$(0,4)$ Projective Superspaces I:
Interacting Linear Sigma Models

Note that there is more than one solution to the equation $\varepsilon_{ab}\widetilde{u}^{a}{u}^{b}=1$. If $\widetilde{u}_{0}^{b}$ is one solution, then so is $\widetilde{u}_{0}^{b}+\omega u^{b}$ for any $\omega\in\mathbf{C}$. Thus there is a shift symmetry on the $\widetilde{u}^{a}$:

$$\widetilde{u}^{a}\to\widetilde{u}^{a}+\omega u^{a}\ ,\quad\text{for}\quad\omega\in\mathbf{C}\ .$$

(2.8)

There is a similar shift symmetry for the conjugate doublet $\widetilde{v}^{a^{\prime}}$.

Consider the derivatives

$$\mathbf{D}_{a^{\prime}+}:=u^{a}\mathrm{D}_{aa^{\prime}+}\ ,\quad\widetilde{\mathbf{D}}_{a^{\prime}+}:=\widetilde{u}^{a}\mathrm{D}_{aa^{\prime}+}\ ,\quad\mathbf{D}_{a+}:=v^{a^{\prime}}\mathrm{D}_{aa^{\prime}+}\ ,\quad\widetilde{\mathbf{D}}_{a+}:=\widetilde{v}^{a^{\prime}}\mathrm{D}_{aa^{\prime}+}\ ,$$

(2.9)

where $\widetilde{u}^{a}$ and $\widetilde{v}^{a^{\prime}}$ are any solutions to the equations (2.7). The algebra of the derivatives (2.9) is obtained from (2.3):

		$\displaystyle\{\mathbf{D}_{a^{\prime}+}\,,\mathbf{D}_{b^{\prime}+}\}=0\ ,\quad\{\widetilde{\mathbf{D}}_{a^{\prime}+}\,,\widetilde{\mathbf{D}}_{b^{\prime}+}\}=0\ ,\quad\{\mathbf{D}_{a^{\prime}+}\,,\widetilde{\mathbf{D}}_{b^{\prime}+}\}=-2\text{i}\varepsilon_{a^{\prime}b^{\prime}}\partial_{++}\ ,$
		$\displaystyle\{\mathbf{D}_{a+}\,,\mathbf{D}_{b+}\}=0\ ,\quad\{\widetilde{\mathbf{D}}_{a+}\,,\widetilde{\mathbf{D}}_{b+}\}=0\ ,\quad\{\mathbf{D}_{a+}\,,\widetilde{\mathbf{D}}_{b+}\}=-2\text{i}\varepsilon_{ab}\partial_{++}\ .$		(2.10)

Note that the shift symmetry (2.8) shifts the derivatives $\widetilde{\mathbf{D}}_{a^{\prime}+}$ by $\omega\mathbf{D}_{a^{\prime}+}$ but it leaves the algebra (2.2) unchanged. We shall see below that the action is also invariant under the shift symmetry up to total derivative terms.

We also introduce the fully contracted derivative

$$\mathbf{D}_{+}=u^{a}v^{a^{\prime}}\mathrm{D}_{aa^{\prime}+}=u^{a}\mathbf{D}_{a+}=v^{a^{\prime}}\mathbf{D}_{a^{\prime}+}\quad\text{which satisfies}\quad\mathbf{D}_{+}^{2}=0\ ,$$

(2.11)

due to the anticommutation relations (2.3). We can recover the algebra in (2.2) by writing $\mathbf{D}_{+}^{2}=u^{a}u^{b}\{\mathbf{D}_{a+},\mathbf{D}_{b+}\}$ or $\mathbf{D}_{+}^{2}=v^{a^{\prime}}v^{b^{\prime}}\{\mathbf{D}_{a^{\prime}+},\mathbf{D}_{b^{\prime}+}\}$.

An $F$-projective superfield $\bm{\Phi}(\underline{z},u)$ is a function of the superspace coordinates $\underline{z}=(x_{\pm\pm},\theta^{aa^{\prime}+})$ and the $\mathbf{CP}^{1}$ coordinates $u^{a}$ which satisfy the following:

1. (1)

   $\bm{\Phi}$ is holomorphic in a domain in $\mathbf{CP}^{1}$,

2. (2)

   $\bm{\Phi}$ satisfies the projective constraints $\mathbf{D}_{a^{\prime}+}\bm{\Phi}(\underline{z},u)=0$,

3. (3)

   $\bm{\Phi}$ may be in non-trivial representations of the R-symmetry group $\text{SU}(2)\times\text{SU}(2)^{\prime}$ and the Lorentz group $\text{SO}(1,1)$.

An $F^{\prime}$-projective superfield is analogously a function of the superspace coordinates $\underline{z}$ and the $\mathbf{CP}^{1}{}^{\prime}$ coordinate $v^{a^{\prime}}$ and is annihilated by $\mathbf{D}_{a+}$. We discuss the different types of projective superfields in Section 2.4.

The $F$-projective constraints $\mathbf{D}_{a^{\prime}+}\bm{\Phi}(\underline{z},u)=0$ can be encoded more economically in terms of the fully contracted derivative (2.11) $\mathbf{D}_{+}=v^{a^{\prime}}\mathbf{D}_{a^{\prime}+}$:

$$\mathbf{D}_{+}\bm{\Phi}=0\ .$$

(2.12)

Since $\bm{\Phi}$ depends only on $u$ and not on $v$, $\mathbf{D}_{+}\bm{\Phi}=v^{a^{\prime}}\mathbf{D}_{a^{\prime}+}\bm{\Phi}$ implies $\mathbf{D}_{a^{\prime}+}\bm{\Phi}=0$. The advantage of (2.12) is that it takes the same form for $F^{\prime}$-projective superfields $\bm{\Phi}(\underline{z},v)$ as well, since we can now recover $\mathbf{D}_{a+}\bm{\Phi}=0$ using $\mathbf{D}_{+}=u^{a}\mathbf{D}_{a+}$. We frequently use the derivative $\mathbf{D}_{+}$ in the paper.

The constraints $\mathbf{D}_{a^{\prime}+}\bm{\Phi}=0$ on a projective superfield $\bm{\Phi}$ imply that $\bm{\Phi}$ depends on only half of the Grassmann coordinates. The appropriate superspace measure which ensures $(0,4)$ invariance of an action composed of projective superfields is then quadratic in the derivatives $\widetilde{\mathbf{D}}_{a^{\prime}+}$, i.e., $\widetilde{\mathbf{D}}_{1^{\prime}+}\widetilde{\mathbf{D}}_{2^{\prime}+}$. The $(0,4)$ supersymmetric action is then given by

$$\mathcal{S}[\bm{\Phi}]=\int\text{d}^{2}x\left(\frac{1}{2\pi\text{i}}\oint_{\gamma}\varepsilon_{ab}u^{a}\text{d}u^{b}\ \widetilde{\mathbf{D}}_{1^{\prime}+}\widetilde{\mathbf{D}}_{2^{\prime}+}\ {\bm{K}}_{--}(\bm{\Phi})\right)_{\boldsymbol{\rvert}}\ ,$$

(2.13)

where

1. 1.

   ${\boldsymbol{\rvert}}$ sets all the Grassmann coordinates to zero (we frequently omit the ${\boldsymbol{\rvert}}$ from our expressions).

2. 2.

   ${\bm{K}}_{--}$ is the superspace Lagrangian which satisfies $\mathbf{D}_{a^{\prime}+}{\bm{K}}_{--}=0$. It must carry the $--$ Lorentz representation (left-moving part of a vector) in order to compensate the $++$ in the projective superspace measure.

3. 3.

   The contour $\gamma\in\mathbf{CP}^{1}$ is chosen to avoid possible singularities in $\widetilde{\mathbf{D}}_{1^{\prime}+}\widetilde{\mathbf{D}}_{2^{\prime}+}{\bm{K}}_{--}$.

The action is invariant (up to total spacetime derivatives) under the shift symmetry (2.8) $\widetilde{\mathbf{D}}_{a^{\prime}+}\to\widetilde{\mathbf{D}}_{a^{\prime}+}+\omega\mathbf{D}_{a^{\prime}+}$ since the Lagrangian ${\bm{K}}_{--}$ satisfies $\mathbf{D}_{a^{\prime}+}{\bm{K}}_{--}=0$. Since the superspace measure $\varepsilon_{ab}u^{a}\text{d}u^{b}\widetilde{\mathbf{D}}_{1^{\prime}+}\widetilde{\mathbf{D}}_{2^{\prime}+}$ is invariant under $F$ and $F^{\prime}$, the action (2.13) is manifestly invariant under $F$ and $F^{\prime}$ if the superspace Lagrangian is invariant.

Suppose a projective superfield $\Phi_{s}$ is in the spin $s$ representation of the Lorentz group $\text{SO}(1,1)$. The requirement that $\mathbf{D}_{a^{\prime}+}\bm{\Phi}_{s}=0$ can be relaxed to have a non-zero right hand side:

$$\mathbf{D}_{a^{\prime}+}\bm{\Phi}_{s}={\bm{S}}_{a^{\prime},s+1}\ ,$$

(2.14)

where ${\bm{S}}_{a^{\prime},s+1}$ is a function of other superfields in the model and is in the spin $s+1$ representation of $\text{SO}(1,1)$. This allows us to introduce interactions (the so-called $E$-terms) as we will see later in Section 5:

The modified constraints (2.14) are consistent with the algebra $\{\mathbf{D}_{a^{\prime}+},\mathbf{D}_{b^{\prime}+}\}=0$ only if the function satisfies

$$\mathbf{D}_{a^{\prime}+}{\bm{S}}_{b^{\prime},s+1}+\mathbf{D}_{b^{\prime}+}{\bm{S}}_{a^{\prime},s+1}=0\ .$$

(2.15)

To ensure $(0,4)$ invariance of the action, we require that the superspace Lagrangian ${\bm{K}}_{--}(\bm{\Phi})$ satisfies $\mathbf{D}_{a^{\prime}+}{\bm{K}}_{--}=0$ even if $\mathbf{D}_{a^{\prime}+}\bm{\Phi}$ is not zero. This further constrains the ${\bm{S}}_{a^{\prime},s+1}$.

Thus, any $(0,4)$ supersymmetric model must satisfy the following constraints:

1. 1.

   The $(0,4)$ algebra $\mathbf{D}_{+}^{2}=0$ must be satisfied on every superfield in the model,

2. 2.

   The superspace Lagrangian ${\bm{K}}_{--}$ must satisfy $\mathbf{D}_{a^{\prime}+}{\bm{K}}_{--}=0$ to ensure $(0,4)$ supersymmetry of the action.

These criteria place stringent constraints on the superfield content and the interactions in a model.

The projective space $\mathbf{CP}^{1}$ is constructed as the quotient space $\{\mathbf{C}^{2}\smallsetminus 0\}/\sim$, where $\sim$ is the following equivalence relation on the coordinates of $\mathbf{C}^{2}$: $(u^{1},u^{2})\sim(\lambda u^{1},\lambda u^{2})$, $\lambda\in\mathbf{C}^{\star}$. We describe $\mathbf{CP}^{1}$ in terms of two charts $\mathsf{U}_{1}$ and $\mathsf{U}_{2}$:

$$\mathsf{U}_{a}:=\{(u^{1},u^{2})\in\mathbf{C}^{2}\ |\ u^{a}\neq 0\}\ .$$

(2.16)

The map $\mathcal{S}\in\text{SU}(2)$ which acts on the homogeneous coordinates as

$$\mathcal{S}:\ \begin{pmatrix}u^{1}\\ u^{2}\end{pmatrix}\longmapsto\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}u^{1}\\ u^{2}\end{pmatrix}=\begin{pmatrix}-u^{2}\\ u^{1}\end{pmatrix}\ ,$$

(2.17)

interchanges the two charts. Using the equivalence $(u^{1},u^{2})\sim(\lambda u^{1},\lambda u^{2})$, $\lambda\in\mathbf{C}^{\star}$, we can scale out the non-zero coordinate in each of the charts and obtain a description in terms of inhomogeneous coordinates:

$$\mathsf{U}_{1}=\{(1,-\rho)\ |\ \rho\in\mathbf{C}\}\ ,\quad\mathsf{U}_{2}=\{(\zeta,1)\ |\ \zeta\in\mathbf{C}\}\ ,$$

(2.18)

with $\rho=-u^{2}/u^{1}$ and $\zeta=u^{1}/u^{2}$. On the intersection $\mathsf{U}_{12}:=\mathsf{U}_{1}\cap\mathsf{U}_{2}$, the local coordinates $\zeta$ and $\rho$ are related by the $\mathcal{S}$ map (2.17)

$$\mathcal{S}:\ \zeta\longmapsto-1/\zeta=\rho\ .$$

(2.19)

We can thus express all our results exclusively in terms of one of the inhomogeneous coordinates, say $\zeta$, by appending the point $\zeta=\infty$ to the chart $\mathsf{U}_{2}$. We frequently adopt this usage to avoid cluttering of notation.

We next express the derivatives $\mathbf{D}_{a^{\prime}+}$ and $\widetilde{\mathbf{D}}_{a^{\prime}+}$ in terms of the local coordinates $\zeta$ and $\rho$ in the charts $\mathsf{U}_{2}$ and $\mathsf{U}_{1}$ respectively. In the chart $\mathsf{U}_{2}$, we have $u^{2}\neq 0$ and $u^{a}=(u^{2})(\zeta,1)$. Thus, we can choose $\widetilde{u}^{a}=(u^{2})^{-1}(1,0)$ which indeed satisfies $\widetilde{u}{}^{a}u^{b}\varepsilon_{ab}=1$. Using the scale invariance $u^{a}\to\lambda u^{a}$, we can set $u^{2}=1$ as discussed above (2.18). The derivatives $\mathbf{D}_{a^{\prime}+}=u^{a}\mathrm{D}_{aa^{\prime}+}$ and $\widetilde{\mathbf{D}}_{a^{\prime}+}=\widetilde{u}{}^{a}\mathrm{D}_{aa^{\prime}+}$ are then given by

$\displaystyle\text{In $\mathsf{U}_{2}$}:\quad$

$\displaystyle\mathbf{D}_{1^{\prime}+}=\zeta\mathrm{D}_{+}+\mathrm{Q}_{+}\ ,\quad\mathbf{D}_{2^{\prime}+}=-\zeta\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}+\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ ,\quad\widetilde{\mathbf{D}}_{1^{\prime}+}=\mathrm{D}_{+}\ ,\quad\widetilde{\mathbf{D}}_{2^{\prime}+}=-\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ ,$

(2.20)

where we have used the expressions (2.5) for $\mathrm{D}_{aa^{\prime}+}$. A similar description can be obtained in the chart $\mathsf{U}_{1}$ in which $u^{1}\neq 0$. Writing $u^{a}=u^{1}(1,-\rho)$, choosing $\widetilde{u}^{a}=(u^{1})^{-1}(0,-1)$ and setting $u^{1}=1$ by scale invariance, we have

$$\text{In $\mathsf{U}_{1}$}:\quad\mathbf{D}_{1^{\prime}+}=\mathrm{D}_{+}-\rho\mathrm{Q}_{+}\ ,\quad\mathbf{D}_{2^{\prime}+}=-\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}-\rho\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ ,\quad\widetilde{\mathbf{D}}_{1^{\prime}+}=-\mathrm{Q}_{+}\ ,\quad\widetilde{\mathbf{D}}_{2^{\prime}+}=-\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ .$$

(2.21)

Observe that, in the intersection $\mathsf{U}_{12}$, the derivatives $\mathbf{D}_{a^{\prime}+}(\rho)$ defined in $\mathsf{U}_{1}$ are related to the $\mathbf{D}_{a^{\prime}+}(\zeta)$ defined in $\mathsf{U}_{2}$ as

$$\mathbf{D}_{a^{\prime}+}(\rho)=(-\rho)\mathbf{D}_{a^{\prime}+}(\zeta(\rho))\ ,$$

(2.22)

which is the gluing rule for a global section of the line bundle $\mathcal{O}(1)\to\mathbf{CP}^{1}$ (we have used that $\zeta(\rho)=-\rho^{-1}$ on the overlap).

Similarly, the derivatives $\widetilde{\mathbf{D}}_{a^{\prime}+}(\rho)$ in $\mathsf{U}_{1}$ and $\widetilde{\mathbf{D}}_{a^{\prime}+}(\zeta)$ in $\mathsf{U}_{2}$ are related on the overlap as

$$\widetilde{\mathbf{D}}_{a^{\prime}+}(\rho)=(-\rho)^{-1}\widetilde{\mathbf{D}}_{a^{\prime}+}(\zeta(\rho))+\rho^{-1}\mathbf{D}_{a^{\prime}+}(\rho)=(-\rho)^{-1}\widetilde{\mathbf{D}}_{a^{\prime}+}(\zeta(\rho))-\mathbf{D}_{a^{\prime}+}(\zeta(\rho))\ ,$$

(2.23)

where we have used (2.22) in going to the last expression. The transformation (2.23) can be viewed as the usual transformation of a section of $\mathcal{O}(-1)$ plus a shift term proportional to $\mathbf{D}_{a^{\prime}+}$ generated by the shift symmetry (2.8). This allows us to define $\widetilde{\mathbf{D}}_{a^{\prime}+}$ globally on $\mathbf{CP}^{1}$, not as a section of $\mathcal{O}(-1)$ but as a section of the affine bundle modelled on $\mathcal{O}(-1)$.

Note: There is an alternate way of writing the $(0,4)$ algebra using the derivatives $\mathbf{D}_{a^{\prime}+}$ and $\frac{\partial}{\partial\zeta}$:

$$\{\mathbf{D}_{a^{\prime}+}\,,[\partial_{\zeta}\,,\mathbf{D}_{b^{\prime}+}]\}=-2\text{i}\varepsilon_{a^{\prime}b^{\prime}}\partial_{++}\ .$$

(2.24)

Thus, one may use the derivatives $\mathbf{D}_{a^{\prime}+}$ and $\partial/\partial\zeta$ instead of $\mathbf{D}_{a^{\prime}+}$ and $\widetilde{\mathbf{D}}_{a^{\prime}+}$ in describing projective superspace. Observe that $\partial_{\zeta}\mathbf{D}_{b^{\prime}+}$ coincides with $\widetilde{\mathbf{D}}_{b^{\prime}+}$ in $\mathsf{U}_{1}$ and $\partial_{\rho}\mathbf{D}_{b^{\prime}+}$ coincides with $\widetilde{\mathbf{D}}_{b^{\prime}+}$ in $\mathsf{U}_{2}$. Further, $\partial_{\zeta}\mathbf{D}_{a^{\prime}+}$ also satisfies the rule (2.23). However, this is expected since the derivative of a section of $\mathcal{O}(1)$ transforms as a section of the affine bundle modelled on $\mathcal{O}(-1)$.

Plugging in the derivatives (2.20) in the action (2.13), we get

$$\mathcal{S}[\bm{\Phi}]=\int\text{d}^{2}x\oint_{\gamma}\frac{\text{d}\zeta}{2\pi\text{i}}\mathrm{D}_{+}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}{\bm{K}}_{--}(\bm{\Phi})\ .$$

(2.25)

We can also rewrite the above action in $(0,2)$ superspace. Using $-\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}=-\zeta^{-1}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}+\zeta^{-1}\mathbf{D}_{2^{\prime}+}$ and $\mathbf{D}_{a^{\prime}+}{\bm{K}}_{--}=0$, we get

$$\mathcal{S}[\bm{\Phi}]=\int\text{d}^{2}x\ \oint_{\gamma}\frac{\text{d}\zeta}{2\pi\text{i}\zeta}\mathrm{D}_{+}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}{\bm{K}}_{--}(\bm{\Phi})=\int\text{d}^{2}x\ \mathrm{D}_{+}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\oint_{\gamma}\frac{\text{d}\zeta}{2\pi\text{i}\zeta}{\bm{K}}_{--}(\bm{\Phi})\ .$$

(2.26)

For completeness, we explicitly describe some analogous aspects of $F^{\prime}$-projective superspace. We have the inhomogeneous coordinate $\zeta^{\prime}$ for the $\mathbf{CP}^{1}$ corresponding to the $F^{\prime}$ doublet $v^{a^{\prime}}$. We then choose $v^{a^{\prime}}=(\zeta^{\prime},1)$ and $\widetilde{v}{}^{a^{\prime}}=(1,0)$ using the scale invariance $v^{a^{\prime}}\to\lambda^{\prime}v^{a^{\prime}}$, $\lambda^{\prime}\in\mathbf{C}^{\star}$. The $F^{\prime}$-projective derivatives $\mathbf{D}_{a+}$ and $\widetilde{\mathbf{D}}_{a+}$ are then

$\displaystyle\mathbf{D}_{1+}=\zeta^{\prime}\mathrm{D}_{+}-\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ ,\quad\mathbf{D}_{2+}=\zeta^{\prime}\mathrm{Q}_{+}+\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ ,\quad\widetilde{\mathbf{D}}_{1+}=\mathrm{D}_{+}\ ,\quad\widetilde{\mathbf{D}}_{2+}=\mathrm{Q}_{+}\ .$

(2.27)

A $(0,4)$ supersymmetric action in $(0,2)$ superspace for $F^{\prime}$-projective superfields $\bm{\Phi}^{\prime}$ is given by

$$\mathcal{S}[\bm{\Phi}^{\prime}]=-\int\text{d}^{2}x\ \mathrm{D}_{+}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\oint_{\gamma^{\prime}}\frac{\text{d}\zeta^{\prime}}{2\pi\text{i}\zeta^{\prime}}{\bm{K}}^{\prime}_{--}(\bm{\Phi}^{\prime})\ .$$

(2.28)

The actions we consider in this paper will only have a single contour integral over either $\zeta$ or $\zeta^{\prime}$.

The fully contracted derivative $\mathbf{D}_{+}=u^{a}v^{a^{\prime}}\mathrm{D}_{aa^{\prime}+}$ (2.11) in terms of $\zeta$ and $\zeta^{\prime}$ is

$$\mathbf{D}_{+}=\zeta\zeta^{\prime}\mathrm{D}_{11^{\prime}+}+\zeta\mathrm{D}_{12^{\prime}+}+\zeta^{\prime}\mathrm{D}_{21^{\prime}+}+\mathrm{D}_{22^{\prime}+}=\zeta\zeta^{\prime}\mathrm{D}_{+}-\zeta\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}+\zeta^{\prime}\mathrm{Q}_{+}+\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}_{+}\ .$$

(2.29)

The superfield is a homogeneous polynomial in the $u^{a}$ of degree $p>0$:

$$\bm{\eta}(\underline{z},u)={\eta}_{a_{1}\cdots a_{p}}(\underline{z})u^{a_{1}}\cdots u^{a_{p}}=\sum_{i=0}^{p}{\eta}_{i}(\underline{z})(u^{1})^{i}(u^{2})^{p-i}\ .$$

(2.30)

The components $\eta_{a_{1}\cdots a_{p}}(\underline{z})$ are ordinary $(0,4)$ superfields, i.e., functions on $\mathbf{R}^{1,1|0,4}$. Note that $\bm{\eta}$ is a global section of the line bundle $\mathcal{O}(p)\to\mathbf{CP}^{1}$. We thus call such superfields $\mathcal{O}(p)$ superfields. In the chart $\mathsf{U}_{2}$ where $u^{2}\neq 0$ we can write $\bm{\eta}$ as

$$\bm{\eta}(\underline{z},u)=(u^{2})^{p}\bm{\eta}(\underline{z},\zeta)=(u^{2})^{p}\sum_{j=0}^{p}\eta_{j}(\underline{z})\,\zeta^{j}\ ,$$

(2.31)

which becomes a polynomial in the inhomogeneous coordinate $\zeta=u^{1}/u^{2}$ when we set $u^{2}=1$.

The $\mathcal{O}(n)$ superfields discussed above are global holomorphic sections of $\mathcal{O}(n)\to\mathbf{CP}^{1}$. We can consider more general superfields which are only local sections of $\mathcal{O}(n)$ and cannot be extended to all of $\mathbf{CP}^{1}$. A familiar class of examples are the meromorphic sections of $\mathcal{O}(n)$ which are rational functions of $u^{a}$:

$$\bm{\eta}(\underline{z},u)=\frac{\bm{P}(\underline{z},u)}{\bm{Q}(\underline{z},u)}=\frac{P_{i_{1}\cdots i_{p}}(\underline{z})u^{i_{1}}\cdots u^{i_{p}}}{Q_{i_{1}\cdots i_{q}}(\underline{z})u^{i_{1}}\cdots u^{i_{q}}}\ ,$$

(2.32)

where $\bm{P}$ and $\bm{Q}$ are homogeneous polynomials of degree $p$ and $q$ respectively. The domain of definition $\mathcal{D}_{\bm{\eta}}$ of $\bm{\eta}$ on $\mathbf{CP}^{1}$ is restricted to the open set where $\bm{Q}(\underline{z},u)\neq 0$. The degree of homogeneity of $\bm{\eta}$ is then $n=p-q$ and thus $\bm{\eta}$ is a local section of $\mathcal{O}(p-q)\to\mathbf{CP}^{1}$ defined on $\mathcal{D}_{\bm{\eta}}$. In terms of the inhomogeneous coordinate $\zeta$, we have

$$\bm{\eta}(\underline{z},\zeta)=\frac{a_{0}(\underline{z})+a_{1}(\underline{z})\zeta+\cdots+a_{p}(\underline{z})\zeta^{p}}{b_{0}(\underline{z})+b_{1}(\underline{z})\zeta+\cdots+b_{q}(\underline{z})\zeta^{q}}\ ,$$

(2.33)

where the $a_{i}(\underline{z})$ are appropriate combinations of $P_{i_{1}\cdots i_{p}}(\underline{z})$ and similarly, $b_{i}(\underline{z})$ are combinations of the $Q_{i_{1}\cdots i_{q}}$.

Consider superfields which are formal power series in $\zeta$ or $\zeta^{-1}$ or both. These appear as series expansions of local holomorphic sections in the neighbourhoods of $\zeta=0$, $\zeta=\infty$ or in the annulus $\mathbf{CP}^{1}\smallsetminus\{0,\infty\}$. Consider a power series in $\zeta$:

$${\mathbf{\Upsilon}}(\underline{z},\zeta)=\sum_{j=0}^{\infty}\mathnormal{\Upsilon}_{j}(\underline{z})\zeta^{j}\ .$$

(2.34)

Such superfields shall be termed arctic since they are well-defined at the north pole $\zeta=0$ of $\mathbf{CP}^{1}$ (and possibly in a neighbourhood of $\zeta=0$ as well). Similarly, a superfield which is a power series in $\zeta^{-1}$ is designated antarctic.

Finally, a superfield which is defined in the annulus and is real under the extended complex conjugation given below in Section 2.6 is called equatorial.