# Chaotic properties of an FIR filtered Hénon map

Vinícius S. Borges<EMAIL_ADDRESS>Magno T. M. Silva<EMAIL_ADDRESS>Marcio Eisencraft<EMAIL_ADDRESS>Escola Politécnica,
University of São Paulo, Brazil

###### Abstract

When chaotic signals are used in practical communication systems, it is
essential to control and eventually limit the spectral bandwidth occupied by
these signals. One way to achieve this goal is to insert a discrete-time
filter into a nonlinear map that generates chaotic signals. However, this can
completely change the dynamic properties of the original map. Considering this
situation, this paper presents a series of numerical experiments aimed at
obtaining the Lyapunov exponents of the signals generated by the two-
dimensional Hénon map with a set of prototypical finite impulse response (FIR)
filters added in the feedback loop. Our results show that the number of filter
coefficients and the location of the zeros have a significant and complex
impact on the behavior of the generated signals. Therefore, FIR filters should
be carefully designed to preserve or suppress chaos in practical applications.

###### keywords:

Chaotic signals, dynamical systems, nonlinear systems, discrete-time filters.

††journal: Communications in Nonlinear Science and Numerical
Simulationmycorrespondingauthormycorrespondingauthorfootnotetext:
Corresponding author11footnotemark: 122footnotemark: 2

##  1 Introduction

A chaotic signal has three main characteristics: it is bounded, aperiodic, and
exhibits a sensitive dependence on initial conditions [1]. These properties
often result in large bandwidth and have stimulated proposals for the
application of chaotic signals in Telecommunications and Signal Processing,
such as modulation [2, 3, 4], image encryption [5, 6, 7], 5G technology
standards [8, 9], ultra-wideband communications [10], among many others.

Since transmission channels are usually limited in bandwidth [11], knowing and
controlling the spectrum of transmitted signals is a relevant problem when
using chaotic signals in communication systems. With this in mind, the authors
of [12] proposed the use of a discrete-time finite impulse response (FIR)
filter [13] in a chaotic signal generator as a way to control the spectral
content of the generated signals.

Linear filters have been used to denoise or detect chaotic signals [15, 16,
17, 18, 19]. But in general, their output is not fed back into the generating
system. Thus, they can be designed and analyzed in a way that is essentially
separated from the chaos generator itself. However, in [12, 14] the filter is
inserted into the chaos generator, and its output is fed back through the
nonlinear system and through the filter itself. In this way, although the
filter is actually linear, the resulting system is highly nonlinear, and the
general behavior of the filtered chaos generator cannot be easily predicted.

Subsequently, it was shown that insertion of this filter does not affect
chaotic synchronization [14], which is essential for chaos-based
communication. However, an important question has remained open: Since the
inserted filters significantly alter the original map, are the generated
signals still chaotic?

After the results of [14], the authors of [20] analyzed the case of a two-
coefficient FIR filter inserted in the feedback loop of a Hénon map. Using
Lyapunov exponents, the authors have shown that, despite the simplicity of the
filter considered, it can induce complicated phenomena such as cascades of
bifurcations, coexistence of attractors, crises, and “shrimps” [21, 22]. Thus,
the design of filters combined with the chaotic map can show dynamics that are
difficult to predict and handle.

In this paper, we extend the research of [20] by considering filters with a
larger number of coefficients and investigating the effect of order and zero
positions on the resulting dynamics. Due to the methodological challenges of
considering many parameters in a dynamical system, we divide our analysis into
five numerical experiments considering different prototypical FIR filters.

The paper is organized as follows. The dynamical system under consideration is
introduced in Section 2. The computation of its fixed points and a discussion
of the stability of the orbits are presented in Section 3. The numerical
experiments with different FIR filters are presented in Section 4. Section 5
closes the paper with the main conclusions.

##  2 The filtered Hénon map

The Hénon map is used as a paradigm to generate chaotic two-dimensional
discrete-time signals [23, 24]. It can be expressed by

$\begin{cases}x_{1}(n+1)=\alpha-(x_{1}(n))^{2}+\beta x_{2}(n)\\\
x_{2}(n+1)=x_{1}(n)\end{cases},$ (1)

where $\alpha$ and $\beta$ are real parameters and $x_{1}(n)$ and $x_{2}(n)$
are the state variables. In particular, in this paper, we consider
$\alpha=1.4$ and $\beta=0.3$ as in [24]. These parameter values are known to
produce chaotic signals, as shown in Figure 1(a), where the state variable
$x_{1}(n)$ is plotted considering two close initial conditions. As can be
seen, the two signals shown in Figure 1(a) are bounded, aperiodic and present
a sensitive dependence on initial conditions [1]. Figures 1(b) and (c) show
the phase space and the power spectral density of the orbit with initial
condition equal to zero, respectively, highlighting the presence of a strange
attractor and the aperiodic nature of the orbit.

Figure 1: (a) Example of chaotic signals generated by (1) with $\alpha=1.4$
and $\beta=0.3$. In solid line $x_{1}(0)=x_{2}(0)=0$ and in dashed line
$x_{1}(0)=x_{2}(0)=0.0001$; (b) phase space; and (c) power spectral density of
the orbit with zero initial conditions.

To limit the spectral content of the signals generated by the Hénon map, a
causal FIR filter [13] with $N_{z}$ zeros and coefficients $c_{j}$,
$j=0,1,\cdots,N_{z}$, $N_{z}\geq 1$, $c_{0}\neq 0$, can be applied to
$x_{1}(n)$, providing

$x_{3}(n)=\sum_{j=0}^{N_{z}}c_{j}x_{1}(n-j).$ (2)

This filtered signal $x_{3}(n)$ is then fed back into the dynamic system,
resulting in the filtered Hénon map [12, 20]

$\begin{cases}x_{1}(n+1)=\alpha-(x_{3}(n))^{2}+\beta x_{2}(n)\\\
x_{2}(n+1)=x_{1}(n)\\\
x_{3}(n+1)=\sum_{j=0}^{N_{z}}c_{j}x_{1}(n-j+1)\end{cases},$ (3)

where the state vector is
$\mathbf{x}(n)\triangleq\left[x_{1}(n),x_{2}(n),x_{3}(n)\right]$.

In the analysis performed in [20], the simplest case $N_{z}=1$ was considered
in detail, which already leads to complex phenomena, e.g., islands of
periodicity in a sea of chaos, known as “shrimps” [21, 22]. Our goal here is
to further extend the results presented there to more general FIR filters. The
task is not straightforward due to the intrinsic nonlinearity of the system
and the increase in dimensionality with $N_{z}$.

To systematize the analysis, we will write the FIR filter in factored form and
define its gain. Thus, the transfer function $H(z)$ of the FIR filter (2) can
be written as [13]

$H(z)\triangleq\sum_{j=0}^{N_{z}}c_{j}z^{-j}=c_{0}\prod_{k=1}^{N_{z}}\frac{(z-z_{k})}{z},$
(4)

where $z_{k}$, $k=1,2,\cdots,N_{z}$ are the zeros of the filter, i.e., the
roots of the polynomial $z^{N_{z}}H(z)$. Defining the _filter gain_ as

$G\triangleq H(1)=\sum_{j=0}^{N_{z}}c_{j}=c_{0}\prod_{k=1}^{N_{z}}{(1-z_{k})}$
(5)

and assuming that $z_{k}\neq 1$, $1\leq k\leq N_{z}$, it will be convenient
for our analysis to express (4) as

$H(z)=G\prod_{k=1}^{N_{z}}\frac{(z-z_{k})}{z(1-z_{k})}.$ (6)

We begin our analysis by studying the fixed points of (3) and their stability
in the following section.

##  3 Fixed points and their stability

To compute the fixed points $\mathbf{p}=\left[p_{1},p_{2},p_{3}\right]$ of
(3), we have to solve for

$\begin{cases}p_{1}&=\alpha-(p_{3})^{2}+\beta p_{2}\\\ p_{2}&=p_{1}\\\
p_{3}&=p_{1}\sum_{j=0}^{N_{z}}c_{j}\end{cases}.$ (7)

Using (5) and replacing $p_{2}$ and $p_{3}$ in the first equation, we get

$G^{2}p_{1}^{2}+(1-\beta)p_{1}-\alpha=0.$ (8)

Thus, for $G\neq 0$,

$p_{1}^{+,-}=\frac{-(1-\beta)\pm\sqrt{(1-\beta)^{2}+4\alpha G^{2}}}{2G^{2}}$
(9)

where $p_{1}^{+}>0$ and $p_{1}^{-}<0$. Replacing (9) in (7), we obtain the
fixed points

$\mathbf{p}^{+}=\left[p_{1}^{+},p_{1}^{+},Gp_{1}^{+}\right]$ (10)

and

$\mathbf{p}^{-}=\left[p_{1}^{-},p_{1}^{-},Gp_{1}^{-}\right].$ (11)

Interestingly, fixed points depend only on the map parameters ($\alpha$ and
$\beta$) and on the filter gain $G$, and not on the individual filter
coefficients. The dependence of fixed points on the sum of the coefficients
$G$ of (5) agrees with the results of [20] where a special case of (9) was
obtained for $N_{z}=1$. We also note that for the particular case $G=0$, (8)
reduces to a linear equation, and there is only one fixed point at
$\left[p_{0},p_{0},0\right]$ where

$p_{0}=\frac{\alpha}{1-\beta}=2,$ (12)

for $\alpha=1.4$ and $\beta=0.3$, as we consider here. Figure 2 shows
$p_{1}^{+}$ and $p_{1}^{-}$ as a function of $G$.

Figure 2: Plots of $p_{1}^{+}$ and $p_{1}^{-}$ of (9). Note that $p_{1}^{-}$
diverges and $p_{1}^{+}\rightarrow p_{0}$ for $G=0$.

To determine the linear stability of the fixed points, we need to transform
(3) into a system of $N_{z}+1$ first-order difference equations. Then, it can
be obtained from the absolute value of the eigenvalues of the associated
Jacobian matrix $\mathbf{J}_{N_{z}}\left(\mathbf{x}(n)\right)$ evaluated at
$\mathbf{p}^{+}$ and $\mathbf{p}^{-}$ [1, 25]. The calculation of
$\mathbf{J}_{N_{z}}\left(\mathbf{x}(n)\right)$ is presented in A.

Although the coordinates of the fixed points depend only on $G$, our
calculations in A show that $\mathbf{J}_{N_{z}}\left(\mathbf{x}(n)\right)$,
and consequently their linear stability, is more subtle, depending on the full
set of filter coefficients. Our numerical analysis shows that $\mathbf{p}^{-}$
is always linearly unstable. This is not the case for $\mathbf{p}^{+}$.
Therefore, in Section 4, we numerically investigate its stability by
considering different prototype filters that lead to different sets of
coefficients. If $\mathbf{p}^{+}$ is unstable, the orbits of (3) can converge
to a periodic or chaotic attractor or diverge to infinity. To distinguish
bounded orbits as periodic or chaotic, we numerically estimate their largest
Lyapunov exponent $\Lambda$ using the tangent map technique [1]. An orbit with
$\Lambda>0$ is considered a chaotic orbit, and an orbit with $\Lambda<0$ is
considered to converge to a periodic or quasiperiodic attractor.

##  4 Numerical Experiments

In this section, a series of numerical experiments333The codes used in all
numerical experiments are avaiable at Github: https://github.com/vinicius-s-
borges/Chaotic-properties-of-an-FIR-filtered-Henon-map are presented to
illustrate the properties of the orbits of the filtered Hénon system (3). All
our experiments were performed with double precision, which we consider
sufficient to avoid the influence of digital noise and truncation.

We consider some specific cases of the FIR filter (6) with (I) $N_{z}$ zeros
equally spaced in the unit circle; (II) a pair of complex conjugate zeros at
different frequencies on the unit circle; (III) same as (II) but now with a
third zero at $z=-1$; (IV) $N_{z}$ zeros at $z=-1$; and (V) zeros designed
using Hamming windows so that the filter is lowpass [13]. These chosen cases
can shed some light on the general dependence of the orbits of (3) on $N_{z}$,
$G$, and $z_{k}$ in (6).

In all experiments, the largest Lyapunov exponent $\Lambda$ was calculated
considering $3\,000$ iterations of (3), eliminating the first 500.
Furthermore, the largest Lyapunov exponent was averaged over 25 different
initial conditions, uniformly chosen in a sphere of radius $0.01$ centered on
the fixed point $\textbf{p}^{+}$ of (10). Our numerical experiments have shown
that these quantities of initial conditions and iterations are sufficient to
determine $\Lambda$ of the attractor with an accuracy higher than $10^{-2}$.

We categorize each parameter set according with the assimptotic properties of
the orbits it can generate into four groups shown in different colors in the
section figures: (a) convergence to $\mathbf{p}^{+}$ in black; (b) convergence
to a periodic orbit in blue; (c) chaotic orbits in red; and (d) divergence to
infinity in gray.

A parameter set is of type (a) if the largest absolute value $\lambda$ of the
eigenvalues of the Jacobian matrix
$\mathbf{J}_{N_{z}}\left(\mathbf{p}^{+}\right)$ (see A) is less than $1$, that
is, $\lambda<1$. Type (b) is characterized for $\lambda>1$ and the generation
of bounded orbits with the maximum Lyapunov exponent $\Lambda<0$. Type (c)
also has $\lambda>1$ but $\Lambda>0$. Finally, if one of the random initial
conditions tested generates an orbit that diverges in less than $3\,000$
iterations, it is considered type (d).

For cases (b) and (c), it is worth emphasizing that we are considering the
average of the largest Lyapunov exponents calculated over a set of initial
conditions. Thus, not all initial conditions necessarily produce the expected
periodic or chaotic behavior. Our classification means that we have found
orbits of this type for a given set of parameters. In some cases, it was even
possible to detect the coexistence of periodic and chaotic attractors with
large attraction basins, as in Experiment III, described in Section 4.3. This
kind of complexity has also been reported in the simplified scenario of [20].

###  4.1 Experiment I: $N_{z}$ equally spaced zeros on the unit circle

In this first experiment, we consider a filter with equally spaced zeros in
the unit circle. In this case, the $N_{z}$ zeros are spaced by $2\pi/N_{z}$
and given by [13]

$z_{k}=e^{j\frac{(2k-1)\pi}{N_{z}}},\;\;\;\;k=1,\ldots,N_{z}.$ (13)

Note that with this choice, for an even $N_{z}$ we have $N_{z}/2$ complex
conjugated pairs of zeros and for an odd $N_{z}$ we have $(N_{z}-1)/2$ complex
conjugated pairs of zeros and a zero at $z=-1$. As an example, the zeros for
$N_{z}=4$ are shown in Figure 3a), and the magnitude of the corresponding
frequency response $\left|H\left(e^{j\omega}\right)\right|$ for $G=1$ is shown
in Figure 3b). It has spectral nulls at the frequencies where the zeros are
located.

Figure 3: Experiment I - $N_{z}$ equally spaced zeros on the unit circle.
Example for $N_{z}=4$ and $G=1$: (a) pole-zero plot and (b) magnitude of the
frequency response in decibels.

The transfer function of such filters is given by

$H_{1}(z)=\begin{cases}G\displaystyle\prod_{k=1}^{N_{z}/2}{\frac{z^{2}-2\cos\left(\frac{(2k-1)\pi}{N_{z}}\right)z+1}{2\left[1-\cos\left(\frac{(2k-1)\pi}{N_{z}}\right)\right]z^{2}}},&\text{even
$N_{z}$}\\\ \\\
G\displaystyle\frac{(z+1)}{2z}\prod_{k=1}^{(N_{z}-1)/2}\frac{z^{2}-2\cos\left(\frac{(2k-1)\pi}{N_{z}}\right)z+1}{2\left[1-\cos\left(\frac{(2k-1)\pi}{N_{z}}\right)\right]z^{2}},&\text{odd
$N_{z}$}\end{cases}.$ (14)

Since $H_{1}(z)$ is only a function of $N_{z}$ and $G$, we can use (3) to
numerically analyze the properties of the generated orbits as they vary. The
results obtained are shown in Figure 4.

Typical orbits are exemplified in Figure 5 along with the phase space and
power spectral density for the chaotic case. They can be compared with Figures
1(b) and (c). One can clearly see that the insertion of the filter
qualitatively changes the phase space, as expected, since the filter is in the
feedback loop of the chaotic map generator. This change reflects the increase
in the dimensionality of the global system caused by the filter. In this case,
the phase space is a 2D projection of a higher dimensional trajectory. The
power spectral density also changes significantly. One should note, however,
that this cannot be directly interpreted as a filtering of the original map
signal by the filter (14), since it is in the feedback loop of a nonlinear
system.

A bifurcation diagram for $N_{z}=1$ is illustrated in Figure 6. The strategy
of following the attractor was used: for each value of $G$, the initial
condition was a point of the attractor obtained from its previous value.

Figure 4: Experiment I - Orbit classification as a function of the filter gain
$G$ and the number of zeros $N_{z}$. Typical orbits for the parameters marked
with $\color[rgb]{.75,0,.25}\times$ are shown in Figure 5. Figure 5:
Experiment I - Examples of orbits converging toward a: (a) stable fixed point
($N_{z}=15$, $G=0.5$, and $\mathbf{x}(0)=[1,1,1]$ in black and
$\mathbf{x}(0)=[-1,-1,-1]$ in blue); (b) period-2 orbit ($N_{z}=17$, $G=0.65$,
and $\mathbf{x}(0)=[0.5,0.5,0.5]$ in black and
$\mathbf{x}(0)=[-0.5,-0.5,-0.5]$ in blue); (c) chaotic attractor ($N_{z}=15$,
$G=1$, and $\mathbf{x}(0)=[0.5,0.5,0.5]$ in black, and
$\mathbf{x}(0)=[0.51,0.51,0.51]$ in blue); (d) phase space; and (e) power
spectral density for case (c).  Figure 6: Experiment I - (a) Bifurcation
diagram for $N_{z}=1$ and (b) Zoom on the rectangle highlighted in (a).

We can observe in Figure 4 that as the gain increases from $G=0$, there is an
evolving pattern of orbits, starting with convergence to a stable fixed point,
passing through a set of periodic orbits, and arriving at chaotic orbits.
Finally, as $G$ continues to increase, the orbits diverge. This pattern can
also be seen for $N_{z}=1$ in the bifurcation diagram in Figure 6. The width
of the range of $G$ values in which the chaotic behavior is present does not
change systematically with $N_{z}$ for $N_{z}>1$. The value of $G$ has more
influence on the asymptotic properties of the orbits than $N_{z}$. For
example, $G=1$ results in chaotic orbits for all $N_{z}>1$ and for $G<0.5$ the
fixed point is stable independently of $N_{z}$.

Another interesting aspect of Figure 4 is that, in general, for odd $N_{z}>1$
when $\mathbf{p}^{+}$ becomes unstable, a window of chaotic orbits appears
immediately. For even $N_{z}$, the loss of stability of the fixed point is
followed by periodic orbits.

Finally, note that for $N_{z}=1$ the only zero is $z_{1}=-1$. Thus, from (6),

$H_{1}(z)=G\frac{z+1}{2z}$ (15)

and the last equation in (3) is $x_{3}(n+1)=c_{0}x_{1}(n+1)+c_{1}x_{1}(n)$
with $c_{0}=c_{1}=G/2$. Thus, the dynamics of this particular case can be
directly compared to Figures 1 and 2 of [20]. As expected, our sequence of
stable fixed-point, periodic orbits, chaotic orbits, and divergence agrees
with that earlier work.

###  4.2 Experiment II: a pair of complex conjugate zeros on the unit circle

In this experiment, we consider only $N_{z}=2$ zeros in $z_{1,2}=e^{\pm
j\omega_{0}}$ with $0<\omega_{0}\leq\pi$. Thus, from (6),

$H_{2}(z)=G\frac{\left(z^{2}-2\cos\omega_{0}z+1\right)}{2z\left(1-\cos\omega_{0}\right)}.$
(16)

Figure 7 shows the pole-zero plot and the magnitude of the frequency response
of (16) for $\omega_{0}=\pi/4$ and $G=1$. This filter is known in the
literature as a notch filter, since its frequency response has only one
spectral null at $\omega=\omega_{0}$ [13].

Figure 7: Experiment II - A pair of complex conjugate zeros on the unit
circle. Example for $\omega_{0}=\pi/4$ and $G=1$: (a) pole-zero plot and (b)
magnitude of the frequency response in decibels.

Varying the frequency $\omega_{0}$ and the gain $G$, we compute the largest
Lyapunov exponent for each set of parameters, allowing us to classify it
according to the orbits, as shown in Figure 8. We can observe in this figure
that for higher $\omega_{0}$, there is a larger set of $G$ values that results
in a stable fixed point. On the other hand, unbounded orbits occur more often
for lower $\omega_{0}$. The region of periodic orbits is larger than the
region of chaotic orbits that appears between the periodic and divergent
orbits for $\omega_{0}>0.2\pi$.

Figure 8:  Experiment II - Classification of each parameter set according to
the orbit as a function of ${\omega_{0}}/{\pi}$ and $G$.

The bifurcation diagram as a function of $G$ for $\omega_{0}=\pi/2$ is shown
in Figure 9. As in Figure 6, we employed the strategy of following the
attractor. We can observe that the system has a stable fixed point for
$G\in[0.00,0.51]$ and a stable period-2 orbit for $G\in[0.51,0.77]$. For
$0.77<G<1.10$ the diagram presents chaotic orbits with periodic windows. For
$G\geq 1.10$ the system has unbounded orbits. We should note that the transfer
function $H_{2}(z)$ for $\omega_{0}=\pi/2$ is the same as $H_{1}(z)$ of (14)
with $N_{z}=2$.

Figure 9: Experiment II - Bifurcation diagram for $\omega_{0}/\pi=0.5$.

###  4.3 Experiment III: a pair of complex conjugate zeros on the unit circle
and a zero at $z=-1$

In this experiment, we have $N_{z}=3$ zeros, which are $z_{1,2}=e^{\pm
j\omega_{0}}$, $0<\omega_{0}\leq\pi$, and $z_{3}=-1$. Thus, replacing these
zeros in (6), we obtain the transfer function

$H_{3}(z)=G\frac{\left(z^{2}-2\cos\omega_{0}z+1\right)(z+1)}{4z\left(1-\cos\omega_{0}\right)}.$
(17)

Figure 10 shows the pole-zero plane and the magnitude of the frequency
response of (17) for $\omega=\pi/4$ and $G=1$.

Figure 10: Experiment III - A pair of complex conjugate zeros at different
frequencies on the unit circle and a zero at $z=-1$. Example for
$\omega_{0}=\pi/4$ and $G=1$: (a) pole-zero plot and (b) magnitude of the
frequency response in decibels.

This experiment is similar to Experiment II with the notable difference that
we have an odd $N_{z}$. As we noted in Experiment I, the parity of $N_{z}$
influences the dynamics (Figure 4). The frequency response has two spectral
nulls at $\omega=\omega_{0}$ and at $\omega=\pi$.

Again, we can study the generated signals as a function of $\omega_{0}$ and
$G$. The results are presented in Figure 11. Comparing Figures 8 and 11, we
can see an increase in the chaotic region with an advance of the chaotic orbit
regions over the regions where periodic orbits predominate, which was not the
case in Experiment II.

Figure 11: Experiment III - Classification of each parameter set according to
the orbit as a function of ${\omega_{0}}/{\pi}$ and $G$.

###  4.4 Experiment IV: $N_{z}$ zeros at $z=-1$

In the fourth experiment, we analyze the influence of the number of zeros in
$z=-1$. In this case, the transfer function of the FIR filter assumes the form

$H_{4}(z)=\frac{G}{2^{N_{z}}}\frac{(z+1)^{N_{z}}}{z^{N_{z}}}.$ (18)

Figure 12 shows the pole-zero plot and the magnitude of the frequency response
of (18) for $N_{z}=1$, $N_{z}=10$, and $N_{z}=20$ and $G=1$. We can observe
that the larger $N_{z}$, the faster the magnitude of the frequency response
decays to zero at $\omega=\pi$.

Figure 12: Experiment IV - $N_{z}$ zeros at $z=-1$. (a) Pole-zero plot and (b)
Magnitude of frequency response. Examples with $N_{z}=1$, $N_{z}=10$ e
$N_{z}=20$ and $G=1$.

In Figure 13, it can be seen that for $N_{z}\leq 3$ there is a growth of the
region where there is a stable fixed point, however, as $N_{z}$ increases the
region with divergent orbits gets larger, and the chaotic and periodic regions
are limited to a gain range between $0.5$ and $1$. In the case of $G\leq 0.5$,
there is a guarantee of finding stable fixed points, as in Experiment I,
Figure 4. Note that the case $N_{z}=1$ is the same for Experiments I and IV.

Figure 13: Experiment IV - Classification of each parameter set according to
the orbit as a function of $N_{z}$ and $G$.

###  4.5 Experiment V: Low-pass filter designed using the Hamming window

To deal with more practical filters, in this section, we consider lowpass FIR
filters designed with different numbers of zeros, cutoff frequency
$\omega_{c}$, $0<\omega_{c}<\pi$, $G=1$, and using the Hamming window [13].
The transfer function of such filters is given by [13]

$H_{5}(z)=\frac{1}{z^{N_{z}}}\displaystyle\frac{\displaystyle\sum_{j=0}^{N_{z}}{\rm
sinc}\,[\omega_{c}(j-N_{z}/2)]\cdot\left[0.54-0.46\cos\left(2\pi
j/N_{z}\right)\right]z^{N_{z}-j}}{\displaystyle\sum_{j=0}^{N_{z}}{\rm
sinc}\,[\omega_{c}(j-N_{z}/2)]\cdot\left[0.54-0.46\cos\left(2\pi
j/N_{z}\right)\right]},$ (19)

where ${\rm sinc}(x)\triangleq\sin(x)/x.$ Figure 14 shows the pole-zero plot
and the magnitude of the frequency response of (19) for $N_{z}=19$ and
$\omega_{c}=0.5\pi$. We can observe that the zeros of the filter that are not
in the unit circle occur in conjugate reciprocal pairs, which is a necessary
condition for obtaining linear-phase FIR filters [13]. Regarding the magnitude
of the frequency response, this example corresponds to a gain in the passband
between $0$ dB and $0.054$ dB and a stopband attenuation of at least $-49$ dB
[13].

Varying $\omega_{c}$ and $N_{z}$, we again calculate the largest Lyapunov
exponent for each set of parameters, leading to the orbit classification shown
in Figure 15. For $N_{z}=1$, $H_{5}(z)$ does not depend on $\omega_{c}$. Thus,
the whole case $N_{z}=1$ in Figure 15 corresponds in fact to the single filter
(15) with $G=1$. The stable fixed point agrees with the result for $N_{z}=1$
and $G=1$ in Figure 4. For $N_{z}=2$, we observe the convergence to a periodic
orbit over the entire range of $\omega_{c}$. As $N_{z}$ increases, the range
of parameters that leads to periodic orbits decreases, and these orbits appear
for low $\omega_{c}$. Chaotic regions appear for $N_{z}\geq 3$ and this is the
most observed case for $N_{z}=6$ and $N_{z}=7$. For $8\leq N_{z}\leq 40$, the
convergence to chaotic orbits is less frequent while divergent orbits
predominate for $N_{z}\geq 8$.

Figure 14: Experiment V - (a) Pole-zero plot and (b) Magnitude of frequency
response. Example with $N_{z}=19$, $\omega_{c}=0.5\pi$, and $G=1$. Figure 15:
Experiment V - Classification of each parameter set according to the orbit as
a function of $N_{z}$ and $\omega_{c}$.

##  5 Conclusions

In recent years, several applications of dynamic systems that generate chaotic
signals have appeared in engineering and applied physics. In some of them,
limiting the frequency content of chaotic signals is fundamental, which can be
achieved by inserting linear filters in their description. However, the
influence of these filters on the asymptotic behavior of the trajectories is
not fully understood. In this sense, in this work, we present numerical
experiments using the recently proposed filtered Hénon map [20] to evaluate
which characteristics (gain, position, parity, and number of zeros) of an FIR
filter are relevant in terms of asymptotic dynamics.

Interestingly, we found that the fixed points of the filtered Hénon map using
a generic FIR filter depend only on the filter coefficients through the filter
gain $G$. Specific values of $c_{j}$ do not affect them. However, their
stability is more subtle and depends on the composition of the filter
coefficients. Our numerical experiments show that for the different filters
tested, there is a range of $G$ that produces chaotic and periodic orbits.
Above this range, the orbits diverge and below it, the fixed points are
stable.

In summary, the incorporation of linear filters into a dynamical system can
significantly change the nature of the generated signals. When using maps that
produce signals known to be chaotic, it cannot be assumed that this property
will be maintained when filters, even linear ones, are introduced in the
system. This variation in dynamics as a function of the filters employed warns
of the need to carefully evaluate proposals for bandlimited chaos-based
communication systems.

This article leaves several possibilities of future research. For instance: i)
the effects of the introduction of filters on the properties associated with
the generation of pseudorandom sequences using the Hénon map [26, 27]; ii) the
possibility of using a symmetric version of the Hénon map [28, 29]; iii) the
influence of digital noise and truncation on the chaotic properties of the
orbits; and iv) the effects of the insertion of infinite impulse response
(IIR) or nonlinear filters into the chaos generator.

##  Appendix A Jacobian matrices

In this appendix, we obtain the Jacobian used to perform the stability
calculations of the numerical experiments of Section 4.

For $N_{z}>3$, we define the following

$\left\\{\begin{aligned} x_{4}(n+1)&\triangleq x_{2}(n)\\\
x_{5}(n+1)&\triangleq x_{4}(n)\\\ x_{6}(n+1)&\triangleq x_{5}(n)\\\ &\vdots\\\
x_{N_{z}+1}(n+1)&\triangleq x_{N_{z}}(n)\end{aligned}\right..$ (20)

Then, replacing $x_{1}(n+1)$ into the last equation of (3) and using (20), we
rewrite the system as

$\left\\{\begin{aligned} x_{1}(n+1)=&\alpha-(x_{3}(n))^{2}+\beta x_{2}(n)\\\
x_{2}(n+1)=&x_{1}(n)\\\ x_{3}(n+1)=&c_{0}\left[\alpha-(x_{3}(n))^{2}+\beta
x_{2}(n)\right]+c_{1}x_{1}(n)\\\
&+c_{2}x_{2}(n)+c_{3}x_{4}(n)+c_{4}x_{5}(n)\\\
&+\cdots+c_{N_{z}}x_{N_{z}+1}(n)\\\ &\vdots\\\ x_{4}(n+1)=&x_{2}(n)\\\
x_{5}(n+1)=&x_{4}(n)\\\ x_{6}(n+1)=&x_{5}(n)\\\ &\vdots\\\
x_{N_{z}+1}(n+1)=&x_{N_{z}}(n)\end{aligned}\right..$ (21)

The Jacobian matrix allows the calculation of the Lyapunov exponents of the
bandlimited Hénon map [1]. For the system described in (21) the Jacobian is
given by

$\mathbf{J}_{N_{z}}\left(\mathbf{x}(n)\right)=\begin{bmatrix}0&\beta&-2x_{3}(n)&0&\cdots&0&0\\\
1&0&0&0&\ldots&0&0\\\
c_{1}&c_{0}\beta+c_{2}&-2c_{0}x_{3}(n)&c_{3}&\ldots&c_{N_{z}-1}&c_{N_{z}}\\\
0&1&0&0&\ldots&0&0\\\ 0&0&0&1&\ldots&0&0\\\ &&\vdots&&\ddots&&\vdots\\\
0&0&0&0&\ldots&1&0\end{bmatrix}_{\left(N_{z}+1\right)\times\left(N_{z}+1\right)}.$
(22)

In addition, the Jacobian matrices for the cases $N_{z}=1$, $N_{z}=2$, and
$N_{z}=3$ can not be obtained directly from (21) and are addressed in the
sequel. For $N_{z}=1$ the system (3) is rewritten as

$\left\\{\begin{array}[]{l}x_{1}(n+1)=\alpha-\left(c_{0}x_{1}(n)+c_{1}x_{2}(n)\right)^{2}+\beta
x_{2}(n)\\\ x_{2}(n+1)=x_{1}(n)\end{array},\right.$ (23)

and its Jacobian matrix is given by

$\mathbf{J}_{1}\left(\mathbf{x}(n)\right)=\left[\begin{array}[]{cc}-2c_{0}^{2}x_{1}(n)-2c_{0}c_{1}x_{2}(n)&-2c_{0}c_{1}x_{1}(n)-2c_{1}^{2}x_{2}(n)+\beta\\\
1&0\end{array}\right]_{2\times 2}.$ (24)

For $N_{z}=2$, the system (21) is rewritten as

$\left\\{\begin{array}[]{l}x_{1}(n+1)=\alpha-(x_{3}(n))^{2}+\beta x_{2}(n)\\\
x_{2}(n+1)=x_{1}(n)\\\ x_{3}(n+1)=c_{0}\left[\alpha-(x_{3}(n))^{2}+\beta
x_{2}(n)\right]+c_{1}x_{1}(n)+c_{2}x_{2}(n)\end{array}\right.,$ (25)

and its Jacobian matrix is given by

$\mathbf{J}_{2}\left(\mathbf{x}(n)\right)=\left[\begin{array}[]{ccc}0&\beta&-2x_{3}(n)\\\
1&0&0\\\ c_{1}&c_{0}\beta+c_{2}&-2c_{0}x_{3}(n)\end{array}\right]_{3\times
3}.$ (26)

Finally, for $N_{z}=3$, the system (21) is rewritten as

$\left\\{\begin{array}[]{l}x_{1}(n+1)=\alpha-(x_{3}(n))^{2}+\beta x_{2}(n)\\\
x_{2}(n+1)=x_{1}(n)\\\ x_{3}(n+1)=c_{0}\left[\alpha-(x_{3}(n))^{2}+\beta
x_{2}(n)\right]+c_{1}x_{1}(n)+c_{2}x_{2}(n)+c_{3}x_{4}(n)\\\
x_{4}(n+1)=x_{2}(n)\end{array}\right.,$ (27)

and its Jacobian matrix given by

$\mathbf{J}_{3}\left(\mathbf{x}(n)\right)=\left[\begin{array}[]{cccc}0&\beta&-2x_{3}(n)&0\\\
1&0&0&0\\\ c_{1}&c_{0}\beta+c_{2}&-2c_{0}x_{3}(n)&c_{3}\\\
0&1&0&0\end{array}\right]_{4\times 4}.$ (28)

## Acknowledgments

This study was financed by CNPq-Brazil (grants 140081/2022-4, 303826/2022-3,
311039/2019-7, and 404081/2023-1), FAPESP (grant 2021/02063-6), and CAPES-
Brazil (Finance Code 001).

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aainstitutetext: HSE University, 6 Usacheva str., Moscow 119048,
Russiabbinstitutetext: P.N. Lebedev Physical Institute of the Russian Academy
of Sciences, 53 Leninskiy pr., Moscow 119991, Russiaccinstitutetext:
Institute for Theoretical and Mathematical Physics, Moscow State University,
119991, Leninskie Gory, GSP-1, Moscow, Russiaddinstitutetext: Skolkovo
Institute of Science and Technology, Krichever Center for Advanced Studies,
Bolshoy Boulevard 30, bld. 1, Moscow 143026, Russiaeeinstitutetext: Landau
Institute for Theoretical Physics, 1A Akademika Semenova av., Chernogolovka
142432, Russia

# On $\beta$-function of $N=2$ supersymmetric integrable sigma-models

Mikhail Alfimov c Ivan Kalinichenko d,e and Alexey Litvinov<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study regularization scheme dependence of Kähler ($N=2$) supersymmetric
sigma models. At the one-loop order the metric $\beta$ function is the same as
in non-supersymmetric case and coincides with the Ricci tensor. First
correction in MS scheme is known to appear in the fourth loop Grisaru:1986px ;
Grisaru:1986dk . We show that for certain integrable Kähler backgrounds, such
as complete $T-$dual of $\eta$-deformed $\mathbb{CP}(n)$ sigma models, there
is a scheme in which the fourth loop contribution vanishes.

##  1 Introduction

Two-dimensional sigma models

$S[\boldsymbol{X}]=\frac{1}{4\pi}\int\left(G_{ij}(\boldsymbol{X})\partial_{a}X^{i}\partial_{a}X^{j}+\dots\right)\,d^{2}\boldsymbol{x}\,.$
(1)

are renormalizable in the weak sense, meaning that RG flow engage in general
an infinite number of coupling constants. Staying withing the class of pure
metric bosonic sigma-models the RG flow equation takes the form Ecker:1971xko
; Friedan:1980jm

$\dot{G}_{ij}+\nabla_{i}V_{j}+\nabla_{j}V_{i}=-\beta_{ij},\qquad\beta_{ij}=R_{ij}+O(R^{2}).$
(2)

With the higher loop terms dropped (of order $O(R^{2})$) this equation is
nothing else, but the celebrated Ricci flow equation. From the analysis of
this non-linear equation one obtains that typically starting from any smooth
initial metric the solution develops singularities both forward (IR) and
backward (UV) in the RG time. However, it is well known that this equation
also possesses ancient solutions that exist on semi-interval starting from
$t=-\infty$. Such solutions correspond to asymptotically free quantum field
theories and thus are of primary interest.

The most well known examples of ancient solutions correspond to sigma-models
on compact symmetric spaces $G/H$ which are integrable on both classical and
quantum level (provided that $H$ is simple). Recent studies of integrable
deformations of $2D$ sigma-models (see Hoare:2021dix  for pedagogical review
to the subject) provide new examples of ancient solutions. These new solutions
are expected to correspond to integrable quantum field theories. In the case
of the so called $\eta-$deformed $O(N)$ sigma-models this has been conjectured
and checked in Fateev:2018yos ; Litvinov:2018bou  (see also Fateev:2019xuq ;
Litvinov:2019rlv ; Alfimov:2020jpy  for other models). One of the main
outcomes of Fateev:2018yos ; Litvinov:2018bou ; Fateev:2019xuq ;
Litvinov:2019rlv ; Alfimov:2020jpy  is that $\eta-$deformed theory is
integrable QFT whose factorized $S$-matrix coincides with trigonometric
version of Zamolodchikov’s $S-$matrix for $O(N)$ theory Zamolodchikov:1978xm .

The approach advocated in Fateev:2018yos ; Litvinov:2018bou ; Fateev:2019xuq ;
Litvinov:2019rlv ; Alfimov:2020jpy  is somewhat indirect. It uses as an
assumption that the classical action of $\eta-$deformed sigma-model gets
promoted to the full quantum action, when all higher loop orders of the RG
equation are taken into account. In particular it implies that there should
exist ancient solution to the full RG equation (2)111In fact its version with
the non-trivial $B$-field contributions which are present in the case of
$\eta-$deformed sigma-models.. The question of the existence of such solutions
is far from being trivial. One of the main obstacles lies in the fact that the
right hand side of (2) – the $\beta$-function is not known in general form.
Saying it more accurately, what is not known is the normal form of the
$\beta-$function to which it can be reduced by the choice of regularization
scheme. This is to be compared to the situation happening for the theories
with finite number of couplings where, say for the theory with one classically
marginal coupling, the normal form of the $\beta$-function is given by first
two loop orders.

The question of higher loop corrections to integrable sigma-model backgrounds
has been considered first in Hoare:2019ark ; Hoare:2019mcc ; Hoare:2020fye .
It has been noticed that classical background require quantum corrections in
order to satisfy (2) in higher loops. In Alfimov:2021sir  two of the present
authors have studied regularization scheme dependence of the $\beta$-function
for sigma models with two-dimensional target space. It has been shown that at
four-loop approximation one can pick a scheme in which the $\beta$-function
for bosonic sigma-model has the form222We note that in the case of two-
dimensional target space there are simplifications due to the relation
$R_{ijkl}=\frac{R}{2}\left(G_{ik}G_{jl}-G_{il}G_{jk}\right)\,,$ (3) which
allows to express all terms in the $\beta$-function through the scalar
curvature and the metric.

$\beta_{ij}=\left(\frac{R}{2}+\frac{R^{2}}{4}+\frac{3R^{3}}{16}+\frac{5R^{4}}{32}-\frac{2+\zeta(3)}{64}\nabla^{2}\left(R^{3}+2R\nabla^{2}R-\frac{1}{2}\nabla^{2}R^{2}\right)+\dots\right)G_{ij}-\\\
-\left(\frac{1}{16}+\frac{5R}{32}+\dots\right)\nabla_{i}R\nabla_{j}R+\ldots\,.$
(4)

It is important that the $\zeta(3)$ terms in the $\beta-$function can not be
exterminated by the choice of scheme. However, as it was shown in
Alfimov:2021sir , for integrable sigma-models such as $\eta-$deformed
$SU(2)/U(1)$ model (also known as sausage model Fateev:1992tk ), and the
corresponding $\lambda$-deformed model Sfetsos:2013wia ; Hollowood:2014rla ,
this term vanishes due to the special relation valid for the classical action

$R^{3}+2R\nabla^{2}R-\frac{1}{2}\nabla^{2}R^{2}=\hbar^{3}\big{(}\kappa-\kappa^{-1}\big{)}^{2}\big{(}\kappa+\kappa^{-1}\big{)},$
(5)

where $\kappa$ and $\hbar$ are parameters of the classical action. Using this
fact, the authors of Alfimov:2021sir  proposed ”exact” expressions for $\eta$
and $\lambda$ deformed backgrounds which solve (2) in the four-loop
approximation.

This paper is aimed to generalize the results of Alfimov:2021sir  and in
particular the relation (5) to higher dimensional target spaces. We shall
consider the case of $N=2$ supersymmetric pure metric sigma-models. This
choice is dictated by simplification reasons. First of all, the
$\beta$-function of supersymmetric model (even for $N=1$ model) does not have
two and three loop contributions in the MS scheme Grisaru:1986px ;
Grisaru:1986dk

$\beta_{ij}^{\scriptscriptstyle{\textrm{SUSY}}}=R_{ij}+T_{ij}^{\scriptscriptstyle{\textrm{$4$-loop}}}+\dots$
(6)

Moreover $N=2$ supersymmetry requires the target space manifold to be Kähler.
In this case the number of scheme parameters reduces drastically and makes the
analysis relatively simple. We will show that there exist a special scheme
(see eqs. (26) and (27) below) in which the fourth-loop contribution vanishes
for a class of integrable sigma-models known as complete $T-$dual of
$\eta-$deformed $SU(n)/U(n-1)$ sigma-models.

It is important to note that the supersymmetric version of the complete
$T-$dual of $\eta-$deformed $SU(n)/U(n-1)$ sigma-model, contrary to its
bosonic version, is integrable at the quantum level. This model corresponds to
integrable perturbation of $N=2$ Kazama-Suzuki coset CFT Kazama:1988qp .
Moreover, one can make arguments justifying that there should exist a scheme
in which this model is one-loop exact (see section 3).

This paper is organized as follows. In section 2 we discuss scheme dependence
of the $\beta$-function in $N=2$ sigma-models. We will emphasize the
importance of using only those metric redefinitions that correspond to the so
called Kähler tensors AlvarezGaume:1980dk . We also provide an explicit
scheme, which we call integrable scheme, that does not alter two and three
loop contributions, but changes the fourth loop one. In section 3 we show that
in integrable scheme the fourth-loop contribution vanishes for a class of
integrable Kähler sigma-models known as complete $T-$dual of $\eta-$deformed
(Yang-Baxter deformed) $SU(n)/U(n-1)$ sigma-model. We also provide some
arguments in favor of one-loop exactness of the supersymmetric version of this
model. In section 4 we give the concluding remarks as well as list some open
questions.

##  2 Beta-function for $N=2$ sigma-models

Let us remind basic concepts of $\beta-$functions in pure metric sigma-models.
The $\beta$-function $\beta_{ij}$ admits the covariant loop expansion

$\beta_{ij}=\beta_{ij}^{(1)}+\beta_{ij}^{(2)}+\beta_{ij}^{(3)}+\ldots\,,$ (7)

where the $L$-th loop order $\beta$-function coefficient $\beta_{ij}^{(L)}$ is
a linear combination of tensors with given degree in the Planck constant
$\hbar$. Namely, since the metric $G_{ij}\sim\hbar^{-1}$ is proportional to
the inverse of $\hbar$, it provides natural scaling properties of all tensors

$G_{ij}\sim\hbar^{-1}\implies
G^{ij}\sim\hbar,\;\Gamma^{k}_{ij}\sim\hbar^{0},\;\nabla_{i}\sim\hbar^{0},\;R_{ijk}^{\quad
l}\sim\hbar^{0},\;R_{ij}\sim\hbar^{0}\quad\text{etc}.\,,$ (8)

Withing this scaling the $L-$th loop coefficient is a linear combination of
tensors with

$\beta_{ij}^{(L)}\sim\hbar^{L-1}\,.$ (9)

For bosonic sigma-models first four coefficients $\beta_{ij}^{(L)}$ have been
computed in the MS scheme Ecker:1971xko ; Friedan:1980jm ; Graham:1987ep ;
Foakes:1987ij ; Jack:1989vp

$\begin{gathered}\beta_{ij}^{(1)}=R_{ij}\,,\quad\beta_{ij}^{(2)}=\frac{1}{2}R_{iklm}R_{j}^{\,\,klm}\,,\\\
\begin{multlined}\beta_{ij}^{(3)}=\frac{1}{8}\nabla_{k}R_{ilmn}\nabla^{k}R_{j}^{\,\,lmn}-\frac{1}{16}\nabla_{i}R_{klmn}\nabla_{j}R^{klmn}-\\\
-\frac{1}{2}R_{imnk}R_{jpq}^{\quad
k}R^{mqnp}-\frac{3}{8}R_{iklj}R^{kmnp}R^{l}_{\,\,mnp}\,.\end{multlined}\beta_{ij}^{(3)}=\frac{1}{8}\nabla_{k}R_{ilmn}\nabla^{k}R_{j}^{\,\,lmn}-\frac{1}{16}\nabla_{i}R_{klmn}\nabla_{j}R^{klmn}-\\\
-\frac{1}{2}R_{imnk}R_{jpq}^{\quad
k}R^{mqnp}-\frac{3}{8}R_{iklj}R^{kmnp}R^{l}_{\,\,mnp}\,.\end{gathered}$ (10)

and

$\beta_{ij}^{(4)}=T_{ij}+\zeta(3)\tilde{T}_{ij},$ (11)

where the explicit forms of the tensors $T_{ij}$ and $\tilde{T}_{ij}$ are
given in Jack:1989vp .

The $N=1$ supersymmetric $\beta$-function is also known in four loops and has
much simpler form AlvarezGaume:1980dk ; Alvarez-Gaume:1981exa ; Grisaru:1986dk
; Grisaru:1986px ; Grisaru:1986wj

$\beta_{ij}^{(1)}=R_{ij}\,,\quad\beta_{ij}^{(2)}=0\,,\quad\beta_{ij}^{(3)}=0\,\quad\text{and}\quad\beta_{ij}^{(4)}=\zeta(3)\tilde{T}_{ij}.$
(12)

We see that the second and the third loop contributions vanish, while the
fourth loop coefficient $\beta_{ij}^{(4)}$ coincides exactly with the
corresponding $\zeta(3)$ term in the bosonic $\beta-$function.

The $N=2$ supersymmetry requires that the target space metric is Kähler
Zumino:1979et . The $\beta$-function is exactly the same as (12), but
specialized to the Kähler geometry. In the following, we will use Latin
indexes $(i,j,\dots)$ for real coordinates and Greek
$(\mu,\nu,\bar{\mu},\bar{\nu},\dots)$ for complex ones. By definition of the
Kähler geometry there is a choice of the complex structure such that the
metric has the form

$G_{\mu\bar{\nu}}=\partial_{\mu}\partial_{\bar{\nu}}K=\nabla_{\mu}\nabla_{\bar{\nu}}K,$
(13)

where $K$ is the Kähler potential. Then Grisaru:1986px ; Grisaru:1986dk

$\beta_{\mu\bar{\nu}}^{(1)}=R_{\mu\bar{\nu}}\,,\quad\beta_{\mu\bar{\nu}}^{(4)}=\nabla_{\mu}\nabla_{\bar{\nu}}\Delta
K\,,$ (14)

where333Note that, as pointed out in Grisaru:1986kw ,
$\zeta(3)\tilde{T}_{\mu\bar{\nu}}$ differ from
$\nabla_{\mu}\nabla_{\bar{\nu}}\Delta K$ by a diffeomorphism term.

$\Delta
K=\frac{\zeta(3)}{24}R_{ijkl}R^{i}\,_{mn}\,{}^{l}\left(R^{jnmk}+R^{jkmn}\right).$
(15)

In terms of the Kähler potential the RG equation takes the form

$\dot{K}=\frac{1}{2}\log\det G-\Delta K+\mathcal{O}(\hbar^{4})\,.$ (16)

The higher loop coefficients $\beta_{ij}^{(L)}$ for $L>1$ are regularization
scheme dependent. In the covariant approach different schemes are related by
covariant metric redefinitions

$G_{ij}\rightarrow
G_{ij}+\sum_{k=0}^{\infty}G_{ij}^{(k)}\,,\quad\text{where}\quad
G_{ij}^{(k)}\sim\hbar^{k}\,.$ (17)

Using (17) one can try to bring the $\beta$-function to some normal form. This
form however is expected to be quite complicated, having non-trivial
contributions at all loops. This is supported by the analysis performed at
lower orders Metsaev:1987zx  which shows that there are tensor structures in
the $\beta$-function that are scheme independent. In Alfimov:2021sir  it has
been shown that the four-loop bosonic $\beta-$function for $D=2$ target space
can be reduced to the form (4). In particular, the $\zeta(3)$ term is one of
scheme invariants that can not be exterminated by the replacement (17).

In Alfimov:2021sir  we noticed that for integrable sigma-models with $2D$
target space the $\zeta(3)$ term vanishes. One can argue that physically
acceptable backgrounds should not contain transcendental terms $\zeta(n)$.
This is confirmed by the exact form of the cigar metric Dijkgraaf:1991ba  and
by the exact metric for the sausage model found in Hoare:2019ark ;
Alfimov:2021sir . Thus it is tempting to check this conjecture for higher
dimensional backgrounds ($D>2$). Here however, one faces computational problem
since the number of scheme parameters is very large444Here we count only the
number of physically acceptable scheme parameters which correspond to
decomposable tensors. Namely, assume that the metric is decomposable
$ds^{2}=A_{ij}(\boldsymbol{X})dX^{i}dX^{j}+B_{ij}(\boldsymbol{Y})dY^{i}dY^{j},$
then $T_{ij}$ is called decomposable if it satisfies
$T_{ij}=T_{ij}^{A}+T_{ij}^{B}$. Then there is one parameter
$G_{ij}^{(1)}=\xi_{1}R_{ij}$ at the order $\hbar$, $5$ parameters at the order
$\hbar^{2}$
$G^{(2)}_{ij}=b_{1}R_{iklm}R_{j}\,^{klm}+b_{2}R_{ikjl}R^{kl}+b_{3}R_{ik}R_{j}^{k}+b_{4}\nabla^{2}R_{ij}+b_{5}\nabla_{i}\nabla_{j}R$
and more that $100$ parameters at the order $\hbar^{3}$!. In order to
facilitate this problem we consider $N=2$ supersymmetric version of this
problem.

General change of scheme which preserves $N=2$ supersymmetry corresponds to
the following change of the Kähler potential555See Fulling:1992vm  for the
basis for the scalars up to order $\hbar^{4}$.

$K\rightarrow K+c_{1}\log\det
G+c_{2}R+\underbrace{c_{3}R^{2}+c_{4}\bigl{(}R_{ijkl}\bigr{)}^{2}+c_{5}\bigl{(}R_{ij}\bigr{)}^{2}+c_{6}\nabla^{2}R}_{\text{$4$
scalars of order $\hbar^{2}$}}+\\\
+\underbrace{c_{7}R^{3}+\dots+c_{23}\nabla^{2}\nabla^{2}R}_{\text{$17$ scalars
of order $\hbar^{3}$
}}+\underbrace{c_{8}R^{4}+\dots+c_{99}\nabla^{2}\nabla^{2}\nabla^{2}R}_{\text{$92$
scalars of order $\hbar^{4}$ }}+\dots,$ (18)

where the scalars of order $\hbar^{L}$ contribute to the change of the
$\beta$-function starting from the $(L+2)$th loop order. Thus working withing
the four loop approximation one has to consider only the first line in (18).

On the other side, since any $N=2$ sigma-model is a specialization of $N=1$
one, it looks natural to consider only universal schemes, which do not require
any tensors other than the Riemanian metric in their construction. In the
current case, one has to retain only the scalars $\Phi$ in (18) whose second
derivative $\nabla_{\mu}\nabla_{\bar{\nu}}\Phi$ correspond to tensors
$T_{ij}^{\Phi}$ in real coordinates _without explicit use of the complex
structure_

$2\nabla_{\mu}\nabla_{\bar{\nu}}\Phi
dz^{\mu}d\bar{z}^{\bar{\nu}}=T_{ij}^{\scriptscriptstyle{\Phi}}dx^{i}dx\,.$
(19)

where $T_{ij}^{\scriptscriptstyle{\Phi}}$ is a function of the curvature and
its derivatives. It can also be rephrased as follows: we are interested in
those metric redefinitions $g_{ij}\rightarrow g_{ij}+T_{ij}$ which have only
mixed indexes when restricted to Kähler manifolds, and these mixed components
have the potential form $T_{\mu\bar{\nu}}=\nabla_{\mu}\nabla_{\bar{\nu}}\Phi$.
In that case the metric redefinition can be interpreted as a redefinition of
the Kähler potential $K\rightarrow K+\Phi$. Similar question has been
addressed in AlvarezGaume:1980dk ; Alvarez-Gaume:1981exa ; Pope:1986hg  while
attempting to classify all possible counter-terms in $N=2$ sigma-models.
Following AlvarezGaume:1980dk ; Alvarez-Gaume:1981exa ; Pope:1986hg  we will
call such tensors _Kähler tensors._

First two terms in (18) correspond to one and two loop counter-terms of order
$1/\epsilon$ and $1/\epsilon^{2}$ respectively Grisaru:1986px ; Grisaru:1986dk
and hence are expected to be Kähler. Indeed, one has

$T_{ij}^{\scriptscriptstyle{\log\det G}}=-2R_{ij},\quad
T_{ij}^{\scriptscriptstyle{R}}=\nabla^{2}R_{ij}+2R_{ikjl}R^{kl}-2R_{ik}R_{j}^{k}.$
(20)

Moreover, in appendix A we show that

$T_{ij}^{\scriptscriptstyle{R_{ij}^{2}}}=2\nabla^{k}R_{il}\nabla_{k}R_{j}\,^{l}-2\nabla^{k}R_{il}\nabla^{l}R_{jk}+\nabla_{i}R_{kl}\nabla^{k}R_{j}\,^{l}+\nabla_{j}R_{kl}\nabla^{k}R_{i}\,^{l}+\\\
+2R^{kl}\nabla_{k}\nabla_{l}R_{ij}+4R_{kilj}R^{k}\,_{m}R^{lm}+2R_{klmi}R^{k}\,_{j}R^{lm}+2R_{klmj}R^{k}\,_{i}R^{lm},$
(21)

and

$T_{ij}^{\scriptscriptstyle{\nabla^{2}R}}=\nabla^{2}T_{ij}^{\scriptscriptstyle{R}}-\frac{1}{2}\left(R_{i}\,^{k}T_{kj}^{\scriptscriptstyle{R}}+R_{j}\,^{k}T_{ki}^{\scriptscriptstyle{R}}\right)+R_{i}\,^{k}\,{}_{j}\,^{l}T_{kl}^{\scriptscriptstyle{R}}.$
(22)

We were unable to find similar expressions for $R^{2}$ and $R_{ijkl}^{2}$ and
we believe that the corresponding tensors depend explicitly on the complex
structure666We note also that the $R^{2}$ term can be eliminated from the
point of view that it corresponds to undecomposable tensor.. Using this fact
as an assumption, we drop the corresponding terms and consider the following
change of scheme

$K\rightarrow K+c_{1}\log\det
G+c_{2}R+c_{3}\bigl{(}R_{ij}\bigr{)}^{2}+c_{4}\nabla^{2}R,$ (23)

which can be written as the following change of the metric

$g_{ij}\rightarrow g_{ij}+c_{1}T_{ij}^{\scriptscriptstyle{\log\det
G}}+c_{2}T_{ij}^{\scriptscriptstyle{R}}+c_{3}T_{ij}^{\scriptscriptstyle{R_{ij}^{2}}}+c_{4}T_{ij}^{\scriptscriptstyle{\nabla^{2}R}}.$
(24)

Taking

$c_{2}=-\frac{c_{1}^{2}}{2},\quad
c_{3}=-\frac{2c_{1}^{3}}{3}+\frac{5\zeta(3)}{48}\quad\text{and}\quad
c_{4}=-\frac{c_{1}^{3}}{4}+\frac{\zeta(3)}{48},$ (25)

and using tedious, but straightforward calculation, one shows that in the new
scheme the Kähler potential satisfies

$\dot{K}=\frac{1}{2}\log\det G-\Delta\tilde{K}+\mathcal{O}(\hbar^{4})\,,$ (26)

with777First equality in (25) provides that the $\beta$-function does not get
corrected in the second and third loop orders.

$\Delta\tilde{K}=\frac{\zeta(3)}{24}\left(R_{ijkl}R^{i}\,_{mn}\,{}^{k}\left(R^{jnml}+R^{jlmn}\right)-6R^{ij}R^{kl}R_{ikjl}+\right.\\\
\left.+R^{ij}R_{ik}R^{k}\,_{j}-3R^{ij}\nabla^{2}R_{ij}+\frac{3}{4}\nabla^{2}R_{ij}^{2}\right)\,.$
(27)

We shall call the corresponding choice of parameters in (24) the integrable
scheme.

The explicit form of the scalar (27) is not very illuminating. Nevertheless,
one can show that for $D=2$ it reduces to the left hand side of (5) up to a
factor. Our main result states that $\Delta\tilde{K}$ given by (27) is a
constant for integrable Kähler sigma-models such as complete $T-$dual of
$\eta-$deformed $\mathbb{CP}(n)$ model and thus does not give corrections to
the metric $\beta-$function.

##  3 Metrics of complete $T-$dual to $\eta-$deformed $\mathbb{CP}(n-1)$ model
and the corresponding integrable sigma-model

The bosonic action of the $\eta-$deformed $G/H$ sigma-model has the form
Klimcik:2008eq ; Delduc:2013fga

$\mathcal{S}=\frac{\kappa}{4\pi\hbar}\int\textrm{Tr}\left(\left(\mathbf{g}\partial_{+}\mathbf{g}^{-1}\right)^{(\textrm{c})}\,\frac{1}{1-i\kappa\mathcal{R}_{\mathbf{g}}\circ\mathrm{P}_{\textrm{c}}}\,\left(\mathbf{g}\partial_{-}\mathbf{g}^{-1}\right)^{(\textrm{c})}\right)d^{2}x\,,$
(28)

where $\mathcal{R}$ is the Drinfel’d-Jimbo solution to the modified YB
equation $\mathbf{g}\in G$,
$\mathcal{R}_{\mathbf{g}}=\textrm{Ad}\,\mathbf{g}\circ\mathcal{R}\circ\textrm{Ad}\,\mathbf{g}^{-1}$
and $\mathrm{P}_{\textrm{c}}$ is the projector on the coset space. In the case
when $G/H$ is a symmetric space, the model (28) is known to be one loop
renormalizable with only one running coupling Hoare:2021dix

$\dot{\kappa}=\frac{c_{2}(G)\hbar(\kappa^{2}-1)}{4},$ (29)

where $c_{2}(G)$ is the dual Coxeter number of $G$.

We consider the theory (28) for $G=SU(n)$ and $H=U(n-1)$, which corresponds to
deformation of $\mathbb{CP}(n-1)$ sigma model. In this case $C_{2}(G)=n$ and
the solution of (29) has the form

$\kappa=\frac{1-e^{\frac{n\hbar}{2}t}}{1+e^{\frac{n\hbar}{2}t}}$ (30)

The theory contains both the metric and the $B-$field (except for the case of
$n=1$). However, as it was observed in Litvinov:2019rlv ; Bykov:2020llx , the
$T-$duality along all the isometric directions (complete $T-$dual) eliminates
the $B-$field completely and makes the metric rather simple. Moreover in
Bykov:2020llx  it has been shown that the corresponding geometry is Kähler.

The corresponding metrics from Litvinov:2019rlv  and from Bykov:2020llx  are
written in different coordinates. The metric found in Litvinov:2019rlv  has a
nice form of the flat metric perturbed by the ”graviton-like” exponential
terms (here for simplicity $\hbar=1$)

$ds^{2}=|d\boldsymbol{z}|^{2}+\frac{2}{e^{nt}-1}\sum_{k=1}^{n}f_{k}(\boldsymbol{x})(\boldsymbol{h}_{k}\cdot
d\boldsymbol{z})^{2},\quad\boldsymbol{z}=\boldsymbol{x}+i\boldsymbol{y},\quad\boldsymbol{z}=\big{(}z_{1},\dots,z_{n-1}\big{)},$
(31)

where

$f_{k}(\boldsymbol{x})=\sum_{l=1}^{n}e^{lt}e^{((\boldsymbol{\alpha}_{k}+\dots+\boldsymbol{\alpha}_{k+l-1})\cdot\boldsymbol{x})}\,.$
(32)

Here $\boldsymbol{\alpha}_{k},\,k=1,\dots,n-1$ are the simple roots of
$\mathfrak{sl}(n)$

$\boldsymbol{\alpha}_{n}=-\sum_{k=1}^{n-1}\boldsymbol{\alpha}_{k},\quad\boldsymbol{\alpha}_{n+k}\equiv\boldsymbol{\alpha}_{k}\,,$
(33)

and $\boldsymbol{h}_{k}$ are the weights

$\boldsymbol{h}_{1}=\boldsymbol{\omega}_{1},\quad\boldsymbol{h}_{2}=\boldsymbol{\omega}_{1}-\boldsymbol{\alpha}_{1},\quad\boldsymbol{h}_{2}=\boldsymbol{\omega}_{1}-\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2},\quad\dots,\quad(\boldsymbol{\omega}_{i},\boldsymbol{\alpha}_{j})=\delta_{ij}.$
(34)

This metric was constructed in Litvinov:2019rlv  as a conjectured dual to the
bosonic $\mathfrak{gl}(n|n)$ Toda field theory. Among other things it was
constructed so that it solves Ricci flow equation (2).

The metric from Bykov:2020llx  is Kähler and provided by the potential

$K(\boldsymbol{\zeta},\bar{\boldsymbol{\zeta}})=\sum_{k=2}^{n-1}i(\boldsymbol{\zeta}_{k}\bar{\boldsymbol{\zeta}}_{k-1}-\boldsymbol{\zeta}_{k-1}\bar{\boldsymbol{\zeta}}_{k})+2\sum_{k=1}^{n}P(\boldsymbol{t}_{k}-\boldsymbol{t}_{k-1}+2\tau),\quad\boldsymbol{\zeta}=\boldsymbol{\xi}+i\boldsymbol{\psi},$
(35)

where $P(\rho)\equiv
i\left(\text{Li}_{2}(e^{i\rho})\,+\,\frac{\rho(2\pi-\rho)}{4}\right)$,
$\boldsymbol{t}_{k}=\boldsymbol{\zeta}_{k}+\bar{\boldsymbol{\zeta}}_{k}$ for
$k=1\dots n-1$ and $\boldsymbol{t}_{0}=-\boldsymbol{t}_{n}$. It is also
convenient to make the following change, in order to get rid of
$\boldsymbol{t}_{n}$:

$\boldsymbol{\xi}_{k}\rightarrow\boldsymbol{\xi}_{k}+\boldsymbol{t}_{n}\left(\frac{k}{n}-\frac{1}{2}\right),\quad\tau\rightarrow\tau-\frac{\boldsymbol{t}_{n}}{n}.$
(36)

We have found the following simple change of variables, which relates two
metrics

$e^{(\boldsymbol{\alpha}_{k}\cdot\boldsymbol{x})}=\frac{e^{t}e^{2i\boldsymbol{\xi}_{k}}-e^{2i\boldsymbol{\xi}_{k-1}}}{e^{t}e^{2i\boldsymbol{\xi}_{k-1}}-e^{2i\boldsymbol{\xi}_{k-2}}},\quad(\boldsymbol{h}_{k}\cdot\boldsymbol{z})=i\boldsymbol{\zeta}_{k-1}-i\boldsymbol{\zeta}_{k-2},\quad
k=1\dots n-1,$ (37)

where

$\boldsymbol{\xi}_{n+k}\equiv\boldsymbol{\xi}_{k},\quad\boldsymbol{\zeta}_{n+k}\equiv\boldsymbol{\zeta}_{k},\quad\boldsymbol{\zeta}_{0}=\boldsymbol{\xi}_{0}=0,\quad\tau=-\frac{it}{2}\quad\sum_{k=1}^{n}\boldsymbol{\alpha}_{k}=0.$
(38)

Using explicit calculation with the help of Mathematica package xAct
Nutma:2013zea , one finds that $\Delta\tilde{K}$ given by (27) is indeed a
constant for the background (31). Namely, it can be shown that (compare to
(5))

$\left(R_{ijkl}R^{i}\,_{rs}\,{}^{k}\big{(}R^{jsrl}+R^{jlrs}\big{)}-6R^{ij}R^{kl}R_{ikjl}+R^{ij}R_{ik}R^{k}\,_{j}-3R^{ij}\nabla^{2}R_{ij}+\frac{3}{4}\nabla^{2}R_{ij}^{2}\right)=\\\
=-\frac{3}{16}n^{2}(n-1)\hbar^{3}\big{(}\kappa-\kappa^{-1}\big{)}^{2}\big{(}\kappa+\kappa^{-1}\big{)},$
(39)

where $\kappa$ is the parameter of the metric (30). This formula generalizes
(5) for $D>2$.

Now we present some arguments that the supersymmetric sigma-model with the
metric (31) is one-loop exact. First, we comment on the bosonic model
corresponding to (31). While being classically integrable, it is anomalous at
the quantum level similarly to the undeformed $\mathbb{CP}(n-1)$ model
Abdalla:1980jt . However, as it was shown in Litvinov:2019rlv ; Fateev:2019xuq
, quantum integrability is restored if one adds additional bosonic field that
decouples in the classical limit. The corresponding metric that includes this
degree of freedom admits the UV expansion Litvinov:2019rlv

$ds^{2}=\sum_{i=1}^{n-1}dx_{i}^{2}+\sum_{i=1}^{n}dy_{i}^{2}+e^{t}\sum_{k=1}^{n}\Big{|}\big{(}\boldsymbol{h}_{k}\cdot
d\boldsymbol{x}\big{)}-i\sqrt{1+b^{-2}}\big{(}\boldsymbol{\mathfrak{h}}_{k}\cdot
d\boldsymbol{y}\big{)}\Big{|}^{2}e^{\frac{(\boldsymbol{\alpha}_{k}\cdot\boldsymbol{x})}{b}}+O(e^{2t}),$
(40)

where $\boldsymbol{\alpha}_{k}$ and $\boldsymbol{h}_{k}$ are given by
(33)-(34) and

$(\boldsymbol{\mathfrak{h}}_{1},\dots,\boldsymbol{\mathfrak{h}}_{n})\in\mathbb{R}^{n}:\qquad\big{(}\boldsymbol{\mathfrak{h}}_{i}\cdot\boldsymbol{\mathfrak{h}}_{j}\big{)}=\delta_{ij}-\frac{1}{1+b^{-2}}\frac{1}{n}.$
(41)

It is important that the leading coefficients in (40) are exact in the Planck
constant $\hbar\overset{\text{def}}{=}b^{-2}$. We note that in the
semiclassical limit the basis $\boldsymbol{\mathfrak{h}}_{k}$ is degenerate

$\big{(}\boldsymbol{\mathfrak{h}}_{i}\cdot\boldsymbol{\mathfrak{h}}_{j}\big{)}=\delta_{ij}-\frac{b^{2}}{1+b^{2}}\frac{1}{n}\,\longrightarrow\delta_{ij}-\frac{1}{n}\implies\Big{(}\sum_{k=1}^{n}\boldsymbol{\mathfrak{h}}_{k}\cdot\boldsymbol{\mathfrak{h}}_{i}\Big{)}\rightarrow
0.$ (42)

and hence after rescaling $\boldsymbol{x}\rightarrow b\boldsymbol{x}$,
$\boldsymbol{y}\rightarrow b\boldsymbol{y}$ one degree of freedom decouples
and one arrives to (31) after simple change of variables. The form of the
leading coefficient in (40) is constrained by integrability. The corresponding
integrable system is defined by the screening operators for bosonic
$\mathfrak{gl}(n|n)$ Toda QFT Litvinov:2019rlv ; Fateev:2019xuq

$\mathcal{S}_{2r-1}=\oint
e^{b(\boldsymbol{h}_{k}\cdot\boldsymbol{x}(z))-i\sqrt{1+b^{2}}(\boldsymbol{\mathfrak{h}}_{k}\cdot\boldsymbol{y}(z))}dz,\quad\mathcal{S}_{2r}=\oint
e^{i\sqrt{1+b^{2}}(\boldsymbol{\mathfrak{h}}_{k}\cdot\boldsymbol{y}(z))-b(\boldsymbol{h}_{k+1}\cdot\boldsymbol{x}(z))}dz\,,$
(43)

with $r=1,\dots,n$.

Contrary, the corresponding supersymmetric model888 The supersymmetric action
is defined in terms of $N=1$ super field as ususal  $S=\frac{1}{4\pi}\int
G_{ij}(\mathbb{X})\mathcal{D}\mathbb{X}^{i}\bar{\mathcal{D}}\mathbb{X}^{j}\,d^{2}\boldsymbol{x}\,d^{2}\theta.$
where
$\mathbb{X}^{i}(z,\bar{z},\theta,\bar{\theta})\overset{\text{def}}{=}X^{i}(z,\bar{z})+\theta\psi^{i}(z,\bar{z})+\bar{\theta}\bar{\psi}^{i}(z,\bar{z})+\theta\bar{\theta}F^{i}(z,\bar{z}),\quad\mathcal{D}\overset{\text{def}}{=}\frac{\partial}{\partial\theta}-\theta\partial,\quad\bar{\mathcal{D}}\overset{\text{def}}{=}\frac{\partial}{\partial\bar{\theta}}-\bar{\theta}\bar{\partial}\,.$
is integrable as it is, without auxiliary degrees of freedom. The metric has
the UV expansion

$ds^{2}=\sum_{i=1}^{n-1}dx_{i}^{2}+\sum_{i=1}^{n-1}dy_{i}^{2}+e^{t}\sum_{k=1}^{n}\Big{|}\big{(}\boldsymbol{h}_{k}\cdot
d\boldsymbol{x}\big{)}-i\big{(}\boldsymbol{h}_{k}\cdot
d\boldsymbol{y}\big{)}\Big{|}^{2}e^{\frac{(\boldsymbol{\alpha}_{k}\cdot\boldsymbol{x})}{b}}+O(e^{2t}),$
(44)

which is controlled by the set of screening fields for supersymmetric
$\mathfrak{gl}(n|n)$ Toda theory Olshanetsky:1982sb ; Evans:1990qq ;
Ito:1991wb

$\tilde{\mathcal{S}}_{2r-1}=\oint
e^{b(\boldsymbol{h}_{k}\cdot\boldsymbol{x}(z,\theta))-ib(\boldsymbol{h}_{k}\cdot\boldsymbol{y}(z,\theta))}dzd\theta,\quad\mathcal{S}_{2r}=\oint
e^{ib(\boldsymbol{h}_{k}\cdot\boldsymbol{y}(z,\theta))-b(\boldsymbol{h}_{k+1}\cdot\boldsymbol{x}(z,\theta))}dz\theta\,.$
(45)

We note that the set of screening fields (45) defines integrable perturbation
of Kazama-Suzuki $N=2$ coset model for $SU(n)/U(n-1)$ Kazama:1988qp .

Comparing the systems (43) and (45) one finds that the later system coincides
with the limit $\hbar=b^{-2}\rightarrow 0$ of the former. Moreover the form of
(44) and (45) suggests that the corresponding sigma-model is one-loop exact.
On the other side (40) and (43) show that the bosonic action will receive
corrections at every loop order. Of course the conjecture that the
supersymmetric sigma-model governed by the UV expansion (44) is one-loop exact
has to be confirmed by explicit calculations. In this paper we have shown,
that there exits a scheme in which the fourth loop coefficient of the
$\beta-$function vanishes.

##  4 Conclusion

In these notes we studied scheme dependence of $N=2$ supersymmetric pure
metric sigma-models. We have shown that for integrable backgrounds such as
complete $T-$dual of $\eta-$deformed $\mathbb{CP}(n-1)$ sigma-models there is
a regularization scheme in which the fourth-loop contribution to the
$\beta-$function vanishes, thus making it one loop exact to this order. There
are several immediate open questions, which we mention

###  $\lambda$-model.

In our previous paper Alfimov:2021sir  it has been also shown that (5) holds
also for $\lambda$-deformed model Sfetsos:2013wia ; Hollowood:2014rla  for
$SU(2)/U(1)$. It is natural to expect that higher dimensional generalization
of (5), namely (39), holds for $SU(n)/U(n-1)$ $\lambda$-model. We have checked
explicitly that this is the case for $n=3$. Let us briefly comment on this
calculation.

It is well known that $\eta$ and $\lambda$ deformations of $G/H$ sigma-model
are related by Poisson-Lie duality with respect to group $G$ and certain
analytic continuation Hoare:2017ukq . There is another relation between two
models noticed in Hoare:2015gda  for $SO(n+1)/SO(n)$. Namely, taking certain
infinite limits of the $\lambda-$model one recovers the complete abelian
$T-$dual of the $\eta-$model. It is not automatically clear that for any coset
$G/H$ it is always possible to find limits that provide similar relation. Thus
in general $\lambda-$model provides independent check of our conjecture.

The action of the $G/H$ $\lambda-$model has the form (here $g\in G$)

$S=\int\textrm{Tr}\Bigl{(}-\frac{1}{2}(g^{-1}\partial
g)(g^{-1}\bar{\partial}g)+J\left(\textrm{Ad}_{g}-1+\lambda\mathbb{P}\right)^{-1}\bar{J}\Bigr{)}d^{2}\xi+S_{WZ},$
(46)

where $\mathbb{P}$ is the projection on the coset space, in our case
$SU(3)/U(2)$, and

$J=g^{-1}\cdot\partial g\,,\quad\bar{J}=\bar{\partial}g\cdot g^{-1}\,.$

We have taken the following parametrization of the group element

$g=e^{i\alpha t_{1}}e^{i\beta t_{6}}e^{i\gamma t_{3}}e^{i\delta t_{8}},$ (47)

where $t_{k}$ are the standard Gell-Mann matrices, one finds that the model
has a non-vanishing, but constant $B-$field

$B=-\frac{1}{6}d\gamma\wedge d\delta.$ (48)

The corresponding metric is very complicated to be presented here. However, we
have checked that this metric is Kähler and $\Delta\tilde{K}=\text{const}$. It
would be interesting to repeat this calculation for general $n$.

### Fifth loop computations

It will be interesting to extend our results to fifth loop. The corresponding
contribution to the $\beta$-function has been computed in Grisaru:1986wj

$\beta_{\mu\bar{\nu}}^{(5)}=\frac{\zeta(4)}{48}\nabla_{\mu}\nabla_{\bar{\mu}}\nabla^{2}R_{ijkl}R^{i}\,_{mn}\,{}^{l}\left(R^{jnmk}+R^{jkmn}\right).$
(49)

In order to study the scheme dependence at this order one has to consider more
general redefinitions of the Kähler potential which include scalars of order
$\hbar^{3}$ in (18). For generic Riemanian metric there are $17$ such scalars
Fulling:1992vm 999Note that for a given $D$ there might be more relations (see
e.g. Xu:1987pz  for $D=4$). Note that in the case of Kähler geometry there is
the relation (see appendix A for details)

$R^{ij}\nabla_{i}\nabla_{j}R=R^{ij}\nabla^{2}R_{ij}+2R^{ij}R_{ikjl}R^{kl}-2R^{ij}R_{ik}R^{k}\,_{j}$
(50)

which reduces the number of independent scalars to $16$.

In order to deal with universal scheme redefinitions, one has to consider only
the scalars whose mixed derivatives correspond to Kähler tensors. It is
expected that one reduces the fifth loop beta function to a form similar to
(27), which vanishes for integrable backgrounds. Apart from trivial
contributions of the form $\nabla^{2}\Delta\tilde{K}$, that vanish due to
(39), we expect high order relations. In the case of $D=2$ we have found at
least one relation at every order in $\hbar$ starting from $\hbar^{3}$. In
paeticular, at the order $\hbar^{4}$ one has

$R^{4}+3R^{2}\nabla^{2}R-\frac{1}{6}R\nabla^{2}R^{2}-\frac{1}{9}\nabla^{2}R^{3}+\frac{2}{3}\left(\big{(}\nabla^{2}R\big{)}^{2}+R\nabla^{2}\nabla^{2}R\right)-\frac{1}{3}\nabla^{2}\left(R\nabla^{2}R\right)=\\\
=\frac{\hbar^{4}}{3}\big{(}\kappa-\kappa^{-1}\big{)}^{2}\big{(}3\kappa^{2}+3\kappa^{-2}+2\big{)}\,,$
(51)

valid for integrable $\eta$ and $\lambda$-deformed $SU(2)/U(1)$ models.

## Acknowledgments

We acknowledge discussions with Alexey Rosly. A.L. has been supported by the
Russian Science Foundation under the grant 22-22-00991 and by Basis
foundation. A.L. is also grateful to Abubakir Koshek for collaboration at the
early stages of this project.

##  Appendix A Kähler tensors

Kähler metric satisfies special properties. In particular the only
non–vanishing components of the curvature are the mixed ones

$R_{\alpha\bar{\beta}\gamma\bar{\delta}}\quad\text{etc}\,,$ (52)

which leads to the following identities

$\nabla_{\mu}R_{\alpha\bar{\alpha}}=\nabla_{\alpha}R_{\mu\bar{\alpha}},\quad\nabla_{\bar{\mu}}R_{\alpha\bar{\alpha}}=\nabla_{\bar{\alpha}}R_{\alpha\bar{\mu}}\,.$
(53)

Consider a scalar $\Phi$ that satisfies the Kähler property (19). We have

$\nabla_{\mu}\nabla_{\bar{\nu}}\nabla^{2}\Phi=2g^{\alpha\bar{\alpha}}\nabla_{\mu}\nabla_{\bar{\nu}}\nabla_{\alpha}\nabla_{\bar{\alpha}}\Phi=2g^{\alpha\bar{\alpha}}\nabla_{\mu}\nabla_{\bar{\alpha}}\nabla_{\alpha}\nabla_{\bar{\nu}}\Phi=\\\
=2g^{\alpha\bar{\alpha}}\left(\nabla_{\bar{\alpha}}\nabla_{\alpha}\nabla_{\mu}\nabla_{\bar{\nu}}\Phi+R_{\mu\bar{\alpha}\alpha}\,^{p}\nabla_{p}\nabla_{\bar{\nu}}\Phi+R_{\mu\bar{\alpha}\bar{\nu}}\,^{p}\nabla_{\alpha}\nabla_{p}\Phi\right)=\\\
=2g^{\alpha\bar{\alpha}}\nabla_{\bar{\alpha}}\nabla_{\alpha}\nabla_{\mu}\nabla_{\bar{\nu}}\Phi-
R_{\mu}\,^{p}\nabla_{p}\nabla_{\bar{\nu}}\Phi+R_{\bar{\nu}k\mu
l}\nabla^{k}\nabla^{l}\Phi,$ (54)

and at the same time exchanging the role of indexes $\mu$ and $\bar{\nu}$ we
have

$\nabla_{\mu}\nabla_{\bar{\nu}}\nabla^{2}\Phi=2g^{\alpha\bar{\alpha}}\nabla_{\alpha}\nabla_{\bar{\alpha}}\nabla_{\mu}\nabla_{\bar{\nu}}\Phi-
R_{\bar{\nu}}\,^{p}\nabla_{p}\nabla_{\mu}\Phi+R_{\mu
k\bar{\nu}l}\nabla^{k}\nabla^{l}\Phi.$ (55)

Taking symmetric combination, we obtain

$\nabla_{\mu}\nabla_{\bar{\nu}}\nabla^{2}\Phi=\nabla^{2}\nabla_{\mu}\nabla_{\bar{\nu}}\Phi-\frac{1}{2}\left(R_{\mu}\,^{p}\nabla_{p}\nabla_{\bar{\nu}}\Phi+R_{\bar{\nu}}\,^{p}\nabla_{p}\nabla_{\mu}\Phi\right)+R_{\mu
k\bar{\nu}l}\nabla^{k}\nabla^{l}\Phi.$ (56)

Finally, using (19) we find the relation

$T_{ij}^{\scriptscriptstyle{\nabla^{2}\Phi}}=\nabla^{2}T_{ij}^{\scriptscriptstyle{\Phi}}-\frac{1}{2}\left(R_{i}\,^{k}T_{kj}^{\scriptscriptstyle{\Phi}}+R_{j}\,^{k}T_{ki}^{\scriptscriptstyle{\Phi}}\right)+R_{i}\,^{k}\,{}_{j}\,^{l}T_{kl}^{\scriptscriptstyle{\Phi}}$
(57)

Taking for $\Phi=\log\det g$ and for $\Phi=R$ we obtain exactly (20) and (22).
We note that contracting (20) with $R^{ij}$, one finds (50).

Now we prove Kähler identity for $R_{ij}^{2}$. Consider

$\nabla_{\mu}\nabla_{\bar{\nu}}R_{ij}^{2}=2\nabla_{\mu}R_{ij}\nabla_{\bar{\nu}}R^{ij}+2\left(\nabla_{\mu}\nabla_{\bar{\nu}}R_{ij}\right)R^{ij}=I_{1}+I_{2}.$
(58)

The first tensor in (59) can be transformed either as

$I_{1}=4\nabla_{\mu}R_{\alpha\bar{\alpha}}\nabla_{\bar{\nu}}R^{\alpha\bar{\alpha}}\overset{\eqref{Identity-1}}{=}4\nabla_{\alpha}R_{\mu\bar{\alpha}}\nabla^{\alpha}R_{\bar{\nu}}\,^{\bar{\alpha}}=4\nabla_{k}R_{\mu
l}\nabla^{k}R_{\bar{\nu}}\,^{l}-4\nabla_{\bar{\beta}}R_{\mu\bar{\alpha}}\nabla^{\bar{\beta}}R_{\bar{\nu}}\,^{\bar{\alpha}}\overset{\eqref{Identity-1}}{=}\\\
\overset{\eqref{Identity-1}}{=}4\nabla_{k}R_{\mu
l}\nabla^{k}R_{\bar{\nu}}\,^{l}-4\nabla_{\bar{\alpha}}R_{\mu\bar{\beta}}\nabla^{\bar{\beta}}R_{\bar{\nu}}\,^{\bar{\alpha}}=4\nabla_{k}R_{\mu
l}\nabla^{k}R_{\bar{\nu}}\,^{l}-4\nabla_{k}R_{\mu
l}\nabla^{l}R_{\bar{\nu}}\,^{k},$ (59)

or as

$I_{1}=4\nabla_{\mu}R_{\alpha\bar{\alpha}}\nabla_{\bar{\nu}}R^{\alpha\bar{\alpha}}\overset{\eqref{Identity-1}}{=}2\nabla_{\alpha}R_{\mu\bar{\alpha}}\nabla_{\bar{\nu}}R^{\alpha\bar{\alpha}}+2\nabla_{\mu}R_{\alpha\bar{\alpha}}\nabla^{\alpha}R_{\bar{\nu}}\,^{\bar{\alpha}}=\\\
=2\nabla_{\mu}R_{kl}\nabla^{k}R_{\bar{\nu}}\,^{l}+2\nabla_{\bar{\nu}}R_{kl}\nabla^{k}R_{\mu}\,^{l},$
(60)

Taking symmetric combination of (59) and (60) one finds the first line in (21)
written in real coordinates. On the other hand, using the identity (53), one
can easily show that this tensor has only mixed components. The second term
$I_{2}$ in (59) we can represent ways

$I_{2}=4\left(\nabla_{\mu}\nabla_{\bar{\nu}}R_{\alpha\bar{\alpha}}\right)R^{\alpha\bar{\alpha}}\overset{\eqref{Identity-1}}{=}4\left(\nabla_{\mu}\nabla_{\bar{\alpha}}R_{\alpha\bar{\nu}}\right)R^{\alpha\bar{\alpha}}=\\\
=4\left(\nabla_{\bar{\alpha}}\nabla_{\alpha}R_{\mu\bar{\nu}}+R_{\bar{\nu}\alpha\mu}\,^{p}R_{p\bar{\alpha}}+R_{\bar{\nu}\alpha\bar{\alpha}}\,^{p}R_{\mu
p}\right)R^{\alpha\bar{\alpha}},$ (61)

and

$I_{2}=4\left(\nabla_{\bar{\nu}\nabla_{\mu}}R_{\alpha\bar{\alpha}}\right)R^{\alpha\bar{\alpha}}\overset{\eqref{Identity-1}}{=}4\left(\nabla_{\bar{\nu}}\nabla_{\alpha}R_{\mu\bar{\alpha}}\right)R^{\alpha\bar{\alpha}}=\\\
=4\left(\nabla_{\alpha}\nabla_{\bar{\alpha}}R_{\mu\bar{\nu}}+R_{\mu\bar{\alpha}\alpha}\,^{p}R_{p\bar{\nu}}+R_{\mu\bar{\alpha}\bar{\nu}}\,^{p}R_{\alpha
p}\right)R^{\alpha\bar{\alpha}}.$ (62)

Taking symmetric combination of (61) and (62) we find the second line in (21)
which also has only mixed components.

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