# Axial anomaly in nonlinear conformal electrodynamics

Francisco Colipí-Marchant<EMAIL_ADDRESS>Departamento de
Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
Blanco Encalada 2008, Santiago, Chile  Cristóbal Corral<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Avenida
Playa Brava 3256, 1111346, Iquique, Chile  Facultad de Ciencias, Universidad
Arturo Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile  Daniel
Flores-Alfonso<EMAIL_ADDRESS>Instituto de Ciencias Exactas y Naturales,
Universidad Arturo Prat, Avenida Playa Brava 3256, 1111346, Iquique, Chile
Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón
2120, 1110939, Iquique, Chile  Leonardo Sanhueza<EMAIL_ADDRESS>Departamento de Física, Universidad de Concepción, Casilla, 160-C, Concepción,
Chile

###### Abstract

We study the axial anomaly of Dirac spinors on gravitational instanton
backgrounds in the context of nonlinear electrodynamics. In order to do so, we
consider Einstein gravity minimally coupled to a recently proposed conformal
electrodynamics that enjoys duality transformation invariance. These
symmetries allow us to generalize the Eguchi-Hanson configuration while
preserving its geometry. We then compute the Dirac index of the nonlinearly
charged Eguchi-Hanson and Taub-NUT configurations. We find that there is an
excess of positive chiral Dirac fermions over the negative ones which triggers
the anomaly.

##  I Introduction

The axial anomaly is a remarkable phenomenon that occurs when the axial
symmetry of a classical system is broken due to quantum effects, providing one
of the most interesting examples where the nontrivial topology of fiber
bundles has direct and measurable consequences in physics. It has been
observed through the anomalous decay $\pi^{0}\to\gamma\gamma$ in the standard
model, which would have been otherwise forbidden at the classical level since
a two-photon state does not couple to an axial current Steinberger (1949);
Fukuda and Miyamoto (1949); Schwinger (1951); Adler (1969); Bell and Jackiw
(1969); Bardeen (1969). On the other hand, in condensed matter systems, the
axial anomaly is a universal phenomenon that appears in three-dimensional
metals in presence of electric and magnetic fields, producing a longitudinal
magnetoresistance which is absent in the classical theory of magnetotransport
Goswami  _et al._ (2015). Additionally, it has been shown that the topological
transport and the chiral magnetic effect can be understood in terms of the
axial anomaly in Weyl semimetals Zyuzin and Burkov (2012). Indeed, the latter
has been detected in zirconium pentatelluride Li  _et al._ (2016) and particle
colliders have presented experimental evidence in favor of their existence by
analyzing the quark-gluon plasma Belmont (2014); Adamczyk  _et al._ (2015). In
topological insulators Volkov and Pankratov (1985); Pankratov  _et al._
(1987); König  _et al._ (2007); Kane and Mele (2005); Bernevig and Zhang
(2006), the effective field theory describing the electromagnetic responses
are governed by the axial anomaly Ryu  _et al._ (2012). These materials have
exotic properties such as spin-momentum locking Hsieh  _et al._ (2009), the
presence of Majorana states in the superconducting phase Fu and Kane (2008),
and the quantum spin hall effect Kane and Mele (2005); crucial features for
spintronic devices and dissipationless transistors in quantum computing.

The origin of the axial anomaly can be traced back to the imbalance of the
number of spinors with different chirality, produced by their interaction with
gauge fields possessing nontrivial topology; this usually happens in presence
of instantons Jackiw and Rebbi (1976, 1977). For instance, the Atiyah-Patodi-
Singer theorem Atiyah and Singer (1968); Atiyah  _et al._ (1973) provides a
deep connection between two topological indices of differential operators
acting on spinors, giving a concrete prescription to know if the background
fields would break the axial symmetry. This fact led Fujikawa to conclude that
the axial anomaly is a nonperturbative effect that arises from the nontrivial
transformation of the fermionic measure in the path-integral formulation of
gauge theories Fujikawa (1979, 1980).

Charap and Duff noticed that when gauge theories are coupled to gravity, the
topology of the matter content may be dictated by the background Charap and
Duff (1977a). Moreover, different instantons exist on spacetimes with
inequivalent topologies. In particular, there are solutions of the Yang-Mills
equations which cannot exist in flat space or would otherwise be singular
Charap and Duff (1977b). These conclusions were reached by analyzing
(anti-)self-dual fields on Euclidean space. In such a scenario, the energy-
momentum tensor vanishes by virtue of the matter’s conformal symmetry in
addition to (anti-)self duality. Their lack of backreaction on these
backgrounds renders their gravitational effect quite subtle. Some physical
aspects that involve nontrivial gravitational and matter fields are even less
obvious, such is the case of the axial anomaly as showcased by Pope in Refs.
Pope (1978, 1981).

In this paper, we study the axial anomaly in electromagnetism beyond Maxwell’s
theory. On the one hand, it is well-known that the effective theory of quantum
electrodynamics induces nonlinear effects when the fermionic degrees of
freedom are integrated out, capturing quantum effects at one-loop level
produced by virtual fermions Heisenberg and Euler (1936). On the other hand,
nonlinear constitutive relations have also been constructed to render the
self-energy of charged point particles finite Born (1933, 1934); Born and
Infeld (1934). Moreover, when coupled to gravity, nonlinear electrodynamics
has been considered an interesting framework to study black holes Ayon-Beato
and Garcia (1998); Bronnikov (2001); Ayon-Beato and Garcia (2000); Cataldo
_et al._ (2000); Burinskii and Hildebrandt (2002); Hassaine and Martinez
(2007); Miskovic and Olea (2011a); Cisterna  _et al._ (2020); Bravo-Gaete  _et
al._ (2022); Díaz García (2022); Ayón-Beato (2022), conserved charges Miskovic
and Olea (2011b), holographic superconductors Jing  _et al._ (2011); Aranguiz
and Miskovic (2014) and black strings Cisterna  _et al._ (2021), to name a
few. A standard reference is Plebański’s work Plebański (1970), while a recent
review is given in Ref. Sorokin (2022).

Given that the nature of the axial anomaly has proved to be fine-drawn, in
this work, we opt to carry out our exploration on (anti-)self-dual fields for
a conformal nonlinear electrodynamics which, by definition, satisfy the
electrodynamic equations of motion automatically. However, a duality
transformation does not necessarily leave the constitutive relation of the
theory invariant. As it turns out, if one imposes duality rotation invariance
on arbitrary nonlinear electrodynamics it can be concluded that there are
infinitely many of them, as many as real functions of a single variable
Salazar  _et al._ (1987); Gibbons and Rasheed (1995). Among this class of
theories, only one enjoys conformal symmetry. The latter is described by the
recently proposed one-parameter family of Lagrangians dubbed _ModMax_ Bandos
_et al._ (2020) that includes Maxwell electrodynamics as a particular case.
The theory has been studied in the context of black holes, birefringence,
supersymmetry, gravitational instantons, wormholes and the ’t Hooft anomaly,
among other applications Flores-Alfonso  _et al._ (2021a); Amirabi and Habib
Mazharimousavi (2021); Bandos  _et al._ (2021); Chatzistavrakidis  _et al._
(2021); Barrientos  _et al._ (2022a); Ballon Bordo  _et al._ (2021); Flores-
Alfonso  _et al._ (2021b).

In this work, we show that gravitational instantons coupled to nonlinearly
charged (anti-)self-dual fields trigger the axial anomaly of charged Dirac
spinors. This represents the first concrete example of the axial anomaly in
ModMax theory. To this end, we study the field equations of the Einstein-
ModMax system when the background has an underlying nontrivial circle
fibration over a Kähler manifold. Such is the case of the Taub-NUT and Eguchi-
Hanson spaces. We begin reviewing the former case as it has already been
discussed in the literature Ballon Bordo  _et al._ (2021); Flores-Alfonso  _et
al._ (2021b). We then formulate an ansatz of Eguchi-Hanson type and find a new
self-gravitating ModMax configuration. Both the fields and the curvature allow
for (anti-)self duality. The difference between positive and negative chiral
charged fermions is computed through the Atiyah-Patodi-Singer theorem taking
into account the boundary contribution. Particular attention is paid to the
computation of the $\eta_{D}$-invariant that measures the difference between
positive and negative eigenvalues of the Dirac operator evaluated on the
tangential components of the boundary. We find that the nonlinearly charged
Eguchi-Hanson and Taub-NUT instantons induce an imbalance between Dirac
spinors of different chirality, triggering the anomaly. We comment on possible
applications regarding other types of anomalies.

The article is organized as follows: in Sec. II, we briefly review the
nonlinear electrodynamics that we shall couple to Einstein gravity. The
Einstein-ModMax system constitute our main theoretical setup. Then, in Sec.
III, we study gravitational instantons as configurations that solve these
equations. We analyze their properties paying close attention to (anti-)self-
duality. Section IV is devoted to computing the index of the Dirac operator in
presence of nonlinear electromagnetic fields. Final comments and discussions
are given in Sec. V.

##  II Nonlinear electrodynamics

Departing from the superposition principle of electromagnetic fields leads to
a generalization of the source-free Maxwell’s equations given by

$\displaystyle\nabla_{\mu}P^{\mu\nu}=0\;\;\;\;\;\mbox{and}\;\;\;\;\;\nabla_{\mu}\tilde{F}^{\mu\nu}=0\,,$
(1)

where $P^{\mu\nu}=P^{\mu\nu}(F^{\alpha\beta})$ has a nonlinear dependence on
the Maxwell’s field strength and
$\tilde{F}_{\mu\nu}=\tfrac{1}{2}\varepsilon_{\mu\nu\lambda\rho}F^{\lambda\rho}$.
The nonlinear electrodynamics dubbed ModMax is the unique nonlinear extension
of Maxwell theory that preserves both conformal symmetry and $SO(2)$ duality
invariance Bandos  _et al._ (2020). Originally the theory was formulated in a
Legendre dual formulation and its Lagrangian was derived afterwards, However,
a straightforward derivation has been given in Ref. Kosyakov (2020). An
additional symmetry of ModMax theory is that it is Legendre self-dual Flores-
Alfonso  _et al._ (2021a), similar to Born-Infeld theory Gibbons and Herdeiro
(2001). In this sense, both electrodynamics are somewhat similar, however, the
Born-Infeld case is among the few theories where birefringence is absent Russo
and Townsend (2023). Indeed, the ModMax electrodynamics exhibits birefringence
of electromagnetic waves when the field equations are linearized around non-
vacuum configurations of constant electromagnetic fields Bandos  _et al._
(2020). Nevertheless, the theory is continuously connected to Maxwell
electrodynamics in the linear limit.

It should be kept in mind that further below we use ModMax theory as the
matter source of the Einstein equations. Thus, we find it convenient to
formulate it using the following action principle

$\displaystyle I_{\rm
M}=-\int_{\mathcal{M}}\text{d}^{4}x\sqrt{|g|}\,\mathscr{L}_{\rm M}\,,$ (2)

where $g=\det g_{\mu\nu}$ is the determinant of the metric and the Lagrangian
is given by

$\displaystyle\mathscr{L}_{\rm
M}=X\cosh\gamma-\sqrt{X^{2}-Y^{2}}\,\sinh\gamma\,.$ (3)

Here, $X=\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and
$Y=\tfrac{1}{4}\tilde{F}_{\mu\nu}F^{\mu\nu}$ are the two standard independent
Euclidean invariants that can be constructed with the field strength and its
dual.111In the Lorentzian version of the theory one should change
$Y^{2}\to-Y^{2}$ in Eq. (2) and variations thereof. Notice that $X$ has even
parity, while, $Y$ is odd. Thus, ModMax theory is a parity preserving
nonlinear extension of Maxwell’s; yet another similarity with the Born-Infeld
case. Additionally, the parameter $\gamma$ measures the degree of nonlinearity
of the electromagnetic fields. When $\gamma=0$ the theory becomes the standard
linear Maxwell electrodynamics. Notice that the parameter is dimensionless, as
required by conformal symmetry. In the Lorentzian formulation, causality
constraints impose that $\gamma\geq 0$ Bandos  _et al._ (2020).

As usual, the Bianchi identity allows for the gauge potential
$A=A_{\mu}\text{d}x^{\mu}$ to be defined by
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Moreover, by
performing arbitrary variations of the action (2) with respect to $A_{\mu}$,
yields Eq. (1) with the constitutive relation

$\displaystyle P_{\mu\nu}$
$\displaystyle=\left(\cosh\gamma-\frac{X\,\sinh\gamma}{\sqrt{X^{2}-Y^{2}}}\right)F_{\mu\nu}+\frac{Y\sinh\gamma}{\sqrt{X^{2}-Y^{2}}}\,\tilde{F}_{\mu\nu}\,.$
(4)

A straightforward calculation shows that the latter satisfies
$\tilde{F}_{\mu\nu}F^{\mu\nu}=\tilde{P}_{\mu\nu}P^{\mu\nu}$, a sufficient
condition for it to be invariance under duality rotations as shown in Ref.
Gibbons and Rasheed (1995). The stress-energy tensor, on the other hand, is
defined as the variation of the ModMax action with respect to the metric; in
particular, using the definitions in Eqs. (3) and (4), it gives

$\displaystyle T_{\mu\nu}=P_{(\mu}^{\ \
\lambda}F_{\nu)\,\lambda}-g_{\mu\nu}\,\mathscr{L}_{\rm M}\,.$ (5)

Notice that the latter reduces to the standard Maxwell stress-energy tensor in
the limit $\gamma\to 0$. Since the theory is conformal, the trace of Eq. (5)
vanishes. Whenever this condition is broken at the quantum level it is known
as the trace anomaly Capper and Duff (1974); Deser  _et al._ (1976); Duff
(1977).

To couple the theory with the Einstein-Hilbert action, we consider

$\displaystyle
I[g_{\mu\nu},A_{\mu}]=\kappa\int_{\mathcal{M}}\text{d}^{4}x\sqrt{|g|}\left(R-2\Lambda\right)+I_{\rm
M}\,,$ (6)

where $\kappa=(16\pi G)^{-1}$ with $G$ being Newton’s constant, $\Lambda$ is
the cosmological constant and $R=g^{\mu\nu}R^{\lambda}{}_{\mu\lambda\nu}$ is
the Ricci scalar. In this case, the field equations for the metric are

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda
g_{\mu\nu}=\frac{1}{2\kappa}T_{\mu\nu}\,.$ (7)

Taking the trace on the latter and using the fact that $T^{\mu}_{\mu}=0$ one
obtains $R=4\Lambda$. Therefore, all solutions to the field equations of
Einstein-ModMax theory have constant Ricci scalar and Eq. (7) reduces to
$R_{\mu\nu}=\Lambda g_{\mu\nu}+8\pi GT_{\mu\nu}$. Thus, the system of second-
order non-linear partial differential equations given by (1) and (7)
determines the dynamics of the theory.

##  III (Anti-)self-dual instantons

Central to our calculation of the Dirac index are systems with non-trivial
Pontryagin invariants. Indeed, whenever the electric and magnetic components
of the Weyl tensor are orthogonal, the Pontryagin density vanishes. Thus
backgrounds with (anti-)self-dual curvature are of particular interest to us.
In this section, we study nonlinearly charged self-gravitating systems with a
nonvanishing Pontryagin term. These configurations solve Einstein’s field
equations (7) and are described geometrically as inhomogeneous metrics on
complex line bundles over Kähler manifolds Page and Pope (1987). The gauge
potential is aligned along the fiber which allows the eletromagnetic fields to
simultaneously satisfy the ModMax constitutive relation (4) and field
equations (1).

In particular, we focus on metrics of the Taub-NUT Taub (1951); Newman  _et
al._ (1963) and Eguchi-Hanson Eguchi and Hanson (1978, 1979a) class. These
Euclidean spaces are known as gravitational instantons Hawking (1977); Eguchi
and Freund (1976); Gibbons and Hawking (1978); Hawking and Pope (1978);
Gibbons and Hawking (1979); Eguchi and Hanson (1978, 1979a, 1979b); Eguchi
_et al._ (1980) due to their close resemblance with Yang-Mills pseudo-particle
configurations.

###  III.1 Taub-NUT

The Lorentzian Taub-NUT spacetime is a one-parameter generalization of the
Schwarzschild black hole that is usually interpreted as a gravitational dyon,
in the sense that it is endowed with an electric and magnetic mass; see for
instance Refs. Lynden-Bell and Nouri-Zonoz (1998); Araneda  _et al._ (2016).
They source the electric and magnetic components of the Weyl tensor,
respectively. In Euclidean signature, metrics of this type are described by

$\displaystyle\text{d}s^{2}=f(r)\left(\text{d}\psi+2n\mathcal{B}_{(k)}\right)^{2}+\frac{\text{d}r^{2}}{f(r)}+\left(r^{2}-n^{2}\right)\text{d}\Sigma^{2}_{(k)}\,,$
(8)

where $f(r)$ is determined from the field equations. Here, $\mathcal{B}_{(k)}$
is the Kähler potential $1$-form that defines the Hermitian symplectic exact
form $\Omega_{(k)}=\text{d}\mathcal{B}_{(k)}$ associated to the Einstein-
Kähler base manifold of constant curvature $\text{d}\Sigma_{(k)}^{2}$. For
$k=1,0,-1$, the base manifold is topologically $\mathbb{S}^{2}$,
$\mathbb{T}^{2}$, and $\mathbb{H}^{2}$, respectively, and their Kähler
potential and line element can be parameterized as shown in Table 1.

| $k=1$ | $k=0$ | $k=-1$
---|---|---|---
$\mathcal{B}_{(k)}$ | $\cos\vartheta\text{d}\varphi$ | $\vartheta\text{d}\varphi$ | $\cosh\vartheta\text{d}\varphi$
$\text{d}\Sigma_{(k)}^{2}$ | $\text{d}\vartheta^{2}+\sin^{2}\vartheta\text{d}\varphi^{2}$ | $\text{d}\vartheta^{2}+\text{d}\varphi^{2}$ | $\text{d}\vartheta^{2}+\sinh^{2}\vartheta\text{d}\varphi^{2}$
Table 1: Line element of the transverse sections and Kähler potential $1$-form
for $\mathbb{S}^{2}$, $\mathbb{T}^{2}$, and $\mathbb{H}^{2}$ topologies,
respectively.

The first charged metric in the Taub-NUT class was found by Brill in Ref.
Brill (1964). More recently, higher-dimensional versions have been constructed
in Ref. Awad (2006). In these configurations, the Maxwell field is assumed to
be proportional to their associated Kähler potential $1$-form, namely,

$\displaystyle
A=A_{\mu}\text{d}x^{\mu}=\alpha(r)\left(\text{d}\psi+2n\mathcal{B}_{(k)}\right)\,.$
(9)

From hereon, we shall assume the same structure for the metric and Maxwell
fields as those given in Eqs. (8) and (9), respectively.

Inserting these ansätze into the field equations (1) and (7), the linearly
independent ordinary differential equations are

$\displaystyle 0$
$\displaystyle=\alpha^{\prime\prime}+\frac{2r\alpha^{\prime}}{\left(r^{2}-n^{2}\right)}-\frac{4n^{2}e^{-2\gamma}\alpha}{\left(r^{2}-n^{2}\right)^{2}}\,,$
(10) $\displaystyle 0$
$\displaystyle=4\kappa\left[rf^{\prime}+\left(\frac{r^{2}+n^{2}}{r^{2}-n^{2}}\right)f+\Lambda\left(r^{2}-n^{2}\right)-k\right]$
$\displaystyle-\left(r^{2}-n^{2}\right)e^{\gamma}\alpha^{\prime
2}+\frac{4n^{2}e^{-\gamma}\alpha^{2}}{\left(r^{2}-n^{2}\right)}\,,$ (11)

respectively, where $\alpha=\alpha(r)$ and $f=f(r)$, with prime denoting
differentiation with respect to the radial coordinate. The system is solved by
integrating first Eq. (10) and then replacing its solution into Eq. (11) to
solve for the metric function; this yields

$\displaystyle f(r)$
$\displaystyle=k\left(\frac{r^{2}+n^{2}}{r^{2}-n^{2}}\right)-\frac{2mGr}{r^{2}-n^{2}}-\frac{e^{-\gamma}}{4\kappa}\left(\frac{q^{2}-p^{2}}{r^{2}-n^{2}}\right)$
$\displaystyle-\frac{\Lambda}{3}\left(\frac{r^{4}-6n^{2}r^{2}-3n^{4}}{r^{2}-n^{2}}\right)\,,$
(12a) $\displaystyle\alpha(r)$
$\displaystyle=\frac{1}{2n}\left[q\sinh\Phi(r)+p\cosh\Phi(r)\right]\,,$ (12b)

where $m$, $q$, and $p$ are integration constants and we have defined

$\displaystyle\Phi(r)=e^{-\gamma}\ln\left(\frac{r-n}{r+n}\right)\,.$ (13)

This solution was first found in Refs. Ballon Bordo  _et al._ (2021); Flores-
Alfonso  _et al._ (2021b) and it is generically endowed with a conical defect
wherever the metric function $f$ vanishes. Moreover, when $k=1$ an additional
defect is present at $\vartheta=0$ and $\vartheta=\pi$ known as a Misner
string Misner (1963). In Lorentzian signature, the latter may lead to closed
time-like curves. Due to this, it has been extensively studied in Euclidean
signature; where it bears a close resemblance with Yang-Mills instantons.
Remarkably, it has been established that the presence of the Misner string
contributes to the entropy in a nontrivial way Mann (1999, 2000); Hawking  _et
al._ (1999); Mann and Stelea (2005); Ciambelli  _et al._ (2021); Garfinkle and
Mann (2000). The reason for this is that it represents an obstruction to the
foliation of the space with a scalar function that represents the Hamiltonian
evolution of the system Hawking and Hunter (1999). Nevertheless, recently the
Lorentzian case has attracted a lot of interest since it has been shown that
the Misner string is transparent for geodesic observers Clément  _et al._
(2015). This, in turn, has lead to a new thermodynamic treatment of the
spacetime, where the nut charge behaves as a genuine thermodynamic variable
with its respective conjugated pair, enlarging the phase space of the system
Durka (2022); Hennigar  _et al._ (2019); Bordo  _et al._ (2019); Ballon Bordo
_et al._ (2019); Frodden and Hidalgo (2022).

Let us focus on the case when $k=1$. First, notice that by performing the
coordinate transformation $\psi\to\tfrac{\psi}{2n}$, the line element (8) can
be written as

$\displaystyle\text{d}s^{2}$
$\displaystyle=\frac{\text{d}r^{2}}{f(r)}+\left(r^{2}-n^{2}\right)\left[\sigma_{1}^{2}+\sigma_{2}^{2}+\frac{4n^{2}f(r)\sigma_{3}^{2}}{\left(r^{2}-n^{2}\right)}\right]\,,$
(14)

where $\sigma_{i}$ are the left-invariant Maurer-Cartan one-forms of $SU(2)$
satisfying
$\text{d}\sigma_{i}+\tfrac{1}{2}\epsilon_{ijk}\sigma_{j}\wedge\sigma_{k}=0$.
Using the basis of Euler angles with $0\leq\vartheta\leq\pi$,
$0\leq\varphi\leq 2\pi$ and $0\leq\psi\leq\beta_{\psi}$, they can be
explicitly represented as

$\displaystyle\sigma_{1}$
$\displaystyle=\cos\psi\,\text{d}\vartheta+\sin\vartheta\,\sin\psi\,\text{d}\varphi\,,$
(15a) $\displaystyle\sigma_{2}$
$\displaystyle=-\sin\psi\,\text{d}\vartheta+\sin\vartheta\,\cos\psi\,\text{d}\varphi\,,$
(15b) $\displaystyle\sigma_{3}$
$\displaystyle=\text{d}\psi+\cos\vartheta\,\text{d}\varphi\,.$ (15c)

In presence of a negative cosmological constant, say
$\Lambda=-\tfrac{3}{\ell^{2}}$, the metric (14) possesses an (anti-)self-dual
Weyl tensor, i.e. $W_{\mu\nu\lambda\rho}=\pm\tilde{W}_{\mu\nu\lambda\rho}$.
Moreover, if the conditions

$\displaystyle p=\pm
q\;\;\;\;\;\mbox{and}\;\;\;\;\;mG=1-\frac{4n^{2}}{\ell^{2}}\,,$ (16)

are met, then the Killing vector field $\partial_{\psi}$ acquires a zero-
dimensional set of fixed points at $r=n$ and the metric function becomes

$\displaystyle f_{\rm nut}(r)$
$\displaystyle=\frac{r-n}{r+n}+\frac{\left(r-n\right)^{2}\left(r+3n\right)}{\ell^{2}(r+n)}\,,$
(17) $\displaystyle\alpha_{\rm nut}(r)$
$\displaystyle=\frac{q}{2n}\left(\frac{r-n}{r+n}\right)^{e^{-\gamma}}\,.$ (18)

This is the (anti-)self-dual Taub-NUT-AdS space coupled to a ModMax field.
Notice that the latter does not backreact on the metric since $T_{\mu\nu}$
vanishes identically, even though the field strength is nontrivial. This is a
unique feature of conformal theories whenever the fields are (anti-)self dual.
In nonlinear electrodynamics, the latter condition is described by

$\displaystyle F_{\mu\nu}=\pm\tilde{P}_{\mu\nu}\,.$ (19)

In this case, the range of coordinates is $n\leq r<\infty$,
$0\leq\vartheta\leq\pi$ and $0\leq\varphi\leq 2\pi$. The absence of conical
singularities, in turn, implies that $0\leq\psi\leq 4\pi$. Then, this metric
is topologically equivalent to $\mathbb{R}^{4}$ near the zero-dimensional set
of fixed points. Additionally, in the limit $\gamma\to 0$, one recovers the
charged Euclidean (anti-)self-dual Taub-NUT solution of Ref. Brill (1964).
Moreover, the ModMax field is asymptotically Maxwell, e.g., $\alpha(r)\to
q/(2n)$ as $r\to\infty$. Thus, although the nonlinearly charged ModMax Taub-
NUT solution is locally inequivalent to the Einstein-Maxwell one, they have
the same asymptotic behavior.

Another instanton within the same class as discussed above is the one called
Taub-Bolt. It is obtained when the Killing vector field $\partial_{\psi}$ has
a two-dimensional set of fixed points at $r=r_{b}$, with $r_{b}>n$, defined as
the largest root of the polynomial $f(r_{b})=0$. Such set of fixed points are
known as bolts, in contrast to the zero-dimensional variety which are called
nuts. The ModMax case has been studied in detail before but not its
contribution to the axial anomaly. Here, however, we shall focus on the
globally (anti-)self-dual Taub-NUT solution presented in Eq. (17) for the sake
of simplicity. Notwithstanding, outside of the Taub-NUT class there is another
configuration with a bolt that interests us and we devote the following
section to it.

###  III.2 Eguchi-Hanson

Having reviewed a known nonlinearly charged gravitational instanton, we are
now in a better position to present a new one. Our objective in this section
is to generalize the Eguchi-Hanson solution to the Einstein-Maxwell equations
found in Ref. Eguchi and Hanson (1979a). That configuration is strictly
Euclidean, that is, it has no Lorentzian analog. Its underlying topology is
that of the sphere’s tangent bundle and so its asymptotic boundary is a lens
space. Geometrically, the background is described by a $U(1)$ fibration over a
$2$-dimensional Kähler manifold. Thus, we consider an Eguchi-Hanson type line
element given by

$\displaystyle\text{d}s^{2}=\frac{r^{2}f(r)}{4}\left(\text{d}\psi+\mathcal{B}_{(k)}\right)^{2}+\frac{\text{d}r^{2}}{f(r)}+\frac{r^{2}}{4}\text{d}\Sigma^{2}_{(k)}\,,$
(20)

where $\mathcal{B}_{(k)}$ and $\text{d}\Sigma^{2}_{(k)}$ are defined in Table
1. Once more, it is fruitful to assume that the gauge field is proportional to
the Kähler one-form potential, namely,

$\displaystyle A\equiv
A_{\mu}\text{d}x^{\mu}=\alpha(r)\left(\text{d}\psi+\mathcal{B}_{(k)}\right)\,.$
(21)

The linearly independent differential equations obtained from the ModMax and
Einstein field equations are

$\displaystyle 0$
$\displaystyle=\alpha^{\prime\prime}+\frac{\alpha^{\prime}}{r}-\frac{4\alpha
e^{-2\gamma}}{r^{2}}\,,$ (22) $\displaystyle 0$
$\displaystyle=\kappa\left[rf^{\prime}+4\left(f-k\right)+\Lambda
r^{2}\right]-\alpha^{\prime
2}e^{\gamma}+\frac{4\alpha^{2}e^{-\gamma}}{r^{2}}\,,$ (23)

respectively. The solution of this system is obtained following a similar
prescription as in the Taub-NUT case, giving

$\displaystyle f(r)$
$\displaystyle=k-\frac{a^{4}}{r^{4}}-\frac{8pqe^{-\gamma}}{\kappa
r^{2}}-\frac{\Lambda r^{2}}{6}\,,$ (24a) $\displaystyle\alpha(r)$
$\displaystyle=q\,r^{-2e^{-\gamma}}+p\,r^{2e^{-\gamma}}\,,$ (24b)

with $a$, $p$, and $q$ being integration constants. This solution has a bolt
at the largest root of the polynomial $f(r_{b})=0$, subject to the condition
$r_{b}\in\mathbb{R}_{>0}$.

Similar to the Taub-NUT case, when $k=1$, the metric (20) can be written in
terms of the left-invariant Maurer-Cartan forms of $SU(2)$ [cf. Eq. (15)]

$\displaystyle\text{d}s^{2}=\frac{\text{d}r^{2}}{f(r)}+\frac{r^{2}}{4}\left(\sigma_{1}^{2}+\sigma^{2}_{2}+{f(r)}\sigma_{3}^{2}\right)\,.$
(25)

Notice that the hypersurfaces of constant radial coordinate are conformally
equivalent to the Berger’s sphere. Although this solution has a bolt, the Weyl
tensor of this space is globally (anti-)self dual only if $\Lambda=0$ for
arbitrary values of $p$ and $q$. This is different from the Taub-Bolt solution
with ModMax electrodynamics, which is asymptotically (anti-)self dual even in
presence of a nonvanishing cosmological constant Flores-Alfonso  _et al._
(2021b). Nevertheless, the gauge field is (anti-)self dual in the sense of Eq.
(19) if and only if $p=0$. Thus, we shall focus on the case $p=\Lambda=0$ in
Eq. (24) from hereon, where the bolt radius becomes $r_{b}=a$. In that case,
the range of coordinates is $a\leq r<\infty$, $0\leq\vartheta\leq\pi$, and
$0\leq\varphi\leq 2\pi$. On the other hand, the absence of conical
singularities at the bolt implies that $0\leq\psi\leq 2\pi$. Therefore, the
degenerate hypersurface $r=a$ is topologically
$\mathbb{S}^{3}/\mathbb{Z}^{2}=\mathbb{RP}^{3}$, while the asymptotic behavior
is $\mathbb{R}^{4}/\mathbb{Z}^{2}$ being asymptotically locally Euclidean.

Before closing this section, let us recall that the Eguchi-Hanson space has
been used to solve orbifold singularities of Calabi-Yau spaces in string
theory Polchinski (2007). It has also been used as a seed metric to study
gravitational solitons possessing nontrivial topology on the hypersurfaces of
constant time Clarkson and Mann (2006a, b); Chng  _et al._ (2006); Wong and
Mann (2012); Hendi  _et al._ (2012); Durgut  _et al._ (2022). Additionally,
higher-curvature corrections to the Eguchi-Hanson metric have been studied in
Refs. Corral  _et al._ (2021, 2022) and the backreaction of conformally
coupled scalar fields was considered in Ref. Barrientos  _et al._ (2022b).
Thus, future applications of the present configuration are plausible.

In the next section, we study how the nonlinearly charged gravitational
instantons presented here contribute to the axial anomaly of Dirac spinors. We
compute the Dirac index explicitly, taking care of the boundary contributions.

##  IV Index theorem and Axial anomaly

The Atiyah-Singer index theorem for the Dirac operator
$\not{D}\equiv\gamma^{a}E^{\mu}_{a}D_{\mu}$ relates the number of positive and
negative chiral spinors labeled by $N_{\pm}$ with topological invariants of
the Pontryagin class Atiyah and Singer (1968).222Here, we denote by
$E^{\mu}_{a}$ the inverse vierbein of $e^{a}_{\mu}$ which itself is defined
through $g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\delta_{ab}$, and it satisfies
$e^{a}_{\mu}E^{\mu}_{b}=\delta^{a}_{b}$ and
$e^{a}_{\mu}E^{\nu}_{a}=\delta^{\nu}_{\mu}$. If the index of the Dirac
operator is different from zero, one concludes that the nontrivial topology of
the gauge connection induces a difference between the number chiral spinors.
The aim of this section is to see whether the gravitational instantons studied
in Sec. III induce the axial anomaly or not.

For noncompact manifolds, the Atiyah-Singer theorem is modified by boundary
integrals of the Chern-Simons form and nonlocal terms related to the spectrum
of the Dirac operator Atiyah  _et al._ (1973). Moreover, if the spinors are
charged under some gauge group, the index theorem receives additional
contributions coming from their second Pontryagin class. For example, in four
dimensions, the index theorem of the Dirac operator coupled to $U(1)$ gauge
fields, i.e.,
$\not{D}=\gamma^{a}E^{\mu}_{a}(\partial_{\mu}+\frac{1}{4}\omega^{ab}{}_{\mu}\gamma_{ab}+iA_{\mu})$,
yields Eguchi  _et al._ (1980); Franchetti (2018)

$\displaystyle N_{+}-N_{-}$
$\displaystyle=-\frac{1}{24}\left[\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\,\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}R_{\sigma\tau}^{\lambda\rho}R^{\sigma\tau\mu\nu}-\frac{1}{4\pi^{2}}\int_{\partial\mathcal{M}}\text{d}^{3}x\sqrt{|h|}\;n^{\mu}\varepsilon_{\mu\nu\lambda\rho}K^{\sigma\nu}\nabla^{\lambda}K^{\rho}_{\sigma}\right]$
$\displaystyle+\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}F^{\mu\nu}F^{\lambda\rho}-\frac{1}{2}\left[\eta_{D}\left(\partial\mathcal{M}\right)+h_{D}\left(\partial\mathcal{M}\right)\right]\,,$
(26)

where $\omega^{ab}=\omega^{ab}{}_{\mu}\,\text{d}x^{\mu}$ is the spin
connection $1$-form, $\gamma_{ab}\equiv\gamma_{[a}\gamma_{b]}$ are the
generators of the Lorentz group in the spinorial representation,
$h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$ is the induced metric at the boundary
$\partial\mathcal{M}$ with space-like unit normal $n^{\mu}$ and we have
defined the extrinsic curvature as
$K_{\mu\nu}=h^{\lambda}_{\mu}h^{\rho}_{\nu}\nabla_{\lambda}n_{\rho}$. Here,
$\eta_{D}(\partial\mathcal{M})$ is the Atiyah-Patodi-Singer invariant that
measures the difference between positive and negative eigenvalues of the Dirac
operator on $\partial\mathcal{M}$ while $h_{D}(\partial\mathcal{M})$ captures
the zero eigenvalues thereof Atiyah  _et al._ (1973). The former has been
recently interpreted as the axial charge of physical states Kobayashi and
Yonekura (2021) and it can be obtained by performing the analytic continuation
of the meromorphic function $\eta_{D}(s)$ to $s=0$, namely
$\eta_{D}(\partial\mathcal{M})=\eta_{D}(s)|_{s=0}$, where

$\displaystyle\eta_{D}(s)=\sum_{\lambda\neq
0}\text{sign}(\lambda)|\lambda|^{-s}\,,$ (27)

with $\lambda$ being the eigenvalues of the Dirac operator $\not{D}$ evaluated
on the tangential components of $\partial\mathcal{M}$. The integral
representation of Eq. (27) is given by

$\displaystyle\eta_{D}(s)=\frac{1}{\Gamma\left(\frac{s+1}{2}\right)}\int_{0}^{\infty}\text{d}x\;x^{\frac{s-1}{2}}\,\text{tr}\left(\not{D}\,e^{-x\not{D}^{2}}\right)\,.$
(28)

Then, if $\text{Index}(\not{D})=0$, there is no axial anomaly whatsoever. This
is the case, for instance, of Dirac fermions on the (anti-)self-dual Taub-NUT
background of general relativity Eguchi  _et al._ (1978). Nevertheless, it was
shown by Pope that this is no longer true in presence of (anti-)self-dual
Maxwell fields Pope (1978, 1981). This implies that the axial symmetry is
broken at the quantum level for charged Dirac spinors on the (anti-)self-dual
Taub-NUT space. We will see here that this is the case in presence of
nonlinear electrodynamics as well.

Equivalently, the Atiyah-Patodi-Singer $\eta_{D}$-invariant can be computed
geometrically through the Hitchin formula Hitchin (1974). This can be done by
writing the induced boundary metric on a hypersurface of constant radial
coordinate, say $r=r_{0}$, conformally as the Berger’s sphere, namely,

$\displaystyle\text{d}s^{2}$
$\displaystyle=\Omega(r_{0})\left[\sigma_{1}^{2}+\sigma_{2}^{2}+\mu^{2}(r_{0})\sigma_{3}^{2}\right]\,,$
(29)

where $\Omega(r_{0})$ and $\mu(r_{0})$ are positive definite functions of
$r_{0}$. It is worth mentioning that $\eta_{D}(s)$ is invariant under a
constant rescaling of the eigenvalues $\lambda$, as it can be seen explicitly
from its definition in Eq. (27). Since the Dirac operator scales uniformly
under conformal transformations of the boundary metric (29), one can set
$\Omega(r_{0})=1$ without loss of generality Bakas and Lust (2011). Then, in
vacuum, the Hitchin’s formula stablishes that the $\eta_{D}$-invariant can be
computed from the boundary metric (29) through Hitchin (1974)

$\displaystyle\eta_{D}\left(\partial\mathcal{M}\right)+h_{D}\left(\partial\mathcal{M}\right)=\frac{1}{6}\left(1-\mu^{2}_{0}\right)^{2}\,,$
(30)

where $\mu_{0}=\mu(r_{0})$. This provides a convenient prescription to obtain
the axial anomaly of Dirac spinors from a geometrical viewpoint.

To compute the $\eta_{D}$-invariant in presence of ModMax fields, we consider
the induced boundary metric (29) over a hypersurface of constant radius. Since
the Dirac operator is invariant under a constant rescaling of the eigenvalues,
without loss of generality we can choose a dreibein basis $1$-form
$e^{a}=e^{a}_{\mu}\text{d}x^{\mu}$ as

$\displaystyle e^{1}$ $\displaystyle=\sigma_{1}\,,$ $\displaystyle e^{2}$
$\displaystyle=\sigma_{2}\,,$ $\displaystyle e^{3}$
$\displaystyle=\mu_{0}\,\sigma_{3}\,.$ (31)

The dual vector basis $E_{a}=E^{\mu}_{a}\partial_{\mu}$ can be computed from
the orthogonality condition with the dreibein one-forms, that is, $\langle
e^{a},E_{b}\rangle=\delta^{a}_{b}$. Then, we obtain

$\displaystyle E_{1}$ $\displaystyle=\Sigma_{1}\,,$ $\displaystyle E_{2}$
$\displaystyle=\Sigma_{2}\,,$ $\displaystyle E_{3}$
$\displaystyle=\mu^{-1}_{0}\Sigma_{3}\,,$ (32)

where $\Sigma_{i}=\Sigma_{i}^{\mu}\partial_{\mu}$ with $i=1,2,3$ are the dual
vector basis to the left-invariant Maurer-Cartan one-forms of $SU(2)$ (15);
that is,

$\displaystyle\Sigma_{1}$
$\displaystyle=-\cot\vartheta\sin\psi\,\partial_{\psi}+\cos\psi\,\partial_{\vartheta}+\frac{\sin\psi}{\sin\vartheta}\,\partial_{\varphi}\,,$
(33a) $\displaystyle\Sigma_{2}$
$\displaystyle=-\cot\vartheta\cos\psi\,\partial_{\psi}-\sin\psi\,\partial_{\vartheta}+\frac{\cos\psi}{\sin\vartheta}\,\partial_{\varphi}\,,$
(33b) $\displaystyle\Sigma_{3}$ $\displaystyle=\partial_{\psi}\,.$ (33c)

On the other hand, the nontrivial components of the spin connection can be
obtained by solving the torsion-free condition
$\text{d}e^{a}+\omega^{a}{}_{b}\wedge e^{b}=0$, giving

$\displaystyle\omega^{12}$
$\displaystyle=\left(\frac{\mu^{2}_{0}-2}{2\mu_{0}}\right)e^{3}\,,$
$\displaystyle\omega^{13}$ $\displaystyle=\frac{\mu_{0}\,e^{2}}{2}\,,$
$\displaystyle\omega^{23}$ $\displaystyle=-\frac{\mu_{0}\,e^{1}}{2}\,.$ (34)

For the Dirac matrices, we choose the representation in a co-dimension $1$
hypersurface of constant radius as

$\displaystyle\gamma^{1}$ $\displaystyle=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ $\displaystyle\gamma^{2}$
$\displaystyle=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},$
$\displaystyle\gamma^{3}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,.$ (35)

Additionally, we denote the $U(1)$ connection $1$-form evaluated at $r=r_{0}$
by $A=b(r_{0})\sigma_{3}\equiv b_{0}\sigma_{3}$.333In the Taub-NUT case, this
definition implies that $b(r)=2n\alpha(r)$ provided the coordinate
transformation $\psi\to\frac{\psi}{2n}$, while for Eguchi-Hanson we have
$b(r)=\alpha(r)$. Then, defining the self-adjoint operator $K_{i}=i\Sigma_{i}$
and the ladder operator $K_{\pm}=K_{1}\pm iK_{2}$, one can show that they
satisfy the Lie algebra of angular momentum, i.e.
$\left[K_{i},K_{j}\right]=i\epsilon_{ijk}K^{k}$,
$\left[K_{3},K_{\pm}\right]=\pm K_{\pm}$, and
$\left[K_{+},K_{-}\right]=K_{3}$. Hence, the Dirac operator acting on a two-
component Dirac spinor expressed in matrix notation can be written as

$\displaystyle\not{D}\Psi$
$\displaystyle=\begin{pmatrix}\mu^{-1}\left(K_{3}+b_{0}\right)&K_{-}\\\
K_{+}&-\mu^{-1}_{0}\left(K_{3}+b_{0}\right)\end{pmatrix}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}$
$\displaystyle\quad+\frac{(\mu^{2}_{0}+2)}{4\mu_{0}}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}=0\,.$ (36)

In order to compute the spectrum of the Dirac operator, we follow Ref. Pope
(1981) and denote $|s\rangle\equiv\,_{s}Y_{lm}(\vartheta,\varphi)$, where
${}_{s}Y_{lm}(\vartheta,\varphi)$ stands for the spin-$s$ spherical harmonics
Goldberg  _et al._ (1967). Then, the self-adjoint operator acting on the state
$|s\rangle$ gives

$\displaystyle K_{\pm}\,|s\rangle$ $\displaystyle=\sqrt{(l\mp s)(l\pm
s+1)}\;|s\pm 1\rangle\,,$ (37) $\displaystyle K_{3}\,|s\rangle$
$\displaystyle=s\,|s\rangle\,,$ (38)

for $|s|\leq l$ and $|m|\leq l$. The eigenvectors of $\not{D}$ can be written
as $\Psi_{1}=c_{1}\,|s\rangle$ and $\Psi_{2}=c_{2}\,|s+1\rangle$, where
$c_{1}$ and $c_{2}$ are constants. Thus, for $-l\leq s\leq l-1$, the
eigenvalues $\lambda$ of the Dirac operator $\not{D}$ are Pope (1981)

$\displaystyle\lambda=\frac{\mu_{0}}{2}\pm\mu_{0}^{-1}\sqrt{(2s+1+2b_{0})^{2}+4\mu^{2}_{0}(l-s)(l+s+1)}\,,$
(39)

where, for each value of $s$, these eigenvalues have a degeneracy of $d=2l+1$.
Notice that there are two special cases: (i) if $s=l$, or (ii) if $s=-(l+1)$
where $c_{2}$ and $c_{1}$ vanish, respectively. Then, their eigenvalues become

$\displaystyle\mbox{(i)}\;\;\;\;\;\lambda$
$\displaystyle=\frac{\mu_{0}}{2}\pm\mu_{0}^{-1}\left|2l+1+2b_{0}\right|\,,$
(40a) $\displaystyle\mbox{(ii)}\;\;\;\;\;\lambda$
$\displaystyle=\frac{\mu_{0}}{2}\pm\mu_{0}^{-1}\left|2l+1-2b_{0}\right|\,.$
(40b)

We are interested in evaluating the $\eta_{D}$-invariant at the asymptotic
boundary, that is, as $r_{0}\to\infty$. For the sake of simplicity, from
hereon we focus on the case with vanishing cosmological constant or,
equivalently, $\ell\to\infty$. Then, for the (anti-)self-dual Taub-NUT (TN)
and Eguchi-Hanson (EH) spaces, we have

TN: $\displaystyle\lim_{r_{0}\to\infty}\mu(r_{0})$ $\displaystyle=0$
$\displaystyle\wedge$ $\displaystyle\lim_{r_{0}\to\infty}b(r_{0})$
$\displaystyle=q\,,$ (41) EH: $\displaystyle\lim_{r_{0}\to\infty}\mu(r_{0})$
$\displaystyle=1$ $\displaystyle\wedge$
$\displaystyle\lim_{r_{0}\to\infty}b(r_{0})$ $\displaystyle=0\,,$ (42)

respectively. Notice that the ModMax field approaches to its Maxwellian
counterpart in the charged Euclidean Taub-NUT metric, as mentioned in Sec.
III. On the other hand, in the case of the Eguchi-Hanson instanton, the ModMax
field decays sufficiently fast towards the asymptotic boundary. Therefore, the
$\eta_{D}$-invariant of the latter coincides with the one computed in Ref.
Eguchi and Hanson (1979a), giving

$\displaystyle\eta_{D}^{\rm(EH)}\left(\partial\mathcal{M}\right)+h_{D}^{\rm(EH)}\left(\partial\mathcal{M}\right)=\frac{1}{4}\,.$
(43)

Although the boundary contributions to the Dirac index in Eq. (26) vanishes
for the charged Eguchi-Hanson solution of Sec. III.2, the bulk part gives

$\displaystyle\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\,\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}R_{\sigma\tau}^{\lambda\rho}R^{\sigma\tau\mu\nu}$
$\displaystyle=-3$ (44)
$\displaystyle\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}F^{\mu\nu}F^{\lambda\rho}$
$\displaystyle=-\frac{q^{2}}{a^{4e^{-\gamma}}}\,.$ (45)

In this case, the topology of the bolt introduces a screening factor on the
topological charge (45) that can be controlled with the nonlinear parameter
$\gamma$. Indeed, in the limit $\gamma\to\infty$, the latter becomes equal to
$-q^{2}$. Then, for a finite $\gamma$, we find that the Dirac index for the
charged Eguchi-Hanson solution of Einstein-Modmax theory is

$\displaystyle N_{+}-N_{-}=-\frac{q^{2}}{a^{4e^{-\gamma}}}\,,$ (46)

which is nonvanishing for $q\neq 0$ with $\gamma\in\mathbb{R}$. Thus, we
conclude that this particular instantonic configuration does contribute to the
axial anomaly.

The computation of the $\eta_{D}$-invariant in the case of the (anti-)self-
dual Taub-NUT instanton is more subtle, since the eigenvalues (39) are
divergent as $r_{0}\to\infty$. Nevertheless, one can exploit the fact that the
$\eta_{D}(s)|_{s=0}$ is invariant under the global rescaling
$\lambda\to\mu_{0}\lambda$ to obtain finite eigenvalues. In order to do so, we
first perform the rescaling of the eigenvalues and then take the limit
$r_{0}\to\infty$. This procedure yields Pope (1981)

$\displaystyle\lambda=\pm|2s+1+2b_{0}|\,.$ (47)

The symmetry between the positive and negative eigenvalues derived from this
equation implies that their sum will not contribute to the
$\eta_{D}$-invariant. However, this is not the case if $s=l$ or if $s=-(l+1)$,
which leads to the particular values in Eq. (40). Thus, after rescaling the
eigenvalues as $\lambda\to\mu_{0}\lambda$ and taking the limit
$r_{0}\to\infty$, one obtains

$\displaystyle\mbox{(i)}\;\;\;\;\;\lambda$ $\displaystyle=2l+1+2b_{0}\,,$
(48a) $\displaystyle\mbox{(ii)}\;\;\;\;\;\lambda$
$\displaystyle=2l+1-2b_{0}\,,$ (48b)

both with degeneracy $d=2l+1$. These exceptional cases are the main
contribution to the $\eta_{D}$-invariant. To see this, let us focus on the
case when $b_{0}$ is an integer with $b_{0}>0$ without loss of generality; the
case for noninteger $b_{0}$ in Einstein-Maxwell theory has been studied in
Ref. Pope (1981). Then, the eigenvalue (48a) is positive definite, while Eq.
(48b) could take negative values if $2l+1<b_{0}$. Thus, considering the
contributions coming from the positive and negative eigenvalues of the Dirac
operator to the $\eta_{D}$-invariant in Eq. (27), the latter can be expressed
as

$\displaystyle\eta_{D}^{\rm(TN)}\left(\partial\mathcal{M}\right)$
$\displaystyle=\lim_{s\to
0}\left[\sum_{l=0}^{\infty}\left(2l+1\right)\left(2l+1+2b_{0}\right)^{-s}-\sum_{l=0}^{2b_{0}-1}\left(2l+1\right)\left(2l+1-2b_{0}\right)^{-s}+\sum_{l=2b_{0}}^{\infty}\left(2l+1\right)\left(2l+1-2b_{0}\right)^{-s}\right]$
$\displaystyle=-\frac{1}{6}-2q\,,$ (49)

where we have used the fact that $h_{D}^{\rm(TN)}(\partial\mathcal{M})=0$ and
the definition of the Riemann zeta function
$\zeta(s)=\sum_{p=1}^{\infty}p^{-s}$. Additionally, some of its particular
values has been taken into account, e.g. $\zeta(-1)=-1/12$ and
$\zeta(0)=-1/2$. Then, inserting Eq. (49) into the Dirac index (26) and using
the bulk integrals

$\displaystyle\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\,\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}R_{\sigma\tau}^{\lambda\rho}R^{\sigma\tau\mu\nu}=2\,,$
(50)
$\displaystyle\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\text{d}^{4}x\sqrt{|g|}\,\varepsilon_{\mu\nu\lambda\rho}F^{\mu\nu}F^{\lambda\rho}=2q^{2}\,,$
(51)

together with the fact that the boundary integral of the Chern-Simons form
vanishes for the (anti-)self-dual Taub-NUT instanton, we find

$\displaystyle N_{+}-N_{-}=2q\left(q+\frac{1}{2}\right)\,.$ (52)

Thus, we conclude that, in the case of the (anti-)self-dual Taub-NUT solution
of Einstein-ModMax theory, the axial anomaly is sourced by the electric charge
of the ModMax field. Indeed, its value does not depend on the new parameter
$\gamma$ that measures the nonlinearity of the electromagnetic fields and it
coincides with the Maxwell case Pope (1981). This is related to the fact that
the asymptotic behavior of the ModMax field does not modify the
$\eta_{D}$-invariant in comparison to the linear Maxwell case. Moreover, the
$U(1)$ Chern-Pontryagin index (51) has no dependence on the $\gamma$-parameter
whatsoever, in contrast to the Eguchi-Hanson case. This is related to the fact
that the latter is endowed with a bolt and we expect that it will not be
necessarily the case in the Taub-Bolt instanton either that we aim to study in
the future.

##  V Discussion

In this work, we have shown that gravitational instantons sourced by nonlinear
conformal electrodynamics induce the axial anomaly when coupled to Einstein
gravity. For doing so, we first review the Taub-NUT solution found in this
theory in Refs. Ballon Bordo  _et al._ (2021); Flores-Alfonso  _et al._
(2021b). In the asymptotic regime, we verify that the solution of the ModMax
field equations approaches its Maxwellian counterpart in this background.
Moreover, the former is continuously connected to the latter as $\gamma\to 0$;
this parameter measures the degree of nonlinearity of the $U(1)$ gauge fields.
Then, we obtain a novel Eguchi-Hanson instantonic solution in Einstein-ModMax
theory. Similar to the Taub-NUT case, we find that the $\gamma$-parameter
screens the electric and magnetic charge of the dyonic nonlinear $U(1)$
fields. By demanding (anti-)self duality and the absence of conical
singularities, we find a nonlinearly charged, geodesically complete lens space
that is asymptotically locally Euclidean.

Then, by using the Atiyah-Patodi-Singer theorem, we compute the index of the
Dirac operator for nonlinearly charged spinors. Although there is no
contribution of the ModMax field to the $\eta_{D}$-invariant in the Eguchi-
Hanson instanton, its nontrivial topology at the bolt induces a nonvanishing
$U(1)$ Chern-Pontryagin index that contributes to the anomaly, detecting the
presence of the nonlinear theory at the level of the topological charge. This
effect of the nonlinear electrodynamics is somewhat similar to charge
screening. Additionally, in the (anti-)self-dual Taub-NUT case, the gauge
potential contributes nontrivially to the $\eta_{D}$-invariant while the
Chern-Pontryagin index is trivial. We compute the former by performing the
analytic continuation of the meromorphic function (27) to $s=0$ and evaluating
the difference between the number of positive and negative eigenvalues of the
Dirac operator. Then, we conclude that both instantons induce an excess of
positive chiral spinors in comparison with the negative ones [cf. Eqs. (46)
and (52)].

Interesting questions remain open. First, studying harmonic forms and spinors
for the nonlinearly charged Taub-bolt is certainly of interest, similar to the
analysis performed in Ref. Franchetti (2019). Additionally, even though the
computation in presence of a negative cosmological constant is more involved,
it could provide important insights regarding the role of nonlinear $U(1)$
fields in the axial anomaly from a conformal field theory viewpoint Maldacena
(1998); Gubser  _et al._ (1998); Witten (1998). Some advances in this
direction have been done in Ref. Franchetti and Galán (2022) where the space
of $SU(2)$-invariant harmonic 2-forms was determined for Taub-NUT-AdS solution
in Einstein-Maxwell theory. Finally, it is worth analyzing whether quantum
effects would break the conformal symmetry of ModMax fields. This is of
interest to us and we postpone a deeper study of this point to the future.

###### Acknowledgements.

We thank Giorgos Anastasiou, Ignacio J. Araya, Eloy Ayón-Beato, Fabrizio
Canfora, Gastón Giribet, Olivera Mišković, Rodrigo Olea, Francisco Rojas, and
Omar Valdivia for insightful comments and discussions. The work of F.C-M. is
supported by Agencia Nacional de Investigación y Desarrollo (ANID) through
FONDECYT No. 1210876. C.C. is partially supported by through FONDECYT grants
No. 11200025 and 1210500. D.F-A. is supported by ANID under FONDECYT grant No.
3220083. L.S. is supported by Beca de Doctorado Nacional ANID No. 21221813.

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