Induced topological gravity and anomaly inflow from Kaehler-Dirac fermions in odd dimensions

We show that the effective action that results from integrating out massive Kaehler-Dirac fermions propagating on a curved three dimensional space is a topological gravity theory of Chern-Simons type. In the presence of a domain wall, massless, two dimensional Kaehler-Dirac fermions appear that are localized to the wall. Potential gravitational anomalies arising for these domain wall fermions are cancelled via anomaly inflow from the bulk gravitational theory. We also study the invariance of the theory under large gauge transformations. The analysis and conclusions generalize straightforwardly to higher dimensions.


I. INTRODUCTION
An alternative to the Dirac equation for describing fermions was proposed many years ago by Kähler [1]. It is based on the simple observation that a natural square root of the Laplacian is the operator d − d † where d is the exterior derivative and d † its adjoint. A key difference that distinguishes the Kähler-Dirac operator from its Dirac cousin is the fact that the former can be defined without reference to a local frame and spin connection. Thus Kähler-Dirac fermions are not globally equivalent to Dirac fermions and are well defined on any smooth manifold. Nevertheless there is a close connection between the two in flat space. The Kähler-Dirac equation in D-dimensional flat space takes the form where Φ = (ω, ω µ , ω µν , . . . , ω µ1...µ D ) is a collection of pform fields with p running from 0 to D. In this paper we will work in Euclidean space. It is straightforward to show that in even dimensions this can be mapped into a Dirac equation describing 2 D/2 degenerate Dirac spinors corresponding to the columns of a 2 D/2 × 2 D/2 matrix Ψ [2,3] (γ µ ∂ µ − m)Ψ = 0 (2) where Ψ = D µ=0 ω µ1...µp γ µ1 · · · γ µp .
Kähler-Dirac fields arise naturally in twisted supersymmetric theories [4] and are closely related to staggered fermions [2,5]. Recently, there has been renewed interest in them in connection with Dai-Freed anomalies [6,7], topological insulators [8] and symmetric mass generation in staggered fermion lattice models [9][10][11][12][13]. They have also been proposed as an ingredient in the construction of chiral lattice theories [14]. One consequence of this work has been the realization that massless Kähler-Dirac theories in even dimensions suffer from a gravitational anomaly, which breaks a global U (1) symmetry, unique to Kähler-Dirac fermions, down to Z 4 [15,16]. In four dimensions this anomaly is given by the Euler density abcd R ab ∧ R cd in contrast to the usual gravitational anomaly of Dirac fermions given by R ab ∧ R ab with R the Riemann tensor [17,18]. It should be noted that since Kähler-Dirac fermions can be decomposed into Dirac fermions in flat space they do not possess conventional γ 5 anomalies.
Remarkably this new anomaly survives discretization since it depends only on the topology of the background which can be captured exactly in a simplicial approximation to the space. This Z 4 symmetry prohibits bare mass terms but allows for four fermion interactions which can gap fermions without breaking symmetries for multiples of two Kähler-Dirac fields [16]. In flat space each such Kähler-Dirac field can be decomposed into 2 D/2+1 Majorana spinors and we deduce that such theories contain eight and sixteen Majorana spinors in two and four dimensions respectively. These magic fermion numbers that allow for symmetric mass generation are in agreement with the cancellation of certain discrete anomalies for Weyl fermions -chiral fermion parity in two dimensions and spin-Z 4 symmetry in four [6,19].
In this paper we will show that massive Kähler-Dirac fermions in odd dimensions exhibit further interesting properties; they yield gravitational Chern-Simons theories at low energies. Furthermore, in the presence of domain walls, these theories contain massless Kähler-Dirac fields localized to the domain wall. We show that potential anomalies for these domain wall fermions, of the type discussed above, are cancelled via anomaly inflow from the bulk gravitational theory.

II. KÄHLER-DIRAC FERMIONS IN THREE DIMENSIONS
Following our earlier discussion the massless Kähler-Dirac (KD) action in three dimensions can be written arXiv:2201.00750v2 [hep-th] 13 Jul 2022 where Φ = (φ, φ µ , φ µν , φ µνλ ) is a collection of p-forms (antisymmetric tensors). Notice that such a field possesses eight (complex) components in three dimensions. This action is invariant under a U (1) symmetry of the form where the linear operator Γ acts on the component pforms φ p according to whether it carries an even or odd number of indices φ p → (−1) p φ p . This property implies that Γ anticommutes with the Kähler-Dirac operator which then ensures the U (1) symmetry of the action. Furthermore, the operator Γ can be used to construct projectors P ± = 1 2 (I ± Γ) which act naturally on a Kähler-Dirac field to yield a pair of so-called reduced Kähler-Dirac fields Φ ± = P ± Φ. The Kähler-Dirac operator maps between Φ + to Φ − and vice versa.
If we want to map three dimensional Kähler-Dirac fermions into a set of spinors we will need the analog of the matrix expansion given in eqn. 3. Clearly one cannot map the eight component fields of a Kähler-Dirac fermion in three dimensions using just the minimal Dirac matrices corresponding to the Pauli matrices. Instead one must double the number of components of the spinor with the resulting matrix representation of the three dimensional Kähler-Dirac field employing 4 × 4 gamma matrices 1 . Naively such a field carries sixteen degrees of freedom but this can be reduced to eight using the projection operators P ± described earlier. These can be implemented in the matrix representation as The use of this four dimensional representation allows one to write down a massive three dimensional Kähler-Dirac action which preserves the U (1) symmetry provided the mass term is taken proportional to γ 4 : This action is invariant under a global spin(3) Lorentz symmetry L and a global spin(4) flavor symmetry F which act on the fields as 1 Three dimensional fermions of this type are called reducible fermions and correspond to a sum of the two irreducible spinor representations for spin(3) -see [20,21] Notice that F should contain L as a subgroup to reflect the Kähler-Dirac nature of the fermions since under a Lorentz transformation a Kähler-Dirac field must transform as a sum of tensor representations. However, to facilitate the computation of the effective action in the next section we will treat both symmetries as independent when gauging the action and impose the Kähler-Dirac condition relating the corresponding gauge fields only after the fermion integration. On a curved space and having gauged the flavor symmetry the action is modified to [22] whereÊ µ is a 3-frame corresponding to the background metric and D µ the associated covariant derivative which acts on the field Ψ as where Ω µ is the three dimensional spin connection andΩ µ is a spin(4) flavor gauge field. Notice that while Kähler-Dirac fermions do not require the use of a spin connection it is necessary to introduce such an object to do calculations in the matrix basis where the Kähler-Dirac field is represented in terms of Ψ.

III. INTEGRATING OUT THE FERMIONS
We will focus on deriving an effective action forΩ µ perturbatively in the limit M → ∞. If we integrate out the fermions we obtain an effective action which can be written Expanding the logarithm the leading term is corresponding to the diagram in fig 1. Here / V = V L + V R where the subscripts indicate whether the gauge field acts on the left or right of the matrix fermion Ψ: In momentum space this gives 2 If we focus on the contribution that is linear in the external momentum p we find Notice that a non-vanishing contribution comes only from employing V R at both vertices. The integral I is given by .
For M p and rescaling k/M → k Employing the identityÊ ABCÊ µ AÊ δ BÊ ν C = µδν we find a contribution to the effective action of the form This is not gauge invariant. There is however a contribution coming from next order in the expansion of the logarithm: 2 For more details see appendix Extracting the leading term which again uses only V R yields where .
where I is given by eqn. 18. In real space this yields Combining eqn. 25 and eqn. 19 gives the effective action where F is the spin(4) curvature. It is the unique term in the effective action that survives the large M limit. Notice that while this piece of the effective action comes from a U.V convergent integral this is not true of other terms arising in S CS ef f at finite M . Employing a Pauli-Villars regulator with mass Λ leads to the replacement 26. This modification plays an important role in our later discussion of domain wall physics and invariance of the effective action under large gauge transfromations in section VII.
We now impose the condition that the original Lorentz symmetry be a subgroup of the spin(4) flavor symmetry by settinĝ where Ω µ is the original spin connection with T AB = 1 4 [γ A , γ B ] the generators while E µ are the additional gauge fields needed for spin (4). In the next section we will see that E µ can be interpreted as a dynamical frame for an emergent geometry. Eq. 26 is hence a Chern-Simons term that ensures the effective action on a manifold without boundary is invariant under gauge transformations of the spin connection that can be smoothly deformed to the identity.

IV. GRAVITY INTERPRETATION
We can decompose the spin(4) curvature also under the original Lorentz group by computing the commutator of the corresponding spin(4) covariant derivative [D µ , D ν ] and expanding the resultant expression on the generators in a manner similar to that given in eqn. 27. This leads to the following expression: where we have discarded boundary terms. Clearly the action rewritten in these variables contains both an Einstein-Hilbert and cosmological constant term as expected of a gravity theory [23,24]. However, the relative coefficients of these terms have been fixed by the requirement that the theory actually enjoys a local spin(4) symmetry now interpreted as a local de Sitter gauge symmetry. Notice that the equation of motion for the Chern-Simons theory F µν = 0 now implies the pair of equations T µν = 0 corresponding to classical Euclidean de Sitter space and a torsion free connection. Of course this identification between Einstein Hilbert and Chern-Simons theory is still problematic at the non-perturbative level since in the path integral the latter necessarily includes degenerate metrics with vanishing frame. This is the origin of the topological character of the gravity theory as discussed in [25]. It is not a surprise that integrating out fermions in odd dimensions leads to a Chern-Simons theory -this is well known in the case of Dirac fermions transforming under some internal symmetry. What is new here is that if those fermions are taken to be of Kähler-Dirac type propagating on a curved background geometry then the induced Chern-Simons theory is actually a (topological) theory of gravity.

V. DOMAIN WALL CONSTRUCTION
In the previous section we assumed that the three dimensional manifold was compact. It is interesting to ask what happens in the presence of a boundary or equivalently if a domain wall is introduced in the system. Our argument parallels the original discussion by Callan and Harvey for Dirac fermions and later employed by Kaplan in his construction of domain wall lattice fermions [26,27].
Let us imagine a manifold of the form M × R with coordinates (x µ , z) where x µ parameterize position on the domain wall. Let us also allow the fermion mass M to change sign as a function of the flat coordinate z as shown in fig. 3 M One expects that massless states appear at z = 0. To One can find zero mode solutions of this equation of the form with γ 3 γ µ D µ ψ = 0 and ψ(x) an eigenvector of the hermitian operator H = −iγ 3 γ 4 with eigenvalue +1. The function χ(z) must then satisfy Thus one finds χ(z) = e −M0|z| corresponding to zero modes exponentially localized to the domain wall at z = 0. Notice that Ψ DW contains just four degrees of freedom -the original reduced field Ψ + contained eight degrees of freedom while the restriction to fields with H = +1 further halves the number of degrees of freedom. Four degrees of freedom corresponds to the field content of a two dimensional Kähler-Dirac field propagating on the wall. We can verify this explicitly by going to a (Euclidean) chiral basis for the gamma matrices corresponding to where σ µ = (iσ i , I) and σ µ = (−iσ i , I). This implies that Ψ + takes the 2 × 2 block form: and the matrix H takes the form: The additional requirement that Ψ DW be an eigenstate of H with eigenvalue +1 shows that ψ 1 contains two right handed two-dimensional Weyl spinors while ψ 2 contains two left handed spinors.
The constraint H = 1 for the domain wall fermions also breaks the original gauge symmetry to spin(2)×spin(2) 3 . The first factor corresponds to the generator 1 4 [γ 1 , γ 2 ] and is associated with the two dimensional spin connec-tionΩ 12 µ needed to enforce local Lorentz invariance for the domain wall modes. The second factor corresponds to H itself. The covariant derivative associated with rotations generated by H takes the form Using HΨ DW = Ψ DW this can be rewritten as If we define the domain wall chiral operatorγ 5 = γ 5 H the covariant derivative associated to H becomes where the domain wall reduced field Ψ ± DW is given by Thus we find that the gauge fieldΩ 34 couples only to a two dimensional reduced Kähler-Dirac field on the wall. A similar feature is seen in the interaction of the domain wall fermion with the two dimensional spin con-nectionΩ 12 . The corresponding term in the covariant derivative takes the form: Thus all gauge interactions on the domain wall couple only to the reduced Kähler-Dirac field Ψ − DW . To summarise we find that the low energy excitations of the three dimensional Kähler-Dirac theory in the presence of a domain wall are massless two dimensional Kähler-Dirac fermions Ψ DW localized to the wall and described by a Lorentz invariant action possessing an additional U (1) symmetry generated by an opera-torΓ =γ 5 ⊗γ 5 . The operatorΓ anticommutes with the two dimensional Kähler-Dirac operator describing the domain wall fermions and allows the Kähler-Dirac field to be projected into two independent reduced Kähler-Dirac fields Ψ − DW and Ψ + DW . Only one of these components Ψ − DW participates in the remaining spin(2)×spin(2) gauge symmetry.

VI. ANOMALY INFLOW FOR KÄHLER-DIRAC FERMIONS
At first glance the structure of the domain wall fermion action appears problematic since it is known that massless Kähler-Dirac fields in even dimensions suffer from a gravitational anomaly [15,16] that breaks this U (1)Γ symmetry down to Z 4 . The two dimensional domain wall action we derived in the previous section includes a gauged version of this symmetry (since the gauge field couples to aΓ reduced fermion) and hence one naively expects an anomaly induced breaking of gauge invariance in the low energy theory. In this section we will show that there is an additional contribution which arises from the bulk action which restores gauge invariance via an anomaly inflow mechanism.
To show in detail how this occurs we first include a brief review of the derivation of the anomaly specialized to the case of two dimensional domain wall fermions. Under a U (1)Γ rotation with parameter α(x) the measure for the Expanding / D 2 we obtain where F contains the surviving non-zero components of the spin(4) curvature corresponding to the symmetry spin(2)×spin (2). Expanding the exponential to O(1/M 2 ) to get a non-zero result for the trace over spinor and flavor indices and acting with e where R µν corresponds to the curvature of the spin con-nectionΩ 12 . We have employed the result e ab e µ a e ν b = µν in the last line. Hence, under a U (1) transformation the measure for a reduced Kähler-Dirac field transforms as 4 This naively breaks gauge invariance. However, the anomaly we have computed for the domain wall fermions is not the whole story. We showed earlier that the bulk contains also an induced Chern-Simons term. In general this also undergoes a non-zero change under a gauge transformation. In general the variation of the bulk Chern-Simons action takes the form: where F is the spin(4) curvature. Under a gauge transfor-mationΩ AB µ →Ω AB µ + D µ ζ AB the effective action changes: where a, b = {1, 2} while c, d = {3, 4} and these indices are to be contracted using two independent twodimensional symbols corresponding to the product of the two invariant tensors for spin(2)×spin(2) -the surviving symmetry on the domain wall. Taking ζ 34 = −ζ 43 = α(x) we find Thus the gauge transformation of the Chern-Simons term in the presence of the domain wall generates a contribution that is equal in magnitude but opposite in sign to that coming from the anomalous variation of the measure for the domain wall fermions -eqn. 47. Thus the bulk and boundary variations cancel and the full theory is gauge invariant. This is anomaly inflow in action for Kähler-Dirac fields. That this should occur is guaranteed by the fact that the Euler characteristic of the bulk theory is zero if it is taken to be a product of a two dimensional space and a circle since χ(S 1 ) = 0 5 . One might worry that the previous argument ignores the fact that the Chern-Simons term was computed for constant mass which is certainly not the situation close to the domain wall. However, it is possible to avoid this problem if one simply computes the change in the Chern-Simons current J 34 µ = δS ef f δA 34 µ between z = ∞ and z = −∞. One then finds where the second factor of two arises from the double counting associated with the fact that A AB µ = −A BA µ . Comparing this to the divergence of the U (1) current arising from the domain wall fermions ∂ µ J 34 µν we see that the net flow of charge off the domain wall is accounted for by the Chern-Simons current.

VII. INVARIANCE UNDER LARGE GAUGE TRANSFORMATIONS
It is of course interesting to ask about the invariance of the theory under large gauge transformations. To facilitate this analysis it is convenient to again adopt a Euclidean chiral basis for the gamma matrices. The spin(4) connection becomeŝ while the fermion field takes the form The Kähler-Dirac action then separates into two independent contributions where tr denotes a trace over a two dimensional block. Each such block will then generate its own SU (2) Chern-Simons term on integration over the fermions : where The relative minus sign in eqn. 55 arises because of differing signs of the mass in the two blocks [25]. Under a gauge transformation A µ → g(x)A µ g −1 (x) + g(x)∂ µ g −1 (x), each Chern-Simons term transforms, up to a boundary term, according to where the winding number n = π 3 (SU (2)) = Z 6 . Thus naively the level number of the CS term is k = ±1/2 with the partition function changing sign for odd n. However, once one regulates the theory with a Pauli-Villars field corresponding to a z-independent cutoff mass Λ the coefficients of the Chern Simons terms (the level numbers) are shifted to k = 0 and k = 1 in the two regions z < 0 and z > 0 respectively. Thus we find that the theory is in fact also invariant under large gauge transformations.

VIII. SUMMARY
We have shown that integrating out massive Kähler-Dirac fermions in a curved three dimensional background yields a Chern-Simons term. This Chern-Simons term corresponds to a topological theory of gravity in which both spin connection and frame emerge from an extended gauge symmetry -in this case Euclidean de Sitter symmetry. Gravity theories of this type were proposed many years ago [23,24,28] and generalize Witten's old observation that three dimensional gravity can be formally written as a Chern-Simons gauge theory [29].
In the presence of a domain wall we have shown that massless two dimensional Kähler-Dirac fermions appear on the wall. These are described by a single Kähler-Dirac field which can be decomposed into two independent components called reduced Kähler-Dirac fields which carry half the number of degrees of freedom. We find that just one of these reduced fields participates in the gauge interactions on the domain wall. Furthermore although the reduced Kähler-Dirac fermions on the wall suffer from a gravitational anomaly there is no violation of local gauge invariance because of anomaly inflow from the bulk gravitational Chern-Simons term.
It is not hard to generalize this construction to higher dimensions. For example, the effective long distance action for massive Kähler-Dirac fermions in five dimensions is also a topological gravity theory of Chern-Simons type [23,24,28] with gauge group spin (6) in Euclidean space. Using the same arguments as for three dimensions it is clear that massless four dimensional Kähler-Dirac fermions invariant under local spin(4) Lorentz transformations and an additional local U (1) symmetry would then arise in the presence of a domain wall in such a theory. Again, the domain wall action will contain a coupling of the U (1) gauge field to a single reduced Kähler-Dirac field. As in three dimensions gauge invariance of the theory remains intact since the gauge variation of the five dimensional Chern-Simons term cancels the potentially anomalous variation arising from the four dimensional reduced fermions.
One of the conclusions one can draw from our work is that coupling reduced Kähler-Dirac fermions to gravity in some even dimensional space is inconsistent due to a (mixed) gravitational anomaly unless the theory lives on a domain wall or boundary of a space of one higher dimension. If this additional dimension is finite there will necessarily be an anti-domain wall which localizes another reduced Kähler-Dirac fermion with the opposite eigenvalue of Γ. In this scenario the Chern-Simons current naturally flows between the two walls and the low energy theory is manifestly well defined.
Much of our discussion for Kähler-Dirac fermions has paralleled existing arguments for Dirac fermions. In this paper we have focused on perturbative anomalies and anomaly inflow. But in [16] it was shown that Kähler-Dirac fermions also exhibit discrete 't Hooft anomalies. Cancelling these anomalies is a necessary condition for symmetric mass generation and requires multiples of two Kähler-Dirac fields or four reduced Kähler-Dirac fields. If we take the flat space limit this constraint translates into the requirement that the theory contain sixteen Majorana spinors in four dimensions in perfect agreement with results for gapping edge modes in four dimensional topological insulators which require cancellation of a seemingly unrelated 't Hooft anomaly for a spin-Z 4 symmetry acting on Weyl fermions. This makes it plausible that theories of Weyl fermions, which are free of all 't Hooft anomalies and hence capable of symmetric mass generation, can be written in terms of Kähler-Dirac fermions. Furthermore since the anomalies of Kähler-Dirac fermions survive intact under discretization this suggests that they may be important for constructing lattice mirror models that target chiral theories in the continuum limit. Indeed numerical simulations of two flavors of interacting staggered fermions (which are obtained by discretization of Kähler-Dirac fermions) show evidence for the existence of a massive symmetric phase [30]. Further work is needed to understand these issues in more detail. measures manifest. We can now choose locally flat coordinates to evaluate the k-space integral. This allows us to reduce the calculation of S ef f to an equivalent flat space problem: