#  DC conductance and memory in 3D gravity

Márcela<EMAIL_ADDRESS>Departamento de Física,
Universidad de Santiago de Chile, Avenida Ecuador 3493, Santiago, Chile.
Hernán A. González<EMAIL_ADDRESS>Facultad de Artes Liberales,
Universidad Adolfo Ibáñez, Diagonal Las Torres 2640, Santiago, Chile.
Kristiansen Lara<EMAIL_ADDRESS>Departamento de Física,
Universidad de Santiago de Chile, Avenida Ecuador 3493, Santiago, Chile.
Miguel<EMAIL_ADDRESS>Departamento de Física, Universidad de
Santiago de Chile, Avenida Ecuador 3493, Santiago, Chile. Center for
Interdisciplinary Research in Astrophysics and Space Exploration (CIRAS),
Universidad de Santiago de Chile, Av. Libertador Bernardo O’Higgins 3363,
Santiago, Chile.

###### Abstract

Transport properties are investigated in the two-dimensional dual dynamics of
AdS3 gravity. By providing boundary conditions that deform the Lagrange
multipliers, we construct a lower dimensional model comprising two copies of
chiral boson excitations with anisotropic scaling symmetry. Using
bosonization, an electric current is identified. By means of the Kubo formula,
we find a DC conductance depending on the level of the theory and the
dynamical exponents. The bulk realization of the linear response is related to
a type of gravitational memory emerging in the context of near-horizon
boundary conditions.

###### Contents

  1. 1 Introduction
  2. 2 Asymptotic conditions and Hamiltonian evolution
    1. 2.1 Kinetic term
    2. 2.2 Boundary Hamiltonian
    3. 2.3 Anisotropic chiral boson spectrum
  3. 3 Linear response and the DC limit
    1. 3.1 Response function
    2. 3.2 Two-terminal conductance
  4. 4 The gravitational side
    1. 4.1 Bulk external source
    2. 4.2 Linear response for near horizon boundary conditions
    3. 4.3 Near horizon memory and chiral bosons
  5. 5 Future prospects
  6. A Gravity as a Chern-Simons theory
  7. B Bosonization and normal ordering
  8. C Kubo Formula

##  1 Introduction

The Chern-Simons description of the Einstein-Hilbert action in three
dimensions[1, 2] stresses the fact that some fundamental properties of gravity
are completely specified by the topology of the manifold and the presence of
boundaries. Therefore, the dynamical content of 3D gravity is determined once
boundary conditions are proposed. It is then possible to find the asymptotic
group of symmetries respecting these configurations, which from a holographic
point of view, defines the global symmetries of the dual theory.

Recently, the specification of field-dependent Lagrange multipliers at
infinity has permitted to propose novel boundary dynamics, via imposition of
integrability conditions consistent with the gauge symmetries [3, 4, 5, 6]. In
the case of asymptotically AdS3 spaces, this procedure deforms the Hamiltonian
of the boundary theory, describing mechanical systems with symmetries other
than conformal.

The holographic representation of three-dimensional gravity finds interesting
applications in the spirit of the AdS/CMT correspondence (for a review on the
subject see e.g. [7, 8]), where observables sensible for condensed matter
systems can be recognized from the symmetries of the gravitational
description. The implementation of the asymptotic symmetric group on the
boundary fixes some of these observables to be described in terms of the
leading components of the gravitational field.

In this work, we would like to compute transport coefficients for three-
dimensional gravity endowed with a particular choice of the Lagrange
multipliers. This specification makes contact with Lifshitz symmetry, that has
shown to be relevant in condensed matter systems [9]. We construct the
corresponding dual theory which describes anisotropic chiral boson excitations
[10], admitting a conservation law based on a residual $U(1)$ symmetry. The
resulting conserved current can be reinterpreted under the light of the
bosonization555See [11] for a discussion from the point of view of condensed
matter systems., permitting the identification of an electronic charge density
and its corresponding electric current. By means of the Kubo formula, we
obtain the linear response associated to the charge-current correlator. This
computation has been previously carried out in the relativistic chiral boson
case by Kane and Fisher [12]. In the DC limit, this correlator yields an Ohm’s
law when the physical setup is characterized by a source representing a two-
terminal reservoir. Here, we present a generalization to [12] in the case of
anisotropic chiral states. Noteworthy, the specification of the spatial and
temporal Chern-Simons gauge fields components is shown to play a precise role
in the functional expression of the DC conductance.

Two-point functions in a 2D CFT play a fundamental role in the calculation of
transport coefficients [13, 14], where the underlying symmetry fixes its value
to a constant. For the energy-momentum operator, this constant corresponds to
the central charge, while for an abelian conserved current is associated to
the $\mathfrak{u}(1)$ Kac-Moody level. The present work can be seen as an
extension of these previous results, including now conductivity coefficients
for theories with Lifshitz scaling.

The external source used to calculate edge-state transport coefficient
describes, from the bulk perspective, a dynamical process allowing the study
of a time-dependent metric evolution. Here, we choose boundary conditions
representing spacetimes near a black hole horizon [15, 16]. Based on this
geometry, we give a complementary bulk description of the charge density and
electric current operators. In this case, they are shown to control the
horizon area and the angular velocity measured by an observer hovering just
outside the horizon region.

We show that the boundary linear response theory has a geometric counterpart
in the bulk, showing an appealing relation with memory effects. In four-
dimensional gravity, these correspond to a physical displacement between a set
of detectors occasioned by the passage of a gravitational wave [17, 18]. In
spite that 3D Einstein gravity is devoid of local gravitational excitations,
the information from the spacetime deformation can be retrieved in terms of an
improper gauge transformation. Remarkably, this transformation encodes the DC
transport coefficient showing a correspondence with the boundary computation.

This paper is organized as follows: In section 2, we revisit the construction
of the boundary dynamics from a Chern-Simons action. We provide boundary
conditions making contact with the model proposed in [10]. More precisely, the
dynamics turns out to be described by two copies of anisotropic chiral boson.
The particular choice of the Lagrange multipliers confers Lifshitz symmetry to
the model whenever the dynamical exponents are the same on both chiralities.
In section 3, we study transport properties of the anisotropic chiral boson.
Using bosonization, we identify an electric charge and current operator. We
obtain the expectation value of these operators in frequency space. In the DC
limit, we show that the two-terminal conductance receives contributions from
the coupling constants characterizing both chiralities. In section 4, we
rederive this conductance from the perspective of an observer in the near-
horizon region and discuss a gravitational memory interpretation. In Section
5, we propose some prospects of this work.

##  2 Asymptotic conditions and Hamiltonian evolution

This section reviews the construction of boundary dynamics in three-
dimensional gravity using the Chern-Simons formulation. In this setup, the
action is given by the difference of two Chern-Simons functionals
$I_{CS}[A^{+}]-I_{CS}[A^{-}]$, where the connections $A^{\pm}$ are
$\mathfrak{sl}(2,\mathbb{R})$ valued one-forms. The spacetime has the topology
$\mathbb{R}\times\Sigma$, where the real line is parameterized by time $t$,
while $\Sigma$ is a disc endowed with coordinates $r>0$ and $0\leq x<2\pi l$,
where $l$ stands for the AdS radius. The level is $K=\frac{l}{4G}$ where $G$
is the Newton constant. See appendix A, for details on conventions and
explicit formulas.

The boundary dynamics is completely defined by the specification of a
symplectic structure and a Hamiltonian. In the following, it is discussed how
to construct this dynamical system from asymptotic boundary conditions
sensible to three-dimensional gravity. The seminal example of this
construction was performed in [19], where it is shown that the Brown-Henneaux
boundary conditions give rise to a theory of two decoupled chiral bosons.
Here, novel boundary conditions are presented, extending Brown-Henneaux and
making contact with the dynamics of a recently introduced model dubbed as
anisotropic chiral boson [10].

###  2.1 Kinetic term

The boundary symplectic structure can be obtained by solving the bulk
constraints and replacing them back in the three-dimensional action. Since the
theory is devoid of local degrees of freedom, the Chern-Simons kinetic term
becomes a total derivative that can be expressed as a local two-dimensional
functional [20, 19].

Here, an alternative path is taken for constructing the reduced kinetic term
of the model. In doing so, it is necessary to study the symmetries allowed by
the field configurations satisfying a specific boundary condition. In the
Chern-Simons formulation, asymptotic conditions are encoded in the spatial
component of the gauge field

$\displaystyle A^{\pm}_{x}=a^{\pm}_{x}+\cdots\,,$ (2.1)

where the ellipsis represents subleading terms close to the boundary. The set
of gauge transformations $\varepsilon^{\pm}$ preserving the asymptotic form of
$A^{\pm}_{x}$ are global as long as they produce non-vanishing surfaces
charges [21]. In that case, they are given by

$\displaystyle\delta Q(\varepsilon^{+},\varepsilon^{-})=-\frac{K}{2\pi}\int
dx\,\langle\varepsilon^{+}\delta a^{+}_{x}-\varepsilon^{-}\delta
a^{-}_{x}\rangle\,.$ (2.2)

Without imposing any further restriction on $a^{\pm}_{x}$, the asymptotic
symmetries are generated by the $\mathfrak{sl}(2,\mathbb{R})$ current algebra
[22, 23]. We consider the simplest set of connections within the
$\mathfrak{sl}(2,\mathbb{R})$ family spanned by only one generator [16]

$\displaystyle a^{\pm}_{x}=\pm\frac{4\pi}{K}\mathcal{J}_{\pm}(x,t)\;L_{0}\,.$
(2.3)

Gauge parameters respecting the above boundary conditions are of the form
$\varepsilon^{\pm}=\varepsilon^{\pm}(x,t)L_{0}$. The corresponding charge
densities are $\mathcal{J}_{\pm}$, giving rise to $\mathfrak{u}(1)$ current
algebras under Dirac brackets

$\displaystyle\left\\{\mathcal{J}_{\pm}(x),\mathcal{J}_{\pm}(x^{\prime})\right\\}_{\rm
D.B.}=\pm\frac{K}{4\pi}\partial_{x}\delta(x-x^{\prime})\,.$ (2.4)

The latter algebra can be used to define the canonical variables at the
boundary. In fact, as shown in [24], the kinetic term producing commutation
relations (2.4) is

$\displaystyle I_{\rm
kinetic}[\mathcal{J}_{+},\mathcal{J}_{-}]=\frac{2\pi}{K}\int
dxdydt\;\epsilon(x-y)\left[\mathcal{J}_{+}(x)\partial_{t}\mathcal{J}_{+}(y)-\mathcal{J}_{-}(x)\partial_{t}\mathcal{J}_{-}(y)\right]\,,$
(2.5)

where $\epsilon(x)$ is the inverse of operator $\partial_{x}$. In order to
obtain a local description, we use fields $\phi_{\pm}$ that are defined by

$\displaystyle\mathcal{J}_{\pm}=\pm\frac{K}{4\pi}\partial_{x}\phi_{\pm}\,,$
(2.6)

and the kinetic term becomes

$\displaystyle I_{\rm kinetic}[\phi_{+},\phi_{-}]=\frac{K}{8\pi}\int
dxdt\;\left[\partial_{x}\phi_{+}\partial_{t}\phi_{+}-\partial_{x}\phi_{-}\partial_{t}\phi_{-}\right]\,.$
(2.7)

###  2.2 Boundary Hamiltonian

To define the time evolution generator of the system, we consider boundary
charges conjugated to the gauge transformations
$\varepsilon^{\pm}=-a^{\pm}_{t}$, where $a^{\pm}_{t}$ is the leading term of
the temporal component $A^{\pm}_{t}$. By using (2.2), we find that

$\displaystyle\delta H=\frac{K}{2\pi}\int dx\langle a^{+}_{t}\,\delta
a^{+}_{x}-a^{-}_{t}\,\delta a^{-}_{x}\rangle\,.$ (2.8)

A local expression for the Hamiltonian can be obtained provided the leading
terms $a^{\pm}_{t}$ fulfill a condition of the form

$\displaystyle a^{\pm}_{t}=\frac{\delta\mathcal{H}_{\pm}(a^{\pm}_{x})}{\delta
a^{\pm}_{x}}\,,$ (2.9)

where $\mathcal{H}_{\pm}(a^{\pm}_{x})$ is local functional of $a^{\pm}_{x}$.
However, the choice of the latter functional is not completely arbitrary since
time evolution of the system must also preserve the asymptotic behavior of
$A^{\pm}_{x}$. To ensure this condition, one then chooses $a^{\pm}_{t}$ to
have the same functional form than the allowed gauge parameters
$\varepsilon^{\pm}$ [3]. For instance, in the case of Brown-Henneaux boundary
conditions without chemical potentials [19], the choice $a^{\pm}_{t}=\pm
a^{\pm}_{x}$ singles out a Hamiltonian that is quadratic in $a^{\pm}_{x}$ and
produces modes evolving in the $t\pm x$ directions. Another possible choice
has been considered in [4, 6] in connection with integrable systems. In that
case, the selected functional $\mathcal{H}_{\pm}(a^{\pm}_{x})$ is given by
AKNS charges and the corresponding Hamiltonian evolution of the modes
encompasses the AKNS hierarchy [25].

With the purpose of studying time-dependent perturbations in a family of 2D
field theories, we focus on the particular class

$\displaystyle a^{\pm}_{t}=\pm
v(-1)^{\frac{z-1}{2}}\partial^{z-1}_{x}a^{\pm}_{x}+\mu L_{0}\,,$ (2.10)

where $z$ is an odd number containing the Brown-Henneaux case for $z=1$, while
$v>0$ is a coupling constant with dimensions of $[{\rm length}]^{z-1}$. We
introduced a non-varying source $\mu(x,t)$ giving an expectation value to
$a^{\pm}_{t}$. The latter is motivated by the AdS/CFT correspondence, where
the expectation value of observables in the dual theory are obtained by taking
functional derivatives with respect to $\mu$. See [26] for an application in
three-dimensional Chern-Simons theory.

Using condition (2.10) in (2.8), we can integrate the Hamiltonian, yielding

$\displaystyle H=H_{0}[\phi_{+}]+H_{0}[\phi_{-}]+\Delta H\,,$ (2.11)

where

$\displaystyle H_{0}[\phi]=\frac{Kv}{8\pi}\int
dx\,\left(\partial^{\frac{z+1}{2}}_{x}\phi\right)^{2}\,,$ (2.12)

representing the free Hamiltonian of the boundary theory, and

$\displaystyle\Delta H=\frac{K}{4\pi}\int
dx\,\mu(x,t)\left(\partial_{x}\phi_{+}-\partial_{x}\phi_{-}\right)\,,$ (2.13)

being the perturbation associated to the external source $\mu$. Note that we
have used $a^{\pm}_{x}=\partial_{x}\phi_{\pm}L_{0}$ to express the Hamiltonian
in terms of the local field $\phi_{\pm}$.

Summarizing, the action of the full model is

$\displaystyle I[\phi_{+},\phi_{-};\mu]=I_{\rm
kinetic}[\phi_{+},\phi_{-}]-\int dt\,H\,.$ (2.14)

###  2.3 Anisotropic chiral boson spectrum

Let us describe the theory spectrum without the external source ($\mu=0$)
which realizes anisotropic scaling symmetry $t\to\lambda^{z}t$ and
$x\to\lambda x$ with an odd dynamical exponent $z$. Excitations are chiral in
the sense that parity transformation ($x\to-x$) swaps the dynamics of
$\phi_{\pm}$ into $\phi_{\mp}$.

The equations of motion corresponds to

$\displaystyle\partial_{x}\left(\partial_{t}\phi_{\pm}\pm(-1)^{\frac{z+1}{2}}v\,\partial_{x}^{z}\phi_{\pm}\right)=0\,.$
(2.15)

Since this is a linear equation, solutions are given by plane waves modes

$\displaystyle\phi_{\pm}=\exp[i(kx\pm\omega_{k}t)]\,,\quad\omega_{k}=vk^{z}\,.$
(2.16)

Furthermore, due to parameterization (2.6) the system inherits zero modes
responsible for the gauge symmetry $\phi_{\pm}\to\phi_{\pm}+f_{\pm}(t)$. In
what follows, we consider these to be gauge fixed and therefore they will not
be present in the subsequent analysis.

Another important feature is that the model (2.14) exhibits a
$\mathfrak{u}(1)$ current symmetry on each sector. Under the infinitesimal
transformation $\delta\phi_{\pm}=\eta^{\pm}_{k}\exp[i(kx\pm\omega_{k}t)]$,
with $\eta^{\pm}_{k}\ll 1$, the Noether theorem yields the expected currents
(2.6).

For the forthcoming analysis, we describe the fields in terms of their Fourier
modes. In the remainder of the paper, we will be concerned with local
properties of the theory, hence $x$ is regarded as a coordinate defined on the
whole real line. This allows to expand the field operator as

$\displaystyle\phi_{\pm}(x,t)=\int^{+\infty}_{-\infty}\frac{dk}{k}e^{i(kx\pm\omega_{k}t)}b_{\pm,k}\,,$
(2.17)

where the condition $b_{\pm,k}^{\dagger}=-b_{\pm,-k}$ ensures hermiticity of
$\phi_{\pm}$.

The Dirac bracket can be read off from the kinetic term in (2.14). The quantum
version yields the equal time commutation relation666In the quantum theory we
use the rule $\left[,\right]\to i\\{,\\}_{\rm DB}$.

$\displaystyle\left[\phi_{\pm}(t,x),\partial_{x^{\prime}}\phi_{\pm}(t,x^{\prime})\right]=\pm\frac{4\pi}{K}i\delta(x-x^{\prime})\,.$
(2.18)

In terms of the $b_{\pm,k}$ operators, they become

$\displaystyle\left[b_{\pm,k},b_{\pm,k^{\prime}}\right]=\pm\frac{2}{K}k\delta(k+k^{\prime})\,.$
(2.19)

##  3 Linear response and the DC limit

This section explores the dynamical evolution of the boundary fields
$\phi_{\pm}$ driven by $\mu$, that is modeled in a specific way in order to
calculate electronic transport coefficients.

###  3.1 Response function

We start by identifying a conservation law using the charges expressed in
terms of the chiral boson (2.6)

$\displaystyle\partial_{t}\mathcal{J}_{\pm}+\partial_{x}\mathcal{I}_{\pm}=0\,,$
(3.1)

implying that $\mathcal{I}_{\pm}$ is

$\displaystyle\mathcal{I}_{\pm}=\mp\frac{K}{4\pi}\partial_{t}\phi_{\pm}\,.$
(3.2)

Equation (3.1) associates $\mathcal{J}_{\pm}$ with a density and
$\mathcal{I}_{\pm}$ with a current. Additionally, in the context of
bosonization, they can be understood as carriers of fermionic degrees of
freedom, so that $\mathcal{J}_{\pm}$ is an electric charge density and
$\mathcal{I}_{\pm}$ corresponds to an electric current flowing along the
boundary. For details on the precise relation between $\mathcal{J}_{\pm}$ and
fermionic operators see appendix B.

We wish to evaluate the linear response of the net current $\mathcal{I}_{\rm
tot}=\mathcal{I}_{+}+\mathcal{I}_{-}$, induced by the deformation of the
theory

$\displaystyle\Delta
H=\int\,dx\,\mu(x,t)\,\left[\mathcal{J}_{+}(x,t)+\mathcal{J}_{-}(x,t)\right]\,,$
(3.3)

that is nothing but an electric potential energy. The source $\mu$ produces a
voltage creating a current in a one-dimensional conductor. In [12], this has
been interpreted as edge states flowing at the boundary of a two-dimensional
material.

Considering that the operators $\mathcal{I}_{\pm}$ and $\mathcal{J}_{\pm}$ are
defined in the interaction picture, we analyze the expectation value by means
of the Kubo formula in frequency space (we refer the reader to appendix C for
conventions)

$\displaystyle\delta\langle\widetilde{\mathcal{I}}_{\pm}(\omega,x)\rangle$
$\displaystyle=\int\,dx^{\prime}\,\widetilde{\mu}(x^{\prime},\omega)\widetilde{\chi}^{\pm}_{\mathcal{I},\mathcal{J}}(\omega;x,x^{\prime})\,,$
(3.4)

where the susceptibility $\widetilde{\chi}^{\pm}_{\mathcal{I},\mathcal{J}}$ is

$\displaystyle\widetilde{\chi}^{\pm}_{\mathcal{I},\mathcal{J}}(\omega;x,x^{\prime})=-i\int
dt\,\Theta(t)e^{i\omega
t}\left\langle\left[\mathcal{I}_{\pm}(x,t),\mathcal{J}_{\pm}(x^{\prime},0)\right]\right\rangle\,.$
(3.5)

After some manipulations it can be expressed as

$\displaystyle\widetilde{\chi}^{\pm}_{\mathcal{I},\mathcal{J}}(\omega;x,x^{\prime})=\frac{K}{4\pi}\left[\pm\delta(x-x^{\prime})\mp(\omega+i\epsilon)F^{\pm}(x-x^{\prime},\omega)\right]\,.$
(3.6)

The functions $F^{\pm}$ have been obtained after expressing the Heaviside
function $\Theta(t)$ as an integral in frequency space, inducing the shift
$\omega\to\omega+i\epsilon$. They become

$\displaystyle(\omega+i\epsilon)F^{\pm}(y,\omega)=i\Theta(-y)\sum_{k^{\pm}_{p}\in{\rm
Im}_{<}}\frac{\omega(k^{\pm}_{p})e^{ik^{\pm}_{p}y}}{\omega^{\prime}(k^{\pm}_{p})}-i\Theta(y)\sum_{k^{\pm}_{p}\in{\rm
Im}_{>}}\frac{\omega(k^{\pm}_{p})e^{ik^{\pm}_{p}y}}{\omega^{\prime}(k^{\pm}_{p})}\,,$
(3.7)

where ${\rm Im}_{\gtrless}$ correspond to the upper and lower complex plane.
The values $k^{\pm}_{p}=k^{\pm}_{p}(\omega)$ are solutions to the equation
$\omega+i\epsilon\pm\omega(k^{\pm}_{p})=0$.

###  3.2 Two-terminal conductance

The situation of interest is modeled by an external source that generates a
flow between two reservoirs located at positions $x=x_{L}$ and $x=x_{R}$ [12]
that perturbs the system for a very short time interval. In order to reproduce
these features we use

$\displaystyle\widetilde{\mu}(x,\omega)=V_{L}\,\Theta(x_{L}-x)+V_{R}\,\Theta(x-x_{R})\,.$
(3.8)

Notice that time localization of the perturbation is ensured as the inverse
Fourier transform of $\widetilde{\mu}$ yields a temporal Dirac delta.

In the presence of this source, the total variation of the electric current on
each sector becomes

$\langle\delta\widetilde{\mathcal{I}}_{\pm}\rangle=\mp\frac{K}{4\pi}\Bigg{\\{}V_{L}\left[\Theta(x-x_{L})\Delta^{\pm}_{>}(x-x_{L})-\Theta(x_{L}-x)\Delta^{\pm}_{<}(x-x_{L})\right]\\\
-V_{R}\left[\Theta(x-x_{R})\Delta^{\pm}_{>}(x-x_{R})-\Theta(x_{R}-x)\Delta^{\pm}_{<}(x-x_{R})\right]\Bigg{\\}}\,,$
(3.9)

where

$\displaystyle\Delta^{\pm}_{\gtrless}(x)=\frac{1}{z^{\pm}}\sum_{k^{\pm}_{p}\in{\rm
Im}_{\gtrless}}e^{ik^{\pm}_{p}x}\,.$ (3.10)

with, for sake of generality, we have considered a Hamiltonian (2.12) with
different dynamical exponents on each sector, given by $z_{+}$ and $z_{-}$
respectively.

In the zero frequency limit, we obtain the DC current contribution. Noticing
that $k^{\pm}_{p}(\omega\to 0)=0$, we see that $\Delta^{\pm}_{\gtrless}$
becomes a constant that essentially counts the number of zeros of the equation

$\displaystyle\omega+i\epsilon\pm v(k^{\pm}_{p})^{z^{\pm}}=0\,,$

on the upper and lower complex plane. Causality, represented by the
$i\epsilon$-prescription, implies that the $\pm$ sector has $\tfrac{z^{\pm}\mp
1}{2}$ zeros on the upper plane and $\tfrac{z^{\pm}\pm 1}{2}$ zeros on the
lower one. Therefore, functions $\Delta^{\pm}_{\gtrless}$ in DC limit read

$\displaystyle\Delta^{\pm}_{>}=\frac{z^{\pm}\mp
1}{2z^{\pm}}\,,\quad\Delta^{\pm}_{<}=\frac{z^{\pm}\pm 1}{2z^{\pm}}\,.$ (3.11)

Using these expressions, the variation of the net current
$\langle\widetilde{\mathcal{I}}_{\rm
tot}\rangle=\langle\widetilde{\mathcal{I}}_{+}+\widetilde{\mathcal{I}}_{-}\rangle$
satisfies Ohm’s law. Indeed, by denoting the voltage as $\Delta
V=V_{L}-V_{R}$, we find that the two-terminal conductance $\sigma$ [12] is
given by

$\displaystyle\quad\sigma=\frac{\langle\delta\widetilde{\mathcal{I}}_{\rm
tot}\rangle}{\Delta
V}=\frac{K}{8\pi}\left(\frac{1}{z_{+}}+\frac{1}{z_{-}}\right)\,.$ (3.12)

The above expression have two contributions that are worth commenting on.
Firstly, there is a factor coming from the level $K=\frac{l}{4G}$ associated
to the infinite-dimensional $\mathfrak{u}(1)$ current symmetry (2.4). This
symmetry is universal in the sense that is controlled by the residual
transformation of the $a^{\pm}_{x}$. Lastly, the conductivity is also modified
by the dynamical exponents of the theory $(z_{+},z_{-})$ that are associated
to the higher derivative Hamiltonians (2.12). This contribution is model
dependent as it emerges from the choice of $a^{\pm}_{t}$.

##  4 The gravitational side

###  4.1 Bulk external source

So far we have considered a boundary source $\mu$ that is included in the set
of asymptotic conditions (2.10). In order to analyze gravitational properties
associated to the DC response, we could proceed with this asymptotic behavior,
construct a metric and identify the corresponding bulk observables of the
preceding calculation. However, we would like to present a complementary set
up that realizes the previous results from a bulk perspective, making contact
with a type of gravitational memory effect. To do so, we construct field
equations where the potential $\mu$ is represented by an external current
instead of being part of the Lagrange multiplier (2.10). This leads us to
consider777The action principle giving rise to these field equations is
discussed in appendix A. The orientation of the manifold is fixed by
$\epsilon^{rxt}=1$.

$\displaystyle\frac{K}{4\pi}\epsilon^{\alpha\mu\nu}F^{\pm}_{\mu\nu}=J^{\alpha}_{\pm}\,,$
(4.1)

allowing us to identify a physical process in the bulk where an external
source modifies the Chern-Simons fields $A^{\pm}$. These gauge connections are
related with their asymptotic values $a^{\pm}$ through the decomposition,

$\displaystyle A^{\pm}=b_{\pm}^{-1}a^{\pm}b_{\pm}+b_{\pm}^{-1}db_{\pm}\,.$
(4.2)

Here, $b_{\pm}$ is an SL$(2,\mathbb{R})$ group element depending on the radial
coordinate $r$. The longitudinal gauge connections are
$a_{\pm}=a^{\pm}_{x}dx+a^{\pm}_{t}dt$, with $a^{\pm}_{x}$ given by (2.3),
while the temporal component is

$\displaystyle a^{\pm}_{t}=\mp\frac{4\pi}{K}\mathcal{I}_{\pm}\,L_{0}\,.$ (4.3)

The source is defined through a current pointing in the radial direction

$\displaystyle
J^{r}_{\pm}=-\frac{K}{2\pi}\partial_{x}\mu\,\,b_{\pm}^{-1}L_{0}b_{\pm}\,,\quad
J^{t}_{\pm}=J^{x}_{\pm}=0\,,$ (4.4)

where its precise dependence on the group element $b_{\pm}$ ensures that the
conservation law
$\partial_{\alpha}J^{\alpha}_{\pm}+[A^{\pm}_{\alpha},J^{\alpha}_{\pm}]=0$
holds. The functional dependence of $\mu$ is chosen so that (4.1) describes
the boundary dynamics of the previous section. Using (3.8), the radial
component reads

$\displaystyle
J^{r}_{\pm}=\frac{K}{2\pi}\delta(t)\Big{[}V_{L}\,\delta(x-x_{L})-V_{R}\,\delta(x-x_{R})\Big{]}\,b_{\pm}^{-1}L_{0}b_{\pm}\,.$
(4.5)

The temporal localization of the external source models a process where the
initial and final configurations correspond to solutions of vacuum Einstein’s
equation. In fact, the evolution between those configurations is controlled by

$\displaystyle\partial_{t}\mathcal{J}_{\pm}+\partial_{x}\mathcal{I}_{\pm}=\pm\frac{K}{4\pi}\delta(t)\Big{[}V_{L}\,\delta(x-x_{L})-V_{R}\,\delta(x-x_{R})\Big{]}\,,$
(4.6)

that has been obtained after using the explicit form of connections (4.2).
These are precisely the field equations obtained from the variational
principle (2.14) when the electric currents $\mathcal{I}_{\pm}$ become

$\displaystyle\mathcal{I}_{\pm}=\mathcal{I}_{\pm}^{\rm in}\pm
v(-)^{\frac{z+1}{2}}\partial_{x}^{z-1}\mathcal{J}_{\pm}\,,$ (4.7)

where $\mathcal{I}_{\pm}^{\rm in}$ are arbitrary functions of $t$ that will be
fixed using initial data. The latter choice is consistent with anisotropic
chiral excitations and corresponds to the boundary condition (2.10) in the
presence of $\mu$ now incorporated in $J^{\alpha}_{\pm}$.

We now compute the retarded response using (4.6) and (4.7) subjected to
initial conditions $\mathcal{J}_{\pm}(x,t=-\infty)=\mathcal{J}^{\rm
in}_{\pm}$, where $\mathcal{J}^{\rm in}_{\pm}$ are constants. The net electric
current in frequency space is then given by

$\displaystyle\delta\widetilde{\mathcal{I}}_{\rm tot}=\int
dx^{\prime}\,\widetilde{\mu}(x^{\prime},\omega)(\widetilde{\chi}^{+}_{\mathcal{I},\mathcal{J}}+\widetilde{\chi}^{-}_{\mathcal{I},\mathcal{J}})\,,$
(4.8)

where the susceptibilities $\widetilde{\chi}^{\pm}_{\mathcal{I},\mathcal{J}}$
are shown to coincide with the boundary computation (3.6). This is not a
surprise since retarded Green functions plays the role of susceptibilities in
the linear response formalism. However, the way results have been derived in
this section will be useful to interpret the disturbance in the observables
$\mathcal{J}_{\pm}$ and $\mathcal{I}_{\pm}$ as a memory effect on the metric
field.

###  4.2 Linear response for near horizon boundary conditions

At this point, we would like to discuss the effect of these induced currents
on the spacetime metric. Thus, we introduce the radial dependence by choosing
a specific form of the group element $b_{\pm}$. Here, we use

$\displaystyle b_{\pm}(r)=\exp\left[{\pm\frac{r}{2l}(L_{1}-L_{-1})}\right]\,,$
(4.9)

rendering boundary conditions for spacetimes close to a Rindler horizon [15,
16]. The geometry close to $r=0$ can be expressed in terms of the net electric
current $\mathcal{I}_{\rm tot}=\mathcal{I}_{+}+\mathcal{I}_{-}$ and the net
charge density $\rho_{\rm tot}=\mathcal{J}_{+}+\mathcal{J}_{-}$, yielding 888
The metric field can be constructed from
$g_{\mu\nu}=\frac{l^{2}}{2}\left\langle(A^{+}_{\mu}-A^{-}_{\mu}),(A^{+}_{\nu}-A^{-}_{\nu})\right\rangle\,.$

$\displaystyle ds^{2}=\left(\frac{2\pi l}{K}\right)^{2}(\mathcal{I}_{\rm
tot}^{2}dt^{2}-2\mathcal{I}_{\rm tot}\rho_{\rm tot}dtdx+\rho_{\rm
tot}^{2}dx^{2})+dr^{2}+O(r^{2})\,,$ (4.10)

where the $O(r^{2})$ terms stand for subleading components. From the above
equation one can readily see that the quantities $\mathcal{I}_{\rm tot}$ and
$\rho_{\rm tot}$ induce a change in the angular velocity and the horizon area
measured by an observer placed in a rotating frame close to $r=0$.

Let us briefly discuss the DC response of the total charge $\rho_{\rm tot}$
from the bulk point view. We first notice that the Fourier transform of the
equation of motion (4.6) implies

$\displaystyle i\omega\widetilde{\rho}_{\rm
tot}-\partial_{x}\widetilde{\mathcal{I}}_{\rm tot}=0\,.$ (4.11)

Since in the DC limit $\delta\widetilde{\mathcal{I}}_{\rm
tot}=\sigma(V_{L}-V_{R})+O(\omega^{\frac{1}{z}})$, we see that
$\delta\widetilde{\rho}_{\rm tot}$ behaves as $O(\omega^{\frac{1}{z}-1})$
which is divergent as $\omega\to 0$. Therefore, generically there is no well-
defined DC response for the variation of the total charge. However, the
operator

$\displaystyle\mathcal{A}\equiv\int dx\,\rho_{\rm tot}\,,$ (4.12)

representing the horizon area, possesses a more natural interpretation in the
context of linear response. In fact, by approximating the integration domain
of (4.12) to the whole real line, one finds that

$\displaystyle\delta\mathcal{A}=0\,,$ (4.13)

where we have used (4.11) and the fact that the electric current vanishes at
the boundaries $x\to\pm\infty$, as indicated by (3.9). From the
thermodynamical viewpoint, the constancy of $\mathcal{A}$ implies that the
process sourced by the two-terminal potential (3.8) is adiabatic, provided the
entropy is given by Bekenstein-Hawking formula.

###  4.3 Near horizon memory and chiral bosons

The spacetime metric with a fully incorporated radial dependence (4.9)
provides a further interpretation of the boundary degrees of freedom. The
evolution process presented in section 4.1 can be understood as a memory
effect arising from an improper gauge transformation. The precise form of this
transformation renders AdS3 spacetime in a particular frame where the chiral
bosons are taken as coordinates.

Let us consider a situation where the source $\mu$ is turned off. By virtue of
(2.6) and (3.2), one finds

$\displaystyle\pm\frac{4\pi}{K}\left(\mathcal{J}_{\pm}dx-\mathcal{I}_{\pm}dt\right)=d\phi_{\pm}\,,$
(4.14)

allowing to express the spacetime metric as

$\displaystyle
ds^{2}=-\frac{l^{2}}{4}\sinh^{2}\left(\frac{r}{l}\right)(d\phi_{+}+d\phi_{-})^{2}+dr^{2}+\frac{l^{2}}{4}\cosh^{2}\left(\frac{r}{l}\right)(d\phi_{+}-d\phi_{-})^{2}\,.$
(4.15)

This foliation corresponds to a double-Wick rotated version of global AdS3.
Furthermore, the diffeomorphism (4.14) makes evident that the degrees of
freedom yielding $\mathcal{J}_{\pm}$ and $\mathcal{I}_{\pm}$ belong to the set
of large gauge transformations. We are now in a position to analyze a process
where the spacetime metric evolves from an initial to a final configuration,
both of which satisfy Einstein’s equation in vacuum. We start by setting an
initial state characterized by

$\displaystyle\phi^{\rm in}_{\pm}(x,t)=\frac{4\pi}{K}(t\pm x)\mathcal{J}^{\rm
in}_{\pm}\,,$ (4.16)

representing a BTZ black hole with inner and outer horizons respectively
denoted by $\frac{2\pi l^{2}}{K}(\mathcal{J}^{\rm in}_{+}\pm\mathcal{J}^{\rm
in}_{-})$ (see e.g. [27]). In order to make this choice consistent with (4.7),
we set the initial value of the electric current to be $\mathcal{I}_{\pm}^{\rm
in}=\mp\mathcal{J}_{\pm}^{\rm in}$.

In presence of the source $\mu$, the configuration modifies according to
(4.6). The net effect is encoded in a new chiral boson $\phi^{\rm out}_{\pm}$
parameterizing the charge density and current after the perturbation $\mu$ has
ceased. Since the source is localized in time, for $t>0$, they are given by

$\displaystyle\mathcal{J}^{\rm
out}_{\pm}=\pm\frac{K}{4\pi}\partial_{x}\phi^{\rm
out}_{\pm}\,,\quad\mathcal{I}^{\rm
out}_{\pm}=\mp\frac{K}{4\pi}\partial_{t}\phi^{\rm out}_{\pm}\,.$ (4.17)

The field $\phi^{\rm out}_{\pm}$ can be found after solving the equation
(4.6), obtaining

$\displaystyle\phi^{\rm out}_{\pm}(x,t)=\phi^{\rm in}_{\pm}(x,t)\mp
V_{L}\,{\psi}_{z}\left(\pm\frac{x-x_{L}}{(vz_{\pm}t)^{1/z_{\pm}}}\right)\pm
V_{R}\,{\psi}_{z}\left(\pm\frac{x-x_{R}}{(vz_{\pm}t)^{1/z_{\pm}}}\right)\,,$
(4.18)

where ${\psi}_{z}(x)$ is a function admitting the following integral
representation

$\displaystyle{\psi}_{z}(x)=\frac{1}{2\pi
i}\mathcal{P}\int^{\infty}_{-\infty}\frac{du}{u}\exp\left[i\left(\frac{u^{z}}{z}+xu\right)\right]\,,$
(4.19)

and $\mathcal{P}$ is the Cauchy principal value999For $z>1$, this function
satisfies $\partial_{x}{\psi}_{z}(x)={\rm Ai}_{z}(x)$ with ${\rm Ai}_{z}$ the
$z$-order Airy function, ${\rm
Ai}_{z}(x)=\frac{1}{2\pi}\int^{\infty}_{-\infty}du\exp\left[i\left(\frac{u^{z}}{z}+x\,u\right)\right]\,$.
See e.g. [28] for more details on higher-order Airy functions..

The final state is given again by the AdS3 geometry (4.15) but now coordinates
$\phi_{\pm}$ contain physically relevant information about the evolution
process. From the point of view of the canonical charges, the chiral boson
$\phi^{\rm out}_{\pm}$ is an improper gauge transformation as
$\mathcal{J}_{\pm}$ undergoes a transition from a constant value to a function
of $x$ and $t$. Despite the presence of higher-order Fourier modes in the
charges, it can be verified that the area operator (4.12) does not change in
the process.

Another appealing feature codified in the chiral bosons is that the electric
current DC response can be recovered from (4.18). Indeed, the conductivity,
defined as the time-average value of the electric current over a voltage gives

$\displaystyle\sigma=-\frac{K}{4\pi\Delta
V}\int^{\infty}_{-\infty}dt\,\partial_{t}(\delta\phi_{+}-\delta\phi_{-})=\frac{K}{4\pi}\left[{\psi}_{z^{+}}(0)+{\psi}_{z^{-}}(0)\right]\,,$
(4.20)

where $\delta\phi_{\pm}=\phi^{\rm out}_{\pm}-\phi^{\rm in}_{\pm}$. The result
(3.12) is recovered after using ${\psi}_{z}(0)=\tfrac{1}{2z}$.

We have found that chiral bosons act as reparameterization modes on AdS space,
modifying the charges and storing DC information of the boundary electric
current. These permanent imprints in the solution space emerge as a three-
dimensional version of the 4D gravitational memory, where the passage of a
gravitational wave induces a permanent displacement of the detectors [17, 18].

It is interesting to compare the present process to the one studied in [29,
30], where a Schwarzschild black hole is hit by a shockwave. Notably, after
the wave has fallen into the black hole, the spacetime metric is deformed by
an infinitesimal supertranslation that codifies the information of the
perturbation. The situation presented here is analogous to [29, 30]. Moreover,
the chiral bosons $\phi_{\pm}$ can be seen as the finite version of the
supertranslation mode controlling the memory in the bulk for any value of the
radial coordinate and not only restricted to the near boundary region.

##  5 Future prospects

The present work opens some possible lines of future research. For example, it
has been shown that edge modes describe fractional and integer quantum Hall
states (for reviews on the subject, see e.g. [31, 32]). It would be
interesting to investigate a connection between chiral anisotropic excitations
and those appearing in the quantum Hall effect. In the present context,
quantization of the conductivity is guaranteed due to the quantization of the
level $K$. For SL$(2,\mathbb{R})$ Chern-Simons, it takes values
$K=\tfrac{n}{4}$ with $n\in\mathbb{N}$ [33]. Thus, the conductance obtained
here is quantized as an integer multiple of

$\frac{1}{32\pi}\left(\frac{1}{z_{+}}+\frac{1}{z_{-}}\right).$

Another avenue of investigation comes from the inclusion of more boundaries to
capture global properties of black holes. As it has been pointed out in [34],
to properly describe the boundary dynamics of this kind of spacetimes it is
necessary to include two boundaries in the Hamiltonian reduction. When the
boundary dynamics is computed for topologies with various boundaries, the zero
modes couple to the degrees of freedom on each boundary. This setup provides a
potential mechanism to intertwine transport phenomena on both boundaries.

Finally, it is desirable to extend the results of this work to different
choices of asymptotic conditions. In particular, those describing non-linear
dynamics at the boundary [4, 35, 36, 6] seem a natural starting point.

#### Acknowledgments

We thank Francisco Rojas and Ricardo Troncoso for useful discussions. This
research has been supported by FONDECYT grants $11190427$, $11190730$,
$1181628$, $1210635$ and the grant ANID Beca Doctorado Nacional 21182110. K.
L. would like to thank the organizers of the workshop “La parte y el todo”,
Afunalhue 2021, for the opportunity to present a preliminar version of this
work. H.G. would like to thank the support of Proyecto de cooperación
internacional 2019/13231-7 FAPESP/ANID.

##  Appendix A Gravity as a Chern-Simons theory

General Relativity can be described in terms of the difference of two Chern-
Simons actions [1, 2]

$I=I_{CS}[A^{+}]-I_{CS}[A^{-}]\,,\quad
I_{CS}[A^{\pm}]=\frac{K}{2\pi}\int\left\langle
A^{\pm},dA^{\pm}+\frac{2}{3}A^{\pm 2}\right\rangle\,,$ (A.1)

where the gauge fields $A^{\pm}$ are valued in $\mathfrak{sl}(2,\mathbb{R})$
algebra. Its generators are

$\displaystyle\begin{split}L^{\pm}_{-1}=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\;,\;L^{\pm}_{0}=\begin{pmatrix}-1/2&0\\\
0&1/2\end{pmatrix},\;L^{\pm}_{1}=\begin{pmatrix}0&-1\\\
0&0\end{pmatrix},\end{split}$ (A.2)

whose commutation relations read

$\left[L^{\pm}_{n},L^{\pm}_{m}\right]=(n-m)L^{\pm}_{n+m}\,.$ (A.3)

The bilinear invariant form is given by

$\displaystyle\left\langle
L^{\pm}_{1},L^{\pm}_{0}\right\rangle=0\quad\mbox{and}\quad\left\langle
L^{\pm}_{0},L^{\pm}_{0}\right\rangle=1/2\,.$ (A.4)

The connection to three-dimensional gravity is achieved by the identification

$\displaystyle A^{\pm}$ $\displaystyle=\omega\pm\frac{e}{l},$ (A.5)

where $e$ is the vielbein and $\omega$ is the spin connection.

In section (4), we have considered Chern-Simons equations in presence of an
external source. At the level of the action principle, the equations of motion
are obtained from

$\displaystyle I_{CS}[A;J]=I_{CS}[A]+\int d^{3}x\langle
A_{\mu},J^{\mu}\rangle\,.$ (A.6)

with

$\displaystyle I=I_{CS}[A^{+};J_{+}]-I_{CS}[A^{-};J_{-}]\,.$ (A.7)

Here, $J_{\pm}$ are $\mathfrak{sl}(2,\mathbb{R})$ valued currents. Then, the
variation of the action (A.7) with respect to the dynamical fields $A^{\pm}$
becomes

$\displaystyle\frac{K}{4\pi}\epsilon^{\mu\nu\rho}F^{\pm}_{\nu\rho}=J_{\pm}^{\mu}\,.$
(A.8)

It is worth noticing that the conservation of the external current
$D^{\pm}_{\mu}J^{\mu}_{\pm}=0$ holds on account of the Bianchi identity
$\epsilon^{\mu\nu\rho}D^{\pm}_{\mu}F^{\pm}_{\nu\rho}=0$. This fact also
ensures gauge invariance of (A.6).

##  Appendix B Bosonization and normal ordering

The interpretation of the charges $\mathcal{J}_{\pm}$ as electronic charge
densities goes as follows. We define the operators [24],

$\displaystyle
c_{\pm}(x)=:e^{-i\sqrt{\frac{K}{2}}\phi_{\pm}(x)}:\quad\text{and}\quad
c^{\dagger}_{\pm}(x)=:e^{i\sqrt{\frac{K}{2}}\phi_{\pm}(x)}:,$ (B.1)

where $:\bf{\mathcal{O}}:$ denotes the normal ordering of the operator
$\bf{\mathcal{O}}$. It is assumed all the fields are on the same time slice,
so the temporal dependence is omitted. Provided the commutation relation
(2.18), we have the boson algebra

$\displaystyle\left[\phi_{\pm}(x),\phi_{\pm}(x^{\prime})\right]=\mp\frac{2\pi}{K}i\,{\rm
sign}(x-x^{\prime})\,.$ (B.2)

In order to properly deal with the normal order prescription, it is useful to
write down the boson fields $\phi_{\pm}$ as

$\phi_{\pm}=\theta_{\pm}+\theta^{\dagger}_{\pm}\,,$ (B.3)

where $\theta_{\pm}$ and $\theta^{\dagger}_{\pm}$ satisfy the commutation
algebra,

$\left[\theta_{\pm}(x),\theta^{\dagger}_{\pm}(x^{\prime})\right]=\pm\frac{2}{K}\log(-2\pi
i(x-x^{\prime}+i\eta))\,,$ (B.4)

and $\eta$ is an infinitesimal positive regulator. Notice that given the sign
difference in the algebra (B.4) for each copy, the creation and annihilation
operators appear interchanged. The pair $(\theta^{\dagger}_{-},\theta_{-})$
acts following the standard convention, namely, as creation and annihilation
operators. On the other hand, creation and annihilation operators corresponds
to $(\theta_{+},\theta^{\dagger}_{+})$ in the plus sector, respectively. The
normal order rule imposes creation operators to be placed to the left of all
annihilation operators in the product. Indeed, taking (B.3) and evaluating
them on (B.1), one should consider

$\displaystyle
c_{+}(x)=e^{-i\sqrt{\frac{K}{2}}\theta_{+}(x)}e^{-i\sqrt{\frac{K}{2}}\theta^{\dagger}_{+}(x)}\quad\text{and}\quad
c_{-}(x)=e^{-i\sqrt{\frac{K}{2}}\theta^{\dagger}_{-}(x)}e^{-i\sqrt{\frac{K}{2}}\theta_{-}(x)}.$
(B.5)

Considering the above and (B.2), one can show that the fields $c_{\pm}$
satisfy the anticommutation relations
$\\{c_{\pm}(x),c_{\pm}^{\dagger}(x^{\prime})\\}=\\{c_{\pm}(x),c_{\pm}(x^{\prime})\\}=\\{c_{\pm}^{\dagger}(x),c_{\pm}^{\dagger}(x^{\prime})\\}=0$
for $x\neq x^{\prime}$. Using the latter, and dealing correctly with the
normal ordering rule, one can express $\mathcal{J}_{\pm}$ defined in (2.6), as
fermionic occupation number operators

$\mathcal{J}_{+}(x)=\sqrt{\frac{K}{2}}:c_{+}(x)c_{+}^{\dagger}(x^{\prime}):\quad\text{and}\quad\mathcal{J}_{-}(x)=\sqrt{\frac{K}{2}}:c_{-}^{\dagger}(x)c_{-}(x^{\prime}):,$
(B.6)

taking the limit $x-x^{\prime}\rightarrow 0$. This justifies calling
$\mathcal{J}_{\pm}$ an electric charge density.

##  Appendix C Kubo Formula

Here we rederive the expectation value of an operator due to a small
perturbation. We consider the following Hamiltonian

$\displaystyle H=H_{0}+\lambda\int dx\,\mu(x)B(x)\,,$ (C.1)

where $\lambda$ is a small dimensionless quantity, $B$ is a time-independent
deformation and $\mu$ is an external source. In the Heisenberg picture,
operators evolve with the full Hamiltonian $H$

$\displaystyle B(t,x)=e^{iH(t-t_{0})}B(x)e^{-iH(t-t_{0})}\equiv
U^{\dagger}(t,t_{0})B_{I}(t,x)U(t,t_{0})\,,$ (C.2)

where we have expressed the perturbation in terms of the operators $U$ and
$B_{I}$ given by

$\displaystyle U(t,t_{0})=e^{iH_{0}(t-t_{0})}e^{-iH(t-t_{0})},\quad
B_{I}(x,t)=e^{iH_{0}(t-t_{0})}B(x)\exp^{-iH_{0}(t-t_{0})}\,.$ (C.3)

The subscript $I$ stands for “interaction picture” which defines the evolution
in terms of the free theory. In our setting, these are the operators we
control in perturbation theory.

In order to compute the expectation value, we need to find the evolution of
the quantum state. Assuming that for $t\leq t_{0}$ the state is given by
$\left|0\right.\rangle$, the wave function for $t>t_{0}$ is dictated by

$\displaystyle\left|\Psi\right.\rangle=e^{-iH(t-t_{0})}\left|0\right.\rangle=e^{-iH_{0}(t-t_{0})}U\left|0\right.\rangle\,.$
(C.4)

Now we consider $\langle\left.\Psi\right|A\left|\Psi\right.\rangle$ and we
compute it to first order in $\lambda$. This can be done by noticing that $U$
satisfies the equation

$\displaystyle i\frac{\partial}{\partial t}U=\lambda\left(\int
dx^{\prime}\,\mu(t,x^{\prime})B_{I}(t,x^{\prime})\right)U\,,$ (C.5)

therefore it can be perturbatively solved in terms of $\lambda$,

$\displaystyle
U(t,t_{0})=1-i\lambda\int^{t}_{t_{0}}dt^{\prime}dx^{\prime}\mu(t,x^{\prime})B_{I}(t,x^{\prime})+O(\lambda^{2})\,.$
(C.6)

Then

$\displaystyle\langle A\rangle=\langle A_{I}\rangle_{0}+i\lambda\int
dx^{\prime}\int^{t}_{t_{0}}dt^{\prime}\mu(t,x)\langle[A_{I}(t,x),B_{I}(t^{\prime},x^{\prime})]\rangle_{0}+O(\lambda^{2})\,.$
(C.7)

Placing the initial state in the far past, $t_{0}\to-\infty$, dropping
$O(\lambda^{2})$ terms and setting $\lambda=1$, we get

$\displaystyle\delta\langle A\rangle=\int
dx^{\prime}\,dt^{\prime}\,\,\chi_{AB}(t-t^{\prime};x,x^{\prime})\mu(t^{\prime},x^{\prime})\,,$
(C.8)

with the response function $\chi_{AB}$ given by

$\displaystyle\chi_{AB}(t-t^{\prime};x,x^{\prime})=-i\Theta(t-t^{\prime})\langle[A_{I}(t,x),B_{I}(t^{\prime},x^{\prime})]\rangle\,.$
(C.9)

Through out this work we have used (C.8) in terms of the frequency. By Taking
Fourier transform one finds

$\displaystyle\delta\widetilde{\langle A\rangle}=\int
dx^{\prime}\,\widetilde{\mu}(\omega,x^{\prime})\,\widetilde{\chi}_{AB}(\omega;x,x^{\prime})\,,$
(C.10)

where the Fourier transform of a function $O$ is defined by
$\widetilde{O}\equiv\int^{\infty}_{-\infty}\,dte^{i\omega t}O$.

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which, by the arbitrariness of $\vf$ implies $c_{\alpha_0,\alpha_1}=c_{\alpha_0} -c_{\alpha_1}$.

The above discussion implies that $h$ is exact if and only if $\bar C$ is a $1$-coboundary. This implies the $\cB^{p,q,1}$ cohomology is isomorphic to the Čech cohomology, which is isomorphic to the de Rham cohomology.

§.§ Conclusion and comparisons

While the results for the simple case studied in Section <ref> are fully satisfactory, the results in Section <ref>-<ref> are still partial.
Indeed, we show that the preset approach yields rather sharp results for the operator associated to the measure of maximal entropy, but less information is obtained, e.g., for the operator associated to the SRB measure.
It is possible that considering the commutation of different operators with the transfer operator more information can be obtained, but this requires further work.

Also, in sections <ref> and <ref> we consider only one dimensional maps, yet the present approach seems amenable to extension to the higher dimensional setting.
In particular, the arguments of section <ref> should allow to considerably improve [24], at least for small potentials.

In the case of two dimensional hyperbolic maps, presented in section <ref>, our approach reproduces in a unified manned all the known results.
Theorems  <ref> and <ref> are a refinement of <cit.>, which contains slightly stronger results than [41].
In addition, for the application to toral parabolic flows, we can obtain the exact equivalent of <cit.> which is sharper than the corresponding results in [41]. Indeed, if $h_t$ is the unit speed flow along the stable manifold of an Anosov map $f$ then our results yield (see [48] for details)

\[
\left|\int_0^T g\circ h_t (x)dt-T\mu_{\operatorname{top}}(g)\right|\leq \Const \|g\|_\infty
\]

which implies that the ergodic average either grows linearly, or $g$ is a cocycle.
(See also [22] for a very recent and short proof of a logarithmic bound in a more general setting.)

We have thus seen that the present approach both reproduces the results in  [8], and enlightens  the connection with the action in cohomology  (already present, in some form, in [41, 42]).

In conclusion, the present strategy unifies and refines the existing results in all the cases we have presented. In addition, it appears amenable to further generalisation. In particular, it seems possible to extend it to the higher dimensional case.

Another promising direction would be to apply it to Anosov flows where some hints of the relevance of some type of cohomology already exists (e.g. see [79]). Along the same lines, it is reasonable that our ideas can yield relevant results if applied to pseudo-Anosov and partially hyperbolic maps.

§ PROOF OF THEOREM <REF>

Before proving Theorem <ref> we need a few preliminary lemmata.

In this case it is convenient to define $\psi(g)(x)=\int_0^x g(y)dy$ and

\[
\cL_+ g=\cL_2 g+\cL_1(D_{f}\cdot \psi(g)).
\]

Note that $\cL_+$ is a positive operator: if $g\geq 0$, then

\begin{equation}\label{eq:cL+positive}
\cL_+g\geq \cL_1(D_{f} \cdot \psi(g))\geq 0
\end{equation}

(here we use the assumption that $D_{f}\geq 0$).
This facilitates the study of its spectrum. There is an obvious connection with the operator we are interested in:

\begin{equation}\label{eq:conenction}
\cL_{\star}g = \cL_{+} g- (\cL_1 D_{f})\cdot \int_0^1(1-y)g(y)dy,
\end{equation}

that is $\cL_\star$ is a rank one perturbation of $\cL_+$.

Before proceeding further we need some information on $\cL_+$.
Recall that, as defined in the Theorem, $\mu_*=\frac {1}{f'(1)}$.

The spectral radius of $\cL_+$, acting on $L^1$, is $\mu_*$. Moreover, $\mu_*$ is an eigenvalue of $\cL_+'$ (the dual operator to $\cL_+$), acting on $L^\infty$ with eigenvector given by the constant function one.

Note that, for all $g\in L^1$,

\begin{equation}\label{eq:average}
\begin{split}
\int_0^1 1\cdot \cL_+g(y)dy&=\int_0^1\left[\frac{g(y)}{f'(y)} +\left(\frac 1{f'(y)}\right)'\psi(g)(y)\right] dy=\int_{0}^{1} \left(\frac {\psi(g)}{f'}\right)'\!\!\!(y) \;dy\\
&=\frac 1{f'(1)}\psi(g)(1)-\frac 1{f'(0)}\psi(g)(0)=\frac 1{f'(1)}\int_0^11 \cdot g(y) \ dy.
\end{split}
\end{equation}

Hence, $\frac 1{f'(1)}$ is an eigenvalue of the dual of $\cL_+$ and hence it belongs to the spectrum of $\cL_+$.
The lemma follows since

\begin{equation}
\label{eq:L1}
\begin{split}
\int_0^1\left|\cL_+g(y)\right|dy&\leq \int_0^1\cL_+|g|(y)dy=\frac 1{f'(1)}\int_0^1 |g(y)| \ dy.
\end{split}
\end{equation}

Note that the above Lemma implies that the space $\bV_0=\{h\in L^1\;|\; \int_0^1 h=0\}$ is invariant under $\cL_+$. However, this does not give much information on the spectrum. To learn more it is convenient to study the operator $\cL_+$ acting on $W^{1,1}$.

For all $g\in W^{1,1}$ we have

\[
\begin{split}
& \|\cL_+g\|_{L^1}\leq\mu_* \|g\|_{L^1}\\
&  \|\cL_+g\|_{W^{1,1}}\leq  \mu_*^{2}\|g\|_{W^{1,1}}+(3{\|D_{f}\|}_\infty+{\|D_{f}'\|}_{L^1}+{\|D_{f}^2\|}_{L^1}+\mu_*)\|g\|_{L^1}.
\end{split}
\]

The first inequality follows from (<ref>). Next, for each $g\in W^{1,1}$, using again (<ref>), we have

\begin{equation}\label{eq:oneder}
(\cL_+g)'=\cL_3 g'+3\cL_2 D_{f}g+\cL_2D_{f}'\psi(g)+\cL_1 D_{f}^2\psi(g).
\end{equation}

Thus  (note that \(D_{f}\geq0\) implies that \(f''\leq 0\) and so \(f'(0)\geq f(x) \geq f'(1)\)), using also the estimates of Lemma <ref>,

\[
\|(\cL_+g)'\|_{L^1}\leq  \mu_*^{2}{\|g'\|}_{L^1}+(3{\|D_{f}\|}_\infty+{\|D_{f}'\|}_{L^1}+{\|D_{f}^2\|}_{L^1}){\|g\|}_{L^1}.
\]

The Lemma follows using again (<ref>).

$\mu_*$ is a simple eigenvalue of $\cL_+$.
Moreover, \(h_{+}\), the eigenvector associated to \(\mu_{*}\), is strictly positive, i.e., \(h_{+}>0\).
In addition, there exists $\mu_1<\mu_*$ such that $\sigma_{W^{1,1}}(\cL_+)\subset \{\mu_*\}\cup\{z\in\bC\;:\; |z|\leq \mu_1\}$.

Lemma <ref> and [51] (see also <cit.>) imply that the essential spectrum of $\cL_+$ is contained in a disk of size $\mu_*^2$. Thus, since $\mu_*$ is an eigenvalue of $\cL_+'$, $\mu_*$ must be an eigenvalue of $\cL_+$. Moreover, Lemma <ref> implies that $\{\mu_*^n\cL_+^n\}$ is uniformly bounded when acting on $W^{1,1}$, hence by <cit.> $\mu_*$ is a semi-simple eigenvalue (no Jordan blocks).

Let $h_+\in W^{1,1}\setminus\{0\}\subset\cC^0$ be a corresponding eigenvector.
Next, suppose that $\cL_+ g=\mu_*e^{i\theta}g$, then $\mu_*|g|\leq \cL_+|g|$, but then

\[
\int_0^1 \cL_+|g|-\mu_*|g|=0
\]

thus $\mu_*|g|= \cL_+|g|$. Accordingly, we can assume that $h_+\geq 0$. But then it must be $h_+>0$. Indeed, if there exists $\bar x$ such that $h_+(\bar x)=0$, then, calling $y$ the maximal element in $f^{-1}(\bar x)$,

\[
0=\mu_*h_+(\bar x)\geq \frac1{f'(y)}\left(\frac 1{f'}\right)'\!\!\!(y)\int_0^y h_+.
\]

Hence $h_+(x)=0$ for all $x\leq y$. Iterating the argument we have that $h_+(x)=0$ for all $x<1$, and, by continuity, $h_+\equiv 0$, contrary to the assumption. Accordingly, if there exists another $h$ such that $\cL_+ h=\mu_* h$, then it cannot be zero anywhere otherwise $|h|$, which is also an eigenvector, would be identically zero. But then there exists $\alpha\in\bR$ such that $\alpha h_+- |h|$ has a zero and hence, since $\alpha h_+- |h|$ also is an eigenvalue, $h=\alpha h_+$. Hence, $\mu_*$ is simple (the corresponding eigenspace has dimension one).

Therefore, if $e^{i\theta}\mu_* g= \cL_+ g$ then there must exist $\vartheta\in \cC^0$ such that $g=e^{i\vartheta} h_+$. It follows

\[
\begin{split}
0=&\mu_* h_+-e^{-i\theta-i\vartheta}\cL_+(e^{i\vartheta}h_+) =\cL_+ h_+-e^{-i\theta-i\vartheta}\cL_+(e^{i\vartheta}h_+)\\
&=\cL_2\left[ 1-e^{-i\theta-i\vartheta\circ f+i\vartheta}\right] h_+ +\cL_1D_{f}\left[\psi(g)- e^{-i\theta-i\vartheta\circ f}\psi(e^{i\vartheta} h_+)\right].
\end{split}
\]

Taking the real part and integrating yields

\[
0= \int_0^1\frac{1-\cos[\theta+\vartheta\circ f-\vartheta]}{f'}h_+ +\int_0^1\!\!\!\!dx D_{f}(x)\int_0^x \!\!\!\!dy \left[1- \cos[\theta+\vartheta\circ f(x)-\vartheta(y)\right]h_+(y).
\]

Since both terms are positive, the only possibility is $\theta+\vartheta\circ f(x)-\vartheta(y)=k\pi$. This implies that $\vartheta$ is constant and hence $g$ is proportional to $h_+$, hence it must be $\theta=0$. This proves that $\mu_*$ is the only peripheral eigenvalue and the spectral gap.

We can now conclude our argument.

Equation (<ref>) and Lemma <ref> imply the bound on the essential spectral radius.

Since  \(f'\) is continuous on \([0,1]\) we know that

\[
\int_0^1|(\tfrac 1{f'(y)})'|  dy
=  \frac{1}{f'(1)} -   \frac {1}{f'(0)}=:\Delta.
\]

This means that the first statement of the theorem follows from Theorem <ref> where

\[
\tau=\lambda^{-1} +\int_0^1\left|\left(\tfrac 1{f'(y)}\right)'\right|  dy
\tfrac 2{f'(1)}-\tfrac 1{f'(0)}=\Delta+\mu_*.
\]

It remains to show the absence of eigenvalues in the sets $A_0,\ldots,A_4$.
By equation (<ref>) we have that if $\cL_\star g=zg$ then

\begin{equation}\label{eq:projected}
(z-\cL_+)g=-\cL_1 D_{f}\int_0^1(1-y)g(y) \ dy.
\end{equation}

Recall  $\mu_1<\mu_*$  from Lemma <ref>.
Let  $|z|>\mu_1$ and suppose, for sake of contradiction, that the right hand side of the above equation is zero. This would mean, by Lemma <ref>, that $g=h_+$ and $z=\mu_*$. However this would then imply that the integral on the right was strictly positive and, since $D_{f}\not\equiv 0$, would contradict the assumption.
This means that the  right hand side of the above equation cannot be zero.
Moreover, $z\neq \mu_*$ since, if it were,
\int_0^1(\mu_*-\cL_+)g=0
by Lemma <ref>, and this would imply that the right hand side of (<ref>) is zero, contrary to what we have seen.

It follows that, possibly after a normalization, for $|z|>\mu_1$ we can write $g=(z-\cL_+)^{-1}\cL_1 D_{f}$, and substituting in (<ref>) we have

\[
(z-\cL_+)^{-1}\cL_1 D_{f} (x)=-(z-\cL_+)^{-1}\cL_1 D_{f}(x)\int_0^1(1-y) (z-\cL_+)^{-1}\cL_1 D_{f}(y)dy.
\]

Accordingly, if we define

\begin{equation}
\label{eq:defXi}
\Xi(z):=1+\int_0^1(1-y) (z-\cL_+)^{-1}\cL_1 D_{f}(y)dy
\end{equation}

we have that $z$ is an eigenvalue of \(\cL_\star\) if and only if $\Xi(z)=0$.
In the following we will repeatedly use the following facts: By Lemma <ref>, the spectral radius of $\cL_+$ is $\mu_*$, and, for each $\vf\in L^1$, $\int_0^1\cL_+\vf=\mu_*\int_0^1\vf$, while, by definition, $\int_0^1\cL_1\vf=\int_0^1\vf$. Also, by (<ref>), $\cL_+$ is a positive operator and so is, obviously, $\cL_1$. Moreover we set \(\int_0^1 D_{f} = \Delta\).

Since by Lemma <ref> we have $\|\cL_+\|_{L^1}\leq \mu_*$,  for $|z|\geq \mu_*$ we can do the trivial estimate

\begin{equation}\label{eq:integral}
\begin{split}
&\left | \int_0^1(1-y) (z-\cL_+)^{-1}\cL_1 D_{f}(y)dy\right|\leq \sum_{n=0}^\infty\int_0^1(1-y) |z|^{-n-1}\cL_+^{n}\cL_1 D_{f}(y)dy\\
&\leq \sum_{n=0}^\infty\int_0^1 |z|^{-n-1}\cL_+^{n}\cL_1 D_{f}(y)dy=\frac{\Delta}{|z|-\mu_*}.
\end{split}
\end{equation}

The above implies that $\Xi(z)\neq 0$ for all $|z|>\tau$, which we know already. Hence, to gain further informations we have to analyse (<ref>) more in depth.

For $z >\mu_*$,

\[
\begin{split}
\int_0^1(1-y) (z-\cL_+)^{-1}\cL_1 D_{f}(y)dy&= \sum_{n=0}^\infty \int_0^1(1-y) z^{-n-1}\cL_+^n\cL_1 D_{f}(y)dy> 0.
\end{split}
\]

which implies $\Xi(z)>1$, hence non zero. Moreover, by Lemma <ref> we get the spectral representation $\cL_+ h=\mu_*h_+\int_0^1 h+Qh$ where for all $\mu\in(\mu_1,\mu_*)$ there exists $C_\mu$ such that $\|Q^n\|_{W^{1,1}}\leq C_\mu \mu^n$. Thus, for $z \in (\mu, \mu_*)$,

\[
\begin{split}
\int_0^1(1-y) (z-\cL_+)^{-1}\cL_1 D_{f}(y)dy=& \int_0^1(1-y) (z-\mu_*)^{-1}h_+(y) dy \left(\int_0^1\cL_1 D_{f}\right)\\
&+\sum_{n=0}^\infty \int_0^1(1-y) z^{-n-1}Q^n\cL_1 D_{f}(y)dy.
\end{split}
\]

Thus, for some $C>0$,

\[
\left|\Xi(z)-1- (z-\mu_*)^{-1}\left[\frac{1}{f'(1)}-\frac{1}{f'(0)}\right]\int_0^1(1-y) h_+\right|\leq C_\mu C(|z|-\mu)^{-1}
\]

Hence, there exists $\mu_2\in (\mu_1,\mu_*)$ such that $\Xi(z)=0$ has no solution for $z>\mu_2$.[ With some further work one could estimate $\mu_2$, but we believe the above suffices to show how to proceed.] This establishes the fact that the set $A_0$ does not belong to the spectrum.

To study non positive $z$, note that $\overline{\Xi(z)}=\Xi(\bar z)$  and that \(( z-\cL_+)^{-1} - (\bar z-\cL_+)^{-1} = (\bar z - z)( z-\cL_+)^{-1}(\bar z-\cL_+)^{-1}\). Hence, if $z=a+ib$, we have

\begin{equation}\label{eq:complex0}
\begin{split}
\Im(\Xi(z))&=ib\int_0^1(1-y) (\bar z-\cL_+)^{-1}(z-\cL_+)^{-1}\cL_1 D_{f}(y)dy\\
&= ib\int_0^1(1-y) ([a-\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy\\
\Re(\Xi(z))&=1+\int_0^1(1-y)(a-\cL_+) ([a-\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy.
\end{split}
\end{equation}

Note that if $a> \mu_*$, then, by Lemma <ref> and (<ref>), $[a-\cL_+]^{-1}=\sum_{n=0}^\infty a^{-n-1}\cL_+^n$ is a positive operator. Hence, if $b\neq 0$, $\Xi(z)=0$ implies

\begin{equation}\label{eq:uff}
\begin{split}
0&=1+\int_0^1(1-y) (\Id+[a-\cL_+]^{-2}b^2)^{-1}[a-\cL_+]^{-1}\cL_1 D_{f}(y)dy\\
&=1+\int_0^1(1-y) [a-\cL_+]^{-1}\cL_1 D_{f}(y)dy\\
-\int_0^1(1-y) (\Id+[a-\cL_+]^{-2}b^2)^{-1}[a-\cL_+]^{-3}b^2\cL_1 D_{f}(y)dy.
\end{split}
\end{equation}

Next we want to estimate the two integrals on the right hand side of (<ref>). Let us start with the first integral, assuming $a>\mu_*$,[ Note that in this computation we are sacrificing optimality of the result to the simplicity of the formulae.]

\[
\begin{split}
\int_0^1(1-y) [a-\cL_+]^{-1}\cL_1 D_{f}(y)dy&\geq a^{-1}\int_0^1(1-y) \cL_1 D_{f}(y)dy\\
\end{split}
\]

To estimate the second integral note that, for  each $g\geq 0$ and $|b|< a-\mu_*$, we can write

\begin{equation}\label{eq:cancellation2}
\begin{split}
(\Id+[a-\cL_+]^{-2}b^2)^{-1}g&=\sum_{n=0}^\infty (-1)^{n}[a-\cL_+]^{-2n}b^{2n}g\\
&\leq \sum_{n=0}^\infty [a-\cL_+]^{-4n}b^{4n}g.
\end{split}
\end{equation}

\[
\begin{split}
&\int_0^1(1-y) (\Id+[a-\cL_+]^{-2}b^2)^{-1}[a-\cL_+]^{-3}b^2\cL_1 D_{f}(y)dy\leq\\
&\leq \sum_{n=0}^\infty\int_0^1 [a-\cL_+]^{-4n-3}b^{4n+2}\cL_1 D_{f}(y)dy\\
&=\sum_{n=0}^\infty [a-\mu_*]^{-4n-3}b^{4n+2}\Delta=\frac{(a-\mu_*)b^2\Delta}{(a-\mu_*)^4-b^4}.
\end{split}
\]

The above implies that $\Xi(z)=0$ has no solution if $a>\mu_*$, $|b|< a-\mu_*$ and

\[
\]

which is implied by $a>\mu_*$ and

\begin{equation}\label{eq:complex1}
b^2 <\frac{(a-\mu_*)\Delta}{2(1+a^{-1}\Gamma)}\left[\sqrt{1+4\frac{(1+a^{-1}\Gamma)^2(a-\mu_*)^2}{\Delta^2}}-1\right].
\end{equation}

This establishes that $A_1$ is disjoint from the spectrum (apart from $1$).

If  $a<-\mu_*$,  we cannot easily use positivity arguments since

\[
\Re(\Xi(z))=1-\int_0^1(1-y)(|a|+\cL_+) ([|a|+\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy.
\]

We are thus left with the cruder estimate,
$|(|a|+\cL_+)^{-1}g|\leq \sum_{n=0}^\infty |a|^{-n-1}\cL_+^n|g|=(|a|-\cL_+)^{-1}|g|$. Hence, for $|b|\leq |a|-\mu_*$ and recalling the computation for $b=0$, we have

\[
\begin{split}
&\int_0^1(1-y)(|a|+\cL_+) ([|a|+\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy\leq \int_0^1(1-y)(|a|+\cL_+)^{-1} \cL_1 D_{f}(y)dy\\
&+\sum_{n=1}^\infty\int_0^1b^{2n}[|a|-\cL_+]^{-2n-1}\cL_1 D_{f}(y)dy.
\end{split}
\]

The above decomposition allows to still use some positivity argument for the first term after the inequality. Indeed,

\[
\begin{split}
\int_0^1(1-y) (|a|+\cL_+)^{-1}\cL_1 D_{f}(y)dy&= \sum_{n=0}^\infty |a|^{-1-n}(-1)^{n}\int_0^1(1-y) \cL_+^{n}\cL_1 D_{f}(y)dy\\
&\leq \sum_{n=0}^\infty |a|^{-1-2n}\int_0^1 \cL_+^{2n}\cL_1 D_{f}(y)dy\\
&\leq \sum_{n=0}^\infty |a|^{-1-2n}\mu_*^{2n}\Delta=\frac{|a|\Delta}{a^2-\mu_*^2}.
\end{split}
\]

\[
\begin{split}
\int_0^1(1-y)(|a|+\cL_+) ([|a|+\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy\leq &\frac{|a|\Delta}{a^2-\mu_*^2}\\
&+\frac{\Delta b^2}{(|a|-\mu_*)[(|a|-\mu_*)^2-b^2]}.
\end{split}
\]

which implies that $\Xi(x)=0$ has no solutions if $a<-\mu_*$ and

\begin{equation}\label{eq:negative_bound}
b^2< \frac{(|a|-\mu_*)^2(a^2-\mu_*^2-\Delta|a|)}{a^2-\mu_*^2+\Delta|a|}.
\end{equation}

This establishes that $A_2$ is disjoint from the spectrum.

On the other hand, for $b\neq 0$, the equations (<ref>) and $\Xi(z)=0$ imply also

\begin{equation}\label{eq:complex}
0=1-\int_0^1(1-y)\cL_+ ([a-\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy.
\end{equation}

If $|z|^2>2|a|\mu_*$, then

\[
\begin{split}
&\int_0^1(1-y)\cL_+ (|z|^2-2a\cL_++\cL_+^2)^{-1}\cL_1 D_{f}(y)dy= \\
&=\int_0^1(1-y)\cL_+ (\Id+(|z|^2-2a\cL_+)^{-1}\cL_+^2)^{-1}(|z|^2-2a\cL_+)^{-1}\cL_1 D_{f}(y)dy.
\end{split}
\]

Note that if $a\geq 0$, then $(|z|^2-2a\cL_+)^{-1}$ is a positive operator, hence

\[
\begin{split}
&\int_0^1(1-y)\cL_+ ([a-\cL_+]^2+b^2)^{-1}\cL_1 D_{f}(y)dy=\\
&=\sum_{n=0}^\infty(-1)^n\int_0^1(1-y)\cL_+ (|z|^2-2a\cL_+)^{-n-1}\cL_+^{2n}\cL_1 D_{f}(y)dy\\
&\leq\sum_{n=0}^\infty\int_0^1\cL_+ (|z|^2-2a\cL_+)^{-2n-1}\cL_+^{4n}\cL_1 D_{f}(y)dy\\
&=\sum_{n=0}^\infty \mu_* (|z|^2-2a\mu_*)^{-2n-1}\mu_*^{4n}\Delta=\frac{\mu_*\Delta(|z|^2-2a\mu_*)}{(|z|^2-2a\mu_*)^2-\mu_*^4}.
\end{split}
\]

It follows that equation (<ref>) has no solution if $a>0$, $|z-\mu_*|>\mu_*$ and

\[
\mu_*\Delta(|z-\mu_*|^2-\mu_*^2)\leq(|z-\mu_*|^2-\mu_*^2)^2-\mu_*^4,
\]

that is $a>0$ and

\begin{equation}\label{eq:complex2.0}
\end{equation}

This establishes that $A_3$ is disjoint from the spectrum.

If $a<0$, again it is harder to take advantage of the positivity, hence, for simplicity, we content ourselves with a cruder estiamte

\[
\begin{split}
&\int_0^1(1-y)\cL_+ (|z|^2-2a\cL_++\cL_+^2)^{-1}\cL_1 D_{f}(y)dy= \\
&= \sum_{n=0}^\infty |z|^{-2-2n}\int_0^1(1-y)\cL_+ (2a\cL_+-\cL_+^2)^{n}\cL_1 D_{f}(y)dy\\
&\leq \sum_{n=0}^\infty |z|^{-2-2n}\int_0^1\cL_+ (\cL_+^2+2|a|\cL_+)^{n}\cL_1 D_{f}(y)dy\\
&=\sum_{n=0}^\infty |z|^{-2-2n}\mu_*(\mu_*^2+2|a|\mu_*)^{n}\Delta=\frac{\mu_*\Delta}{|z|^2-\mu_*^2-2|a|\mu_*}.
\end{split}
\]

It follows that equation (<ref>) has no solution if

\begin{equation}\label{eq:complex2}
\end{equation}

Finally, this establishes that $A_4$ is disjoint from the spectrum.

Collecting (<ref>), (<ref>), (<ref>) and (<ref>) completes the proof of Theorem <ref>.

The previous proof it is not optimal. We refrain form walking this road further as our purpose was only to show that the study of the spectrum can be reduced to the study of a concrete function (<ref>), which plays the role of a determinant but may be easier to study than the dynamical determinant.

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