so X, Y , Z are Killing vectors.
Our metric (5) can be written as and is therefore left-invariant.
In terms of the exponential coordinates x = ξ, y = η z = ζ + ξη/2 this metric can be made to look more complicated [23] but of course it is just a reparametrization of same space.
In order to obtain a countable set of normalizable Dirac eigenfunctions it is useful to compactify R 3 by introducing periodic boundary conditions.We can do this by quotienting by the left action of a discrete subgroup of He[R].We must use a left action because we want our left-invariant metric to descend to a well-defined metric on the quotient space.
The simplest discrete subgroup of He The right coset Γ\He identifies so functions on Γ\He are functions on R 3 that satisfy The appearance of y in the z entry on the RHS of (60) means that these are "twisted" periodic boundary conditions.If {x} denotes the fractional part of x and ⌊x⌋ the integer part, we can take l = −⌊x⌋ and m = −⌊y⌋, n = −⌊z − ⌊x⌋y⌋ and so replace any group element by an equivalent in which the new x, y, z each lie in [0, 1).Consequently the compactified coset manifold can be identified with the cube [0, 1] 3 ∈ R 3 , but with twisted glueings of the opposite faces.

A.2 Harmonic analysis
The volume form for both systems of coordinates is and provides the measure defining a Hilbert space L 2 [Γ\He].
Our twisted boundary conditions appear to hinder Fourier transforms and mode decompositions on L 2 [Γ\He], but in fact they are no more complicated than -and indeed closely related to -the mode decompositions that are used when solving for the eigenstates of a Schrödinger particle moving in a Landau-gauge uniform magnetic field [16].
What happens is that the Hilbert space L 2 [Γ\He] decomposes into an orthogonal direct sum where Consequently H n consists of functions with discrete z-momenta labelled by n.In particular, H 0 consists of ordinary periodic functions of x and y that do not depend on z.
A function f ∈ H 1 can be obtained from any function ψ(x) ∈ L 2 [R] by a Zak (or Weil-Brezin) transform [17,18] To see that this is so, observe that Indeed all functions in H 1 can be obtained this way because given an element of f ∈ H 1 we can obtain a corresponding ψ(x) via an inverse map W −1 For n = 0, H n can be further decomposed as an orthogonal direct sum Functions in H n,m obey and may be obtained from a ψ(x) (70) There is again an inverse map and so again all functions in H n,m arise in this manner.

A.3 Scalar field Laplacian
The H n and H n,m subspaces correspond to the eigenspace decompositions of the Laplacian In particular, when −∇ 2 acts on functions in f = W n,m [ψ](x, y, z) ∈ H n,m we have We recognize that the expression in parentheses is an ω n = 2π|n| harmonic oscillator whose eigenfunctions are ψ n,k (x + L + m/n) with Here are the normalized ω = 1 wavefunctions.The resulting eigenvalues of −∇ 2 are [7] The n-fold degeneracy arises because the eigenvalues are independent of m = 0, . . ., |n| − 1.
For functions in H 0 we have the usual 2-d periodic spectrum B Dirac equation in curved space-time

B.1 General manifolds
The skew-Hermitian Dirac operator on an Here the e a ≡ e µ a ∂ µ compose an orthonormal vielbein on M and the γ a are Hermitian matrices obeying The object is the covariant derivative acting on the components of a Dirac spinor.It contains the "spin-connection" components ω bca ≡ ω bcµ e µ a defined by the action of the usual covariant derivative on the vielbein vectors e µ a ∇ µ e c = ∇ ea e c = e b ω bca (81) combined with the skew-Hermitian spin-representation generators of so(N) In four space-time dimensions we can take the euclidean-signature gamma matrices to be The space parts of the spin operators are then When the metric is of the form we have where, in flat space, is the Hermitian Hamiltonian of a right-handed Weyl fermion.This remains true in curved space.Similarly −H is the Hamiltonian of a left-handed Weyl fermion.
In Minkowski-metric signature (−, +, +, +) the gamma matrices become Then becomes the pair of Weyl equations The H operator is the same in both Euclidean and Minkowski signatures.

B.2 Weyl Hamiltonian for the Heisenberg manifold
Define the normalized dreibeins require only knowing the commutators appearing in The only non-zero components are together with their ω ijk = −ω jik partners.We reviewed the curved space Dirac operator in section B.1.For the three dimensional case we set e µ a ∂ µ = E a and γ a = σ a so giving When we extend to 4 dimensions −i D will become the Hamiltonian of the right-handed Weyl fermion.
Similar geometric methods can be applied to derive Dirac operators and Hamiltonians on spacetimes related to group manifolds.such as the Bianchi type-IX spaces of deformed 3-spheres [20,21] and their higher-dimensional Berger sphere versions.

B.3 Weyl spectrum for the Heisenberg manifold.
In principle the two-component spinors on which H R acts could have nontrivial spin structures -meaning that they acquire additional factors of ±1 as they pass though the periodic boundary conditions [22].We will consider only the simplest case in which both upper and lower components have the same periodicity properties as the scalar functions in the Laplace eigenproblem.
Acting on states of the form we have and hence eigenvalues On states of the form we have Here there are two different cases to consider: Case n > 0: Set ω n = 2π|n| > 0 then on oscillator states ϕ k ( √ ω n (x + m/n)) we recognize the ladder operators We therefore try spinors of the form whence with eigenvalues Again we have a degeneracy m = 1, . . ., |n| − 1 indicating the existence of Dirac gravitational Landau levels.
The special case k = 0 has ϕ k−1 = 0 and hence eigenvectors with eigenvalues Case n < 0: Again ω = 2π|n|, but â and â † are interchanged so we need whence with eigenvalues that appear the same as before, but the special k = 0 case now has eigenvalues Note that for n > 0 the special eigenvalues are − 2πn b − b 4a 2 and for n < 0 they are 2πn b − b 4a 2 , so the general case is − 2π|n| b − b 4a 2 .The |n| agrees with the spectrum in [22] but does not accord with the spectrum displayed on page 30 of [14], which has n rather than |n|.

C Computing η(0) from the spectrum
We now compute the spectral asymmetry η(0) directly from the spectrum of the Weyl Hamiltonian4 .We do this partly to compare it with the integral of the Â genus, but more importantly so that we can see how much of the vacuum charge arises from the k = 0 "topological" modes, the k > 0 Landaulevels, and the n = 0 modes.Each of these three parts of the spectrum requires different techniques, The simplest is the k = 0 "topological" branch with energies (120) For the other branches with their ± √ . . .we use the method described on page 34 of [20].The idea is to assume that λ = b/4a 2 is small enough that For λ sufficiently small the remainder denoted by ". .." will be analytic and zero at s = 0. Indeed the factors of s ensure that the s → 0 limit of all terms will be zero unless F (s + 1) and F (s + 3) have simple poles at s = 0. We must therefore compute the residues of F (s) at s = 1 and s = 3.
For the periodic spectrum involving we need The spectrum allows all positive or negative n's but in defining F 1 we must exclude the case when both n 1 and n 2 are zero; hence the −1 in the integral.With the n 1 = n 2 = 0 term absent the Gauss sum is exponentially small at large t, but has a small-t asymptotic expansion arising from the Poisson summation identity Corrections to the leading π/t are ∝ e −#/t , and so zero from the viewpoint of asymptotic expansions.The poles in F 1 (s) arise from the small-t divergences and so for the purposes of extracting the residues we may write The upper limit "1" on the integral (127) can be replaced by any convenient number without altering the residues, so To find the spin-connection we can use the Koszul formula which extracts the Levi-Civita spin connection from2g(∇ X Y, Z) = Xg(Y, Z)+Y g(Z, X)−Zg(X, Y )−g(X, [Y, Z])+g(Y, [Z, X])+g(Z, [X, Y ])(93) For our orthonormal frame the first three terms are zero, so the dreibein spin-connection components