FTPI-MINN-24-05, UMN-TH-4314/24

Lie-algebraic Kähler sigma models
with the U(1) isotropy

Chao-Hsiang Sheua and Mikhail Shifmana,b

aDepartment of Physics, University of Minnesota, Minneapolis, MN 55455
and
bWilliam I. Fine Theoretical Physics Institute, University of Minnesota,
Minneapolis, MN 55455

Abstract

We discuss various questions which emerge in connection with the Lie-algebraic
deformation of $\mathbb{CP}^{1}$ sigma model in two dimensions. First we
supersymmetrize the original model endowing it with the minimal ${\cal
N}=(0,2)$ and extended ${\cal N}=(2,2)$ supersymmetries. Then we derive the
general hypercurrent anomaly in the both cases. In the latter case this
anomaly is one-loop but is somewhat different from the standard expressions
one can find in the literature because the target manifold is non-symmetric.
We also show how to introduce the twisted masses and the $\theta$ term, and
study the BPS equation for instantons, in particular the value of the
topological charge. Then we demonstrate that the second loop in the $\beta$
function of the non-supersymmetric Lie-algebraic sigma model is due to an
infrared effect. To this end we use a supersymmetric regularization. We also
conjecture that the above statement is valid for higher loops too, similar to
the parallel phenomenon in four-dimensional ${\cal N}=1$ super-Yang-Mills. In
the second part of the paper we develop a special dimensional reduction –
namely, starting from the two-dimensional Lie-algebraic model we arrive at a
quasi-exactly solvable quantum-mechanical problem of the Lamé type.

###### Contents

  1. 1 Introduction
  2. 2 Extending the deformed $\mathbb{CP}^{1}$ model
    1. 2.1  ${\mathcal{N}}=(2,2)$ and ${\mathcal{N}}=(0,2)$ supersymmetrization
    2. 2.2 Adding twisted masses
    3. 2.3 The $\theta$ term
  3. 3 Analysis of the two-loop beta function
    1. 3.1 The two-loop beta function of the bosonic model from supersymmetry
    2. 3.2 Comparison with the first-order formalism
    3. 3.3 Comment on the literature
  4. 4 Reduction to quantum mechanics
    1. 4.1 Comparison to Dunne-Shifman
    2. 4.2 Generalizations
  5. 5 Conclusions and outlooks
  6. A Conventions on elliptic integrals and Jacobi elliptic function
  7. B Derivation of anomalous axial $U(1)$ currrent
  8. C Comparison between deformed $\mathbb{CP}^{1}$ quantum mechanics and Lamé quantum mechanics
  9. D Details of Lie-algebraic features of supersymmetric Lamé quantum mechanics

##  1 Introduction

In this paper we continue the studies of one(complex)-dimensional sigma models
on Kählerian target spaces which generalize the $\mathbb{CP}^{1}$ model in a
Lie-algebraic way [1, 2, 3]. For practical applications in baby Skyrmions this
model is usually formulated in the from

${\mathcal{L}}=\frac{1}{2g^{2}(S_{3})}(\partial S_{i})(\partial
S_{i})\,,\qquad\vec{S}\vec{S}=1\,.$ (1.1)

where the coupling $g^{2}$ becomes a function of $S_{3}$, the third component
of the isovector $\vec{S}$,

$g^{2}(S_{3})=g^{2}\cdot\left(\textstyle{\frac{1+k}{2}}+\textstyle{\frac{1-k}{2}}\,S_{3}^{2}\right).$
(1.2)

Moreover, $k$ is a numerical parameter defined below in Eq. (1.5). At $k=1$ we
return to the Heisenberg O(3) model. With $k\neq 1$ the round metric is
deformed.

For theoretical applications in two dimensions (2D) it is more convenient to
use the geometric representation,

${\mathcal{L}}=G_{1\overline{1}}\left(\partial_{\mu}\overline{\varphi}\partial^{\mu}\varphi\right),$
(1.3)

where $G_{1\overline{1}}$ is a generalization of the Fubini-Study
metric,111The equality $n_{1}=n_{3}$ can always be achieved by rescaling the
fields $\varphi,\overline{\varphi}$.

$\displaystyle G_{1\overline{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{1}+n_{2}\overline{\varphi}\varphi+n_{3}(\overline{\varphi}\varphi)^{2}}\,,$
(1.4) $\displaystyle n_{1}$ $\displaystyle=$ $\displaystyle
n_{3}=\frac{g^{2}}{2},\quad n_{2}=g^{2}k\,.$ (1.5)

If $k=1$ the metric (1.4) is the standard Fubini-Study metric. In what follows
we will use a simplified notation,

$G_{1\overline{1}}\equiv G\,,\quad G^{1\overline{1}}\equiv G^{-1}\,.$

Other abbreviations are introduced in Eqs. (2.3) and (2.5).

One can consider another deformation of this model, by the so-called twisted
mass term [4, 5]. Then, Eq. (1.3) takes the form

${\mathcal{L}}_{m}=G_{1\overline{1}}\left(\partial_{\mu}\overline{\varphi}\partial^{\mu}\varphi-m^{2}\overline{\varphi}\varphi\right),$
(1.6)

In temrs of representation (1.1) we then have

$\mathcal{L}=\frac{1}{2g^{2}(S_{3})}\Bigl{[}(\partial S_{i})(\partial
S_{i})-\absolutevalue{m}^{2}(1-S_{3}^{2})\Bigr{]}\,.$ (1.7)

Both deformations (i.e. the metric deformation $k\neq 1$ and a “potential”
deformation $m\neq 0$) destroy O(3) invariance of the target space and
introduce a dependence on the orientation of $\vec{S}$ with respect to the
third axis in the isospace, see Fig. 1.

Figure 1: The orientation of $\vec{S}$-vector embedding in a three-Euclidean
space. The contours from inside to outside (i.e. red to blue) correspond to
$k$ equal to 1.0, 9.5, 200, and 1000, respectively.

Perturbation theory in the model at hand have been studied in [1, 2] in the
framework of the so called first-order formalism related to an operator
product expansion (OPE), see the list of references in [1, 2]. The following
phenomenon have been observed there.

At one loop OPE for certain chiral currents have the form

$J_{a}(z)J_{b}(0)=\frac{1}{z}f^{c}_{ab}J_{c}(0)$ (1.8)

where $f^{c}_{ab}$ are the $sl(2)$ algebra structure constants. The above
expression fully reveals the Lie-algebraic structure of (1.4). However, a
straightforward calculation of the second loop produces a term

$\frac{1}{z^{2}}\partial_{i}v_{a}^{j}\partial_{j}v_{b}^{i}$ (1.9)

where $v_{a}^{j}$ are Killing vectors. The above structure is obviously non-
geometric. However, one can show that with a proper regularization within the
first-order formalism the partial derivatives in (1.9) are replaced by
covariant derivatives, and the required “geometricity” is recovered. In Ref.
[2] the regularization method was based on supersymmetry despite the fact that
the calculated $\beta$ function was that of the non-supersymmetric model
(1.3). In [1] the following $H$ hypothesis was formulated: In (non-
supersymmetric) Kählerian sigma models, an anomaly is present in the
calculation of the second and higher loops. In [2] the validity of the $H$
hypothesis was verified in the second loop.

In this paper we reveal an infrared anomaly in the second $\beta$-function
coefficient. We use the standard perturbation theory and standard two-loop
Feynman graphs. Our derivation is based on ${\cal N}=(2,2)$ supersymmetry;
however, the strategy is different from that in [2]. Our analysis has close
parallels with the holonomy anomaly in ${\cal N}=1$ supersymmetric Yang-Mills
(SYM) theory [6] and two-dimemsional sigma models [7, 8]. Just like in the
latter case it is likely that the anomalous effect detected in non-
supersymmetric Kählerian sigma models can be reformulated as a subtlety in the
measure in the corresponding path integral [9].

Our work consist of several parts. First, we supersymmetrize the model (1.3),
(1.4), presenting both ${\cal N}=(2,2)$ and ${\cal N}=(0,2)$ versions. Section
2.1.3 is devoted to the study of the hypercurrents in ${\cal N}=(2,2)$ and
${\cal N}=(0,2)$ versions of the model. The standard expressions known in the
literature have to be modified to take into account the non-symmetric nature
of the target space. Then, in Secs. 2.2 and 2.3 we introduce the twisted mass
and the $\theta$ angle.

In Sec. 3 we calculate the two-loop $\beta$ function coefficient in the non-
supersymmetric sigma model by virtue of a supersymmetric regularization. Our
calculation is transparent and demonstrates the role of the infrared
contribution. Section 4 is devoted to an interesting aspect of reducing the
two-dimensional model under consideration to a Lie-algebraic quantum-
mechanical model [10, 11] presenting the so-called Lamé problem [12]. Under
certain values of quantized free parameters it becomes quasi-exactly solvable,
and, moreover, exhibits duality in the non-supersymmetric case.

##  2 Extending the deformed $\mathbb{CP}^{1}$ model

In this section, we elaborate on supersymmetric extensions of the deformed
$\mathbb{CP}^{1}$ model, incorporating twisted masses and a topological term
in a consistent manner. Applications of the present construction to the
analysis of $\beta$ functions and the reduced quantum mechanical model can be
found in Sec. 3 and 4, respectively.

###  2.1  ${\mathcal{N}}=(2,2)$ and ${\mathcal{N}}=(0,2)$ supersymmetrization

####  2.1.1  ${\mathcal{N}}=(2,2)$

We start with a brief review of the general construction of two-dimensional
${\mathcal{N}}=(2,2)$ and ${\mathcal{N}}=(0,2)$ sigma models. The target space
of the model under consideration is a one (complex) dimensional manifold; it
is Kählerian and admits ${\mathcal{N}}=(2,2)$ structure [13]. Since the basics
of the ${\mathcal{N}}=(2,2)$ model can be found in standard textbooks, we just
quote the results and remind the relevant geometric data. Suppose the target
space is parametrized by the complex coordinates $\varphi$ and
$\overline{\varphi}$. By promoting the scalar field to the corresponding
superfields, $\Phi,\Phi^{\dagger}$, and integrating out the Grassmann
coordinates, one finds the component formulation (see for example [14, 15]),
namely,

$\displaystyle\mathcal{L}_{(2,2)}=G\left[\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}{}+i\overline{\psi}\cancel{\partial}\psi+i\overline{\psi}\gamma^{\mu}(\Gamma\partial_{\mu}\varphi)\psi\right]-\frac{1}{2}R_{1\overline{1}1\overline{1}}\left(\overline{\psi}{}\psi\right)^{2}$
(2.1)

where $\psi$ is a Dirac fermion,

$\displaystyle\psi=\matrixquantity(\psi_{R}\\\
\psi_{L})\,,\quad\overline{\psi}=\psi^{\dagger}\gamma^{0}\,,$ (2.2)

and

$\Gamma\equiv\Gamma^{1}_{11}$ (2.3)

is the Christoffel symbol. The essential geometric data (in addition to Eq.
(1.4)) are

$\displaystyle\Gamma=-\frac{\overline{\varphi}{}(n_{2}+2n_{3}\absolutevalue{\varphi}^{2})}{n_{1}+n_{2}\absolutevalue{\varphi}^{2}+n_{3}\absolutevalue{\varphi}^{4}};$
(2.4a) $\displaystyle
R_{1\overline{1}1\overline{1}}=-\frac{1}{2}G^{2}\mathcal{R}=-\frac{n_{1}n_{2}+4n_{1}n_{3}\absolutevalue{\varphi}^{2}+n_{2}n_{3}\absolutevalue{\varphi}^{4}}{(n_{1}+n_{2}\absolutevalue{\varphi}^{2}+n_{3}\absolutevalue{\varphi}^{4})^{3}}$
(2.4b)

in which $\mathcal{R}$ is the scalar curvature,

${\cal R}=2G^{\overline{n}m}R_{m\overline{n}}\,,\quad
R_{m\overline{n}}=-G^{\overline{j}i}R_{i\overline{j}m\overline{n}}\,.$ (2.5)

####  2.1.2  ${\mathcal{N}}=(0,2)$

As for the ${\mathcal{N}}=(0,2)$ formulation, we limit our discussion to the
so-called minimal model [7]. Following the same lines as in [16, 8, 17], we
introduce an ${\mathcal{N}}=(0,2)$ chiral superfield $A$ which, in terms of
component fields, takes the form

$\displaystyle
A(x,\theta,\theta^{\dagger})=\varphi(x)+\sqrt{2}\theta\psi_{L}(x)+i\theta^{\dagger}\theta\partial_{L}\varphi(x)$
(2.6)

where $\partial_{L}=\partial_{t}+\partial_{z}$ and $\theta,\theta^{\dagger}$
are the Grassmann coordinates. The $(0,2)$ supersymmetric transformation of
$A$ is

$\displaystyle\delta_{\epsilon,\epsilon^{\dagger}}A=\partial_{L}\varphi\cdot
2i\epsilon^{\dagger}\theta+\sqrt{2}\epsilon\psi_{L}\,.$ (2.7)

The Lagrangian can be written as

$\displaystyle\mathcal{L}_{(0,2)}$
$\displaystyle=\frac{1}{4}\int\differential^{2}{\theta}\left[K_{1}(A,A^{\dagger})i\partial_{R}A+\mbox{h.c.}\right]$
$\displaystyle=G\left[\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}+i\overline{\psi}_{L}\partial_{R}\psi_{L}+i\overline{\psi}_{L}(\Gamma\partial_{R}\varphi)\psi_{L}\right].$
(2.8)

Note that $K_{1}$ is the first derivative of the Kähler potential,222The
explicit form of the Kähler potential $K$ of the deformed $\mathbb{CP}^{1}$
model is given in [3].

$\displaystyle
K_{1}\equiv\partialderivative{K}{A}=\frac{2}{A\sqrt{n_{2}^{2}-4n_{1}n_{3}}}\operatorname{arctanh}\left(\frac{AA^{\dagger}\sqrt{n_{2}^{2}-4n_{1}n_{3}}}{2n_{1}+n_{2}AA^{\dagger}}\right).$
(2.9)

By construction, Eq. (2.1.2) is ${\mathcal{N}}=(0,2)$ invariant. In contrast
to the undeformed model, there is no nonlinear transformation of $A$
corresponding to the global rotations other than the $U(1)$ which can be
straightforwardly seen in the above formulation.

As a side remark, we note that the ${\mathcal{N}}=(2,2)$ supersymmetry can be
recovered from Eq. (2.1.2) by introduced another ${\mathcal{N}}=(0,2)$
superfield $B$,

$\displaystyle B(x,\theta,\theta^{\dagger})$
$\displaystyle=\psi_{R}(x)+\sqrt{2}\theta
F(x)+i\theta^{\dagger}\theta\partial_{L}\psi_{R}(x)$ (2.10)

obeying the transformation

$\displaystyle\delta_{\epsilon,\epsilon^{\dagger}}B=\partial_{L}\psi_{R}\cdot
2i\epsilon^{\dagger}\theta+\sqrt{2}\epsilon F\,.$ (2.11)

The corresponding Lagrangian is

$\displaystyle\mathcal{L}_{B}$
$\displaystyle=\frac{1}{2}\int\differential^{2}{\theta}\left[G(A,A^{\dagger})B^{\dagger}B\right]$
$\displaystyle=G\left[i\overline{\psi}_{R}\partial_{L}\psi_{R}+i\overline{\psi}_{R}(\Gamma\partial_{L}\varphi)\psi_{R}\right]-\frac{1}{2}R_{1\overline{1}1\overline{1}}(\overline{\psi}\psi)^{2}$
(2.12)

where $G(A,A^{\dagger})$ is the metric obtained by promoting
$\varphi,\overline{\varphi}$ to $A,A^{\dagger}$, respectively, in (2.4a). Note
that we have integrated out the auxiliary $F$ field. One can then see that the
combination of (2.1.2) and (2.1.2) leads to Eq. (2.1). The enhancement of the
supersymmetry from ${\mathcal{N}}=(0,2)$ to ${\mathcal{N}}=(2,2)$ was first
demonstrated in [18] for the undeformed $\mathbb{CP}^{1}$ case.

####  2.1.3 Hypercurrent multiplet

In the following, we analyze the hypercurrent multiplet $\mathcal{J}_{\mu}$
(see [5, 19, 20, 21] for review and examples) of the deformed
$\mathbb{CP}^{1}$ model. Our discussion on the case of ${\mathcal{N}}=(2,2)$
will run parallel to that of [5]. This supermultiplet contains a $R$-current
$v_{\mu}$, a supercurrent $s_{\mu\alpha}$, and the energy-momentum tensor
$\vartheta_{\mu\nu}$,

$\displaystyle\mathcal{J}_{\mu}=v_{\mu}+\left[\theta\gamma^{0}s_{\mu}+\mbox{H.c.}\right]-2\overline{\theta}\gamma^{\nu}\theta\,\vartheta_{\mu\nu}+\cdots\,$
(2.13)

where the $\gamma$ matrices are defined as

$\displaystyle\gamma^{0}=\sigma_{2}\,,\quad\gamma^{1}=i\sigma_{1}\,,\quad\gamma_{5}=\sigma_{3}\,.$
(2.14)

Here the Grassmannian coordinate has two complex components
$\theta=(\theta^{1},\theta^{2})$ in contrast to the case of
${\mathcal{N}}=(0,2)$ which has only one relevant Grassmannian coordinate. The
lowest component $v_{\mu}$ in the hypercurrent is the vector U(1) current.
Although classically $v_{\mu}$ is algebraically related to the axial current
discussed in Sec. 2.3) at the quantum level they are different – the axial
current has an anomaly.333In fact, there are three independently conserved
U(1) currents in this model. The first is the axial current presented in Eq.
(2.35). It is generated by the transformation (2.36), conserved at the
classical level and acquires a one-loop anomaly in
$\partial_{\mu}J_{5}^{\mu}$. The second conserved current $J^{\varphi}_{\mu}$
is purely bosonic, it is generated by the transformation $\varphi\to
e^{i\beta}\varphi$ and $\overline{\varphi}\to e^{-i\beta}\overline{\varphi}$.
Needless to say, it is anomaly-free. The third is purely fermionic vector
current, also anomaly-free. In the spinorial notation, it takes the form

$\displaystyle\mathcal{J}_{\alpha\beta}=\left(\gamma^{0}\gamma^{\mu}\right)_{\alpha\beta}\mathcal{J}_{\mu}=G\overline{D}_{\alpha}\overline{\Phi}D_{\beta}\Phi$
(2.15)

where $D_{\alpha}$ and $\overline{D}_{\beta}$ are superderivatives and $\Phi$
and $\overline{\Phi}$ are the chiral superfields with the lowest components
$\varphi$ and $\overline{\varphi}{}$, respectively. At the classical level,
the spinorial components $\mathcal{J}_{11}$ and $\mathcal{J}_{22}$ are
conserved, namely,

$\displaystyle\left[\overline{D}_{2}\mathcal{J}_{11}\right]_{\mbox{\scriptsize
classical}}=\left[\overline{D}_{1}\mathcal{J}_{22}\right]_{\mbox{\scriptsize
classical}}=0\,.$ (2.16)

In $\mathcal{J}_{\mu}$, only two diagonal components of
$\mathcal{J}_{\alpha\beta}$ are relevant.

Quantum mechanically, the hypercurrent expressions in (2.16) are anomalous The
anomaly is exhausted by the one-loop effect. For $\mathbb{CP}^{1}$ the anomaly
equations were derived in [5]. In our deformed model the anomaly in the right-
hand side takes the form

$\displaystyle\overline{D}_{2}\mathcal{J}_{11}=\frac{1}{4\pi}\overline{D}_{1}\left[\frac{1}{2}G\mathcal{R}\overline{D}_{2}\overline{\Phi}D_{1}\Phi\right]\,,\quad\overline{D}_{1}\mathcal{J}_{22}=\frac{1}{4\pi}\overline{D}_{2}\left[\frac{1}{2}G\mathcal{R}\overline{D}_{1}\overline{\Phi}D_{2}\Phi\right]\,.$
(2.17)

Note the emergence of the scalar curvature $\mathcal{R}$ on the right-hand
side.

The coefficient in (2.17) can be verified through the scale anomaly of the
energy-momentum tensor i.e. the anomaly in $\gamma_{\mu}s^{\mu}_{\alpha}$ [5,
22],

$\displaystyle\left(\vartheta^{\mu}_{\mu}\right)_{\rm anom}$
$\displaystyle=\frac{1}{4\pi}G\mathcal{R}\left(\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}+i\overline{\psi}\gamma^{\mu}\nabla_{\mu}\psi\right),$
(2.18a) $\displaystyle\left(\gamma_{\mu}s^{\mu}\right)_{\rm anom}$
$\displaystyle=\frac{1}{4\pi}G\mathcal{R}(\partial_{\mu}\overline{\varphi})\gamma^{\lambda}\psi\,.$
(2.18b)

Note that the hypermultiplet $\mathcal{J}_{\mu}$ falls in the class of the
$R_{V}$-multiplets in [21] since $\partial_{\mu}\mathcal{J}^{\mu}=0$. In fact,
Eq. (2.17) can be recast in the standard form of the hypercurrent multiplet
proposed in [20, 21]. Namely,444For a general ${\cal N}=(2,2)$ $\sigma$ model,
the anomalies of the hypercurrent take the form
$\displaystyle\chi_{\beta}=\overline{D}_{\beta}\left(-\frac{1}{8\pi}D^{\alpha}\overline{D}_{\alpha}\log\det
G_{i\overline{j}}\right)\,.$ (2.19) This issue was previously discussed in
[20] for the case of the symmetric Kähler manifolds. Since in our deformed
$\mathbb{CP}^{1}$ model the target space is non-symmetric, the expression in
the right-hand side of (2.19) is somewhat different from that in [20].

$\displaystyle\overline{D}^{\alpha}\mathcal{J}_{\beta\alpha}=\chi_{\beta}\mbox{\quad
with\quad}\chi_{\beta}=\overline{D}_{\beta}\left(-\frac{1}{4\pi}D^{\alpha}\overline{D}_{\alpha}\log
G\right)$ (2.20)

for which we use the fact that only the twisted chiral (antichiral) part of
$D^{\alpha}\overline{D}_{\alpha}$ contributes in the first (second) equations
in (2.17).

*****

The hypercurrent for the $\mathbb{CP}^{1}$ ${\cal N}=(0,2)$ model was
discussed in [7, 8]. Taking into account our Lie-algebraic extension we arrive
at the classical expressions

$\displaystyle\mathcal{J}_{2}$
$\displaystyle=\frac{1}{2}\evaluated{\mathcal{J}_{22}}_{\theta^{1}=0}=\frac{1}{2}G\overline{D}A^{\dagger}DA\,,$
(2.21a) $\displaystyle\widetilde{\mathcal{T}}_{1111}$
$\displaystyle=-\frac{1}{2}[\overline{D}_{1},D_{1}]\evaluated{\mathcal{J}_{11}}_{\theta^{1}=0}=G\partial_{R}A^{\dagger}\partial_{R}A\,,$
(2.21b)

where $\mathcal{J}_{2}$ and $\widetilde{\mathcal{T}}_{1111}$ stand for two
components in the hypercurrent in the ${\cal N}=(0,2)$ model, and $A$ is the
${\mathcal{N}}=(0,2)$ superfield defined in Sect. 2.1.2. In the
${\mathcal{N}}=(0,2)$ superspace, the reduced superderivatives are

$\displaystyle
D=\partialderivative{\theta}-i\theta^{\dagger}\partial_{L}\,,\qquad\overline{D}=-\partialderivative{\theta^{\dagger}}+i\theta\partial_{L}$
(2.22)

Here the lowest component of $\mathcal{J}_{2}$ is the chiral $U(1)$ current
$G\psi^{\dagger}_{L}\psi_{L}$ and is _not_ conserved as the quantum
corrections are taken into account. Also, the bosonic component of
$\widetilde{\mathcal{T}}_{1111}$ is the part of the energy-momoentum tensor,
$T_{1111}$.

In general, like in the ${\mathcal{N}}=(2,2)$ case, the hypercurrent (2.21) is
conserved classically and becomes anomalous due to one-loop corrections. In
other words, the general anomaly equations turn out to be

$\displaystyle\begin{aligned}
&\partial_{R}\mathcal{J}_{2}=-\frac{1}{2}D_{2}X+\frac{1}{2}\overline{D}_{2}\overline{X}\,,\qquad\overline{D}_{2}\widetilde{\mathcal{T}}_{1111}=\partial_{R}X\,,\\\\[5.69054pt]
&X\equiv-\frac{1}{8\pi}G\mathcal{R}(\partial_{R}A)\overline{D}A^{\dagger}\end{aligned}$
(2.23)

where $X$ encodes the anomalous part of two real supermultiplets
$\mathcal{J}_{2}$ and $\widetilde{\mathcal{T}}_{1111}$. Note that the
coefficient of Eq. (2.23) can be fixed by the anomalous chiral $U(1)$ current
$G\psi^{\dagger}_{L}\psi_{L}$ in parallel to the consideration of the axial
$U(1)$ current in Appendix B,

$\displaystyle\partial_{R}(G\psi^{\dagger}_{L}\psi_{L})=2\cdot\left(\frac{i}{8\pi}G\mathcal{R}\epsilon^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\overline{\varphi}\right)$
(2.24)

where the prefactor $2$ indicates the number of the fermion zero modes in the
instanton background, which is half of the number in ${\mathcal{N}}=(2,2)$
theory, see also (2.37).

Important warning: in the ${\mathcal{N}}=(0,2)$ case, the $\beta$ function is
not exhausted by one loop, see Sect. 3. Therefore, the one-loop anomaly
expression given in (2.23) should be understood as an operator expressions
subject to further infrared multi-loop corrections, just in the same way as in
${\mathcal{N}}=1$ super-Yang-Mills (see [6] and Secs. 10.16.1-10.16.4 in
[23]). In our problem, the latter conjecture is not yet proven.

###  2.2 Adding twisted masses

As observed in [4, 24, 25, 26], one can introduce the twisted mass parameter
consistent with the underlying supersymmetries in the presence of the $U(1)$
isometry of the system. That is, the infinitesimal transformations read

$\displaystyle\delta\varphi=it_{1}\varphi\,,\quad\delta\overline{\varphi}=-it_{1}\overline{\varphi}$
(2.25)

where $t_{1}$ is the variable parametrizing the isometry. Notice that for the
deformed $\mathbb{CP}^{1}$ model, such an isometry can be summarized by the
Killing potential $D(\varphi,\overline{\varphi})$,

$\displaystyle D(\varphi,\overline{\varphi})$
$\displaystyle=\frac{1}{2\sqrt{k^{2}-1}}\log(\frac{\absolutevalue{\varphi}^{2}+k-\sqrt{k^{2}-1}}{\absolutevalue{\varphi}^{2}+k+\sqrt{k^{2}-1}})$
(2.26)

generating the Killing vectors, namely,

$\displaystyle\derivative{\varphi}{t_{1}}=-iG^{-1}\partialderivative{D}{\overline{\varphi}{}}\,,\quad\derivative{\overline{\varphi}{}}{t_{1}}=-iG^{-1}\partialderivative{D}{\varphi}\,.$
(2.27)

Here the Killing potential is defined up to an additive constant.

One can then introduce a constant auxiliary vector multiplet $V$ parametrized
by the twisted masses, $m$ and $\overline{m}$, to modify the $U(1)$ invariant
combination $\absolutevalue{\Phi}^{2}$ in the associated Kähler potential as
$\Phi^{\dagger}e^{V}\Phi$. In the following, we directly quote the resulting
Lagrangian. The interested readers can find a concise review in Sec. 2 of [5].
The deformed $\mathbb{CP}^{1}$ model with the twisted masses is formulated as
follows 555Generally speaking, for a sigma model with a complex target space
with the Killing vectors $X^{i},\overline{X}^{\overline{j}}$ for the $U(1)$
isometries the associated Lagrangian takes the form
$\displaystyle\mathcal{L}_{m}=G_{i\overline{j}}\left[\partial_{\mu}\varphi^{i}\partial^{\mu}\overline{\varphi}^{\overline{j}}-\absolutevalue{m}^{2}X^{i}\overline{X}^{\overline{j}}+i\overline{\psi}^{\overline{j}}\cancel{\nabla}\psi^{i}-i(D_{k}X^{i})\overline{\psi}^{\overline{j}}\psi^{k}\right]-\frac{1}{2}R_{i\overline{j}k\overline{l}}\overline{\psi}^{\overline{j}}\psi^{i}\overline{\psi}^{\overline{l}}\psi^{k}$
where $D_{k}X^{i}=\partial_{k}X^{i}+\Gamma^{i}_{kj}X^{j}$ is the covariant
derivative on the target space.

$\displaystyle\mathcal{L}_{m}$
$\displaystyle=G\left[\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}-\absolutevalue{m}^{2}\varphi\overline{\varphi}{}+i\overline{\psi}\cancel{\nabla}\psi-\left(1+\Gamma\varphi\right)\overline{\psi}\widetilde{\mu}\psi\right]-\frac{1}{2}R_{1\overline{1}1\overline{1}}\left(\overline{\psi}\psi\right)^{2}$
(2.28)

where $\nabla_{\mu}$ is the covariant derivative

$\displaystyle\nabla_{\mu}\psi=\partial_{\mu}\psi+(\Gamma\partial_{\mu}\varphi)\psi$
(2.29)

and

$\displaystyle\widetilde{\mu}\equiv
m\frac{1-\gamma_{5}}{2}+\overline{m}\frac{1+\gamma_{5}}{2}\,.$ (2.30)

Similarly, one can introduce the twisted mass for the ${\mathcal{N}}=(0,2)$
model by replacing the aforementioned constant auxiliary ${\mathcal{N}}=(2,2)$
vector multiplet with a ${\mathcal{N}}=(0,2)$ one. As a result, the
${\mathcal{N}}=(0,2)$ model with twisted mass takes the form

$\displaystyle\mathcal{L}_{m,(0,2)}=G\left[\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}-m^{2}\varphi\overline{\varphi}{}+i\overline{\psi}_{L}\nabla_{R}\psi_{L}-m\left(1+\Gamma\varphi\right)\overline{\psi}_{L}\psi_{L}\right]\,$
(2.31)

where $m$ is real in the ${\mathcal{N}}=(0,2)$ case.

###  2.3 The $\theta$ term

The $\theta$ term can be added in a straightforward manner [5, 22],

$\displaystyle\mathcal{L}_{\theta}$
$\displaystyle=\frac{i\theta}{8\pi}G\mathcal{R}\differential{\varphi}\wedge\differential{\overline{\varphi}}\,.$
(2.32)

Note that the theta term is topological and invariant under $2\pi\mathbb{Z}$
translation since the topological charge defined as

$\displaystyle Q\equiv\frac{1}{8\pi}\int
G\mathcal{R}\differential^{2}{\varphi}\in\mathbb{Z}\,.$ (2.33)

In particular, $\absolutevalue{Q}=1$ for the (anti-)instanton solution. For
completeness, the theta term can also be expressed as a total derivative

$\displaystyle
R_{1\overline{1}}\differential{\varphi}\wedge\differential{\overline{\varphi}}=\differential\left(-\frac{2n_{1}+n_{2}\absolutevalue{\varphi}^{2}}{n_{1}+n_{2}\absolutevalue{\varphi}^{2}+n_{3}\absolutevalue{\varphi}^{4}}\cdot\differential{\log{\varphi}}\right)\,.$
(2.34)

This also indicates that (2.33) saturates at the small field configuration.

The theta term (2.32) can be further utilized to detect the number of fermion
zero modes and then the number of the bosonic zero modes via supersymmetry. To
this end, let us consider the divergence of the axial $U(1)$ current

$\displaystyle J_{5}^{\mu}\equiv G\overline{\psi}\gamma^{\mu}\gamma_{5}\psi$
(2.35)

generated by the $U(1)_{A}$ transformation

$\displaystyle\psi\to
e^{i\alpha\gamma_{5}}\,\psi\,,\quad\overline{\psi}\to\overline{\psi}\,e^{i\alpha\gamma_{5}}$
(2.36)

where $\alpha$ is the variable parametrizing the transformation. Classically,
$j_{5}^{\mu}$ is conserved, but it becomes anomalous as the quantum effects
are taken into account. Namely, in ${\mathcal{N}}=(2,2)$,

$\displaystyle\partial_{\mu}J_{5}^{\mu}=4\cdot\left(-\frac{i}{8\pi}G\mathcal{R}\epsilon^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\overline{\varphi}\right)$
(2.37)

implying the number of the fermion zero modes is four and therefore the same
for the bosonic sector through supersymmetry. See Appendix B for further
detail. Comparing with Eq. (2.32), one sees that $\theta\to\theta-4\alpha$
under the axial $U(1)_{A}$ rotation (2.36), which breaks $U(1)_{A}$ into
$\mathbb{Z}_{4}$.

Furthermore, it is worth pointing out how the instanton action is related to
the topological charge defined in (2.33). According to the standard BPS
argument, the action satisfies

$\displaystyle S$ $\displaystyle=\int
G\left[\frac{1}{2}\absolutevalue{\partial_{\mu}\varphi\pm\epsilon_{\mu\nu}\partial^{\nu}\varphi}^{2}\mp\epsilon_{\mu\nu}\partial^{\mu}\varphi\partial^{\nu}\overline{\varphi}{}\right]\differential^{2}{x}$
$\displaystyle\geq\frac{2\pi\log\left[\frac{n_{2}}{2n_{1}n_{3}}\left(n_{2}+\sqrt{n_{2}^{2}-4n_{1}n_{3}}\right)-1\right]}{\sqrt{n_{2}^{2}-4n_{1}n_{3}}}\cdot\absolutevalue{Q}$
(2.38)

where $Q$ is the topological charge. The action saturates the BPS bound for
the instanton configuration. In the non-degenerate case (i.e.
$n_{1},n_{3}\neq\infty,0$), the overall coefficient in front of $|Q|$ in (2.3)
is

$\displaystyle\frac{4\pi\operatorname{arccosh}{k}}{g^{2}\sqrt{k^{2}-1}}\mbox{\quad
for\quad}k\geq 1$ (2.39)

as was previously derived in [3]. The $\mathbb{CP}^{1}$ expression can be
obtained by further taking the $k\to 1$ limit.

##  3 Analysis of the two-loop beta function

###  3.1 The two-loop beta function of the bosonic model from supersymmetry

We will start from the universal fact that the second (and all higher)
coefficients of two-dimensional ${\cal N}=(2,2)$ sigma models vanish [27, 28],
see also [29, 30] for recent discussions.666 In the mid-1980s this fact was
questioned by Grisaru, van de Ven and Zanon [31] who analyzed the $\beta$
functions in two-dimensional Kähler $\sigma$ models up to four loops. In the
case of the Ricci-flat manifolds they arrived at the conclusion that there is
a non-vanishing contribution to the $\beta$ function cubic in the Riemann
curvature of the target space at the fourth loop. This result is in the direct
contradiction with those reported in [32, 33]. The class of models we study is
not Ricci-flat. Morever, in Sec. 3.3 we will argue that the result [31] cannot
be applied to the model under discussion. Our argument is based on the Dorey’s
exact solution [26] for the mass spectrum in $\mathbb{CP}^{N-1}$ à la Seiberg-
Witten type [34, 35].  In supersymmetric sigma models the two-loop
contributions to the $\beta$ function can be separated into the (purely)
bosonic $\beta_{2,b}$ and fermionic $\beta_{2,f}$ parts. In other words,

$\displaystyle\beta^{(2)}=\beta_{b}^{(2)}+\beta_{f}^{(2)}\overset{!}{=}0$
(3.1)

which implies, in turn, that the purely bosonic component can be extracted
from the fermionic part (which is much more amenable for loop calculations),

$\displaystyle\beta_{b}^{(2)}=-\beta_{f}^{(2)}\,.$ (3.2)

Note that $\beta_{b}^{(2)}$ is identical to the beta function of the non-
supersymmetric case.

To confirm the previous assertion, let us apply the background field method.
It suffices to consider the two-loop fermionic diagram in Fig. 2 – the only
non-trivial diagram with the requred logarithmic divergence.

Figure 2: The two-loop fermion diagram in a background field calculation.

Here $\partial\varphi,\partial\overline{\varphi}$ are the external legs, and
$q$, $\psi$, and their complex conjugates are the quantum scalar field and
fermions, respectively. To proceed, we consider the background expansion via
the Kähler coordinates [36, 37]. The only relevant interactions in Fig. 2 are
the 3-vertices, namely,

$\displaystyle
iR_{1\overline{1}1\overline{1}}\left(\overline{q}\partial_{\mu}\varphi-q\partial_{\mu}\overline{\varphi}\right)\left(\overline{\psi}{}\gamma^{\mu}\psi\right)$
(3.3)

where
$R_{1\overline{1}1\overline{1}}=R_{1\overline{1}1\overline{1}}(\varphi,\overline{\varphi})$.
Therefore, the two-loop fermion correction to the Lagrangian (2.1) is

$\displaystyle i\Delta S_{2,f}$
$\displaystyle=\int\differential^{2}{x}\differential^{2}{y}\,(R_{1\overline{1}1\overline{1}})^{2}\partial_{\mu}\varphi\partial_{\nu}\overline{\varphi}{}\expectationvalue{\left(\overline{q}\overline{\psi}{}\gamma^{\mu}\psi\right)_{x}\left(q\overline{\psi}{}\gamma^{\mu}\psi\right)_{y}}$
$\displaystyle=-\frac{i}{8\pi^{2}\epsilon}\int\differential^{2}{x}(R_{1\overline{1}1\overline{1}})^{2}G^{-3}\partial_{\mu}\varphi\partial^{\nu}\overline{\varphi}{}+\cdots$
(3.4)

where the ellipses stand for non-logarithmic divergences and $\epsilon=2-D$ in
dimensional regularization. In accordance with the renormalization group
equation (see e.g. [38, 27]), we then have

$\displaystyle\beta_{f}^{(2)}=2\cdot\left(-\frac{1}{8\pi^{2}}\right)(R_{1\overline{1}1\overline{1}})^{2}G^{-3}=-\frac{1}{16\pi^{2}}G\mathcal{R}^{2}\,.$
(3.5)

Note that to get the second equality, the first equation in (2.4b) is used.

Now, invoking (3.2) we arrive at

$\displaystyle\beta_{b}^{(2)}=-\beta_{f}^{(2)}=\frac{1}{16\pi^{2}}G\mathcal{R}^{2}\,$
(3.6)

which matches with the general formula in the purely bosonic model [38],

$\displaystyle\beta_{b}^{(2)}=-\frac{1}{4\pi^{2}}R_{1\overline{\mu}\nu\overline{\lambda}}R_{\overline{1}}^{~{}\overline{\mu}\nu\overline{\lambda}}=\frac{1}{16\pi^{2}}G\mathcal{R}^{2}\,.$
(3.7)

In the present case, all the Greek indices $\mu,\nu,\lambda$ are 1. As a
consistent check, taking the limit $k=1$ in (1.5) we recover the
$\mathbb{CP}^{1}$ result 777This calculation for $\beta_{b}^{(2)}$ in
$\mathbb{CP}^{1}$ was first presented in [23], page 265, see also subsection
3.1.1.

$\displaystyle\beta^{(2)}\left(\frac{2}{g^{2}}\right)=\frac{g^{2}}{2\pi^{2}}\implies\beta^{(2)}(g^{2})=-\frac{g^{6}}{4\pi^{2}}\,.$
(3.8)

As explained in detail in [23] on pages 674-676, the calculation of the graph
in Fig. 2 becomes transparent if we first deal with the fermion loop keeping
fixed the momentum flowing through the dashed (bosonic) line. As is clear from
Eq. (3.3) the fermion loop is exactly the same as in the two-dimensional
Schwinger model. It has no logarithms and is saturated in the infrared.
Including the bosonic loop provides us with the first power of $\log\mu$. This
is exactly what is expected in the two-loop graph for the $beta$ function. The
Schwinger “anomaly” is crucial.

####  3.1.1 Verification around the origin

If we limit ourselves to the vicinity of the origin in the target space and
forget for a short while about the target space invariance, the proof of our
assertion can be greatly simplified. Indeed, because the overall structure of
the field dependence is constrained by the target space geometry of the
deformed $\mathbb{CP}^{1}$ model, we can accept that the second coefficient of
the beta function in the bosonic model takes the form

$\displaystyle\beta^{(2)}=c_{2}G\mathcal{R}^{2}\,.$ (3.9)

Plugging the explicit expression for the geometric data given in Eq. (2.4),
one sees that

$\displaystyle\beta^{(2)}\to c_{2}\cdot\frac{4n_{2}^{2}}{n_{1}}$ (3.10)

for $\varphi,\overline{\varphi}\approx 0$. In the same approximation, the
leading terms in the Lagrangian are

$\displaystyle\mathcal{L}_{(2,2)}$
$\displaystyle=\frac{1}{n_{1}}\left(\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}+i\overline{\psi}\cancel{\partial}\psi\right)-i\left(\frac{n_{2}}{n_{1}^{2}}\right)\overline{\varphi}\partial_{\mu}\varphi\left(\overline{\psi}\gamma^{\mu}\psi\right)+\cdots$
(3.11)

Then, considering the same two-loop diagram in Fig. 2, we obtain the two-loop
Lagrangian

$\displaystyle\Delta\mathcal{L}$
$\displaystyle=\left[-2\cdot\left(\frac{n_{2}}{n_{1}^{2}}\right)^{2}T\left\\{\left(\overline{\varphi}\overline{\psi}\gamma^{\mu}\psi\right),\left(\varphi\overline{\psi}\gamma^{\nu}\psi\right)\right\\}\right]\partial_{\mu}\varphi\partial_{\nu}\overline{\varphi}$
$\displaystyle=-\left(\frac{2n_{2}^{2}}{n_{1}}\right)\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}\cdot\frac{1}{8\pi^{2}}\log\frac{M}{\mu}$
(3.12)

where $M$ and $\mu$ are the ultraviolet and infrared cutoffs, respectively.
The constant $c_{2}$ turns out to be

$\displaystyle c_{2}=-\frac{1}{16\pi^{2}}$ (3.13)

consistent with the covariant derivation given in (3.5).

####  3.1.2 The two-loop beta function of ${\mathcal{N}}=(0,2)$ extension

Based on the result of Eq. (3.5), the second coefficient of the beta function
for minimal ${\mathcal{N}}=(0,2)$ sigma models with a one-complex-dimensional
target space can be readily identified. To see that this is the case, note
that the fermion sector in ${\mathcal{N}}=(2,2)$ models consists of two Weyl
fermions (one left-handed fermion and one right-handed) while in
${\mathcal{N}}=(0,2)$ models, there exists only one Weyl fermion. This
indicates that the contribution from fermions at the two-loop level is half of
Eq. (3.5) in the ${\mathcal{N}}=(0,2)$ case. Consequently, combining with the
bosonic contribution, we obtain the second coefficient of the beta function
that

$\displaystyle\beta^{(2)}_{(0,2)}=\left(1-\frac{1}{2}\right)\cdot\frac{1}{16\pi^{2}}G\mathcal{R}^{2}=\frac{1}{32\pi^{2}}G\mathcal{R}^{2}\,.$
(3.14)

Going through the same process around (3.8), one would see the second
coefficient of the ${\mathcal{N}}=(0,2)$ $\mathbb{CP}^{1}$ model

$\displaystyle\beta^{(2)}_{(0,2)}(\mathbb{CP}^{1})=-\frac{g^{6}}{8\pi^{2}}$
(3.15)

which was first derived in [7] through the superfield calculation.

###  3.2 Comparison with the first-order formalism

Our results (3.6) for the bosonic model coincides with that obtained in [2] by
virtue of the first-order formalism. The regularization procedure used in [2]
was as follows. We start from the ${\mathcal{N}}=(2,2)$ theory. In first-order
formalism it is obvious that all loops in $\beta$ beyond the first loop vanish
– there is no anomaly and holomorphy is preserved. Then we endow the fermion
field with a mass term $m_{f}$ and compute the $\beta$ function coefficients
with large but fixed value $m_{f}$. In the limit $m_{f}\to\infty$ we discover
that some “extra” terms do not vanish. It is just these extra terms which are
responsible for the transition from
$\partial_{i}v_{a}^{j}\partial_{j}v_{b}^{i}$ in (1.9) to
$\nabla_{i}v_{a}^{j}\nabla_{j}v_{b}^{i}$ where $\nabla_{\ell}$ stands for the
covariant derivative. This procedure can be viewed as an ultraviolet
derivation of the anomaly.

###  3.3 Comment on the literature

We started Sec. 3.1 from the statement that “the second (and all higher)
coefficients of two-dimensional ${\cal N}=(2,2)$ sigma models vanish.” In this
subsection we will discuss this statement in more detail. A series of papers
on this subject was published in [32, 33, 39] in the early 1980s. Then this
issue was revisited 1985-86, approximately simultaneously with the publication
[31].

The authors of [31] state the opposite – that the $\beta$ functions in two-
dimensional Ricci flat Kähler $\sigma$ models have a non-vanishing
contribution at four loops cubic in the Riemann curvature of the target space,
see their Eq. (5.16).

The class of Lie-algebraic models we are interested in (it includes, in
particular $\mathbb{CP}^{N-1}$ models) is not not Ricci-flat. In 37 years that
elapsed since the publication of Grisaru et. al. a significant progress
happened in understanding both perturbation theory and exact solutions in
$\mathbb{CP}^{N-1}$ models (and their extensions) and, in “parallel” to them,
exact solutions in Yang-Mills theories with various degrees of supersymmetry.

If supersymmetry is minimal, the $\beta$ functions are indeed multi-loop, but
are exactly calculable. For ${\cal N}=2$ supersymmetry the perturbative
$\beta$ functions are exhausted by the first loop. This is seen from the
analysis of the holomorphy properties with regards to the complexified
coupling constant

$1/g^{2}_{\rm holom}=1/g^{2}+i\theta/(8\pi^{2})$

in super-Yang-Mills and

$1/g^{2}_{\rm holom}=1/g^{2}+i\theta/(4\pi)$

in 2D $\mathbb{CP}^{N-1}$ models (see below). Moreover, this statement is
confirmed by the exact solutions.

The exact solution for the mass spectrum of the Seiberg-Witten ${\cal N}=2$
super-Yang-Mills [34, 35] (say, for SU(2)${}_{\rm gauge})$, parametrized by a
single modular invariant $u$, being expanded in the ratio $u/\Lambda^{2}$
exhibits the first order in $\log(u/\Lambda)$ plus all powers of

$(u/\Lambda)^{4n},\quad n=1,2,3,4,...$ (3.16)

The power terms of the expansion (3.16) do not contain logarithms and come
from instantons (this series can be – and in fact, has been – obtained by
using the Nekrasov localization [40]).

Next, in 1998 Dorey published a paper [26] in which he obtained the exact
solution à la Seiberg-Witten for the ${\cal N}=(2,2)$ $\mathbb{CP}^{N-1}$
models with twisted masses (in $\mathbb{CP}^{1}$ there is only one twisted
mass parameter). Dorey’s method repeats the Seiberg-Witten’s analysis [34, 35]
in ${\cal N}=2$ Yang-Mills step by step.

If one replaces the twisted mass of $\mathbb{CP}^{N-1}$ model by the modular
parameters $u_{i}$ of the Seiberg-Witten derivation in Yang-Mills, then the
formula for the spectrum in Yang-Mills in four dimensions is exactly the same
as the Dorey’s formulae in $\mathbb{CP}^{N-1}$ in two dimensions (see Eqs.
(112) in the general case and (117) for a particular case of $\mathbb{CP}^{1}$
in [26].)

Dorey’s observation [26] can be summarized as follows:

The mass spectrum on the Coulomb branch ($\xi=0$ where $\xi$ is the Fayet-
Iliopoulos term) of the Seiberg-Witten theory, with unconfined ‘t Hooft-
Polyakov-like monopoles and dyons coincides with that of $\mathbb{CP}^{N-1}$
models emerging on the vortex string [41, 42] in the Higgs phase (i.e.
$\xi\neq 0$). In [41] it was proved that the central charges cannot depend on
the non-holomorphic parameter $\xi$ in the BPS sector. This established a one-
to-one correspondence between the mass spectra of the two seemingly different
theories. They prove to be identical in the BPS sectors, hence, $2D$-$4D$
correspondence.

Dorey’s formula for the BPS mass spectrum depends on the ratio $m/\Lambda$,
where $m$ is the twisted mass, and $\Lambda$ is the scale parameter of the
theory, obtained through the dimensional transmutation (as in any
asymptotically free theory). Let us assume that this parameter is large and
expand the Dorey’s exact solution in the ratio $m/\Lambda$. In parallel with
${\cal N}=2$ Yang-Mills the Dorey expansion contains the first order in
$\log(m/\Lambda)$ plus powers of

$(m/\Lambda)^{2nN},\quad n=1,2,3,4,...$

where $N$ comes from $\mathbb{CP}^{N-1}$, with nothing else.

We emphasize, that the perturbative term is $\log(m/\Lambda)$ is unique. There
are no terms $\log^{2}$ or $\log\log$, etc. in the expansion of the exact
formula. Since the masses are physically observable, their expression must be
consistent with the $\beta$ function. This can only happen if the perturbative
$\beta$ function is purely one-loop in the class of models under
consideration.

Just for completeness, let us mention that in minimal supersymmetries (such as
${\cal N}=1$ Yang-Mills or ${\cal N}=(0,2)$ $\mathbb{CP}^{1}$) the $\beta$
functions contain all loops. However, if it were not for holomorphic anomaly
[43, 44], all coefficients, starting from the two-loop coefficient would
vanish – only the one-loop coefficient would survive. The break down of
holomorphy is an infrared effect [43, 44], which is well understood. This is
best illustrated by the instanton formula (IR is automatically regularized in
the instantion background). Its general form valid both for super-Yang-Mills
and $\mathbb{CP}^{N-1}$ (without matter fields) is as follows,

$\beta(\alpha)=-\left(n_{b}-\frac{n_{f}}{2}\right)\,\frac{\alpha^{2}}{2\pi}\left[1-\frac{\left(n_{b}-n_{f}\right)\,\alpha}{4\pi}\right]^{-1}\,,$
(3.17)

where $n_{b}$ and $n_{f}$ are the numbers of the bosonic and fermionic zero
modes, respectively. Above, $\alpha=g^{2}/(4\pi)$ in super-Yang-Mills and
$\alpha=g^{2}/2$ in $\mathbb{CP}^{N-1}$.

All coefficients in the $\beta$ function (3.17) are integers and, moreover, of
a purely geometric nature. They are in one-to-one correspondence with the
number of symmetries nontrivially realized on the BPST or BP instanton.
Equation (3.17) is valid for ${\cal N}=2$ and $4$ ($n_{b}=n_{f}$ and
$n_{b}=\frac{1}{2}n_{f}$, respectively).

For the minimal supersymmetriy (${\cal N}=1$ in $4D$ Yang-Mills) it stays
valid too and presents an all-loop $\beta$ function in the form of the
geometric progression [45, 46, 47, 48]. Minimal supersymmetry in the class of
$\sigma$ models is ${\cal N}=(0,2)$. Only $\mathbb{CP}^{1}$ and its
generalizations can be considered in this class since $\mathbb{CP}^{N-1}$ with
$N\geq 3$ do not allow the minimal ${\cal N}=(0,2)$ super-extension because of
the geometric anomalies (see [8, 49, 50] and references therein). In Ref. [3]
it is demonstrated in detail that the term in the square brackets in Eq.
(3.17) remains intact in the Lie-algebraic deformation of $\mathbb{CP}^{1}$
(see Eq. (49) in [3]).

##  4 Reduction to quantum mechanics

In this section, we explore quantum mechanics (QM) associated with the
deformed $\mathbb{CP}^{1}$ model, derived through compactification along the
spatial dimension. Under a particular scheme of compactifications, the
resulting quantum mechanical system is the Lamé QM problem which is Lie-
algebraic and quasi-exactly solvable. It can also be viewed as the
interpolation between two solvable quantum mechanics, the sine-Gordon and the
Pöschl-Teller systems [51] 888The Pöschl-Teller case, i.e. $k\to\infty$, is
_not_ periodic.. For additional insights regarding the connection to other
integrable and Lie-algebraic models, interested readers are referred to
earlier discussions from the 90s [52, 53, 54].

For the time being, let us consider only the bosonic version of the deformed
$\mathbb{CP}^{1}$ model with non-singular parameters, i.e. $n_{1},n_{3}$ are
neither zero nor infinity. Eq. (1.3) can be recast via the field redefinition
[3] such that the Lagrangian reads

$\displaystyle\mathcal{L}_{b}=\frac{2}{g^{2}_{2d}}\frac{\partial_{\mu}\varphi\,\partial^{\mu}\overline{\varphi}}{1+2k\absolutevalue{\varphi}^{2}+\absolutevalue{\varphi}^{4}}$
(4.1)

where the parameters $n_{i}$ are

$\displaystyle n_{1}=n_{3}=\frac{g^{2}_{2d}}{2}\mbox{\quad
and\quad}n_{2}=g^{2}_{2d}k$ (4.2)

in which $k\in[1,\infty)$. Then to understand the connection between the
deformed $\mathbb{CP}^{1}$ model and the Lamé equation, one can take the
following reparametrization of our main model. Namely,

$\displaystyle\varphi(t,z)=\frac{\sqrt{1-\kappa}\operatorname{sd}(\theta(t,z)|\kappa)}{1+\operatorname{cd}(\theta(t,z)|\kappa)}e^{i\alpha(t,z)}\,,\quad\overline{\varphi}(t,z)=\frac{\sqrt{1-\kappa}\operatorname{sd}(\theta(t,z)|\kappa)}{1+\operatorname{cd}(\theta(t,z)|\kappa)}e^{-i\alpha(t,z)}$
(4.3)

where $\operatorname{sd}(\theta|\kappa)$ and
$\operatorname{cd}(\theta|\kappa)$ are two kinds of Jacobi elliptic function
and $\alpha$ is the azimuthal angle. The parameter $\theta$ is defined on
$[0,2K(\kappa))$ where $K(\kappa)$ is the complete elliptic integral of the
first kind and the other parameter $\kappa$ is associated with the original
elongation factor $k$ in the way

$\displaystyle\kappa\equiv\frac{k-1}{k+1}\in[0,1)\,.$ (4.4)

The conventions and further properties of the Jacobi elliptic functions and
their integrals used in this paper are summarized in Appendix A. Plugging
(4.3) into the Lagrangian (4.1), we can write down the bosonic Lagrangian in
terms of $\theta$ and $\alpha$,

$\displaystyle\mathcal{L}_{b}=\frac{2}{g^{2}_{2d}(1+k)}\Bigl{[}\partial_{\mu}\theta\,\partial^{\mu}\theta+\operatorname{sn}^{2}(\theta|\kappa)\partial_{\mu}\alpha\,\partial^{\mu}\alpha\Bigr{]}\,.$
(4.5)

As shown in (4.5), we already observe some signals of the emergence of Lamé
potential as the coefficient of the kinetic term of $\alpha$.

Next, we can apply the Scherk-Schwarz dimensional reduction [55] such that the
underlying spacetime is $\mathbb{R}\times S^{1}_{L}$ where $L$ is the
circumference of the compactified circle and the spacetime dependences of
$\theta$ and $\alpha$ fields are restricted

$\displaystyle\theta(t,z)=\theta(t)\,,\quad\alpha(t,z)=\alpha_{0}-\alpha_{1}z\,,$
(4.6)

where $\alpha_{0}$ and $\alpha_{1}$ are real time-independent constants and
the latter one is constraint by the boundary condition along $S^{1}_{L}$. For
example, the periodic boundary condition on $\varphi,\overline{\varphi}$

$\displaystyle\alpha(t,z+L)=\alpha_{0}-\alpha_{1}(z+L)\overset{!}{=}\alpha_{0}-\alpha_{1}+2\pi
n\,,\quad n\in\mathbb{Z}$ (4.7)

implies that

$\displaystyle\alpha_{1}=\frac{2\pi n}{L}\,.$ (4.8)

Similar arguments can be applied to the anti-periodic boundary condition and
the twisted boundary conditions. Note that only a subset of the field
configurations in the deformed $\mathbb{CP}^{1}$ model aligns with the
condition specified in Eq. (4.6). For instance, mutli-fractional instanton
solutions are not compatible with the Scherk-Schwarz scenario while composites
with pairs of fractional and anti-fractional instantons satisfies the
assumption.999 This observation can be straightforwardly deduced from the
findings related to the $\mathbb{CP}^{1}$ model [56]. As demonstrated in [3],
the instanton equation for the deformed $\mathbb{CP}^{1}$ model is identical
to that of the original $\mathbb{CP}^{1}$ model, leading to equivalent
(fractional/anti-fractional) instanton solutions. Consequently, the
discussions pertinent to the $\mathbb{CP}^{1}$ model are equally applicable to
the deformed variant.

This phenomenon was initially identified in the comparison of the
$\mathbb{CP}^{1}$ model with the sine-Gordon model [56], demonstrating that
not all field configurations from the two-dimensional model are preserved
under the Scherk-Schwarz reduction. A more detailed comparison of deformed
$\mathbb{CP}^{1}$ quantum mechanics and Lamé quantum mechanics is provided in
Appendix C.

To proceed with the dimensional reduction, we then insert (4.6) into the two-
dimensional Lagrangian and integrate over the $z$-direction, which leads to

$\displaystyle\mathcal{L}_{1}=\frac{2L}{g^{2}_{2d}(1+k)}\left[\left(\derivative{\theta}{t}\right)^{2}-\alpha_{1}^{2}\operatorname{sn}^{2}(\theta|\kappa)\right]\,.$
(4.9)

For what follows it is convenient to denote the one-dimensnional coupling
constant

$\displaystyle\frac{1}{g^{2}}\equiv\frac{2L}{g^{2}_{2d}(1+k)}$ (4.10)

and rescale the time variable $t\to 2t/g^{2}$. The system of Eq. (4.9) can be
quantized adhering to the standard lore of the quantum mechanics, wherein the
time-independent Schrödinger equation is expressed as

$\displaystyle\frac{1}{g^{2}}\left[-\derivative[2]{\theta}+\alpha_{1}^{2}\operatorname{sn}^{2}(\theta|\kappa)\right]\Phi(\theta)=E\Phi(\theta)$
(4.11)

in which $\Phi(\theta)$ is the corresponding wave-function as a function of
the compact coordinate $\theta$. Eq. (4.11) is recognized in the literature as
the Lamé model [10, 12, 57, 58].

#### Two limits of the Lamé equation

In the ordinary construction [1, 2, 3], we have seen that the two-dimensional
deformed $\mathbb{CP}^{1}$ model is a (Lie-algebraic) generalization of the
classic $\mathbb{CP}^{1}$ model. This can also be realized in its one-
dimensional reduction (4.11). The figure 3 shows the transition of the
potentials from the sine-Gordon model through to the Lamé one and finally to
the Pöschl-Teller system.

Figure 3: The demonstration of the potential of three quantum mechanical
systems, sine-Gordon, Lamé, and Pöschl-Teller one. The elliptic modulus of the
Lamé potential $\kappa$ is 0.7. The blue, orange, and green curves represent
the potentials of sine-Gordon, Lamé, and Pöschl-Teller systems, respectively.

Starting with the limit $\kappa$ approaching zero, one has
$\operatorname{sn}(\theta|\kappa)\to\sin{\theta}$ and the Hamiltonian in this
case

$\displaystyle
g^{2}H_{\kappa=0}=-\derivative[2]{\theta}+\alpha_{1}^{2}\sin^{2}{\theta}$
(4.12)

in which $\theta\in[0,\pi)$. This Hamiltonian is precisely the one of sine-
Gordon quantum mechanics whose potential is periodic.

On the other hand, the one-dimensional model is also non-trivial in the other
limit $\kappa$ reaching the unity. Before proceeding to the reduced quantum
mechanics, we briefly review some basic results of the deformed model in the
large $k$ limit. From the two-dimensional perspective, the deformed
$\mathbb{CP}^{1}$ model turns out to be the sausage/cigar model [59, 60] as
the elongation $k$ becomes large [3]. And in the exact limit $\kappa\to 1$, or
equivalently $k\to\infty$, the target space of the complex fields
$\varphi,\overline{\varphi}$ deforms to a cylinder [29] in the present limit.
Namely,

$\displaystyle\mathcal{L}\sim\frac{\partial_{\mu}\varphi\,\partial^{\mu}\overline{\varphi}}{\absolutevalue{\varphi}^{2}}~{}\xleftrightarrow{~{}~{}~{}\varphi=e^{u}~{}~{}~{}}~{}\partial_{\mu}u\,\partial^{\mu}\overline{u}\,.$
(4.13)

On the one dimensional side, the Schrödinger equation (4.11) can be written
as101010We have used the fact that ${\displaystyle\lim_{\kappa\to
1}\operatorname{sn}(\theta|\kappa)=\tanh{\theta}}$.

$\displaystyle\frac{1}{g^{2}}\left[-\derivative[2]{x}+\alpha_{1}^{2}(1-\sech^{2}{x})\right]\Phi(x)=E\Phi(x)$
(4.14)

in which $x$ now is defined on the half-real line $\mathbb{R}_{\geq 0}$.
Certainly, the (4.14) is reflection invariant, and we can extend the domain
from $x\in\mathbb{R}_{\geq 0}$ to $x\in\mathbb{R}$. Note that Eq. (4.14) is
recognized as the Pöschl-Teller system [61] and is also quasi-exactly solvable
for judiciously chosen coefficient $\alpha_{1}$ (cf. [10, 11].)

###  4.1 Comparison to Dunne-Shifman

As well-known in the literature, the Lamé model is Lie-algebraic and quasi-
exactly solvable [11, 10, 57, 58, 12]. In other words, the associated
Hamiltonian can be expressed as a matrix-valued function of a certain Lie
algebra in some representation and a subset of the spectrum can be solved
exactly. In [12], the Lamé system is studied in detail and shown that there
exists a duality between bands and gaps in the spectrum. In the following, we
detail the condition when our system is algebraic and its connection to the
boundary condition imposed in the Scherk-Schwarz reduction. Without loss of
generality, we may set $g^{2}$ and the circumference of the compactified
dimension $L$ to be the unity for convenience.

To this end, recall that in the original construction of the Lie-algebraic
sigma model [1, 2, 3], the differential representation of $sl(2)$-algebra in
the spin-$j$ representation is

$\displaystyle T^{+}=2j\eta-\eta^{2}\partial_{\eta}\,,\quad
T^{0}=-j\eta+\eta\partial_{\eta}\,,\quad T^{-}=\partial_{\eta}$ (4.15)

where $j$ is a semi-integer and

$\displaystyle\eta\equiv 1-\operatorname{sn}^{2}(\theta|\kappa)$ (4.16)

following [12]. Then (4.11) is recast in the form

$H=4\left[(-1+\kappa)T^{0}T^{-}+(-1+2\kappa)T^{+}T^{-}-\kappa
T^{+}T^{0}\right]\\\\[5.69054pt] +2\left[-(1+6j)\kappa
T^{+}-2(1+2j)(-1+2\kappa)T^{0}+(1+2j)(-1+\kappa)T^{-}\right]\\\\[5.69054pt]
-4j(1+2j)(-1+2\kappa)+\alpha_{1}^{2}+\eta\left[4j(4j+1)\kappa-\alpha_{1}^{2}\right]\,.$
(4.17)

For (4.17) to be Lie-algebraic, there should be no dependence on the variable
$\eta$ in the Hamiltonian. Hence, one requires that111111The dimension of
$\alpha_{1}$ can be recovered by taking $\alpha_{1}\to\alpha_{1}/g^{2}$.

$\displaystyle\alpha_{1}^{2}=4j(4j+1)\kappa$ (4.18)

Note that with the condition (4.18), the coefficient of the potential term in
(4.11) is a multiple of $\kappa$ satisfying the quasi-exactly solvable
condition [12, 11]. As discussed in the dimensional reduction process, the
constant $\alpha_{1}$ depends on the boundary condition. Unlike (4.8) obtained
under periodic boundary condition, the condition specified in Eq. (4.18)
necessitates the imposition of the twisted boundary condition

$\displaystyle\varphi(t,z+L)=e^{\pm i\sqrt{4j(4j+1)\kappa}}\varphi(t,z)\,.$
(4.19)

###  4.2 Generalizations

So far, we have provided a comprehensive discussion on the relation between
quasi-exactly solvable quantum mechanics and the Lamé quantum mechanics, as
derived from dimensional reduction through a specific scheme. In Sec. 2, we
also see some generalizations to the bosonic deformed $\mathbb{CP}^{1}$ model.
Let us examine how these additional elements in the extended deformed model
influence the corresponding quantum mechanics.

####  4.2.1 Reduction of the deformed $\mathbb{CP}^{1}$ model with twisted
masses

In the case of the deformed model with twisted mass (2.28), one has

$\displaystyle\mathcal{L}_{m,b}=G\Bigl{[}\partial_{\mu}\varphi\partial^{\mu}\overline{\varphi}-\absolutevalue{m}^{2}\varphi\overline{\varphi}\Bigr{]}$
(4.20)

in which the fermionic part is ignored for the time being. Then, taking the
same elliptic parametrization (4.3) and going through the same dimensnional
reduction process, one deduces the Hamiltonian with twisted mass

$\displaystyle
H_{m}=-\derivative[2]{\theta}+\left(\alpha_{1}^{2}+\absolutevalue{m}^{2}\right)\operatorname{sn}^{2}(\theta|\kappa)\,.$
(4.21)

By introducing an additional mass parameter, we can relax the twisted boundary
condition as specified in Eq. (4.19), and instead, define a quantization
condition for the mass parameter to satisfy the Lie-algebraic criterion. For
example, let us keep the periodic boundary condition (4.8) intact. Then the
quantization condition for the system to be QES is

$\displaystyle\absolutevalue*{m}^{2}=J(J+1)\kappa-\left(\frac{2\pi
n}{L}\right)^{2}$ (4.22)

for $J\in\mathbb{N}$ and $n\in\mathbb{Z}$. Similar treatments can be applied
to anti-periodic and other boundary conditions along the compactified
dimension.

####  4.2.2 Supersymmetric Lamé model

(a) $\alpha=2$

(b) $\alpha=6$

Figure 4: The potential $V_{-}(\theta)$ of the supersymmetric Lamé model
defined in (4.25). The blue and orange lines correspond to the potential with
$\kappa$ equal to $0.5$ and $0.8$, respectively. $V_{+}(\theta)$ has the
identical structure to $V_{-}(\theta)$, but is shifted by a half period.

Another direction to generalize the Lamé model is to supersymmetrize it.
Generally speaking, a supersymmetric quantum mechanics deformed by a potential
term takes the form (see for example [14])

$\displaystyle
H_{o}=-\derivative[2]{\theta}+(W^{\prime}(\theta))^{2}+W^{\prime\prime}(\theta)\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ (4.23)

where $W(\theta)$ is the superpotential and the matrix representation of
fermions is adopted. By projecting onto the subspaces of different fermion
numbers, the Hamiltonian (4.23) can be further simplified to an effective
bosonic system, namely,

$\displaystyle H=-\derivative[2]{\theta}+(W^{\prime}(\theta))^{2}\mp
W^{\prime\prime}(\theta)\,.$ (4.24)

In particular, in the supersymmetric Lamé quantum mechanics, we have

$\displaystyle
H=-\derivative[2]{\theta}+V_{\mp}(\theta)=-\derivative[2]{\theta}+\alpha^{2}\kappa\operatorname{sn}^{2}(\theta|\kappa)\mp\alpha\sqrt{\kappa}\operatorname{cn}(\theta|\kappa)\operatorname{dn}(\theta|\kappa)$
(4.25)

with the Schrödinger equation $H\Phi(\theta)=E\Phi(\theta)$. Note that the
superpotential121212Here the parameter $\alpha_{1}=\alpha\sqrt{\kappa}$
comparing to the previous case. $W(\theta)$ in our case is

$\displaystyle
W(\theta)=-\alpha\operatorname{arctanh}(\sqrt{\kappa}\operatorname{cd}(\theta|\kappa))\,.$
(4.26)

With the potential $V_{\pm}(\theta)$ in Eq. (4.25) at hand, the supersymmetry
is manifest, but this is not the case of Lie-algebraicity. The later part of
this section is devoted to this issue.

Before moving on to the discussion on the Lie-algebraic structure of the
supersymmetric Lamé QM, let us have a closer look at other interesting
features of this system. First, we note that (4.25) is compatible with the 2D
construction (2.1.2) and the dimensional reduction scheme. To see this is the
case, in addition to (4.6), one also needs the reduction of the fermions

$\displaystyle\psi(t,z)=\psi(t)e^{i(\alpha_{0}-\alpha_{1}z)}$ (4.27)

with the same boundary condition as the scalar fields. As a result, the
reduced one-dimensional Lagrangian is

$\displaystyle
g^{2}\widetilde{\mathcal{L}}_{(0,2)}=\frac{1}{2}\left[\dot{\theta}(t)^{2}-\alpha_{1}^{2}\operatorname{sn}^{2}(\theta|\kappa)\right]+i\overline{\chi}\dot{\chi}-\alpha_{1}\operatorname{cn}(\theta|\kappa)\operatorname{dn}(\theta|\kappa)\overline{\chi}\chi\,,$
(4.28)

where $\chi$ is the normalized fermion field via velbein. This Lagrangian is
obtained from (2.1.2) by substituting $\varphi$ and $\psi$ within the
compactification scheme (4.6) and (4.27) and integration over $z$. It implies
that the Hamiltonian indeed matches with the one proposed in (4.25) by taking
$\overline{\chi}$ and $\chi$ to be $\sigma^{+}$ and $\sigma^{-}$,
respectively, in the matrix representation. The ordering of the fermions is
fixed such that the original Hamiltonian is $\\{Q,\overline{Q}\\}/2$ where
$Q,\overline{Q}$ are supercharges.

Lastly, the potential of the SUSY quantum mechanical model is depicted in
figure 4. It is clear that the potential is periodic with period $4K(\kappa)$
due to it elliptic function nature. As the value of $\alpha$ increases, it is
observed that the period is halved compared to cases with smaller $\alpha$.
This is because the portion of the potential modified by the supersymmetric
effect (or, equivalently, the presence of Weyl fermions) is proportional to
$\alpha$, whereas the non-supersymmetric part of the potential is proportional
to $\alpha^{2}$ and has a period of $2K(\kappa)$. Furthermore, if one intend
to conduct analysis via the WKB perturbation, it is noteworthy that there
exist two distinct types of instanton events in the inverted potential: one
involves tunneling through a tall barrier, while the other occurs via a low
barrier.

#### Lie-algebraic features of supersymmetric Lamé Hamiltonian

To advance our understanding on the Lie-algebraic nature of the supersymmetric
Lamé model, let us analyze the potential terms in the Hamiltonian in detail.
In accordance with the argument in [58], the Hamiltonian (4.25), especially
the last term, does not match the general form of the quasi-exactly solvable
model with a double-periodic potential. However, this is not the end the
story. The system can be transformed into a Lie-algebraic form via an
appropriate coordinate transformation, though the quasi-exactly solvable
condition requires separate verification. The aforementioned assertion is
elaborated as follows. Consider the coordinate transformation

$\displaystyle\theta^{\prime}=i\left(\theta-K-iK^{\prime}\right)$ (4.29)

where

$\displaystyle K^{\prime}\equiv K(1-\kappa)=K(\kappa^{\prime})\,.$ (4.30)

Utilizing the identities of Jacobi elliptic functions, we rephrase the
corresponding Schrödinger equation of (4.25) in terms of dual variables
$\theta^{\prime},\kappa^{\prime}$

$\displaystyle\widetilde{H}\Phi=\left[-\derivative[2]{{\theta^{\prime}}}+\alpha^{2}\kappa^{\prime}\operatorname{sn}^{2}(\theta^{\prime}|\kappa^{\prime})\pm
i\alpha\kappa^{\prime}\operatorname{sn}(\theta^{\prime}|\kappa^{\prime})\operatorname{cn}(\theta^{\prime}|\kappa^{\prime})\right]\Phi=\left(\alpha^{2}-E\right)\Phi$
(4.31)

which fits into the Lie-algebraic form given in [58]. The Hamiltonian (4.31)
is not the in the canonical form of SUSY QM. However, we can use the identity
of the Jacobi elliptic functions presented in Appendix A such that the dual
superpotential is

$\displaystyle\widetilde{W}^{\prime}(\theta^{\prime})=i\operatorname{dn}(\theta^{\prime}|\kappa^{\prime})\,.$
(4.32)

Then, the Lie-algebraic feature of the system is realized by the similar
differential representation in (4.15), but with a different variable

$\displaystyle\xi\equiv\frac{\operatorname{sn}(\theta^{\prime}|\kappa^{\prime})}{\operatorname{cn}(\theta^{\prime}|\kappa^{\prime})}\,.$
(4.33)

Note that the Hamiltonian $\widetilde{H}$ is formulated in the general form

$\displaystyle\widetilde{H}=-\sum_{a,b=0,\pm}C_{ab}T^{a}T^{b}-\sum_{a=0,\pm}C_{a}T^{a}-d$
(4.34)

with

$\displaystyle\begin{aligned} C_{++}&=(1-\kappa^{\prime})\,,\quad
C_{00}=1+\kappa^{\prime}\,,\quad C_{--}=1\,,\quad C_{\pm 0}=C_{0\pm}=0\,,\quad
C_{\pm\mp}=0\\\\[5.69054pt]
d&=\frac{1}{4\kappa^{\prime}}\left[C_{-}^{2}-\left(C_{0}^{2}+2C_{+}C_{-}\right)+\frac{C_{+}^{2}}{1-\kappa^{\prime}}\right]-2j(j+1)\,.\end{aligned}$
(4.35)

The other coefficients $C_{\pm}$ and $C_{0}$ can be derived from the
consistency condition (D.1) given in [58]. The derivation is tedious, but
straightforward and we left further illustrations in Appendix D. The complete
set of solutions to (D.1) can be categorized into three groups, as summarized
in Table 1. Qualitative discussions on this set of solutions and some examples
of specific representations are provided as follows.

Table 1: A set of solutions to consistency equations of QES.
$\begin{array}[]{ccccc}\hline\cr\mbox{Group}&C_{+}&C_{-}&C_{0}&\alpha\rule{0.0pt}{11.19443pt}\rule[-3.87495pt]{0.0pt}{0.0pt}\\\
\hline\cr\hline\cr~{}~{}(\mbox{A1}){}{}&\mp i(1-\kappa^{\prime})&\pm
i&-2j\kappa^{\prime}&\mp\frac{1}{2}(1+4j)\rule{0.0pt}{11.19443pt}\rule[-3.87495pt]{0.0pt}{0.0pt}\\\\[5.69054pt]
~{}~{}(\mbox{B1}){}{}&\mp
i\left(\sqrt{1-\kappa^{\prime}}-1+\kappa^{\prime}\right)&\pm
i\left(-1+\sqrt{1-\kappa^{\prime}}\right)&-(2j+1)\kappa^{\prime}&\pm(2j+1)\rule{0.0pt}{11.19443pt}\rule[-3.87495pt]{0.0pt}{0.0pt}\\\\[5.69054pt]
~{}~{}(\mbox{B2}){}{}&\mp
i\left(\sqrt{1-\kappa^{\prime}}+1-\kappa^{\prime}\right)&\pm
i\left(1+\sqrt{1-\kappa^{\prime}}\right)&-(2j+1)\kappa^{\prime}&\mp(2j+1)\rule{0.0pt}{11.19443pt}\rule[-3.87495pt]{0.0pt}{0.0pt}\\\\[5.69054pt]
\hline\cr\end{array}$

To start with, note that the group (A1) in Table 1 with $j=0$ conincides with
the supersymmetric ground state. To see this is the case, for $j=0$, there are
no contributions from $T^{\pm}$ and $T^{0}$, but from the constant term, which
is $1/4$. On the other hand, we have from the right-handed side of (4.31) that

$\displaystyle\frac{1}{4}=\left[\left(\frac{1+4j}{2}\right)^{2}-E_{j}\right]_{j=0}$
(4.36)

implying the vanishing of the ground state energy.

Generally speaking, due to the existence of the solutions in Table 1 to
consistency equations, one would hence conclude that the supersymmetric is
quasi-exactly solvable. In other words, the tasks of solving the differential
equation for the eigenstates and eigenenergy are then translated to the
problem of solving eigenvalues and eigenvectors of $(2j+1)\times(2j+1)$
matrices of the spin-$j$ representations of $sl(2)$ algebra. The explicit
matrix form of the dual Hamiltonian can be found by inserting the solutions
given in table 1 to the general expression (4.34). However, this observation
does not hold universally across all values of $\kappa^{\prime}$ as some
eigenvalues emerge as complex for these specific $\kappa^{\prime}$ values. We
consider some numerical investigations on the eigenvalues of the dual
Hamiltonian for different sets of solutions in Table 1.

Here we give the examples of quasi-exact solvability of some lower spin
representations, for instance, $j=1/2$ and $1$. For the spin-$1/2$ sector, the
Hamiltonian takes the forms

$\displaystyle H_{j=1/2}^{(A1)}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{3}{4}(2+\kappa^{\prime})&\pm
i(1-\kappa^{\prime})\\\\[5.69054pt] \mp
i&\displaystyle\frac{1}{4}(6-\kappa^{\prime})\end{pmatrix}$ (4.37a)
$\displaystyle H_{j=1/2}^{(B1)}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{1}{4}(7+8\kappa^{\prime}-2\sqrt{1-\kappa^{\prime}})&\pm
i(1-\kappa^{\prime}-\sqrt{1-\kappa^{\prime}})\\\\[5.69054pt] \mp
i\left(1-\sqrt{1-\kappa^{\prime}}\right)&\displaystyle\frac{1}{4}(7-2\sqrt{1-\kappa^{\prime}})\end{pmatrix}$
(4.37b) $\displaystyle H_{j=1/2}^{(B2)}$
$\displaystyle=H_{j=1/2}^{(B1)}(\sqrt{1-\kappa^{\prime}}\to-\sqrt{1-\kappa^{\prime}})$
(4.37c)

with eigenvalues

$\displaystyle E_{j=1/2}^{(A1)}$
$\displaystyle=\frac{6+\kappa^{\prime}}{4}\pm\frac{2-\kappa^{\prime}}{2}$
(4.38a) $\displaystyle E_{j=1/2}^{(B1)}$
$\displaystyle=\frac{1}{4}\left(7+4\kappa^{\prime}-2\sqrt{1-\kappa^{\prime}}\mp\sqrt{\kappa^{\prime
2}+\sqrt{1-\kappa^{\prime}}(\kappa^{\prime}-2)+2(1-\kappa^{\prime})}\right)$
(4.38b) $\displaystyle E_{j=1/2}^{(B2)}$
$\displaystyle=E_{j=1/2}^{(B1)}(\sqrt{1-\kappa^{\prime}}\to-\sqrt{1-\kappa^{\prime}})$
(4.38c)

The energy of the group (A1) with $j=1/2$ is real for all $\kappa^{\prime}$.
Regarding case (B1) and (B2), it is observed that the elements in the square
roots of Eq. (4.38b) and (4.38c) remains non-negative for
$0<\kappa^{\prime}<1$. This signifies that the system is fully quasi-exactly
solvable across the entire interval for all three parametrization. In the
spin-1 sector, a slightly different observation is made as in the spin-$1/2$
scenario, where the associated energy within the group (B2) is real over a
specific interval of $\kappa^{\prime}$. Meanwhile, the energy for groups (A1)
and (B1) remains real across the entire range of $\kappa^{\prime}$. This is
most effectively illustrated by figure 5, given that the expressions for the
dual Hamiltonians are lengthy and their physical meaning is not immediately
clear from their precise formulations. The energy of the eigenstates of the
dual Hamiltonian for $j=1/2,1,...,5/2$ are depicted in figure 6.

(a) $E^{(A1)}_{j=1}$

(b) $E^{(B1)}_{j=1}$

Figure 5: Eigenenergy of group (A1) and (B1) in spin-1 representation.

Figure 6: Energy of eigenstates of dual Hamiltonian for $j=1/2,3/2,2,5/2$. The
case of $j=1$ was presented in figure 5.

To wrap up the current section, we comment on the supersymmetric Lamé model
with twisted masses. By employing the same steps as outlined earlier, one can
similarly derive the supersymmetric Lamé model. The configuration of the
potential terms remains nearly identical to that of the model without twisted
masses, with the primary distinction being in the associated coefficients.
Namely,

$\displaystyle\alpha^{2}\to\alpha^{2}+\absolutevalue{m}^{2}\,,\quad\alpha\to\alpha+m$
(4.39)

in which we assume $m$ is real for simplicity. Also, we can expect that this
system is Lie-algebraic and quasi-exactly solvable because the twisted mass as
an additional parameter increases the degree of freedom of (D.1).

##  5 Conclusions and outlooks

In our study, we explored various aspects related to the Lie-algebraic
deformation of the $\mathbb{CP}^{1}$ sigma model in two dimensions. We stated
from generalizing the original model endowing it with heterotic ${\cal
N}=(0,2)$ and extended ${\cal N}=(2,2)$ supersymmetries. Then, we identified
the hypercurrent anomaly in both scenarios. Notably, in the extended
supersymmetry case, the anomaly, while being one-loop, deviates from
conventional formulations found in the literature [20] due to the non-
symmetric nature of the target Kähler manifold. We further elucidated the
incorporation of twisted masses and the $\theta$ term and examined the BPS
equation for instantons, focusing on its connection to the topological charge.

Moreover, we established that the second-loop contribution to the $\beta$
function in the non-supersymmetric Lie-algebraic sigma model arises from an
infrared phenomenon. We used a supersymmetric regularization to substantiate
our findings. We suggest that this inference extends to higher loops, drawing
a parallel with similar behaviors observed in four-dimensional ${\cal N}=1$
super-Yang-Mills theory.

In the second half of the paper, we point out the relationship between the
Lie-algebraically deformed $\mathbb{CP}^{1}$ model and Lamé-type quantum
mechanics, achieved through the Scherk-Schwarz dimensional reduction
technique. This result is then further generalized to the case of
${\mathcal{N}}=(0,2)$ supersymmetry. Upon certain additional requirements, the
supersymmetric quantal problem obtained in this way proves to be quasi-exactly
solvable. Further research into the link between two-dimensional Lie-algebraic
models and quasi-exactly solvable quantum mechanics could focus on the
deformation of 2D sigma models via specific potentials and the generalization
of Lamé quantum mechanics, including the study of the associated Lamé
equation.

## Acknowledgments

M.S. is grateful to A. Turbiner and O. Gamayun for useful discussions. This
work is supported in part by DOE grant DE-SC0011842 and the Simons Foundation
Targeted Grant 920184 to the Fine Theoretical Physics Institute.

## Appendices

##  Appendix A Conventions on elliptic integrals and Jacobi elliptic function

In this appendix, we provide a summary of the definitions and several useful
identities of Jacobi elliptic functions that are used in the main text.
Further properties can be found in, for example, [62, 63]. First, the elliptic
integral of the first kind is defined as

$\displaystyle
F(\kappa|\phi)\equiv\int_{0}^{\phi}\frac{\differential{t}}{\sqrt{1-\kappa\sin^{2}{t}}}\,,$
(A.1)

where the complete elliptic integral $K(\kappa)=F(\kappa|\pi/2)$. To define
Jacobi elliptic functions, it suffices to consider the inverse of the
incomplete integral $F(\kappa|\phi)$

$\displaystyle\phi\equiv\operatorname*{a.m.}(F|\kappa)\,.$ (A.2)

Then the Jacobi elliptic functions are represented as

$\displaystyle\begin{aligned}
&\operatorname{sn}(F|\kappa)\equiv\sin{\phi}\,,\quad\operatorname{cn}(F|\kappa)\equiv\cos{\phi}\,,\quad\operatorname{dn}(F|\kappa)\equiv\sqrt{1-\kappa\sin^{2}{\phi}}\\\\[5.69054pt]
&\operatorname{sd}(F|\kappa)\equiv\frac{\sin{\phi}}{\sqrt{1-\kappa\sin^{2}{\phi}}}\,,\quad\operatorname{cd}(F|\kappa)\equiv\frac{\cos{\phi}}{\sqrt{1-\kappa\sin^{2}{\phi}}}\,.\end{aligned}$
(A.3)

with the elliptic modulus $\kappa\in[0,1)$ and $\kappa^{\prime}\equiv
1-\kappa$.

The following identities are used in discussion of the Lamé systems, in
particular, in Sec. 4 for the dual transformations,

$\displaystyle\begin{aligned}
\kappa\operatorname{sn}^{2}(\theta,\kappa)&=1-\kappa^{\prime}\operatorname{sn}^{2}(\theta^{\prime},\kappa^{\prime})\,,\\\\[5.69054pt]
\sqrt{\kappa}\operatorname{cn}(\theta,\kappa)\operatorname{dn}(\theta,\kappa)&=i\kappa^{\prime}\operatorname{sn}(\theta^{\prime},\kappa^{\prime})\operatorname{cn}(\theta^{\prime},\kappa^{\prime})\,.\end{aligned}$
(A.4)

##  Appendix B Derivation of anomalous axial $U(1)$ currrent

In this section, we outline the derivation of the connection between the
divergence of the anomalous axial current and the theta term of the deformed
$\mathbb{CP}^{1}$ model.

Parallel to the discussion in [64], one has that

$\displaystyle\partial_{\mu}\left(G\overline{\psi}{}\gamma^{\mu}\gamma_{5}\psi\right)$
$\displaystyle=\partial_{\mu}\left(\overline{\chi}\gamma^{\mu}\gamma_{5}\chi\right)=2i\Tr\gamma_{5}f\left(\frac{\cancel{D}^{2}}{\Lambda^{2}}\right)$
(B.1)

where $f(x)$ is the regularization function with cutoff $\Lambda$ such that

$\displaystyle f(0)=1\mbox{\quad and\quad}\lim_{x\to\infty}f(x)=0\,.$

Here we consider the vielbein to decompose the metric, say,

$\displaystyle\overline{\chi}=\sqrt{G}\overline{\psi}{}\,,\quad\chi=\sqrt{G}\psi\,,$
(B.2)

and the covariant derivative $D_{\mu}$ under this frame is

$\displaystyle
D_{\mu}\equiv\frac{1}{2}\left(\Gamma\partial_{\mu}\varphi-\overline{\Gamma}\partial_{\mu}\overline{\varphi}{}\right)\,.$

Consequently, the trace turns out to be

$\displaystyle\Tr\gamma_{5}f\left(\frac{\cancel{D}^{2}}{\Lambda^{2}}\right)$
$\displaystyle=\Lambda^{2}\tr\int\frac{\differential^{2}{k}}{(2\pi)^{2}}\gamma_{5}f\left(-k^{2}+\frac{2i(k\cdot
D)}{\Lambda}+\frac{D^{2}}{\Lambda^{2}}-\frac{1}{4\Lambda^{2}}[\gamma^{\mu},\gamma^{\nu}]R_{(2),\mu\nu}\right)$
$\displaystyle\to\frac{1}{2\pi}R_{1\overline{1}}\epsilon^{\mu\nu}\partial_{\mu}\overline{\varphi}{}\partial_{\nu}\varphi\,.$
(B.3)

To get the second line of (B), we take the limit $\Lambda\to\infty$ and employ
the commutation relation

$\displaystyle[D_{\mu},D_{\nu}]$ $\displaystyle=-R_{(2),\mu\nu}$

in which

$\displaystyle R_{(2),\mu\nu}\equiv
R_{1\overline{1}}\left(\partial_{\mu}\overline{\varphi}{}\partial_{\nu}\varphi-\partial_{\nu}\overline{\varphi}{}\partial_{\mu}\varphi\right)\,.$
(B.4)

Therefore, combining (B.1) and (B), we have

$\displaystyle\partial_{\mu}\left(G\overline{\psi}{}\gamma^{\mu}\gamma_{5}\psi\right)=-\frac{i}{\pi}R_{1\overline{1}}\epsilon^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\overline{\varphi}{}\,.$
(B.5)

Together with (2.5), we arrive at (2.37).

##  Appendix C Comparison between deformed $\mathbb{CP}^{1}$ quantum mechanics
and Lamé quantum mechanics

Here the distinction between different dimensional reduction scenarios are
detailed. In particular, we compare the resultant quantum mechanics from the
Kaluza–Klein (KK) and the Scherk-Schwarz reductions.

In the Kaluza–Klein framework, one assumes the spacetime dependence of the
field to be

$\displaystyle\varphi(t,z)=\sum_{n=0}^{\infty}\varphi_{(n)}(t)\exp(i\frac{2\pi
nz}{L})\,,\quad\overline{\varphi}(t,z)=\sum_{n=0}^{\infty}\overline{\varphi}_{(n)}(t)\exp(-i\frac{2\pi
nz}{L})\,.$ (C.1)

As substituting this into (1.3), integrating along the $z$-direction, and
keeping only the lowest mode, we have

$\displaystyle\mathcal{L}_{KK}=G\dot{\varphi}\dot{\overline{\varphi}}=\dot{\theta}^{2}+\operatorname{sn}^{2}(\theta|\kappa)\dot{\alpha}^{2}$
(C.2)

from which we can see that there will be some additional pieces in the
deformed $\mathbb{CP}^{1}$ Hamiltonian, comparing with the Lamé equation.
Indeed, according to Eq. (C.2), the Hamiltonian of the deformed
$\mathbb{CP}^{1}$ model is

$\displaystyle
H_{d\mathbb{CP}^{1}}=-\frac{1}{4}\Delta=-\frac{1}{4}\left[\derivative[2]{\theta}+\frac{1}{\operatorname{sn}(\theta|\kappa)}\derivative{\theta}+\frac{1}{\operatorname{sn}^{2}(\theta|\kappa)}\derivative[2]{\alpha}\right]$
(C.3)

where $\Delta$ is the Laplace operator. Note that (C.2) is a _two-dimensional_
quantum mechanical system while the Lamé model (4.11) is of one dimension
which depends only on $\theta$. Thus even we consider the zero-angular
momentum sector of the Hilbert space (i.e.
$\partialderivative*{\Phi}{\alpha}=0$), there is still an additional linear
contribution in $\theta$ in the deformed $\mathbb{CP}^{1}$ Hamiltonian than
the Lamé one. If one keeps a higher mode rather than the lowest one, it will
introduce an extra mass term, but can still do nothing with eliminating the
linear differential part in $\theta$. The further discussion on the deformed
$\mathbb{CP}^{1}$ quantum mechanics derived from the KK reduction can be found
in [65]. In fact, the similar issue between the $\mathbb{CP}^{1}$ model and
the sine-Gordon model was discussed in [66] from the perspective of resurgence
analysis.

##  Appendix D Details of Lie-algebraic features of supersymmetric Lamé
quantum mechanics

The sufficient and necessary condition for rendering the system quasi-exactly
solvable is that there exists non-trivial solutions $(C_{\pm},C_{0},\alpha)$
to the equations of consistency131313These consistent equations arise from the
change of variables from $\xi$ to $\theta^{\prime}$. given in [58], namely,

$\displaystyle\kappa^{\prime}j(j+1)-\frac{C_{0}}{2}(2j+1)+\frac{1}{4\kappa^{\prime}}\left[C_{0}^{2}-(C_{+}-C_{-})^{2}\right]=\alpha^{2}\kappa^{\prime}\,,$
(D.1a)
$\displaystyle\frac{1}{2\kappa^{\prime}}\left(C_{+}-C_{-}\right)\left[\kappa^{\prime}(2j+1)-C_{0}\right]=i\alpha\kappa^{\prime}\,,$
(D.1b)
$\displaystyle\frac{1}{2\kappa^{\prime}}\left[C_{+}-(1-\kappa^{\prime})C_{-}\right]\left[\kappa^{\prime}(2j+1)+C_{0}\right]=0\,,$
(D.1c)
$\displaystyle\kappa^{\prime}j(j+1)+\frac{C_{0}}{2}(2j+1)+\frac{1}{4\kappa^{\prime}}\left[C_{0}^{2}-\frac{(C_{+}-(1-\kappa^{\prime})C_{-})^{2}}{1-\kappa^{\prime}}\right]=0\,.$
(D.1d)

For group (A1), it corresponds to taking $C_{+}=(1-\kappa^{\prime})C_{-}$
while taking $C_{0}=\kappa^{\prime}(2j+1)$ is related to group (B1) and (B2).

As reducing to the (bosonic) Lamé case, we have the vanishing right-handed
side of (D.1b). The solution to this reduced case is
$C_{\pm}=0,C_{0}=-2jk,\alpha=2j(1+2j)$ as claimed in [12] since $j$ is a semi-
integer.

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