††institutetext: a Birla Institute of Technology and Science, Pilani Campus,
Rajasthan 333031, India.
b Indian Institute of Technology Kanpur, Kanpur, 208018, India.

# Tensionless Strings in a Kalb-Ramond Background

Aritra Banerjee a, Ritankar Chatterjee b, Priyadarshini Pandit b
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate tensionless (or null) bosonic string theory with a Kalb-Ramond
background turned on. In analogy with the tensile case, we find that the Kalb-
Ramond field has a non-trivial effect on the spectrum only when the theory is
compactified on an $\left(S^{1}\right)^{\otimes d}$ background with $d\geq 2$.
We discuss the effect of this background field on the tensionless spectrum
constructed on three known consistent null string vacua. We elucidate further
on the intriguing fate of duality symmetries in these classes of string
theories when the background field is turned on.

##  1 Introduction

String theory, for past many decades, continues to hold significant prominence
as a potential quantum theory of gravity. At its core, string theory relies on
an inherent length scale identified as the tension ($T$) of the fundamental
string, which determines its behavior. By considering two extreme limits based
on this tension, we can understand different aspects of the theory. When $T$
becomes extremely large ($T\to\infty$), string theory converges to the point
particle limit, where it effectively reduces to a theory of (super)gravity.
Conversely, when $T$ approaches zero ($T\to 0$), string length becomes
extremely large, thereby enabling the exploration of ultra high energy regime
of the theory. This intriguing ultra high energy null string sector was first
analyzed by Schild Schild:1976vq . In this limit, the string scattering
amplitudes are found to exhibit a remarkably simple behavior Gross:1987kza ;
Gross:1987ar , characterized by an infinite number of linear relations.
Comparing with the massless limit of the point particle, all this suggests the
presence of a possible higher symmetry structure Gross:1988ue  within this
tensionless regime. Tensionless strings are also expected to exhibit massless
higher spin symmetries Vasiliev:2003cph ; Sagnotti:2003qa ; Bonelli:2003kh ,
resulting in connections to holographic dualities Sezgin:2002rt ;
Gaberdiel:2014cha .

Tensionless strings arise in various scenarios, providing valuable insights
into the behavior of strings in extreme situations, for example near generic
event horizons. It has been demonstrated in Bagchi:2021ban ; Bagchi:2020ats
that a tensile closed string gradually becomes tensionless when it encounters
the event horizon of a Schwarzschild black hole. One significant manifestation
of tensionless strings occurs near the Hagedorn temperature $\mathcal{T}_{H}$
PhysRevD.26.3735 ; Olesen:1985ej ; Atick:1988si , the very high-temperature
limit of string theory. An interesting observation in support of this is the
discovery of a closed-to-open string transition when the string tension is
tuned to zero Bagchi:2019cay . This observation highlights the significance of
studying the tensionless limit of string theory to gain a deeper understanding
of how strings perceive generic null surfaces, and perhaps by extension,
spacetime singularities. Furthermore, tensionless or null strings have
recently been employed to develop a quantum model of black hole horizons,
specifically BTZ black holes in AdS3 Bagchi:2022iqb , in a setup reminiscent
of the black hole membrane paradigm. By counting the microstates of null
strings, it has been possible to obtain insights into the entropy and
logarithmic corrections associated with these black holes.

Nevertheless, developments in the formalism front associated to null string
theory is still only in a nascent state. Since the early work of Schild
Schild:1976vq , there have been several studies on tensionless strings or null
strings in flat spacetime Karlhede:1986wb ; Lizzi:1986nv ; Gamboa:1989zc ;
Gamboa:1989px ; Gustafsson:1994kr ; Lindstrom:2003mg . In recent years,
however, the study of tensionless string theory has been pursued, in a rather
streamlined way, through two distinct approaches: the intrinsic approach
Isberg:1993av  and the limiting approach Bagchi:2013bga ; Bagchi:2015nca . The
intrinsic approach involves constructing a worldsheet action that incorporates
the degenerate metric structure of tensionless strings, while preserving
invariance under a gauge symmetry, namely worldsheet diffeomorphism
invariance. However, this gauge symmetry can only be partially fixed, similar
to the case of tensile theory. After gauge fixing, the action still remains
invariant under a residual gauge symmetry, which remarkably results in the
emergence of the BMS3 (3d Bondi-Metzner-Sachs) algebra Bagchi:2013bga ,
replacing the two copies of the Virasoro algebra present in the tensile case.
Notably, the BMS algebra represents the asymptotic symmetry algebra of
Minkowski spacetime at null infinity Bondi:1962px ; Sachs:1962wk ;
Ashtekar:1996cd . And it comes as no surprise that the same algebra appears in
the context of symmetries associated to generic null hypersurfaces Lévy1965 ;
NDS ; Henneaux:1979vn . This algebra has contributed significantly to our
understanding of holographic description of flat spacetime Bagchi:2010zz ;
Bagchi:2012xr ; Bagchi:2012cy ; Bagchi:2012yk ; Bagchi:2014iea ;
Bagchi:2016bcd ; Asadi:2016plj ; Jiang:2017ecm , black holes Donnay:2019jiz ,
fluid dynamics near high rapidity surfaces Bagchi:2023ysc ; Bagchi:2023rwd ;
Armas:2023dcz , physics of Fractons Bidussi:2021nmp ; Figueroa-
OFarrill:2023vbj , and flat bands in condensed matter physics Bagchi:2022eui ;
Marsot:2022imf , to only name a few.

The alternative approach, known as the limiting approach, involves taking a
specific limit on the worldsheet coordinates from the tensile string theory
Bagchi:2013bga ; Bagchi:2015nca . This limit corresponds to an ultra-
relativistic regime or Carrollian limit Duval:2014lpa  where the worldsheet
speed of light tends to zero. In terms of the worldsheet coordinates
($\tau,\sigma$), this limit can be expressed as
{$\tau\to\epsilon\tau,\sigma\to\sigma$}, with $\epsilon\to 0$. As expected,
under this scaling, the two copies of the Virasoro algebra transforms into the
BMS3 algebra, establishing the consistency between the limiting approach and
the intrinsic approach 111In fact, the Carrollian Conformal Algebra in two
dimensions (CCA2), the Carrollian contraction of two copies of Virasoro, is
isomorphic to the BMS3 Duval:2014lpa .. Furthermore, investigations have been
carried out to explore the supersymmetric extension of tensionless string
theory Bagchi:2016yyf ; Bagchi:2017cte ; Bagchi:2018wsn , both intrinsically
and from limits, where the residual gauge symmetry is that of a Super-BMS
algebra.

Significant advancement has been achieved in recent years in the quantization
of tensionless strings as well, leading to the discovery of three consistent
quantum theories based on different BMS invariant vacua: Oscillator, Induced,
and Flipped Bagchi:2020fpr ; Bagchi:2019cay ; Bagchi:2021rfw . The tensionless
limit of the conventional tensile quantum theory based on highest weight
representation connects to the Induced vacuum, which in turn corresponds to
the induced representation of the BMS algebra. The Flipped vacuum, which
belongs to the highest weight representation of the BMS algebra itself,
corresponds to the tensionless limit of a twisted string theory Casali:2016atr
, a cousin of usual tensile string theory. These two theories although
classically equivalent, are quantum mechanically very different Casali:2016atr
; Lee:2017utr . The Oscillator vacuum on the other hand, exhibits intriguing
connections to the Induced vacuum through Bogoliubov transformation. The light
cone quantization and path integral quantization of all the three tensionless
Bosonic theories have also been studied in Bagchi:2021rfw ; Chen:2023esw .

In this paper, we would like to continue understanding the formal structure of
a null string theory in better detail. As we know, tensile string theory
requires extra dimensions to be compactified on a manifold Polchinski:1998rq ;
Blumenhagen:2013fgp  to write down a four dimensional effective theory.
Compactification introduces quantum numbers like winding number and quantized
momentum, leading to a spectrum of massless and massive vector and scalar
states. Recently, this study of compactified string theory has been extended
to tensionless regime Banerjee:2023ekd  by the present set of authors. The
compactified spectrum for all three consistent string vacua has been explored
in detail. Unsurprisingly, it has also been noted that the impact of a
constant Kalb-Ramond field switched on the tensionless string theory target
space can only reflect in the spectrum for a theory with multiple dimension
compactification (toroidal compactification on $T^{d}$). This perfectly fits
in with the effect of constant $B$ field in compactification of tensile string
theory Polchinski:1998rq ; Blumenhagen:2013fgp ; Becker:2006dvp .

So it makes perfect sense to study in-depth the impact of a constant $B$ field
in the tensionless regime of string theory. Furthermore such a study opens up
the possibility of exploring the fate of string dualities on the tensionless
spectrum. In the earlier work Banerjee:2023ekd  we noted that the structure of
T-duality along the compactified directions apparently seems to be lost as
$\alpha^{\prime}\to\infty$ in this limit. The present work will also take a
close look on whether the presence of a background field can remedy this
issue. Consequently we will also look at the generalized symmetry structure of
our spectrum in comparison to the tensile counterpart thereof. For different
consistent vacua, investigation into the symmetries of mass spectrum will lead
to interesting conclusions. We will also have ample comparisons between
tensile and tensionless physics whenever applicable in our discussions.

### Plan of the paper

The paper is organized as follows: In Section (2), we present an intrinsic
formulation of the tensionless bosonic string action with a constant Kalb-
Ramond field along with its symmetries, obtained from the Nambu-Goto version
of the tensile theory with constant $B$ field. We also revisit the canonical
quantization of the tensionless string theory, leading to three distinct
quantum theories based on different vacua. The machinery for studying
compactification on a $d$-dimensional torus in this case is discussed in
Section (3). Sections (4),(5) and (6) delve into the detailed effects of the
constant $B$ field on the level matching conditions, mass spectrum and
symmetries thereof for the three quantum theories. Finally, in Section (7), we
summarize our findings and suggest potential future directions. The appendices
contain additional computations and supplementary discussions.

##  2 Tensionless strings with constant B field: Classical analysis

We start with a tensile string propagating in a Lorentzian flat target space
with an antisymmetric field $B_{\mu\nu}$. Note that one can easily write a
Nambu Goto version of the usual tensile string action with constant Kalb-
Ramond field (see Gangopadhyay:2006gmx  for example). The Polyakov version of
the same can be derived by constructing an interpolating Lagrangian. We start
off with the same action 222In Gangopadhyay:2006gmx , they have taken the
tension of the string to be unity. However, since our actual intention is to
take tensionless limit, we need to work with a finite tension $T$.

$\displaystyle S=T\int
d^{2}\xi\left[\sqrt{-\text{det}\gamma_{\alpha\beta}}+B_{\mu\nu}\dot{X}^{\mu}X^{\prime\nu}\right],$
(1)

where $\xi^{\alpha}$ ($\sigma,\tau$) are the worldsheet coordinates and
$\gamma_{\alpha\beta}$ is the induced metric on the worldsheet given by

$\displaystyle\gamma_{\alpha\beta}=\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}.$
(2)

Here $X^{\mu}$ denotes the spacetime coordinates, where $\mu\in
0,1,\cdots,D-1$ and $\eta_{\mu\nu}$ is the background flat metric. The
equation of motion of the above action can be calculated as,

$\displaystyle\dot{\Pi}^{\mu}+K^{\prime\mu}=0.$ (3)

Here the expression for $\Pi_{\mu}$ and $K_{\mu}$ are given by

$\begin{split}\Pi_{\mu}=\frac{\partial{\mathcal{L}}}{\partial{\dot{X}^{\mu}}}=T\frac{(\dot{X}\cdot
X^{\prime})X^{\prime}_{\mu}-X^{\prime 2}\dot{X}_{\mu}}{\sqrt{(\dot{X}\cdot
X^{\prime})^{2}-\dot{X}^{2}X^{\prime 2}}}+TB_{\mu\nu}X^{\prime\nu},\\\
K_{\mu}=\frac{\partial{\mathcal{L}}}{\partial{X^{\prime\mu}}}=T\frac{(\dot{X}\cdot
X^{\prime})\dot{X}_{\mu}-\dot{X}^{2}X^{\prime}_{\mu}}{\sqrt{(\dot{X}\cdot
X^{\prime})^{2}-\dot{X}^{2}X^{\prime
2}}}-TB_{\mu\nu}\dot{X}^{\nu},\end{split}$ (4)

where (′) represents differentiation with respect to $\sigma$ and dot
($\cdot$) represents differentiation with respect to $\tau$. The canonical
momentum $\Pi_{\mu}$ satisfies the following constraints

$\Pi_{\mu}X^{\prime\mu}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(\Pi_{\mu}-TB_{\mu\nu}X^{\prime\nu})^{2}+T^{2}X^{\prime 2}=0.$ (5)

Now, before taking tensionless limit of this action let us have a look at the
action in (1). In the tensionless limit, this action seemingly vanishes.
However, following the method outlined in Isberg:1993av , we can go to the
Hamiltonian formalism to achieve a form of the Polyakov action with a finite
$T\to 0$ limit. In Gangopadhyay:2006gmx , a similar method has been used to
achieve a form of Polyakov version of the flat space tensile string with
constant $B$ field. In the remaining part of the section we will see how does
it work for tensionless limit.

The canonical Hamiltonian of the action in (1) can be calculated using (4) as

$\displaystyle\Pi_{\mu}\dot{X}^{\mu}-\mathcal{L}=0.$ (6)

The result can also be viewed from the reparameterization invariance of the
action (1). Hence the total Hamiltonian can be written as a linear combination
of the constraints given in (5).

$\displaystyle\mathcal{H}_{T}=\rho\leavevmode\nobreak\
\Pi_{\mu}X^{\prime\mu}+\frac{\lambda}{2}\left\\{\left(\Pi_{\mu}-TB_{\mu\nu}X^{\prime\nu}\right)^{2}+T^{2}X^{\prime
2}\right\\}.$ (7)

Here $\lambda$ and $\rho$ are Lagrange multipliers. The interpolating
Lagrangian is calculated as

$\begin{split}&\mathcal{L}_{I}=\Pi_{\mu}\dot{X}^{\mu}-\mathcal{H}_{T}\\\
=\Pi_{\mu}\dot{X}^{\mu}-\rho\Pi_{\mu}X^{\prime\mu}-\frac{\lambda}{2}&\left[\Pi^{2}+T^{2}X^{\prime
2}-2TB_{\mu\nu}\Pi^{\mu}X^{\prime\nu}+T^{2}B_{\mu\nu}B^{\mu}_{\rho}X^{\prime\nu}X^{\prime\rho}\right].\end{split}$
(8)

In this interpolating Lagrangian, $\lambda$ and $\rho$ are treated as scalar
fields. Since the canonical momentum $\Pi_{\mu}$ is an auxiliary field, we
integrate it out from the interpolating Lagrangian. Finally we obtain the
following expression for Lagrangian

$\displaystyle\mathcal{L}_{I}=\frac{1}{2\lambda}\left[\dot{X}^{2}-2\rho(\dot{X}\cdot
X^{\prime})+(\rho^{2}-\lambda^{2}T^{2})X^{\prime 2}+2\lambda
TB_{\mu\nu}\dot{X}^{\mu}X^{\prime\nu}\right].$ (9)

We now identify the coefficients of $\dot{X}^{2}$, $\dot{X}\cdot X^{\prime}$
and $X^{\prime 2}$ in the form of a metric as

$\displaystyle g^{\alpha\beta}=\begin{bmatrix}1&-\rho\\\
-\rho&\leavevmode\nobreak\ \leavevmode\nobreak\
\rho^{2}-\lambda^{2}T^{2}\end{bmatrix},\hskip
14.22636pt\sqrt{-g}=\sqrt{-\text{det}g_{\alpha\beta}}=\frac{1}{\lambda
T}\leavevmode\nobreak\ .$ (10)

This results in the Polyakov version of the action given by,

$\displaystyle S_{I}=\frac{T}{2}\int d\tau
d\sigma\left(\sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}+\epsilon^{\alpha\beta}B_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\right).$
(11)

Now taking the tensionless limit, viz. $T\to\epsilon T$, $\epsilon\to 0$ on
this action 333Note that this can be equivalently thought of as
$\alpha^{\prime}\to\frac{c^{\prime}}{\epsilon}$ with a finite $c^{\prime}$,
which we will be using later on., we can redefine our worldsheet variables to
take care of the degenerate structure 444This kind of limit from tensile
string with a background field to its tensionless counterpart has been
discussed earlier in a recent work Gomis:2023eav . In the notation there,
these strings are called M(-1)T strings. Moreover, in Bagchi:2024unl , the
example of taking ultrarelativistic (Carroll) limit on a theory of two scalars
with a $J^{1}\wedge\bar{J}^{2}$ marginal deformation has been considered,
which boils down to the same physical situation.,

$\displaystyle T\sqrt{-g}g^{\alpha\beta}\to V^{\alpha}V^{\beta}\hskip
8.53581pt\text{and}\hskip
8.53581ptB_{\mu\nu}\to\frac{B_{\mu\nu}}{\epsilon},\hskip
14.22636pt\text{where}\hskip
14.22636ptV^{\alpha}=\frac{1}{\sqrt{\lambda}}(1,-\rho).$ (12)

Here, $V^{\alpha}$ is a timelike vielbein density of the worldsheet. Let us
define the scaled antisymmetric field $\mathcal{B}_{\mu\nu}$ as

$\displaystyle\mathcal{B}_{\mu\nu}=\frac{1}{\alpha^{\prime}}B_{\mu\nu},\hskip
14.22636ptT=\frac{1}{2\pi\alpha^{\prime}}$ (13)

The limit on the $B$ field makes sure that $\mathcal{B}_{\mu\nu}$ remains
finite. Hence the action of tensionless string with a $B$ field will be

$\displaystyle S=\int
d^{2}\xi\left(V^{\alpha}V^{\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}+\frac{1}{2\pi}\epsilon^{\alpha\beta}\mathcal{B}_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\right).$
(14)

This will be the action we will be mostly concerned with throughout this work.
Note that this is nothing but an extension of the so-called ILST action
Isberg:1993av  for tensionless strings.

###  2.1 Symmetries of the action

It can be shown that the action (14) is invariant under diffeomorphism
($\xi^{\alpha}\to\xi^{\alpha}+\epsilon^{\alpha}$), under which, the vector
density $V^{\alpha}$ and the scalars $X^{\mu}(\sigma,\tau)$ transform in the
following way:

$\begin{split}\delta
V^{\alpha}=-&V^{\beta}\partial_{\beta}\epsilon^{\alpha}+\epsilon^{\beta}\partial_{\beta}V^{\alpha}+\frac{1}{2}(\partial_{\beta}\epsilon^{\beta})V^{\alpha}\\\
&\delta
X^{\mu}(\sigma,\tau)=\epsilon^{\alpha}\partial_{\alpha}X^{\mu}(\sigma,\tau)\end{split}$
(15)

The action (14) with $\mathcal{B}_{\mu\nu}=0$ is known to be invariant under
the transformation (15) (see Isberg:1993av ; Bagchi:2015nca ). We focus on the
extra term, and variation of the $B$ field part gives us just a total
derivative term

$\delta(\epsilon^{\alpha\beta}\mathcal{B}_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu})=\partial_{\alpha}(\epsilon^{\alpha\beta}\mathcal{B}_{\mu\nu}\epsilon^{\gamma}\partial_{\gamma}X^{\mu}\partial_{\beta}X^{\nu}).$
(16)

Hence the worldsheet diffeomorphism leave the full action (14) invariant as
well. This enables us to do gauge fixing analogous to conformal gauge in case
of tensile string theory. Since $V^{\alpha}$ of this action transforms like
(15), we can make a gauge choice $V^{\alpha}=(v,0)$, where $v$ is an arbitrary
parameter. For our convenience we make the following choice of
parameterization for the constant $v$:

$\displaystyle v=\sqrt{\frac{1}{2\pi c^{\prime}}},\hskip
14.22636pt\alpha^{\prime}=\frac{c^{\prime}}{\epsilon},\hskip
14.22636pt\epsilon\to 0.$ (17)

Similar to its tensile counterpart, there is a residual symmetry left over
after the above gauge fixing. The gauge fixed action will remain invariant
under the following form of $\epsilon^{\alpha}$

$\epsilon^{\alpha}=\left(f^{\prime}(\sigma)\tau+g(\sigma),f(\sigma)\right).$
(18)

Under this set of transformations, a free function $G(\xi^{\alpha})$
transforms as:

$\delta
G=\left[f^{\prime}(\sigma)\tau\partial_{\tau}+f(\sigma)\partial_{\sigma}+g(\sigma)\partial_{\tau}\right]G=\left[L(f)+M(g)\right]G.$
(19)

Therefore, the symmetry generators can be defined as:

$\begin{split}L(f)=f^{\prime}&(\sigma)\tau\partial_{\tau}+f(\sigma)\partial_{\sigma}=\sum_{n}a_{n}e^{in\sigma}(\partial_{\sigma}+in\tau\partial_{\tau})=-i\sum_{n}a_{n}L_{n},\\\
&M(g)=g(\sigma)\partial_{\tau}=\sum_{n}b_{n}e^{in\sigma}\partial_{\tau}=-i\sum_{n}b_{n}M_{n}.\end{split}$
(20)

Here $f$ and $g$ are expanded in Fourier modes as
$f=\sum_{n}a_{n}e^{in\sigma},\leavevmode\nobreak\
g=\sum_{n}b_{n}e^{in\sigma}$. The symmetry algebra can be written as:

$[L_{m},L_{n}]=(m-n)L_{m+n},\leavevmode\nobreak\ \leavevmode\nobreak\
[L_{m},M_{n}]=(m-n)M_{m+n},\leavevmode\nobreak\ \leavevmode\nobreak\
[M_{m},M_{n}]=0.$ (21)

The above algebra forms the classical part of the $\text{BMS}_{3}$ algebra,
which arises here replacing the two copies of Virasoro (Witt) algebra in the
case of tensile strings. Hence, the worldsheet residual gauge symmetry of the
tensionless string theory with a $B$ field remains that of $\text{BMS}_{3}$.
This is what one expects for generic null worldsheet theories Isberg:1993av ;
Bagchi:2015nca .

###  2.2 Energy-Momentum tensor

We now move on to compute the energy-momentum tensor for the action (14). In
order to do that, let us recall the definition of energy momentum tensor in
terms of vielbeins as in Hartong:2015xda ; Bergshoeff:2017btm ; Bagchi:2022eav
:

$\displaystyle T^{\alpha}_{\hskip
2.84526pt\beta}=\frac{e^{\alpha}_{A}}{2e}\frac{\delta S}{\delta
e^{\beta}_{A}}$ (22)

In the above $e^{\alpha}_{A}$ $(A=0,1)$ are the timelike and spacelike
vielbeins repectively. $e$ is given by the defining relations:

$\displaystyle
e=\det(e^{A}_{\alpha})=\frac{1}{2}\epsilon_{AB}e^{A}_{\alpha}e^{B}_{\beta}\epsilon^{\alpha\beta},\hskip
14.22636pte^{A}_{\alpha}e^{\alpha}_{B}=\delta^{A}_{B},\hskip
14.22636pte^{A}_{\alpha}e^{\beta}_{A}=\delta^{\beta}_{\alpha}.$ (23)

Looking at the action (14) one can identify that $e^{\alpha}_{0}=V^{\alpha}$
for our case. One can also see that there is no spacelike vielbein
$e^{\alpha}_{1}$ in the action. Hence, using (22), it is easy to find the
following energy-momentum tensor:

$T^{\alpha}_{\beta}=V^{\alpha}V^{\gamma}\partial_{\gamma}X^{\mu}\partial_{\beta}X_{\mu}-\frac{1}{2}V^{\rho}V^{\gamma}\partial_{\rho}X^{\mu}\partial_{\gamma}X_{\mu}\delta^{\alpha}_{\beta}.$
(24)

This EM tensor turns out to be identical to the EM tensor for tensionless
string without $B$ field Isberg:1993av . We recall that in the tensile
counterpart of the theory too, the constant $B$ field part did not affect the
EM tensor Blumenhagen:2013fgp  since it is a boundary term. In the gauge (17),
the non-trivial components of the (EM) tensor are:

$T^{0}_{0}=-T^{1}_{1}=\frac{1}{4\pi c^{\prime}}\dot{X}^{2}\equiv
T_{2}(\sigma,\tau),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T^{0}_{1}=\frac{1}{2\pi c^{\prime}}\dot{X}\cdot
X^{\prime}\equiv T_{1}(\sigma,\tau).$ (25)

The zeroth component of the Noether current can be integrated to be,

$Q=\int d\sigma J^{0}=\int
d\sigma\left[T_{1}f+T_{2}(f^{\prime}\tau+g)\right].$ (26)

We now expand $f$ and $g$ in Fourier modes and obtain

$Q=\sum_{n}a_{n}\int d\sigma\left(T_{1}+in\tau
T_{2}\right)e^{in\sigma}+\sum_{n}b_{n}\int d\sigma
T_{2}e^{in\sigma}=\sum_{n}a_{n}L_{n}+\sum_{n}b_{n}M_{n}.$ (27)

Thus, inverting the above relation we can write the expression for EM tensor
in terms of generators of BMS algebra as

$T_{1}(\tau,\sigma)=\frac{1}{2\pi}\sum_{n}(L_{n}-in\tau
M_{n})e^{-in\sigma},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
T_{2}(\tau,\sigma)=\frac{1}{2\pi}\sum_{n}M_{n}e^{-in\sigma}.$ (28)

###  2.3 Equations of motions and mode expansions

Now, we calculate the equations of motion for the tensionless string with
constant $B$ field. It turns out that, just like the tensile theory, here too,
the $B$ field dependent part in the action doesn’t have any effect on the
equations of motion and we get

$\partial_{\alpha}(V^{\alpha}V^{\beta}\partial_{\beta}X^{\mu})=0$ (29a)
$V^{\beta}\gamma_{\alpha\beta}=0,$ (29b)

where $\gamma_{\alpha\beta}$ is the induced metric on the worldsheet defined
in (2). Equations (29a), (29b) are identical to the equations of motion
derived for free tensionless strings in Isberg:1993av ; Bagchi:2015nca .
Equation (29a) is the equation of motion for $X^{\mu}$. Equation (29b) yields
two constraints on $X^{\mu}$. In the chosen gauge $V^{\alpha}=(v,0)$, our
equation of motion of $X^{\mu}$ becomes

$\ddot{X}^{\mu}=0.$ (30)

The equations of motion (30) with a closed string boundary condition
$X^{\mu}(\tau,\sigma)=X^{\mu}(\tau,\sigma+2\pi)$, can be solved by the
following mode expansion:

$X^{\mu}(\tau,\sigma)=x^{\mu}+\sqrt{\frac{c^{\prime}}{2}}B^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}(A^{\mu}_{n}-in\tau B^{\mu}_{n})e^{-in\sigma}.$ (31)

In the same gauge ($V^{\alpha}=(v,0)$), the constraints (29b) will reduce to

$\displaystyle\dot{X}^{2}=0,\hskip 14.22636pt\dot{X}\cdot X^{\prime}=0.$ (32)

Following Isberg:1993av ; Bagchi:2015nca , we identify them as the components
of the stress-energy tensor ($T^{0}_{0},T^{0}_{1}$). We use the mode expansion
(31) on the two constraints (32) and equate the result with the expression for
the EM tensor defined in terms of generators of the residual symmetry algebra
in (28). After some algebra, this readily gives,

$L_{n}=\frac{1}{2}\sum_{m}A_{-m}\cdot B_{m+n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
M_{n}=\frac{1}{2}\sum_{m}B_{-m}\cdot B_{m+n}.$ (33)

We now determine the algebra satisfied by these oscillator modes ($A,B$). In
the case of tensile string theory with a constant $B$ field, we have seen that
the only aspect affected by the presence of $B$ field is the canonical
momentum Polchinski:1998rq ; Blumenhagen:2013fgp ; Becker:2006dvp . Here too,
we are going to see the same. The canonical momentum in our case turns out to
be

$\displaystyle\Pi_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{X}^{\mu}}=\frac{\dot{X}^{\nu}}{2\pi
c^{\prime}}\eta_{\nu\mu}+\frac{X^{\prime\nu}}{2\pi}\mathcal{B}_{\mu\nu}.$ (34)

In the above equation, we take $\epsilon^{01}=1$. However, the equal time
Poisson bracket between $X(\sigma,\tau)$ and $\dot{X}(\sigma^{\prime},\tau)$
will still be intact. As a result, the Poisson brackets between the
oscillators $A$ and $B$ will be same as they are for tensionless strings
without $B$ field (obtained, for example, in Isberg:1993av )

$\\{A_{m}^{\mu},A_{n}^{\nu}\\}_{P.B}\leavevmode\nobreak\ =\leavevmode\nobreak\
\\{B_{m}^{\mu},B_{n}^{\nu}\\}_{P.B}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\\{A_{m}^{\mu},B_{n}^{\nu}\\}_{P.B}=-2im\delta_{m+n}\eta^{\mu\nu}.$ (35)

The above algebra is not the harmonic oscillator algebra. Hence we define new
modes which are linear combination of these $(A,B)$ oscillator modes
Bagchi:2020fpr  satisfying the harmonic oscillator algebra. These new modes
can be written as,

$C_{n}^{\mu}=\frac{1}{2}(A_{n}^{\mu}+B_{n}^{\nu}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{C}_{n}^{\mu}=\frac{1}{2}(-A_{-n}^{\mu}+B_{-n}^{\mu}).$ (36)

These modes form the oscillator basis of the tensionless string. They satisfy
the algebra given by,

$\\{C_{m}^{\mu},C_{n}^{\nu}\\}=-im\delta_{m+n,0}\eta^{\mu\nu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\\{\tilde{C}_{m}^{\mu},\tilde{C}_{n}^{\nu}\\}=-im\delta_{m+n,0}\eta^{\mu\nu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\\{C_{m}^{\mu},\tilde{C}_{n}^{\nu}\\}=0.$ (37)

Now we can write the mode expansion (31) in terms of these new modes as

$\displaystyle
X^{\mu}(\tau,\sigma)=x^{\mu}+2\sqrt{\frac{c^{\prime}}{2}}C^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\left[(C^{\mu}_{n}-\tilde{C}^{\mu}_{-n})-in\tau(C^{\mu}_{n}+\tilde{C}^{\mu}_{-n})\right]e^{-in\sigma},$
(38)

where we have used the periodicity of $X^{\mu}$. We can write the mode
expansion given above in terms of “left” and “right” moving modes
Bagchi:2020fpr  as:

$\displaystyle X^{\mu}_{L}$
$\displaystyle=\frac{x^{\mu}}{2}+\sqrt{\frac{c^{\prime}}{2}}C^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}(C^{\mu}_{n}-in\tau C^{\mu}_{n})e^{-in\sigma}$ (39)
$\displaystyle X^{\mu}_{R}$
$\displaystyle=\frac{x^{\mu}}{2}+\sqrt{\frac{c^{\prime}}{2}}\tilde{C}^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}(\tilde{C}^{\mu}_{n}-in\tau\tilde{C}^{\mu}_{n})e^{in\sigma},$
(40)

where the zeroth modes are related to the momentum of the tensionless string
as

$C_{0}^{\mu}=\tilde{C}_{0}^{\nu}=\sqrt{\frac{c^{\prime}}{2}}k^{\mu}.$

Equation (34) leads us to the following centre of mass momentum

$\displaystyle\pi_{\mu}=\int_{0}^{2\pi}d\sigma\Pi_{\mu}=k^{\nu}\eta_{\nu\mu}+\frac{1}{2\pi}\Big{(}X^{\nu}(\sigma+2\pi,\tau)-X^{\nu}(\sigma,\tau)\Big{)}\mathcal{B}_{\mu\nu},$
(41)

where $k^{\mu}$ is the centre of mass momentum of the tensionless string
without background field. Now, in the case of closed strings in a non-
compactified background, we have
$X^{\nu}(\sigma+2\pi,\tau)=X^{\nu}(\sigma,\tau)$ and hence there is no
modification in the centre of mass momentum. However, in case of compactified
background, a winding number ($W$) appears in (41). Hence, effect of constant
$B$ field can only be observed in case of compactified background spacetime.
This prompts us to now move on to quantize the classical bosonic closed
tensionless strings with $B$ field, albeit in a compact target space.

###  2.4 Three possible quantization schemes

After a detailed discussion on the classical picture of tensionless strings
with constant Kalb-Ramond field, we now move on to quantize it using the usual
method of covariant quantization, analogous to quantization of bosonic
tensionless strings without $B$ field. We begin with the tensionless action
with $B$ field (14), and notice that the equations of motion and constraints
obtained from this action in the gauge $V^{\alpha}=(v,0)$, are identical to
the case of tensionless strings without $B$ field. Hence, the commutation
relations between the operators $X^{\mu}$ and its canonical momenta $P^{\mu}$
also takes the same form:

$\begin{split}&\leavevmode\nobreak\
\left[X^{\mu}(\tau,\sigma),P_{\nu}(\tau,\sigma^{\prime})\right]=i\delta(\sigma-\sigma^{\prime})\delta^{\mu}_{\nu}.\end{split}$
(42)

We use the above relations in the mode expansion (38) to compute the
commutators of $C$ oscillators satisfying harmonic oscillator algebra, which
rather unsurprisingly reads as,

$\displaystyle[C^{\mu}_{m},C^{\nu}_{n}]=[\tilde{C}^{\mu}_{m},\tilde{C}^{\nu}_{n}]=m\eta^{\mu\nu}\delta_{m+n},\hskip
14.22636pt[C^{\mu}_{m},\tilde{C}^{\nu}_{n}]=0.$ (43)

Now we move on to define putative vacua of the theory using $C$ oscillators
and build a Hilbert space of physical states on it. The spectrum of physical
states however demands imposition of constraints on the Hilbert space. There
are different ways to apply these constraints, as elucidated in Bagchi:2020fpr
, resulting in different quantum theories, as will be discussed below.

### Imposing quantum constraints

Here we focus on different ways of applying constraints on the Hilbert space
to pick out the physical state spectrum. Classically we restrict the Hilbert
space simply by demanding vanishing of the component of the energy momentum
tensor (29b). However, when we quantize the theory, we promote these
components of energy momentum tensor $T_{1}$ and $T_{2}$ to operators and
demand vanishing of all its matrix element on the physical states,

$\bra{phys^{\prime}}T_{1}\ket{phys}=\bra{phys^{\prime}}T_{2}\ket{phys}=0.$
(44)

Using the relation (28), the above constraints can be rewritten in terms of
generators of BMS3 algebra as

$\bra{phys^{\prime}}L_{n}\ket{phys}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\bra{phys^{\prime}}M_{n}\ket{phys}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\forall n\in\mathbb{Z}.$ (45)

In order to impose the above constraints, we assume the vacuum state to be a
physical state at least for $n\neq 0$,

$\bra{0}L_{n}\ket{0}=\bra{0}M_{n}\ket{0}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall n\neq
0.$ (46)

For $n=0$, normal ordering ambiguities takes part in filtering out the
physical spectrum on the Hilbert space. Based on equation (45), a total of
nine possible ways might exist for imposing constraints on physical states.
However, on detailed analysis, it was found that only three out of these nine
possibilities align consistently with the BMS3 algebra Bagchi:2020fpr ,

$\displaystyle L_{n}\ket{phys}\neq 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ M_{n}\ket{phys}\neq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\forall\leavevmode\nobreak\ n),$
(47a) $\displaystyle L_{n}\ket{phys}\neq 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
M_{n}\ket{phys}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\forall\leavevmode\nobreak\ n\neq 0),$ (47b)
$\displaystyle L_{n}\ket{phys}=\leavevmode\nobreak\ \leavevmode\nobreak\
M_{n}\ket{phys}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\forall\leavevmode\nobreak\ n>0).$ (47c)

The three conditions mentioned above give rise to three inequivalent quantum
theories based on three distinct vacua namely, Oscillator, Induced and Flipped
vacuum. In the rest of this work, we will gradually examine the impact of
having a constant $B$ field on each of these cases in detail in separate
sections.

##  3 Compactification on torus $T^{d}$

Before moving on to the spectra, in this section, we study the effects of a
constant $B$ field, which generically apply to all the three quantum theories
mentioned earlier. We make two choices to start with: the components of metric
along the compactified coordinates are Euclidean, i.e. $G_{IJ}=\delta_{IJ}$
and the only non-zero components of the $B$ field will be along the
compactified directions. Let us consider the compactification radius of $I$-th
compactified direction to be $R_{I}$. In order to compactify $d$ dimensions,
we make the following identification555For compactified indices $I$, repeated
indices are not summed over unless explicitly mentioned.

$\displaystyle X^{I}\sim X^{I}+2\pi\omega^{I}R_{I}\hskip
14.22636ptI\in\\{1,2,....,d\\}.$ (48)

This leads us to the following closed string boundary conditions for
compactified coordinates

$\displaystyle
X^{I}(\sigma+2\pi,\tau)=X^{I}(\sigma,\tau)+2\pi\omega^{I}R_{I}\hskip
14.22636ptI\in\\{1,2,....,d\\},$ (49)

where $\omega^{I}$ are the winding numbers. Following Becker:2006dvp , we use
the following coordinate rescaling on the compactified coordinates

$\displaystyle\widetilde{X}^{I}=\frac{X^{I}}{R_{I}}.$ (50)

The components of the metric and the antisymmetric $B$ field along the
compactified coordinates will change in the following way (Einstein’s
summation convention not applied in order to avoid confusion, there is no
summation over repeated indices)

$\begin{split}&G_{IJ}=\frac{\partial
X^{K}}{\partial\widetilde{X}^{I}}\frac{\partial
X^{L}}{\partial\widetilde{X}^{J}}\delta_{KL}=R_{I}R_{J}\sum_{K=1}^{d}\sum_{L=1}^{d}\delta^{K}_{I}\delta^{L}_{J}\delta_{KL}=R^{2}_{I}\delta_{IJ}\\\
&\mathbb{B}_{IJ}=\frac{\partial
X^{K}}{\partial\widetilde{X}^{I}}\frac{\partial
X^{L}}{\partial\widetilde{X}^{J}}\mathcal{B}_{KL}=R_{I}R_{J}\sum_{K=1}^{d}\sum_{L=1}^{d}\delta^{K}_{I}\delta^{L}_{J}\mathcal{B}_{KL}=R_{I}R_{J}\mathcal{B}_{IJ}\end{split}$
(51)

The expression of inverse metric $G^{-1}$ along compactified directions
becomes

$\displaystyle G^{IJ}=\frac{1}{R^{2}_{I}}\delta^{IJ}.$ (52)

For the new scaled coordinates $\widetilde{X}^{I}$, the identification (48)
looks like

$\displaystyle\widetilde{X}^{I}\sim\widetilde{X}^{I}+2\pi\omega^{I}.$ (53)

As a result, the new periodic conditions satisfied by $\widetilde{X}^{I}$ will
be
$\widetilde{X}^{I}(\sigma+2\pi,\tau)-\widetilde{X}^{I}(\sigma,\tau)=2\pi\omega^{I}$.
In terms of these new coordinates the relevant part of the gauge fixed action
in (14) becomes

$\displaystyle\int
d^{2}\xi\left(v^{2}\dot{\widetilde{X}}^{I}\dot{\widetilde{X}}^{J}G_{IJ}+\epsilon^{\alpha\beta}\mathbb{B}_{IJ}\partial_{\alpha}\widetilde{X}^{I}\partial_{\beta}\widetilde{X}^{J}\right),$
(54)

where $v$ is the parameter given in (17). The canonical momentum along the
compactified directions is given by

$\displaystyle\widetilde{\Pi}_{I}=\frac{1}{c^{\prime}}\sum_{J=1}^{d}\dot{\widetilde{X}}^{J}G_{IJ}+\sum_{J=1}^{d}\mathbb{B}_{IJ}\widetilde{X}^{\prime
J}.$ (55)

Hence, the centre of mass momentum will be a combination of compactified
momenta and winding number, thus including the contribution from the $B$
field:

$\displaystyle\widetilde{\pi}_{I}=\frac{1}{2\pi}\int_{0}^{2\pi}d\sigma\widetilde{\Pi}_{I}=\sum_{J=1}^{d}K^{J}G_{IJ}+\sum_{J=1}^{d}\mathbb{B}_{IJ}\omega^{J}.$
(56)

Now since the canonical momentum is the generator of translation, we would
like the quantum mechanical wave function containing the factor
$\exp\left[{i\sum_{I=1}^{d}\widetilde{\pi}_{I}\widetilde{X}^{I}}\right]$ to be
periodic in $\widetilde{X}^{I}\sim\widetilde{X}^{I}+2\pi\omega^{I}$ as well.
As a result we need to impose the extra condition
$\omega^{I}\widetilde{\pi}_{I}\in\mathbb{Z}$ on the system. The implication is
that $\pi_{I}$ has to be quantized as winding number ($\omega^{I}$) is an
integer:

$\widetilde{\pi}_{I}=k_{I},\hskip 14.22636ptk_{I}\in\mathbb{Z}.$ (57)

Just like in our earlier work Banerjee:2023ekd , here again defining a
dimensionless field $Y^{I}$ will make our calculation more convenient. $Y^{I}$
is defined as

$\displaystyle\widetilde{X}^{I}=\sqrt{\frac{c^{\prime}}{2}}Y^{I}.$ (58)

Using (31), $Y^{I}$ can be expressed in terms of oscillators $A$ and $B$ as

$\displaystyle Y^{I}=y^{I}+A^{I}_{0}\sigma+B^{I}_{0}\tau+i\sum_{n\neq
0}\frac{1}{n}(A^{I}_{n}-in\tau B^{I}_{n})e^{-in\sigma},$ (59)

where both zero modes are non-zero in the compactified theory. From (53), (56)
and (57) we obtain the following form of those:

$\displaystyle(B_{0})_{I}=\sqrt{2c^{\prime}}\Big{(}k_{I}-\sum_{J=1}^{d}\mathbb{B}_{IJ}\omega^{J}\Big{)},\hskip
28.45274ptA^{I}_{0}=\sqrt{\frac{2}{c^{\prime}}}\omega^{I}.$ (60)

We can also express $Y^{I}$ in terms of the oscillators $C$ and $\tilde{C}$ as
before:

$\begin{split}Y^{I}_{L}&=y^{I}_{L}+K^{I}_{L}(\tau+\sigma)+i\sum_{n\neq
0}\frac{1}{n}(C^{\mu}_{n}-in\tau C^{\mu}_{n})e^{-in\sigma},\\\
Y^{I}_{R}&=y^{I}_{R}+K^{I}_{R}(\tau-\sigma)+i\sum_{n\neq
0}\frac{1}{n}(\tilde{C}^{\mu}_{n}-in\tau\tilde{C}^{\mu}_{n})e^{in\sigma}.\end{split}$
(61)

Here $(K^{I})_{L,R}$ are the left and right moving momenta of the string along
compactified dimensions. Applying (48) and (57) on (61) gives us

$\displaystyle(k_{I})_{L,R}=\frac{1}{\sqrt{2}}\bigg{\\{}\sqrt{c^{\prime}}\Big{(}k_{I}-\sum_{J=1}^{d}\mathbb{B}_{IJ}\omega^{J}\Big{)}\pm\frac{1}{\sqrt{c^{\prime}}}\sum_{J=1}^{d}G_{IJ}\omega^{J}\bigg{\\}},$
(62)

which is the main input in generating the spectra of the compact theory. With
our basic ingredients in place, we now go forward with the discussion of the
three vacua separately in what follows.

##  4 Oscillator vacuum

In this section, we explore the tensionless theory resulting from the first
and the weakest condition in (47). The physical state conditions for a theory
constructed upon this vacuum are imposed in pure “sandwich” form of stress
tensor components:

$\displaystyle\bra{phys^{\prime}}L_{n}-a_{L}\delta_{n,0}\ket{phys}=\bra{phys^{\prime}}M_{n}-a_{M}\delta_{n,0}\ket{phys}=0.$
(63)

Here, $L_{n}$ and $M_{n}$ are the generators of the worldsheet BMS algebra and
$a_{\\{L,M\\}}$ are the ordering ambiguities associated to the zero modes.
Note that these generators can be written in terms of the oscillators $C$ and
$\tilde{C}$ from (33) as

$\begin{split}L_{n}&=\frac{1}{2}\sum_{m}\left[C_{-m}\cdot
C_{m+n}-\tilde{C}_{-m}\cdot\tilde{C}_{m-n}\right],\\\
M_{n}&=\frac{1}{2}\sum_{m}\left[C_{-m}\cdot
C_{m+n}+\tilde{C}_{-m}\cdot\tilde{C}_{m-n}+2C_{-m}\cdot\tilde{C}_{-m-n}\right].\end{split}$
(64)

The vacuum on which this theory is based is called oscillator vacuum which is
consequently defined as

$\displaystyle
C^{\mu}_{n}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{c}\leavevmode\nobreak\
=\leavevmode\nobreak\
\tilde{C}^{\mu}_{n}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{c}=0\hskip
14.22636pt\forall n>0.$ (65)

The name reflects the similarity with tensile highest weight string vacuum
defined likewise using the oscillators $\alpha_{n},\tilde{\alpha}_{n}$. To
familiarise the reader with the notation used here, note that using a string
of $C_{-n}$ and $\tilde{C}_{-n}$, we can construct a generic excited state on
this Oscillator vacuum as follows,

$\displaystyle\ket{r,s,k^{\mu},k_{I},\omega^{I}}=\sum_{j}\rho_{j}\Bigg{(}\prod_{i=1}^{p}C^{a_{i}}_{-m_{i}}\prod_{j=1}^{q}\tilde{C}^{b_{j}}_{-n_{j}}\Bigg{)}_{j}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{c},$
(66)

where $a_{i}$ is the power of $C_{-m_{i}}$ and $b_{j}$ is the power of
$\tilde{C}_{-n_{j}}$ oscillators. Here the “levels” $r$ and $s$ are given by

$\displaystyle r=\sum_{i}^{p}a_{i}m_{i},\hskip
14.22636pts=\sum_{i}^{q}b_{i}n_{i}.$ (67)

From (64), we obtain the expression of $L_{0}$ in terms of the momenta
$k_{I\hskip 0.7113ptL,R}$ as

$\begin{split}L_{0}=\mathcal{N}-\widetilde{\mathcal{N}}+\frac{1}{2}\sum_{I=1}^{d}\big{(}K^{I}_{L}&K_{I\hskip
0.7113ptL}-K^{I}_{R}K_{I\hskip
0.7113ptR}\big{)}=\mathcal{N}-\widetilde{\mathcal{N}}+\sum_{I,J=1}^{d}\Big{(}k_{I}-\mathbb{B}_{IJ}\omega^{J}\Big{)}\omega^{I},\\\
&=\mathcal{N}-\widetilde{\mathcal{N}}+\sum_{I=1}^{d}k_{I}\omega^{I}.\end{split}$
(68)

where the operators are defined as

$\displaystyle\mathcal{N}=\sum_{m>0}C_{-m}\cdot C_{m}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\hskip
14.22636pt\widetilde{\mathcal{N}}=\sum_{m>0}\tilde{C}_{-m}\cdot\tilde{C}_{m}.$
(69)

Here these $\mathcal{N}$ and $\widetilde{\mathcal{N}}$ are nothing but number
operators for the $C$ and $\tilde{C}$. To go from first line to second line in
(68) we have used the fact that $\mathbb{B}_{IJ}$ is antisymmetric in $I,J$
indices and hence the term $\mathbb{B}_{IJ}\omega^{I}\omega^{J}=0$. Similarly,
the expression for $M_{0}$ in terms of $(k_{I})_{L,R}$ can be obtained from
(64) as 666Note that there is a cross term in canonical momenta here, which
isn’t present in the tensile case. This makes sure the canonical basis in
$\\{k_{I},\omega_{I}\\}$ space takes an antisymmetric structure spanned by
$\Big{(}k_{I}-\mathbb{B}_{IJ}\omega^{J}\Big{)}$. :

$\begin{split}&M_{0}=\frac{1}{2}\sum_{I=1}^{d}\big{(}K^{I}_{L}K_{I\hskip
0.7113ptL}+K^{I}_{R}K_{I\hskip 0.7113ptR}+2K^{I}_{L}K_{I\hskip
0.7113ptR}\big{)}+c^{\prime}\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}+\mathcal{N}+\widetilde{\mathcal{N}}+X+X^{\dagger},\\\
&=c^{\prime}\sum_{I,J,K,L=1}^{d}\Big{(}k_{I}-\mathbb{B}_{IJ}\omega^{J}\Big{)}G^{IL}\Big{(}k_{L}-\mathbb{B}_{LK}\omega^{K}\Big{)}+c^{\prime}\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}+\mathcal{N}+\widetilde{\mathcal{N}}+X+X^{\dagger}.\end{split}$
(70)

where

$\displaystyle m^{2}=\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ X=\sum_{m>0}C_{m}\cdot\tilde{C}_{m}$ (71)

The state $\ket{r,s}$ described in (66) is an eigenstate of $\mathcal{N}$ and
$\widetilde{\mathcal{N}}$ by definition, with eigenvalues $r$ and $s$
respectively. The $X$ operators are special and need to be dealt with
separately under sandwiches as in Bagchi:2020fpr . These eigenstates of
$\mathcal{N}$ and $\widetilde{\mathcal{N}}$ forms the basis of the tensionless
Hilbert space.

###  4.1 Level matching condition and mass spectrum

We now move on to investigate the modification in the level matching condition
due to presence of this constant $B$ field. It is known that the level
matching condition for a physical state is derived from the physical state
condition for $L_{0}$ with two identical states, which in this case is given
by

$\displaystyle\bra{r,s}L_{0}-a_{L}\ket{r,s}=0$ (72)

It has been shown in Bagchi:2020fpr ; Bagchi:2021rfw  that compliance with the
algebra demands $a_{L}=0$ in the present case. Applying (72) with the
expression of $L_{0}$ as given in (68) on a generic state
$\ket{r,s,k^{\mu},k_{I},\omega^{I}}$ we end up with the following level
matching condition for physical state

$\displaystyle s-r=\sum_{I=1}^{d}k_{I}\omega^{I}.$ (73)

This level matching condition turns out to be identical to the level matching
condition for a compactified theory without $B$ field, further in line with
expectation coming from tensile theory.

The mass spectrum on the other hand is computed from the $M_{0}$ condition
Bagchi:2020fpr , given in the sandwiched version:

$\displaystyle\bra{r,s}M_{0}-a_{M}\ket{r,s}=0$ (74)

We again recall that $a_{M}=2$ for this theory. Following the analysis in
Bagchi:2020fpr  we can see that the expression of $M_{0}$ in (74) assigns the
following mass squared matrix to the particular state
$\ket{r,s,k^{\mu},k_{I},\omega^{I}}$

$\begin{split}m^{2}&={{\sum}}_{I,J,K,L=1}^{d}\Big{(}k_{I}-\mathbb{B}_{IJ}\omega^{J}\Big{)}G^{IL}\Big{(}k_{L}-\mathbb{B}_{LK}\omega^{K}\Big{)}+\frac{1}{c^{\prime}}(r+s-2).\\\
&=k^{T}G^{-1}k-2k^{T}G^{-1}\mathbb{B}\omega-\omega^{T}\mathbb{B}G^{-1}\mathbb{B}\omega+\frac{1}{c^{\prime}}(r+s-2)\end{split}$
(75)

In the last line we have expressed the winding and discrete momentum part of
the mass spectrum in matrix notation where $B$ field and inverse metric has
been expressed as square matrices $(\mathbb{B},G)$ and $k$ and $\omega$ are
expressed as column matrices

$\displaystyle k=\begin{bmatrix}k_{1}\\\ k_{2}\\\ \vdots\\\
k_{d}\end{bmatrix},\hskip 14.22636pt\omega=\begin{bmatrix}\omega_{1}\\\
\omega_{2}\\\ \vdots\\\ \omega_{d}\end{bmatrix}.$ (76)

One should note that although we started from a diagonal metric as given in
(51), while deriving the mass spectrum (75) from the action (54) we did not
use the fact that $G_{IJ}$ is diagonal. Hence, the mass spectrum (75) is valid
for any symmetric $G_{IJ}$. The terms in the mass spectrum (75) involving the
internal momentum $k_{I}$ and winding number $\omega^{I}$ can be rewritten as

$\displaystyle\big{[}\omega^{T}\hskip 8.53581ptk^{T}\big{]}\hskip
5.69054pt\mathcal{G}\begin{bmatrix}\omega\\\ k\end{bmatrix},$ (77)

where $\mathcal{G}$ by definition is given by a $2d\times 2d$ matrix defined
along the compactified directions,

$\displaystyle\mathcal{G}=\begin{bmatrix}-\mathbb{B}G^{-1}\mathbb{B}&\mathbb{B}G^{-1}\\\
-G^{-1}\mathbb{B}&G^{-1}\end{bmatrix}$ (78)

If we switch off the $B$ field in (75) we obtain the mass spectrum

$\displaystyle
m^{2}=\sum_{I,J=1}^{d}k_{I}G^{IJ}k_{J}+\frac{1}{c^{\prime}}(r+s-2).$ (79)

This is identical to the mass spectrum for tensionless string theory
constructed on oscillator vacuum in a multi-dimension compactified target
space, albeit with a diagonal metric Banerjee:2023ekd . Note that the above
was a purely KK momentum dominated spectrum, whereas the inclusion of $B$
field makes sure (75) we have non-trivial winding contribution as well. This
will be an important point in our later discussions.

One thing to immediately note about the matrix $\mathcal{G}$, often called the
generalised metric Aldazabal:2013sca , is that it is not invertible unlike the
corresponding matrix in tensile case Becker:2006dvp , which makes all the
difference. In a separate section dedicated to the theory constructed on
Induced vacuum we will discuss how the corresponding matrix in the tensile
case explicitly reduces to the degenrate $\mathcal{G}$ at tensionless limit.
Now the degeneracies in the generalised metric is not uncommon, and in the
literature one can find instances (see for example Morand:2017fnv  and follow
ups thereof) where embedding of non-Riemannian target spaces in Double Field
Theory (DFT) has been considered. We just would like to point out to the
reader, that for our case the compactified target space is still non-
degenerate ($\sim\delta^{IJ}$), however the worldsheet is Carrollian, which
leads to the degeneration of the whole generalised metric itself. Having said
that, it is also fair to point out that a zero determinant generalised metric
can also appear from a degenerate compact target space, something we will see
in a future section.

###  4.2 Symmetries of the mass spectrum

Let us now look closely at the detailed symmetries of the Tensionless mass
spectrum in our case. For the tensile case with constant background metric and
antisymmetric field, the symmetries of string mass spectrum compactified on a
$d$-Torus can be generalized to the group $O(d,d;\mathbb{Z})$ Becker:2006dvp ;
Blumenhagen:2013fgp . The ${O(d,d;\mathbb{Z})}$ group is a group of
transformations matrices ($O$) in $d+d$ dimensions which satisfy the following

$\displaystyle O^{T}\begin{bmatrix}0&\mathbb{1}_{d}\\\
\mathbb{1}_{d}&0\end{bmatrix}O=\begin{bmatrix}0&\mathbb{1}_{d}\\\
\mathbb{1}_{d}&0\end{bmatrix}.$ (80)

This symmetry of tensile string mass spectrum, written in terms of the
generalised metric, has been well-known Aldazabal:2013sca . In what follows,
we will carefully analyze the fate of all of these symmetries in our
degenerate case.

####  4.2.1 Fate of the $O(d,d;\mathbb{Z})$ Symmetry

In general, the $O(d,d;\mathbb{Z})$ symmetries in the tensile spectrum could
be thought of as symmetries of the $O(d,d)$ invariant generalised metric, the
tensile cousin of (78). In this metric, momentum and winding modes are $d$
dimensional vectors as written in (76), which can be embedded into $2d$
dimensional generalised momentum, so that the compactified contribution to the
mass formula becomes an inner product of generalised momenta w.r.t. the
$O(d,d)$ metric. The symmetry group $O(d,d)$ can further be decomposed as
successive products of subgroup symmetries, and we will give a description of
these subgroups in the tensionless version of the theory.

##### Inversion symmetries

We start with the part of the mass spectrum in (77) along with the definition
(78). The first question we ask: is it possible to find a transformation for
$\mathcal{G}$ which, along with the duality transformation
$\omega\leftrightarrow k$, would make the mass spectrum (77) invariant? It is
not hard to see that if we make $\omega\leftrightarrow k$ transformation on
(77) then it can remain invariant, but only if we make the following
transformation on $\mathcal{G}$

$\displaystyle\mathcal{G}\to\mathcal{G^{\prime}}\hskip
14.22636pt\mathcal{G^{\prime}}=\begin{bmatrix}-\mathbb{B}^{\prime}G^{\prime-1}\mathbb{B}^{\prime}&\mathbb{B}^{\prime}G^{\prime-1}\\\
-G^{\prime-1}\mathbb{B}^{\prime}&G^{\prime-1}\end{bmatrix}=\begin{bmatrix}G^{-1}&-G^{-1}\mathbb{B}\\\
\mathbb{B}G^{-1}&-\mathbb{B}G^{-1}\mathbb{B}\end{bmatrix}.$ (81)

However, there is one catch! Does the transformation (81) make sense for our
case? Clearly for $\mathbb{B}=0$ it doesn’t. At $\mathbb{B}=0$, $\mathcal{G}$
and $\mathcal{G}^{\prime}$ respectively become

$\displaystyle\mathcal{G}=\begin{bmatrix}0_{d\times d}&0_{d\times d}\\\
0_{d\times d}&G^{-1}\end{bmatrix}\hskip
14.22636pt\mathcal{G}^{\prime}=\begin{bmatrix}G^{-1}&0_{d\times d}\\\
0_{d\times d}&0_{d\times d}\end{bmatrix}.$

If we try to write this transformation in terms of compactification radii, it
would very weirdly look like

$\displaystyle\frac{1}{R_{i}}\leftrightarrow 0.$ (82)

Which could be interpreted as the $\alpha^{\prime}\to\infty$ fate of the
$R\to\frac{\alpha^{\prime}}{R}$ duality prevalent in the tensile case. In
order to make the transformation (81) sensible, the $B$ field we switched on
($\mathbb{B}$ as well as $\mathbb{B}^{\prime}$) must satisfy certain
conditions. In fact, (81) demands the following equations to be simultaneously
satisfied

$G^{\prime-1}=-\mathbb{B}G^{-1}\mathbb{B}$ (83a)
$-G^{\prime-1}\mathbb{B}^{\prime}=\mathbb{B}G^{-1}$ (83b)

Applying (83a) on (83b) one gets the following

$\displaystyle\mathbb{B}G^{-1}\mathbb{B}\mathbb{B}^{\prime}=\mathbb{B}G^{-1}\implies\mathbb{B}\mathbb{B}^{\prime}=\mathbb{1}_{d}$

Clearly the transformation in (81) is valid only if
$\mathbb{B}^{\prime}=\mathbb{B}^{-1}$. The implication is that the
transformation (83a) can be consistently defined only for a $B$ field that is
invertible. Hence the inversion symmetry can be observed only for explicitly
invertible $B$ fields in particular subspaces. Although for $\mathbb{B}=0$ the
transformation could be trivially defined, as we have just seen, it does not
make sense.

Let us compare this to the known tensile case. In case of tensile string
theory, $\mathcal{G}$ is invertible and the symmetry was
$\omega\leftrightarrow k,\hskip 2.84526pt\mathcal{G}\to\mathcal{G}^{-1}$.
Here, however, $\mathcal{G}$ is not invertible and consequently one can see:

$\displaystyle\mathcal{G}\mathcal{G}^{\prime}=0.$ (84)

Hence we should readily expect some of the symmetries of the parent $O(d,d)$
to be broken for null string spectrum.

##### Basis transformations

As discussed in Giveon:1994fu , one other component of ${O(d,d;\mathbb{Z})}$
group is basis change matrices, given by

$\displaystyle g_{C}=\begin{bmatrix}C&0\\\ 0&(C^{T})^{-1}\end{bmatrix},$ (85)

where $C\in GL(d,\mathbb{Z})$. The effect of this transformation is more or
less straightforward. Applying the transformation $g_{C}$ on the tensionless
generalised metric, $\mathcal{G}$ we get the following:

$\displaystyle\mathcal{G}^{\prime}$ $\displaystyle=\begin{bmatrix}C&0\\\
0&(C^{T})^{-1}\end{bmatrix}\begin{bmatrix}-\mathbb{B}G^{-1}\mathbb{B}&\mathbb{B}G^{-1}\\\
-G^{-1}\mathbb{B}&G^{-1}\end{bmatrix}\begin{bmatrix}C^{T}&0\\\
0&C^{-1}\end{bmatrix}=\begin{bmatrix}-\mathbb{B}^{\prime}G^{\prime-1}\mathbb{B}^{\prime}&\mathbb{B}^{\prime}G^{\prime-1}\\\
-G^{\prime-1}\mathbb{B}^{\prime}&G^{\prime-1}\end{bmatrix},$ (86)

where $\mathbb{B}^{\prime}$ and $G^{\prime}$ are respectively given by

$\displaystyle\mathbb{B}^{\prime}=C\mathbb{B}(C^{T})\hskip
14.22636ptG^{\prime}=CG(C^{T}).$ (87)

The compactified part of the mass spectrum corresponding to the new
generalised metric $\mathcal{G^{\prime}}$ is given by

$\displaystyle m^{2}=[\omega^{T}\hskip 8.53581ptk^{T}]\hskip
5.69054pt\mathcal{G}^{\prime}\begin{bmatrix}\omega\\\
k\end{bmatrix}=[\omega^{\prime T}\hskip 8.53581ptk^{\prime T}]\hskip
5.69054pt\mathcal{G}\begin{bmatrix}\omega^{\prime}\\\
k^{\prime}\end{bmatrix},\hskip 14.22636pt\begin{bmatrix}\omega^{\prime}\\\
k^{\prime}\end{bmatrix}=\begin{bmatrix}C^{T}&0\\\
0&C^{-1}\end{bmatrix}\begin{bmatrix}\omega\\\ k\end{bmatrix}$ (88)

Hence, a tensionless mass spectrum with metric $G^{\prime}$ and $B$ field
$\mathbb{B}^{\prime}$ with winding number and internal momenta given by the
$2d$ vector $\\{\omega,k\\}$ is equal to a tensionless mass spectrum with
metric $G$ and $B$ field $\mathbb{B}$ with winding number $C^{T}\omega$ and
internal momenta $C^{-1}k$.

##### Sectorised T-duality

We have already seen in the preceding sections that for invertible $B$ fields
the mass spectrum (77) is symmetric under the inversion transformation
$\omega\leftrightarrow k$ and $\mathcal{G}\leftrightarrow\mathcal{G}^{\prime}$
where $\mathcal{G}^{\prime}$ is given by (81). Note that this transformation
in terms of matrix notation can be given as:

$\mathcal{G^{\prime}}\to A\leavevmode\nobreak\ \mathcal{G}\leavevmode\nobreak\
A^{T}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \begin{bmatrix}\omega^{\prime}\\\
k^{\prime}\end{bmatrix}\longrightarrow A\leavevmode\nobreak\
\begin{bmatrix}\omega\\\ k\end{bmatrix},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ A=\begin{bmatrix}0&\mathbb{1}_{d}\\\
\mathbb{1}_{d}&0\end{bmatrix}.$ (89)

sectorised T-duality is a generalisation of the circle inversion duality and
acts on a specifically chosen compact direction, instead of all directions at
once. In retrospect, the sectorised duality matrices are given by

$\displaystyle g_{D_{i}}=\begin{bmatrix}\mathbb{1}_{d}-e_{i}&e_{i}\\\
e_{i}&\mathbb{1}_{d}-e_{i}\end{bmatrix},$ (90)

where $e_{i}$ is a matrix with $1$ in the $ii$ position and all the remaining
elements are zeros. The sectorised duality matrix $g_{D_{i}}$, when applied on
$\\{\omega,k\\}$, will interchange a particular $\omega_{i}$ and $k_{i}$, with
$i\in\\{I\\}$, leaving the rest of the $\omega$’s and $k$’s intact. If we
apply this matrix on our degenerate $\mathcal{G}$ as
$\mathcal{G}^{\prime}=g_{D_{i}}\mathcal{G}g_{D_{i}}^{T}$, remarkably it leads
us to find that $\mathcal{G}^{\prime}$ cannot be consistently expressed in
terms of a metric and a $B$ field in the form given in (78) like we did in
case of $g_{C}$. Hence, even though the mass spectrum is still
symmetric777Since $g_{D_{i}}^{T}g_{D_{i}}=\mathbb{1}$, if we apply
$\mathcal{G}^{\prime}=g_{D_{i}}\mathcal{G}g_{D_{i}}^{T}$ and
$\begin{bmatrix}\omega^{\prime}\\\
k^{\prime}\end{bmatrix}=g_{D_{i}}\begin{bmatrix}\omega\\\ k\end{bmatrix}$ then
the mass spectrum will still be same., the symmetry would not close to a
different action for tensionless string with a different $B$ field. One could
say this indicates in general the closure of sectorised T-dual theory is not
defined in the same compact moduli space for the tensionless string, owing to
the fact that generalised metric is degenerate. In the coming section we will
work out the example of $d=2$, where we will see this effect even more
explicitly.

Note that the inversion transformation as written in (89) is generated by the
sectorised duality matrices in the following way

$\displaystyle A=\begin{bmatrix}0&\mathbb{1}_{d}\\\
\mathbb{1}_{d}&0\end{bmatrix}=\begin{bmatrix}\mathbb{1}_{d}-\sum_{i=1}^{d}e_{i}&\sum_{i}^{d}e_{i}\\\
\sum_{i}^{d}e_{i}&\mathbb{1}_{d}-\sum_{i}^{d}e_{i}\end{bmatrix}=\prod_{i=1}^{d}g_{D_{i}}.$
(91)

As we have seen, although $g_{D_{i}}$s individually do not lead to any mapping
to a new tensionless action with different $G$ and $B$, the inversion
symmetry, which is generated by product of all $g_{D_{i}}$s, does lead us to
an action with $G^{\prime}=-\mathbb{B}^{-1}G\mathbb{B}^{-1}$ and
$\mathbb{B}^{\prime}=\mathbb{B}^{-1}$ (provided that $\mathbb{B}$ is
invertible along all the $i$th legs).

##### Discrete shift symmetry

Let us recall the level matching condition and mass spectrum of the physical
states constructed on oscillator vacuum. We write them out explicitly in the
following suggestive way

$\begin{split}s&-r=\frac{1}{2}\sum_{I=1}^{d}\big{(}K^{I}_{L}K_{I\hskip
0.7113ptL}-K^{I}_{R}K_{I\hskip 0.7113ptR}\big{)},\\\
m^{2}=\frac{1}{2c^{\prime}}&\sum_{I=1}^{d}(K^{I}_{L}+K^{I}_{R})(K_{I\hskip
0.7113ptL}+K_{I\hskip 0.7113ptR})+\frac{1}{c^{\prime}}(r+s-2).\end{split}$
(92)

Let us reiterate that $(K_{I})_{L,R}$ are as obtained in (62)

$\displaystyle(K_{I})_{\hskip
0.7113ptL,R}=\frac{1}{\sqrt{2}}\bigg{\\{}\sqrt{c^{\prime}}\Big{(}k_{I}-\sum_{J=1}^{d}\mathbb{B}_{IJ}\omega^{J}\Big{)}\pm\frac{1}{\sqrt{c^{\prime}}}\sum_{J=1}^{d}G_{IJ}\omega^{J}\bigg{\\}}.$

It can be easily seen that $(K_{I})_{\hskip 0.7113ptL,R}$ are invariant under
the following discrete transformations

$\displaystyle\mathbb{B}_{IJ}\to\mathbb{B}_{IJ}+\mathcal{N}_{IJ},\hskip
22.76219ptk_{I}\to k_{I}+\mathcal{N}_{IJ}\omega^{J},\hskip
22.76219pt\omega^{I}\to\omega^{I},\hskip
22.76219pt\mathcal{N}_{IJ}=-\mathcal{N}_{JI},$ (93)

where $\mathcal{N}_{IJ}\in\mathbb{Z}$. This can be expressed in matrix
notation as:

$\displaystyle g_{N}=\begin{bmatrix}\mathbb{1}_{d}&0\\\
N_{IJ}&\mathbb{1}_{d}\end{bmatrix}$ (94)

If we apply the $g_{N}$ transformation on the generalised metric, then
$\mathcal{G}^{\prime}=g_{N}\mathcal{G}g_{N}^{T}$ can be consistently expressed
in terms of $G^{\prime}_{IJ}=G_{IJ}$ and
$\mathbb{B}^{\prime}_{IJ}=\mathbb{B}_{IJ}+\mathcal{N}_{IJ}$ in the form given
in (78). Since the part of the mass spectrum (92) depending on
$\\{\mathbb{B}_{IJ},k_{I},\omega^{I}\\}$ comes entirely from $(k_{I})_{L,R}$,
it is apparent that the tensionless mass spectrum (77), like the tensile
counterpart, is invariant under the transformations (93).

###  4.3 Concrete examples of compactification

####  Even dimension compactified ($d=2$)

Now, we look into the mass spectrum of tensionless string with constant $B$
field in a spacetime with only two of the dimensions compactified on a two
torus. The reader may compare the contents in this section with the mass
spectrum symmetries of tensile strings with a $B$ field, having two
compactified target space directions, as discussed in Appendix (A). When we
compactify only two dimensions and demand that the constant $B$ field has non-
zero components only along those two compactified directions, we immediately
arrive at a unique choice of the $B$ field which is

$\displaystyle\mathcal{B}_{IJ}=b\hskip 2.84526pt\epsilon_{IJ},\hskip
14.22636ptI,J\in\\{1,2\\}$ (95)

where $\epsilon_{IJ}$ is the antisymmetric tensor in two dimensions. We choose
$\epsilon_{12}=1$ here. After the coordinate scaling defined in (51), the
rescaled $\mathbb{B}_{IJ}$ becomes

$\mathbb{B}_{IJ}=bR_{I}R_{J}\epsilon_{IJ}.$ (96)

The above $\mathbb{B}_{IJ}$ is invertible, and hence, we can expect a sensible
inversion symmetry to exist for the resulting mass spectrum. We find that for
the $B$ field (95), equation (75) gets simplified to

$\displaystyle
m_{2}^{2}=\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+2b\Bigg{(}\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\Bigg{)}+b^{2}(R_{1}^{2}\omega_{1}^{2}+R_{2}^{2}\omega_{2}^{2})+\frac{1}{c^{\prime}}(r+s-2).$
(97)

In the above, we have made a slight abuse of notation. In order to make the
expression less messy we chose to write {$\omega_{1},\omega_{2}$} instead of
{$\omega^{1},\omega^{2}$}. Hereafter in the explicit form of mass spectrum
(i.e. not in the matrix form), we continue to follow this notation. We have
seen that the mass spectrum in (75) is symmetric under the transformations:

$\displaystyle k_{I}\leftrightarrow\omega^{I},\hskip
14.22636ptR_{I}\leftrightarrow\frac{1}{bR_{I}},\hskip 14.22636ptb\to-b\hskip
14.22636ptI\in\\{1,2\\}$ (98)

It can be seen that for $d=2$, $G^{-1}$ and $B$ field given in (52) and (97)
respectively, the transformation defined on the generalised metric
$\mathcal{G}$ defined in (81) will boil down to (98). Hence we see that indeed
the mass-spectrum displays a sensible inversion symmetry, which very evidently
would not happen when the $B$ field is turned off. This non-trivial effect of
the antisymmetric field in reinstating a version of T-duality remains a very
crucial point in our discussion.

Now, we want to see whether this mass spectrum displays the full
$O(2,2;\mathbb{Z})$ symmetry, in line with our discussion before. It can be
easily seen that the discrete shift symmetry is preserved for the mass
spectrum: for $d=2$, the transformation (93) in terms of $b$, $R_{1}$ and
$R_{2}$ becomes the following

$\displaystyle b\to b+\frac{n}{R_{1}R_{2}}\hskip 14.22636ptk_{I}\to
k_{I}+n\epsilon_{IJ}\omega^{J}\hskip 14.22636ptn\in\mathbb{Z}.$ (99)

One can further check that the mass spectrum is symmetric under the basis
change transformation where the transformation of $G$ and $\mathbb{B}$ will be
dictated by (86) and (87).

Since we discussed that sectorised dualities are non-trivial in this case, we
can try to find the impact of $g_{D_{i}}$ on $\mathcal{G}$. Let us first
express it for $d=2$ and $\mathbb{B}_{IJ}$ as written in (96) in the form of a
$4\times 4$ matrix:

$\displaystyle\mathcal{G}=\begin{bmatrix}b^{2}R^{2}_{1}&0&0&\frac{bR_{1}}{R_{2}}\\\
0&b^{2}R^{2}_{2}&-\frac{bR_{2}}{R_{1}}&0\\\
0&-\frac{bR_{2}}{R_{1}}&\frac{1}{R^{2}_{1}}&0\\\
\frac{bR_{1}}{R_{2}}&0&0&\frac{1}{R^{2}_{2}}\end{bmatrix}$ (100)

Let us consider the sectorised transformation $g_{D_{2}}$. It will explicitly
interchange $k_{2}$ and $\omega_{2}$ and matrix form of this is given by

$\displaystyle\begin{bmatrix}1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
0&1&0&0\end{bmatrix}$ (101)

Now we would like to study how $\mathcal{G}$ evolves for the $g_{D_{2}}$
transformation. It is easy to arrive at:

$\displaystyle\mathcal{G}^{\prime}=g_{D_{2}}\mathcal{G}g_{D_{2}}^{T}=\begin{bmatrix}b^{2}R^{2}_{1}&\frac{bR_{1}}{R_{2}}&0&0\\\
\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}&0&0\\\
0&0&\frac{1}{R^{2}_{1}}&-\frac{bR_{2}}{R_{1}}\\\
0&0&-\frac{bR_{2}}{R_{1}}&b^{2}R^{2}_{2}\end{bmatrix}$ (102)

If we recall the expression of $\mathcal{G}$ in terms of $G$ and $\mathbb{B}$
from (78), we see that the off-diagonal blocks seem to suggest that
$\mathbb{B^{\prime}}=0$, however, the diagonal blocks suggest that
$\mathbb{B^{\prime}}\neq 0$. Hence $\mathcal{G}^{\prime}$ cannot be put into
the form of a generalised metric derived solely from any tensionless string
action. Although the mass spectrum is still invariant under this
transformation, this transformation cannot be called a symmetry of tensionless
action since it does not lead to any mapping between two different tensionless
theories. Note that this property is very unique to the tensionless string
sigma model, and hints at the same mass spectrum being shared by some other
theory, not covered by the structure of (78). We will comment more on this
when we study the limiting spectrum in the Induced vacuum section.

It is important to note that although the transformations $g_{D_{1}}$ and
$g_{D_{2}}$ individually do not lead us to any new tensionless action, the
inversion symmetry as given in (98) is generated by both of them combined, as
the field is invertible along these two compact legs of the target space 888In
fact in two dimensional compact space, the sectorised T-duality group in the
tensile string case comprises of
$\\{\mathbb{I},g_{D_{1}},g_{D_{2}},g_{D_{1}}g_{D_{2}}\\}$, while the analogue
in the null case reads $\\{\mathbb{I},g_{D_{1}}g_{D_{2}}\\}$ i.e. only even
duality transforms work. This can be thought of as a breaking of the symmetry
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\to\mathbb{Z}_{2}$.. Moreover as we have
seen, it connects the theory to an action with a different $B$ field. Again,
remember that this depends crucially on having an invertible $B$ field along
both of the compact directions.

#### Example of odd dimensions compactified ($d=3$)

Next, to understand these dualities better, we find the mass spectrum of
tensionless strings in a target spacetime with three dimensions compactified
with constant $B$ fields and see whether we can find similar forms of symmetry
in the spectrum as we found in the $2d$ case. Now since we have more than one
directional option to put the legs of our two-form field on, the dynamics is
surely more interesting. The most general constant $3\times 3$ $B$ field is
given by

$\displaystyle\mathbb{B}_{IJ}=\begin{bmatrix}0&&aR_{1}R_{2}&&bR_{1}R_{3}\\\
-aR_{1}R_{2}&&0&&cR_{2}R_{3}\\\ -bR_{1}R_{3}&&-cR_{2}R_{3}&&0\\\
\end{bmatrix}$ (103)

The mass spectrum obtained from the above $B$ field is given by

$\begin{split}m_{3}^{2}=&\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+\frac{k_{3}^{2}}{R_{3}^{2}}+2a\left(\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\right)+2b\left(\frac{R_{1}}{R_{3}}k_{3}\omega_{1}-\frac{R_{3}}{R_{1}}k_{1}\omega_{3}\right)\\\
&+2c\left(\frac{R_{2}}{R_{3}}k_{3}\omega_{2}-\frac{R_{3}}{R_{2}}k_{2}\omega_{3}\right)+\left(aR_{1}\omega_{1}-cR_{3}\omega_{3}\right)^{2}+(aR_{2}\omega_{2}+bR_{3}\omega_{3})^{2}\\\
&+(bR_{1}\omega_{1}+cR_{2}\omega_{2})^{2}+\frac{1}{c^{\prime}}(r+s-2).\end{split}$
(104)

Here $a,b,c$ are constants and $R_{I}$ are the relevant compact radii. Clearly
this mass spectrum won’t be symmetric under the duality transformation of the
type given in (98) if all of these fields are turned on. In fact if we
interchange $k_{I}$ and $\omega_{I}$ in the mass spectrum (104), there is no
transformation on $R_{I}$ which will take us from quadratic terms of $k$’s to
the quadratic terms of $\omega$’s. The reason is that the quadratic terms in
$\omega$ involves three cross terms like $\omega_{K}\omega_{J}$, and there is
no such cross terms in the quadratic terms of $k$’s. The only way we can get
rid of all three cross terms in the quadratic terms of $\omega$’s is to set at
least two of the three parameters $a,b,c$ to zero, i.e. working with one of
the following three $B$ fields with two legs among the three compact
directions:

$\displaystyle\begin{split}\mathbb{B}_{IJ}^{1}=&\begin{bmatrix}0&&aR_{1}R_{2}&&0\\\
-aR_{1}R_{2}&&0&&0\\\ 0&&0&&0\\\ \end{bmatrix},\hskip
14.22636pt\mathbb{B}_{IJ}^{2}=\begin{bmatrix}0&&0&&bR_{1}R_{3}\\\ 0&&0&&0\\\
-bR_{1}R_{3}&&0&&0\\\ \end{bmatrix},\\\ &\hskip
42.67912pt\mathbb{B}_{IJ}^{3}=\begin{bmatrix}0&&0&&0\\\ 0&&0&&cR_{2}R_{3}\\\
0&&-cR_{2}R_{3}&&0\\\ \end{bmatrix}.\end{split}$ (105)

However, whenever we try to work on one of the above three $B$ fields we end
up getting a mass spectrum where in order to define a T-dual transformation of
the type (98), we need to make a nonsensical transformation of type (82) on
one of the compactified radius. For example, if we work with the first $B$
field where $a\neq 0,b=c=0$, the quadratic terms of $\omega$ will be

$\displaystyle a^{2}(R^{2}_{1}\omega^{2}_{1}+R^{2}_{2}\omega^{2}_{2}).$ (106)

The absence of the $\omega^{2}_{3}R^{2}_{3}$ term in the mass spectrum now
implies the transformation of type (82) on $R_{3}$. Hence, for $d=3$, we
cannot define the full T-duality, i.e. the full inversion symmetry involving
all compact directions.

It can be easily seen that the discrete shift symmetry is preserved for the
mass spectrum (104) for $d=3$, where the transformation (93) in terms of
$a,b,c,R_{1},R_{2}$ and $R_{3}$ becomes the following

$\displaystyle a\to a+\frac{l}{R_{1}R_{2}},\hskip 8.53581ptb\to
b+\frac{m}{R_{1}R_{3}},\hskip 8.53581ptc\to c+\frac{n}{R_{2}R_{3}}\hskip
8.53581ptk_{I}\to k_{I}+\mathcal{N}_{IJ}\omega^{J}\hskip
8.53581ptl,m,n\in\mathbb{Z}$ (107)

where $\mathcal{N}_{IJ}$ is given by a $3\times 3$ matrix:

$\displaystyle\mathcal{N}_{IJ}=\begin{bmatrix}0&&l&&m\\\ -l&&0&&n\\\
-m&&-n&&0\\\ \end{bmatrix}.$ (108)

Now, sectorised duality transformations, as defined in (90), are not respected
by the mass spectrum (104). However, if we use any one of the
$\mathbb{B}_{IJ}^{i}$ for $(i=1,2,3)$ as given in (105), then we observe that
the mass spectrum will be symmetric under the following combination of
transformations respectively:

$B$ field | Even sectorised duality
---|---
$\mathbb{B}_{IJ}^{1}$ | $g_{D_{1}}g_{D_{2}}$
$\mathbb{B}_{IJ}^{2}$ | $g_{D_{1}}g_{D_{3}}$
$\mathbb{B}_{IJ}^{3}$ | $g_{D_{2}}g_{D_{3}}$

Table 1: Configuration of symmetries for $B$ field along pairwise legs of the
compact manifold.

This indeed mirrors our discussions in the $2d$ compact case, where such even
sectorised dualities were evident along the $B$ field directions. Let us
consider $\mathcal{G}$ defined in (78) constructed using $\mathbb{B}_{IJ}^{1}$
and $G_{IJ}=R_{I}^{2}\delta_{IJ}$, which corresponds to the mass spectrum
(104) with $b=c=0$. It can be shown that the transformation
$g_{D_{1}}g_{D_{2}}$ connects this one to a different action involving
$\mathbb{B}^{\prime},G^{\prime}$ given below

$\displaystyle\mathbb{B}_{IJ}^{\prime 1}=$
$\displaystyle\begin{bmatrix}0&&-\frac{1}{aR_{1}R_{2}}&&0\\\
\frac{1}{aR_{1}R_{2}}&&0&&0\\\ 0&&0&&0\\\ \end{bmatrix},\hskip
14.22636ptG_{IJ}^{\prime 1}=\begin{bmatrix}\frac{1}{a^{2}R_{1}^{2}}&&0&&0\\\
0&&\frac{1}{a^{2}R_{2}^{2}}&&0\\\ 0&&0&&R_{3}^{2}\\\ \end{bmatrix}$ (109)

For detailed calculation, the reader is referred to appendix (B). For
$\mathbb{B}_{IJ}^{2}$ and $\mathbb{B}_{IJ}^{3}$, the transformations listed in
Table (1) is given by

$\mathbb{B}_{IJ}^{2}\;\xrightarrow{g_{D_{1}}g_{D_{3}}}{}\;\mathbb{B}_{IJ}^{\prime
2}=\begin{bmatrix}0&&0&&-\frac{1}{bR_{1}R_{3}}\\\ 0&&0&&0\\\
\frac{1}{bR_{1}R_{3}}&&0&&0\\\ \end{bmatrix},\leavevmode\nobreak\
\leavevmode\nobreak\
G_{IJ}^{2}\;\xrightarrow{g_{D_{1}}g_{D_{3}}}{}\;G_{IJ}^{\prime
2}=\begin{bmatrix}\frac{1}{b^{2}R_{1}^{2}}&&0&&0\\\ 0&&R_{2}^{2}&&0\\\
0&&0&&\frac{1}{b^{2}R_{3}^{2}}\\\ \end{bmatrix}$
$\mathbb{B}_{IJ}^{3}\;\xrightarrow{g_{D_{2}}g_{D_{3}}}{}\;\mathbb{B}_{IJ}^{\prime
3}=\begin{bmatrix}0&&0&&0\\\ 0&&0&&-\frac{1}{cR_{2}R_{3}}\\\
0&&\frac{1}{cR_{2}R_{3}}&&0\\\ \end{bmatrix},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
G_{IJ}^{3}\;\xrightarrow{g_{D_{2}}g_{D_{3}}}{}\;G_{IJ}^{\prime
3}=\begin{bmatrix}R_{1}^{2}&&0&&0\\\ 0&&\frac{1}{c^{2}R_{2}^{2}}&&0\\\
0&&0&&\frac{1}{c^{2}R_{3}^{2}}\\\ \end{bmatrix}$

Looking at the above three transformation of $B$ fields we observe one common
feature in all of them. All $\mathbb{B}_{IJ}^{i}$’s listed in (105) acts like
invertible two-form $B$ fields in a 2 dimensional sub-space ($T^{2}$ $\subset$
$T^{3}$). The transformations listed in the table are restricted, in a
pairwise manner, to those sub-spaces only and in those sub-spaces, they
transform those $B$’s to their respective inverses.

We also tried sectorised duality transformations on linear combinations of the
two-form fields and found that the mass spectrum constructed on different $B$
fields will be symmetric under different combinations of $g_{D_{i}}$’s. They
are given below in Table (2).

$B$ field | Even set of sectorised dualities
---|---
$l_{1}\mathbb{B}_{IJ}^{1}+l_{2}\mathbb{B}_{IJ}^{2}$ |  $g_{D_{1}}g_{D_{2}}$, $g_{D_{1}}g_{D_{3}}$, $g_{D_{3}}g_{D_{1}}g_{D_{2}}g_{D_{3}}$, $g_{D_{1}}g_{D_{2}}g_{D_{3}}g_{D_{2}}$
$l_{2}\mathbb{B}_{IJ}^{2}+l_{3}\mathbb{B}_{IJ}^{3}$ |  $g_{D_{1}}g_{D_{3}}$, $g_{D_{2}}g_{D_{3}}$, $g_{D_{1}}g_{D_{2}}g_{D_{3}}g_{D_{1}}$, $g_{D_{1}}g_{D_{2}}g_{D_{3}}g_{D_{2}}$
$l_{1}\mathbb{B}_{IJ}^{1}+l_{3}\mathbb{B}_{IJ}^{3}$ |  $g_{D_{1}}g_{D_{2}}$, $g_{D_{2}}g_{D_{3}}$, $g_{D_{3}}g_{D_{1}}g_{D_{2}}g_{D_{3}}$, $g_{D_{1}}g_{D_{2}}g_{D_{3}}g_{D_{1}}$

Table 2: Configurations of linear combinations of $B$ field along the compact
manifold. Here $l_{i}$’s are constant parameters.

Based on our observation of the effects of inversion transformation and
sectorised dualities, both for $T^{2}$ and $T^{3}$ compactifications, we can
now make some generalised comments on the impact of sectorised duality in any
number of compact dimensions. Operating sectorised duality an odd number of
times never lead us to any mass spectrum of tensionless strings which can be
constructed from some different null string theory compatible with a
$\mathbb{B}^{\prime}$ and $G^{\prime}$. However for appropriate $\mathbb{B}$’s
applying sectorised duality only even number of times gives tensionless mass
spectrum with different $\mathbb{B}^{\prime}$ and $G^{\prime}$.

For even dimension compactification, the inversion transformation itself
becomes an example of even number of sectorised duality transformation, where
we need an invertible $B$ field to make it work. However when we compactify
odd number of dimensions, the full inversion symmetry consists of odd number
of sectorised duality transformations, which by virtue of the discussion
above, does not work for any $B$ field. But still in odd dimensions, if we
chose a (or a collection of) $B$ field(s) which acts like an invertible $B$
field in an even (two) dimensional subspace, then within that even dimensional
subspace, operating sectorised duality will invert that $B$ field
consistently. The degeneracy of the $\mathcal{G}$ thus significantly
constrains the allowed forms of symmetries.

###  4.4 States arising due to compactification

In this section we look into the new states arising due to compactification.
We study how the presence of a constant $B$ field affects the masses of those
states. We also study the nature of the particles corresponding to those
states. In order to visualise the effects of Kalb- Ramond field more clearly,
we will work out the example of two compactified dimensions.

#### Vacuum

In our earlier work Banerjee:2023ekd  it was found that when we consider
oscillator vacuum in a target space with one dimension compactified, the
vacuum state $\ket{0,0,k^{\mu},K,W}$ can be physical only if $KW=0$, i.e.
either $K$ or $W$ is zero, which is the consequence of the level matching
condition. However, when more than one dimensions are compactified, we can
have oscillator vacuum $\ket{0,0,k^{\mu},k_{I},\omega^{I}}$ with both
$k_{I}\neq 0$ and $\omega^{I}\neq 0$. The only restriction coming from the
level matching condition (73) is that the internal momentum on the torus
$T^{d}$ is always orthogonal to the winding number vector i.e.

$\displaystyle\sum_{I=1}^{d}k_{I}\omega^{I}=0.$ (110)

Mass of these vacuum states are given by

$\displaystyle
m_{vac,d}^{2}=k^{T}G^{-1}k-2k^{T}G^{-1}\mathbb{B}\omega-\omega^{T}\mathbb{B}G^{-1}\mathbb{B}\omega-\frac{2}{c^{\prime}}.$

For the special case of $d=2$, the mass-squared of the vacuum will be given by

$\begin{split}m_{vac,2}^{2}&=\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+2b\Bigg{(}\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\Bigg{)}+b^{2}(R_{1}^{2}\omega_{1}^{2}+R_{2}^{2}\omega_{2}^{2})-\frac{2}{c^{\prime}}\\\
&=\left(\frac{k_{1}}{R_{1}}-bR_{2}\omega_{2}\right)^{2}+\left(\frac{k_{2}}{R_{2}}+bR_{1}\omega_{1}\right)^{2}-\frac{2}{c^{\prime}}.\end{split}$
(111)

Here we will see that the presence of the $B$ field severely constraints the
range of parametric values of {$k_{I},\omega^{I},R_{I}$} for which mass of the
vacuum becomes tachyonic. Let’s do a ballpark calculation: For sake of
simplicity we take $R_{1}=R_{2}=R$. The mass of the state
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}$ in that case becomes

$\displaystyle
m_{vac,2}^{2}=\frac{1}{R^{2}}(k_{1}^{2}+k_{2}^{2})+2b(k_{2}\omega_{1}-k_{1}\omega_{2})+b^{2}R^{2}(\omega_{1}^{2}+\omega_{2}^{2})-\frac{2}{c^{\prime}}.$
(112)

The constraint satisfied by {$k_{I},\omega^{I}$} for this case, as per level
matching, is

$k_{1}\omega_{1}+k_{2}\omega_{2}=0.$ (113)

The following Table (3) provides us with the details of $m^{2}$ of vacuum for
given values of $k_{1},k_{2},\omega_{1}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \omega_{2}$ so that the above constraint is
satisfied. For the sake of the reader, a heuristic plot for the same is
provided in fig (1).

We also can get tachyons for the states {$r=1,s=0$} and {$r=0,s=1$}, and for
detailed analysis the reader is referred to Appendix (D). However, we observe
that there would be no tachyons for the higher level states {$r\geq 1,s\geq
1$}. The expression for mass formula for the state {$r=1,s=1$} can be
expressed as:

$m^{2}=\left(\frac{k_{1}}{R}+bR\omega_{2}\right)^{2}+\left(\frac{k_{2}}{R}+bR\omega_{1}\right)^{2}$
(114)

which is always $\geq 0$. This indicates the absence of tachyons at and above
level two spectrum. However massless condition can be achieved in this case,
with $k_{1}=-b\omega_{2}R^{2}$ and $k_{2}=b\omega_{1}R^{2}$. If we switch off
the $B$ field from the expression of mass of the ground state, then as
expected, the mass will be independent of the winding number given by

$\displaystyle
m_{B=0}^{2}=\frac{1}{R^{2}}\sum_{I,J=1}^{d}k_{I}g^{IJ}k_{J}-\frac{2}{c^{\prime}}.$
(115)

This perfectly matches with the result derived in Banerjee:2023ekd .

$k_{1},\omega_{1}$ | $k_{2},\omega_{2}$ | $m^{2}$ |  Condition for $m^{2}<0$
---|---|---|---
$k_{1}=\omega_{1}=0$ | $k_{2}=\omega_{2}=0$ | $m^{2}=-\frac{2}{c^{\prime}}$ | Always
$k_{2}=0,\omega_{2}\neq 0$ | $m^{2}=b^{2}R^{2}\omega^{2}_{2}-\frac{2}{c^{\prime}}$ | $\omega_{2}<\frac{1}{bR}\sqrt{\frac{2}{c^{\prime}}}$
$k_{2}\neq 0,\omega_{2}=0$ | $m^{2}=\frac{k_{2}^{2}}{R^{2}}-\frac{2}{c^{\prime}}$ | $k_{2}<R\sqrt{\frac{2}{c^{\prime}}}$
$k_{1}=0,\leavevmode\nobreak\ \omega_{1}\neq 0$ | $k_{2}=\omega_{2}=0$ | $m^{2}=b^{2}R^{2}\omega_{1}^{2}-\frac{2}{c^{\prime}}$ | $\omega_{1}<\frac{1}{bR}\sqrt{\frac{2}{c^{\prime}}}$
$k_{2}=0,\omega_{2}\neq 0$ | $m^{2}=b^{2}R^{2}(\omega_{1}^{2}+\omega^{2}_{2})-\frac{2}{c^{\prime}}$ | $\omega_{1}^{2}+\omega^{2}_{2}<\frac{2}{c^{\prime}b^{2}R^{2}}$
$k_{2}\neq 0,\omega_{2}=0$ | $m^{2}=\left(\frac{k_{2}}{R}-bR\omega_{1}\right)^{2}-\frac{2}{c^{\prime}}$ | $\left(\frac{k_{2}}{R}-bR\omega_{1}\right)^{2}<\frac{2}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}=0$ | $k_{2}=\omega_{2}=0$ | $m^{2}=\frac{k_{1}^{2}}{R^{2}}-\frac{2}{c^{\prime}}$ | $k_{1}<R\sqrt{\frac{2}{c^{\prime}}}$
$k_{2}=0,\omega_{2}\neq 0$ | $m^{2}=\left(\frac{k_{1}}{R}+bR\omega_{2}\right)^{2}-\frac{2}{c^{\prime}}$ | $\left(\frac{k_{2}}{R}+bR\omega_{1}\right)^{2}<\frac{2}{c^{\prime}}$
$k_{2}\neq 0,\omega_{2}=0$ | $m^{2}=\frac{k_{1}^{2}+k_{2}^{2}}{R^{2}}-\frac{2}{c^{\prime}}$ | $k_{1}^{2}+k_{2}^{2}<\frac{2R^{2}}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}\neq 0$ |  |  | $k_{2}=\omega_{1}bR^{2}$
$k_{2}\neq 0,\leavevmode\nobreak\ \omega_{2}\neq 0$ | $m^{2}=(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}$ | or
| $-\frac{2}{c^{\prime}}$ | $(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}<\frac{2}{c^{\prime}}$

Table 3: Mass square eigenvalues for vacuum state $r=s=0$ for given values of
$k_{I},\omega^{I}$. Note that the notation has been simplified here.
Conditions for getting tachyonic states are shown separately.

Figure 1: Heuristic plot of $m_{2}^{2}$ in (112) vs Radius ($R$) for different
values of constant $B$ field. Red is for $b=0$, blue for $b=0.5$, orange for
$b=1$, purple for $b=1.5$, magenta is for $b=2$. Here we have considered
$\omega_{2}=0,\omega_{1}=1,k_{1}=0\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ k_{2}=1$ with $c^{\prime}=1$. Minima for these
curves signifies the position of tachyon. Note that the effective tachyonic
region reduces as a the $B$ field is dialled to higher values.

#### Vector states

Let us consider the following vector states built on a vacuum
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}$.

$\displaystyle\ket{V^{\mu\hskip
1.42262ptI}_{k_{I},\omega^{I}}}=C^{\mu}_{-1}\tilde{C}^{I}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}\hskip
14.22636pt\ket{\tilde{V}^{\mu\hskip
1.42262ptI}_{k_{I},\omega^{I}}}=\tilde{C}^{\mu}_{-1}C^{I}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}$
(116)

As we have seen, this state will be physical if $k_{I}\omega^{I}=0$. Looking
at (77) one can see that for $k_{I}=\omega^{I}=0$, this state will be
massless. (77) also makes it clear that if we switch off the $B$ field, then
for $k_{I}=0$, these states will be massless for any {$\omega^{I}$}.

#### Scalar states

We can also construct $d^{2}$ number of scalar states on a ground state
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}$ as below

$\displaystyle\ket{\Phi^{IJ}}=C^{I}_{-1}\tilde{C}^{J}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}$
(117)

For $B=0$, the states with zero internal momenta $k_{I}$ will be massless for
all {$\omega^{I}$}.

###  4.5 Take home summary

Let us summarize the findings of this section for the sake of the reader,
since the lessons will be important going forward.

  * •

Presence of constant $B$ field does not affect the level matching condition
just like tensile theory (see (73)).

  * •

The mass spectrum (75) however gets contribution from $B$ field for compact
target space. Without $B$ field, the mass spectrum is independent of winding
number $\omega^{I}$. Physically, this meant that due to the large string
length (much greater than compactification radii)999It is a known result in
string theory that when compactification radius becomes much shorter than the
string length, the mass spectrum becomes purely dependent on the momentum
modes only. In the tensionless sector, similar situation arises by keeping the
compactification radii intact and making the string infinitely long., one
could wind the string along compact coordinates without requiring extra
energy. However, the existence of a topological $B$ field term along the
compactified directions results into creating an equilibrium, where one needs
a finite amount of energy proportional to the $B$ field. 101010It is important
to recall here that for string, the role of $B$ field is analogous to the role
played by vector potential $A_{\mu}$ for point particles in electromagnetism..

  * •

The generalised metric $\mathcal{G}$, on which the mass spectrum is based,
turns out to be degenerate. This points towards a degenerate version of Double
Field Theory when one considers the similar limit on the inverse metric.

  * •

The new mass spectrum (75) satisfies a partially broken $O(d,d;\mathbb{Z})$
symmetry, where only even number of sectorised T-dualities acting along the
compact legs of the $B$ field preserves the structure of a null $\mathcal{G}$.

  * •

Looking at the effects of inversion transformation and sectorised dualities on
$T^{2}$ and $T^{3}$ explicitly, we can say that only even number of operations
of sectorised duality can lead us to mass spectrum of purely tensionless
strings constructed from different $\mathbb{B}^{\prime}$ and $G^{\prime}$.

  * •

From the above observation we can say that when we compactify even number of
dimensions where the full inversion transformation consists of even number of
sectorised duality operations, we can get an invertible $B$ field to make it
work.

  * •

However, for odd dimension compactification, only a particular choice (or a
collection) of $B$ field(s), which acts like an invertible $B$ field in an
even dimensional subspace, operating sectorised duality in all those even
subspaces will invert the $B$ field.

  * •

As shown in our earlier work Banerjee:2023ekd , we obtain infinite number of
new physical states for each level for multi-dimension compactification. The
presence of $B$ field in the theory constraints the range of parametric values
for which mass of the vacuum becomes tachyonic.

##  5 Induced vacuum

This section deals with the study of effect of constant $B$ field on the mass
spectrum for Induced vacuum of the tensionless string theory. This vacuum is
intriguing, since intrinsically derived quantum mechanical results can be
arrived at by taking ultrarelativistic limits on the highest weight states of
tensile string theory. In what follows, we will see whether the same works
with a constant $B$ field on both regimes of the theory. Our basic setup
remains the same as in the last section, only the action of the oscillators
will change.

###  5.1 Intrinsic mass spectrum and level matching

We begin by reminding the reader the mode expansion of the dimensionless field
$Y^{I}$ (58) in terms of oscillators $A$’s and $B$’s as in (59)

$\displaystyle Y^{I}=y^{I}+A^{I}_{0}\sigma+B^{I}_{0}\tau+i\sum_{n\neq
0}\frac{1}{n}\big{(}A^{I}_{n}-in\tau B^{I}_{n}\big{)}e^{-in\sigma}.$

Here $A^{I}_{0}$ and $B^{I}_{0}$ are given in (60). It is better to work in
this language for the Induced vacuum, as the tensile string states will
directly reduce to states constructed on this vacuum in the tensionless limit.
We also use here the equations defined in (56) and (57). Using (47) one can
see the action of all $M_{n},\leavevmode\nobreak\ n\neq 0$ annihilates the
vacuum, which is a physical state. This in turn signfies that all
$B_{n},\leavevmode\nobreak\ n\neq 0$ annihilates the physical vacuum
Bagchi:2020fpr , which becomes the defining relation.

To focus on the zero modes, we apply the physical state condition $M_{0}$ with
$a_{M}=0$ (as $B$’s here commute among each other, there is no ordering
ambiguity) on the vacuum $\ket{0,0,k^{\mu},k_{I},\omega^{I}}$ to get,

$\displaystyle\hskip 36.98866pt\sum_{m}B_{-m}\cdot
B_{m}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0$
$\displaystyle\implies\Big{(}\sum_{m\neq 0}B_{-m}\cdot
B_{m}+B^{2}_{0}\Big{)}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0$
$\displaystyle\implies$ $\displaystyle
B^{2}_{0}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=2c^{\prime}\Bigg{(}k^{2}+\sum_{I=1}^{d}K^{I}K_{I}\Bigg{)}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0.$
(118)

Hence, using (56) the mass squared eigenvalue of the state
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}$ is given by

$\displaystyle
m^{2}=\sum_{I=1}^{d}K_{I}G^{IJ}K_{J}=\sum_{I,J,K=1}^{d}\Big{(}k_{I}-\mathbb{B}_{IK}\omega^{K}\Big{)}G^{IJ}\Big{(}k_{J}-\mathbb{B}_{JL}\omega^{L}\Big{)}$
(119)

Using the matrix notation introduced earlier in (75) and (76) we rewrite the
expression of this mass spectrum as

$\displaystyle
m^{2}=k^{T}G^{-1}k-2k^{T}G^{-1}\mathbb{B}\omega-\omega^{T}\mathbb{B}G^{-1}\mathbb{B}\omega.$
(120)

We immediately notice two important structures in this mass spectrum. Firstly,
the $(\omega,k)$ lattice structure in case of induced vacuum theory is
identical to that of the oscillator vacuum theory. Secondly, we recall that in
case of tensionless string without $B$ field Banerjee:2023ekd , the mass
spectrum did not have any contribution from the winding number. In fact, in
that case the mass spectrum is just rest energy contributed by the internal
momenta. However, for the constant $B$ field case, we do have finite
contribution in the mass spectrum from the winding sector. Switching off the
$B$ field in (120) would lead us to the following mass spectrum

$\displaystyle
m^{2}=\sum_{I,J=1}^{d}k_{I}G^{IJ}k_{J}=\sum_{I}\frac{k^{2}_{I}}{R^{2}_{I}}$
(121)

If we set all the compactification radii to be equal then this mass spectrum
will reduce to the one for induced vacuum derived in our earlier work
Banerjee:2023ekd . This purely momentum dominated spectrum can also be reached
via a direct limit on the tensile mass spectra.

On the other hand, the physical state constraint coming from $L_{n}$ will be
given by

$\begin{split}&\leftindex_{I}{\bra{0,0,k^{\mu},k_{I},\omega^{I}}}L_{n}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0\hskip
14.22636pt\forall n\\\
\implies&\leftindex_{I}{\bra{0,0,k^{\mu},k_{I},\omega^{I}}}\Big{(}A^{\mu}_{n}B_{\mu\hskip
2.84526pt0}+\sum_{I}A^{I}_{n}B_{I\hskip
2.84526pt0}\Big{)}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0\hskip
14.22636pt\forall n.\end{split}$ (122)

For $n=0$, we recall that the normal ordering ambiguity for $L_{0}$ i.e.
$a_{L}$ vanishes as shown in Bagchi:2021rfw ; Chen:2023esw , and that leads us
to the following:

$\begin{split}&\leftindex_{I}{\bra{0,0,k^{\mu},k_{I},\omega^{I}}}\Big{(}A^{\mu}_{0}B_{\mu\hskip
2.84526pt0}+\sum_{I}A^{I}_{0}B_{I\hskip
2.84526pt0}\Big{)}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}=0\\\
\implies&\sum_{I}k_{I}\omega^{I}\leftindex_{I}{\braket{0,0,k^{\mu},k_{I},\omega^{I}}{0,0,k^{\mu},k_{I},\omega^{I}}}_{I}=0\\\
\implies&\sum_{I}k_{I}\omega^{I}=0.\end{split}$ (123)

This can be thought of as the reminiscent of the level matching condition of
the tensile strings with multiple dimensions compactified.

###  5.2 Limit from tensile string

####  5.2.1 Tensionless limit of tensile perturbative states

As we know from Bagchi:2015nca  and Bagchi:2020fpr , the quantised tensionless
string theory built on induced vacuum naturally emerges from the tensionless
limit of the tensile string theory. In Bagchi:2019cay , it has been shown that
when we take direct tensionless limit on the tensile perturbative physical
states, at $\epsilon\to 0$ they condense explicitly on the induced vacuum. We
briefly review that before discussing the fate of tensile states for our case
with the $B$ field.

Let us recall that for non-compactified tensile string the perturbative states
are level matched. Hence we consider the following excited state created via
the action of creation-annihilation operators:

$\displaystyle\ket{\zeta_{n}}=\rho_{\mu\nu}\alpha^{\mu}_{-n}\tilde{\alpha}^{\mu}_{-n}\ket{0}_{\alpha},$
(124)

where $\rho_{\mu\nu}$ is a polarisation tensor. Around tensionless limit i.e.
$\epsilon=0$, the tensile string vacuum can be perturbatively expanded in
terms of $\epsilon$ in following way :

$\displaystyle\ket{0}_{\alpha}=\ket{0}_{I}+\epsilon\ket{I_{1}}+\epsilon^{2}\ket{I_{2}}+\cdots$
(125)

We recall the definition of tensile string vacua which is

$\displaystyle\alpha_{n}\ket{0}_{\alpha}=\tilde{\alpha}_{n}\ket{0}_{\alpha}=0\hskip
14.22636pt\forall n>0.$ (126)

We also recall how the oscillators $\alpha_{n}$ and $\tilde{\alpha}_{n}$ are
connected to tensionless oscillators $A_{n}$ and $B_{n}$111111See Appendix (C)
for details of the limiting computation of modes.

$\displaystyle\alpha_{n}=\frac{1}{2}\Big{[}\sqrt{\epsilon}A_{n}+\frac{1}{\sqrt{\epsilon}}B_{n}\Big{]},\hskip
14.22636pt\tilde{\alpha}_{n}=\frac{1}{2}\Big{[}-\sqrt{\epsilon}A_{-n}+\frac{1}{\sqrt{\epsilon}}B_{-n}\Big{]}.$
(127)

Applying (127) and (125) to the definition (126) one gets the following

$\displaystyle\ket{\zeta_{n}}=\frac{1}{\epsilon}\rho_{\mu\nu}\Big{(}B^{\mu}_{-n}+\epsilon
A^{\mu}_{-n}\Big{)}\Big{(}B^{\nu}_{n}-\epsilon
A^{\nu}_{n}\Big{)}\Big{(}\ket{0}_{I}+\epsilon\ket{I_{1}}+\epsilon^{2}\ket{I_{2}}+\cdots\Big{)}.$
(128)

Using the algebra satisfied by $A$ and $B$ one can see that the new
perturbative state in the limit:

$\displaystyle\ket{\zeta_{n}}=\Xi\ket{0}_{I},\hskip
14.22636pt\Xi=2n\eta^{\mu\nu}\rho_{\mu\nu}.$ (129)

In this section we will study the fate of the perturbative physical states of
the tensile string theory with background Kalb-Ramond field and see whether
they reduce to the physical states of tensionless string theory built on
induced vacuum. We recall that for tensile string theory in a target spacetime
with $d$ compactified dimensions, the level matching condition 121212It is
important to recall here that in tensile theory too, the constant $B$ field
does not have any effect on the level matching condition. is modified to:

$\displaystyle\widetilde{N}-N=\sum_{I}k_{I}\omega^{I}=l.$ (130)

Figure 2: Fate of physical states in the tensionless limit in the Induced
case.

Let us consider a tensile physical perturbative string state satisfying the
above level matching condition

$\displaystyle\ket{\zeta_{n}}=\rho_{\mu\nu}\alpha^{\mu}_{-n}\tilde{\alpha}^{\mu}_{-n-l}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{\alpha}.$
(131)

In our previous work Banerjee:2023ekd  we considered a target spacetime with
only one dimension compactified on a circle. There we considered the tensile
state

$\displaystyle\ket{\xi_{n}}=\rho_{\mu\nu}\alpha^{\mu}_{-n}\tilde{\alpha}^{\mu}_{-n-KW}\ket{0,0,k^{\mu},K,W}_{\alpha},$

and we examined its fate under the explicit tensionless limit using the
limiting procedure outlined above. To remind the reader, our findings were as
given below

  * •

For $KW=0$, $\ket{\xi_{n}}$ condense at the induced vacuum
$\ket{0,0,k^{\mu},K,W}_{I}$. Note that this condensation is nothing but the
Bose-Einstein like condensation described in Bagchi:2019cay .

  * •

For $KW>0$, $\ket{\xi_{n}}$ go to unphysical state.

  * •

For $KW<0$ and $n\geq|KW|$, $\ket{\xi_{n}}$ goes to unphysical state again.

  * •

For $KW<0$ and $n<|KW|$, $\ket{\xi_{n}}$ altogether vanishes at tensionless
limit.

The generalisation to a target spacetime with $d$ dimensions compactified, as
in the present case, is straightforward131313The steps of the calculation are
identical to the same of one-dimension compactified and hence, not reproduced
here. One can see Banerjee:2023ekd  for details of the calculation..

  * •

For $l=0$, $\ket{\zeta_{n}}$ condenses at
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}$.

  * •

For $l>0$, $\ket{\zeta_{n}}$ tends to unphysical state.

  * •

For $l<0$, $n\geq|l|$ too, $\ket{\zeta_{n}}$ tends to unphysical state.

  * •

For $l<0$ and $n<|l|$ , $\ket{\zeta_{n}}$ vanishes at tensionless limit.

Hence, if a perturbative tensile physical state still remains a physical state
in the tensionless limit, it will always condense on the induced vacuum, or
else become unphysical or even vanish. A visualisation of the fate of the
states is shown in figure (2).

For completeness, let us also comment on the fate of non-perturbative string
states. As discussed in Bagchi:2020fpr , the physical state conditions for the
induced vacuum tensionless string theory allows the following non-perturbative
states in the spectrum of physical states.

$\ket{\phi}=\text{exp}\Bigg{(}i\sum_{n,m}\omega_{n}A_{n-m}\cdot
B_{m}\Bigg{)}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{I}$ (132)

In the light of the fact that both $A_{0}$ and $B_{0}$ commute with all
$A_{n}$’s and $B_{n}$’s one can see that $\ket{\phi}$ must have the same
momentum and winding number as the vacuum state it has been constructed on

$B^{\mu}_{0}\ket{\phi}=k^{\mu}\ket{\phi}\hskip 14.22636ptB_{I\hskip
2.84526pt0}\ket{\phi}=k_{I}\ket{\phi}\hskip
14.22636ptA^{I}_{0}\ket{\phi}=\sqrt{\frac{2}{c^{\prime}}}\omega^{I}\ket{\phi}.$
(133)

Consequently mass of this state also has to be same as the vacuum state it has
been constructed upon, i.e.

$\displaystyle
m^{2}\ket{\phi}=(k^{T}G^{-1}k-2k^{T}G^{-1}\mathbb{B}\omega-\omega^{T}\mathbb{B}G^{-1}\mathbb{B}\omega)\ket{\phi}.$
(134)

####  5.2.2 Limit on mass spectra and the generalised metric

Now we will discuss what happens to the mass spectrum in the explicit
tensionless limit. We already learnt that a subset of the tensile physical
perturbative states condense to the induced vacuum in the tensionless limit
and the rest simply become unphysical in the new theory. Hence, if we want the
limit from tensile string theory to be consistent with the theory
intrinsically constructed on the induced vacuum, it is important to show that
the mass of induced vacuum in (120) is identical to the tensionless limit of
the mass formula for particular tensile states which remain physical (viz.
$l=0$) in the tensionless limit.

Let us consider the Polyakov action of the tensile string with a constant $B$
field (11) in a compactified spacetime with $d$ number of compact dimensions,
where $B_{\mu\nu}\neq 0$ only along the compact dimensions. We recall that the
mass spectrum for the tensile physical perturbative states is given by (see
for example, Becker:2006dvp ; Blumenhagen:2013fgp )

$\displaystyle m^{2}=k^{T}G^{-1}k+\frac{1}{\alpha^{\prime
2}}\omega^{T}(G-bG^{-1}b)\omega-\frac{2}{\alpha^{\prime}}k^{T}bG^{-1}\omega+\frac{1}{\alpha^{\prime}}(N_{L}+N_{R}-2).$
(135)

In the above equation, the metric along compact directions
$G_{IJ}=R_{I}^{2}\delta_{IJ}$ and the antisymmetric tensor $b_{IJ}$ is related
to the tensor $B_{IJ}$ in exactly the same way as in (51), i.e.

$b_{IJ}=\frac{\partial X^{K}}{\partial\widetilde{X}^{I}}\frac{\partial
X^{L}}{\partial\widetilde{X}^{J}}B_{KL}=R_{I}R_{J}\sum_{K=1}^{d}\sum_{L=1}^{d}\delta^{K}_{I}\delta^{L}_{J}B_{KL}=R_{I}R_{J}B_{IJ}.$
(136)

We also have the level matching condition in terms of tensile number operators
$N_{R}-N_{L}=\sum_{I=1}^{d}k_{I}\omega^{I}$. From (13) and (51), one can see
that

$\displaystyle\frac{1}{\alpha^{\prime}}b_{IJ}=\mathbb{B}_{IJ}.$ (137)

Writing the part of mass spectrum involving momenta and winding,
$\\{k_{I},\omega^{I}\\}$ of (135) in the matrix notation we used earlier, we
get

$\displaystyle\Big{[}\omega^{T}\hskip
11.38109ptk^{T}\Big{]}\begin{bmatrix}\cfrac{1}{\alpha^{\prime
2}}\left(G-bG^{-1}b\right)&\leavevmode\nobreak\ \leavevmode\nobreak\
\cfrac{1}{\alpha^{\prime}}\leavevmode\nobreak\ bG^{-1}\\\
-\cfrac{1}{\alpha^{\prime}}\leavevmode\nobreak\
G^{-1}b&G^{-1}\end{bmatrix}\hskip 5.69054pt\begin{bmatrix}\omega\\\ \\\
k\end{bmatrix},$ (138)

Taking tensionless limit corresponds to taking
$\alpha^{\prime}\to\frac{c^{\prime}}{\epsilon}$ with $\epsilon\to 0$, as
explained earlier. We recall the limit taken on $B$ field in tensionless limit
as given in (12) and the definition of $\mathcal{B}_{\mu\nu}$ in (13).
Together, these limits on $\alpha^{\prime}$ and $b_{IJ}$ keeps the term
$\mathbb{B}_{IJ}$ intact as we take $\epsilon\to 0$. As a result, the terms in
the mass spectrum (135) involving $\frac{b}{\alpha^{\prime}}$ survive the
tensionless limit. However, the terms only involving
$\frac{1}{\alpha^{\prime}}$ becomes suppressed due to the limit
$\alpha^{\prime}\to\frac{\alpha^{\prime}}{\epsilon}$. Consequently, at the
tensionless limit, (135) will reduce to (120). Therefore, the limit from
tensile mass spectrum (i.e., limiting approach) turns out to be consistent
with the intrinsically derived mass spectrum.

Let us now focus on the tensile mass spectrum as given in (138). The
generalised momentum here belongs to a $2d$ dimensional spacetime where the
$2d\times 2d$ dimensional matrix in (138) acts as inverse metric. The metric
in the spacetime becomes inverse of the matrix, given by

$\displaystyle\begin{bmatrix}G^{-1}&\leavevmode\nobreak\ \leavevmode\nobreak\
-\cfrac{1}{\alpha^{\prime}}\leavevmode\nobreak\ G^{-1}b\\\
\cfrac{1}{\alpha^{\prime}}\leavevmode\nobreak\
bG^{-1}&\cfrac{1}{\alpha^{\prime 2}}\left(G-bG^{-1}b\right)\end{bmatrix},$
(139)

In this space a manifestly $O(d,d)$ symmetric field theory can be defined,
often called as double field theory (DFT) (see Aldazabal:2013sca ). Taking a
consistent tensionless limit ($\alpha^{\prime}\to\infty$) on the inverse
generalised metric in (138) we get:

$\displaystyle\mathcal{G}=\begin{bmatrix}-\mathbb{B}G^{-1}\mathbb{B}&\mathbb{B}G^{-1}\\\
-G^{-1}\mathbb{B}&G^{-1}\end{bmatrix}$ (140)

It turns out that the new metric is indeed the degenerate one we presented in
(78), and a similar limit on (139) will give us the degenerate DFT metric. The
degeneracy at tensionless limit becomes even more manifest when we work with
the special case of $b=0$. For that case, at $\alpha^{\prime}\to\infty$, we
can easily see how the entire winding sector gets suppressed, a degeneracy of
order $d$ appears, and the mass spectrum depends only on the compactified
momenta.

A well known DFT result is that when the compactification scale becomes much
smaller than the string length scale, the DFT only depends on dual/momentum
coordinates Aldazabal:2013sca . Since the tensionless limit happens to be the
ultra-relativistic limit on the worldsheet level, the string here becomes
extremely long, even much larger than the compactification radius. Hence,
taking tensionless limit can be considered to be an alternative way to achieve
the “compactification scale $\ll$ string length scale” scenario for Double
Field Theory.

####  5.2.3 Narain Lattice at the limit

The mass spectrum obtained in (135) comes from the following left and right
moving momentum for the tensile string

$\displaystyle(P_{I})_{\hskip
0.7113ptL,R}=\frac{1}{\sqrt{2}}\bigg{\\{}\sqrt{\alpha^{\prime}}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}\sum_{J=1}^{d}b_{IJ}\omega^{J}\Big{)}\pm\frac{1}{\sqrt{\alpha^{\prime}}}\sum_{J=1}^{d}G_{IJ}\omega^{J}\bigg{\\}}.$
(141)

As is well known, the vector $P=\big{(}(P_{I})_{\hskip
0.7113ptL},(P_{I})_{\hskip 0.7113ptR}\big{)}$ belongs to a $2d$-dimensional
even self-dual lattice of Lorentzian signature
$\left((+1)^{d},(-1)^{d})\right)$, also called Narain lattice Becker:2006dvp .
The Lorentzian length of the vector $P$ on this lattice space is given by

$P^{2}=(P_{I})_{\hskip 0.7113ptL}G^{IJ}(P_{I})_{\hskip
0.7113ptL}-(P_{I})_{\hskip 0.7113ptR}G^{IJ}(P_{I})_{\hskip 0.7113ptR}$ (142)

In matrix notation we can write this as

$\displaystyle P^{2}=\Big{[}(P_{I})_{\hskip 0.7113ptL}\hskip
5.69054pt(P_{I})_{\hskip 0.7113ptR}\Big{]}\begin{bmatrix}G^{IJ}&0\\\
0&-G^{IJ}\end{bmatrix}\begin{bmatrix}(P_{I})_{\hskip 0.7113ptL}\\\
(P_{I})_{\hskip 0.7113ptR}\end{bmatrix}.$ (143)

We can also write this in following way by rotating the momentum basis of the
lattice

$\displaystyle P^{2}=\frac{1}{2}\Big{[}\big{(}(P_{I})_{\hskip
0.7113ptL}+(P_{I})_{\hskip 0.7113ptR}\big{)}\hskip
8.53581pt\big{(}(P_{I})_{\hskip 0.7113ptL}-(P_{I})_{\hskip
0.7113ptR}\big{)}\Big{]}\begin{bmatrix}0&G^{IJ}\\\
G^{IJ}&0\end{bmatrix}\begin{bmatrix}\big{(}(P_{I})_{\hskip
0.7113ptL}+(P_{I})_{\hskip 0.7113ptR}\big{)}\vspace{0.1cm}\\\
\big{(}(P_{I})_{\hskip 0.7113ptL}-(P_{I})_{\hskip
0.7113ptR}\big{)}\end{bmatrix}.$ (144)

Taking tensionless limit ($\alpha^{\prime}\to c^{\prime}/\epsilon$) on
$(P_{I})_{\hskip 0.7113ptL,R}$ as written in (62), one can clearly see that
the basis vectors are inhomogeneously scaled up and down by $\epsilon$,

$\displaystyle\big{(}(P_{I})_{\hskip 0.7113ptL}+(P_{I})_{\hskip
0.7113ptR}\big{)}=\frac{1}{\sqrt{\epsilon}}\big{(}(k_{I})_{\hskip
0.7113ptL}+(k_{I})_{\hskip 0.7113ptR}\big{)},\hskip
14.22636pt\big{(}(P_{I})_{\hskip 0.7113ptL}-(P_{I})_{\hskip
0.7113ptR}\big{)}=\sqrt{\epsilon}\big{(}(k_{I})_{\hskip
0.7113ptL}-(k_{I})_{\hskip 0.7113ptR}\big{)}.$ (145)

Evidently, at tensionless limit $\epsilon\to 0$, one can see that

$\displaystyle P^{2}=\frac{1}{2}\left[\big{(}(k_{I})_{\hskip
0.7113ptL}+(k_{I})_{\hskip 0.7113ptR}\big{)}\hskip
8.53581pt\big{(}(k_{I})_{\hskip 0.7113ptL}-(k_{I})_{\hskip
0.7113ptR}\big{)}\right]\begin{bmatrix}0&\frac{\sqrt{\epsilon}}{\sqrt{\epsilon}}G^{IJ}\\\
\frac{\sqrt{\epsilon}}{\sqrt{\epsilon}}G^{IJ}&0\end{bmatrix}\begin{bmatrix}\big{(}(k_{I})_{\hskip
0.7113ptL}+(k_{I})_{\hskip 0.7113ptR}\big{)}\vspace{0.1cm}\\\
\big{(}(k_{I})_{\hskip 0.7113ptL}-(k_{I})_{\hskip
0.7113ptR}\big{)}\end{bmatrix}.$ (146)

This implies that at $\epsilon\to 0$, although the left-right basis vectors
are scaled, the volume of the Narain lattice remains effectively intact,
keeping the length of the vector unchanged. This also makes sure that we have
the same level matching conditions as in tensile string theory.

###  5.3 Concrete example: Two dimensional compact space

Now, we look into the mass spectrum of tensile string with constant $B$ field
in a spacetime with only two of the dimensions compactified. We choose
$\mathbb{B}$ in the following way

$\displaystyle\mathbb{B}_{IJ}=b\hskip 2.84526pt\epsilon_{IJ},\hskip
14.22636ptI,J\in\\{1,2\\}$ (147)

where $\epsilon_{IJ}$ is the antisymmetric tensor in two dimensions. Choosing
$\epsilon_{12}=1$, we find that for $d=2$, equation (120) gets simplified to

$\displaystyle
m^{2}=\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+2b\Bigg{(}\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\Bigg{)}+b^{2}(R_{1}^{2}\omega_{1}^{2}+R_{2}^{2}\omega_{2}^{2})$
(148)

Just like in the case of oscillator vacuum, this mass spectrum is symmetric
under the following set of transformations:

$\displaystyle k_{I}\leftrightarrow\omega^{I},\hskip
14.22636ptR_{I}\leftrightarrow\frac{1}{bR_{I}},\hskip 14.22636ptb\to-b\hskip
14.22636ptI\in\\{1,2\\}.$ (149)

Now, let us consider the tensile counterpart of this theory, i.e. a tensile
string with a $B$ field compactified on a two-torus. In that case the mass
spectrum is

$\displaystyle
m^{2}=\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+\frac{2b}{\alpha^{\prime}}\Bigg{(}\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\Bigg{)}+\frac{(1+b^{2})}{\alpha^{\prime
2}}(R_{1}^{2}\omega_{1}^{2}+R_{2}^{2}\omega_{2}^{2})+\frac{1}{\alpha^{\prime}}(N_{L}+N_{R}-2).$
(150)

As we can clearly see, direct tensionless limit
($\alpha^{\prime}\to\frac{\alpha^{\prime}}{\epsilon}$,
$b\to\frac{b}{\epsilon}$, $\epsilon\to 0$) taken on this mass spectrum will
lead us to the intrinsically derived mass-spectrum in (148).

Since (148) is identical to the compact part of the mass spectrum we found in
oscillator vacuum (see (97)), the symmetries satisfied (and broken) by this
mass spectrum too, is the same, which means all the discussions in the example
of oscillator vacuum compactified on $T^{2}$ is applicable here as well. Let
us recall the transformation of generalised metric under sectorised duality
transformation $g_{D_{2}}$ as shown in (153), where we saw that the
transformed generalised metric is no longer corresponding to the tensionless
sector. In order to make more sense of this transformation let us look back at
the similar scenario in the tensile case where the same sectorised duality
transformation is applied on the generalised metric constructed from Polyakov
action with $G_{IJ}=R^{2}_{I}\delta_{IJ}$ and $B$ field
$B_{IJ}=B\epsilon_{IJ}.$ It can be shown that the inverse generalised metric
for tensile string after applying $g_{D_{2}}$ can be rewritten as below
141414For detailed calculations for tensile string calculation see Appendix
(A). We use the subscript $\alpha^{\prime}$ to distinguish the tensile case.

$\displaystyle\mathcal{G}^{\prime-1}_{\alpha^{\prime}}=g_{D_{2}}\mathcal{G}^{-1}_{\alpha^{\prime}}g_{D_{2}}^{T}=\begin{bmatrix}(1+b^{2})R^{2}_{1}&\frac{bR_{1}}{R_{2}}&0&0\\\
\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}&0&0\\\
0&0&\frac{1}{R^{2}_{1}}&-\frac{bR_{2}}{R_{1}}\\\
0&0&-\frac{bR_{2}}{R_{1}}&(1+b^{2})R^{2}_{2}.\end{bmatrix}$ (151)

It can be easily seen that the two diagonal blocks are inverse of each other.
Hence if we consider the following metric on a torus $T^{2}$

$\displaystyle(G_{\alpha^{\prime}})_{IJ}=\begin{bmatrix}(1+b^{2})R^{2}_{1}&\frac{bR_{1}}{R_{2}}\\\
\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}\end{bmatrix}$ (152)

Then the transformed inverse generalised metric can be rewritten as below

$\displaystyle\mathcal{G}^{\prime-1}_{\alpha^{\prime}}=\begin{bmatrix}G&0\\\
0&G^{-1}\end{bmatrix}.$ (153)

Looking at the expression of tensile generalised metric one can see that this
new generalised metric correspond to a Polyakov action with $G_{IJ}$ given in
(152) and $B_{IJ}=0$. Now, in comparison with this, let us rewrite the
diagonal blocks in the $\mathcal{G}^{\prime}$ from (102):

$\displaystyle(G_{c^{\prime}})_{IJ}=\begin{bmatrix}b^{2}R^{2}_{1}&\frac{bR_{1}}{R_{2}}\\\
\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}\end{bmatrix}\hskip
14.22636pt(G^{\prime}_{c^{\prime}})_{IJ}=\begin{bmatrix}\frac{1}{R^{2}_{1}}&-\frac{bR_{2}}{R_{1}}\\\
-\frac{bR_{2}}{R_{1}}&b^{2}R^{2}_{2}\end{bmatrix}.$ (154)

One can see that $detG_{c^{\prime}}=detG^{\prime}_{c^{\prime}}=0$, and also
that $G_{c^{\prime}}G^{\prime}_{c^{\prime}}=0$. The implication is that
$G_{c^{\prime}}$ and $G^{\prime}_{c^{\prime}}$ can be considered to be the
metric and inverse metric of a degenerate torus. In fact one can see that the
degenerate torus having metric $(G_{c^{\prime}})_{IJ}$ comes from the
following equivalent limit of the non-degenerate torus having metric
$G_{\alpha^{\prime}}$ in (152)151515i.e. one cycle of the two torus is
“pinched”.:

$\displaystyle R_{1}\to\epsilon R_{1},\hskip
8.53581ptb\to\frac{b}{\epsilon}\hskip 8.53581ptR_{2}\to R_{2}.$ (155)

Consequently, $\mathcal{G^{\prime}}$ in (102) can be thought of to be
constructed from a different limit of string theory where the tensile string
is compactified on a degenerate torus without any $B$ field. This seems to
indicate that sectorised duality connects tensionless strings on a non-
degenrate compact manifold to a different corner of string theory. We have
found similar connections between tensionless limit and degenerate torus limit
for compactification on $T^{3}$ as well. However, a general proof of such
connection for compactification on $T^{d}$ is beyond the scope of the current
work.

We must mention here, this kind of connections between different corners of
string theory is far from new. It is known for a long time that non-
relativistic closed string theory is connected to DLCQ limit of string theory
Gomis:2000bd . Later many such connections between different limits of string
theories has been explored. For detailed and updated picture of this duality
web the reader is referred to Gomis:2023eav ; Blair:2023noj . Although the
connections we have found in the current work probably do not correspond to
any existing duality connection, one could conjecture that this might lead to
an extension of the duality web.

##  6 Flipped vacuum

###  6.1 Detour: Effect of constant $B$ field on twisted tensile string

As we know from earlier works Bagchi:2020fpr , the quantum tensionless theory
constructed on the flipped vacuum can be arrived at by taking direct
tensionless limit on tensile version of twisted string theory. Hence, before
studying the effect of a constant $B$ field on tensionless limit of the
theory, we should first learn the effect of constant $B$ field on a tensile
twisted string theory. This analysis itself is novel, and can be considered as
generalization of Casali:2017mss ; Lee:2017crr , where $d$ ($>1$) number of
dimensions are compactified and a constant $B$ field is present in the target
spacetime.

#### Classical theory and mode expansions

Classically, this twisted string theory is identical to the usual string
theory with $B$ field and hence the action is same as the Polyakov action with
a $B$ field:

$\displaystyle S=\frac{T}{2}\int
d^{2}\xi\Big{(}\sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}+\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{\mu\nu}\Big{)}.$
(156)

After conformal gauge fixing, i.e. putting
$g^{\alpha\beta}=\eta^{\alpha\beta}$, the equations of motion, constraints,
the commutation relations and the symmetries of this action can be found to be
the same as that for the action without $B$ field. The equation of motion
along with the closed string boundary conditions ((49) for compact coordinates
and $X^{\nu}(\sigma+2\pi,\tau)=X^{\nu}(\sigma,\tau)$ for non-compact
coordinates) leads us to the following familiar mode expansions:

$\begin{split}&X^{\mu}=X^{\mu}_{L}+X^{\mu}_{R}\\\
X^{\mu}_{L}=x^{\mu}+\sqrt{\frac{\alpha^{\prime}}{2}}&\alpha^{\mu}_{0}(\tau+\sigma)+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\alpha^{\mu}_{n}e^{-in(\tau+\sigma)}\\\
X^{\mu}_{R}=x^{\mu}+\sqrt{\frac{\alpha^{\prime}}{2}}&\tilde{\alpha}^{\mu}_{0}(\tau-\sigma)+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\sigma)},\\\ \end{split}$

where $\\{\alpha,\tilde{\alpha}\\}$ satisfy the harmonic oscillator algebra.
For compactified coordinates we use the same scaling we used before, namely
(50). Consequently we have the new metric as in (51), along with new $B$ field
given by (136). We also have new compactified boundary conditions exactly like
in (53). In terms of the scaled coordinates we now rewrite the compactified
part of the gauge fixed action

$\displaystyle S_{C}=\frac{T}{2}\int
d^{2}\xi\Big{(}\eta^{\alpha\beta}\partial_{\alpha}\widetilde{X}^{I}\partial_{\beta}\widetilde{X}^{J}G_{IJ}+\epsilon^{\alpha\beta}\partial_{\alpha}\widetilde{X}^{I}\partial_{\beta}\widetilde{X}^{J}b_{IJ}\Big{)}.$
(157)

As we all know, the canonical momentum gets modified due to the presence of
$B$ field as
$\Pi_{I}=\frac{1}{2\pi\alpha^{\prime}}\Big{(}\dot{\widetilde{X}^{J}}G_{IJ}+b_{IJ}\widetilde{X}^{\prime
J}\Big{)}$, and the internal momentum of the string becomes

$\displaystyle\pi_{I}=\int_{0}^{2\pi}d\sigma\Pi_{I}=\sum_{J}G_{IJ}K^{J}+\frac{1}{\alpha^{\prime}}\sum_{J}b_{IJ}W^{J}.$
(158)

Here again, following section (5), we define the dimensionless field $Y^{I}$
as $X^{I}=\sqrt{\frac{\alpha^{\prime}}{2}}Y^{I}$, where the mode expansion of
$Y^{I}$ (splitting into left and right part) is given in terms of twisted
tensile oscillators:

$\begin{split}Y^{I}_{L}&=y^{I}_{L}+\big{(}P^{I}\big{)}_{L}(\tau+\sigma)+i\sum_{n\neq
0}\frac{1}{n}\alpha^{\mu}_{n}e^{-in(\tau+\sigma)},\\\
Y^{I}_{R}&=y^{I}_{R}+\big{(}P^{I}\big{)}_{R}(\tau-\sigma)+i\sum_{n\neq
0}\frac{1}{n}\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\sigma)}.\end{split}$ (159)

#### Quantization

Classically, the residual gauge symmetry of this theory gives us two copies of
the Virasoro algebra as well. We can also construct the same algebra from the
classical constraints in the same way we did for usual string theory. However,
the quantum theory for twisted strings is constructed on the flipped
representation of the Virasoro algebra, which constitues an inner automorphism
of the same. This makes it distinct from the usual tensile string theory,
which is constructed on the highest weight representation of the Virasoro
algebra. The physical state condition for this twisted theory turns out to be
Bagchi:2020fpr :

$\begin{split}\big{(}\mathcal{L}_{n}-a\delta_{n,0}\big{)}\ket{phys}=\big{(}\bar{\mathcal{L}}_{-n}-\bar{a}\delta_{n,0}\big{)}\ket{phys}=0\hskip
14.22636pt\forall n\geq 0\end{split}$ (160)

where $\\{\mathcal{L}_{n},\bar{\mathcal{L}}_{n}\\}$ are the (anti)holomorphic
generators of the Virasoro algebra and {$a,\bar{a}$} are associated normal
ordering constants. The vacuum in this theory is defined as a combination of
highest weight and lowest weight conditions on the $\alpha$,

$\displaystyle\alpha_{n}\ket{0}=\tilde{\alpha}_{-n}\ket{0}=0,\hskip
14.22636pt\forall n>0$ (161)

Since in the right modes the role of creation and annihilation operators are
flipped, the definition of normal ordering for the right modes also change.
The new definition of normal ordering for right modes is

$\displaystyle:\tilde{\alpha}^{\mu}_{-m}\tilde{\alpha}^{\nu}_{n}:=\tilde{\alpha}^{\nu}_{n}\tilde{\alpha}^{\mu}_{-m}.$
(162)

The number operators in this theory are defined as

$\displaystyle N=\sum_{n=1}^{\infty}:\alpha_{-n}\cdot\alpha_{n}:\hskip
14.22636pt\widetilde{N}=-\sum_{n=1}^{\infty}:\tilde{\alpha}_{-n}\cdot\tilde{\alpha}_{n}:$
(163)

The entire Hilbert space of this theory can be spanned by the eigenbasis of
the number operators, and the basis states of the eigenbasis can be
constructed by acting the oscillators $\\{\alpha_{n},\tilde{\alpha}_{-n}\\}$
on the vacuum defined in (161). In order to make the wave function single
valued, canonical momentum $\pi_{I}$ must be quantized i.e.
$\pi_{I}=k_{I},\hskip 2.84526ptk_{I}\in\mathbb{Z}$. These result into the
following, again familiar, expression for $\big{(}P_{I}\big{)}_{L,R}$:

$\displaystyle\big{(}P_{I}\big{)}_{L,R}=\frac{1}{\sqrt{2}}\bigg{\\{}\sqrt{\alpha^{\prime}}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}\sum_{J=1}^{d}b_{IJ}\omega^{J}\Big{)}\pm\frac{1}{\sqrt{\alpha^{\prime}}}\sum_{J=1}^{d}G_{IJ}\omega^{J}\bigg{\\}}.$
(164)

Now, the zero modes $\mathcal{L}_{0}$ and $\bar{\mathcal{L}_{0}}$ can be
expressed in terms of the number operators, windings and momenta as

$\begin{split}\mathcal{L}_{0}&=\frac{\alpha^{\prime}}{4}\sum_{I.J=1}^{d}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}+\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}+\frac{\alpha^{\prime}}{4}\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}+N\\\
\bar{\mathcal{L}}_{0}&=\frac{\alpha^{\prime}}{4}\sum_{I.J=1}^{d}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}-\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}+\frac{\alpha^{\prime}}{4}\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}-\widetilde{N}.\end{split}$
(165)

Using $\mathcal{L}_{0}$ and $\bar{\mathcal{L}}_{0}$ conditions in (160) with
$a=-\bar{a}=1$ and adding and subtracting them, we have the following
constraints on physical states of level $(r,s)$, where $r,s$ are eigenvalues
of number operators $N$ and $\widetilde{N}$ respectively,

$\begin{split}&r+s+\frac{\alpha^{\prime}}{4}\sum_{I.J=1}^{d}\Bigg{[}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}+\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}-\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}-\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}\Bigg{]}-2=0\\\
r&-s+\frac{\alpha^{\prime}}{4}\sum_{I.J=1}^{d}\Bigg{[}\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}+\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}+\Big{(}k_{I}-\frac{1}{\alpha^{\prime}}b_{IJ}\omega^{J}-\frac{1}{\alpha^{\prime}}G_{IJ}\omega^{J}\Big{)}^{2}\Bigg{]}-\frac{\alpha^{\prime}}{2}m^{2}=0.\end{split}$

In the above, we have used $m^{2}=-\sum_{\mu=0}^{25-d}k_{\mu}k^{\mu}$,
considering bosonic twisted tensile string theories are also consistent in
$D=26$ . Simplifying these constraints one gets the level matching condition
and the mass formula:

$r+s+\sum_{I=1}^{d}k_{I}\omega^{I}=2,$ (166a)
$m^{2}=k^{T}G^{-1}k+\frac{1}{\alpha^{\prime
2}}\omega^{T}(G-bG^{-1}b)\omega-\frac{2}{\alpha^{\prime}}k^{T}bG^{-1}\omega+\frac{2}{\alpha^{\prime}}(r-s).$
(166b)

There are immediately two important things to note as a consequence of these
constraints:

  * •

For non-compact target spacetime, the level matching condition implied that
only states of level 2 could be physical. In Casali:2017mss ; Lee:2017crr ;
Banerjee:2023ekd  it was found that single compact dimension allows more
freedom: states of any level can be compactified for appropriate values of
internal momentum and winding number. In (166a) we see that for $d>1$, the
freedom is extended even more, in this case for each level we will have
infinite number of possible physical states.

  * •

Comparing the mass spectrum (166b) with the mass spectrum for usual tensile
string (135), one finds that the part of the expression arising due to
compactification happens to be exactly same as that of tensile string. Hence,
from this, one can conclude that this mass spectrum has all the discrete
symmetries satisfied by the usual tensile mass spectrum. In other words the
twisted mass spectrum will also have an explicit $O(d,d,\mathbb{Z})$ symmetry.

###  6.2 Tensionless flipped strings with $B$ field

In this section we analyze the effect of constant $B$ field on the null
flipped or “ambitwistor” string theory, the tensionless progeny of the twisted
theory we discussed in the last subsection. It is well known that this theory
inherently possesses the highest weight representation of the BMS3 algebra.
The physical state conditions leading to this quantum theory are given by just
the usual right action of the stress tensor constraints:

$(L_{n}-a_{L}\delta_{n,0})\ket{phys}=0\hskip 14.22636pt\forall n\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (M_{n}-a_{M}\delta_{n,0})\ket{phys}=0\hskip
14.22636pt\forall n\geq 0.$ (167)

This theory also involves left and right moving oscillators like the tensile
parent thereof, much like the theory defined on the oscillator vacuum.
However, here the right moving oscillators are redefined, to comply with the
parent vacuum conditions Bagchi:2020fpr , as below:

$\displaystyle\tilde{\mathcal{C}}_{n}=\tilde{C}_{-n}.$ (168)

The flipped vacuum for a compactified theory, on which the ambitwistor string
theory (defined by the subscript $A$) is based on, is defined somewhat like
the condition (161), i.e.

$\displaystyle
C^{\mu}_{n}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}=\tilde{\mathcal{C}}^{\mu}_{n}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}=0\hskip
14.22636pt\forall n>0.$ (169)

As we can see, just like the tensile counterpart of the theory, here too, the
role of the creation and annihilation operators have been flipped. The algebra
satisfied by the newly defined oscillators turns out to be the following:

$\displaystyle[C^{\mu}_{m},C^{\nu}_{n}]=m\eta^{\mu\nu}\delta_{m+n}\hskip
14.22636pt[\tilde{\mathcal{C}}^{\mu}_{m},\tilde{\mathcal{C}}^{\nu}_{n}]=-m\eta^{\mu\nu}\delta_{m+n},\hskip
14.22636pt[C^{\mu}_{m},\tilde{\mathcal{C}}^{\nu}_{n}]=0.$ (170)

The generators $L_{n}$ and $M_{n}$ can be expressed in terms of
$\\{C,\tilde{\mathcal{C}}\\}$ as below

$\begin{split}L_{n}&=\frac{1}{2}\sum_{m}\left[C_{-m}\cdot
C_{m+n}-\tilde{\mathcal{C}}_{m}\cdot\tilde{\mathcal{C}}_{-m+n}\right],\\\
M_{n}&=\frac{1}{2}\sum_{m}\left[C_{-m}\cdot
C_{m+n}+\tilde{\mathcal{C}}_{m}\cdot\tilde{\mathcal{C}}_{-m+n}+2C_{-m}\cdot\tilde{\mathcal{C}}_{m+n}\right].\end{split}$
(171)

Here again we use the equations (56)-(58) and obtain the expression for the
canonical momenta $(K_{I})_{L,R}$ which is exactly similar as we had in the
case of oscillator vacuum, given by

$\displaystyle(k_{I})_{\hskip
0.7113ptL,R}=\frac{1}{\sqrt{2}}\bigg{\\{}\sqrt{c^{\prime}}\Bigg{(}k_{I}-\sum_{J=1}^{d}\mathcal{B}_{IJ}\omega^{J}\Bigg{)}\pm\frac{1}{\sqrt{c^{\prime}}}\sum_{J=1}^{d}G_{IJ}\omega^{J}\bigg{\\}}.$
(172)

The normal ordered form of the zero modes $L_{0}$ and $M_{0}$ in terms of
$(k_{I})_{\hskip 0.7113ptL,R}$ are given by

$\displaystyle\begin{split}L_{0}=N+\bar{N}+\frac{1}{2}\sum_{I=1}^{d}\big{(}(K^{I})_{L}(k_{I})_{\hskip
0.7113ptL}-(K^{I})_{R}(k_{I})_{\hskip
0.7113ptR}\big{)}=N+\bar{N}+\sum_{I=1}^{d}k_{I}\omega^{I}\end{split}$ (173)

and

$\begin{split}M_{0}&=\frac{1}{2c^{\prime}}\sum_{I=1}^{d}(K^{I}_{L}+K^{I}_{R})(K_{I\hskip
0.7113ptL}+K_{I\hskip
0.7113ptR})+c^{\prime}\sum_{\mu={0}}^{25-d}k_{\mu}k^{\mu}+N-\bar{N}+X+Y\\\
=&c^{\prime}\sum_{I,J=1}^{d}\Big{(}k_{I}-\mathcal{B}_{IJ}\omega^{J}\Big{)}G^{IJ}\Big{(}k_{I}-\mathcal{B}_{IK}\omega^{K}\Big{)}+c^{\prime}\sum_{\mu={0}}^{25-d}k_{\mu}k^{\mu}+N-\bar{N}+X+Y.\end{split}$
(174)

The physical state conditions for tensionless string on flipped vacuum is
given in (167). As we have already seen before Bagchi:2021rfw , this theory
has $a_{L}=2$ and $a_{M}=0$. Using $a_{L}=2$ on the $L_{0}$ condition gives us
the following physical state condition for a generic state of level
$\\{r,s\\}$

$\displaystyle r+s+\sum_{i=1}^{d}k_{I}\omega^{I}=2$ (175)

Here we note that the level matching condition we obtained in (175) is
identical to that in (166a), indicating towards a one to one mapping from
tensile and tensionless states through the limit. In Banerjee:2023ekd , we
have seen that when we compactify one dimension, we can have physical states
in any possible level for suitable choice of internal momentum ($K$) and
winding number ($W$). However, for states with $r+s\neq 2$, we could have only
finite number of physical states at each level. For example, for level zero,
($r+s=0$), there would be only four possible states (since there are only four
possible integer combinations of $K$ and $W$ which can ensure that $KW=2$).
Similarly we can find only two states for level one. However, for more compact
dimensions $d\geq 2$, even for $r+s\neq 2$, we will be able to get infinite
number of possible combinations of $\\{k_{I},\omega^{I}\\}$ satisfying (175).
As a result, for $d\geq 2$ we have infinite number of physical states at any
level.

Now, we will apply the physical state conditions (167) (keeping in mind that
$a_{M}=0$) along with (175) on level 0, level 1 and level 2 states and study
the constraints to be satisfied by them in order to become explicit physical
states.

### Level 0 states

The level 0 states are essentially flipped vacuum states defined in (169),
i.e. $\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}$. The $L_{0}$ and $M_{0}$
physical state conditions are the only non-trivial physical state conditions
for this level. Since for vacuum both $r=s=0$, the $L_{0}$ condition leads us
to:

$\displaystyle\sum_{I=1}^{d}k_{I}\omega^{I}=2.$ (176)

As we have already discussed, there are infinite number of states satisfying
(176). Now let us look at the $M_{0}$ condition, which gives us the mass

$\begin{split}m^{2}&=-\sum_{\mu={0}}^{25-d}k_{\mu}k^{\mu}=\sum_{I,J,L=1}^{d}\Big{(}k_{I}-\mathcal{B}_{IJ}\omega^{J}\Big{)}G^{IL}\Big{(}k_{L}-\mathcal{B}_{LK}\omega^{K}\Big{)}\\\
&=k^{T}G^{-1}k-2k^{T}G^{-1}\mathbb{B}\omega-\omega^{T}\mathbb{B}G^{-1}\mathbb{B}\omega\end{split}$
(177)

### Level 1 states

As discussed in Bagchi:2020fpr , the physical state conditions given in (167)
for $n>1$ will be trivially satisfied by a state of level 1. Hence we have
only four nontrivial physical state conditions, namely the $L_{0}$, $M_{0}$,
$L_{1}$ and $M_{1}$ conditions. A generic state of level 1 is given by

$\displaystyle\ket{1,k_{I},\omega^{I}}_{A}=a_{\mu}C^{\mu}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}+b_{\mu}\tilde{\mathcal{C}}^{\mu}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}.$
(178)

The $L_{0}$ condition simply gives us $\sum_{I=1}^{d}k_{I}\omega^{I}=1$, which
leads us to infinite number of level 1 states. The other non-trivial physical
state conditions are then given by

$\displaystyle
L_{1}\ket{1,k_{I},\omega^{I}}_{A}=M_{1}\ket{1,k_{I},\omega^{I}}_{A}=M_{0}\ket{1,k_{I},\omega^{I}}_{A}=0$
(179)

They lead us to the following constraints for
$\\{k^{\mu},k_{I},\omega^{I},a_{\mu},b_{\mu}\\}$.

$\displaystyle\Bigg{[}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(a_{I}+b_{I})+\frac{1}{c^{\prime}}\sum_{I=1}^{d}\omega^{I}(a_{I}-b_{I})$
$\displaystyle\hskip
166.44879pt+\sum_{\mu=0}^{25-d}k^{\mu}(a_{\mu}+b_{\mu})\Bigg{]}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}=0$
$\displaystyle\Bigg{[}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(a_{I}-b_{I})+\sum_{\mu=0}^{25-d}k^{\mu}(a_{\mu}-b_{\mu})\Bigg{]}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}=0$
(180)
$\displaystyle\Big{[}\big{(}A+1\big{)}a_{\mu}-b_{\mu}\Big{]}C^{\mu}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}+\Big{[}\big{(}A-1\big{)}b_{\mu}+a_{\mu}\Big{]}\tilde{\mathcal{C}}^{\mu}_{-1}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}=0.$

where we have used: