# Exotic Origins of Tensionless Superstrings

Arjun Bagchi  Indian Institute of Technology Kanpur, Kalyanpur, Kanpur 208016.
INDIA.
Aritra Banerjee  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, CHINA.
Shankhadeep Chakrabortty  Indian Institute of Technology Ropar, Rupnagar,
Punjab 140001, INDIA.
Pulastya Parekh  Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. INDIA.

###### Abstract

A new type of tensionless superstring theory, called the Inhomogeneous
tensionless superstring, has been recently introduced. This is characterised
by a residual symmetry algebra on the worldsheet richer in structure than the
previously known symmetry algebra related to Homogeneous tensionless
superstring. In this paper, we investigate theories of tensile superstring
that the Inhomogeneous tensionless superstring with manifestly real fermions
could arise from. We provide two different candidates for these parent tensile
theories. The nature of the limit dictates that these theories have some
exotic features.

##  1 Introduction

String theory presently is the most viable of the formulations for a theory of
quantum gravity. The theory is endowed with a characteristic length-scale
$\alpha^{\prime}$, – the length of the fundamental string. In the limit
$\alpha^{\prime}\to 0$, the string reduces to a point particle and we recover
known physics, i.e. low-energy supergravity, making it possible to explore the
classical regime of gravity from string theory. The opposite limit,
$\alpha^{\prime}\to\infty$ is expected to explore the extreme quantum sector
of gravity.

The tension of the string is proportional to the inverse of $\alpha^{\prime}$
and hence the $\alpha^{\prime}\to\infty$ is the limit where the tension of the
string goes to zero. This tensionless limit has been of interest since [1] due
to a variety of reasons. In this limit, the masses of all the string states go
to zero and it is expected that there would be an emergent symmetry much
larger than the usual symmetries of string theory. This has been explored in
the early days by Gross and Mende from the point of view of string scattering
[2, 3, 4], and of late has been related to higher spin holographic dualities
[5, 6, 7]. Another, less explored but possibly more intriguing connection, is
with the physics of Hagedorn transitions [8]. It was noted in [9, 10] that the
tension of the string decreases to zero as it approaches the Hagedorn
temperature.

Our previous construction of tensionless strings has focussed on utilising the
power of symmetries on the worldsheet. In [11], it was shown that the 3d
Bondi-Metzner-Sachs (BMS3) algebra or equivalently the 2d Galilean Conformal
Algebra, arises on the worldsheet of the tensionless string as residual gauge
symmetries in the equivalent of the conformal gauge. This is principally
because the tensionless string is the 1-d extended analogue of the massless
point particle and the metric on the worldsheet becomes null. A limiting
procedure on the worldsheet was proposed which took one from the symmetries of
the tensile worldsheet (Virasoro $\otimes$ Virasoro) to the BMS algebra. In
[12], aspects of the bosonic tensionless theory were further developed and the
power of the mapping on the worldsheet was further clarified. The move to
superstrings threw open new and interesting possibilities, which we briefly
review below. The most striking of this was the discovery of a completely new
tensionless superstring, which we will refer to as the Inhomogeneous
tensionless superstring [13], as opposed to the Homogeneous tensionless
superstring described earlier in [14, 15, 16] and also understood via limiting
procedures in [17].

An intriguing connection between tensionless superstrings and ambitwistors has
recently been unearthed [18]. The version of the tensionless superstring that
has been used is the Homogeneous one. But our formulation of the Inhomogeneous
version points to the tantalising prospect of the discovery of a richer
variant of the ambitwistor string. A very important question in the context of
ambitwistors is tracing their origin to a parent superstring. Some attempts
have been made in [19]. Keeping this in mind, and also with an eye out to
applications of tensionless strings to the physics of the Hagedorn phase
transitions, in the present paper, we attempt to answer the question what
parent theory our tensionless strings arise from. We will be mostly focused on
our newly discovered Inhomogeneous version in the current note.

##  2 Tensionless Superstrings

The tensionless equivalent of the RNS action [14] for the spacetime bosonic
fields $X^{\mu}$ and its supersymmetric partners $\psi^{\mu}$ is given by

$S=\int
d^{2}\sigma\Big{[}V^{a}V^{b}\partial_{a}X\cdot\partial_{b}X+i\bar{\psi}\cdot\rho^{a}\partial_{a}\psi\Big{]}.$
(1)

In the tensionless limit, the worldsheet metric becomes degenerate and is
replaced by a product of two vector densities $V^{a}$:

$T\sqrt{-g}g^{ab}\to V^{a}V^{b},\quad\mbox{as}\quad T\to 0.$ (2)

The equivalent of the conformal gauge in tensile strings in this case is the
choice $V^{a}=(1,0)$. In the action (1), $\rho^{a}$ are the 2d gamma matrices
which now satisfy a modified Clifford algebra:

$\\{\rho^{a},\rho^{b}\\}=2V^{a}V^{b}.$ (3)

The above equation (3) is central to the formulation of different versions of
tensionless superstrings. The equation admits two (and only two) inequivalent
classes of solutions for $\rho^{a}$ that gives rise to the two inequivalent
theories.

#### Homogeneous Tensionless Superstrings:

The simplest possible choice is $\rho^{a}=V^{a}\mathbb{I}_{2\times 2}$, which
in our gauge reads $\rho^{0}=\mathbb{I},~{}\rho^{1}=\mathbb{O}$. The action
(1) in this case becomes

$S=\int d^{2}\sigma\Big{[}\dot{X}^{2}+i\bar{\psi}\cdot\dot{\psi}\Big{]}.$ (4)

This describes what we call the Homogeneous Tensionless Superstring [17]. One
can also consider the action in the worldsheet superspace [16], where the
generators of the symmetry transformations give rise to the following residual
symmetry algebra:

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},\qquad[L_{m},M_{n}]=(m-n)M_{m+n},$
$\displaystyle[L_{m},Q^{\alpha}_{r}]=\left(\frac{m}{2}-r\right)Q^{\alpha}_{m+r},\qquad\\{Q^{\alpha}_{r},Q^{\alpha^{\prime}}_{s}\\}=\delta^{\alpha\alpha^{\prime}}M_{r+s}.$
(5)

Here the spinor index $\alpha$ denotes $\pm$. The bosonic indices $m,n$ take
integer values while fermionic indices $r,s$ have half integer values. This
algebra is the classical part of a supersymmetric extension of the (3d) Bondi-
Metzner-Sachs (BMS) algebra. This algebra, where the two fermionic generators
behave in a similar manner, is known as the homogeneous version of Super BMS
(SBMSH). This algebra has interestingly also arisen in the context of
asymptotic symmetries of supergravity in 3d flat spacetimes [20].

#### Inhomogeneous Tensionless Superstrings:

The other inequivalent class of the modified gamma matrices that solve the
deformed Clifford algebra (3) is given by [13]

$\rho^{0}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\rho^{1}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}.$ (6)

For this class of matrices, the associated fermions ($\psi^{\mu}$) need not be
Majorana spinors. In [13] they were considered to be complex, however in this
work we will work with real fermions. Considering the above choice of gamma
matrices the action (1) takes the form

$S=\int
d^{2}\sigma\Big{[}\dot{X}^{2}+i(\psi_{1}\cdot\dot{\psi}_{0}+\psi_{0}\cdot\dot{\psi}_{1}-\psi_{0}\cdot\psi^{\prime}_{0})\Big{]},$
(7)

where $\psi^{\mu}_{0}$ and $\psi^{\mu}_{1}$ are the components of the spinor
$\psi^{\mu}$ which we can take to be real (Majorana). The residual symmetry
algebra associated to this action can also be calculated in a manner similar
to [13]. The symmetries that leaves the above action invariant comprises of
local diffeomorphisms (paramterized by $\xi$) and supersymmtery
transformations ($\epsilon$) given by

$\displaystyle\delta_{\xi}\psi_{0}$ $\displaystyle=$
$\displaystyle\xi^{a}\partial_{a}\psi_{0}+\frac{1}{4}\partial_{a}\xi^{a}\psi_{0},\quad\delta_{\xi}\psi_{1}=\xi^{a}\partial_{a}\psi_{1}+\frac{1}{4}\partial_{a}\xi^{a}\psi_{1}+\frac{1}{2}\partial_{1}\xi^{0}\psi_{0}$
(8a) $\displaystyle\delta_{\epsilon}\psi_{0}$ $\displaystyle=$
$\displaystyle-\epsilon^{1}\dot{X},\quad\delta_{\epsilon}\psi_{1}=-\epsilon^{0}\dot{X}-\epsilon^{0}X^{\prime},~{}~{}\delta_{\xi}X=\xi^{a}\partial_{a}X,\quad\delta_{\epsilon}X=i(\epsilon^{0}\psi_{0}+\epsilon^{1}\psi_{1})~{}~{}$
(8b)

where the parameters obey the conditions (Lorentz indices have been
suppressed):

$\displaystyle\partial_{0}\xi^{0}=\partial_{1}\xi^{1},\quad\partial_{0}\xi^{1}=0,~{}~{}\partial_{0}\epsilon^{0}=\partial_{1}\epsilon^{1},\quad\partial_{0}\epsilon^{1}=0.$
(9)

These transformations can be realised on a modified superspace associated with
the string worldsheet, parameterised by Grassmanian coordinates $\alpha$ and
$\chi$. A general superfield $Y$ on this superspace can then be expanded
as111We should mention here, in general we could also have a term containing
an auxiliary field of the form $\alpha\chi B(\sigma^{a})$, however without any
loss of generality we can put $B=0$ for our case.

$\displaystyle Y(\sigma^{a},\alpha,\chi)$ $\displaystyle=$ $\displaystyle
X(\sigma^{a})+i\alpha\psi_{0}(\sigma^{a})+i\chi\psi_{1}(\sigma^{a}).$ (10)

Symmetry transformations on superspace coordinates are 222The Grassmanian
parameters $\epsilon^{a}$ as well as the coordinates $\alpha$ and $\chi$ are
manifestly real here and above.:

$\delta\tau=\xi^{0}+i(\epsilon^{0}\chi+\epsilon^{1}\alpha),\,\delta\sigma=\xi^{1}+i\epsilon^{1}\chi,\,\delta\alpha=\epsilon^{0}+\frac{1}{4}\partial_{a}\xi^{a}\alpha+\frac{1}{2}\partial_{1}\xi^{0}\chi,\,\delta\chi=\epsilon^{1}+\frac{1}{4}\partial_{a}\xi^{a}\chi.$
(11)

The differential generators of the above transformations read

$\displaystyle L_{n}$ $\displaystyle=$ $\displaystyle
ie^{in\sigma}\Big{[}\partial_{\sigma}+in\Big{(}\frac{\alpha\partial_{\alpha}+\chi\partial_{\chi}}{2}+\tau\partial_{\tau}\Big{)}-\frac{n^{2}}{2}\tau\chi\partial_{\alpha}\Big{]},\
M_{n}=ie^{in\sigma}\Big{[}\partial_{\tau}+\frac{in}{2}\chi\partial_{\alpha}\Big{]},$
$\displaystyle G_{r}$ $\displaystyle=$ $\displaystyle
e^{ir\sigma}\Big{[}\partial_{\chi}+i(\alpha\partial_{\tau}+\chi\partial_{\sigma})+ir\tau\partial_{\alpha}-r\chi\tau\partial_{\tau}\Big{]},\quad\
\ \ H_{r}=e^{ir\sigma}\Big{[}\partial_{\alpha}+i\chi\partial_{\tau}\Big{]}.$
(12)

All of these generators are manifestly hermitian and close to form:

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},\quad[L_{m},M_{n}]=(m-n)M_{m+n},$
$\displaystyle[L_{m},G_{r}]=\Big{(}\frac{m}{2}-r\Big{)}G_{m+r},\quad[M_{m},G_{r}]=\Big{(}\frac{m}{2}-r\Big{)}H_{m+r},\
\\{G_{r},G_{s}\\}=2L_{r+s},$
$\displaystyle[L_{m},H_{r}]=\Big{(}\frac{m}{2}-r\Big{)}H_{m+r},\quad\\{G_{r},H_{s}\\}=2M_{r+s}.$
(13)

This algebra clearly has more non-zero commutators and hence a richer
structure than SBMSH (2). The fermionic generators ($G_{r}$, $H_{r}$) here
behave differently to the Homogeneous case. Hence this avatar of the Super-BMS
algebra is also called the Inhomogeneous SBMS or SBMSI and the associated
string was called Inhomogeneous Tensionless Superstring. This algebra has also
been found to be the asymptotic symmetries of a twisted version of
supergravity in 3d asymptotically flat spacetimes [20].

The equations of motion obtained by varying (7) with respect to $X^{\mu}$ and
$\psi^{\mu}$ reads:

$\ddot{X}^{\mu}=0,\quad\dot{\psi}_{0}^{\mu}=0,\quad\dot{\psi}_{1}^{\mu}={\psi^{\prime}}_{0}^{\mu}.$
(14)

We can solve these equations with closed string boundary conditions to obtain

$\displaystyle X^{\mu}(\tau,\sigma)$ $\displaystyle=$ $\displaystyle
x^{\mu}+\sqrt{2c^{\prime}}B^{\mu}_{0}\tau+i\sqrt{2c^{\prime}}\sum_{n\neq
0}\frac{1}{n}(A^{\mu}_{n}-in\tau B^{\mu}_{n})e^{-in\sigma},$
$\displaystyle\psi_{0}^{\mu}(\sigma,\tau)$ $\displaystyle=$
$\displaystyle\sqrt{c^{\prime}}\sum_{r}\beta^{\mu}_{r}e^{-ir\sigma},~{}~{}~{}\psi_{1}^{\mu}(\sigma,\tau)=\sqrt{c^{\prime}}\sum_{r}[\gamma^{\mu}_{r}-ir\tau\beta^{\mu}_{r}]e^{-ir\sigma}.$
(15)

Here, $n\in\mathbb{Z}$, $r\in\mathbb{Z}+\frac{1}{2}$ (we focus of the NS-NS
sector). $c^{\prime}$ is a constant with dimensions of (mass)-1. The
hermiticity of the generators must follow from the above mode expansions. This
in turn implies the hermiticity of the modes $\gamma$ and $\beta$, i.e.
$\gamma^{\dagger}_{r}=\gamma_{-r},\quad\beta^{\dagger}_{r}=\beta_{-r}.$ It is
straightforward to write down the generators in terms of oscillator modes, and
check that the constraint algebra closes to SBMSI (2).

Upon quantization, the commutators of the modes are given by

$[A_{m}^{\mu},B_{n}^{\nu}]=2m\delta_{m+n}\eta^{\mu\nu},\quad\\{\gamma_{r}^{\mu},\beta_{s}^{\nu}\\}=2\delta_{r+s}\eta^{\mu\nu}.$
(16)

We would like to have an oscillator construction of modes hence transform to a
new basis:

$C^{\mu}_{n}=\frac{1}{2}\Big{(}A^{\mu}_{n}+B^{\mu}_{n}\Big{)},\,\tilde{C}^{\mu}_{n}=\frac{1}{2}\Big{(}-A^{\mu}_{-n}+B^{\mu}_{-n}\Big{)};\quad\omega^{\mu}_{r}=\frac{1}{2}\Big{(}\gamma^{\mu}_{r}+\beta^{\mu}_{r}\Big{)},\,\
\tilde{\omega}^{\mu}_{r}=\frac{i}{2}\Big{(}-\gamma^{\mu}_{-r}+\beta^{\mu}_{-r}\Big{)}.$
(17)

These in turn follow the canonical commutation relations for harmonic
oscillators:

$\displaystyle[C_{m}^{\mu},C_{n}^{\nu}]$ $\displaystyle=$
$\displaystyle[\tilde{C}_{m}^{\mu},\tilde{C}_{n}^{\nu}]=m\delta_{m+n}\eta^{\mu\nu},~{}~{}~{}\\{\omega_{r}^{\mu},\omega_{s}^{\nu}\\}=\\{\tilde{\omega}_{r}^{\mu},\tilde{\omega}_{s}^{\nu}\\}=\delta_{r+s}\eta^{\mu\nu}.$
(18)

Since we demand $A,B,\gamma,\beta$ all to be manifestly hermitian, we find
$\omega_{r}$ is hermitian, $\tilde{\omega}_{r}$ is anti-hermitiant:

$\omega^{\dagger}_{r}=\omega_{-r};\quad\tilde{\omega}^{\dagger}_{r}=-\tilde{\omega}_{-r}$
(19)

While the commutation relations (18) still hold, we also have the extra input,
$\\{\tilde{\omega}_{r},\tilde{\omega}^{\dagger}_{r}\\}=-1.$ This
unconventional commutator allows us to define creation and annihilation
operators in a consistent way. We will be able to define a vacuum
$|00\rangle_{C}$ with the conditions

$C_{n}|00\rangle_{C}=\tilde{C}_{n}|00\rangle_{C}=\omega_{r}|00\rangle_{C}=\tilde{\omega}_{r}|00\rangle_{C}=0\
\ \ \forall\ n,r>0.$ (20)

Negative modes of the oscillators can be used to build up states on the vacuum
in the usual way.

##  3 Exotic Parent I: Maverick Superstrings

Now that we have a well-defined tensionless theory, we will attempt to trace
its origins to parent superstring theories. We will construct two different
avenues of obtaining such theories. We first start with what we will call a
maverick superstring theory.

Let us consider the following gauge-fixed action for a tensile superstring
with worldsheet bosonic fields $X$ and its supersymmetric partners $\psi$:

$S=\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\Big{[}2\partial_{+}X\cdot\partial_{-}X+i(\psi_{+}\cdot\partial_{-}\psi_{+}+\psi_{-}\cdot\partial_{+}\psi_{-})\Big{]}.$
(21)

Here $\psi_{\pm}$ are the two spinorial components of $\psi$. We assume
$\psi_{+}$ to be hermitian and $\psi_{-}$ to be anti-hermitian. We stress that
this does not spoil the hermiticity of the action. Under local diffeomorphism
($\xi$) and supersymmetry transformations ($\epsilon$)

$\delta_{\xi}X=\xi^{a}\partial_{a}X,\,\delta_{\xi}\psi_{\pm}=\xi^{a}\partial_{a}\psi_{\pm}+\frac{1}{2}\partial_{\pm}\xi^{\pm}\psi_{\pm},\quad\delta_{\epsilon}X=i(\epsilon^{+}\psi_{+}+\epsilon^{-}\psi_{-}),\,\delta_{\epsilon}\psi_{\pm}=-2\epsilon^{\pm}\partial_{\pm}X.$
(22)

leave the action invariant. The infinitesimal supersymmetry parameters
$\epsilon^{\pm}$ form a two-component spinor where $\epsilon^{-}$ is anti-
hermitian keeping $X$ and $\delta X$ real. Invariance of the action (21) also
implies the consistency conditions on $\xi$ and $\epsilon$:
$\partial_{\pm}\xi^{\mp}=\partial_{\pm}\epsilon^{\mp}=0$, which are same as in
the usual tensile superstring theory. The superspace coordinates
$\theta,\bar{\theta}$ also form a two component spinor where one component is
anti-hermitian. The transformations in superspace that keep the action (21)
invariant are the same as in the usual tensile string and we can find the
generators in a differential form:

$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$ $\displaystyle
ie^{-in\sigma^{+}}\Big{(}\partial_{+}+\frac{in}{2}\theta\partial_{\theta}\Big{)},\
\ \ \
\mathcal{Q}_{r}=e^{-ir\sigma^{+}}(\partial_{\theta}+i\theta\partial_{+}),$
$\displaystyle\bar{\mathcal{L}}_{n}$ $\displaystyle=$ $\displaystyle
ie^{-in\sigma^{-}}\Big{(}\partial_{-}+\frac{in}{2}\bar{\theta}\partial_{\bar{\theta}}\Big{)},\
\ \ \
\bar{\mathcal{Q}}_{r}=e^{-ir\sigma^{-}}(\partial_{\bar{\theta}}+i\bar{\theta}\partial_{-}).$
(23)

These close to form the usual Super-Virasoro algebra:

$[{\mathcal{L}}_{n},{\mathcal{L}}_{m}]=(n-m){\mathcal{L}}_{n+m},\quad[{\mathcal{L}}_{n},\mathcal{Q}_{r}]=\left(\frac{n}{2}-r\right)\mathcal{Q}_{n+r},\quad\\{\mathcal{Q}_{r},\mathcal{Q}_{s}\\}=2{\mathcal{L}}_{r+s},$
(24)

(and similarly for the anti-holomorphic part). It is interesting to note that
the anti-hermicity of the maverick superstrings does not manifest itself in
the symmetry algebra, even though the generator $\bar{\mathcal{Q}}$ is anti-
hermitian.

The same algebra can also be obtained from the constraints. The solutions of
the equations of motion $\partial_{-}\partial_{+}X=0$ and
$\partial_{\pm}\psi_{\mp}=0$, can be constructed in terms of mode expansions
in the NS-NS sector:

$\displaystyle X^{\mu}(\tau,\sigma)$ $\displaystyle=$ $\displaystyle
x^{\mu}+2\sqrt{2\alpha^{\prime}}\alpha^{\mu}_{0}\tau+i\sqrt{2\alpha^{\prime}}\sum_{n\neq
0}\frac{1}{n}\Big{[}\alpha^{\mu}_{n}e^{-in(\tau+\sigma)}+\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\sigma)}\Big{]},$
(25a) $\displaystyle\psi_{+}^{\mu}(\sigma,\tau)$ $\displaystyle=$
$\displaystyle\sqrt{2\alpha^{\prime}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}b^{\mu}_{r}e^{-ir(\tau+\sigma)},\quad\psi_{-}^{\mu}(\sigma,\tau)=\sqrt{2\alpha^{\prime}}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\tilde{b}^{\mu}_{r}e^{-ir(\tau-\sigma)}.$
(25b)

The canonical commutation relation of $X^{\mu}$ and $\psi_{\pm}^{\mu}$ is:

$[X^{\mu}(\sigma),\dot{X}^{\nu}(\sigma^{\prime})]=\eta^{\mu\nu}\delta(\sigma-\sigma^{\prime}),\quad\\{\psi^{\mu}_{\alpha}(\sigma),\psi^{\nu}_{\alpha^{\prime}}(\sigma^{\prime})\\}=\eta^{\mu\nu}\delta_{\alpha\alpha^{\prime}}\delta(\sigma-\sigma^{\prime}).$
(26)

The above brackets translate into (anti-)commutation relations between the
modes,

$[\alpha_{m}^{\mu},\alpha_{n}^{\nu}]=[\tilde{\alpha}_{m}^{\mu},\tilde{\alpha}_{n}^{\nu}]=m\delta_{m+n}\eta^{\mu\nu},~{}~{}\\{b_{r}^{\mu},b_{s}^{\nu}\\}=\\{\tilde{b}_{r}^{\mu},\tilde{b}_{s}^{\nu}\\}=\delta_{r+s}\eta^{\mu\nu}.$
(27)

Anti-hermiticity of the $\psi_{-}$ implies
$\tilde{b}^{\dagger}_{r}=-\tilde{b}_{-r}$. This in turn implies
$\\{\tilde{b}_{r},\tilde{b}^{\dagger}_{r}\\}=-1$. Now, by using the components
of energy momentum tensor and the supercurrent, the constraints can be written
down in terms of the mode expansions. Let us write here the antiholomorphic
ones explicitly, which will be important for us later;

$\displaystyle\bar{{\mathcal{L}}}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{m}\tilde{\alpha}_{-m}\cdot\tilde{\alpha}_{m+n}+\frac{1}{4}\sum_{r}(2r+n)\tilde{b}_{-r}\cdot\tilde{b}_{r+n},\quad\bar{\mathcal{Q}}_{r}=\sum_{m}\tilde{\alpha}_{-m}\cdot\tilde{b}_{m+r}.$
(28)

The anti-hermiticity of $\tilde{b}$ implies $\bar{\mathcal{Q}}_{r}$ is anti-
hermitian. The constraints again generate the Super-Virasoro algebra (24), and
we see that that the anti-hermiticity does not affect the construction at the
level of the algebra.

#### Tensionless limit on Worldsheet:

The tensionless limit on the string worldsheet [11] can be understood as a
singular limit where the spatial co-ordinate $\sigma\to\infty$ keeping the
worldsheet time $\tau$ fixed. Equivalently, in the bosonic case, this is:

$\sigma\rightarrow\sigma\ \mbox{and}\ \tau\rightarrow\varepsilon\tau,\ \
\varepsilon\rightarrow 0.$ (29)

This amounts to taking the worldsheet speed of light effectively to zero. The
limit is thus an ultra-relativistic or a Carrollian limit on the worldsheet,
where the bosonic generators of the Virasoro algebra change in the following
way:

$L_{n}={\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{-n},\quad
M_{n}=\varepsilon({\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{-n}).$ (30)

The symmetry algebra obeyed by these contracted generators is the 3d BMS (or
equivalently the 2d GCA) [21]. Since we want the tensionless limit to be
consistent, the string length $\alpha^{\prime}$ should also scale as
$\alpha^{\prime}\rightarrow c^{\prime}\varepsilon$, where $c^{\prime}$ is the
finite parameter defined earlier.

In the supersymmetric case, there are two possible scaling of the fermionic
degrees of freedom. For the homogeneous contraction, all generic fermionic
objects $\zeta$ scale as $\zeta\rightarrow\sqrt{\varepsilon}\zeta$. In the
inhomogeneous case, two components of worldsheet spinors contract in different
ways:

$\psi_{0}\rightarrow\frac{1}{\varepsilon}\psi_{0},~{}~{}\psi_{1}\rightarrow\psi_{1}$
(31)

where the new spinors are linear combinations of tensile spinors:
$\psi_{0,1}=\frac{1}{\sqrt{2}}(\psi_{+}\pm i\psi_{-})$. For consistency, the
SUSY parameter $\epsilon$ and the superspace coordinates $\theta,\bar{\theta}$
scale similarly:

$(\epsilon^{0},\epsilon^{1})=\frac{1}{\sqrt{2}}(\epsilon^{+}\mp
i\epsilon^{-}),\ \ (\alpha,\chi)=\frac{1}{\sqrt{2}}(\theta\mp
i\bar{\theta});\quad(\epsilon^{0},\epsilon^{1})\rightarrow(\varepsilon\epsilon^{0},\epsilon^{1});\,(\alpha,\chi)\rightarrow(\varepsilon\alpha,\chi).$
(32)

Starting from (21) we can take the UR limit at every step till (28). The
bosonic and fermionic modes transform under this limit as

$\displaystyle X$ $\displaystyle=$ $\displaystyle
x+\sqrt{2c^{\prime}}\sqrt{\varepsilon}(\alpha_{0}+\tilde{\alpha}_{0})\tau+i\sqrt{2c^{\prime}}\sum_{n\neq
0}\frac{1}{n}\Big{[}\frac{1}{\sqrt{\varepsilon}}(\alpha_{n}-\tilde{\alpha}_{-n})-in\tau\sqrt{\varepsilon}(\alpha_{n}+\tilde{\alpha}_{-n})\Big{]}e^{-in\sigma},$
$\displaystyle\psi_{0}$ $\displaystyle=$
$\displaystyle\sqrt{c^{\prime}}\sum_{r}\sqrt{\varepsilon}(b_{r}+i\tilde{b}_{-r})e^{-ir\sigma};~{}~{}\psi_{1}=\sqrt{c^{\prime}}\sum_{r}\Big{[}\frac{b_{r}-i\tilde{b}_{-r}}{\sqrt{\varepsilon}}-ir\tau\sqrt{\varepsilon}(b_{r}+i\tilde{b}_{-r})\Big{]}e^{-ir\sigma}.$

A comparison between the tensionless (2) and the tensile oscillator modes
gives us

$\displaystyle
A_{n}={\varepsilon}^{-1/2}(\alpha_{n}-\tilde{\alpha}_{-n}),B_{n}={\varepsilon}^{1/2}(\alpha_{n}+\tilde{\alpha}_{-n});\,\gamma_{r}={\varepsilon}^{-1/2}(b_{r}-i\tilde{b}_{-r}),\beta_{r}={\varepsilon}^{1/2}(b_{r}+i\tilde{b}_{-r}).$
(33)

The tensionless generators (2) are related to that of the parent theory by

$\displaystyle
L_{n}={\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{-n},\,M_{n}=\varepsilon({\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{-n}),\quad
G_{r}=\mathcal{Q}_{r}-i\bar{\mathcal{Q}}_{-r},\,H_{r}=\varepsilon(\mathcal{Q}_{r}+i\bar{\mathcal{Q}}_{-r}).$
(34)

Needless to say, it can be easily checked that these satisfy SBMSI (2) and all
of them are hermitian. Thus at a classical level, we have successfully found a
parent theory from which we can consistently take the (Inhomogeneous)
tensionless limit and find a theory with well defined oscillators and
hermitian generators.

#### Bogolioubov transformations and vacua:

In (17), we made a change of basis so that the commutation relations obey the
usual harmonic oscillator algebra. In terms of the changed oscillators the
above relation between tensile/tensionless modes now has a rather suggestive
form:

$\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\alpha_{n}\cosh\varphi+\tilde{\alpha}_{-n}\sinh\varphi,\quad\omega_{r}=b_{r}\cosh\varphi+i\tilde{b}_{-r}\sinh\varphi;$
(35a) $\displaystyle\tilde{C}_{n}$ $\displaystyle=$
$\displaystyle\alpha_{-n}\sinh\varphi+\tilde{\alpha}_{n}\cosh\varphi,\quad\tilde{\omega}_{r}=-ib_{-r}\sinh\varphi+\tilde{b}_{r}\cosh\varphi,$
(35b)

where $\tanh\varphi=\frac{\varepsilon-1}{\varepsilon+1}$. These are Bogoliubov
transformations on the worldsheet parameterised by $\varphi$ since the
canonical commutation relations are preserved in the course of this
transformation. This gives us an explicit evolution of the tensile theory to
its tensionless offspring by dialling $\varphi$ or equivalently $\varepsilon$.
The generators of the Bogoliubov transformations (35) for the bosonic and the
fermionic parts can be written in the following form:

$\displaystyle G_{B}(\varphi)$ $\displaystyle=$ $\displaystyle
i\sum_{n>0}\varphi\Big{[}a_{-n}\cdot\tilde{a}_{-n}-a_{n}\cdot\tilde{a}_{n}\Big{]},~{}~{}G_{F}(\varphi)=\sum_{r>0}\varphi\Big{[}b_{-r}\cdot\tilde{b}_{-r}-b_{r}\cdot\tilde{b}_{r}\Big{]},$
(36a)

where $a_{n}=\frac{1}{\sqrt{n}}\alpha_{n}$ are normalised oscillator modes. It
can be checked that both of these generators are hermitian in our case. In
terms of these generators, the transformations the two sets of oscillators are
given by:

$C_{n}=e^{-iG_{B}}\alpha_{n}e^{iG_{B}},\,\tilde{C}_{n}=e^{-iG_{B}}\tilde{\alpha}_{n}e^{iG_{B}},\quad\omega_{r}=e^{-iG_{F}}b_{r}e^{iG_{F}},\,\tilde{\omega}_{r}=e^{-iG_{F}}\tilde{b}_{r}e^{iG_{F}}.$
(37)

In [12], the bosonic version of this was considered. It was shown that the
transformation implies a mapping between the tensionless vacuum
$|0\rangle_{C}$ and the tensile one $|0\rangle_{\alpha}$ where one can be
written as a squeezed state over the other.

$\displaystyle|0\rangle_{C}=e^{iG_{B}(\varphi)}|0\rangle_{\alpha}={\sqrt{\cosh\varphi}}\prod_{n>0}\exp\Big{[}\tanh\varphi\
a_{-n}\cdot\tilde{a}_{-n}\Big{]}|0\rangle_{\alpha}.$ (38)

So the new tensionless vacuum is a highly energised state in comparison to the
tensile vacuum. Notice that there is a normalisation constant which sits in
front of the expression. When the tensionless limit is taken, this
normalisation constant blows up, indicating something like a phase transition.
In [22], this was further explored and it was shown that in the tensionless
limit, starting with closed strings one ends up with a Neumann boundary state,
indicating that open strings (or space-filling D-branes) arise at the end
point of this evolution. It was also conjectured that there is a Bose-Einstein
like condensation on the worldsheet forming this open string from the closed
string excitations.

In the supersymmetric case here, using the generators (36) the vacuum of the
tensile and tensionless theories can again be mapped to one another as
follows:

$|00\rangle_{C}=e^{iG_{B}(\varphi)}e^{iG_{F}(\varphi)}|00\rangle_{\alpha}\\\
=\prod_{n,r>0}\exp\Big{[}\tanh\varphi(a_{-n}\cdot\tilde{a}_{-n}+ib_{-r}\cdot\tilde{b}_{-r})\Big{]}|00\rangle_{\alpha}.$

It is interesting to note that unlike in the bosonic case, there is no
infinite normalisation factor outside the exponential [12] as we take the
tensionless limit. In fact we can explicitly put $\varepsilon=0$ in the above
expression by recalling the relation between $\tanh\varphi$ and $\varepsilon$,
and obtain

$|00\rangle_{C}=\prod_{n,r>0}\exp\Big{[}-(a^{\dagger}_{n}\cdot\tilde{a}^{\dagger}_{n}-ib^{\dagger}_{r}\cdot\tilde{b}^{\dagger}_{r})\Big{]}|00\rangle_{\alpha}.$
(39)

The above supersymmetric state, in analogy to the bosonic one, is a highly
squeezed state with respect to the $|00\rangle_{\alpha}$ vacuum. The above
expression is in the form of a fermionic boundary state and there should be a
similar story of the formation of an open string from closed strings that
emerges in this context as well. But the absence of the infinite normalisation
constant indicates that there would also be significant departures from the
bosonic picture. We leave a detailed discussion of this for future work.

##  4 Exotic Parent II: Flipped Superstrings

In our first construction of a parent theory, we played with the hermiticity
of one of the modes of the fermion to generate our tensionless theory. We now
put forward a different theory, a string theory with an unconventional vacuum
state.

#### An automorphism of the Super-Virasoro:

Let us consider the following transformation on the anti-holomorphic sector of
the tensile modes

$\tilde{\alpha}_{n}\rightarrow\tilde{\alpha}^{\prime}_{n}=\pm
i\tilde{\alpha}_{-n},\quad\tilde{b}_{r}\rightarrow\tilde{b}^{\prime}_{r}=\pm\tilde{b}_{-r}.$
(40)

Here, the unprimed modes represent the usual hermitian Super-Virasoro modes,
while the primed ones represent the changed modes. There is a clear flip
between the creation and annihilation operators. A bit of algebra gives us the
associated transformation on the constraints

$\bar{\mathcal{L}}_{n}\rightarrow\bar{\mathcal{L}}^{\prime}_{n}=-\bar{\mathcal{L}}_{-n};\quad\bar{\mathcal{Q}}_{r}\rightarrow\bar{\mathcal{Q}}^{\prime}_{r}=\pm
i\bar{\mathcal{Q}}_{-r};$ (41)

Intriguingly, if we start from the usual Super-Virasoro constraints and
perform the above transformation, the algebra would remain preserved. In the
centrally extended version, the algebra will again remain invariant under
above transformations provided we have, $\bar{c}\to\bar{c}^{\prime}=-\bar{c}.$
Hence this set of transformations is an automorphism of the Super-Virasoro
algebra.

#### Tensile Flipped Vacuum:

Without any loss of generality we can consider the case
$\tilde{\alpha}^{\prime}_{n}=i\tilde{\alpha}_{-n}$ and
$\tilde{b}^{\prime}_{r}=-\tilde{b}_{-r}$. Then we can see that the vacuum of
the theory $|00\rangle_{f}$, is defined such that

$\alpha_{n}|00\rangle_{f}=b_{r}|00\rangle_{f}=0;~{}~{}~{}~{}\tilde{\alpha}_{-n}|00\rangle_{f}=\tilde{b}_{-r}|00\rangle_{f}=0\
\forall\ n,r>0.$ (42a)

The last line is condition for a flipped vacuum, where the nature of the
right-handed oscillators (and their actions on the vacuum) have been flipped.
This asymmetric vacuum is reminiscent of works in the context of Ambitwistor
string theory 333For related introduction to the Ambitwistor strings, the
reader should consult the seminal work [23] and references therein.. A similar
left-right asymmetric vacuum was first put forward briefly in [24], and later
revisited in a series of works including [25, 26, 18]. A closed bosonic
tensile string theory built on this vacuum was recently studied in [19].

The tensionless counterpart: We saw that for the maverick tensile strings, the
tensile vacuum evolved to the tensionless one via worldsheet Bogoliubov
transformations. However, for the flipped case, the situation is quite
different. Using the relations between the tensile and tensionless modes, and
following the UR limit on the worldsheet, the condition for the flipped vacuum
(42) in terms of the oscillators $C_{n}$, $\tilde{C}_{n}$, $\omega_{r}$ and
$\tilde{\omega}_{r}$ becomes

$C_{n}|00\rangle_{F}=\tilde{C}_{-n}|00\rangle_{F}=\omega_{r}|00\rangle_{F}=\tilde{\omega}_{-r}|00\rangle_{F}=0\
\ \ \forall\ n,r>0$ (43)

This means that the vacuum $|00\rangle_{F}$ remains the vacuum in the
tensionless theory. Of course, there may be pieces of the initial state that
drop off in the tensionless limit. One way of envisioning such a case is by
looking near the $\varepsilon\to 0$ limit. Here we would be able to write

$|00\rangle_{f}=|00\rangle_{F}+\varepsilon|01\rangle+\varepsilon^{2}|02\rangle+\ldots$
(44)

The limit clearly gives us

$\lim_{\varepsilon\to 0}|00\rangle_{f}=|00\rangle_{F}$ (45)

We can quantise the limiting theory around the same vacuum as the parent one.
To this end, we will discuss the representation theory underlying this
construction.

#### Tensionless Representations:

Once we define a vacuum of the theory, we can build up states by acting
oscillator modes on it. We are going to define the physical string states
appropriately, depending on the choice of vacua. Let us first start with the
usual tensile case. The states are labelled by their conformal weights:

${\mathcal{L}}_{0}|h,\bar{h}\rangle=h|h,\bar{h}\rangle,\quad\bar{{\mathcal{L}}}_{0}|h,\bar{h}\rangle=\bar{h}|h,\bar{h}\rangle$
(46)

A primary state $|h,\bar{h}\rangle_{p}$ is defined by

${\mathcal{L}}_{n}|h,\bar{h}\rangle=\bar{{\mathcal{L}}}_{n}|h,\bar{h}\rangle=0,\quad\mathcal{Q}_{r}|h,\bar{h}\rangle=\bar{\mathcal{Q}}_{r}|h,\bar{h}\rangle=0\
\forall\ n,r>0,$ (47)

The highest weight representations are built on the primary states by acting
raising operators
${\mathcal{L}}_{-n},\bar{{\mathcal{L}}}_{-n},\mathcal{Q}_{-r},\bar{\mathcal{Q}}_{-r}$.
This builds the Hilbert space of the usual tensile string.

Now, let us focus on the SBMSI. We will define states by labelling them with
$L_{0}$ and $M_{0}$444We work in the NS sector and so the SUSY generators are
half-integer moded and hence don’t contribute to the Cartan sub-algebra..

$L_{0}|h_{L},h_{M}\rangle=h_{L}|h_{L},h_{M}\rangle,\quad
M_{0}|h_{L},h_{M}\rangle=h_{M}|h_{L},h_{M}\rangle.$ (48)

The primary states are defined a way analogous to the above:

$L_{n}|h_{L},h_{M}\rangle_{p}=M_{n}|h_{L},h_{M}\rangle_{p}=0,\quad
G_{r}|h_{L},h_{M}\rangle_{p}=H_{r}|h_{L},h_{M}\rangle_{p}=0\ \forall\ n,r>0$
(49)

The highest-weight representations of the SBMSI are built out of modules
constructed in turn from these primary states by raising operators.

We also define another class of representations, familiar from the Poincare
algebra, the so-called induced representations. The states are still labelled
by $L_{0}$ and $M_{0}$.

$M_{0}|M,s\rangle_{i}=M|M,s\rangle_{i},\quad
L_{0}|M,s\rangle_{i}=s|M,s\rangle_{i}.$ (50)

But now the action of the generators for non-zero modes on the states is
different. These are given by

$M_{n}|M,s\rangle_{i}=0,\quad H_{r}|M,s\rangle_{i}=0\ \forall\ n,r\neq 0.$
(51)

Generic states are constructed by acting on these states with $L_{n}$ and
$G_{r}$ for all positive and negative values of the modes. For more details,
the readers are pointed to [27, 28] for a discussion of the bosonic algebra
and the homogeneous supersymmetric algebra.

#### Representations, Limit and Flip:

Let us rewrite (34) in a form that will be helpful in understanding the action
of the limit on the representation theory.

$\displaystyle
2{\mathcal{L}}_{n}=\frac{1}{\varepsilon}M_{n}+L_{n},\,2\bar{{\mathcal{L}}}_{n}=\frac{1}{\varepsilon}M_{-n}-L_{-n};\quad
2\mathcal{Q}_{r}=G_{r}+\frac{1}{\varepsilon}H_{r},\,2\bar{\mathcal{Q}}_{r}=i\left(G_{-r}-\frac{1}{\varepsilon}H_{-r}\right)$
(52)

It is clear from the above that in the UR limit, the primary state conditions
of Super-Virasoro (47) do not map to primary conditions of SBMSI (49) since
there is a mixing of positive and negative modes of the generators in the UR
limit. The highest weight representations of the Super-Virasoro, in the limit,
then do not yield highest-weight representations of the SBMSI, but instead
lead to the induced representation (51):

$\displaystyle{\mathcal{L}}_{n}|\varphi\rangle=0\,\Rightarrow
M_{n}|\phi\rangle=0,\quad\bar{{\mathcal{L}}}_{n}|\varphi\rangle=0\,\Rightarrow
M_{-n}|\phi\rangle=0,\quad\forall n>0;$
$\displaystyle\mathcal{Q}_{r}|\varphi\rangle=0\,\Rightarrow
H_{r}|\phi\rangle=0,\quad\bar{\mathcal{Q}}_{r}|\varphi\rangle=0\,\Rightarrow
H_{-r}|\phi\rangle=0,\quad\forall r>0.$ (53)

Notice that the labels on the state, however, do map to each other. The labels
are related by

$|\varphi\rangle=|h,\bar{h}\rangle,\,\,|\phi\rangle=|M,s\rangle,\quad\mbox{where}\quad
M=\varepsilon(h+\bar{h}),\,s=h-\bar{h}.$ (54)

On the other hand, there exists another limit from the Super-Virasoro algebra
to the SBMSI that can be thought of as a non-relativistic (NR) limit where the
speed of light goes to infinity instead of zero. The generators in this limit
scale in the following fashion

$L_{n}={\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{n},\quad
M_{n}=\varepsilon({\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{n}),\quad
G_{r}=Q_{r}+\bar{\mathcal{Q}}_{r},\quad
H_{r}=\varepsilon(Q_{r}-\bar{\mathcal{Q}}_{r}).$ (55)

Notice now that there is no mixing between positive and negative modes in this
limit. Hence, following steps outlined above, we can see that the NR limit
would yield highest-weight representations of SBMSI when you start with the
Super-Virasoro highest-weights. The weights are now related by

$h_{L}=h+\bar{h},\,h_{M}=\varepsilon(h-\bar{h}).$ (56)

Notice that the two contractions (34) and (55) are related by the automorphism
of the Virasoro algebra (41) discussed at the beginning of the section555This
seems to be at the heart of recent discussions of the isomorphism between NR
and UR conformal algebras discussed in the context of 3d Minkowskian
holography [29]. Some pertinent comments regarding the bosonic version of this
automorphism can be found in [30]..

In the case of the flipped vacuum, starting with the conditions (42), we can
once again construct primary states, with a sign on the modes of the anti-
holomorphic oscillators. We will need to modify (47) by the following
expression

$\displaystyle{\mathcal{L}}_{n}|h,\bar{h}\rangle=\bar{{\mathcal{L}}}_{-n}|h,\bar{h}\rangle=0,$
$\displaystyle\mathcal{Q}_{r}|h,\bar{h}\rangle=\bar{\mathcal{Q}}_{-r}|h,\bar{h}\rangle=0\
\forall\ n,r>0,$ (57a) $\displaystyle({\mathcal{L}}_{0}-h)|h,\bar{h}\rangle$
$\displaystyle=$
$\displaystyle(\bar{{\mathcal{L}}}_{0}+\bar{h})|h,\bar{h}\rangle=0.$ (57b)

This effectively means that we are imposing different representations of the
super-Virasoro on left and right handed sides. We have one highest weight
representation tensored with a lowest weight representation.

Applying the contraction as in (34), we see that for the UR limit on the
“flipped” vacuum effectively acts as a NR limit. We adapt the mantra:

$\mbox{Flipped}\oplus\mbox{UR limit}=\mbox{NR limit}$ (58)

Under the UR limit, the primaries of Super-Virasoro in this flipped sector map
to the primaries of the SBMSI. These states are now

$\displaystyle L_{n}|h_{L},h_{M}\rangle=M_{n}|h_{L},h_{M}\rangle=0,$
$\displaystyle G_{r}|h_{L},h_{M}\rangle=H_{r}|h_{L},h_{M}\rangle=0\ \forall\
n,r>0,$ (59a) $\displaystyle(L_{0}-h_{L})|h_{L},h_{M}\rangle$ $\displaystyle=$
$\displaystyle(M_{0}-h_{M})|h_{L},h_{M}\rangle=0.$ (59b)

In the above, $h_{L}=h-\bar{h}$ and $h_{M}=\varepsilon(h+\bar{h})$. The vacuum
of the tensionless limit of this theory would satisfy the above relations.
Therefore, just by tweaking one sector, not only will we have well-defined
hermitian constraints, but also a highest weight representation of SBMSI.

#### Connections with Ambitwistors

It is rather intriguing to note that tracing back the origins of inhomogeneous
tensionless strings naturally gives rise to such flipped string vacua as a
consequence. The recent connection between the ambitwistor string and the
tensionless limit proposed in [18] asserted that the ambitwistor strings arose
from an alternative gauge choice on the tensionless worldsheet theory. The
fact that a flipped vacuum also exists for the ambitwistor string points to
two interesting consequences. Firstly, we note that the symmetry of the
ambitwistor superstring in its present avatar is governed by the SBMSH. Our
current construction hints at a new inhomogeneous counterpart of the
ambitwistor superstring. Secondly, we have identified a parent theory and this
should be very useful in the endeavour of constructing the origin of the
ambitwistor superstring.

Let us add something more about why the inhomogeneous algebra would be a good
candidate for the ambitwistor superstring. The ambitwistor string is a chiral
object and the bosonic version is governed by a single Virasoro algebra. This
is so even though the symmetry that arises on the worldsheet is BMS3. By
considering the analysis of null vectors in the highest weight representations
performed in [21], which reduces the BMS3 to its Virasoro subalgebra, [31]
proposed how to reconcile the worldsheet symmetries of the ambitwistor string
to its chiral nature. The SBMSI contains a single Super-virasoro sub-algbera,
unlike the SBMSH, and one can perform a truncation of the SBMSI down to this
subalgebra in the case where the central charge $c_{M}$ vanishes [32, 13]. It
is thus very likely that there would exist a supersymmetric version of the
Ambitwistor string which has the SBMSI as its underlying algebra and the
chiral nature of ambitwistor superstring would manifest itself in a way very
similar to the bosonic case.

##  5 Conclusions

In this paper, we revisited the construction of the inhomogeneous tensionless
superstring introduced in [13]. We demanded reality of spinors and this led us
to manifestly hermitian generators for the underlying residual symmetry
algebra, the SBMSI. We then attempted to answer what properties of a tensile
theory this version of the tensionless superstring arose from and found two
distinct ways of implementing this. The first was the Maverick superstring
where fermions in the parent theory has certain intriguing properties. Armed
with this knowledge, as a natural extension of the bosonic analysis in [12],
we constructed the Bogoliubov transformations that connect the tensile and
tensionless oscillators and thereby came up with the mapping between the two
superstring vacua. The other parent tensile superstring, which we called the
Flipped superstring was one with an unconventional vacuum. Our construction
was based on a discrete automorphism of the parent Super-virasoro algebra.
This observation led us to this different vacuum, which has been considered
recently in [18] in the context of Ambitwistor strings. We made several
comments on the underlying representation theory and possible emergence of a
new ambitwistor superstring.

### Acknowledgements

We thank Punit Sharma for early collaboration on aspects of this work. We also
thank Rudranil Basu, David Skinner and Wei Song for useful discussions. This
research is supported by the following grants: DST-Inspire faculty fellowship
(Arjun Bagchi [AB]), Mathematical Research Impact Centric Support by SERB
(AB), the Max Planck visiting fellowship (AB, PP), the Chinese Academy of
Sciences (CAS) Hundred-Talent Program (Aritra Banerjee [ArB]), Key Research
Program of Frontier Sciences, CAS (ArB), Project 11647601 supported by NSFC
(ArB), ISIRD grant 9-252/2016/IITRPR/708 (SC), Grant No.
09/936(0098)/2013-EMR-I, SERB (PP).

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