Vertex operators for the superstring with manifest $d=6$ $\mathcal{N}=1$ supersymmetry

After performing a field redefinition of the gauge-fixed RNS variables Berkovits:1994vy Berkovits:1999im , the worldsheet fields of the six-dimensional part consist of six conformal weight zero bosons $x^{\underline{a}}$, $\underline{a}=\{0$ to $5\}$, and a canonically conjugate left-moving pair of fermions $\{p_{\alpha},\theta^{\alpha}\}$ of conformal weight one and zero, respectively, together with its right-moving part $\{\widehat{p}_{\widehat{\alpha}},\widehat{\theta}^{\widehat{\alpha}}\}$, where $\alpha,\widehat{\alpha}=\{1$ to $4\}$.

In a flat six-dimensional background the worldsheet action in conformal gauge takes the form¹¹1The OPEs between our fundamental worldsheet fields $\{p_{\alpha},\theta^{\alpha},\rho,\sigma\}$ are given by eqs. (20), with the replacement of $\alpha j\rightarrow\alpha$.

$$S=\int d^{2}z\,\bigg{(}\frac{1}{2}\partial x^{\underline{a}}\overline{\partial}x_{\underline{a}}+p_{\alpha}\overline{\partial}\theta^{\alpha}+\widehat{p}_{\widehat{\alpha}}\partial\widehat{\theta}^{\widehat{\alpha}}\bigg{)}+S_{\rho,\sigma}+S_{C}\,,$$

(1)

where $\frac{\partial}{\partial z}=\partial$, $\frac{\partial}{\partial\overline{z}}=\overline{\partial}$, $S_{\rho,\sigma}$ is the part of the action characterizing the chiral bosons $\rho$ and $\sigma$, as well as their anti-chiral counterparts, to be defined by their OPEs and stress-tensor below, and $S_{C}$ corresponds to the four-dimensional compactification variables. These variables can be taken to be any $c=6$ $\mathcal{N}=2$ superconformal field theory describing the compactification manifold, which can be either $\rm K3$ or $\rm T^{4}$ Berkovits:1999im .

For the Type-IIB (Type-IIA) superstring, an up $\alpha$ index and an up (down) $\widehat{\alpha}$ index transform as a Weyl spinor of SU(4), a down $\alpha$ index and a down (up) $\widehat{\alpha}$ index transform as an anti-Weyl spinor of SU(4). In this case, note that Weyl and anti-Weyl spinors are not related by complex conjugation. Also, we will only discuss the open string part of the worldsheet theory in what follows.

To define physical states, one needs to supplement the action (1) with the twisted $c=6$ $\mathcal{N}=2$ constraints Berkovits:1994vy

	$\displaystyle T_{\rm hyb}$	$\displaystyle=-\frac{1}{2}\partial x^{\underline{a}}\partial x_{\underline{a}}-p_{\alpha}\partial\theta^{\alpha}-\frac{1}{2}\partial\rho\partial\rho-\frac{1}{2}\partial\sigma\partial\sigma+\frac{3}{2}\partial^{2}(\rho+i\sigma)+T_{C}\,,$		(2a)
	$\displaystyle G^{+}_{\rm hyb}$	$\displaystyle=-(p)^{4}e^{-2\rho-i\sigma}+\frac{i}{2}p_{\alpha}p_{\beta}\partial x^{\alpha\beta}e^{-\rho}-\frac{1}{2}\partial x^{\underline{a}}\partial x_{\underline{a}}e^{i\sigma}-p_{\alpha}\partial\theta^{\alpha}e^{i\sigma}$
		$\displaystyle-\frac{1}{2}\partial(\rho+i\sigma)\partial(\rho+i\sigma)e^{i\sigma}+\frac{1}{2}\partial^{2}(\rho+i\sigma)e^{i\sigma}+G^{+}_{C}\,,$		(2b)
	$\displaystyle G^{-}_{\rm hyb}$	$\displaystyle=e^{-i\sigma}+G^{-}_{C}\,,$		(2c)
	$\displaystyle J_{\rm hyb}$	$\displaystyle=\partial(\rho+i\sigma)+J_{C}\,,$		(2d)

where $(p)^{4}=\frac{1}{24}\epsilon^{\alpha\beta\gamma\delta}p_{\alpha}p_{\beta}p_{\gamma}p_{\delta}$, $x_{\alpha\beta}=\sigma^{\underline{a}}_{\alpha\beta}x_{\underline{a}}$ and $\sigma^{\underline{a}}_{\alpha\beta}$ are the six-dimensional Pauli matrices, which are $4\times 4$ antisymmetric in the spinor indices. In this work, we use the same conventions for the six-dimensional Pauli matrices as in ref. (Daniel:2024kkp, , Appendix A).

Note that $\{T_{C},G^{\pm}_{C},J_{C}\}$ represent a twisted $c=6$ $\mathcal{N}=2$ superconformal field theory describing the compactification manifold, so that $\{T_{\rm hyb}-T_{C},G^{\pm}_{\rm hyb}-G^{\pm}_{C},J_{\rm hyb}-J_{C}\}$ describe a $c=0$ $\mathcal{N}=2$ superconformal algebra (SCA). The generators $\{T_{C},G^{\pm}_{C},J_{C}\}$ have no poles with the six-dimensional worldsheet variables and no poles with the chiral bosons $\{\rho,\sigma\}$. For the closed string, we also have the right-moving piece of the above algebra.

The operators $e^{m\rho+ni\sigma}$ are conformal tensors and have conformal weight $\frac{1}{2}(-m^{2}+3m+n^{2}-3n)$. The definition of normal-ordering used in eqs. (2), and in the rest of this work, is presented in Appendix B. In particular, notice that we can write the first two terms in the second line of (2) in a more compact form as

$\displaystyle-\frac{1}{2}\partial(\rho+i\sigma)\partial(\rho+i\sigma)e^{i\sigma}+\frac{1}{2}\partial^{2}(\rho+i\sigma)e^{i\sigma}$

$\displaystyle=\big{(}\partial e^{-\rho-i\sigma},e^{\rho+2i\sigma}\big{)}\,,$

(3)

using our normal-ordering prescription.

Correspondingly, any twisted $c=6$ $\mathcal{N}=2$ SCA (2) can be extended to a twisted small $c=6$ $\mathcal{N}=4$ SCA Berkovits:1994vy by adding two bosonic currents and two supercurrents, as detailed in Appendix A. The additional $\mathcal{N}=4$ superconformal generators in the six-dimensional hybrid formalism are

	$\displaystyle\widetilde{G}^{+}_{\rm hyb}$	$\displaystyle=e^{\rho}J^{++}_{C}-e^{\rho+i\sigma}\widetilde{G}^{+}_{C}\,,$		(4a)
	$\displaystyle\widetilde{G}^{-}_{\rm hyb}$	$\displaystyle=\bigg{(}-(p)^{4}e^{-3\rho-2i\sigma}+\frac{i}{2}p_{\alpha}p_{\beta}\partial x^{\alpha\beta}e^{-2\rho-i\sigma}-\frac{1}{2}\partial x^{\underline{a}}\partial x_{\underline{a}}e^{-\rho}-p_{\alpha}\partial\theta^{\alpha}e^{-\rho}$
		$\displaystyle-\big{(}\partial e^{-\rho-i\sigma},e^{i\sigma}\big{)}\bigg{)}J^{--}_{C}+e^{-\rho-i\sigma}\widetilde{G}^{-}_{C}\,,$		(4b)
	$\displaystyle J^{++}_{\rm hyb}$	$\displaystyle=-e^{\rho+i\sigma}J^{++}_{C}\,,$		(4c)
	$\displaystyle J^{--}_{\rm hyb}$	$\displaystyle=e^{-\rho-i\sigma}J^{--}_{C}\,,$		(4d)

where $\{\widetilde{G}^{\pm}_{C},J^{\pm\pm}_{C}\}$, that together with $\{T_{C},G^{\pm}_{C},J_{C}\}$, form a twisted small $c=6$ $\mathcal{N}=4$ SCA which has no poles with the six-dimensional worldsheet variables and also no poles with the chiral bosons $\{\rho,\sigma\}$.

The spacetime supersymmetry charges in the six-dimensional hybrid formalism are given by Berkovits:1999im

$\displaystyle Q^{\rm hyb}_{\alpha 1}$

$\displaystyle=\oint p_{\alpha}\,,$

$\displaystyle Q^{\rm hyb}_{\alpha 2}$

$\displaystyle=\oint\big{(}e^{-\rho-i\sigma}p_{\alpha}+i\partial x_{\alpha\beta}\theta^{\beta}\big{)}\,,$

(5)

and satisfy the spacetime SUSY algebra $\{Q^{\rm hyb}_{\alpha 1},Q^{\rm hyb}_{\alpha 2}\}=-i\oint\partial x_{\alpha\beta}$. Note that the charge $Q^{\rm hyb}_{\alpha 2}$ has the presence of the $\{\rho,\sigma\}$-ghosts and, for that reason, it is called the “non-standard” supersymmetry generator Berkovits:1999im .

The superconformal generators (2) are manifestly invariant under the SUSY charge $Q^{\rm hyb}_{\alpha 1}$. Invariance under $Q^{\rm hyb}_{\alpha 2}$ is difficult to check for the supercurrent $G^{+}_{\rm hyb}$. However, the latter can be made manifest by noting that one can write the supercurrent as

$\displaystyle G^{+}_{\rm hyb}$

$\displaystyle=-\frac{1}{24}\epsilon^{\alpha\beta\gamma\delta}Q^{\rm hyb}_{\alpha 2}Q^{\rm hyb}_{\beta 2}Q^{\rm hyb}_{\gamma 2}Q^{\rm hyb}_{\delta 2}e^{2\rho+3i\sigma}+G^{+}_{C}\,,$

(6)

which is a property that also holds in an $\rm AdS_{3}\times S^{3}$ background including the normal-ordering contributions Daniel:2024kkp .

Note that we are denoting operators defined throughout this section with the subscript/superscript “$\rm hyb$”, so as to not cause confusion with the generators to be introduced in Section 3.

Following refs. Berkovits:1999im Daniel:2024kkp , physical states $V_{\rm hyb}$ of the theory are defined to satisfy the equation of motion²²2For a holomorphic operator $\mathcal{O}$ with conformal dimension $h$, $(\mathcal{O})_{r}$ is defined by the usual mode expansion in the plane, namely, $\mathcal{O}(z)=\sum_{r}(\mathcal{O})_{r}z^{-r-h}$.

$\displaystyle(G^{+}_{\rm hyb})_{0}(\widetilde{G}^{+}_{\rm hyb})_{0}V_{\rm hyb}$

$\displaystyle=0\,,$

(7)

so that the vertex operator $V_{\rm hyb}$ is defined up to the gauge transformation

$\displaystyle\delta V_{\rm hyb}$

$\displaystyle=(G^{+}_{\rm hyb})_{0}\Lambda+(\widetilde{G}_{\rm hyb})_{0}\Omega\,,$

(8)

for some $\Lambda$ and some $\Omega$. Moreover, it is consistent to impose the additional gauge-fixing conditions

$\displaystyle(G^{-}_{\rm hyb})_{0}V_{\rm hyb}$

$\displaystyle=(\widetilde{G}^{-}_{\rm hyb})_{0}V_{\rm hyb}=(T_{\rm hyb})_{0}V_{\rm hyb}=(J_{\rm hyb})_{0}V_{\rm hyb}=0\,.$

(9)

As an example, let us consider the massless compactification-independent states in six dimensions for the open superstring. The most general vertex operator with conformal weight zero and no poles with the $\rm U(1)$-current $J_{\rm hyb}$ has the form

$\displaystyle V_{\rm hyb}$

$\displaystyle=\sum_{n=0}^{\infty}V_{n}e^{n(\rho+i\sigma)}\,.$

(10)

The conditions of no double poles or higher with $G^{-}_{\rm hyb}$ and with $\widetilde{G}^{-}_{\rm hyb}$ imply that $V_{n}=0$ for $n\geq 2$ and $n\leq-2$, respectively. From the remaining equations coming from $(\widetilde{G}^{-}_{\rm hyb})_{0}V_{\rm hyb}=0$, together with the gauge transformations (8), one can gauge-fix $V_{\rm hyb}$ to the form

$\displaystyle V_{\rm hyb}$

$\displaystyle=V_{1}e^{\rho+i\sigma}+V_{0}\,,$

(11)

where

	$\displaystyle V_{1}$	$\displaystyle=\theta^{\alpha}\chi_{\alpha 2}+\frac{i}{2}(\theta\sigma^{\underline{a}}\theta)a_{\underline{a}}-(\theta^{3})_{\alpha}\psi^{\alpha 2}\,,$		(12a)
	$\displaystyle V_{0}$	$\displaystyle=\theta^{\alpha}\chi_{\alpha 1}\,,$		(12b)

with $\psi^{\alpha j}=\epsilon^{jk}i\partial^{\alpha\beta}\chi_{\beta k}$ the gluino and $a_{\underline{a}}$ the gluon. The two-dimensional Levi-Civita symbol takes the values $\epsilon_{12}=\epsilon^{21}=1$. Even though $\chi_{\alpha j}$ is not gauge-invariant, we have that $\delta\psi^{\alpha j}=0$ under a gauge transformation. In our conventions, we are using

$\displaystyle(\theta^{3})_{\alpha}$

$\displaystyle=\frac{1}{6}\epsilon_{\alpha\beta\gamma\delta}\theta^{\beta}\theta^{\gamma}\theta^{\delta}\,,$

$\displaystyle(\theta^{4})$

$\displaystyle=\frac{1}{24}\epsilon_{\alpha\beta\gamma\delta}\theta^{\alpha}\theta^{\beta}\theta^{\gamma}\theta^{\delta}\,,$

(13)

where $\epsilon_{\alpha\beta\gamma\delta}$ is the Levi-Civita symbol with $\epsilon_{1234}=1$.

The superfield $V_{1}$ satisfies the equation of motion $\partial^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}V_{1}=0$ which, together with the gauge transformations, imply that the component fields obey

	$\displaystyle\partial^{\underline{a}}\partial_{\underline{a}}a_{\underline{b}}=\partial^{\underline{a}}a_{\underline{a}}$	$\displaystyle=0\,,$	$\displaystyle\delta a_{\underline{a}}$	$\displaystyle=\partial_{\underline{a}}\lambda\,,$		(14a)
	$\displaystyle\partial_{\alpha\beta}\psi^{\beta j}$	$\displaystyle=0\,,$	$\displaystyle\delta\psi^{\alpha j}$	$\displaystyle=0\,,$		(14b)

for some $\lambda$. The gauge transformation of $a_{\underline{a}}$ comes from choosing $\Lambda=(\theta)^{4}\lambda$ in (8).

Eqs. (14) are the field content of $d=6$ super-Yang-Mills (SYM). It is also important to note that all degrees of freedom are contained in the superfield $V_{1}$ of eq. (12a). In Section 3, we will see how one can describe superstring vertex operators for the SYM states in terms of the usual superfields of $d=6$ $\mathcal{N}=1$ superspace Howe:1983fr .

Furthermore, by considering (10) in the gauge where $(G_{\rm hyb}^{+})_{0}(\widetilde{G}_{\rm hyb}^{+})_{0}V_{\rm hyb}=0$, we find that the integrated vertex operator for the open superstring compactification-independent massless states is

	$\displaystyle W_{\rm hyb}$	$\displaystyle=\int(G^{+}_{\rm hyb})_{0}(G^{-}_{\rm hyb})_{-1}V_{\rm hyb}$
		$\displaystyle=\int\bigg{(}-e^{-\rho-i\sigma}p_{\alpha}(\nabla^{3})^{\alpha}-\frac{i}{2}\partial x^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}+ip_{\alpha}\partial^{\alpha\beta}\nabla_{\beta}\bigg{)}V_{1}+p_{\alpha}(\nabla^{3})^{\alpha}V_{2}\,.$		(15)

Even though the six-dimensional hybrid formalism presented above preserves manifest $\rm SO(1,5)$ Lorentz invariance, only half of the eight supersymmetries of $d=6$ $\mathcal{N}=1$ superspace are manifest, i.e., act geometrically in the target superspace. This can be observed by the fact that only four left-moving $\theta$s are present in the worldsheet action (1) as fundamental fields.

However, one can proceed as in ref. Berkovits:1999du and add four more left-moving $\theta$s and four right-moving $\widehat{\theta}$s, as well as their conjugate momenta as fundamental worldsheet variables to the action. This doubling of fermionic degrees of freedom can be accomplished by appending the index $j=\{1,2\}$ to $\{p_{\alpha},\theta^{\alpha}\}$, so that we end up with

$$S=\int d^{2}z\,\bigg{(}\frac{1}{2}\partial x^{\underline{a}}\overline{\partial}x_{\underline{a}}+p_{\alpha j}\overline{\partial}\theta^{\alpha j}+\widehat{p}_{\widehat{\alpha}j}\partial\widehat{\theta}^{\widehat{\alpha}j}\bigg{)}+S_{\rho,\sigma}+S_{C}\,,$$

(16)

where $p_{\alpha j}=\epsilon_{jk}p^{k}_{\alpha}$, $\epsilon_{12}=-\epsilon^{12}=1$, $\epsilon_{jk}\epsilon^{kl}=\delta_{j}^{l}$ and repeated indices are summed over.

Consequently, eq. (16) is invariant under the $d=6$ $\mathcal{N}=1$ spacetime supersymmetry transformations generated by the charge

$\displaystyle Q_{\alpha j}$

$\displaystyle=\oint\bigg{(}p_{\alpha j}-\frac{i}{2}\epsilon_{jk}\partial x_{\alpha\beta}\theta^{\beta k}-\frac{1}{24}\epsilon_{\alpha\beta\gamma\delta}\epsilon_{jk}\epsilon_{lm}\theta^{\beta k}\theta^{\gamma l}\partial\theta^{\delta m}\bigg{)}\,,$

(17)

and which satisfy the $d=6$ $\mathcal{N}=1$ SUSY algebra

$\displaystyle\{Q_{\alpha j},Q_{\beta k}\}$

$\displaystyle=-i\epsilon_{jk}\oint\partial x_{\alpha\beta}\,.$

(18)

For the closed string, we also have a left- and a right-moving supersymmetry generator $Q_{\alpha j}$ and $\widehat{Q}_{\widehat{\alpha}j}$, respectively. These charges then generate the $d=6$ $\mathcal{N}=2$ supersymmetry and, hence, for Type II strings the amount of SUSY is doubled.

Beyond that, it is convenient to construct extensions of the worldsheet fields $\{p_{\alpha j},\partial x^{\underline{a}}\}$ that are invariant under the transformations generated by (17). One can easily check that this is achieved by the following on-shell spacetime supersymmetric — or just supersymmetric — worldsheet variables

	$\displaystyle d_{\alpha j}$	$\displaystyle=p_{\alpha j}+\frac{i}{2}\epsilon_{jk}\partial x_{\alpha\beta}\theta^{\beta k}+\frac{1}{8}\epsilon_{\alpha\beta\gamma\delta}\epsilon_{jk}\epsilon_{lm}\theta^{\beta k}\theta^{\gamma l}\partial\theta^{\delta m}\,,$		(19a)
	$\displaystyle\Pi^{\underline{a}}$	$\displaystyle=\partial x^{\underline{a}}-\frac{i}{2}\epsilon_{jk}\sigma^{\underline{a}}_{\alpha\beta}\theta^{\alpha j}\partial\theta^{\beta k}\,.$		(19b)

The six-dimensional worldsheet fields in (16) have the following singularities in their OPEs

	$\displaystyle p_{\alpha j}(y)\theta^{\beta k}(z)$	$\displaystyle\sim\delta^{k}_{j}\delta^{\beta}_{\alpha}(y-z)^{-1}\,,$		(20a)
	$\displaystyle\partial x^{\underline{a}}(y)\partial x^{\underline{b}}(z)$	$\displaystyle\sim-\eta^{\underline{a}\underline{b}}(y-z)^{-2}\,,$		(20b)
	$\displaystyle\rho(y)\rho(z)$	$\displaystyle\sim-\log(y-z)\,,$		(20c)
	$\displaystyle\sigma(y)\sigma(z)$	$\displaystyle\sim-\log(y-z)\,,$		(20d)

where $\eta^{\underline{a}\underline{b}}=\text{diag}(-,+,+,+,+,+)$ and, in turn, eqs. (20) can be used to show that the supersymmetric variables (19) satisfy

	$\displaystyle d_{\alpha j}(y)d_{\beta k}(z)$	$\displaystyle\sim(y-z)^{-1}i\epsilon_{jk}\Pi_{\alpha\beta}(z)\,,$		(21a)
	$\displaystyle d_{\alpha j}(y)\Pi^{\underline{a}}(z)$	$\displaystyle\sim-(y-z)^{-1}i\epsilon_{jk}\sigma^{\underline{a}}_{\alpha\beta}\partial\theta^{\beta k}(z)\,,$		(21b)
	$\displaystyle\Pi^{\underline{a}}(y)\Pi^{\underline{b}}(z)$	$\displaystyle\sim-(y-z)^{-2}\eta^{\underline{a}\underline{b}}\,,$		(21c)
	$\displaystyle d_{\alpha j}(y)\partial\theta^{\beta k}(z)$	$\displaystyle\sim(y-z)^{-2}\delta^{k}_{j}\delta^{\beta}_{\alpha}\,.$		(21d)

Also, notice the following ordering effect using the OPEs of the fundamental fields

	$\displaystyle\oint\frac{dy}{y-z}\,\Pi_{\alpha\beta}(y)d_{\gamma j}(z)$	$\displaystyle=\frac{3i}{4}\epsilon_{jk}\epsilon_{\alpha\beta\gamma\delta}\partial^{2}\theta^{\delta k}(z)\,,$		(22a)
	$\displaystyle\oint\frac{dy}{y-z}\,d_{\gamma j}(y)\Pi_{\alpha\beta}(z)$	$\displaystyle=-\frac{i}{4}\epsilon_{jk}\epsilon_{\alpha\beta\gamma\delta}\partial^{2}\theta^{\delta k}(z)\,.$		(22b)

As we have argued, the worldsheet action (16) is invariant under the $d=6$ $\mathcal{N}=1$ spacetime supersymmetry transformations, however, in order to preserve the description of the original six-dimensional hybrid superstring, one must include a set of constraints which reduce the action (16) to (1). This can be accomplished by the fermionic first-class constraints Berkovits:1999du

$\displaystyle D_{\alpha}$

$\displaystyle=d_{\alpha 2}-e^{-\rho-i\sigma}d_{\alpha 1}\,,$

(23)

and since

$\displaystyle[D_{\alpha},\theta^{\beta 2}]$

$\displaystyle=\delta^{\beta}_{\alpha}\,,$

(24)

one can use (23) to gauge-fix (16) to (1). Therefore, working with the action (16) and the harmonic constraint $D_{\alpha}$, it is possible to manifestly preserve all of the $d=6$ $\mathcal{N}=1$ supersymmetries.³³3We note in passing that there exists a similarity transformation with the property $e^{S}d_{\alpha 2}e^{-S}=D_{\alpha}$ where $S=\theta^{\alpha 2}d_{\alpha 1}e^{-\rho-i\sigma}-\frac{i}{2}\theta^{\alpha 2}\theta^{\beta 2}\Pi_{\alpha\beta}e^{-\rho-i\sigma}+(\theta^{2})^{3}_{\alpha}\partial\theta^{\alpha 1}e^{-\rho-i\sigma}+\frac{1}{2}(\theta^{2})^{4}\partial(\rho+i\sigma)e^{-2\rho-2i\sigma}$.

In this case, the $\mathcal{N}=2$ constraints (2) are modified and can be written in a manifestly spacetime supersymmetric form as Berkovits:1999du

	$\displaystyle T_{\rm hyb}$	$\displaystyle=-\frac{1}{2}\Pi^{\underline{a}}\Pi_{\underline{a}}-d_{\alpha 1}\partial\theta^{\alpha 1}-e^{-\rho-i\sigma}d_{\alpha 1}\partial\theta^{\alpha 2}-\frac{1}{2}\partial\rho\partial\rho-\frac{1}{2}\partial\sigma\partial\sigma$
		$\displaystyle+\frac{3}{2}\partial^{2}(\rho+i\sigma)+T_{C}\,,$		(25a)
	$\displaystyle G^{+}_{\rm hyb}$	$\displaystyle=-(d_{1})^{4}e^{-2\rho-i\sigma}+\frac{i}{2}d_{\alpha 1}d_{\beta 1}\Pi^{\alpha\beta}e^{-\rho}+d_{\alpha 1}\partial\theta^{\alpha 2}\partial(\rho+i\sigma)e^{-\rho}+d_{\alpha 1}\partial^{2}\theta^{\alpha 2}e^{-\rho}$
		$\displaystyle-\frac{1}{2}\Pi^{\underline{a}}\Pi_{\underline{a}}e^{i\sigma}-d_{\alpha 1}\partial\theta^{\alpha 1}e^{i\sigma}-\frac{1}{2}\partial(\rho+i\sigma)\partial(\rho+i\sigma)e^{i\sigma}$
		$\displaystyle+\frac{1}{2}\partial^{2}(\rho+i\sigma)e^{i\sigma}+G^{+}_{C}\,,$		(25b)
	$\displaystyle G^{-}_{\rm hyb}$	$\displaystyle=e^{-i\sigma}+G^{-}_{C}\,,$		(25c)
	$\displaystyle J_{\rm hyb}$	$\displaystyle=\partial(\rho+i\sigma)+J_{C}\,,$		(25d)

which, as in the previous section, still obey a twisted $c=6$ $\mathcal{N}=2$ SCA, and we defined $(d_{1})^{4}=\frac{1}{24}\epsilon^{\alpha\beta\gamma\delta}d_{\alpha 1}d_{\beta 1}d_{\gamma 1}d_{\delta 1}$. Let us mention that when one gauge fix $\theta^{\alpha 2}=0$, the constraints (25) reduce to the ones compatible with the action (1), i.e., eqs. (2). Note that the stress-tensor $T_{\rm hyb}$ is the expected stress tensor when $D_{\alpha}=0$, because

$$-d_{\alpha 1}\partial\theta^{\alpha 1}-e^{-\rho-i\sigma}d_{\alpha 1}\partial\theta^{\alpha 2}=-d_{\alpha j}\partial\theta^{\alpha j}+D_{\alpha}\partial\theta^{\alpha 2}\,.$$

(26)

It is also important to be aware that the $\mathcal{N}=2$ algebra is preserved independently of how one chooses to gauge-fix the local symmetry generated by $D_{\alpha}$. This is because the form of the $\mathcal{N}=2$ generators (25) was chosen so that they have no poles with the harmonic-like constraint (23). The non-trivial part in showing this is for the generator $G^{+}_{\rm hyb}$, nonetheless, it becomes manifest by noting the property that one can write $G^{+}_{\rm hyb}$ as

$$G^{+}_{\rm hyb}=-\frac{1}{24}\epsilon^{\alpha\beta\gamma\delta}[D_{\alpha},\{D_{\beta},[D_{\gamma},\{D_{\delta},e^{2\rho+3i\sigma}\}]\}]+G_{C}^{+}\,,$$

(27)

where the graded bracket $[D_{\alpha},\mathcal{O}\}(z)=\oint dy\,D_{\alpha}(y)\mathcal{O}(z)$ denotes the simple pole in the OPE between $D_{\alpha}$ and $\mathcal{O}$.

The details of the calculation establishing eq. (27) are given in Appendix C. To the knowledge of the author, this is the first time that eq. (27) is proven considering the normal-ordering contributions. Note also the similarity between identities (27) and (6).

For the massless compactification-independent sector of the open superstring, the vertex operator now reads

$\displaystyle V_{\rm hyb}=\sum_{n=-\infty}^{\infty}V_{n}(x,\theta)e^{n(\rho+i\sigma)}\,,$

(28)

which takes the same form as in eq. (10), but now $V_{n}(x,\theta)$ depends on the zero modes of $\{x^{\underline{a}},\theta^{\alpha j}\}$. Therefore, it contains the eight fermionic $\theta$ coordinates of $d=6$ $\mathcal{N}=1$ superspace. Of course, contrasting with the hybrid formalism of the previous section, in the present case the physical states $V_{\rm hyb}$ also have to be annihilated by $D_{\alpha}$.

It is interesting to note what is the effect of imposing the constraint $D_{\alpha}$ for $V_{\rm hyb}$ of eq. (28). To do that, let us first define the new superspace variables

$\displaystyle\theta^{\alpha-}$

$\displaystyle=\frac{1}{2}\big{(}\theta^{\alpha 2}-e^{\rho+i\sigma}\theta^{\alpha 1}\big{)}\,,$

$\displaystyle\theta^{\alpha+}$

$\displaystyle=\frac{1}{2}\big{(}\theta^{\alpha 1}+e^{-\rho-i\sigma}\theta^{\alpha 2}\big{)}\,.$

(29)

The condition that $V_{\rm hyb}$ has no poles with $D_{\alpha}$ implies that

$\displaystyle\big{(}\nabla_{\alpha 2}-e^{-\rho-i\sigma}\nabla_{\alpha 1}\big{)}V_{\rm hyb}$

$\displaystyle=0\,,$

(30)

where $\nabla_{\alpha j}=\frac{\partial}{\partial\theta^{\alpha j}}-\frac{i}{2}\epsilon_{jk}\theta^{\beta k}\partial_{\alpha\beta}$ is the zero mode of $d_{\alpha j}$ acting on $V_{\rm hyb}$. By defining ${x^{\prime}}^{\underline{a}}=x^{\underline{a}}$ and then doing the shift

$\displaystyle{x^{\prime}}^{\underline{a}}+i\theta^{\alpha-}\theta^{\beta+}\sigma^{\underline{a}}_{\alpha\beta}\mapsto x^{\underline{a}}\,,$

(31)

we learn that $V_{\rm hyb}$ is independent of $\theta^{\alpha-}$ and, for that reason, it is a function of only the zero modes of $\{x^{\underline{a}},\theta^{\alpha+}\}$. As a consequence, after identifying $\theta^{\alpha+}=\theta^{\alpha}$ the component fields of $V_{\rm hyb}$ in (28) can be related to the component fields of $V_{\rm hyb}$ in (10), therefore, we recover the usual six-dimensional description of the vertex operator in Section 2.1. Nonetheless, the identification of the component fields is only possible after imposing $D_{\alpha}=0$.

From eqs. (25), one can construct the remaining twisted small $c=6$ $\mathcal{N}=4$ generators, and in the gauge where $(G_{\rm hyb}^{+})_{0}(\widetilde{G}_{\rm hyb}^{+})_{0}V_{\rm hyb}=0$ the integrated vertex operator for the massless compactification-independent states of the open superstring now reads Berkovits:1999du

	$\displaystyle W_{\rm hyb}$	$\displaystyle=\int(G^{+}_{\rm hyb})_{0}(G^{-}_{\rm hyb})_{-1}V_{\rm hyb}$
		$\displaystyle=\int\,\bigg{[}\bigg{(}-\frac{1}{6}e^{-\rho-i\sigma}\epsilon^{\alpha\beta\gamma\delta}d_{\alpha 1}\nabla_{\beta 1}\nabla_{\gamma 1}\nabla_{\delta 1}-\frac{i}{2}\Pi^{\alpha\beta}\nabla_{\alpha 1}\nabla_{\beta 1}$
		$\displaystyle+id_{\alpha 1}\partial^{\alpha\beta}\nabla_{\beta 1}-\partial\theta^{\alpha 2}\nabla_{\alpha 1}\bigg{)}V_{1}+\frac{1}{6}\epsilon^{\alpha\beta\gamma\delta}d_{\alpha 1}\nabla_{\beta 1}\nabla_{\gamma 1}\nabla_{\delta 1}V_{2}\bigg{]}$
		$\displaystyle=\int\,\bigg{[}\frac{1}{6}\epsilon^{\alpha\beta\gamma\delta}\Big{(}d_{\alpha 2}\nabla_{\beta 1}\nabla_{\gamma 1}\nabla_{\delta 2}-d_{\alpha 1}\nabla_{\beta 2}\nabla_{\gamma 2}\nabla_{\delta 1}\Big{)}$
		$\displaystyle+\frac{i}{4}\Pi^{\alpha\beta}[\nabla_{\alpha 1},\nabla_{\beta 2}]-\frac{1}{2}\partial\theta^{\alpha 1}\nabla_{\alpha 1}+\frac{1}{2}\partial\theta^{\alpha 2}\nabla_{\alpha 2}\bigg{]}V_{0}\,,$		(32)

where the supersymmetric derivative $\nabla_{\alpha j}$ satisfy the algebra $\{\nabla_{\alpha j},\nabla_{\beta k}\}=-i\epsilon_{jk}\partial_{\alpha\beta}$. To arrive at the last equality in (2.3), we subtracted a total derivative and used the relations implied by the constraint (23), namely, $\nabla_{\alpha 1}V_{1}=-\nabla_{\alpha 2}V_{0}$ and $\nabla_{\alpha 1}V_{2}=\nabla_{\alpha 2}V_{1}$.

To the worldsheet theory (16), we introduce a bosonic spinor $\lambda^{\alpha}$ of conformal weight zero and its conjugate momenta $w_{\alpha}$ of conformal weight one. As we will momentarily see, the ghost $\lambda^{\alpha}$ will be responsible for relaxing the constraint $D_{\alpha}$. We also include the non-minimal variables $\{\overline{\lambda}_{\alpha},r_{\alpha}\}$ Berkovits:2005bt of conformal weight zero, as well as their conjugate momenta $\{\overline{w}^{\alpha},s^{\alpha}\}$ of conformal weight one. The fields $\{s^{\alpha},r_{\alpha}\}$ are worldsheet fermions and $\{\overline{w}^{\alpha},\overline{\lambda}_{\alpha}\}$ bosons.

The worldsheet action now takes the form

	$\displaystyle S$	$\displaystyle=\int d^{2}z\,\bigg{(}\frac{1}{2}\partial x^{\underline{a}}\overline{\partial}x_{\underline{a}}+p_{\alpha j}\overline{\partial}\theta^{\alpha j}+w_{\alpha}\overline{\partial}\lambda^{\alpha}+s^{\alpha}\overline{\partial}r_{\alpha}+\overline{w}^{\alpha}\overline{\partial}\overline{\lambda}_{\alpha}$
		$\displaystyle+\widehat{p}_{\widehat{\alpha}j}\partial\widehat{\theta}^{\widehat{\alpha}j}+\widehat{w}_{\widehat{\alpha}}\partial\widehat{\lambda}^{\widehat{\alpha}}+\widehat{s}^{\widehat{\alpha}}\partial\widehat{r}_{\widehat{\alpha}}+\widehat{\overline{w}}^{\widehat{\alpha}}\partial\widehat{\overline{\lambda}}_{\widehat{\alpha}}\bigg{)}+S_{\rho,\sigma}+S_{C}\,,$		(33)

where the “hatted” fields are right-moving and, for simplicity, will be ignored in what follows. The singularities in the OPEs of the new variables are

	$\displaystyle w_{\alpha}(y)\lambda^{\beta}(z)$	$\displaystyle\sim-\delta^{\beta}_{\alpha}(y-z)^{-1}\,,$		(34a)
	$\displaystyle\overline{w}^{\alpha}(y)\overline{\lambda}_{\beta}(z)$	$\displaystyle\sim-\delta^{\alpha}_{\beta}(y-z)^{-1}\,,$		(34b)
	$\displaystyle s^{\alpha}(y)r_{\beta}(z)$	$\displaystyle\sim\delta^{\alpha}_{\beta}(y-z)^{-1}\,,$		(34c)

and, unlike in Berkovits:2005bt , $\{\lambda^{\alpha},\overline{\lambda}_{\alpha},r_{\alpha}\}$ are not constrained. Note further that, as opposed to the worldsheet action (16), the stress-tensor of (3.1) has vanishing central charge.

With these additional variables, it is still possible to construct superconformal generators satisfying a twisted $c=6$ $\mathcal{N}=2$ SCA as in Section 2.

In this case, we have

	$\displaystyle T$	$\displaystyle=T_{\rm hyb}-D_{\alpha}\partial\theta^{\alpha 2}-w_{\alpha}\partial\lambda^{\alpha}-\overline{w}^{\alpha}\partial\overline{\lambda}_{\alpha}-s^{\alpha}\partial r_{\alpha}\,,$		(35a)
	$\displaystyle G^{+}$	$\displaystyle=G^{+}_{\rm hyb}-\lambda^{\alpha}D_{\alpha}-\overline{w}^{\alpha}r_{\alpha}\,,$		(35b)
	$\displaystyle G^{-}$	$\displaystyle=G^{-}_{\rm hyb}+w_{\alpha}\partial\theta^{\alpha 2}+s^{\alpha}\partial\overline{\lambda}_{\alpha}\,,$		(35c)
	$\displaystyle J$	$\displaystyle=J_{\rm hyb}-w_{\alpha}\lambda^{\alpha}-s^{\alpha}r_{\alpha}\,,$		(35d)

where $\{T_{\rm hyb},G^{+}_{\rm hyb},G^{-}_{\rm hyb},J_{\rm hyb}\}$ are the $c=6$ $\mathcal{N}=2$ generators of eqs. (25). Note that $T$ is now the usual stress-tensor, because the terms added in (35a) to $T_{\rm hyb}$ precisely cancel the atypical contribution in (26). Explicitly, we now have

	$\displaystyle T$	$\displaystyle=-\frac{1}{2}\Pi^{\underline{a}}\Pi_{\underline{a}}-d_{\alpha j}\partial\theta^{\alpha j}-w_{\alpha}\partial\lambda^{\alpha}-\overline{w}^{\alpha}\partial\overline{\lambda}_{\alpha}-s^{\alpha}\partial r_{\alpha}$
		$\displaystyle-\frac{1}{2}\partial\rho\partial\rho-\frac{1}{2}\partial\sigma\partial\sigma+\frac{3}{2}\partial^{2}(\rho+i\sigma)+T_{C}\,.$		(36)