November, 2020

Higher Derivative Supersymmetric Nonlinear Sigma Models
on Hermitian Symmetric Spaces, and BPS States Therein

Muneto Nitta†aaanitta(at)phys-h.keio.ac.jp and Shin
Sasaki‡bbbshin-s(at)kitasato-u.ac.jp
† Department of Physics, and Research and Education Center for Natural
Sciences,
Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
‡ Department of Physics, Kitasato University
Sagamihara 252-0373, Japan

We formulate four-dimensional $\mathcal{N}=1$ supersymmetric nonlinear sigma
models on Hermitian symmetric spaces with higher derivative terms, free from
the auxiliary field problem and the Ostrogradski’s ghosts, as gauged linear
sigma models. We then study Bogomol’nyi-Prasad-Sommerfield equations
preserving 1/2 or 1/4 supersymmetries. We find that there are distinct
branches, that we call canonical ($F=0$) and non-canonical ($F\neq 0$)
branches, associated with solutions to auxiliary fields $F$ in chiral
multiplets. For the ${\mathbb{C}}P^{N}$ model, we obtain a supersymmetric
${\mathbb{C}}P^{N}$ Skyrme-Faddeev model in the canonical branch while in the
non-canonical branch the Lagrangian consists of solely the ${\mathbb{C}}P^{N}$
Skyrme-Faddeev term without a canonical kinetic term. These structures can be
extended to the Grassmann manifold $G_{M,N}=SU(M)/[SU(M-N)\times SU(N)\times
U(1)]$. For other Hermitian symmetric spaces such as the quadric surface
$Q^{N-2}=SO(N)/[SO(N-2)\times U(1)])$, we impose F-term (holomorphic)
constraints for embedding them into ${\mathbb{C}}P^{N-1}$ or Grassmann
manifold. We find that these constraints are consistent in the canonical
branch but yield additional constraints on the dynamical fields thus reducing
the target spaces in the non-canonical branch.

###### Contents

  1. 1 Introduction
  2. 2 Supersymmetric higher derivative terms in the chiral model
  3. 3 Nonlinear sigma models with D-term constraints
    1. 3.1 $\mathbb{C}P^{N-1}$ model
    2. 3.2 $G_{M,N}$ model
  4. 4 Nonlinear sigma models with D- and F-term constraints
    1. 4.1 Quadric surface $Q^{N-2}$
    2. 4.2 $SO(2N)/U(N)$ and $Sp(N)/U(N)$ models
    3. 4.3 $E_{6}/[SO(10)\times U(1)]$ and $E_{7}/[E_{6}\times U(1)]$ models
  5. 5 Bogomol’nyi-Prasad-Sommerfield states
    1. 5.1 BPS states in canonical branch
    2. 5.2 BPS states in non-canonical branch
  6. 6 Comment on fermionic interactions
  7. 7 Conclusion and discussions

##  1 Introduction

Nonlinear sigma models are typical examples of low-energy effective theories.
When a global symmetry $G$ is spontaneously broken down to its subgroup $H$,
there appear massless Nambu-Goldstone bosons dominant at low energy, and their
low energy dynamics can be described by nonlinear sigma model whose target
space is a coset space $G/H$ [1, 2]. If one wants to study physics at higher
energy, one needs higher derivative correction terms, typically appearing in
the chiral perturbation theory of QCD [3]. Other examples of higher derivative
terms can be found in various contexts such as the Skyrme model, low-energy
effective theories of superstring theory, and worldvolume theories of
topological solitons and branes such as Nambu-Goto and Dirac-Born-Infeld
actions.

In supersymmetric theories, higher derivative terms often bring us troubles.
It is known that constructing derivative terms in the form of
$\partial_{m}\Phi$ suffers from a technical issue called the auxiliary field
problem [4, 5]. This stems from the fact that the equation of motion for the
auxiliary field $F$ in the chiral superfield $\Phi$ ceases to be an algebraic
equation and becomes a kinematical one. As a consequence, it is hard to
integrate out the auxiliary field and the interactions of physical fields are
not apparent. Supersymmetric extensions of the Wess-Zumino-Witten term [6] and
the Skyrme-Faddeev model [7, 8] are such examples with the auxiliary field
problem, while low-energy Lagrangians for supersymmetric gauge theories in
Refs. [9, 10, 11] (see also Refs. [12, 13, 14]), and supersymmetric extensions
of Dirac-Born-Infeld action [15, 16] and K-field theories [17, 18] are free
from this problem. A broad class of supersymmetric derivative terms free from
the auxiliary field problem and the Ostrogradski’s ghost [19] was found as
four-dimensional $\mathcal{N}=1$ supersymmetric higher derivative chiral
models formulated in terms of superfields [20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32].111 Another possibility is to gauge them away [33] if there
are Ostrogradski’s higher derivative ghosts [34, 35].  The model consists of a
Kähler potential $K$ and a superpotential $W$ together with a $(2,2)$ Kähler
tensor $\Lambda$ that determines derivative corrections. It was applied to a
new mechanism of supersymmetry breaking in modulated vacua [36] (see also
Refs. [37]). The general formulation was extended for other superfields: the
most general ghost-free (and tachyon-free) higher derivative ${\cal N}=1$
supersymmetric Lagrangian for vector multiplets (1-form gauge fields) [38] and
3-form gauge field [39]. As for nonlinear sigma models of chiral superfields,
only known examples are the supersymmetric ${\mathbb{C}}P^{1}$ model (or baby
Skyrme model) [24, 25, 27, 40, 41, 42] and the supersymmetric Skyrme model
[30]. When one solves the auxiliary field equations of motion for the
auxiliary field $F$, it in general allows more than one solutions $F=0$ and
$F\neq 0$. The former is called the canonical branch and the latter the non-
canonical branch. The bosonic part of the Lagrangian of the canonical branch
consists of a canonical kinetic term and a four derivative term for the
${\mathbb{C}}P^{1}$ model [43]. This admits lump (sigma model instanton)
solutions identical to those in the ${\mathbb{C}}P^{1}$ model without a higher
derivative term [44], as Bogomol’nyi-Prasad-Sommerfield (BPS) states
preserving 1/2 supersymmetric charges among the original supersymmetry [27].
On the other hand, in the non-canonical branch, the bosonic part of the
Lagrangian consists of only the four derivative term of the Skyrme-Faddeev
type [43] (that is without fourth order time derivatives) without any
canonical kinetic term. This admits a compact baby Skyrmion [45] which is a
BPS state preserving 1/4 supersymmetry [27]. While the general Lagrangian in
superspace [20] can be defined on any Kähler target spaces, solving auxiliary
field equations consisting of more than one chiral multiplets is still an open
problem, except for a single matrix chiral superfield [30].

The purpose of this paper is to present higher derivative supersymmetric
nonlinear sigma models with a wider class of target spaces – Hermitian
symmetric spaces:

$\displaystyle\mathbb{C}P^{N-1}$ $\displaystyle=$ $\displaystyle
SU(N)/[SU(N-1)\times U(1)],$ $\displaystyle G_{M,N}$ $\displaystyle=$
$\displaystyle U(M)/[U(M-N)\times U(N)],$ $\displaystyle Q^{N-2}$
$\displaystyle=$ $\displaystyle SO(N)/[SO(N-2)\times U(1)],$ (1.1)
$\displaystyle SO(2N)/U(N),\quad\quad\quad Sp(N)/U(N),$ $\displaystyle
E_{6}/[SO(10)\times U(1)],\quad E_{7}/[E_{6}\times U(1)],$

for which we solve auxiliary field equations. In the case without higher
derivative terms, supersymmetric nonlinear sigma models on Hermitian symmetric
spaces can be constructed by imposing supersymmetric constraints on gauged
linear sigma models [27]. This formulation is found to help us to solve
auxiliary field equations even with higher derivative terms. We employ the
gauged linear sigma models with higher derivative terms for chiral multiplets
[29] and then take the strong gauge coupling (nonlinear sigma model) limit.
Solving auxiliary field equations yields the canonical and non-canonical
branches for the on-shell actions. For the ${\mathbb{C}}P^{N}$ model, we
obtain a supersymmetric extension of the ${\mathbb{C}}P^{N}$ Skyrme-Faddeev
model [47, 48, 49, 50] in the canonical branch, while the on-shell Lagrangian
in the non-canonical branch consists solely of the ${\mathbb{C}}P^{N}$ Skyrme-
Faddeev term (the term without fourth order time derivatives). We then study
BPS states in these models, and find that lumps (sigma model instantons)
identical to those without higher derivative terms remain 1/2 BPS states in
the canonical branch, while compact baby Skyrmions are 1/4 BPS states in the
non-canonical branch.

The organization of this paper is as follows. In the next section, we
introduce the supersymmetric higher derivative term in the gauged chiral
models. We show that a derivative term determined by a $(2,2)$ Kähler tensor
provides supersymmetric higher derivative terms free from the auxiliary field
problem. We will see that there are two on-shell branches associated with the
solutions to the auxiliary field in the chiral multiplets. In Section 3, we
study the sigma model limit of the gauged chiral models. We first focus on the
sigma models whose target spaces are the Hermitian symmetric spaces
$\mathbb{C}P^{N-1}$ and $G_{M,N}$. They are defined only by the D-term
constraints. We examine the limit in the two distinct branches and write down
the higher derivative terms in the nonlinear sigma models. In Section 4, we
discuss the nonlinear sigma models defined by the F-term constraints in
addition to D-term constraints, which include the target spaces $Q^{N-2}$,
$SO(2N)/U(N)$, $Sp(N)/U(N)$, $E_{6}/[SO(10)\times U(1)]$, $E_{7}/[E_{6}\times
U(1)]$. We will show that the F-term constraints give additional conditions on
the target spaces in the non-canonical branch thus reducing the target spaces,
while they do not in the canonical branch. This distinguishes the situation
from the $\mathbb{C}P^{N-1}$ and $G_{M,N}$ cases.

In Section 5, we discuss the BPS states in the nonlinear sigma models, and in
Section 6, we mention fermionic terms in the models. Section 7 is devoted to
conclusion and discussions. As for a superfield notation, we follow the Wess-
Bagger convention [51].

##  2 Supersymmetric higher derivative terms in the chiral model

In this section, we briefly introduce the supersymmetric higher derivative
term to the chiral model that are free from the auxiliary field problem. The
Lagrangian is given by [20, 29]

$\displaystyle\mathcal{L}=$ $\displaystyle\ \int\\!d^{4}\theta\
K(\Phi,\Phi^{\dagger})+\frac{1}{16}\int\\!d^{4}\theta\
\Lambda_{i\bar{j}k\bar{l},ab}{}^{cd}(\Phi,\Phi^{\dagger})D^{\alpha}\Phi^{ia}D_{\alpha}\Phi^{kb}\bar{D}_{\dot{\alpha}}\Phi^{\dagger\bar{j}}_{c}\bar{D}^{\dot{\alpha}}\Phi^{\dagger\bar{l}}_{d}$
$\displaystyle\ +\left(\int\\!d^{2}\theta\ W(\Phi)+\mathrm{h.c.}\right),$
(2.1)

where $K(\Phi,\Phi^{\dagger})$ is a Kähler potential, $W(\Phi)$ is a
superpotential and $\Lambda_{i\bar{j}k\bar{l}}(\Phi,\Phi^{\dagger})$ is a
$(2,2)$ Kähler tensor defined by the chiral superfields $\Phi,\Phi^{\dagger}$
and their derivatives $D_{\alpha}\Phi$,
$\bar{D}_{\dot{\alpha}}\Phi^{\dagger}$, $\partial_{m}\Phi$, and
$\partial_{m}\Phi^{\dagger}$. We assume that the chiral superfields
$\Phi^{ia}\ (i=1,\ldots,\mathrm{dim}\,G,a=1,\ldots,\mathrm{dim}\,G^{\prime})$
belong to the fundamental representations of the global and gauge symmetries
$G$ and $G^{\prime}$, respectively. The four-dimensional $\mathcal{N}=1$
chiral superfield is expanded, in the chiral basis
$y^{m}=x^{m}+i\theta\sigma^{m}\bar{\theta}$, as

$\displaystyle\Phi^{ia}=\varphi^{ia}(y)+\theta\psi^{ia}(y)+\theta^{2}F^{ia}(y).$
(2.2)

We stress that the pure bosonic component in the fourth derivative part

$\displaystyle
D^{\alpha}\Phi^{ia}D_{\alpha}\Phi^{kb}\bar{D}_{\dot{\alpha}}\Phi^{\dagger\bar{j}}_{c}\bar{D}^{\dot{\alpha}}\Phi^{\dagger\bar{l}}_{d}$
(2.3)

saturates the Grassmann coordinates, and only the bosonic fields in the Kähler
tensor $\Lambda_{i\bar{j}k\bar{l}}$ contribute to this pure bosonic term [20].
Indeed, the component expansion of the term (2.3) can be evaluated as

$\displaystyle
D^{\alpha}\Phi^{i\alpha}D_{\alpha}\Phi^{kb}\bar{D}_{\dot{\alpha}}\Phi^{\dagger\bar{j}}_{c}\bar{D}^{\dot{\alpha}}\Phi^{\dagger\bar{l}}_{d}=$
$\displaystyle\
16\theta^{2}\bar{\theta}^{2}\left[(\partial_{m}\varphi^{ia}\partial^{m}\varphi^{kb})(\partial_{n}\bar{\varphi}^{\bar{j}}_{c}\partial^{n}\bar{\varphi}^{\bar{l}}_{d})\right.$
$\displaystyle\
\left.-\frac{1}{2}\left(\partial_{m}\varphi^{ia}F^{kb}+F^{ia}\partial_{m}\varphi^{kb}\right)\left(\partial^{m}\bar{\varphi}^{\bar{j}}_{c}\bar{F}^{\bar{l}}_{d}+\bar{F}^{\bar{j}}_{c}\partial^{m}\bar{\varphi}^{\bar{l}}_{d}\right)+F^{ia}\bar{F}^{\bar{j}}_{c}F^{kb}\bar{F}^{\bar{l}}_{d}\right]$
$\displaystyle\ +\cdots.$ (2.4)

Here $\cdots$ are terms involving fermions. Due to this remarkable structure,
a large class of supersymmetric models with derivative corrections can be
realized by the Lagrangian (2.1). For example, the $\mathcal{N}=1$
supersymmetric Dirac-Born-Infeld model is given by the Kähler tensor (scalar)
[15, 16]

$\displaystyle\Lambda(\Phi,\Phi^{\dagger})=\frac{1}{1+A+\sqrt{(1+A)^{2}-B}},\qquad
A=\partial_{m}\Phi\partial^{m}\Phi^{\dagger},\qquad
B=\partial_{m}\Phi\partial^{m}\Phi\partial_{n}\Phi^{\dagger}\partial^{n}\Phi^{\dagger}.$
(2.5)

A supersymmetric Skyrme-Faddeev model is given by the Kähler metric
$K_{\varphi\bar{\varphi}}(1+|\varphi|^{2})^{-2}$ and the Kähler scalar [27]

$\displaystyle\Lambda(\Phi,\Phi^{\dagger})=(\partial_{m}\Phi\partial^{m}\Phi\partial_{n}\Phi^{\dagger}\partial^{n}\Phi^{\dagger})^{-1}\frac{1}{(1+\Phi\Phi^{\dagger})^{4}}\left[(\partial_{m}\Phi^{\dagger}\partial^{m}\Phi)^{2}-\partial_{m}\Phi\partial^{m}\Phi\partial_{n}\Phi^{\dagger}\partial^{n}\Phi^{\dagger}\right].$
(2.6)

The other examples include the Galileon inflation and the ghost condensation
models [21, 22] and a superconformal higher derivative nonlinear sigma model
[52].

Now we make the global symmetry $G^{\prime}$ be gauged by introducing the
$\mathcal{N}=1$ vector multiplet $V=V^{\hat{a}}T^{\hat{a}}\
(\hat{a}=0,1,\ldots,\mathrm{dim}\,\mathcal{G}^{\prime})$:

$\displaystyle
V=-(\theta\sigma^{m}\bar{\theta})A_{m}(x)+i\theta^{2}\bar{\theta}\bar{\lambda}(x)-i\bar{\theta}^{2}\theta\lambda(x)+\frac{1}{2}\theta^{2}\bar{\theta}^{2}D(x).$
(2.7)

Here we have employed the Wess-Zumino gauge and $T^{\hat{a}}$ are the
generators of $\mathcal{G}^{\prime}$ satisfying the normalization
$\mathrm{Tr}(T^{\hat{a}}T^{\hat{b}})=k\delta^{\hat{a}\hat{b}}$. In the
derivative corrections, the vector multiplet is introduced in the gauge
covariantization of the supercovariant derivative:

$\displaystyle
D_{\alpha}\Phi^{ia}\to\mathcal{D}_{\alpha}\Phi^{ia}=D_{\alpha}\Phi^{ia}+(\Gamma_{\alpha})^{a}{}_{b}\Phi^{ib}.$
(2.8)

The gauge superconnection $\Gamma_{\alpha}$ is defined by

$\displaystyle(\Gamma_{\alpha})^{a}{}_{b}=e^{-2gV}D_{\alpha}e^{2gV},$ (2.9)

where $g$ is the gauge coupling constant. The gauge invariant derivative term
is given by a $(2,2)$ Kähler tensor of the following structure:

$\displaystyle\Lambda_{i\bar{j}k\bar{l},ab}{}^{cd}=\Lambda_{i\bar{j}k\bar{l}}(\Phi,\Phi^{\dagger})(e^{2gV})^{c}{}_{a}(e^{2gV})^{d}{}_{b},$
(2.10)

where $\Lambda_{i\bar{j}k\bar{l}}$ is composed of gauge invariant quantities
made of $\Phi,\Phi^{\dagger},V$. These include the gauge covariant derivatives
of the chiral superfields $\mathcal{D}_{\alpha}\Phi$,
$\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger}$. We note that terms made of
$\partial_{m}\Phi$, $\partial_{m}\Phi^{\dagger}$, which are allowed in the
ungauged case, are forbidden in the gauged models due to supersymmetry and the
gauge covariance. Then, the gauged Lagrangian can be written as

$\displaystyle\mathcal{L}=$ $\displaystyle\ \int\\!d^{4}\theta\
K(\Phi,\Phi^{\dagger},V)-\frac{1}{16}\int\\!d^{4}\theta\
\Lambda_{i\bar{j}k\bar{l}}(\Phi,\Phi^{\dagger})\left(\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger\bar{j}}e^{2gV}\mathcal{D}^{\alpha}\Phi^{i}\right)\left(\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger\bar{l}}e^{2gV}\mathcal{D}_{\alpha}\Phi^{k}\right)$
$\displaystyle\
+\left(\int\\!d^{2}\theta\,W(\Phi)+\mathrm{h.c.}\right)+\frac{1}{16kg^{2}}\mathrm{Tr}\left[\int\\!d^{2}\theta\
W^{\alpha}W_{\alpha}+(\mathrm{h.c.})\right]-2\kappa g\int\\!d^{4}\theta\
\mathrm{Tr}V,$ (2.11)

where we have introduced the gauge kinetic and Fayet-Iliopoulos (FI) terms
with the FI parameter $\kappa$. The superpotential and the FI term are
necessary to impose the supersymmetric constraints. The bosonic part of the
Lagrangian is given by [29]

$\displaystyle\mathcal{L}_{\rm boson}=$ $\displaystyle\
-\frac{\partial^{2}K}{\partial\bar{\varphi}_{a}^{\bar{j}}\partial\varphi^{ib}}D_{m}\bar{\varphi}_{a}^{\bar{j}}D^{m}\varphi^{ib}-\frac{\partial^{2}K}{\partial\bar{\varphi}_{a}^{\bar{j}}\partial\varphi^{ib}}\bar{F}_{a}^{\bar{j}}F^{ib}+\frac{g}{2}D^{\hat{a}}\left(\bar{\varphi}_{c}^{\bar{j}}(T^{\hat{a}})^{c}{}_{d}\frac{\partial
K}{\partial\bar{\varphi}_{d}^{\bar{j}}}+\frac{\partial
K}{\partial\varphi^{ic}}(T^{\hat{a}})^{c}{}_{d}\varphi^{id}-2\kappa\delta^{\hat{a}}{}_{0}\right)$
$\displaystyle\
+\frac{1}{k}\mathrm{Tr}\left[-\frac{1}{4}F_{mn}F^{mn}+\frac{1}{2}D^{2}\right]+\frac{\partial
W}{\partial\varphi^{ia}}F^{ia}+\frac{\partial\bar{W}}{\partial\bar{\varphi}^{\bar{i}}_{a}}\bar{F}^{\bar{i}}_{a}$
$\displaystyle\
+\Lambda_{ik\bar{j}\bar{l}}(\varphi,\bar{\varphi})\Bigg{[}(D^{m}\bar{\varphi}_{a}^{\bar{j}}D^{n}\varphi^{ia})(D_{m}\bar{\varphi}_{b}^{\bar{l}}D_{n}\varphi^{kb})$
$\displaystyle\
\qquad\qquad\qquad\qquad-\frac{1}{2}(D_{m}\varphi^{ia}F^{kb}+F^{ia}D_{m}\varphi^{kb})(D^{m}\bar{\varphi}_{a}^{\bar{j}}\bar{F}_{b}^{\bar{l}}+\bar{F}_{a}^{\bar{j}}D^{m}\bar{\varphi}_{b}^{\bar{l}})+F^{ia}\bar{F}^{\bar{j}}_{a}F^{kb}\bar{F}^{\bar{l}}_{b}\Bigg{]},$
(2.12)

where $F_{mn}=\partial_{m}A_{n}-\partial_{m}A_{n}+i[A_{m},A_{n}]$ and $T^{0}$
is the $U(1)$ generator in $G^{\prime}$. A notable fact about the Lagrangian
(2.12) is that, due to the derivative corrections, the equation of motion for
the auxiliary field $F$ is not linear:

$\displaystyle-\frac{\partial^{2}K}{\partial\bar{\varphi}^{\bar{j}^{\prime}}_{a}\partial\varphi^{ib}}F^{ib}+\frac{\partial\bar{W}}{\partial\bar{\varphi}^{\bar{j}^{\prime}}_{a}}$
$\displaystyle+\Lambda_{ik\bar{j}\bar{l}}(\varphi,\bar{\varphi})\Bigg{[}-\frac{1}{2}\Big{(}(D_{m}\varphi^{ib}F^{ka}+F^{ib}D_{m}\varphi^{ka})\Big{)}\Big{(}D^{m}\varphi^{\bar{j}}_{b}\delta^{\bar{j}^{\prime}\bar{l}}+\delta^{\bar{j}^{\prime}\bar{j}}D^{m}\bar{\varphi}^{\bar{l}}_{b}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+F^{ia}\delta^{\bar{j}^{\prime}\bar{j}}F^{kb}\bar{F}^{\bar{l}}_{b}+F^{ib}\bar{F}^{\bar{j}}_{b}F^{ka}\delta^{\bar{j}^{\prime}\bar{l}}\Bigg{]}=0.$
(2.13)

This apparently allows several solutions for the auxiliary field. The
solutions provide several distinct on-shell Lagrangians. For example, when
$W=0$, the inhomogeneous term in (2.13) vanishes and it is obvious that $F=0$
is a solution. We call this the canonical branch. In addition, we have another
solution $F\not=0$ and we call the corresponding theory the non-canonical
branch. Both the branches exhibit remarkable features [27, 16, 23, 26].
Although this shows an unusual situation in supersymmetric theories, we stress
that Eq. (2.13) is the algebraic equation, not the kinematical one, which
guarantees that $F$ still keeps the role of the auxiliary field.

In the following sections, we examine the nonlinear sigma model limit
$g\to\infty$ in the gauged linear sigma models. In the limit, the vector
multiplet carries non-propagating degrees of freedom and it will be integrated
out. We in particular focus on the Hermitian symmetric spaces of the type
$G/H$ [46]. This procedure makes us to write down a variety of supersymmetric
nonlinear sigma models with higher derivative terms.

##  3 Nonlinear sigma models with D-term constraints

In this section, we discuss supersymmetric nonlinear sigma models whose target
spaces are the complex projective space $\mathbb{C}P^{N-1}$ and the Grassmann
manifold $G_{M,N}$. It is known that they are obtained in the sigma model
limit $g\to\infty$ of supersymmetric gauge theories with D-term constraints
[53, 57, 46]. The former is obtained from an Abelian gauge theory while the
latter comes from a non-Abelian gauge theory. In the following, we explicitly
integrate out the gauge field both in the canonical and the non-canonical
branches in the sigma model limit $g\to\infty$.

###  3.1 $\mathbb{C}P^{N-1}$ model

We first consider the nonlinear sigma model whose target space is
$\mathbb{C}P^{N-1}=SU(N)/[SU(N-1)\times U(1)]$. We formulate this model as a
gauged linear sigma model with the global symmetry $G=SU(N)$ and gauge
symmetry $G^{\prime}=U(1)$, where the latter is complexified
$G^{\prime}_{\mathbb{C}}=U(1)_{\mathbb{C}}$. We consider the flat Kähler
potential $K=\delta_{i\bar{j}}\Phi^{i}\Phi^{\dagger\bar{j}}=\Phi^{\dagger
i}\Phi^{i}$ and $W=0$. The chiral superfields $\Phi^{i},\Phi^{\dagger
i},(i=1,\ldots,N)$ belong to the fundamental representation $\mathbf{N}$ of
the global $SU(N)$ symmetry, and their $U(1)$ charges are assigned as
$(+1,-1)$. We assume that the $(2,2)$ Kähler tensor is given by the flat
metric as

$\displaystyle\Lambda_{ik\bar{j}\bar{l}}=\delta_{i\bar{j}}\delta_{k\bar{l}}\Lambda(\Phi,\Phi^{\dagger}),$
(3.1)

where $\Lambda(\Phi,\bar{\Phi})$ is a gauge invariant Kähler scalar composed
of the chiral superfields and their gauge covariant derivatives
$\mathcal{D}_{\alpha}\Phi$,
$\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger}$.222 The case of $\Lambda=$
const. was studied in Appendix B of Ref. [50].  Then, the Lagrangian becomes
[53]

$\displaystyle\mathcal{L}=$ $\displaystyle\ \int\\!d^{4}\theta\ \Phi^{\dagger
i}e^{2V}\Phi^{i}+\frac{1}{16}\int\\!d^{4}\theta\
\Lambda(\Phi,\Phi^{\dagger})\delta_{i\bar{j}}\delta_{k\bar{l}}\,e^{4V}\mathcal{D}^{\alpha}\Phi^{i}\mathcal{D}_{\alpha}\Phi^{k}\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger\bar{j}}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger\bar{l}}$
$\displaystyle+\frac{1}{16g^{2}}\left[\int\\!d^{2}\theta\
W^{\alpha}W_{\alpha}+(\mathrm{h.c.})\right]-2\kappa\int\\!d^{4}\theta\ V.$
(3.2)

Here we have rescaled $V\to\frac{1}{g}V$. We have introduced the FI term which
provides the D-term constraints on the component fields in $\Phi^{i}$. Note
that for the Abelian gauge group, we have
$\mathcal{D}_{\alpha}\Phi=D_{\alpha}\Phi+(D_{\alpha}V)\Phi$,
$\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger}=\bar{D}^{\dot{\alpha}}\Phi^{\dagger}+(\bar{D}^{\dot{\alpha}}V)\Phi^{\dagger}$.
The bosonic part of the Lagrangian (3.2) is then given by

$\displaystyle\mathcal{L}_{\rm boson}=$ $\displaystyle\
-D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}+F^{i}\bar{F}^{i}+D(\bar{\varphi}^{i}\varphi^{i}-\kappa)+\frac{1}{2g}D^{2}-\frac{1}{4g^{2}}F_{mn}F^{mn}$
$\displaystyle\
+\delta_{i\bar{j}}\delta_{k\bar{l}}\Lambda(\varphi,\bar{\varphi})\left(\frac{}{}(D_{m}\varphi^{i}D_{n}\bar{\varphi}^{\bar{j}})(D^{m}\varphi^{k}D^{n}\bar{\varphi}^{\bar{l}})-2(D_{m}\varphi^{i}D^{m}\bar{\varphi}^{\bar{j}})(F^{k}\bar{F}^{\bar{l}})+F^{i}\bar{F}^{\bar{j}}F^{k}\bar{F}^{\bar{l}}\right),$
(3.3)

where $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}$ and the gauge covariant
derivative is defined by
$D_{m}\varphi^{i}=\partial_{m}\varphi^{i}+iA_{m}\varphi^{i}$. In the
following, we assume that the gauge invariant Kähler scalar
$\Lambda(\varphi,\bar{\varphi})$ is given by a function of
$X=D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}$ as a typical example. The most
plausible reason for this assumption is that this quantity contains only the
first order time derivative of fields. This guarantees the absence of the
Ostrogradski’s ghost [19] in the theory. This assumption is easily relaxed but
allowing other gauge and Lorentz invariant quantities such as
$X=\varphi^{i}\bar{\varphi}^{i},\,D_{m}\varphi^{i}D^{n}\bar{\varphi}^{i}D^{m}\varphi^{j}D^{n}\bar{\varphi}^{j}$
and so on does not change the following discussions.

In the sigma model limit $g\to\infty$, the gauge kinetic and $D^{2}$ terms
vanish and the gauge field $A_{m}$ becomes an auxiliary field. Before
integrating out these fields, we first integrate out the auxiliary fields in
the chiral superfields. The equation of motion for the auxiliary field
$\bar{F}^{\bar{i}}$ is

$\displaystyle 0=$ $\displaystyle\
\frac{\partial\mathcal{L}}{\partial\bar{F}^{\bar{i}}}=F^{i}-2\Lambda(X)\left((D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})F^{i}-(F^{j}\bar{F}^{j})F^{i}\right).$
(3.4)

As we have noticed, there are two branches associated with the solutions
$F^{i}=0$ and $F^{i}\not=0$. We examine each branch separately.

#### Canonical branch

For the canonical branch corresponding to the solution $F^{i}=0$, the
Lagrangian (3.3) becomes

$\displaystyle\mathcal{L}_{\text{c}}=$ $\displaystyle\
-D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}+\Lambda(D_{m}\varphi^{i}D_{n}\bar{\varphi}^{i})(D^{m}\varphi^{j}D^{n}\bar{\varphi}^{j})+D(\bar{\varphi}^{i}\varphi^{i}-\kappa).$
(3.5)

The equation of motion for the auxiliary field $D$ gives the following
constraint for the scalar fields $\varphi^{i}$:

$\displaystyle\varphi^{i}\bar{\varphi}^{i}=\kappa.$ (3.6)

The equation of motion for the gauge field $A_{m}$ is

$\displaystyle 0=\frac{\partial\mathcal{L}}{\partial A_{m}}=$ $\displaystyle\
-2\kappa\left(i\kappa^{-1}\varphi^{i}\partial^{m}\bar{\varphi}^{i}+A^{m}\right)+i\frac{\partial\Lambda}{\partial
X}\left(\varphi^{i}D^{m}\bar{\varphi}^{i}-\bar{\varphi}^{i}D^{m}\varphi^{i}\right)$
$\displaystyle\
+2i\Lambda(X)\Big{[}(\varphi^{i}D_{p}\bar{\varphi}^{i})(D_{m}\varphi^{j}D^{p}\bar{\varphi}^{j})-(\bar{\varphi}^{i}D_{p}\varphi^{i})(D_{m}\bar{\varphi}^{j}D^{p}\varphi^{j})\Big{]}.$
(3.7)

Using the constraint (3.6), we find that the solution to this equation is
given by

$\displaystyle A_{m}=i\kappa^{-1}\bar{\varphi}^{i}\partial_{m}\varphi^{i}.$
(3.8)

Substituting the solution (3.8) into (3.5) and using the constraint (3.6), we
obtain the Lagrangian in the canonical branch:

$\displaystyle\mathcal{L}_{\text{c}}=$ $\displaystyle\
-\tilde{D}_{m}\bar{\varphi}^{i}\tilde{D}^{m}\varphi^{i}+\Lambda(\tilde{X})(\tilde{D}_{m}\varphi^{i}\tilde{D}_{n}\bar{\varphi}^{i})(\tilde{D}^{m}\varphi^{j}\tilde{D}^{n}\bar{\varphi}^{j}),$
(3.9)

where we have defined
$\tilde{D}\varphi^{i}=\partial_{m}\varphi^{i}-\kappa^{-1}(\bar{\varphi}^{j}\partial_{m}\varphi^{j})\varphi^{i}$
and $\tilde{X}=\tilde{D}_{m}\varphi^{i}\tilde{D}^{m}\bar{\varphi}^{i}$.

It is convenient to solve the constraint (3.6) explicitly by using the
parametrization

$\displaystyle\varphi^{i}=W^{i}\frac{\sqrt{\kappa}}{\sqrt{W^{\dagger}\cdot
W}}.$ (3.10)

By using the $U(1)_{\mathbb{C}}$ gauge symmetry, we can fix $W^{1}=1$. Then,
we have the conventional parametrization of the $\mathbb{C}P^{N-1}$ model:

$\displaystyle\varphi^{i}=\frac{\sqrt{\kappa}}{\sqrt{1+|\vec{u}|^{2}}}\left(\begin{array}[]{c}1\\\
u^{s}\end{array}\right),\qquad\bar{\varphi}^{\bar{i}}=\frac{\sqrt{\kappa}}{\sqrt{1+|\vec{u}|^{2}}}\left(\frac{}{}1,\
\bar{u}^{\bar{s}}\right),\qquad(s,\bar{s}=2,\ldots,N),$ (3.13)

where $u^{s},\bar{u}^{\bar{s}}$ are inhomogeneous coordinates of the
${\mathbb{C}}P^{N-1}$ manifold. By using this form, the Lagrangian in the
canonical branch can be rewritten as

$\displaystyle\mathcal{L}_{\text{c}}=$ $\displaystyle\
-\frac{\kappa}{(1+|\vec{u}|^{2})^{2}}\left[(1+|\vec{u}|^{2})(\partial_{m}\vec{u}^{*}\cdot\partial^{m}\vec{u})-(\vec{u}\cdot\partial_{m}\vec{u}^{*})(\vec{u}^{*}\cdot\partial^{m}\vec{u})\right]$
$\displaystyle\
+\frac{\kappa^{2}}{(1+|\vec{u}|^{2})^{4}}\Lambda(u,\bar{u})\left[\frac{}{}(1+|\vec{u}|^{2})^{2}(\partial_{m}\vec{u}^{*}\cdot\partial_{n}\vec{u})(\partial^{m}\vec{u}^{*}\cdot\partial^{n}\vec{u})\right.$
$\displaystyle\
\qquad\quad\left.\frac{}{}-2(1+|\vec{u}|^{2})(\partial_{m}\vec{u}^{*}\cdot\partial_{n}\vec{u})(\vec{u}\cdot\partial^{m}\vec{u}^{*})(\vec{u}^{*}\cdot\partial^{n}\vec{u})+(\vec{u}\cdot\partial_{m}\vec{u}^{*})^{2}(\vec{u}^{*}\cdot\partial_{n}\vec{u})^{2}\right].$
(3.14)

By using the Fubini-Study metric

$\displaystyle
g_{s\bar{t}}=\kappa\frac{(1+|\vec{u}|^{2})\delta_{s\bar{t}}-u^{s}\bar{u}^{\bar{t}}}{(1+|\vec{u}|^{2})^{2}},$
(3.15)

for the $\mathbb{C}P^{N-1}$ manifold, the Lagrangian (3.14) can be simply
written as

$\displaystyle\mathcal{L}_{\text{c}}=-g_{s\bar{t}}\partial_{m}u^{s}\partial^{m}\bar{u}^{\bar{t}}+g_{s\bar{t}}g_{q\bar{r}}\Lambda(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u})(\partial_{m}u^{s}\partial_{n}\bar{u}^{\bar{t}})(\partial^{m}u^{q}\partial^{n}\bar{u}^{\bar{r}}).$
(3.16)

The first term is the ordinary kinetic term of the $\mathbb{C}P^{N-1}$
nonlinear sigma model and the second is the derivative corrections determined
by the arbitrary scalar function
$\Lambda(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u})$.

Before going to the discussion on the the non-canonical branch, two comments
in this Lagrangian are in order. This Lagrangian with $\Lambda=$const. was
obtained in Ref. [50] as the low-energy effective theory of a BPS non-Abelian
vortex in ${\cal N}=2$ supersymmetric $U(N)$ gauge theory [54]; A non-Abelian
vortex in this case allows the supersymmetric ${\mathbb{C}}P^{N-1}$ model on
the vortex worldsheet, on which 1/2 BPS condition preserves four supercharges
among eight supercharges that the ${\cal N}=2$ supersymmetric theory in the
bulk has. The bosonic part of the four-derivative correction term is precisely
in this form [50].

Without supersymmetry, the ${\mathbb{C}}P^{N-1}$ Skyrme-Faddeev model was
proposed in Refs. [47, 48, 49] as the ${\mathbb{C}}P^{N-1}$ model with four
derivative terms. In this case, there are three kinds of four derivative
terms, and the one in Eq. (3.16) (with $\Lambda=$const.) corresponds to a
particular choice among them. Thus, we call the Lagrangian in Eq. (3.16) (the
bosonic part of) the supersymmetric ${\mathbb{C}}P^{N-1}$ Skyrme-Faddeev
model.

#### Non-canonical branch

We next study the non-canonical branch corresponding to solutions
$F_{i}\not=0$. Again, the equation of motion for $\bar{F}$ is

$\displaystyle 0=$ $\displaystyle\
\frac{\partial\mathcal{L}}{\partial\bar{F}^{\bar{i}}}=F^{i}-2\Lambda(X)\Big{(}(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})F^{i}-(F^{j}\bar{F}^{j})F^{i}\Big{)}.$
(3.17)

The solution $F^{i}\not=0$ is found to be

$\displaystyle
F^{i}\bar{F}^{i}=-\frac{1}{2\Lambda}+D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}.$
(3.18)

Substituting this solution into the Lagrangian (3.3), it becomes

$\displaystyle\mathcal{L}_{\text{nc}}=\Lambda(X)\left[(D_{m}\bar{\varphi}^{i}D_{n}\varphi^{i})(D^{m}\bar{\varphi}^{j}D^{n}\varphi^{j})-(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})^{2}\right]-\frac{1}{4\Lambda(X)}+D(\bar{\varphi}^{i}\varphi^{i}-\kappa).$
(3.19)

where we have assumed the sigma model limit $g\to\infty$. The D-term condition
leads to the constraint (3.6) while the equation of motion for the gauge field
is

$\displaystyle 0=$ $\displaystyle\ i\frac{\partial\Lambda}{\partial
X}\left(\varphi^{k}D^{m}\bar{\varphi}^{k}-\bar{\varphi}^{k}D^{m}\varphi^{k}\right)\left[(D_{m}\bar{\varphi}^{i}D_{n}\varphi^{i})(D^{m}\bar{\varphi}^{j}D^{n}\varphi^{j})-(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})^{2}\right]$
$\displaystyle\
+2i\Lambda(X)\Big{[}\frac{}{}-\bar{\varphi}^{j}D^{q}\varphi^{j}(D^{m}\bar{\varphi}^{i}D_{q}\varphi^{i})+\varphi^{j}D^{p}\bar{\varphi}^{j}(D_{p}\bar{\varphi}^{i}D^{m}\varphi^{i})$
$\displaystyle\
\qquad\qquad\qquad\qquad\qquad-(D_{p}\varphi^{i}D^{p}\bar{\varphi}^{i})(\varphi^{j}D^{m}\bar{\varphi}^{j}-\bar{\varphi}^{j}D^{m}\varphi^{j})\Big{]}$
$\displaystyle\ +4i\Lambda^{-2}(X)\frac{\partial\Lambda}{\partial
X}\left(\varphi^{i}D^{m}\bar{\varphi}^{i}-\bar{\varphi}^{i}D^{m}\varphi^{i}\right).$
(3.20)

By using the constraint (3.6), we find that the equation (3.20) is again
solved by

$\displaystyle A_{m}=i\kappa^{-1}\bar{\varphi}^{i}\partial_{m}\varphi^{i}.$
(3.21)

Substituting this solution into the Lagrangian (3.19), we find the new sigma
model given by

$\displaystyle\mathcal{L}_{\text{nc}}=\Lambda(\tilde{X})\left[(\tilde{D}_{m}\bar{\varphi}^{i}\tilde{D}_{n}\varphi^{i})(\tilde{D}^{m}\bar{\varphi}^{j}\tilde{D}^{n}\varphi^{j})-(\tilde{D}_{m}\varphi^{j}\tilde{D}^{m}\bar{\varphi}^{j})^{2}\right]-\frac{1}{4\Lambda(\tilde{X})}.$
(3.22)

The expression (3.13) enable us to rewrite the Lagrangian (3.22) as

$\displaystyle\mathcal{L}_{\text{nc}}=$ $\displaystyle\
\frac{\kappa^{2}}{(1+|\vec{u}|^{2})^{4}}\Lambda(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u})\left[\frac{}{}(1+|\vec{u}|^{2})^{2}\left\\{(\partial_{m}\vec{u}\cdot\partial_{n}\vec{u}^{*})(\partial^{m}\vec{u}\cdot\partial^{n}\vec{u}^{*})-(\partial_{m}\vec{u}^{*}\cdot\partial^{m}\vec{u})^{2}\right\\}\right.$
$\displaystyle\qquad-2(1+|\vec{u}|^{2})\left\\{(\partial_{m}\vec{u}^{*}\cdot\partial_{n}\vec{u})(\vec{u}\cdot\partial^{m}\vec{u}^{*})(\vec{u}^{*}\cdot\partial^{n}\vec{u})-(\partial_{m}\vec{u}^{*}\cdot\partial^{m}\vec{u})(\vec{u}\cdot\partial_{n}\vec{u}^{*})(\vec{u}^{*}\cdot\partial^{n}\vec{u})\right\\}$
$\displaystyle\qquad\left.+(\vec{u}\cdot\partial_{m}\vec{u}^{*})^{2}(\vec{u}^{*}\cdot\partial_{n}\vec{u})^{2}-\left\\{(\vec{u}\cdot\partial_{m}\vec{u}^{*})(\vec{u}^{*}\cdot\partial^{m}\vec{u})\right\\}^{2}\right]-\frac{1}{4}\Lambda^{-1}(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u}).$
(3.23)

By using the Fubini-Study metric (3.15), this can be simply expressed as

$\displaystyle\mathcal{L}_{\text{nc}}=$ $\displaystyle\
\Lambda(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u})\left[\frac{}{}(g_{s\bar{t}}\partial_{m}u^{s}\partial_{n}\bar{u}^{\bar{t}})(g_{q\bar{r}}\partial^{m}u^{q}\partial^{n}\bar{u}^{\bar{q}})-(g_{s\bar{t}}\partial_{m}u^{s}\partial^{m}\bar{u}^{\bar{t}})^{2}\right]$
(3.24)
$\displaystyle-\frac{1}{4}\Lambda^{-1}(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u}).$

This four derivative term does not contain fourth order time derivatives,
unlike the one in Eq. (3.16) in the canonical branch. It is obvious that the
target space of the scalar fields is the $\mathbb{C}P^{N-1}$ space but the
canonical kinetic term $g_{s\bar{t}}\partial_{m}u^{s}\partial^{m}u^{\bar{t}}$
is absent.

In particular for the $N=2$ case of $\mathbb{C}P^{1}$ , the Lagrangian reduces
to

$\displaystyle\mathcal{L}_{\text{nc}}=$ $\displaystyle\
-\kappa^{2}\Lambda\frac{(\partial_{m}u\partial^{m}\bar{u})^{2}-|\partial_{m}u\partial^{m}u|^{2}}{(1+|u|^{2})^{4}}-\frac{1}{4\Lambda}$
(3.25)

which is known as the supersymmetric baby Skyrme model [24, 25, 27, 40, 41,
42]. One notices that the first term (with $\Lambda=1$) is nothing but the
Skyrme-Faddeev fourth derivative term (without the fourth order time
derivatives) while the second term provides higher derivative corrections
determined by $\Lambda(u,\bar{u},\partial_{m}u,\partial_{m}\bar{u})$. The
Lagrangian in Eq. (3.24) may be called the supersymmetric
${\mathbb{C}}P^{N-1}$ baby Skyrme model.

###  3.2 $G_{M,N}$ model

We next consider the nonlinear sigma model whose target space is the Grassmann
manifold $G_{M,N}=U(M)/[U(M-N)\times U(N)]$. We consider a gauged linear sigma
model with the global symmetry $G=SU(M)$ and gauge symmetry $U(N)$, of which
the chiral superfield $\Phi^{i}_{a}$ ($i=1,\ldots M$; $a=1,\ldots N$) belong
to the $(\mathbf{M},\bar{\mathbf{N}})$ representation. The global symmetry
$G=SU(M)$ and gauge symmetry $U(N)_{\mathbb{C}}=GL(N,{\mathbb{C}})$ act on it
as

$\displaystyle\Phi\to g\Phi e^{i\Theta^{\prime}},\qquad e^{2V}\to
e^{-i\Theta^{\prime}}e^{2V}e^{i\Theta^{\prime\dagger}},$ (3.26)

where $g\in SU(M)$, $\Theta^{\prime}(x,\theta,\bar{\theta})$ is a chiral
superfield of a $U(N)_{\mathbb{C}}$ gauge parameter, and we have introduced
the associated $U(N)$ vector superfield $V(x,\theta,\bar{\theta})$. The
$G\times G^{\prime}$ invariant Lagrangian is therefore [57, 46]

$\displaystyle\mathcal{L}=$ $\displaystyle\
\int\\!d^{4}\theta\,\mathrm{Tr}\left[\Phi
e^{2V}\Phi^{\dagger}\right]-\frac{1}{16}\int\\!d^{4}\theta\,\Lambda(\Phi,\Phi^{\dagger})\mathrm{Tr}\left[\mathcal{D}^{\alpha}\Phi
e^{2V}\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger}\right]\mathrm{Tr}\left[\mathcal{D}_{\alpha}\Phi
e^{2V}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger}\right]$ $\displaystyle\
+\frac{1}{16g^{2}}\left(\int\\!d^{2}\theta\,\mathrm{Tr}[W^{\alpha}W_{\alpha}]+(\mathrm{h.c.})\right)-2\kappa\int\\!d^{4}\theta\,\mathrm{Tr}V.$
(3.27)

The derivative corrections are given in the double trace form and we have
included the FI term which provides the D-term conditions on the chiral
multiplets.

In the nonlinear sigma model limit $g\to\infty$, the bosonic part of the
Lagrangian becomes

$\displaystyle\mathcal{L}_{\rm boson}=$ $\displaystyle\
-\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]+\mathrm{Tr}[\bar{\varphi}D\varphi-\kappa
D]+\mathrm{Tr}[F\bar{F}]$ $\displaystyle\
+\Lambda(\varphi,\bar{\varphi})\Big{\\{}\mathrm{Tr}[D_{m}\varphi
D_{n}\bar{\varphi}]\mathrm{Tr}[D^{m}\varphi
D^{n}\bar{\varphi}]-\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]\mathrm{Tr}[F\bar{F}]$ $\displaystyle\
\qquad\qquad\qquad-\mathrm{Tr}[FD_{m}\bar{\varphi}]\mathrm{Tr}[D^{m}\varphi\bar{F}]+\mathrm{Tr}[F\bar{F}]\mathrm{Tr}[F\bar{F}]\Big{\\}}.$
(3.28)

The gauge covariant derivative is $D_{m}\varphi=\partial_{m}\varphi+i\varphi
A_{m}$. In the following, we assume $\Lambda$ is given by the gauge invariant
quantity $X=\mathrm{Tr}[D_{m}\varphi D^{m}\bar{\varphi}]$. As clarified
before, this condition is easily relaxed allowing for any other gauge
invariant quantities. The equation of motion for the auxiliary field $D$ gives
the constraint:

$\displaystyle\bar{\varphi}\varphi=\kappa\mathbf{1}_{N}.$ (3.29)

On the other hand, the equation of motion for the auxiliary field $\bar{F}$ is
given by

$\displaystyle F+\Lambda(X)\Big{\\{}-\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]F-\mathrm{Tr}[FD_{m}\bar{\varphi}]D^{m}\varphi+2\mathrm{Tr}[F\bar{F}]F\Big{\\}}=0.$
(3.30)

#### Canonical branch

It is obvious that $F=0$ is a solution. In this case, the Lagrangian becomes

$\displaystyle\mathcal{L}_{\mathrm{c}}=$ $\displaystyle\
-\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]+\Lambda(X)\mathrm{Tr}[D_{m}\varphi
D_{n}\bar{\varphi}]\mathrm{Tr}[D^{m}\varphi D^{n}\bar{\varphi}],$ (3.31)

where the scalar fields satisfy the constraint (3.29). The equation of motion
for $A_{m}$ then becomes,

$\displaystyle\bar{\varphi}\partial_{m}\varphi+i\kappa
A_{m}+i\frac{\partial\Lambda}{\partial X}\left(\varphi
D^{m}\bar{\varphi}-D^{m}\varphi\bar{\varphi}\right)$
$\displaystyle\qquad-\Lambda(X)\mathrm{Tr}\Big{[}D_{m}\varphi
D_{n}\bar{\varphi}+D_{n}\varphi
D_{m}\bar{\varphi}\Big{]}\left(\bar{\varphi}\partial^{n}\varphi+i\kappa
A^{n}\right)=0,$ (3.32)

where we have used the constraint (3.29). We again find that the exact
solution to this equation is given by

$\displaystyle A_{m}=i\kappa^{-1}\bar{\varphi}\partial_{m}\varphi.$ (3.33)

Plugging this solution back into the Lagrangian, we obtain the nonlinear sigma
model supplemented by the derivative corrections:

$\displaystyle\mathcal{L}_{\mathrm{c}}=$ $\displaystyle\
-\mathrm{Tr}[\tilde{D}_{m}\varphi\tilde{D}^{m}\bar{\varphi}]+\Lambda(\tilde{X})\mathrm{Tr}[\tilde{D}_{m}\varphi\tilde{D}_{n}\bar{\varphi}]\mathrm{Tr}[\tilde{D}^{m}\varphi\tilde{D}^{n}\bar{\varphi}].$
(3.34)

Here
$\tilde{D}_{m}\varphi=\partial_{m}\varphi-\kappa^{-1}\varphi(\bar{\varphi}\partial_{m}\varphi)$
and $\tilde{X}=\mathrm{Tr}[\tilde{D}_{m}\varphi\tilde{D}^{m}\bar{\varphi}]$.
The constraint (3.29) is solved by the parametrization:

$\displaystyle\varphi=W\frac{\sqrt{\kappa}}{\sqrt{W^{\dagger}W}},$ (3.35)

where $W$ is an $M\times N$ matrix. Substituting this into the Lagrangian, the
first term in (3.34) becomes the kinetic term of the sigma model whose target
space is the Grassmann manifold $G_{M,N}$ while the second term gives
derivative corrections determined by the arbitrary gauge invariant function
$\Lambda(W,W^{\dagger},\partial_{m}W,\partial_{m}W^{\dagger})$.

#### Non-canonical branch

We next examine the non-canonical branch. It is obvious that there are non-
zero solutions $F\not=0$ to the equation (3.30). Compared with the case of the
Abelian gauge symmetry, it is not straightforward to write down explicit
solutions for $F$. However, the gauge invariant solution
$\mathrm{Tr}[F\bar{F}]$ should be given by the gauge and the Lorentz invariant
quantities $Y$ such as

$\displaystyle Y=\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}],\quad\mathrm{Tr}[D_{m}\varphi
D_{n}\bar{\varphi}D^{m}\varphi
D^{n}\bar{\varphi}],\quad\mathrm{Tr}[D_{m}\varphi
D_{n}\bar{\varphi}D^{n}\varphi D^{m}\bar{\varphi}],$ (3.36)

and so on. Using the equation of motion for $F$, the non-canonical Lagrangian
becomes

$\displaystyle\mathcal{L}_{\mathrm{nc}}=-\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]+\Lambda(X)\Big{\\{}\mathrm{Tr}[D_{m}\varphi
D_{n}\bar{\varphi}]\mathrm{Tr}[D^{m}\varphi
D^{n}\bar{\varphi}]-\mathrm{Tr}[F\bar{F}(Y)]^{2}\Big{\\}}.$ (3.37)

Here $\mathrm{Tr}[F\bar{F}(Y)]$ is a solution $F\not=0$ given by $Y$. The
variation of the Lagrangian (3.37) by $A_{m}$ is therefore

$\displaystyle\delta\mathcal{L}_{\mathrm{nc}}=$ $\displaystyle\
i\mathrm{Tr}\left[(\bar{\varphi}\partial^{m}\varphi+i\kappa A^{m})\delta
A_{m}\right]+i\frac{\partial\Lambda}{\partial
X}\mathrm{Tr}\Big{[}\left(\varphi
D^{m}\bar{\varphi}-D^{m}\varphi\bar{\varphi}\right)\delta A_{m}\Big{]}$
$\displaystyle\ -i\Lambda(X)\mathrm{Tr}\Big{[}D^{m}\varphi
D^{n}\bar{\varphi}+D^{n}\varphi
D^{m}\bar{\varphi}\Big{]}\mathrm{Tr}\Big{[}(\bar{\varphi}\partial_{n}\varphi+i\kappa
A_{n})\delta A_{m}\Big{]}$ $\displaystyle\
-2\Lambda(\varphi,\bar{\varphi})\mathrm{Tr}[F\bar{F}]\mathrm{Tr}\left[(F\bar{F})^{\prime}\delta
Y\right].$ (3.38)

Here $(F\bar{F})^{\prime}$ stands for the differentiation with respect to $Y$.
In the last term in (3.38), one finds that the variation of the gauge
invariant quantity $Y$ is proportional to
$\bar{\varphi}\partial^{m}\varphi+i\kappa A^{m}$. Indeed, for
$Y=\mathrm{Tr}[D_{m}\varphi D^{m}\bar{\varphi}]$, we have

$\displaystyle\delta Y=2i\mathrm{Tr}[(\bar{\varphi}\partial^{m}\varphi+i\kappa
A^{m})\delta A_{m}].$ (3.39)

Therefore even in the non-canonical branch, we find that the solution to the
equation of motion for $A_{m}$ is given by
$A_{m}=i\kappa^{-1}\varphi\partial_{m}\bar{\varphi}$. For the other gauge
invariant combinations $Y$ such as (3.36), the result is the same. The
constraint (3.29) is solved by the parametrization (3.35) and the Lagrangian
becomes again the Grassmanian sigma model with the derivative corrections.

We comment on the symmetric nature of the Grassmanian manifold
$G_{M,N}=G_{M,M-N}$. We have constructed the Grassmanian sigma models by
gauging the symmetry $G^{\prime}=U(N)$ but there is an alternative way to
define the same target space by gauging $G^{\prime}=U(M-N)$. It is obvious
that the latter construction is essentially the same with the former one
except the $N=1$ case. In this case, one notices that
$G_{N,1}=G_{N,N-1}=\mathbb{C}P^{N-1}$. This equality implies that we can
construct supersymmetric sigma model whose target space is
$\mathbb{C}P^{N-1}$, either by gauging the non-Abelian $U(N-1)$ symmetry or
the Abelian $U(1)$ symmetry. The former is nothing but the construction
discussed in this section while the latter is the formulation discussed in the
previous section. We find apparently different Lagrangians (3.37) and (3.22)
in the derivative corrections. This means that the equivalence of the target
spaces in different formalism are not taken over to the derivative corrections
in the superfield language. In the next section, we will see a similar
situation in the canonical branch of the nonlinear sigma models defined by the
F-term constraints. We will, among other things, encounter qualitatively
different structures of the target spaces in the non-canonical branch.

##  4 Nonlinear sigma models with D- and F-term constraints

In this section, we discuss the nonlinear sigma models defined by F-term
constraints in addition to the D-term constraints [46, 55]. Such Hermitian
symmetric target spaces include the quadric surface
$Q^{N-2}=SO(N)/[SO(N-2)\times U(1)]$, the quotient spaces of the types
$SO(2N)/U(N)$, $Sp(2N)/U(N)$, $E_{6}/[SO(10)\times U(1)]$ and
$E_{7}/[E_{6}\times U(1)]$. We begin with the target space
$Q^{N-2}=SO(N)/[SO(N-2)\times U(1)]$ as a typical example.

###  4.1 Quadric surface $Q^{N-2}$

We consider a gauged linear sigma model with the global symmetry $G=SO(N)$ and
the gauge symmetry $U(1)$. The chiral superfield $\Phi^{i}$ ($i=1,\ldots,N$)
belongs to the fundamental representation $\mathbf{N}$ of $SO(N)$ and has the
$U(1)$ charge $+1$. The Lagrangian in the nonlinear sigma model limit is

$\displaystyle\mathcal{L}=$ $\displaystyle\
\int\\!d^{4}\theta\,\left(\Phi^{i}e^{2V}\Phi^{\dagger i}-2\kappa
V\right)+\left(\int\\!d^{2}\theta\,\Phi_{0}\vec{\Phi}^{t}J\vec{\Phi}+\mathrm{h.c.}\right)$
$\displaystyle\
+\frac{1}{16}\int\\!d^{4}\theta\,\Lambda(\Phi,\Phi^{\dagger})e^{4V}\mathcal{D}^{\alpha}\Phi^{i}\mathcal{D}_{\alpha}\Phi^{j}\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger
i}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger j},$ (4.1)

where we have introduced the gauge invariant superpotential
$W=\Phi_{0}\vec{\Phi}^{t}J\vec{\Phi}$. This will provide the F-term
constraints on the chiral multiplet. Here $\Phi_{0}$ is the $SO(N)$ singlet
and whose $U(1)$ charge is assigned to $-2$, $V$ is the $U(1)$ vector
multiplet and the matrix $J$ is given by

$\displaystyle J=\left(\begin{array}[]{ccc}0&0&1\\\ 0&{\bf 1}_{N-2}&0\\\
1&0&0\end{array}\right).$ (4.5)

The gauge transformations are given by

$\displaystyle\Phi\to e^{-i\Phi^{\prime}}\Phi,\quad\Phi_{0}\to
e^{2i\Theta^{\prime}}\Phi_{0},\quad e^{2V}\to
e^{-i\Theta^{\prime\dagger}}e^{2V}e^{i\Theta^{\prime}}.$ (4.6)

Then the bosonic part of the Lagrangian is evaluated as

$\displaystyle\mathcal{L}_{\mathrm{boson}}=$ $\displaystyle\
-D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}+F^{i}\bar{F}^{i}+F_{0}(\vec{\varphi}^{t}J\vec{\varphi})+\bar{F}_{0}(\vec{\bar{\varphi}}^{t}J\vec{\bar{\varphi}})+D(\bar{\varphi}^{i}\varphi^{i}-\kappa)$
$\displaystyle\
+\Lambda(X)\Big{\\{}(D_{m}\varphi^{i}D_{n}\bar{\varphi}^{i})(D^{m}\varphi^{j}D^{n}\bar{\varphi}^{j})-2(D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i})F^{j}\bar{F}^{j}+(F^{i}F^{i})^{2}\Big{\\}}$
$\displaystyle\
+2\varphi_{0}\left\\{\varphi^{1}F^{N}+\varphi^{N}F^{1}+\sum_{\hat{i}=2}^{N-1}\varphi^{\hat{i}}F^{\hat{i}}\right\\}+2\bar{\varphi}_{0}\left\\{\bar{\varphi}^{1}\bar{F}^{N}+\bar{\varphi}^{N}\bar{F}^{1}+\sum_{\hat{i}=2}^{N-1}\bar{\varphi}^{\hat{i}}\bar{F}^{\hat{i}}\right\\},$
(4.7)

where we have again assumed that $\Lambda$ is a function of the gauge
invariant quantity $X=D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}$. The equation of
motion for $D$ gives the constraint

$\displaystyle\bar{\varphi}^{i}\varphi^{i}=\kappa.$ (4.8)

Since $\Phi_{0}$ does not propagate, it is also integrated out. The F-term
constraints are given by the equation of motions for $F_{0},\varphi_{0}$:

$\displaystyle\delta F_{0}:\vec{\varphi}^{t}J\vec{\varphi}=0,$ (4.9)
$\displaystyle\delta\varphi_{0}:\varphi^{1}F^{N}+\varphi^{N}F^{1}+\sum_{\hat{i}=2}^{N-1}\varphi^{\hat{i}}F^{\hat{i}}=0.$
(4.10)

These give the constraints for the fields $\varphi^{i},\bar{\varphi}^{i}$. The
equations of motion for $\bar{F}^{i}$ are

$\displaystyle\delta\bar{F}^{1}:F^{1}-2\Lambda(X)\Big{\\{}(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})F^{1}-(F^{j}\bar{F}^{j})F^{1}\Big{\\}}+2\bar{\varphi}_{0}\bar{\varphi}^{N}=0,$
$\displaystyle\delta\bar{F}^{\hat{i}}:F^{\hat{i}}-2\Lambda(X)\Big{\\{}(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})F^{\hat{i}}-(F^{j}\bar{F}^{j})F^{\hat{i}}\Big{\\}}+2\bar{\varphi}_{0}\bar{\varphi}^{\hat{i}}=0,\quad(\hat{i}=2,\ldots,N-1),$
$\displaystyle\delta\bar{F}^{N}:F^{N}-2\Lambda(X)\Big{\\{}(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})F^{N}-(F^{j}\bar{F}^{j})F^{N}\Big{\\}}+2\bar{\varphi}_{0}\bar{\varphi}^{1}=0.$
(4.11)

The equations (4.8), (4.9), (4.10) and (4.11) should be solved simultaneously.
In the following, we solve these equations in the canonical and the non-
canonical branches separately.

#### Canonical branch

One finds that a solution to the equations (4.11) is given by

$\displaystyle\varphi_{0}=0,\quad F^{i}=0.$ (4.12)

This corresponds to the canonical branch. We note that in this branch, the
constraint (4.10) becomes trivial. Then the Lagrangian becomes

$\displaystyle\mathcal{L}_{\mathrm{c}}=$ $\displaystyle\
-D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}+\Lambda(X)(D_{m}\varphi^{i}D_{n}\bar{\varphi}^{i})(D^{m}\varphi^{j}D^{n}\bar{\varphi}^{j}).$
(4.13)

The equation of motion for $A_{m}$ is therefore

$\displaystyle-\kappa(i\kappa^{-1}\varphi^{i}\partial_{m}\bar{\varphi}^{i}+A_{m})+i\frac{\partial\Lambda}{\partial
X}(\varphi^{i}D^{m}\bar{\varphi}^{i}-\bar{\varphi}^{i}D^{m}\varphi^{i})$
$\displaystyle+i\Lambda(\varphi,\bar{\varphi})\Bigg{\\{}(\varphi^{i}D_{n}\bar{\varphi}^{i})(D_{m}\varphi^{j}D^{n}\bar{\varphi}^{j})-(\bar{\varphi}^{i}D^{n}\varphi^{i})(D_{m}\bar{\varphi}^{j}D^{n}\varphi^{j})\Bigg{\\}}=0,$
(4.14)

where we have used the constraint (4.8). We find that the solution is given by

$\displaystyle A_{m}=i\kappa^{-1}\bar{\varphi}^{i}\partial_{m}\varphi^{i}.$
(4.15)

Substituting this solution to the Lagrangian, the covariant derivative
$D_{m}\varphi^{i}$ in the Lagrangian (4.13) is replaced by
$\tilde{D}_{m}\varphi^{i}$. Now we solve the remaining constraints (4.8) and
(4.9). The constraint (4.8) is solved by the following parametrization:

$\displaystyle\varphi^{i}=W^{i}\frac{\sqrt{\kappa}}{\sqrt{W^{\dagger}\cdot
W}}.$ (4.16)

Since we have

$\displaystyle\vec{\varphi}^{t}J\vec{\varphi}=\frac{\kappa}{W^{\dagger}\cdot
W}(2W^{1}W^{N}+(W^{\hat{i}})^{2})=0,\qquad(\hat{i}=2,\ldots,N-1),$ (4.17)

the last constraint (4.9) is solved by

$\displaystyle W^{i}=\left(\begin{array}[]{c}1\\\ u^{\hat{i}}\\\
-\frac{1}{2}(u^{\hat{i}})^{2}\end{array}\right),$ (4.21)

where we have fixed $W^{1}=1$ by the $U(1)_{\mathbb{C}}$ gauge transformation.
Plugging this back into the Lagrangian, we obtain the $Q^{N-2}$ sigma model
whose derivative corrections are completely determined by
$\Lambda(W,W^{\dagger},\partial_{m}W,\partial_{m}W^{\dagger})$.

#### Non-canonical branch

In the non-canonical branch, we find that a solution to the equations (4.11)
is given by

$\displaystyle\varphi_{0}=0,\quad
F^{i}\bar{F}^{i}=-\frac{1}{2\Lambda(X)}+D_{m}\varphi^{i}D^{m}\bar{\varphi}^{i}.$
(4.22)

Then, the Lagrangian becomes

$\displaystyle\mathcal{L}_{\mathrm{nc}}=\Lambda(X)\left[(D_{m}\bar{\varphi}^{i}D_{n}\varphi^{i})(D^{m}\bar{\varphi}^{j}D^{n}\varphi^{j})-(D_{m}\varphi^{j}D^{m}\bar{\varphi}^{j})^{2}\right]-\frac{1}{4\Lambda(X)}.$
(4.23)

The equation of motion for $A_{m}$ is the same with (3.20) and it is again
solved by (3.21). Substituting this into the Lagrangian, the gauge covariant
derivative $D_{m}\varphi^{i}$ is replaced by $\tilde{D}_{m}\varphi^{i}$. The
remaining constraints (4.8) and (4.9) are solved by the parametrizations
(4.16) and (4.21) and the target space is $Q^{N-2}$. However, we should keep
in mind that there is an extra constraint (4.10) given by

$\displaystyle\varphi^{1}F^{N}(\tilde{X})+\varphi^{N}F^{1}(\tilde{X})+\sum_{\hat{i}=2}^{N-1}\varphi^{\hat{i}}F^{\hat{i}}(\tilde{X})=0,$
(4.24)

where $F^{i}(\tilde{X})\not=0$ is the solution to the auxiliary field in the
non-canonical branch. The constraint (4.24) generically contains space-time
derivatives of the fields $\varphi^{i},\bar{\varphi}^{\bar{i}}$ and this has
to be solved together with the following equation of motion:

$\displaystyle\Lambda^{\prime}(\tilde{X})\tilde{D}^{m}\varphi^{i}\left[(\tilde{D}_{m}\bar{\varphi}^{j}\tilde{D}_{n}\varphi^{j})(\tilde{D}^{m}\bar{\varphi}^{k}\tilde{D}^{n}\varphi^{k})-(\tilde{D}_{m}\varphi^{j}\tilde{D}^{m}\bar{\varphi}^{j})^{2}\right]$
$\displaystyle-2\Lambda(\tilde{X})\left[\tilde{D}_{n}\varphi^{i}(\tilde{D}^{m}\bar{\varphi}^{j}\tilde{D}^{n}\varphi^{j})-\tilde{D}_{m}\varphi^{i}(\tilde{D}_{n}\varphi^{j}\tilde{D}^{n}\bar{\varphi}^{j})\right]+\frac{1}{4\Lambda^{2}(\tilde{X})}\Lambda^{\prime}(\tilde{X})\tilde{D}^{m}\varphi^{i}=0,$
(4.25)

where the prime in $\Lambda^{\prime}(\tilde{X})$ denotes the differentiation
with respect to $\tilde{X}$. In general, it is uncertain whether the
simultaneous equations (4.24) and (4.25) admit non-trivial solutions or not.
Instead, the most plausible way to work with these equations is to restrict
the dynamics to a subspace in $Q^{N-2}$. This subspace highly depends on
explicit structures of the solution $F^{i}$. For example, if we consider the
$N=4$ case, one finds that a solution to the equation of motion for $F^{i}$ in
the non-canonical branch is

$\displaystyle F^{2}=F^{3}=F^{4}=0,\qquad
F^{1}=\sqrt{-\frac{1}{2\Lambda(\tilde{X})}+\tilde{D}_{m}\varphi^{i}\tilde{D}^{m}\bar{\varphi}^{i}}.$
(4.26)

Substituting this solution to the constraint (4.24) gives a condition

$\displaystyle\varphi^{4}\sqrt{-\frac{1}{2\Lambda(\tilde{X})}+\tilde{D}_{m}\varphi^{i}\tilde{D}^{m}\bar{\varphi}^{i}}=0.$
(4.27)

In order that this is compatible with equation of motion, we first choose the
subspace $\varphi^{4}=0$ in $Q^{2}$, and then solve the equation (4.25) in
that subspace.

A comment is in order. As for the Grassmann case, we have an equality
$Q^{4}=G_{4,2}$. The constraint (4.24) is necessary in the construction of the
$Q^{4}$ sigma model discussed in this section but such a restriction to a
subspace was absent in the construction of the $G_{4,2}$ model in the previous
section. Apparently, they have different Lagrangians even though there is a
target space isomorphism between them. The same is true for the relation
$Q^{1}=G_{2,1}=\mathbb{C}P^{1}$. In the non-canonical branch, the different
constructions of sigma models result in the different derivative corrections
and the field space that the dynamics occurs, although they give identical
models in the canonical branches.

###  4.2 $SO(2N)/U(N)$ and $Sp(N)/U(N)$ models

Observed that the $Q^{N-2}$ model has remarkable structures in the non-
canonical branch due to the F-term constraint, we next consider nonlinear
sigma models with the target spaces $SO(2N)/U(N)$ or $Sp(N)/U(N)$. The gauged
linear sigma model has the global symmetry $G=SO(2N)$ or $Sp(N)[=USp(2N)]$ and
gauge symmetry $G^{\prime}=U(N)$. The chiral superfield $\Phi$ belongs to the
$(\mathbf{2N},\bar{\mathbf{N}})$ representation of $G\times G^{\prime}$ and it
is expressed as an $2N\times N$ matrix. The constraint is given by the F-term
superpotential

$\displaystyle W=\mathrm{Tr}\left[\Phi_{0}\Phi^{t}J^{\prime}\Phi\right],$
(4.28)

where $\Phi_{0}$ is an $N\times N$ symmetric (anti-symmetric) matrix
superfield satisfying $\Phi_{0}^{t}=\epsilon\Phi_{0}$ for $G=SO(2N)[Sp(N)]$.
Here $\epsilon=1$ for $G=SO(2N)$ and $\epsilon=-1$ for $G=Sp(N)$. The
$U(1)_{D}\subset U(N)$ charge of $\Phi_{0}$ is assigned to $-2$ to cancel the
$U(1)$ charge $+1$ of $\Phi$. The $2N\times 2N$ matrix $J^{\prime}$ is given
by

$\displaystyle J^{\prime}=\left(\begin{array}[]{cc}0&\mathbf{1}_{N}\\\
\epsilon\mathbf{1}_{N}&0\end{array}\right),$ (4.31)

which is an invariant tensor of $SO(2N)$ ($\epsilon=1$) or $Sp(N)$
($\epsilon=-1$).

The equation of motion for $D$ gives the constraint (3.29) while that for
$F_{0}$ gives the constraint

$\displaystyle\varphi^{t}J^{\prime}\varphi=0,$ (4.32)

and that for $\varphi_{0}$ gives a constraint

$\displaystyle\varphi^{t}J^{\prime}F=0.$ (4.33)

The equation for $F$ is (4.11), but the gauge symmetry is non-Abelian. One
finds that $\varphi_{0}=0$ and $F=0$ are a solution to the equation of motion
for $F$. Therefore the constraint (4.33) becomes trivial in the canonical
branch. We find that the gauge field is integrated out by the solution
$A_{m}=i\kappa^{-1}\bar{\varphi}\partial_{m}\varphi$ in the sigma model limit
which is the same with the case without the derivative corrections. Then the
target space of the sigma model is nothing but the $SO(2N)/U(N)$ or
$Sp(N)/U(N)$ and the derivative corrections appear as in the Grassmann case in
Eq. (3.37).

Compared with the $Q^{N-2}$ model, it is not straightforward to write down
solutions $F\not=0$ explicitly due to the non-Abelian gauge symmetry. However,
similar to the Grassmann case, the solution $\mathrm{Tr}[F\bar{F}]$ is given
by the gauge invariant quantities such as $\mathrm{Tr}[D_{m}\varphi
D^{m}\bar{\varphi}]$. Therefore we again find that the gauge field $A_{m}$ is
integrated out by $A_{m}=i\kappa^{-1}\bar{\varphi}\partial_{m}\varphi$. The
subsequent discussion is parallel to the $Q^{N-2}$ case. Due to the extra
constraint (4.33), we find that the target spaces $SO(2N)/U(N)$ or
$Sp(N)/U(N)$ should be restricted to their subspaces in the non-canonical
branches.

###  4.3 $E_{6}/[SO(10)\times U(1)]$ and $E_{7}/[E_{6}\times U(1)]$ models

For the case of the target space $E_{6}/[SO(10)\times U(1)]$, the global
symmetry is $G=E_{6}$ and the $G^{\prime}=U(1)$ symmetry is gauged in the
gauged linear sigma model [46]. The chiral superfield $\Phi$ belongs to the
fundamental representation $\mathbf{27}$ of $E_{6}$. This is decomposed into
the maximal subgroup $SO(10)\times U(1)$. The F-term constraint is given by
the superpotential

$\displaystyle W=\Gamma_{ijk}\Phi_{0}^{i}\Phi^{j}\Phi^{k},$ (4.34)

where $\Gamma_{ijk}$ is a rank-3 symmetric invariant tensor of $E_{6}$. Since
the gauge symmetry is Abelian, the structure of the sigma model is essentially
the same with the $Q^{N-2}$ case. Due to the constraint

$\displaystyle\Gamma_{ijk}\varphi^{i}F^{k}=0$ (4.35)

derived by the superpotential, the target space of the sigma model is
restricted to a subspace of $E_{6}/[SO(10)\times U(1)]$ in the non-canonical
branch.

For the target space $E_{7}/[E_{6}\times U(1)]$, the global symmetry is
$G=E_{7}$ and the $G^{\prime}=U(1)$ symmetry is gauged in the gauge linear
sigma model. The F-term constraint is [46]

$\displaystyle
W=d_{\alpha\beta\gamma\delta}\Phi_{0}^{\alpha}\Phi^{\beta}\Phi^{\gamma}\Phi^{\delta},$
(4.36)

where $d_{\alpha\beta\gamma\delta}$ is a rank-4 symmetric invariant tensor of
$E_{7}$. Since the gauge symmetry is Abelian, the structure of the sigma model
is essentially the same with the $Q^{N-2}$ case. Again, the constraint

$\displaystyle
d_{\alpha\beta\gamma\delta}\varphi^{\alpha}\varphi^{\beta}F^{\gamma}=0$ (4.37)

defines a subspace in $E_{7}/[E_{6}\times U(1)]$ in the non-canonical branch.

##  5 Bogomol’nyi-Prasad-Sommerfield states

In this section, we study the BPS conditions in the nonlinear sigma models
discussed in the previous section. BPS properties of Skyrme type and higher
derivative models have been studied in various contexts [59, 60, 61, 62]. It
has been discussed that the gauged chiral models admit the 1/2 BPS vortex
state and the 1/4 BPS states [29]. In the following, we derive the 1/2 and 1/4
BPS conditions in the nonlinear sigma models in the limit $g\to\infty$ of
those in the gauged chiral models.

###  5.1 BPS states in canonical branch

In the canonical branch, the 1/2 BPS vortex equation for the finite $g$ for
gauged linear sigma model is

$\displaystyle\bar{D}_{z}\varphi^{i}=0,\qquad\frac{1}{g}F_{12}-(\bar{\varphi}\varphi-\kappa\mathbf{1})=0,$
(5.1)

where $z=x^{1}+ix^{2}$ is the complex coordinate in the $(x^{1},x^{2})$-plane.
These are $U(M)$ BPS semilocal vortex equations with $N$-flavors in the case
of Grassmann case [58, 54], for which we have found that higher derivative
corrections do not appear. They reduce to the ordinary BPS vortex equations in
the Abelian-Higgs model for the $U(1)$ gauge theory.

The energy bound for this configuration is [29]333 We comment on the positive-
semi definiteness of the energy in the higher derivative chiral models. It is
not always true that the energy density derived from Eq. (2.1) is positive-
semi definite for arbitrary $\Lambda$ and the Kähler potential $K$. In order
to define the BPS states as minima of energy, we have to choose appropriate
$\Lambda$ and $K$ that makes the energy be positive-semi definite [27].

$\displaystyle\mathcal{E}=-\kappa F^{0}_{12}.$ (5.2)

The configuration is classified by the vortex number
$\int\\!d^{2}x\,F^{0}_{12}$.

In the nonlinear sigma model limit $g\to\infty$, the second condition in (5.1)
gives the D-term constraint while the first one becomes

$\displaystyle\bar{\partial}_{z}\varphi^{i}-\kappa^{-1}(\bar{\varphi}^{j}\bar{\partial}_{z}\varphi^{j})\varphi^{i}=0.$
(5.3)

This implies the following 1/2 BPS lump equation in the sigma models:

$\displaystyle\bar{\partial}_{z}\varphi^{i}=0.$ (5.4)

The bound (5.2) survives in the limit $g\to\infty$ and the result is

$\displaystyle\mathcal{E}=\frac{1}{2}|\partial_{z}\varphi|^{2}.$ (5.5)

This provides the lump charge density in the nonlinear sigma model. We note
that the derivative corrections never show up in the equation (5.4) and the
BPS bound (5.5) in the canonical branch.444 As denoted below Eq. (3.16), for
the case of the ${\mathbb{C}}P^{N-1}$ model, the Lagrangian in Eq. (3.16)
(with $\Lambda=$const.) can be obtained [50] as the low-energy effective
theory of a BPS non-Abelian vortex in ${\cal N}=2$ supersymmetric $U(N)$ gauge
theory [54]. In this context, ${\mathbb{C}}P^{N-1}$ lumps (sigma model
instantons) on the vortex worldsheet represent Yang-Mills instantons in the
bulk theory, and such instanton-vortex composites are 1/4 BPS states [56]. It
was discussed in Ref. [50] that this fact implies that lump solutions should
not have derivative corrections.

###  5.2 BPS states in non-canonical branch

In the non-canonical branch, we have only 1/4 BPS state [29]. The BPS
equations in the gauged linear sigma models are

$\displaystyle\bar{D}_{z}\varphi=-iF,\qquad\frac{1}{g}F_{12}=\bar{\varphi}\varphi-\kappa\mathbf{1}.$
(5.6)

In the nonlinear sigma model limit $g\to\infty$, the BPS equations become

$\displaystyle\bar{D}_{z}\varphi=-iF,\qquad\bar{\varphi}\varphi=\kappa\mathbf{1}.$
(5.7)

The latter equation gives the D-term constraint. From the first equation, we
obtain

$\displaystyle\bar{D}_{z}\varphi D_{z}\bar{\varphi}=F\bar{F}.$ (5.8)

For the $\mathbb{C}P^{N-1}$, $Q^{N-2}$ cases, the gauge group is Abelian and
we find an explicit solution for $F$. Then the BPS condition is rewritten as

$\displaystyle\bar{\tilde{D}}_{z}\bar{\varphi}\tilde{D}_{z}\varphi-\bar{\tilde{D}}_{z}\varphi^{i}\tilde{D}_{z}\bar{\varphi}^{i}=\Lambda^{-1}(\varphi,\bar{\varphi}).$
(5.9)

This is nothing but a compacton type equation. To see this, we choose an
appropriate function $\Lambda$. For example, we consider the following
function in the $\mathbb{C}P^{1}$ model,

$\displaystyle\Lambda=2\left(\frac{|u|^{2}}{1+|u|^{2}}\right)^{-\frac{s}{2}},$
(5.10)

where $s$ is a constant. Then, the BPS equation (5.9) results in the following
compacton equation [45]:

$\displaystyle ng_{y}=-g^{\frac{s}{2}}.$ (5.11)

Here we have assumed the ansatz

$\displaystyle u=e^{in\theta}f(r),\quad 1-g=\frac{1}{1+f^{2}},\quad
y=\frac{r^{2}}{2},$ (5.12)

where $r$ and $\theta$ are the polar coordinates in the $(x^{1},x^{2})$-plane.

##  6 Comment on fermionic interactions

Finally, we comment on the fermionic interactions of the models, while in the
previous sections, we have focused on the bosonic sector of the supersymmetric
models and have shown that the vector field $A_{m}$ is integrated out exactly.
However, the integration of the gaugino is rather awkward from the reason
described below. We illustrate the problem in the $\mathbb{C}P^{N-1}$ model.

Assuming that $\Lambda$ does not contain $V$ for simplicity, the equation of
motion for the vector superfield $V$ in the sigma model limit $g\to\infty$ is
given by

$\displaystyle\quad 2\Phi^{i}e^{2V}\Phi^{\dagger i}-\kappa$
$\displaystyle+\frac{1}{4}\Lambda(\Phi,\Phi^{\dagger})e^{4V}\mathcal{D}^{\alpha}\Phi^{i}\mathcal{D}_{\alpha}\Phi^{j}\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger
i}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger j}$
$\displaystyle+\frac{1}{4}D_{\alpha}\left[\Lambda(\Phi,\Phi^{\dagger})e^{4V}\Phi^{i}\mathcal{D}^{\alpha}\Phi^{j}\bar{\mathcal{D}}_{\dot{\alpha}}\Phi^{\dagger
i}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger j}\right]$
$\displaystyle+\frac{1}{4}\bar{D}_{\dot{\alpha}}\left[\Lambda(\Phi,\Phi^{\dagger})e^{4V}\mathcal{D}_{\alpha}\Phi^{i}\mathcal{D}^{\alpha}\Phi^{j}\Phi^{\dagger
i}\bar{\mathcal{D}}^{\dot{\alpha}}\Phi^{\dagger j}\right]=0.$ (6.1)

If the term in the second line is ignored, we find that the vector superfield
is solved by

$\displaystyle V=-\frac{1}{2}\log(2\kappa^{-1}\Phi^{i}\Phi^{\dagger i}).$
(6.2)

This is easily confirmed if one notices that the following relation holds for
Eq. (6.2):

$\displaystyle\Phi^{\dagger i}\mathcal{D}_{\alpha}\Phi^{i}=\Phi^{\dagger
i}\left[D_{\alpha}\Phi^{i}+2(D_{\alpha}V)\Phi^{i}\right]=\Phi^{\dagger
i}D_{\alpha}\Phi^{i}-\frac{1}{|\Phi|^{2}}\Phi^{\dagger
j}D_{\alpha}\Phi^{j}(\Phi^{i}\Phi^{\dagger i})=0.$ (6.3)

The would-be solution (6.2), whose $\theta\sigma^{m}\bar{\theta}$ component
gives $A_{m}=i\kappa^{-1}\bar{\varphi}^{i}\partial_{m}\varphi^{i}$, precisely
gives the correct Kähler potential $K=-\log(2\kappa^{-1}\Phi\Phi^{\dagger})$
for the $\mathbb{C}P^{N-1}$ model. However, when the term in the second line
in Eq. (6.1) is included, Eq. (6.2) fails to be a solution. This implies that
the expression in Eq. (6.2) is correct only in the bosonic sector. Including
the second line in Eq. (6.1) modifies the equality in Eq. (6.2) in the
fermionic sector.

We note that in solving the auxiliary field $F$, the fermionic fields $\psi$
in the chiral multiplets are introduced perturbatively around the bosonic
solution $F=F(\varphi,\bar{\varphi})$ [20]. The same is true for the vector
multiplet. Since the interaction terms are all expressed by the superfields
explicitly, it is in principle possible to write down the equations for the
fermions. We can integrate out the vector multiplet starting from the bosonic
part of the solution (6.2) and introduce the fermions perturbatively. This
procedure determines the fermionic interactions in the sigma model limit
$g\to\infty$. Even though the fermionic interactions are rather cumbersome, we
stress that the BPS conditions in the nonlinear sigma models are not affected
by details of the fermionic interactions. As we have shown, this is obtained
by those in the gauged chiral models.

##  7 Conclusion and discussions

In this paper, we have constructed $\mathcal{N}=1$ supersymmetric higher
derivative terms in nonlinear sigma models whose target spaces are Hermitian
symmetric spaces. We have considered the sigma model limit of the
supersymmetric gauged linear sigma models involving the derivative terms for
chiral multiplets that are free from the auxiliary field problem and the
Ostrogradski’s ghost. In the sigma model limit, the vector multiplet does not
propagate anymore and can be integrated out. Due to the supersymmetric
derivative corrections, there are two distinct on-shell branches called the
canonical and non-canonical branches. This structure is carried over to the
nonlinear sigma models. We have shown that the gauge field $A_{m}$ is
explicitly integrated out, both in the canonical and the non-canonical
branches, even in the presence of higher derivative terms.

In the canonical branch, where the auxiliary field is given by $F=0$, we
explicitly have written down the Lagrangians of the nonlinear sigma models.
They consist of the canonical kinetic terms and the derivative corrections
characterized by the arbitrary function $\Lambda$. This is a natural
generalization of the sigma model construction discussed in Ref. [46].

For the $\mathbb{C}P^{N-1}$ model, we have obtained the supersymmetric
$\mathbb{C}P^{N-1}$ Skyrme-Faddeev model. In the non-canonical branch, where
the auxiliary field is given by $F\not=0$, the situation changes drastically.
For the $\mathbb{C}P^{N-1}$ model, an Abelian symmetry is gauged and we have
found the explicit solutions of the non-zero auxiliary field $F$. We have
solved all the constraints and have written down the explicit Lagrangians of
the $\mathbb{C}P^{N-1}$ nonlinear sigma model. The canonical quadratic kinetic
term is absent in the non-canonical branch, and there is only the fourth
derivative term as a generalization of the Skyrme-Faddeev term of the
$\mathbb{C}P^{1}$ model. We thus have obtained the supersymmetric
$\mathbb{C}P^{N-1}$ Skyrme-Faddeev model.

For the formulation of the $G_{M,N}$ model, a non-Abelian $U(N)$ symmetry is
gauged in the gauged linear sigma model. Compared with the $\mathbb{C}P^{N-1}$
case for which the gauge symmetry is Abelian in the gauged linear sigma model,
the equation of motion for the auxiliary field in the $G_{M,N}$ model is
rather complicated. Although it is hard to find explicit $F\not=0$ solutions,
we have been able to integrate out the gauge field with the help of the gauge
invariance of the higher derivative terms.

For other Hermitian symmetric spaces, in addition to D-term constraints, we
further impose F-term constraints yielding holomorphic embedding of the target
spaces into $\mathbb{C}P^{N-1}$ or Grassmann manifold, as was done for the
case without higher derivative terms. In the canonical branches, these
constraints are consistent in the presence of higher derivative terms, but we
find that in the non-canonical branch these constraints yield further
additional constraints reducing the target spaces to their submanifolds.

For the formulation of the $Q^{N-2}$ model, an Abelian symmetry is gauged in
the gauged linear sigma model, but we have an extra constraint coming from the
F-term embedding $Q^{N-2}$ to $\mathbb{C}P^{N-1}$. The non-zero solution
$F\not=0$ yields an extra constraint on the scalar fields. We have found that
in order that these constraints are compatible with the equation of motion in
the non-canonical branches, the dynamics of fields should be restricted to a
subspace in the $Q^{N-2}$ manifold, which is a peculiar property of this
model. We have also discussed the sigma models whose target spaces are
$SO(2N)/U(N)$, $Sp(N)/U(N)$, $E_{6}/[SO(10)\times U(1)]$ and
$E_{7}/[E_{6}\times U(1)]$. The first two cases are essentially similar to the
cases of the Grassmann and the $Q^{N-2}$ cases. For the latter two cases,
although the superpotential is complicated compared with the $Q^{N-2}$ case,
the structure of the higher derivative terms is similar to that of the
$Q^{N-2}$ case. We also have found that these constructions provide the
different sigma models even though there are several isomorphisms among the
target spaces such as $G_{N,1}\simeq G_{N,N-1}\simeq\mathbb{C}P^{N-1}$,
$Q^{1}\simeq G_{2,1}\simeq\mathbb{C}P^{1}$, $Q^{4}\simeq G_{4,2}$ and so on.

Finally, we provide a comment on the BPS equations and the integration of the
fermionic terms in the vector multiplet. In the canonical branches, BPS
equations and their solutions are the same with those without higher
derivative terms, while, in the non-canonical branches they give compacton
type configurations.

It would be interesting to study explicit solutions to the BPS equations in
the non-canonical branch. As we have noted, the 1/4 BPS equation in the non-
canonical branch of the $\mathbb{C}P^{N-1}$ model reduces to that of the
compactons. Finding solutions in the other sigma models is interesting. We
will come back to these issues in future studies.

In this paper, we have considered the case that there are no potential terms
in the nonlinear sigma models (although we have introduced superpotentials for
the F-term constraints in the gauged linear sigma models). If we also
introduce superpotentials or twisted masses (from dimensional reductions),
there are more BPS states such as domain walls, their junctions and so on
[27]. Introducing potential terms and studying associated BPS states remain as
a future work.

### Acknowledgments

This work was supported in part by Grant-in-Aid for Scientific Research, JSPS
KAKENHI Grant Numbers JP18H01217 (M. N.) and JP20K03952 (S. S.).

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