# Anomalies and symmetric mass generation for Kähler-Dirac fermions

Nouman Butt  Department of Physics, University of Illinois at Urbana-
Champaign, 1110 W Green St, Urbana, IL 61801  Simon Catterall  Department of
Physics, Syracuse University, Syracuse, NY 13244, USA  Arnab Pradhan
Department of Physics, Syracuse University, Syracuse, NY 13244, USA  Goksu Can
Toga  Department of Physics, Syracuse University, Syracuse, NY 13244, USA

###### Abstract

We show that massless Kähler-Dirac fermions exhibit a mixed gravitational
anomaly involving an exact $U(1)$ symmetry which is unique to Kähler-Dirac
fields. Under this $U(1)$ symmetry the partition function transforms by a
phase depending only on the Euler character of the background space.
Compactifying flat space to a sphere we learn that the anomaly vanishes in odd
dimensions but breaks the symmetry down to $Z_{4}$ in even dimensions. This
$Z_{4}$ is sufficient to prohibit bilinear terms from arising in the fermionic
effective action. Four fermion terms are allowed but require multiples of two
flavors of Kähler-Dirac field. In four dimensional flat space each Kähler-
Dirac field can be decomposed into four Dirac spinors and hence these anomaly
constraints ensure that eight Dirac fermions or, for real representations,
sixteen Majorana fermions are needed for a consistent interacting theory.
These constraints on fermion number agree with known results for topological
insulators and recent work on discrete anomalies rooted in the Dai-Freed
theorem. Our work suggests that Kähler-Dirac fermions may offer an independent
path to understanding these constraints. Finally we point out that this
anomaly survives intact under discretization and hence is relevant in
understanding recent numerical results on lattice models possessing massive
symmetric phases.

##  I Introduction

The Kähler-Dirac equation gives an alternative to the Dirac equation for
describing fermions in which the physical degrees of freedom are carried by
antisymmetric tensors rather than spinors. These tensors transform under a
twisted rotation group that corresponds to the diagonal subgroup of the usual
(Euclidean) Lorentz group and a corresponding flavor symmetry. In flat space
the Kähler-Dirac field can be decomposed into a set of degenerate Dirac
spinors but this equivalence is lost in a curved background since the coupling
to gravity differs from the Dirac case. Indeed, unlike the case of Dirac
fermions, Kähler-Dirac fermions can be defined on an arbitrary manifold
without the need to introduce a frame and spin connection. These facts were
emphasized many years ago by Banks et al. where an attempt was made to
identify the four Dirac fermions residing in each four dimensional Kähler-
Dirac field with the number of generations in the Standard Model Banks et al.
(1982). With precision LEP data ruling out a fourth generation this idea had
to be abandoned. However, in this paper, we will argue that there may be
another natural interpretation for this degeneracy - it is required in order
to write down anomaly free, and hence consistent, theories of interacting
fermions.

These constraints arise because the partition function of a massless free
Kähler-Dirac field in curved space transforms by a phase under a particular
global $U(1)$ symmetry with the phase being determined by the Euler character
$\chi$ of the background. The appearance of $\chi$ shows that the anomaly is
gravitational in origin but is distinct from the usual mixed gravitational
anomaly of Weyl fermions Fujikawa (1980). If we compactify flat space to a
sphere this anomaly breaks the $U(1)$ to $Z_{4}$ in even dimensions which is
sufficient to prohibit the appearance of fermion bilinear terms in the quantum
effective action. Four fermion terms are however allowed and the simplest
operator of this type requires two flavors of Kähler-Dirac fermion. Since a
massless Kähler-Dirac field can be decomposed into two so-called reduced
Kähler-Dirac fields, each carrying half the original degrees of freedom, four
reduced fields are required for the minimal four fermion interaction.

In even dimensions a reduced Kähler-Dirac field, in the flat space limit, can
be decomposed into $2^{D/2}$ Majorana spinors and we learn that consistent
interacting theories in even dimensions possess $2^{D/2+2}$ Majorana fields.
These fermion numbers agree with a series of anomaly cancellation conditions
associated with certain discrete symmetries in dimensions two and four - see
table 1 and García-Etxebarria and Montero (2019); Wan and Wang (2020).

D | Symmetry | Critical number of Majoranas
---|---|---
2 | Chiral fermion parity | 8
4 | Spin-$Z_{4}$ | 16
Table 1: Number of Weyl fermions needed for consistent interacting theories in
$D=2$ and $D=4$

Since cancellation of all ’t Hooft anomalies is a necessary condition for
fermions to acquire mass without breaking symmetries this suggests it may be
possible to build models with precisely this fermion content where all
fermions are gapped in the I.R. In fact there are examples in condensed matter
physics where precisely this occurs see Fidkowski and Kitaev (2010); Ryu and
Zhang (2012); Kapustin et al. (2015); You and Xu (2015); Morimoto et al.
(2015); You et al. (2018); Wang and Wen (2020); Guo et al. (2020). Similar
results have been obtained in staggered fermions in lattice gauge theory Ayyar
and Chandrasekharan (2015); Ayyar and Chandrasekharan (2016a, b); Catterall
(2016); Catterall and Butt (2018); Catterall et al. (2018).

The plan of the paper is as follows. We start in section II by giving a brief
introduction to Kähler-Dirac fermions exhibiting their connection to Dirac
fermions, and showing how, in the case of massless fields, the theory is
invariant under a $U(1)$ symmetry whose generator $\Gamma$ anticommutes with
the Kähler-Dirac operator on any curved background. Using $\Gamma$ one can
project out half the degrees of freedom to obtain a reduced Kähler-Dirac
field. Section III shows that the $U(1)$ symmetry suffers from a gravitational
anomaly in even dimensions and breaks to $Z_{4}$. In section IV we show using
a spectral flow argument that this remaining $Z_{4}$ symmetry suffers from a
global anomaly in the presence of interactions unless the theory contains
multiples of four reduced Kähler-Dirac fields. In section V we point out that
a necessary condition for symmetric mass generation in such theories is that
these anomalies cancel and we give examples of possible four fermion
interactions that might be capable of achieving this. Section VI points out
the connections between these continuum ideas and lattice fermions and a final
summary appears in section VII.

##  II Review of Kähler-Dirac Fields

The Kähler-Dirac equation arises on taking the square root of the Laplacian
operator written in terms of exterior derivatives. We start by defining the
Kähler-Dirac operator $K=d-d^{\dagger}$ with $d^{\dagger}$ the adjoint of $d$.
Clearly $-K^{2}=dd^{\dagger}+d^{\dagger}d=\Box$ since the derivative operators
are nilpotent. This suggests an alternative to the usual Dirac equation Kahler
(1962):

$(K-m)\Phi=0$ (1)

where $\Phi=\left(\phi_{0},\phi_{1},\ldots\phi_{p},\phi_{D}\right)$ is a
collection of $p$-forms. The action of the derivative operators on these forms
is then given by Banks et al. (1982); Catterall et al. (2018)

$\displaystyle\begin{split}d\Phi=\Big{(}0,\partial_{\mu}\phi,\partial_{\mu_{1}}\phi_{\mu_{2}}-\partial_{\mu_{2}}\phi_{\mu_{1}},\ldots\\\
\qquad,\sum_{perms\,\pi}\left(-1\right)^{\pi}\partial_{\mu_{1}}\phi_{\mu_{2}\ldots\mu_{D}}\Big{)}\end{split}$
(2)
$-d^{\dagger}\Phi=\left(\phi^{\nu},\phi^{\nu}_{\mu},\ldots,\phi^{\nu}_{\mu_{1},\ldots,\mu_{D-1}},0\right)_{;\nu}.$
(3)

An inner product of two such Kähler-Dirac fields $A$ and $B$ can be defined as

$\left[A,B\right]=\sum_{p}\frac{1}{p!}a^{{\mu_{1}}\ldots{\mu_{D}}}b_{{\mu_{1}}\ldots{\mu_{D}}}.$
(4)

Using this allows one to obtain the Kähler-Dirac equation by the variation of
the Kähler-Dirac action

$S_{\rm KD}=\int
d^{D}x\sqrt{g}\,\left[\overline{\Phi},\left(K-m\right)\Phi\right]$ (5)

where $\overline{\Phi}$ is an independent (in Euclidean space) Kähler-Dirac
field.

It is easy to see that the Kähler-Dirac operator anticommutes with a linear
operator $\Gamma$ which acts on the $p$-form fields as

$\Gamma:\quad\phi_{p}\to\left(-1\right)^{p}\phi_{p}.$ (6)

This anti-commutation property can be used to construct a $U(1)$ symmetry of
the massless Kähler-Dirac action which acts on $\Phi$ as

$\Phi\to e^{i\alpha\Gamma}\Phi.$ (7)

Furthermore, using $\Gamma$ one can define operators which project out even
and odd form fermions - so-called reduced Kähler-Dirac fields
$\Phi_{\pm}=P_{\pm}\Phi$ with

$P_{\pm}=\frac{1}{2}\left(1\pm\Gamma\right).$ (8)

The Kähler-Dirac operator $K$ couples even to odd forms and hence the massless
Kähler-Dirac action separates into two independent pieces
$S=\int\left(\overline{\Phi}_{+}\,K\Phi_{-}+\overline{\Phi}_{-}\,K\Phi_{+}\right)$.
Retaining just one of these terms one obtains an action for such a reduced
Kähler-Dirac (RKD) field

$S_{\rm RKD}=\int d^{D}x\sqrt{g}\,\left[\overline{\Phi}_{-},K\Phi_{+}\right].$
(9)

Notice that the single flavor reduced theory admits no mass term since
$\overline{\Phi}_{+}\Phi_{-}=0$. Finally if we relabel
$\overline{\Phi}_{-}\to\Phi_{-}$ this reduced action can be rewritten in a
Majorana-like form

$S_{\rm RKD}=\frac{1}{2}\int d^{D}x\sqrt{g}\,\left[\Phi,K\Phi\right].$ (10)

Given that both the Dirac operator and the Kähler-Dirac operator correspond to
square roots of the Laplacian one might imagine that there is a relation
between the Kähler-Dirac field and spinor fields. To exhibit this relationship
we construct a matrix $\Psi$ by combining the p-form components of the Kähler-
Dirac field $\Phi$ with products of Dirac gamma matrices

$\Psi=\sum_{p=0}^{D}\frac{1}{p!}\gamma^{{\mu_{1}}\ldots{\mu_{p}}}\phi_{{\mu_{1}}\ldots{\mu_{p}}}.$
(11)

where
$\gamma^{{\mu_{1}}\ldots{\mu_{p}}}=\gamma^{\mu_{1}}\gamma^{\mu_{2}}\cdots\gamma^{\mu_{p}}$
are constructed using the usual (Euclidean) Dirac matrices
$\gamma^{\mu}=\gamma^{a}e^{\mu}_{a}$ 111We will restrict ourselves to even
dimensions in what follows. Odd dimensions require twice as many spinor
degrees of freedom to match the number of components of a Kähler-Dirac field -
see Hands (2021). In flat space it is straightforward to show that the matrix
$\Psi$ satisfies the usual Dirac equation

$\left(\gamma^{\mu}\partial_{\mu}-m\right)\Psi=0$ (12)

and describes $2^{D/2}$ degenerate Dirac spinors corresponding to the columns
of $\Psi$. This equation of motion can be derived from the action

$S=\int d^{D}x\,{\rm
Tr}\,\left[\overline{\Psi}(\gamma^{\mu}\partial_{\mu}-m)\Psi\right].$ (13)

This action is invariant under a global ${\rm Spin}(D)\times SU(2^{D/2})$
symmetry where the first factor corresponds to Euclidean Lorentz
transformations and the second to an internal flavor symmetry. In even
dimensions we can write a matrix representation of the $U(1)$ generator as
$\Gamma\equiv\gamma_{5}\otimes\gamma_{5}$ where the two factors act by left
and right multiplication on $\Psi$. The matrix representation of a reduced
Kähler-Dirac field is then given by

$\Psi_{\pm}=\frac{1}{2}\left(\Psi\pm\gamma_{5}\Psi\gamma_{5}\right).$ (14)

Similarly the reduced action can be written as

$\int d^{D}x\,{\rm
Tr}\,\left(\overline{\Psi}_{-}\gamma^{\mu}\partial_{\mu}\Psi_{+}\right).$ (15)

The condition $\overline{\Phi}_{-}=\Phi_{-}$ then implies the matrix condition

$\Psi^{*}=B\Psi B^{T}$ (16)

where $B=C\gamma_{5}$ with $C$ the usual charge conjugation matrix. Using this
one can write a Majorana-like matrix representation of the reduced action as

$\frac{1}{2}\int d^{D}x\,{\rm
Tr}\,\left(B\Psi^{T}B^{T}\gamma^{\mu}\partial_{\mu}\Psi\right).$ (17)

Notice that after this reduction the free theory in flat space corresponds to
$2^{D/2-1}$ Dirac or $2^{D/2}$ Majorana spinors.

We can gain further insight into the reality condition eqn. 16 by going to a
chiral basis for the gamma matrices. In four dimensions the full Kähler-Dirac
field $\Psi$ then takes the form

$\Psi=\left(\begin{array}[]{cc}E&O^{\prime}\\\ O&E^{\prime}\end{array}\right)$
(18)

where $O$ and $O^{\prime}$ denote $2\times 2$ blocks of odd form fields while
$E$ and $E^{\prime}$ denote corresponding even form fields. Each block
contains a doublet of Weyl fields which transform in representations of the
$SU(2)_{L}\times SU(2)_{R}$ Lorentz and an $SU(2)\times SU(2)$ flavor
symmetry. The condition $\Psi^{*}=B\Psi B^{T}$ implies
$O^{\prime}=i\sigma_{2}O^{*}i\sigma_{2}$ and
$E^{\prime}=-i\sigma_{2}E^{*}i\sigma_{2}$. This suggests that the operation
$X\to i\sigma_{2}X^{*}i\sigma_{2}$ can be interpreted as a generalized charge
conjugation operator that flips both chirality and flavor representation of a
given Weyl doublet within the Kähler-Dirac field. It also implies that both
$(O,O^{\prime})$ and $(E,E^{\prime})$ constitute doublets of Majorana spinors.

Finally we should note that while the Kähler-Dirac equation eqn. 1 written in
the language of forms does not change in curved space, its matrix
representation takes the modified form

$(e^{\mu}_{a}\gamma^{a}D_{\mu}-m)\Psi=0$ (19)

where $e_{\mu}^{a}$ is the vielbein or frame and
$D_{\mu}\Psi=\partial_{\mu}\Psi+[\omega_{\mu},\Psi]$ is the covariant
derivative associated to the spin connection.

##  III A gravitational anomaly for Kähler-Dirac fields

In the previous section we showed that the Kähler-Dirac action is invariant
under a $U(1)$ symmetry. However in the quantum theory we should also be
careful to examine the invariance of the fermion measure. We will do this for
a generic four dimensional curved background using the matrix representation
of the Kähler-Dirac theory. The curved space action in $D=4$ reads

$S=\int d^{4}x\sqrt{g}\,{\rm
Tr}\,\left(\overline{\Psi}e^{\mu}_{a}\gamma^{a}D_{\mu}\Psi\right)$ (20)

where $D_{\mu}\Psi=\partial_{\mu}\Psi+[\omega_{\mu},\Psi]$ with $\omega_{\mu}$
the spin connection and we have introduced the frame field $e_{\mu}^{a}(x)$ to
translate between flat and curved space indices with
$e_{\mu}^{a}e_{\nu}^{b}\delta_{ab}=g_{\mu\nu}$. In the standard way we start
by expanding $\Psi$ and $\overline{\Psi}$ on a basis of eigenstates of the
Kähler-Dirac operator:

$\gamma^{\mu}D_{\mu}\phi_{n}=\lambda_{n}\phi_{n}$ (21)

with $\int d^{4}x\,e(x){\rm
Tr}\left(\overline{\phi}_{n}(x)\phi_{m}(x)\right)=\delta_{nm}$ and $e={\rm
det}\,(e_{\mu}^{a})$. Thus

$\displaystyle\Psi(x)$ $\displaystyle=\sum_{n}a_{n}\phi_{n}(x)$ (22)
$\displaystyle\overline{\Psi}(x)$
$\displaystyle=\sum_{n}\overline{b}_{n}\overline{\phi}_{n}(x).$ (23)

The measure is then written
$D\overline{\Psi}\,D\Psi=\prod_{n}d\overline{b}_{n}\,da_{n}$ and the variation
of this measure under the $U(1)$ transformation with parameter $\alpha(x)$ is
given by $e^{-2i\int\alpha(x)A(x)}$ where the anomaly $A$ is formally given by

$A(x)={\rm Tr}\,\sum_{n}e\overline{\phi}_{n}\Gamma\phi_{n}(x)$ (24)

where the operator $\Gamma=\Gamma_{5}\otimes\gamma_{5}$ carries a flavor
rotation matrix $\gamma_{5}$ acting on the right of the matrix field and a
curved space chiral matrix $\Gamma_{5}$ acting on the left with
$\Gamma^{5}=\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}e^{1}_{a}e^{2}_{b}e^{3}_{c}e^{4}_{d}=\gamma^{5}e$.
We need a gauge invariant regulator for this expression so we try inserting
the factor

$e^{\frac{1}{M^{2}}\left(\gamma^{\mu}D_{\mu}\right)^{2}}$ (25)

into the expression for $A$. We can write

$\left(\gamma^{\mu}D_{\mu}\right)^{2}=D^{\mu}D_{\mu}+e^{\mu}_{c}e^{\nu}_{d}\sigma^{cd}[D_{\mu},D_{\nu}]$
(26)

where $\sigma^{cd}=\frac{1}{4}[\gamma^{c},\gamma^{d}]$ are the generators of
${\rm Spin}(4)$. Furthermore for KD fermions we have:

$[D_{\mu},D_{\nu}]\psi=[R_{\mu\nu},\psi]$

where $R_{\mu\nu}=\frac{1}{2}R_{\mu\nu}^{ab}\sigma_{ab}$. Plugging this
expression into eqn. 26 yields

$\left(\gamma^{\mu}D_{\mu}\right)^{2}\psi=D^{\mu}D_{\mu}\psi+{\frac{1}{2}}e^{\mu}_{c}e^{\nu}_{d}\sigma^{cd}R_{\mu\nu}^{ab}[\sigma_{ab},\psi].$

The anomaly can then be written

$\displaystyle A(x)$ $\displaystyle=\lim_{M\to\infty}{\rm
Tr}\,\sum_{n}e\left(\overline{\phi}_{n}{\Gamma}e^{\frac{1}{M^{2}}(\gamma^{\mu}D_{\mu})^{2}}\phi_{n}\right)$
(27) $\displaystyle=\lim_{M\to\infty}{\rm Tr}\,\left(\Gamma
e^{\frac{1}{M^{2}}(\gamma^{\mu}D_{\mu})^{2}}\sum_{n}e\phi_{n}\overline{\phi}_{n}\right)$
$\displaystyle=\lim_{x\to x^{{}^{\prime}}}\lim_{M\to\infty}{\rm
Tr}\,\left(\Gamma
e^{\frac{1}{M^{2}}(\gamma^{\mu}D_{\mu})^{2}}\delta(x-x^{{}^{\prime}})\right)$
$\displaystyle\begin{split}&=\lim_{x\to x^{{}^{\prime}}}\lim_{M\to\infty}{\rm
Tr}\,\Big{(}e\gamma^{5}e^{\frac{1}{M^{2}}(D^{\mu}D_{\mu}+{\frac{1}{2}}e^{\mu}_{c}e^{\nu}_{d}{\sigma^{cd}}R_{\mu\nu}^{ab}[\sigma_{ab},.])}\\\
&\qquad\times\delta(x-x^{{}^{\prime}})\gamma_{5}\Big{)}.\end{split}$

Expanding the exponential to $O(1/M^{4})$ to get a non-zero result for the
trace over spinor and flavor indices and acting with
$e^{\frac{1}{M^{2}}D_{\mu}D^{\mu}}$ on the delta function yields 222See
appendix for more details

$\displaystyle\begin{split}A&=\frac{1}{16\pi^{2}}\left(\frac{1}{2!}\right)\left(\frac{1}{4}\right){\rm
tr}\,\left(e\gamma^{5}\sigma^{ab}\sigma^{cd}\right)e^{\mu}_{a}e^{\nu}_{b}e^{\rho}_{c}e^{\lambda}_{d}R_{\mu\nu}^{CD}R_{\rho\lambda}^{EF}\\\
&\qquad\times{\rm
tr}\,\left(\sigma_{CD}\sigma_{EF}\gamma_{5}\right)\end{split}$
$\displaystyle=\frac{1}{128\pi^{2}}\epsilon^{\mu\nu\rho\lambda}\epsilon_{CDEF}R_{\mu\nu}^{CD}R_{\rho\lambda}^{EF}$
(28)

where we have also employed the result:

$e\epsilon^{abcd}e^{\mu}_{a}e^{\nu}_{b}e^{\rho}_{c}e^{\lambda}_{d}=\epsilon^{\mu\nu\rho\lambda}.$

Thus the anomaly $A(x)$ is just the Euler density and we find that the phase
transformation of the partition function under the global $U(1)$ symmetry is
then

$Z\to e^{-2i\alpha\int d^{4}x\,A(x)}=e^{-2i\alpha\chi}Z.$ (29)

This result for the anomaly agrees with a previous lattice calculation that
employed a discretization of the Kähler-Dirac action on simplicial lattices
Catterall et al. (2018). The non-zero value for the anomaly originates in the
existence of exact zero modes of the Kähler-Dirac operator. Such zero modes
are eigenstates of $\Gamma$ and eqn. 24 shows that $\int
d^{4}x\,A(x)=n_{+}-n_{-}$ where $n_{\pm}$ denotes the number of zero modes
with $\Gamma=\pm 1$. Our final result is then a consequence of the index
theorem

$n_{+}-n_{-}=\chi.$ (30)

On the sphere (which can be regarded as a compactification of $R^{4}$) the
presence of this phase breaks the $U(1)$ to $Z_{4}$. This non-anomalous
$Z_{4}$ is then sufficient to prohibit fermion bilinear mass terms from
appearing in the effective action of the theory. This compactification of the
space on a sphere is similar to the strategy that is used to show the
importance of instantons in the QCD path integral. It places constraints on
the terms that can appear in the effective action as the radius of the sphere
is sent to infinity. Four fermion terms are allowed but require at least two
flavors of Kähler-Dirac field to be non-vanishing 333Of course we can go
further and demand that the anomaly be cancelled for manifolds with other
values of $\chi$. For example, on manifolds with $\chi=-4$ only eight fermion
terms are allowed, only twelve fermion terms on spaces with $\chi=-6$ etc.
Other work on higher order multifermion interactions can be found in Wu et al.
(2019); Jian and Xu (2020).. Since each massless Kähler-Dirac field can be
written in terms of two independent reduced fields this implies consistent,
interacting theories require four reduced Kähler-Dirac fields. In four
dimensions, and taking the flat space limit, each such reduced field
corresponds to four Majorana fermions and we learn that such theories contain
sixteen Majorana fermions. It is not hard to generalize this argument to any
even dimension.

##  IV A global $Z_{4}$ anomaly

In the last section we found that a system of free Kähler-Dirac fermions
propagating on an even dimensional space is anomalous and remains invariant
only under a $Z_{4}$ symmetry. In this section we will examine such a theory
in the presence of interactions and show that this residual $Z_{4}$ symmetry
suffers from a global anomaly unless the theory contains multiples of four
Kähler-Dirac fields.

From eqn. 10 it is clear that the effective action for a reduced Kähler-Dirac
fermion is given by a Pfaffian ${\rm Pf}(K)$. From the property
$[\Gamma,K]_{+}=0$ it is easy to show that ${\rm det}\,(K)\geq 0$. However
since ${\rm Pf}\,(K)=\pm\sqrt{\rm det}\,(K)$ there is an ambiguity in the
phase of the Pfaffian.

To analyze this in more detail we will consider the theory on a torus with
periodic boundary conditions and deform the theory to remove a potential zero
mode. The simplest possibility is to couple a pair of such reduced Kähler-
Dirac fields to an auxiliary real scalar field $\sigma$. The fermion operator
is then given by

$M=\delta^{ab}K+\sigma(x)\epsilon^{ab}.$ (31)

We will assume that the total action (including terms involving just $\sigma$)
is invariant under a discrete symmetry which extends the fermionic $Z_{4}$
discussed in the previous section:

$\displaystyle\Phi$ $\displaystyle\to i\Gamma\Phi$ (32) $\displaystyle\sigma$
$\displaystyle\to-\sigma.$ (33)

Notice that this fermion operator is antisymmetric and real and hence all
eigenvalues of $M$ lie on the imaginary axis. Let us define the Pfaffian as
the product of the eigenvalues in the upper half plane in the background of
some reference configuration $\sigma=\sigma_{0}={\rm constant}$. By continuity
we define the Pfaffian to be the product of these same eigenvalues under
fluctuations of $\sigma$. Furthermore, it is easy to see that as a consequence
of the $Z_{4}$ symmetry

$\Gamma M\left(\sigma\right)\Gamma=-M\left(-\sigma\right).$ (34)

This result shows that the spectrum and hence the determinant is indeed
invariant under the $Z_{4}$ transformation $\sigma\to-\sigma$. But this is not
enough to show the Pfaffian itself is unchanged since there remains the
possibility that eigenvalues flow through the origin as $\sigma$ is deformed
smoothly to $-\sigma$ leading to a sign change. To understand what happens we
consider a smooth interpolation of $\sigma$:

$\sigma(s)=s\sigma_{0}\quad s\in\left(-1,+1\right).$ (35)

The question of eigenvalue flow can be decided by focusing on the behavior of
the eigenvalues of the fermion operator closest to the origin at small $s$. In
this region the eigenvalues of smallest magnitude correspond to fields which
are constant over the lattice and satisfy the eigenvalue equation:

$\sigma_{0}s\epsilon^{ab}v^{b}=\lambda v^{a}.$ (36)

The two eigenvalues $\lambda=\pm i\sigma_{0}s$. Clearly these eigenvalues
change sign as as $s$ varies from positive to negative values leading to a
Pfaffian sign change. This can also be seen explicitly from eqn. 34 since

${\rm Pf}\,\left[M(-\sigma)\right]={\rm det}\,\left[\Gamma\right]{\rm
Pf}\,\left[M(\sigma)\right]=-{\rm Pf}\,\left[M(\sigma)\right].$ (37)

We thus learn that the Pfaffian of the 2 flavor system indeed changes sign
under the $Z_{4}$ transformation. On integration over $\sigma$ the value of
any even function of $\sigma$ including the partition function would then
yield zero rendering expectation values of $Z_{4}$ invariant operators ill-
defined. This corresponds to a global anomaly for the discrete $Z_{4}$
symmetry in the interacting theory.

Clearly this global anomaly will be cancelled for any multiple of four reduced
Kähler-Dirac fields provided that they couple via a Yukawa term of the form
$\sigma(x)\Phi^{a}C^{ab}\Phi^{b}$ with $C$ a real, antisymmetric matrix. In
that case $C$ can be brought to a canonical antisymmetric form
$\left(\lambda_{1}i\sigma_{2}\otimes\lambda_{2}i\sigma_{2}\otimes...\right)$
using a non-anomalous orthogonal transformation. Positivity of the Pfaffian
under $\sigma\to-\sigma$ then depends on the Yukawa interaction in $M$
containing an even number of such $2\times 2$ blocks. Decomposing the reduced
Kähler-Dirac field in a flat background into spinors we see that anomaly
cancellation occurs for eight or sixteen Majorana fermions in two and four
dimensions respectively.

The spectral flow argument we have given is similar to the one given by Witten
in showing that a single Weyl fermion in the fundamental representation of
$SU(2)$ is anomalous Witten (1982).

##  V Symmetric mass generation

The cancellation of anomalies is crucial to the problem of giving masses to
fermions without breaking symmetries. Since anomalies originate from massless
states then any phase where all states are massive in the I.R must necessarily
arise from a U.V theory with vanishing anomaly. In particular, it is only
possible to accomplish such symmetric mass generation if one cancels off the
’t Hooft anomalies for all global symmetries Razamat and Tong (2021); Tong
(2021).

In the previous section we have seen that only multiples of four reduced
Kähler-Dirac fields have vanishing $Z_{4}$ anomaly. Thus we require that any
interactions we introduce in the theory respect this symmetry. The simplest
such interaction is a four fermion operator as we have already discussed. It
corresponds to adding a simple $\int\sigma^{2}$ term to the Yukawa action
discussed in the previous section.

We should note however that cancellation of anomalies is a necessary condition
for symmetric mass generation but it may not be sufficient – the fact that
four fermion terms are perturbatively irrelevant operators in dimensions
greater than two may mean that a more complicated scalar action is required –
indeed this was the finding of numerical work in four dimensions where a
continuous phase transition to a massive symmetric phase was found only by
tuning an additional scalar kinetic term Butt et al. (2018).

With this caveat it is useful to give examples of possible four fermion terms
that might lead to symmetric mass generation For example, one can imagine
taking four reduced Kähler-Dirac fields transforming in the fundamental
representation of a $SO(4)$ symmetry and employ the term

$\int d^{D}x\,\sqrt{g}\,\left(\left[\Phi^{a},\Phi^{b}\right]_{+}\right)^{2}$
(38)

where the $+$ subscript indicates that fermion bilinear is projected to the
self-dual $(1,0)$ representation of $SO(4)$. In practice this can be
implemented in flat space via a Yukawa term which is given in the matrix
representation by

$\int d^{D}x\,G\,{\rm
Tr}\,\left(\Psi^{a}(x)\Psi^{b}(x)\right)_{+}\sigma_{ab}(x)+\frac{1}{2}\sigma^{2}_{ab}(x).$
(39)

Notice that this Yukawa interaction is mediated by a scalar $\sigma_{ab}^{+}$
that also transforms in the self-dual representation of $SO(4)$

$\sigma_{ab}^{+}=\frac{1}{2}\left(\sigma_{ab}+\frac{1}{2}\epsilon_{abcd}\sigma_{cd}\right)$
(40)

In the next section we show how Kähler-Dirac theories can be discretized in a
manner which leaves the anomaly structure of the theory intact and results in
theories of (reduced) staggered fermions. The four fermion interaction that
results from integrating over $\sigma_{+}$ in eqn. 39 has been studied using
numerical simulation and the results of this work indeed provide evidence of a
massive symmetric phase in dimensions two and three Ayyar and Chandrasekharan
(2015); Ayyar and Chandrasekharan (2016a, b); Catterall (2016) while an
additional scalar kinetic operator was also needed in four dimensions Butt et
al. (2018).

As another example one can take eight flavors of reduced Kähler-Dirac field
which are taken to transform in the eight dimensional real spinor
representation of ${\rm Spin}(7)$. An appropriate Yukawa term which might be
used to gap those fermions is given by

$\int d^{D}x\,{\rm
Tr}\,\left(\Psi^{a}(x)\Gamma^{ab}_{\mu}\Psi^{b}(x)\right)\sigma_{\mu}(x)$ (41)

where $\Gamma_{\mu},\mu=1\ldots 7$ are the (real) Dirac matrices for ${\rm
Spin}(7)$ You and Xu (2015). This interaction was shown by Fidkowski and
Kitaev to gap out boundary Majorana modes in a (1+1)-dimensional system
without breaking symmetries Fidkowski and Kitaev (2010). This interaction may
also play a role in constructing Kähler-Dirac theories that target GUT models.
For example, if one is able to gap out the $(E,E^{\prime})$ blocks occurring
in eqn. 18 for a reduced Kähler-Dirac field valued in ${\rm Spin}(7)$ the
remaining light fields live in the representation $(8,2,1)$. If the ${\rm
Spin}(7)$ is subsequently Higgsed to ${\rm Spin}(6)=SU(4)$ then this
representation breaks to $(4,2,1)\oplus(\overline{4},1,2)$ which is the field
content of the Pati-Salam theory Pati (1975).

##  VI Exact anomalies for lattice fermions

One of the most important properties of the Kähler-Dirac equation is that it
can be discretized without encountering fermion doubling Rabin (1982); Becher
and Joos (1982). Furthermore this discretization procedure can be done for any
random triangulation of the space allowing one to capture topological features
of the spectrum. The idea is to replace continuum p-forms by p-cochains or
lattice fields living on oriented p-simplices in the triangulation. The
exterior derivative and its adjoint are mapped to co-boundary and boundary
operators which act naturally on these p-simplices and retain much of the
structure of their continuum cousins – for example they are both nilpotent
operators. Homology theory can then be used to show that the spectrum of the
discrete theory evolves smoothly into its continuum cousin as the lattice
spacing is sent to zero – there are no additional lattice modes or doublers
that obscure the continuum limit. Furthermore, the number of exact zero modes
of the lattice Kähler-Dirac operator is exactly the same as found in the
continuum theory. This immediately suggests that the anomaly encountered
earlier which depends only on the topology of the background space can be
exactly reproduced in the lattice theory. This was confirmed in Catterall et
al. (2018) where a lattice calculation revealed precisely the same
gravitational anomaly derived in this paper.

If one restricts to regular lattices with the topology of the torus it is
straightforward to see that the discrete Kähler-Dirac operator discussed above
can be mapped to a staggered lattice fermion operator on a lattice with half
the lattice spacing Banks et al. (1982). One simply maps the p-form components
located in the matrix $\Psi^{a}$ into a set of single component lattice
fermions $\chi^{a}$ via

$\Psi(x)=\sum_{n_{\mu}=0,1}\chi(x+n_{\mu})\gamma^{\left(x+n_{\mu}\right)}$
(42)

where $\gamma^{x}=\prod_{i=1}^{D}\gamma_{i}^{x_{i}}$ and the summation runs
over the $2^{D}$ points in a unit hypercube of a regular lattice. If one
substitutes this expression into the continuum kinetic term, replaces the
continuum derivative with a symmetric finite difference and carries out the
trace operation one obtains the free staggered fermion action. Indeed the
operator $\Gamma$ acting on forms then becomes the site parity operator
$\epsilon(x)=\left(-1\right)^{\sum_{i=1}^{D}x_{i}}$ and the $U(1)$ symmetry of
the massless Kähler-Dirac action is just the familiar $U(1)_{\epsilon}$
symmetry of staggered fermions. Indeed, it is possible to repeat the arguments
of section IV to show that a staggered fermion theory equipped with a four
fermion term is only well-defined for multiples of four reduced staggered
fermions under which the classical $Z_{4}$ symmetry is preserved. This helps
to explain why these theories seem capable of generating a massive symmetric
phase Ayyar and Chandrasekharan (2015); Ayyar and Chandrasekharan (2016a, b);
Catterall (2016); Butt et al. (2018).

##  VII Summary

In this paper we have shown that theories of massless Kähler-Dirac fermions
suffer from a gravitational anomaly that breaks a $U(1)$ symmetry down to
$Z_{4}$ in even dimensions. We derive this anomaly by computing the symmetry
variation of the path integral for free Kähler-Dirac fermions propagating in a
background curved space. The remaining $Z_{4}$ prohibits fermion bilinear mass
terms from arising in the quantum effective action. We then use spectral flow
arguments to argue that multiples of four flavors of Kähler-Dirac are needed
to avoid a further global anomaly in this $Z_{4}$ symmetry in the presence of
interactions. Since four fermion interactions are allowed by these constraints
we argue that they may be capable of gapping such systems without breaking
symmetries.

In flat space each reduced Kähler-Dirac field transforming in a real
representation can be decomposed into $2^{D/2}$ Majorana fermions. Thus
anomaly cancellation in the interacting theory dictates a very specific
fermion content - multiples of eight and sixteen Majorana fermions in two and
four dimensions respectively. Remarkably, this fermion counting agrees with
independent constraints based on the cancellation of the chiral fermion parity
and spin-$Z_{4}$ symmetries of Weyl fermions in two and four dimensions
García-Etxebarria and Montero (2019); Wang and Wen (2020) 444 One can also
decompose a Kähler-Dirac fermion into two and four Majorana spinors in one and
three dimensions respectively. Building four fermion operators for these
Kähler-Dirac fields then yields theories with eight and sixteen Majorana
spinors which is also in agreement with results from odd dimensional
topological insulators..

Finally we discuss how this anomaly can be realized exactly in lattice
realizations of such systems and emphasize how the results in this paper shed
light on the appearance of massive symmetric phases in recent simulations of
lattice four fermion models. The appearance of an anomaly in a lattice system
is notable as it contradicts the usual folklore that anomalies only appear in
systems with an infinite number of degrees of freedom.

While the anomaly vanishes for closed odd dimensional manifolds it is non-zero
for odd dimensional manifolds with boundary. For example the Euler
characteristic of the three ball is $\chi(B^{3})=1$ and the symmetry in the
bulk is hence broken to $Z_{2}$ allowing for the presence of mass terms.
However the boundary fields living on $S^{2}$ possess an enhanced $Z_{4}$
symmetry prohibiting such bilinear terms and we learn that such boundary
fields can instead be gapped using four fermion interactions.

###### Acknowledgements.

This work was supported by the US Department of Energy (DOE), Office of
Science, Office of High Energy Physics under Award Number DE-SC0009998. SC is
grateful for helpful discussions with Erich Poppitz, David Tong and Yuven
Wang.

##  Appendix A Delta function

Following Fujikawa (1980)

$\delta(x-x^{{}^{\prime}})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik_{\mu}D^{\mu}\sigma(x,x^{{}^{\prime}})}$

where $D_{\mu}$ is now the Kähler-Dirac operator and
$\sigma(x,x^{{}^{\prime}})$ is the geodesic biscalar [a generalization of
$\frac{1}{2}(x-x^{{}^{\prime}})^{2}$ in flat space] defined by

$\sigma(x,x^{{}^{\prime}})=\frac{1}{2}g^{\mu\nu}D_{\mu}\sigma(x,x^{{}^{\prime}})D_{\nu}\sigma(x,x^{{}^{\prime}})$

with

$\sigma(x,x)=0$

and

$\lim_{x\to
x^{{}^{\prime}}}D_{\mu}D^{\nu}\sigma(x,x^{{}^{\prime}})=g^{\nu}_{\mu}.$

Now,

$\displaystyle\begin{split}D^{2}\delta(x-x^{{}^{\prime}})&=D^{\nu}\int\frac{d^{4}k}{(2\pi)^{4}}[ik_{\lambda}D_{\nu}D^{\lambda}\sigma\\\
&\qquad+\frac{1}{2!}\\{(D_{\nu}(ik\cdot D\sigma))(ik\cdot D\sigma)\\\
&\quad\qquad+(ik\cdot D\sigma)(D_{\nu}(ik\cdot D\sigma)\\}+...]\end{split}$
(43)
$\displaystyle\begin{split}&=\int\frac{d^{4}k}{(2\pi)^{4}}[D^{\nu}(ik_{\lambda}D_{\nu}D^{\lambda}\sigma)\\\
&\qquad+(ik_{\lambda}D_{\nu}D^{\lambda}\sigma)(ik_{\rho}D^{\nu}D^{\rho}\sigma)+...].\end{split}$
(44)

Taking $\lim_{x\to x^{{}^{\prime}}}$, other terms represented by … vanish, and
we obtain

$\displaystyle D^{2}\delta(x-x^{{}^{\prime}})$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}[(ik_{\lambda}g_{\nu}^{\lambda})(ik_{\rho}g^{\nu\rho})+D^{\nu}(ik_{\lambda}g_{\nu}^{\lambda})]$
(45)
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}[(ik_{\nu})(ik^{\nu})+iD^{\nu}(k_{\nu})]$
(46) $\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}[-k^{2}+iD^{\nu}(k_{\nu})].$
(47)

Which implies

$\lim_{x\to
x^{{}^{\prime}}}e^{D^{2}}\delta(x-x^{{}^{\prime}})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{[-k^{2}+iD^{\nu}(k_{\nu})]}.$
(48)

Hence,

$\displaystyle\lim_{x\to
x^{{}^{\prime}}}e^{D^{2}/M^{2}}\delta(x-x^{{}^{\prime}})$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}e^{[-k^{2}+iD^{\nu}(k_{\nu})]/M^{2}}$
(49)
$\displaystyle=M^{4}\int\frac{d^{4}k}{(2\pi)^{4}}e^{[-k^{2}+iD^{\nu}(k_{\nu})/M]}$
(50) $\displaystyle=M^{4}\frac{1}{16\pi^{2}}.$ (51)

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