${\cal N}=1$ supersymmetry and Non-Riemannian Double Field Theory

Eric Lescano

Division of Theoretical Physics, Rudjer Boskovic Institute
Bijenicka 54, 10000 Zagreb, Croatia

<EMAIL_ADDRESS>
Abstract

We construct the ${\cal N}=1$ supersymmetric extension of Double Field Theory
for Riemannian and the non-Riemannian in a unified approach. The inclusion of
fermions in the double geometry force us to use the generalized frame
formalism to construct the generalized flux components for these geometries.
We focus on the most general prescription required to get the $D=10$ minimal
supergravity model. We study how to consistently avoid the gauge fixing
procedure of the double Lorentz symmetry when $n=\bar{n}$ and $0\leq n\leq 5$,
which gives rise to a bigravity structure (pair of vielbeins producing the
same non-Riemannian degrees of freedom). As an example we show how to to
include fermionic degrees of freedom in the type I torsional Newton-Cartan
(TNC) theory ($n=\bar{n}=1$) which is related to Carrollian geometries and
stringy Newton Cartan through duality rotations and/or null
reductions/uplifts.

##  1 Introduction

Double Field Theory (DFT) [1, 2] is typically understood as a convenient
rewriting of the low energy limit of string theory in terms of $O(D,D)$
multiplets. All the massless fields of the universal NS-NS sector can be
regrouped in terms of a generalized metric, ${\cal H}_{MN}$ ($M,N=0\dots
2D-1$) and a generalized dilaton $d$, while a variational principle can be
defined in terms of these fields in a double space. Inspecting the most
general parametrization of the former fields the authors in [3] showed that
both Riemannian and non-Riemannian geometries can be rewriting in this
framework. Moreover, they classified all the possible scenarios using two
integers $(n,\bar{n})$, where $n+\bar{n}=D$ and $n=\bar{n}=0$ corresponds to
the Riemmanian case.

Effective non-Riemannian geometries are backgrounds compatible with string
theory [4] and, interestingly enough, these backgrounds are related to
Riemannian ones by T-duality transformations (or a null reduction/uplift) [5].
In this sense, the formulation introduced in [3] is a natural proposal for
unifying all these scenarios in a duality invariant way. The ${\cal N}=1$
supersymmetric extension of DFT was introduced in [6] up to two fermions and
in [7] considering the full four fermions interactions. This extension
requires the generalized frame/flux formalism of DFT [8] because of the
presence of fermionic degrees of freedom and a Clifford algebra at the level
of the double geometry.

The main goal of this work is to implement the previous extension for
Riemannian and non-Riemannian scenarios in a unified approach. Our main
results are

  * $-$

The construction of the generalized flux formalism for non-Riemannian DFT, in
agreement with the generalized metric formulation.

  * $-$

The study of the gauge fixing procedure of the double Lorentz transformations
for these scenarios which give rise to a bigravity structure for the degrees
of freedom of the non-Riemannian supergravity.

  * $-$

The computation of the general action and (super) symmetry transformations for
the non-Riemannian fields.

  * $-$

The ${\cal N}=1$ supersymmetric extension of TNC geometry coming from DFT.

In order to achieve the previous results we use the generalized frame
parametrization introduced in [3] in order to construct the different
components of the generalized fluxes that are required by the DFT action
principle. Particularly, only $n=\bar{n}$ scenarios are compatible with the
double Lorentz symmetry and therefore with supersymmetry as firstly noticed in
[3] (see also the last section of [9]). We construct the most general ${\cal
N}=1$ extension to the DFT action principle in order to reproduce the minimal
$D=10$ supergravity model and we read the transformation rules for the degrees
of freedom of the non-Riemannian supergravity in agreement with the DFT
construction.

The bosonic part of this framework was given in [10] considering a generalized
metric formulation, which is equivalent to the generalized flux formalism here
presented. The only difference is that the latter forces $n=\bar{n}$ while the
inclusion of supersymmetry fixes the dimension of the space-time from the very
beginning. Therefore we work in $D=10$, while extensions to other non-
Riemannian supergravities should be easily extracted from our model
considering the proper Clifford algebra and fermionic relations. We will not
assume a factorization for the geometry, but see [11] for a bosonic treatment
between a Riemannian formulation on the external space and a non-Riemannian
one on the internal space.

In this work we also show how this generic supersymmetric extension can be
used in order to include fermionic degrees of freedom in the Torsional Newton-
Cartan (TNC) gravity. This is a convenient example because it was explored at
the (supersymmetric) worldsheet level in [12] and it is a geometry related to
Carrollian and stringy Newton-Cartan using T-duality and/or null
reduction/uplift (See [10] for the specific transformations between TNC and
Carrollian/stringy Newton-Cartan). In the last part of the work we discuss
potential continuations for this line of investigation. Particularly we
discuss about testing this model using perturbative theory and to extend the
proposal to be able to describe the ${\cal N}=1$ supersymmetric extension for
type II TNC.

##  2 Flux formulation for Non-Riemannian Double Field Theory

The construction of the flux formalism [8] of the non-Riemannian version of
DFT 111See [13] for reviews of the topic. can be performed considering four
steps 222Here we mimic the presentation of [14], but for the non-Riemannian
case. These lectures are also useful as an introduction to DFT.. We start by
describing the double geometry and the fundamental fields of the theory, with
their non-relativistic parametrization. Then we inspect the symmetry rule
transformations and finally we discuss about the action principle. Our
conventions for the indices are as follows: $O(D,D)$ indices are given by
$M,N,\dots=0,\dots,2D-1$ where $M=({}_{\dot{\mu}},{}^{\dot{\mu}})$. Their flat
indices (double Lorentz indices) are $A=(\underline{A},\overline{A})$, where
$\underline{A}=\underline{\dot{a}}$ and $\overline{A}=\dot{\overline{a}}$.
Space-time indices are $\dot{\mu}=0,\dots,d-1$.

  1. 1.

Double geometry: We consider a double geometry with coordinates $X^{M}$. We
equip a group invariant metric $\eta_{MN}$,

$\displaystyle{\eta}_{MN}=\left(\begin{matrix}0&\delta_{\dot{\mu}}^{\dot{\nu}}\\\
\delta^{\dot{\mu}}_{\dot{\nu}}&0\end{matrix}\right)\,,$ (2.1)

(and its inverse $\eta^{MN}$) which is used to lower (and raise) double curved
indices. On each point of the double space we consider a double tangent space,
so we are able to define flat vectors $V^{A}$. Then we consider two additional
invariant and flat metrics $\eta_{AB}$ and ${\cal H}_{AB}$. The former is used
to lower flat indices and both of them are used to construct the following
flat projectors,

$\displaystyle P_{AB}=\frac{1}{2}\left(\eta_{AB}-{\cal
H}_{AB}\right)=\left(\begin{matrix}P^{\underline{\dot{a}}\underline{\dot{b}}}&0\\\
0&0\end{matrix}\right)\,,$ (2.2)
$\displaystyle\overline{P}_{AB}=\frac{1}{2}\left(\eta_{AB}+{\cal
H}_{AB}\right)=\left(\begin{matrix}0&0\\\
0&\overline{P}^{\bar{\dot{a}}\bar{\dot{b}}}\end{matrix}\right)\,,$ (2.3)

which satisfy

$\displaystyle{\overline{P}}_{{AB}}{\overline{P}}^{B}{}_{C}={\overline{P}}_{{AC}}\,,$
$\displaystyle\quad{P}_{{AB}}{P}^{B}{}_{C}={P}_{{AC}},$ (2.4)
$\displaystyle{P}_{{AB}}{\overline{P}}^{B}{}_{C}={\overline{P}}_{{AB}}{P}^{B}{}_{C}=0\,,$
$\displaystyle\quad{\overline{P}}_{{AB}}+{P}_{{AB}}=\eta_{{AB}}\,.$

The invariant projectors of DFT are parametrized in the following way

$\displaystyle
P_{\underline{\dot{a}\dot{b}}}=\left(\begin{matrix}-\eta_{\underline{a}\underline{b}}&0&0\\\
0&-\delta_{ij}&0\\\
0&0&\delta_{ij}\end{matrix}\right),\quad\overline{P}_{\overline{\dot{a}\dot{b}}}=\left(\begin{matrix}\eta_{\bar{a}\bar{b}}&0&0\\\
0&-\delta_{\bar{i}\bar{j}}&0\\\
0&0&\delta_{\bar{i}\bar{j}}\end{matrix}\right)\,.$ (2.5)

Thus, $P$ and $\bar{P}$ project onto complementary orthogonal subspaces and
any arbitrary flat vector can be written as

$\displaystyle
V^{A}=P^{A}{}_{B}V^{B}+\overline{P}^{A}{}_{B}V^{B}=V^{\underline{A}}+V^{\overline{A}}\,.$
(2.6)

  2. 2.

Fundamental fields: The fundamental fields of DFT are a generalized frame
$E_{MA}(X)=E_{MA}$ and a generalized dilaton $d(X)=d$. The former is
equivalent to a vielbein for this double geometry and for non-Riemannian
backgrounds its parametrization is

$E^{M}{}_{A}=\left(\begin{matrix}{E}_{\dot{\mu}\underline{\dot{a}}}&{E}^{\dot{\mu}}{}_{\underline{\dot{a}}}\\\
E_{\dot{\mu}\overline{\dot{a}}}&E^{\dot{\mu}}{}_{\overline{\dot{a}}}\end{matrix}\right)\
=\
\frac{1}{\sqrt{2}}\left(\begin{matrix}-k_{\dot{\mu}\dot{\underline{a}}}-B_{\dot{\rho}\dot{\mu}}{h}^{\dot{\rho}}{}_{\dot{\underline{a}}}&h^{\dot{\mu}}{}_{\dot{\underline{a}}}\\\
\bar{k}_{\dot{\mu}\overline{\dot{a}}}-B_{\dot{\rho}\dot{\mu}}{}\bar{h}^{\dot{\rho}}{}_{\overline{\dot{a}}}&\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}\end{matrix}\right)\,,$
(2.7)

where

$\displaystyle
k_{\dot{\mu}\underline{\dot{a}}}=(k_{\dot{\mu}\underline{a}},x_{\dot{\mu}},x_{\dot{\mu}})\,,\quad\bar{k}_{\dot{\mu}\overline{\dot{a}}}=(\bar{k}_{\dot{\mu}\overline{a}},\bar{x}_{\dot{\mu}},\bar{x}_{\dot{\mu}})\,,$
(2.8) $\displaystyle
h^{\dot{\mu}}{}_{\underline{\dot{a}}}=(h^{\dot{\mu}}{}_{\underline{a}},y^{\dot{\mu}},y^{\dot{\mu}})\,,\quad\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}=(\bar{h}^{\dot{\mu}}{}_{\overline{a}},\bar{y}^{\dot{\mu}},\bar{y}^{\dot{\mu}})\,,$
(2.9)

and

$\displaystyle\bar{k}_{\dot{\mu}}{}^{\overline{a}}\bar{k}_{\dot{\nu}\overline{a}}=k_{\dot{\mu}}{}^{\underline{a}}k_{\dot{\nu}\underline{a}}$
$\displaystyle=$ $\displaystyle K_{\dot{\mu}\dot{\nu}}\,,$ (2.10)
$\displaystyle\bar{h}^{\dot{\mu}}{}_{\overline{a}}\bar{h}^{\dot{\nu}\overline{a}}=h^{\dot{\mu}}{}_{\underline{a}}h^{\dot{\nu}\underline{a}}$
$\displaystyle=$ $\displaystyle H^{\dot{\mu}\dot{\nu}}\,.$ (2.11)

In the previous expresion $K_{\dot{\mu}\dot{\nu}}$ and
$H^{\dot{\mu}\dot{\nu}}$ are two symmetric tensors whose kernels are spanned
by $\Big{\\{}x_{\dot{\mu}},\bar{x}_{\dot{\mu}}\Big{\\}}$ and
$\Big{\\{}y^{\dot{\mu}},\bar{y}^{\dot{\mu}}\Big{\\}}$, respectively,

$\displaystyle
H^{\dot{\mu}\dot{\nu}}x_{\dot{\nu}}=H^{\dot{\mu}\dot{\nu}}\bar{x}_{\dot{\nu}}=K_{\dot{\mu}\dot{\nu}}y^{\dot{\nu}}=K_{\dot{\mu}\dot{\nu}}\bar{y}^{\dot{\nu}}=0\,,$
(2.12)

while $B_{\dot{\mu}\dot{\nu}}$ is identified the ordinary B-field.

We define the following completeness relations,

$\displaystyle\delta^{\dot{\mu}}_{\dot{\nu}}$ $\displaystyle=$ $\displaystyle
H^{\dot{\mu}\dot{\rho}}K_{\dot{\nu}\dot{\rho}}+y^{\dot{\mu}}x_{\dot{\nu}}+\bar{y}^{\dot{\mu}}\bar{x}_{\dot{\nu}}\,,$
(2.13)
$\displaystyle\delta_{\dot{\mu}}^{\dot{\nu}}+x_{\dot{\mu}}y^{\dot{\nu}}-\bar{x}_{\dot{\mu}}\bar{y}^{\dot{\nu}}$
$\displaystyle=$
$\displaystyle\bar{k}_{\dot{\mu}\overline{\dot{a}}}\bar{h}^{\dot{\nu}\overline{\dot{a}}}\,,$
(2.14)
$\displaystyle\delta_{\dot{\mu}}^{\dot{\nu}}-x_{\dot{\mu}}y^{\dot{\nu}}+\bar{x}_{\dot{\mu}}\bar{y}^{\dot{\nu}}$
$\displaystyle=$ $\displaystyle
k_{\dot{\mu}\underline{\dot{a}}}h^{\dot{\nu}\underline{\dot{a}}}\,,$ (2.15)
$\displaystyle\delta_{\underline{\dot{a}}}^{\underline{\dot{b}}}$
$\displaystyle=$ $\displaystyle
k_{\dot{\mu}\underline{\dot{a}}}h^{\dot{\mu}\underline{\dot{b}}}\,,$ (2.16)
$\displaystyle\delta_{\overline{\dot{a}}}^{\overline{\dot{b}}}$
$\displaystyle=$
$\displaystyle\bar{k}_{\dot{\mu}\overline{\dot{a}}}\bar{h}^{\dot{\mu}\overline{\dot{b}}}\,.$
(2.17)

With the generalized frame it is possible to construct the generalized metric
and the invariant metric,

$\displaystyle E_{MA}{\cal H}^{AB}E_{NB}$ $\displaystyle=$ $\displaystyle{\cal
H}_{MN}\,,$ $\displaystyle E_{MA}\eta^{AB}E_{NB}$ $\displaystyle=$
$\displaystyle\eta_{MN}\,.$ (2.18)

The conventions for this paper were chosen in order to the generalized metric
coincides with the one in [10],

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle=\begin{pmatrix}K_{\mu\nu}-{B}_{\mu\rho}H^{\rho\sigma}{B}_{\sigma\nu}+2x^{i}_{(\mu}{B}_{\nu)\rho}y^{\rho}_{i}-2\bar{x}^{\bar{\imath}}_{(\mu}{B}_{\nu)\rho}\bar{y}^{\rho}_{\bar{\imath}}&&-H^{\nu\rho}{B}_{\rho\mu}+y^{\nu}_{i}x^{i}_{\mu}-\bar{y}^{\nu}_{\bar{\imath}}\bar{x}^{\bar{\imath}}_{\mu}\\\
-H^{\mu\rho}{B}_{\rho\nu}+y^{\nu}_{i}x^{i}_{\mu}-\bar{y}^{\nu}_{\bar{\imath}}\bar{x}^{\bar{\imath}}_{\mu}&&H^{\mu\nu}\,\end{pmatrix}\,,$
(2.19)

which can be factorized as follows [15]

$\displaystyle\mathcal{H}_{MN}=\begin{pmatrix}1&{B}\\\
0&1\end{pmatrix}\begin{pmatrix}K&Z\\\
Z^{T}&H\end{pmatrix}\begin{pmatrix}1&0\\\ -{B}&1\end{pmatrix}$ (2.20)

where we defined

$Z^{\mu}_{\ \nu}\equiv
y^{\mu}_{i}x_{\nu}^{i}-\bar{y}^{\mu}_{\bar{\imath}}\bar{x}_{\nu}^{\bar{\imath}}.$
(2.21)

Finally, with the help of (2.18) one can define curved projectors,

$\displaystyle P_{MN}=\frac{1}{2}\left(\eta_{MN}-{\cal H}_{MN}\right)\,,\ \ \
\ \ \overline{P}_{MN}=\frac{1}{2}\left(\eta_{MN}+{\cal H}_{MN}\right)\ .$
(2.22)

which satisfy

$\displaystyle{\overline{P}}_{{MQ}}{\overline{P}}^{Q}{}_{N}={\overline{P}}_{{MN}}\,,$
$\displaystyle\quad{P}_{{MQ}}{P}^{Q}{}_{N}={P}_{{MN}},$ (2.23)
$\displaystyle{P}_{{MQ}}{\overline{P}}^{Q}{}_{N}={\overline{P}}_{{MQ}}{P}^{Q}{}_{N}=0\,,$
$\displaystyle\quad{\overline{P}}_{{MN}}+{P}_{{MN}}=\eta_{{MN}}\,.$

  3. 3.

Symmetries: DFT is a T-duality invariant formulation, which can describe both
Riemannian and non-Riemannian backgrounds. Consequently, all the fields and
parameters are written in representations of the duality group and duality
invariance is always guaranteed.

We can also define generalized diffeomorphisms. These are infinitesimal
transformations acting on a generic double vector $V^{M}$ through a
generalized Lie derivative, i.e.,

$\displaystyle\delta_{\hat{\xi}}V^{M}={\mathcal{L}}_{\hat{\xi}}V^{M}=\hat{\xi}^{N}\partial_{N}V^{M}+(\partial^{M}\hat{\xi}_{P}-\partial_{P}\hat{\xi}^{M})V^{P}+\omega\partial_{N}\hat{\xi}^{N}V^{M}\,.$
(2.24)

In the previous expression we consider a generic parameter $\hat{\xi}^{M}$ and
a density weight factor $\omega$. The closure of the generalized
diffeomorphism transformations,

$\displaystyle\Big{[}\delta_{\hat{\xi}_{1}},\delta_{\hat{\xi}_{2}}\Big{]}V^{M}{}_{N}=\delta_{\hat{\xi}_{21}}V^{M}{}_{N}\,,$
(2.25)

is provided by the C-bracket,

$\displaystyle\hat{\xi}^{M}_{12}=\hat{\xi}^{P}_{1}\frac{\partial\hat{\xi}^{M}_{2}}{\partial
X^{P}}-\frac{1}{2}\hat{\xi}^{P}_{1}\frac{\partial\hat{\xi}_{2P}}{\partial
X_{M}}-(1\leftrightarrow 2)\,.$ (2.26)

Moreover the closure is satisfied imposing the strong constraint

$\displaystyle\partial_{M}\star\partial^{M}\star$ $\displaystyle=$
$\displaystyle 0\,,$ (2.27) $\displaystyle\partial_{M}(\partial^{M})$
$\displaystyle=$ $\displaystyle 0\,.$ (2.28)

These constraints are solved using $\partial_{M}=(0,\partial_{\mu})$, which
means that the generalized fields and parameters do not depend on the dual
coordinates. On the other hand, the DFT Jacobiator is not trivial (but it is
given by a trivial parameter) and the algebraic structure of DFT is given by
an $L_{\infty}$ algebra with a non-trivial $l_{3}$ product [16], which
measures the failure of the Jacobi identity in the double geometry.

The generalized metric transforms as a tensor with $\omega=0$ with respect to
generalized diffeomorphisms and, when non-Riemannian backgrounds are
considered, one obtains that $H^{\dot{\mu}\dot{\nu}}$ and
$K_{\dot{\mu}\dot{\nu}}$ transforms as tensors under ordinary diffeomorphisms,

$\displaystyle\delta_{\xi}H^{\dot{\mu}\dot{\nu}}=L_{\xi}H^{\dot{\mu}\dot{\nu}},\quad\delta_{\xi}K_{\dot{\mu}\dot{\nu}}=L_{\xi}K_{\dot{\mu}\dot{\nu}}\,.$
(2.29)

As usually the B-field receives a gauge transformation,

$\displaystyle\delta
B_{\dot{\mu}\dot{\nu}}=L_{\xi}B_{\dot{\mu}\dot{\nu}}+2\partial_{[\dot{\mu}}\zeta_{\dot{\nu}]}\,,$
(2.30)

where $\xi_{M}=(\zeta_{\dot{\mu}},\xi^{\dot{\mu}})$.

The generalized metric (2.19) is invariant under $GL(n)\times GL(\bar{n})$
rotations

$\displaystyle\left(x^{i}_{\mu},y^{\mu}_{i},\bar{x}^{\bar{\imath}}_{\mu},\bar{y}^{\nu}_{\bar{\imath}}\right)\rightarrow\left(x^{j}_{\mu}R^{i}_{j},\left(R^{-1}\right)_{i}^{\hphantom{i}j}y^{\nu}_{j},\bar{x}^{\bar{\jmath}}_{\mu}\bar{R}_{\bar{\jmath}}^{\hphantom{\bar{\jmath}}\bar{\imath}},\left(\bar{R}^{-1}\right)_{\bar{\imath}}^{\hphantom{\bar{\imath}}\bar{\jmath}}\bar{y}^{\nu}_{j}\right)\,,$
(2.31)

and under the generalized shift symmetry

$\displaystyle\begin{split}\left(y^{\mu}_{i}\right)^{\prime}&=y_{i}^{\mu}+V^{\mu}_{i}\\\
\left(\bar{y}^{\mu}_{\bar{\imath}}\right)^{\prime}&=\bar{y}_{\bar{\imath}}^{\mu}+\bar{V}^{\mu}_{\bar{\imath}}\\\
\left(K_{\mu\nu}\right)^{\prime}&=K_{\mu\nu}-2x^{i}_{(\mu}K_{\nu)\rho}V^{\rho}_{i}-2\bar{x}^{\bar{\imath}}_{(\mu}K_{\nu)\rho}\bar{V}^{\rho}_{\bar{\imath}}+\left(x_{\mu}^{i}V_{\rho
i}+\bar{x}^{\bar{\imath}}_{\mu}\bar{V}_{\rho\bar{\imath}}\right)\left(x^{i}_{\nu}V^{\rho}_{i}+\bar{x}^{\bar{\imath}}_{\nu}\bar{V}^{\rho}_{\bar{\imath}}\right)\\\
\left({B}_{\mu\nu}\right)^{\prime}&={B}_{\mu\nu}-2x^{i}_{[\mu}V_{\nu]i}+2\bar{x}^{\bar{\imath}}_{[\mu}\bar{V}_{\nu]\bar{\imath}}+2x^{i}_{[\mu}\bar{x}^{\bar{\jmath}}_{\nu]}\left(y_{i}^{\rho}\bar{V}_{\rho\bar{\jmath}}+\bar{y}_{\bar{\jmath}}^{\rho}V_{\rho
i}+V_{\rho i}\bar{V}^{\rho}_{\bar{\jmath}}\right)\end{split}$

with $V_{\mu i}$ and $\bar{V}_{\mu\bar{\imath}}$ being the transformation
parameters and we defined $V^{\mu}_{i}\equiv H^{\mu\rho}V_{\rho
i},\bar{V}^{\mu}_{\bar{\imath}}\equiv H^{\mu\rho}\bar{V}_{\rho\bar{\imath}}$.

Another symmetry of DFT are double Lorentz transformations, which act as

$\displaystyle\delta_{\Lambda}V^{A}=V^{B}\Lambda_{B}{}^{A}\,,$ (2.32)

on an arbitrary flat vector $V^{A}$. Demanding $\delta_{\Lambda}\eta_{AB}=0$
we have $\Lambda_{AB}=-\Lambda_{BA}$. Moreover, using the
decomposition/notation (2.6), the condition $\delta_{\Lambda}{\cal H}_{AB}=0$
implies

$\displaystyle\Lambda_{\underline{A}\overline{B}}=\Lambda_{\overline{A}\underline{B}}=0\,.$
(2.33)

The generalized dilaton is a double Lorentz invariant, and its transformation
under generalized diffeomorphisms is not covariant. However $e^{-2d}$
transforms as a generalized scalar density with $\omega=1$. The generalized
frame transforms as a vector under double Lorentz transformations, and
therefore,

$\displaystyle\delta_{\Lambda}k_{\dot{\mu}\underline{\dot{a}}}=k_{\dot{\mu}\underline{\dot{b}}}\Lambda^{\underline{\dot{b}}}{}_{\underline{{\dot{a}}}},\quad\delta_{\Lambda}\bar{k}_{\dot{\mu}\overline{\dot{a}}}=\bar{k}_{\dot{\mu}\overline{\dot{b}}}\Lambda^{\overline{\dot{b}}}{}_{\overline{\dot{a}}}\,,$
(2.34)
$\displaystyle\delta_{\Lambda}h^{\dot{\mu}}{}_{\underline{\dot{a}}}=h^{\dot{\mu}}{}_{\underline{\dot{b}}}\Lambda^{\underline{\dot{b}}}{}_{\underline{{\dot{a}}}},\quad\delta_{\Lambda}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}=\bar{h}^{\dot{\mu}}{}_{\overline{\dot{b}}}\Lambda^{\overline{\dot{b}}}{}_{\overline{\dot{a}}}\,.$
(2.35)

The previous transformations will define a bigravity structure since we will
use both $(k,\bar{K})$ and $(h,\bar{h})$ to create the same non-relativistic
degrees of freedom. A similar scenario, but for a relativistic bigravity
theory coming from string theory was studied in [17].

Acting on a generic vector the Lorentz derivative is defined as

$\nabla_{A}V_{B}=E_{A}V_{B}+\omega_{AB}{}^{C}V_{C}\,$ (2.36)

where $E_{A}\equiv\sqrt{2}{E}_{A}{}^{M}{}\partial_{M}$ and $\omega_{AB}{}^{C}$
is a spin connection which satisfies

$\omega_{ABC}=-\omega_{ACB}\,\qquad\mathrm{and}\qquad\omega_{A\overline{B}\underline{C}}=\omega_{A\underline{B}\overline{C}}=0\,.$
(2.37)

Only the totally antisymmetric and trace parts of $\omega_{ABC}$ can be
determined in terms of the fundamental fields of the theory, namely

$\displaystyle\omega_{[{ABC}]}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}{F}_{{ABC}}\,,$ (2.38)
$\displaystyle\omega_{{BA}}{}^{B}$ $\displaystyle=$
$\displaystyle-\sqrt{2}e^{2d}\partial_{M}\left(E^{M}{}_{A}e^{-2d}\right)\equiv-{F}_{A}\,,$
(2.39)

the latter arising from partial integration with the dilaton density

$\int e^{-2d}V\nabla_{A}V^{A}=-\int e^{-2d}V^{A}\nabla_{A}V\,,$ (2.40)

for arbitrary $V$ and $V^{A}$. The relevant components of the generalized
fluxes for computing the bosonic and fermionic heterotic DFT action are

$\displaystyle\Big{\\{}F_{\underline{A}},F_{\underline{A}\underline{B}\underline{C}},F_{\overline{A}\underline{B}\underline{C}},F_{\underline{A}\overline{B}\overline{C}}\Big{\\}}\,.$
(2.41)

The parametrization of the useful components for describing the bosonic sector
of the theory is given by

$\displaystyle F_{\overline{\dot{a}}\underline{{\dot{b}}{\dot{c}}}}$
$\displaystyle=-\frac{1}{2}(\bar{h}^{\dot{\nu}}{}_{\overline{\dot{a}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}k_{\dot{\mu}\underline{\dot{c}}}+\bar{h}^{\dot{\nu}}{}_{\overline{\dot{a}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{b}}}h^{\dot{\mu}}{}_{\underline{\dot{c}}})+h^{\dot{\nu}}{}_{[\underline{\dot{b}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{c}}]}\bar{k}_{\dot{\mu}\overline{\dot{a}}}$
$\displaystyle\quad-h^{\dot{\nu}}{}_{[\underline{\dot{b}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{c}}]}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}-\frac{1}{2}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}h^{\dot{\nu}}{}_{\underline{\dot{b}}}h^{\dot{\rho}}{}_{\underline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}}\,,$
(2.42a) $\displaystyle F_{\underline{\dot{a}}\overline{{\dot{b}}{\dot{c}}}}$
$\displaystyle=\frac{1}{2}(h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{b}}}\bar{k}_{\dot{\mu}\overline{\dot{c}}}+h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}\bar{k}_{\dot{\mu}\overline{\dot{b}}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{c}}})-\bar{h}^{\dot{\nu}}{}_{[\overline{\dot{b}}}\partial_{\dot{\nu}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{c}}]}k_{\dot{\mu}\underline{\dot{a}}}$
$\displaystyle\quad+\bar{h}^{\dot{\nu}}{}_{[\overline{\dot{b}}}\partial_{\dot{\nu}}\bar{k}_{\dot{\mu}\overline{\dot{c}}]}h^{\dot{\mu}}{}_{\underline{\dot{a}}}-\frac{1}{2}h^{\dot{\mu}}{}_{\underline{\dot{a}}}\bar{h}^{\dot{\nu}}{}_{\overline{\dot{b}}}\bar{h}^{\dot{\rho}}{}_{\overline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}}\,,$
(2.42b) $\displaystyle F_{\underline{{\dot{a}}{\dot{b}}{\dot{c}}}}$
$\displaystyle=-\frac{3}{2}(h^{\dot{\nu}}{}_{[\underline{\dot{a}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}k_{\dot{\mu}\underline{\dot{c}}]}+h^{\dot{\nu}}{}_{[\underline{\dot{a}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{b}}}h^{\dot{\mu}}{}_{\underline{\dot{c}}]})-\frac{1}{2}h^{\dot{\mu}}{}_{\underline{\dot{a}}}h^{\dot{\nu}}{}_{\dot{\underline{b}}}h^{\dot{\rho}}{}_{\underline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}}\,,$
(2.42c) $\displaystyle F_{\underline{\dot{a}}}$
$\displaystyle=\partial_{\dot{\mu}}h^{\dot{\mu}}{}_{\underline{\dot{a}}}-2\partial_{\underline{\dot{a}}}(\phi-\frac{1}{2}ln(e))\,,$
(2.42d)

where the H-flux is given by
$H_{\dot{\mu}\dot{\nu}\dot{\rho}}=3\partial_{[\dot{\mu}}B_{\dot{\nu}\dot{\rho}]}$.

  4. 4.

Action principle:

The DFT action is given by

$\displaystyle\int d^{2D}Xe^{-2d}{\cal R}\,,$ (2.43)

where ${\cal R}(E,d)$ is a two derivative scalar under generalized
diffeomorphisms and it is invariant under Lorentz transformations. This object
is known as the generalized Ricci scalar, and can be written in terms of the
generalized fluxes [8],

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle
2E_{\underline{A}}F^{\underline{A}}+F_{\underline{A}}F^{\underline{A}}-\frac{1}{6}{F}_{\underline{ABC}}F^{\underline{ABC}}-\frac{1}{2}{F}_{\overline{A}\underline{BC}}F^{\overline{A}\underline{BC}}\,,$
(2.44)

The full parametrization of the DFT action in terms of $H^{\dot{\mu}\nu}$,
$K_{\dot{\mu}\dot{\nu}}$, their kernels, $B_{\dot{\mu}\dot{\nu}}$ and $\phi$
has been detailed study in [10] for torsional Newton-Cartan theory, Carrollian
theory and string Newton-Cartan theory. The analysis was performed considering
the generalized metric formalism of DFT, which is equivalent to the flux
formalism. In this work we will compute the extra bosonic terms that appear in
the heterotic extension of this theory, as well as the fermionic contributions
to leading order in fermions.

In this work we are focused in obtaining the minimal ${\cal N}=1$
supersymmetric extension for a general non-Riemannian DFT which means that our
goul is to obtain a minimal $D=10$ supergravity defined on a non-Riemannian
background. However, in principle, the inclusion of a Yang-Mills field in the
parametrization of the generalized frame it is possible, for example,

$E^{M}{}_{A}\ =\
\frac{1}{\sqrt{2}}\left(\begin{matrix}-k_{\dot{\mu}\underline{a}}-C_{\dot{\rho}\dot{\mu}}{h}^{\dot{\rho}}{}_{\underline{\dot{a}}}&h^{\dot{\mu}}{}_{\underline{\dot{a}}}&-A_{\dot{\rho}}{}^{\alpha}{h}^{\dot{\rho}}{}_{\underline{\dot{a}}}\,,\\\
\bar{k}_{\dot{\mu}\overline{\dot{a}}}-C_{\dot{\rho}\dot{\mu}}{}\bar{h}^{\dot{\rho}}{}_{\overline{\dot{a}}}&\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}&-A_{\dot{\rho}}{}^{\alpha}\bar{h}^{\dot{\rho}}{}_{\overline{\dot{a}}}\\\
\sqrt{2}A_{\dot{\mu}\beta}e^{\beta}{}_{\overline{\alpha}}&0&\sqrt{2}e^{\alpha}{}_{\overline{\alpha}}\end{matrix}\right)\,,$
(2.45)

where
$C_{\dot{\mu}\dot{\nu}}=B_{\dot{\mu}\dot{\nu}}+\frac{1}{2}A_{\dot{\mu}}{}^{\alpha}\kappa_{\alpha\beta}A_{\dot{\nu}}{}^{\beta}$
mimicking [18] and $\kappa^{\alpha\beta}$ is a Cartan-Killing (inverse) metric
which satisfies

$\displaystyle
e^{\alpha}{}_{\overline{\alpha}}\kappa^{\overline{\alpha}\overline{\beta}}e^{\beta}{}_{\overline{\beta}}=\kappa^{\alpha\beta}\,.$
(2.46)

As usual the indices $\overline{A}$ have to decomposed as
$=(\overline{a},\overline{\alpha})$ and we have to use
$e^{\alpha}{}_{\overline{\alpha}}$ as a constant object to relate the Cartan-
Killing metrics. Using this enlargement of the global symmetry from $O(D,D)$
to $O(D,D+n)$ with $n=496$ and considering the previous generalize frame it is
now possible to construct the generalized metric for Heterotic DFT in non-
Riemannian backgrounds as

$\displaystyle{\cal H}^{\dot{\mu}\dot{\nu}}$ $\displaystyle=$ $\displaystyle
H^{\dot{\mu}\dot{\nu}}$ $\displaystyle{\cal H}^{\dot{\mu}}{}_{\dot{\nu}}$
$\displaystyle=$ $\displaystyle-
C_{\dot{\rho}\dot{\nu}}H^{\dot{\rho}\dot{\mu}}-x_{\dot{\nu}}y^{\dot{\mu}}+\bar{x}_{\dot{\nu}}\bar{y}^{\dot{\mu}}$
$\displaystyle{\cal H}^{\dot{\mu}}{}_{\alpha}$ $\displaystyle=$
$\displaystyle-H^{\dot{\mu}\dot{\rho}}A_{\dot{\rho}\alpha}$
$\displaystyle{\cal H}_{\dot{\mu}\dot{\nu}}$ $\displaystyle=$ $\displaystyle
K_{\dot{\mu}\dot{\nu}}-C_{\dot{\mu}\dot{\sigma}}H^{\dot{\sigma}\dot{\rho}}C_{\dot{\rho}\dot{\nu}}-2x_{(\dot{\mu}}C_{\dot{\nu})\dot{\rho}}y^{\dot{\rho}}+2\bar{x}_{(\dot{\mu}}C_{\dot{\nu})\dot{\rho}}\bar{y}^{\dot{\rho}}-A_{\dot{\mu}\alpha}A_{\dot{\nu}}{}^{\alpha}$
$\displaystyle{\cal H}_{\dot{\mu}\alpha}$ $\displaystyle=$ $\displaystyle
A_{\dot{\rho}\alpha}(C_{\dot{\sigma}\dot{\mu}}H^{\dot{\sigma}\dot{\rho}}+\delta_{\dot{\mu}}^{\dot{\rho}}+x_{\dot{\mu}}y^{\dot{\rho}}-\bar{x}_{\dot{\mu}}\bar{y}^{\dot{\rho}})$
$\displaystyle{\cal H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa_{\alpha\beta}+A_{\dot{\rho}\alpha}H^{\dot{\rho}\dot{\sigma}}A_{\dot{\sigma}\beta}\,.$
(2.47)

This metric can be used 333See also [19] for formulations with explicit double
Yang Mills fields in the double geometry. to include Yang-Mills contributions
to the bosonic sector of the theory, while the generalized frame given in
(2.46) is the one to construct the full heterotic version of this model. In
this work we will keep $A_{\mu}=0$ for simplicity.

##  3 ${\cal N}=1$ supersymmetric extension for a general non-Riemannian
Double Field Theory

###  3.1 General construction

The ${\cal N}=1$ supersymmetric extension of DFT is achieved by fixing the
dimension of the target space ($D=10$) and adding generalized spinor fields
that act as supersymmetric partners of the bosonic fields: the generalized
gravitino $\Psi_{\overline{A}}$ and the generalized dilatino $\rho$.

The covariant derivative of spinor fields acquires an additional term in order
to derive the spinor indices. For instance,

$\displaystyle\nabla_{A}\Psi_{\overline{B}}$ $\displaystyle=$ $\displaystyle
E_{A}\Psi_{\overline{B}}+\omega_{A\overline{B}}{}^{\overline{C}}\Psi_{\overline{C}}-\frac{1}{4}\omega_{A\underline{BC}}\gamma^{\underline{BC}}\Psi_{\overline{B}}\,,$
$\displaystyle\nabla_{A}\rho$ $\displaystyle=$ $\displaystyle
E_{A}\rho-\frac{1}{4}\omega_{A\underline{BC}}\gamma^{\underline{BC}}\rho\,.$
(3.1)

The gamma matrices satisfy a Clifford algebra,

$\left\\{\gamma^{\underline{A}},\gamma^{\underline{B}}\right\\}=-2P^{\underline{AB}}\
,$ (3.2)

and we use the standard convention for antisymmetrization of $\gamma$-matrices
$\gamma^{{\underline{A}\dots\underline{B}}}=\gamma^{[\underline{A}}\dots\gamma^{\underline{B}]}$.

The generalized supersymmetry transformations of the fundamental fields are
parameterized by an infinitesimal Majorana fermion $\epsilon$, that is a
spinor of $O(1,9)_{L}$,

$\displaystyle\delta E^{M}{}_{\underline{A}}{}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\overline{\epsilon}\gamma_{\underline{A}}\Psi^{\overline{B}}E^{M}{}_{\overline{B}}\,,$
(3.3) $\displaystyle\delta E^{M}{}_{\overline{A}}{}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\overline{\epsilon}\gamma^{\underline{B}}\Psi_{\overline{A}}E^{M}{}_{\underline{B}}{}\,,$
(3.4) $\displaystyle\delta d$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\overline{\epsilon}\rho\,,$ (3.5)

from where we read

$\displaystyle\delta h^{\dot{\mu}\underline{\dot{a}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{\epsilon}\gamma^{\underline{\dot{a}}}\psi^{\overline{\dot{b}}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{b}}}\,,\quad\delta\bar{h}^{\dot{\mu}\overline{\dot{a}}}=\frac{1}{2}\bar{\epsilon}\gamma^{\underline{\dot{b}}}\psi^{\overline{\dot{a}}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}$
(3.6) $\displaystyle\delta k_{\dot{\mu}\underline{\dot{a}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{\epsilon}\gamma^{\underline{\dot{c}}}k_{\dot{\mu}\underline{\dot{c}}}\psi^{\overline{\dot{b}}}e_{\underline{\dot{a}}\overline{\dot{b}}}\quad\delta\bar{k}_{\dot{\mu}\underline{\dot{a}}}=\frac{1}{2}\bar{\epsilon}\gamma^{\underline{\dot{a}}}e_{\overline{\dot{a}}\underline{\dot{a}}}\psi^{\overline{\dot{b}}}\bar{k}_{\dot{\mu}\overline{\dot{b}}}\,,$
(3.7)

with
$e_{\underline{\dot{a}}\overline{\dot{b}}}=-e_{\overline{\dot{b}}\underline{\dot{a}}}=h^{\dot{\mu}}{}_{\underline{\dot{a}}}\bar{k}_{\dot{\mu}\overline{\dot{b}}}$.
Interestingly enough, $k_{\dot{\mu}\dot{a}}$ and $\bar{k}_{\dot{\mu}\dot{a}}$
cannot be identified using a gauge fixing procedure unless
$x_{\dot{\mu}}=\bar{x}_{\dot{\mu}}=y^{\dot{\mu}}=\bar{y}^{\dot{\mu}}=0$, and
the same happens with $h^{\dot{\mu}}{}_{\underline{\dot{a}}}$ and
$\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}$ and the double Lorentz
parameters. This is a very interesting point of the construction since,
typically, the ${\cal N}=1$ supersymmetric DFT requires that one of the double
parameters is related to the other up to fermionic contributions (gauge fixing
condition). In these kind of (non-Riemannian) geometries, some of the
supersymmetric transformations depend on the products
$k_{\dot{\mu}\underline{\dot{a}}}\overline{h}^{\dot{\mu}}{}_{\overline{\dot{b}}}$
or
$\overline{k}_{\dot{\mu}\overline{\dot{a}}}\underline{h}^{\dot{\mu}}{}_{\underline{\dot{b}}}$
which are typically Minkowski metrics in Riemannian geometries due to the
identifications of $h\leftrightarrow\bar{h}$ and $k\leftrightarrow\bar{k}$ in
those cases. Although this is not a problem for the minimal supergravity
model, the inclusion of the Yang-Mills sector coming from a Heterotic DFT
could be not straightforward. We will return to this point in the final
section.

On the other hand, the supersymmetric transformation for the B-field is given
by

$\displaystyle\delta
B_{\dot{\mu}\dot{\nu}}=\bar{\epsilon}\gamma^{\underline{\dot{a}}}k_{[\dot{\mu}\underline{\dot{a}}}\psi^{\overline{\dot{b}}}\bar{k}_{\dot{\nu}]\overline{\dot{b}}}\,,$
(3.8)

while the transformations for the fermionic degrees of freedom take the usual
form,

$\displaystyle\delta\Psi_{\overline{\dot{a}}}$ $\displaystyle=$
$\displaystyle\nabla_{\overline{\dot{a}}}\epsilon\,,\quad\delta\rho=-\gamma^{\underline{\dot{a}}}\nabla_{\underline{\dot{a}}}\epsilon\,.$
(3.9)

The symmetry transformation rules leave the following action invariant,

$\displaystyle S_{\mathcal{N}=1\ \textrm{DFT}}$ $\displaystyle=$
$\displaystyle\int d^{20}Xe^{-2d}\Big{(}{\cal
R}+\overline{\Psi}^{\overline{A}}\gamma^{\underline{B}}\nabla_{\underline{B}}\Psi_{\overline{A}}-\overline{\rho}\gamma^{\underline{A}}\nabla_{\underline{A}}\rho+2\overline{\Psi}^{\overline{A}}\nabla_{\overline{A}}\rho\Big{)}\,,$
(3.10)

where ${\cal R}={\cal R}(H,K,B,e,x,y)$ and $e^{-2d}=e$. The fermionic
Lagrangian of the previous expression can be written in terms of the different
components of (2.41) as

$\displaystyle L_{\psi\psi}$ $\displaystyle=$
$\displaystyle\bar{\Psi}^{\overline{\dot{a}}}\gamma^{\underline{\dot{b}}}E_{\underline{\dot{b}}}\Psi_{\overline{\dot{a}}}-\bar{\Psi}^{\overline{\dot{b}}}\gamma^{\underline{\dot{a}}}(\frac{1}{2}(h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{b}}}\bar{k}_{\dot{\mu}\overline{\dot{c}}}+h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}\bar{k}_{\dot{\mu}\overline{\dot{b}}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{c}}})-\bar{h}^{\dot{\nu}}{}_{\overline{\dot{b}}}\partial_{\dot{\nu}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{c}}}k_{\dot{\mu}\underline{\dot{a}}}$
(3.11)
$\displaystyle+\bar{h}^{\dot{\nu}}{}_{\overline{\dot{b}}}\partial_{\dot{\nu}}\bar{k}_{\dot{\mu}\overline{\dot{c}}}h^{\dot{\mu}}{}_{\underline{\dot{a}}}-\frac{1}{2}h^{\dot{\mu}}{}_{\underline{\dot{a}}}\bar{h}^{\dot{\nu}}{}_{\overline{\dot{b}}}\bar{h}^{\dot{\rho}}{}_{\overline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}})\Psi^{\overline{\dot{c}}}$
$\displaystyle-\bar{\Psi}^{\overline{\dot{d}}}\gamma^{\underline{\dot{a}}\underline{\dot{b}}\underline{\dot{c}}}(\frac{1}{8}(h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}k_{\dot{\mu}\underline{\dot{c}}}+h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{b}}}h^{\dot{\mu}}{}_{\underline{\dot{c}}})+\frac{1}{24}h^{\dot{\mu}}{}_{\underline{\dot{a}}}h^{\dot{\nu}}{}_{\dot{\underline{b}}}h^{\dot{\rho}}{}_{\underline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}})\Psi_{\overline{\dot{d}}}$
$\displaystyle+\bar{\Psi}^{\overline{\dot{a}}}\gamma^{\underline{\dot{c}}}(\frac{1}{2}\partial_{\dot{\mu}}h^{\dot{\mu}}{}_{\underline{\dot{c}}}-\partial_{\underline{\dot{c}}}(\phi-\frac{1}{2}ln(e)))\Psi_{\overline{\dot{a}}}\,,$
$\displaystyle L_{\psi\rho}$ $\displaystyle=$ $\displaystyle
2\bar{\Psi}^{\overline{\dot{a}}}E_{\overline{\dot{a}}}\rho+\frac{1}{2}\bar{\Psi}^{\overline{\dot{a}}}\Big{[}-\frac{1}{2}(\bar{h}^{\dot{\nu}}{}_{\overline{\dot{a}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}k_{\dot{\mu}\underline{\dot{c}}}+\bar{h}^{\dot{\nu}}{}_{\overline{\dot{a}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{b}}}h^{\dot{\mu}}{}_{\underline{\dot{c}}})$
(3.12)
$\displaystyle+h^{\dot{\nu}}{}_{\underline{\dot{b}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{c}}}\bar{k}_{\dot{\mu}\overline{\dot{a}}}-h^{\dot{\nu}}{}_{\underline{\dot{b}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{c}}}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}-\frac{1}{2}\bar{h}^{\dot{\mu}}{}_{\overline{\dot{a}}}h^{\dot{\nu}}{}_{\underline{\dot{b}}}h^{\dot{\rho}}{}_{\underline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}}\Big{]}\gamma^{\underline{\dot{b}}\underline{\dot{c}}}\rho\,,$
$\displaystyle L_{\rho\rho}$ $\displaystyle=$
$\displaystyle-\bar{\rho}\gamma^{\underline{\dot{a}}}E_{\underline{\dot{a}}}\rho+\bar{\rho}\gamma^{\underline{\dot{a}}\underline{\dot{b}}\underline{\dot{c}}}(\frac{1}{8}(h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}k_{\dot{\mu}\underline{\dot{c}}}+h^{\dot{\nu}}{}_{\underline{\dot{a}}}\partial_{\dot{\nu}}k_{\dot{\mu}\underline{\dot{b}}}h^{\dot{\mu}}{}_{\underline{\dot{c}}})+\frac{1}{24}h^{\dot{\mu}}{}_{\underline{\dot{a}}}h^{\dot{\nu}}{}_{\dot{\underline{b}}}h^{\dot{\rho}}{}_{\underline{\dot{c}}}H_{\dot{\mu}\dot{\nu}\dot{\rho}})\rho$
(3.13)
$\displaystyle-\bar{\rho}\gamma^{\underline{\dot{b}}}(\frac{1}{2}\partial_{\dot{\mu}}h^{\dot{\mu}}{}_{\underline{\dot{b}}}-\partial_{\underline{\dot{b}}}(\phi-\frac{1}{2}ln(e)))\rho\,.$

So far this is the most general form of the leading order in fermions
Lagrangian coming from the non-relativistic parametrization of ${\cal N}=1$
DFT. In here we are focusing in the minimal D=10 supergravity model which
means that we are not taking into account the contributions coming from the
gauge field $A$ and its supersymmetric partner the gaugino. While the former
should be encoded through the generalized frame the latter should be part of
the generalized gravitino as happens in the Riemannian cases. In the next
subsection we will show the ${\cal N}=1$ supersymmetric extension of type I
TNC geometry, which corresponds to a $(n,\bar{n})=(1,1)$ non-riemannian DFT
using the classification given in [3].

###  3.2 ${\cal N}=1$ supersymmetric type I TNC geometry

TNC geometries are characterized by local Galilean symmetry. When coming from
the flux formulation of DFT we need to think about this theory as a bymetric
gravity with a double tangent space structure. This is not the case in [10]
since the generalized metric formulation is enough to describe the bosonic
sector of the theory.

The non-relativistic spacetime is embedded using a pair of spatial frames
$k^{\underline{a}}_{\mu}$ and $k^{\overline{a}}_{\mu}$ used to define a
spatial (or transverse) metric

$\displaystyle h_{\mu\nu}$
$\displaystyle=k_{\mu}{}_{\underline{a}}k_{\nu}{}^{\underline{a}}=\overline{k}_{\mu}{}_{\overline{a}}\overline{k}_{\nu}{}^{\overline{a}}\,,$
(3.14)

and a temporal frame $\tau_{\mu}$ used as a universal one-form clock. We also
introduce a one-form masses $m_{\mu}$ in order to have the following bosonic
symmetry transformations,

$\displaystyle\begin{split}\delta\overline{k}_{\mu}{}^{\overline{a}}&=\mathcal{L}_{\xi}\overline{k}_{\mu}{}^{\overline{a}}+\Lambda^{\overline{a}}\tau_{\mu}+\Lambda^{\overline{a}}{}_{\overline{b}}\overline{k}_{\mu}{}^{\overline{b}}\,,\\\
\delta\tau_{\mu}&=\mathcal{L}_{\xi}\tau_{\mu}\,,\\\ \delta
m_{\mu}&=\mathcal{L}_{\xi}m_{\mu}+\Lambda_{\underline{a}}k_{\mu}{}^{\underline{a}}+\partial_{\mu}\sigma\,,\end{split}$
(3.15)

with $\Lambda^{\underline{a}}$ and $\Lambda^{\overline{a}}$ local Galilean
boost parameters which satisfy
$\Lambda^{\underline{a}}k_{\mu\underline{a}}=\Lambda^{\overline{a}}\overline{k}_{\mu\overline{a}}$
and $\sigma$ is a $U(1)$ gauge transformation parameter. The square matrices
$(\tau_{\mu},k_{\mu}{}^{\underline{a}})$ and
$(\tau_{\mu},\overline{k}_{\mu}{}^{\overline{a}})$ have inverses
$(-\upsilon^{\mu},h^{\mu}{}_{\underline{a}})$ and
$(-\upsilon^{\mu},\overline{h}^{\mu}{}_{\overline{a}})$ where these inverse
frames are orthogonal

$\upsilon^{\mu}\tau_{\mu}=-1\,,\qquad
k^{\mu}{}_{\underline{a}}\tau_{\mu}=\overline{k}^{\mu}{}_{\overline{a}}\tau_{\mu}=0\,,\qquad
h_{\mu}{}^{\underline{a}}\upsilon^{\mu}=\overline{h}_{\mu}{}^{\overline{a}}\upsilon^{\mu}=0\,,$
(3.16)

and complete

$k_{\nu}{}^{\underline{a}}k^{\mu}{}_{\underline{a}}-\upsilon^{\mu}\tau_{\nu}=\delta^{\mu}_{\nu}\,,\quad\overline{k}_{\nu}{}^{\overline{a}}\overline{k}^{\mu}{}_{\overline{a}}-\upsilon^{\mu}\tau_{\nu}=\delta^{\mu}_{\nu}$
(3.17)

Making use of inverse spatial frames $h^{\mu}{}_{\underline{a}}$ and
$\overline{h}^{\mu}{}_{\overline{a}}$ one can define an inverse spatial metric
as

$\displaystyle h^{\mu\nu}$
$\displaystyle=h^{\mu}{}_{\underline{a}}h^{\nu}{}_{\underline{a}}=\overline{h}^{\mu}{}_{\overline{a}}\overline{h}^{\nu}{}_{\overline{a}}\,.$
(3.18)

From the DFT generalized frame we obtain the following relations,

$\displaystyle-k_{\mu}{}^{\underline{a}}k_{\nu\underline{a}}$ $\displaystyle=$
$\displaystyle\overline{k}_{\mu}{}^{\overline{a}}\overline{k}_{\nu\overline{a}}=h_{\mu\nu}$
(3.19) $\displaystyle-h^{\mu}{}_{\underline{a}}h^{\nu\underline{a}}$
$\displaystyle=$
$\displaystyle\overline{h}^{\mu}{}_{\overline{a}}\overline{h}^{\nu\overline{a}}=h^{\mu\nu}$
(3.20) $\displaystyle-h^{\mu}{}_{\underline{a}}h^{\underline{a}}$
$\displaystyle=$
$\displaystyle\overline{h}^{\mu}{}_{\overline{a}}\overline{h}^{\overline{a}}=h^{\mu\rho}\aleph_{\rho}\,,$
(3.21)

and we set $e^{-2d}=e$, while the B-field and the kernels are given by

$\displaystyle{B}_{\dot{\mu}\dot{\nu}}$
$\displaystyle=\begin{pmatrix}\bar{B}_{\mu\nu}&&-m_{\mu}\\\
m_{\nu}&&0\end{pmatrix}$ (3.22)

and

$\displaystyle x_{\dot{\mu}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\tau_{\mu}-\aleph_{\mu}\\\
1\end{pmatrix},$ $\displaystyle\bar{x}_{\dot{\mu}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\tau_{\mu}+\aleph_{\mu}\\\
-1\end{pmatrix},$ $\displaystyle y^{\dot{\mu}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}-\upsilon^{\mu}\\\
1-\upsilon^{\mu}\aleph_{\mu}\end{pmatrix},$ $\displaystyle\bar{y}^{\dot{\mu}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}-\upsilon^{\mu}\\\
-1-\upsilon^{\mu}\aleph_{\mu}\end{pmatrix}.$ (3.23)

We also define

$b_{\mu\nu}\equiv\partial_{[\mu}\aleph_{\nu]},\qquad\qquad\mathfrak{e}_{\mu}\equiv\hat{\upsilon}^{\rho}b_{\rho\mu}\,.$
(3.24)

In order to read the sypersymmetric extension to the transformations rules we
decompose the generalized gravitino in the following way

$\displaystyle\Psi_{\overline{A}}=\Psi_{\overline{\dot{a}}}=(\Psi_{\overline{a}},\psi,\tilde{\psi})\,,$
(3.25)

while the DFT gamma matrices as
$\gamma_{\underline{\dot{a}}}=(\gamma_{\underline{a}},1,i)$. Furthermore, we
keep $\rho$ as the $10$-dimensional dilatino field. The supersymmetric
extension for the transformations is therefore given by

$\displaystyle\delta\overline{h}^{\mu}{}_{\overline{a}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\epsilon}\gamma^{\nu}\psi_{\overline{a}}(\delta^{\mu}_{\nu}-\aleph_{\nu}v^{\mu})$
(3.26) $\displaystyle\delta v^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bar{\epsilon}\psi^{\nu}(\delta^{\mu}_{\nu}+\aleph_{\nu}v^{\mu})\,$
(3.27)

while the universal clock transforms as

$\displaystyle\delta\tau_{\mu}=\frac{1}{\sqrt{2}}\tau_{\mu}\bar{\epsilon}\psi^{\nu}(\tau_{\nu}-\aleph_{\nu})\,.$
(3.28)

Finally the transformation of the B-field is given by

$\displaystyle\delta
B_{\mu\nu}=\bar{\epsilon}\gamma^{\underline{\dot{a}}}k_{[\mu\underline{\dot{a}}}\psi^{\overline{\dot{b}}}\bar{k}_{\nu]\overline{\dot{b}}}\,.$
(3.29)

The present model constitutes a convenient example to show how the ${\cal
N}=1$ supersymmetric extension of the non-Riemannian DFT works. Since this
geometry is related to Carrollian geometry and string Newton Cartan throught
particular T-duality rotations or a null reduction/uplift, the formalism here
introduced applies also in that scenarios. In the next part of the work we
discuss about potential uses of the present model.

##  4 Future directions

In this work we have explored non-Riemannian geometries using the flux
formalism of DFT and we include fermionic degrees of freedom in order to
construct a ${\cal N}=1$ supersymmetric invariant formulation. From the
supersymmetric transformations rules we have extract the transformations for
the non-Riemannian parametrization of the generalized frame. We have
constructed the generalized fluxes required for the bosonic and fermionic
action and we gave the most general form of the action in order to capture the
supersymmetric extension of the non-Riemannian geometries with $n=\bar{n}$. As
an example we show the supersymmetric extension for type I TNC, which
corresponds to a particular case where $n=\bar{n}=1$. We finish this work with
some possible continuations for this project

  * $-$

Here we have explored the exact form of the ${\cal N}=1$ supersymmetric DFT
compatible with non-Riemannian exact backgrounds. It would be interesting to
consider perturbations around these generic backgrounds starting from the DFT
framework, i.e., to perturbe the generalized frame, dilaton, gravitino and
dilatino (the last two with possible applications to fermionic condensates).
As a simple model it would be nice to perturbe the aforementioned DFT
background fields using the generalized Kerr-Schild ansatz which is a linear
perturbation in the generalized frame using a pair of generalized null vectors
[20]. Using this model one would be able to extract the perturbative form of
the degrees of freedom for the non-Riemannian supergravity and explore notions
of single/zeroth copy for these geometries.

  * $-$

We restrict our supersymmetric extension to minimal non-Riemannian
supergravity models which can be obtained from DFT. The inclusion of the Yang-
Mills sector is another possible continuation, which is straightforward for
type I TNC since in this work we gave the prescription for the generalized
frame. However, in more general scenarios, like type II TNC, it would be
necessary to generalized the method given in [21], take the relativistic limit
in terms DFT fields, and then to take advantage of the dependence of the
generalized frame on the gauge field (and the generalized gravitino on the
gaugino) in order to read the vectorial part of the model. It would be very
interesting to recover this gauge sector of heterotic supergravity defined on
a non-Riemannian geometry from a DFT model.

  * $-$

The inclusion of fermionic degrees of freedom for non-Riemannian geometries
coming from a DFT formulation opens the possibility of non-Riemannian
extensions for the superspace formulation of DFT [22], where a suitable
generalization for the generalized super frame could be the way of regroup all
the degrees of freedom of the theory making use of the orthosymplectic
extension of the $O(D,D)$ group. This procedure could give some insights about
how to obtain the supersymmetric sector of Heterotic supergravity coming from
a non-Riemannian formulation [11].

### Acknowledgements

We thank Sourav Roychowdhury for interesting discussions and Jan Rosseel for
enlightening comments and discussions. We thank IPhT-CEA Saclay for
hospitality during the last stage of this project. We also thank to the
French-Croatian bilateral project “Physics of non-geometric fluxes” of the
Cogito programme 2021-2022 and, particularly, to Mariana Grana and Athanasios
Chatzistavrakidis. E.L is supported by the Croatian Science Foundation project
IP-2019-04-4168.

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##  Appendix A Appendix

###  A.1. Search strategy and objectives

The primary goal of our search strategy was to identify relevant literature on
risks and harms of foundation models and ensure a comprehensive understanding
of their socio-political and environmental dimensions. To achieve this
objective, we conducted searches across five academic electronic databases.
The selected databases include arXiv, ACM Digital Library, IEEE, Scopus, and
Web of Science. These databases were chosen based on their relevance in the
field, their inclusion of a broader range of sources such as conference
proceedings and preprints, and access to the full text of the articles.

To develop the search strategy, we chose a handful of keywords based on their
relevance to the research question and maximize the scope of the search:
“foundation model”, “large language model”, “llm”, “general purpose artificial
intelligence”, “GPAI”, “risk”, “harm”, “ethic”. In addition, we used Boolean
operators to refine search queries (see Table 2 for example). In some cases,
we refined when the result set had a high percentage of irrelevant returns:
AND TI “artificial intelligence” OR TI “AI” OR AB “artificial intelligence” OR
AB “AI”.

Searches were conducted on title and abstract fields. A sample set was
piloted, before a full search was conducted across the selected databases.
Inclusion criteria covered English language journal articles, including
conference papers and pre-prints, but excluding grey literature, monographs,
commentaries, correspondences, and opinion pieces.

The results were added to a Zotero library, which also included our snowball
sample, and the merge of these two libraries was exported into a spreadsheet
for refinement and coding. We then removed duplicates and non-relevant papers,
reaching 167 papers in our sample. The use of an abductive approach supported
establishing clearer links with the research objectives and developing a
codified mapping of key relationships found in the literature results. To code
the papers, firstly, we extracted applicable keyword information from the
abstract of each primary study as part of an initial coding process. The
resulting keywords served largely as summative and process attributes of a
risk or type of harm relevant to foundation models or LLMs. Three researchers
subsequently clustered these attributes (Level 2 – attribute codes) and
assigned them into Level 1 or parent categories. For example, to attribute
code copyright and intellectual property violations we assigned the parent
category legal and regulatory violations. For the attribute codes difficult to
ensure explainability and reliance on proprietary software we assigned the
parent category lock-in and opacity risks (See Table 3 for full list of parent
and attribute codes). The codes were subsequently refined by all co-authors.

Table 2. Example of broad search (with refining keywords): Broad search (*) |  Refining #2
---|---
TI “foundation model*” OR AB “foundation model*” |  AND TI “Risk*” OR TI “Harm*” OR TI “ethic*”OR AB “Risk*” OR AB “Harm*” OR AB “ethic*”
TI “Large Language model*” OR TI “LLM*” OR AB “Large Language model*” OR AB ”LLM*” |  AND TI “Risk*” OR TI “Harm*” OR TI “ethic*” OR AB “Risk*” OR AB “Harm*” OR AB “ethic*”
TI ”general purpose artificial intelligence” OR TI ”general- purpose artificial intelligence” OR TI “GPAI” OR AB “general purpose artificial intelligence” OR AB ”general-purpose artificial intelligence” OR AB “GPAI” |  AND TI “Risk*” OR TI “Harm*” OR TI “ethic*” OR AB “Risk*” OR AB “Harm*” OR AB “ethic*”
Table 3. Parent and attribute codes. Parent Code |  Attribute Code
---|---
Bias and societal prejudices |  Amplifying and perpetuating stereotypes and societal biases
Behavioral biases
Political biases
Misinformation, disinformation and propaganda |  Abusive interactive experiences
Manipulation
Misinformation spread
Unreliable performance |  Accuracy/inaccuracy (outputs)
Harmful or toxic outputs
Language performance gap
Poor performance due to excessive training with synthetic data
Unpredictability of behaviour pre- and post-deployment
Untruthfulness of outputs
Disparate performance
Cybersecurity risks and harms |  Cyber-attacks payload
Data leakage
Data poisoning attacks
Fraudulent services
Goal hijacking and prompt leaking
Prompt injection attacks
Spear phishing
Jailbreaking
Malicious actors
Impersonation attack
Privacy risks and harms |  Privacy risks and harms
Systemic social and economic risks and harms |  Erosion of semantic capital
Misleading narratives about AI
Narrowing of the market/Monopolisation
Outcome homogenization
Perpetuation of inequalities
Effects on labor market
Widening of digital divide
Concentration of authoritative power
Invisibilization and poor working condition of data and content moderation
labor
Legal and regulatory violations |  Copyright and intellectual property violations
Data protection violations
Consumer protection laws violations
Environmental effects and ecological disruption |  Environmental effects and ecological disruption
Misuse |  Misuse
Biological misuse
Dual use
Illegitimate surveillance
Creation of violent or harmful content
Lock-in and opacity risks |  Challenges to benchmarking
Difficult to ensure explainability
Lack of replicability and transparency
Low technological readiness
Reliance on proprietary software
Overdependency in human-computer interaction |  Overdependency in human-computer interaction
Data risks and harms |  Data extractivism
Data quality
Datasets containing toxic data
Degradation of the digital commons
Value misalignment |  Value misalignment
Extreme or catastrophic risks and harms |  Extreme risks
Catastrophic risks

###  A.2. Results of the coding and codes description

Table 4. Papers per risks and harms parent and attribute codes and codes description. Risks and Harms Codes (and Papers) |  Code Description |  Count
---|---|---
Bias and societal prejudices
Amplifying and perpetuating stereotypes and societal biases:
(Abid et al., 2021; Ahuja et al., 2023; Alemany et al., 2023; Birhane et al.,
2021; Awais et al., 2023; Bender et al., 2021; Bommasani et al., 2022;
Anderljung et al., 2023; Au Yeung et al., 2023; Borchers et al., 2022; Brewer
et al., 2023; Chan et al., 2023; Connor and O’Neill, 2023; da Silva et al.,
2021; Huang et al., 2023a; Dacon et al., 2022; de Jager, 2023; Deas et al.,
2023; Fan et al., 2023; Farina and Lavazza, 2023; Felkner et al., 2023;
Gadiraju et al., 2023; Glocker et al., 2022; Guo and Caliskan, 2021; Hossain
et al., 2023; Hosseini and Horbach, 2023; Huang et al., 2023b; Hundt et al.,
2022; Khowaja et al., 2023; Kidd and Birhane, 2023; Kirk et al., 2023; Liao
and Vaughan, 2023; Mattas, 2023; Mhlanga, 2023; Nadeem et al., 2020; Nozza et
al., 2022; O’Neill and Connor, 2023; Ovalle et al., 2023; Piñeiro-Martin et
al., 2023; Qadir, 2022; Rauh et al., 2022; Sebastian, 2023; Shaikh et al.,
2022; Shelby et al., 2023; Simmons, 2023; Solaiman et al., 2019, 2023; Steed
et al., 2022; Sun et al., 2023; Talat et al., 2022; Talboy and Fuller, 2023;
Tan and Celis, 2019; Thakur, 2023; Thieme et al., 2023; van der Wal et al.,
2023; Venkit et al., 2023; Wang et al., 2024; Weidinger et al., 2022, 2021;
Wójcik, 2022; Zhuo et al., 2023b; Zhao et al., 2023; Wu et al., 2023; Jakesch
et al., 2023a)
Behavioral biases:
(Chen et al., 2023)
Political biases:
(Liu et al., 2021; Motoki et al., 2023; Rozado, 2023; Urman and Makhortykh, 2023) |  The perpetuation and amplification of hegemonic views, harmful stereotypes, societal and behavioral biases. |  72
Misinformation, disinformation and propaganda
Abusive interactive experiences:
(Bossert and Hagendorff, 2021; Rosenberg, 2023; Shelby et al., 2023)
Disinformation:
(Anderljung et al., 2023; Barnard et al., 2023; Bian et al., 2023; Christ et
al., 2023; Fecher et al., 2023; Kidd and Birhane, 2023; Liao et al., 2023;
Májovský et al., 2023; Mattas, 2023; Neupane et al., 2023; Pan et al., 2023;
Shelby et al., 2023; Solaiman et al., 2019, 2023; Weidinger et al., 2021,
2022; Bommasani et al., 2022; Li et al., 2023b)
Manipulation:
(Bender et al., 2021; Chan et al., 2023; Kirk et al., 2023; Shelby et al.,
2023; Weidinger et al., 2021, 2022; Shevlane et al., 2023)
Hallucinated information:
(Connor and O’Neill, 2023; Liao and Vaughan, 2023; Májovský et al., 2023;
Deroy et al., 2023; Kang et al., 2023b; Tian et al., 2023)
Misinformation spread:
(Anderljung et al., 2023; Bian et al., 2023; Chan, 2023; Christ et al., 2023; Farina and Lavazza, 2023; Fecher et al., 2023; Gao et al., 2023; Goldstein et al., 2023; Huang et al., 2023a; Khowaja et al., 2023; Kidd and Birhane, 2023; Kirk et al., 2023; Liao et al., 2023; Liao and Vaughan, 2023; Mattas, 2023; Neupane et al., 2023; Piñeiro-Martin et al., 2023; Qadir, 2022; Sebastian, 2023; Shelby et al., 2023; Solaiman et al., 2023; Sun et al., 2023; Rillig et al., 2023; Strasser, 2023; Thapa and Adhikari, 2023; Weidinger et al., 2021, 2022; Zhou et al., 2023a, b; Zhuo et al., 2023b; Li et al., 2023b; Jakesch et al., 2023b; Tang et al., 2023) . |  Unintentional or intentional efforts to disseminate false or misleading information. Related risks and harms include disinformation, misinformation spread, extreme manipulation, and abusive interactive experiences.
|  45
Cybersecurity risks and harms
Cyber-attacks payload:
(Charan et al., 2023; Weidinger et al., 2021, 2022)
Data leakage:
(Bender et al., 2021; Derner and Batistič, 2023; Fan et al., 2023; Huang et
al., 2023a; Jagannatha et al., 2021; Liao and Vaughan, 2023; O’Neill and
Connor, 2023; Piñeiro-Martin et al., 2023; Plant et al., 2022; Sebastian,
2023; Shao et al., 2023; Wang et al., 2024; Weidinger et al., 2021, 2022; Zhuo
et al., 2023b; Zhou et al., 2023a)
Data poisoning attacks:
(Fan et al., 2023; Huang et al., 2023a; Li et al., 2023a; Wu et al., 2023)
Fraudulent services:
(Derner and Batistič, 2023; Farina and Lavazza, 2023; Kirk et al., 2023;
Porsdam Mann et al., 2023; Sebastian, 2023; Strasser, 2023; Weidinger et al.,
2021, 2022)
Goal hijacking and prompt leaking:
(Perez and Ribeiro, 2022)
Prompt injection attacks:
(Greshake et al., 2023; Liu et al., 2023; Wu et al., 2023; Zhuo et al., 2023b)
Spear phishing:
(Anderljung et al., 2023; Derner and Batistič, 2023; Eggmann et al., 2023;
Farina and Lavazza, 2023; Hazell, 2023; Karanjai, 2022; Neupane et al., 2023;
Sebastian, 2023; Solaiman et al., 2023)
Jailbreaking:
(Rao et al., 2023; Wang et al., 2024)
Malicious actors:
(Awais et al., 2023; Bommasani et al., 2022; Eggmann et al., 2023; Farina and
Lavazza, 2023; Kang et al., 2023a; Weidinger et al., 2021, 2022; Wu et al.,
2023; Jakesch et al., 2023b)
Impersonation attack:
(Piñeiro-Martin et al., 2023; Solaiman et al., 2023; Zhou et al., 2023a) |  Unauthorized access and attacks to the foundation model that exploit vulnerabilities or compromise its integrity. Examples of cybersecurity risks and harms include cyber-attacks payload, data leakage, data poisoning attacks, data pollution, fraudulent services, goal hijacking and prompt leaking, offensive cyber capabilities, prompt injection attacks, spear phishing, jailbreaking, and other attacks by malicious actors.
|  37
Risks and Harms Codes (and Papers) |  Code Description |  Count
---|---|---
Unreliable performance
Accuracy/inaccuracy (outputs):
(Au Yeung et al., 2023; Awais et al., 2023; Farina and Lavazza, 2023; Fecher
et al., 2023; Huang et al., 2023a; Kang et al., 2023b; Liao and Vaughan, 2023;
O’Neill and Connor, 2023; Piñeiro-Martin et al., 2023; Shelby et al., 2023;
Strasser, 2023; Zhuo et al., 2023b; Alkaissi and McFarlane, 2023; Borji, 2023;
Rudolph et al., 2023; Uz and Umay, 2023; Ye et al., 2023; Casper et al., 2023)
Harmful or toxic outputs:
(Anderljung et al., 2023; Bender et al., 2021; da Silva et al., 2021;
Deshpande et al., 2023; Huang et al., 2023b; Liao and Vaughan, 2023; Borji,
2023; Ma et al., 2023; Rauh et al., 2022; Shaikh et al., 2022; Shelby et al.,
2023; Solaiman et al., 2023; Strasser, 2023; Sun et al., 2023; Zhuo et al.,
2023a, b; Blodgett and Madaio, 2021; Dinan et al., 2021; Gehman et al., 2020;
Li et al., 2023c; Qi et al., 2023; Triguero et al., 2023; Vashishtha et al.,
2023; Weidinger et al., 2021, 2022; Welbl et al., 2021; Casper et al., 2023)
Language performance gap:
(Alshahrani et al., 2022; Weidinger et al., 2021, 2022; Verma et al., 2022;
Zhuo et al., 2023b)
Poor performance due to excessive training with synthetic data:
(Veselovsky et al., 2023)
Unpredictability of behaviour pre- and post- deployment:
(Anderljung et al., 2023)
Untruthfulness of outputs:
(Borji, 2023; Oviedo-Trespalacios et al., 2023; Rauh et al., 2022)
Disparate performance:
(Bommasani et al., 2022; Shelby et al., 2023; Solaiman et al., 2023) |  Inconsistent or undesirable performance exhibited by foundation models. This includes inaccurate or non-factual outputs, harmful or toxic outputs, language performance gap, performance disparities at group levels, and unpredictaibility of behavior pre- and post-deployment.
|  44
Privacy risks and harms
(Bender et al., 2021; Eggmann et al., 2023; Fecher et al., 2023; Huang et al., 2023a; Jagannatha et al., 2021; Kandpal et al., 2022; Khowaja et al., 2023; Kirk et al., 2023; Liao and Vaughan, 2023; Li et al., 2023a; O’Neill and Connor, 2023; Porsdam Mann et al., 2023; Piñeiro-Martin et al., 2023; Shelby et al., 2023; Solaiman et al., 2023; Strasser, 2023; Tian et al., 2023; Wang et al., 2024; Weidinger et al., 2021, 2022; Wu et al., 2023; Zhou et al., 2023a; Patsakis and Lykousas, 2023; Peris et al., 2023; Yan et al., 2023; Zhang et al., 2023; Lu et al., 2023; Mireshghallah et al., 2022) |  Adverse consequences associated with the collection, storage, processing, and use of data beyond the data pipeline that arise from the increasing capabilities of foundation models and their potential to inadvertently expose private or sensitive data. |  29
Systemic social and economic risks and harms
Erosion of semantic capital:
(Nannini, 2023)
Misleading narratives about AI:
(Ferri and Gloerich, 2023; Nannini, 2023; Shanahan, 2023)
Narrowing of the market/Monopolisation:
(Bommasani et al., 2022; Luitse and Denkena, 2021)
Outcome homogenization:
(Bommasani et al., 2022; Kirk et al., 2023; Porsdam Mann et al., 2023; Thieme
et al., 2023)
Perpetuation of inequalities:
(Bommasani et al., 2022; Fecher et al., 2023; Kirk et al., 2023; Porsdam Mann
et al., 2023; Rillig et al., 2023; Shelby et al., 2023; Solaiman et al., 2023;
Weidinger et al., 2021, 2022; Williams, 2022)
Effects on labor market:
(Farina and Lavazza, 2023; Khowaja et al., 2023; Kirk et al., 2023; Liao and
Vaughan, 2023; Mattas, 2023; Eloundou et al., 2023; Lund et al., 2023; Qadir,
2022; Solaiman et al., 2023; Weidinger et al., 2021, 2022; Zarifhonarvar,
2023)
Widening of digital divide:
(Khowaja et al., 2023; Shelby et al., 2023)
Concentration of authoritative power:
(Khowaja et al., 2023; Solaiman et al., 2023)
Invisibilization and poor working condition of data and content moderation
labor:
(Solaiman et al., 2023) |  Risks and harms that arise from the widespread adoption and impact of foundation models on societal and economic structures. These range from the erosion of semantic capital, outcome homogenization, and misleading narratives about AI, to the widening of digital divide, the narrowing of the market or monopolization, effects on labor market, and the perpetuation of inequalities. |  24
Legal and regulatory violations
Copyright and intellectual property violations:
(Al-Kaswan and Izadi, 2023; Bommasani et al., 2022; Fecher et al., 2023;
Farina and Lavazza, 2023; Khowaja et al., 2023; Lund et al., 2023; O’Neill and
Connor, 2023; Piñeiro-Martin et al., 2023; Porsdam Mann et al., 2023;
Sebastian, 2023; Strasser, 2023; Solaiman et al., 2023; Lee et al., 2023;
Henderson et al., 2023a; Huang and Siddarth, 2023)
Data protection violations:
(Al-Kaswan and Izadi, 2023; Bommasani et al., 2022; Kandpal et al., 2022;
Khowaja et al., 2023; Kirk et al., 2023; O’Neill and Connor, 2023; Patsakis
and Lykousas, 2023; Peris et al., 2023; Piñeiro-Martin et al., 2023; Porsdam
Mann et al., 2023; Sebastian, 2023; Solaiman et al., 2023; Strasser, 2023;
Tian et al., 2023; Zhang et al., 2023)
Consumer protection laws violations:
(Sebastian, 2023) |  Breaches of laws and regulations, including those designed to safeguard individuals’ privacy and secure handling of their personal data, intellectual property, consumer protection, and cybersecurity. |  21
Risks and Harms Codes (and Papers) |  Code Description |  Count
---|---|---
Environmental effects and ecological disruption
(Anthony et al., 2020; Bender et al., 2021; Bommasani et al., 2022; Hagendorff et al., 2023; Huang et al., 2023a; Khowaja et al., 2023; Kirk et al., 2023; Kirkpatrick, 2023; Liao and Vaughan, 2023; Luccioni et al., 2022; O’Neill and Connor, 2023; Rillig et al., 2023; Shelby et al., 2023; Solaiman et al., 2023; Triguero et al., 2023; Weidinger et al., 2022; Patterson et al., 2021; Li et al., 2023d) |  Adverse and negative impacts of foundation models on ecology and animals. |  19
Misuse
Misuse:
(Anderljung et al., 2023)
Biological misuse:
(Cohen, 2023; Sandbrink, 2023)
Dual use:
(Anderljung et al., 2023; Kang et al., 2023a; Weidinger et al., 2021, 2022;
Glukhov et al., 2023; Soice et al., 2023; Henderson et al., 2023b)
Illegitime surveillance:
(Anderljung et al., 2023; Bommasani et al., 2022; Kirk et al., 2023; Weidinger
et al., 2021, 2022)
Creation of violent or harmful content:
(Anderljung et al., 2023; Bender et al., 2021; Bommasani et al., 2022; Kirk et al., 2023; Shelby et al., 2023; Solaiman et al., 2023) |  The misuse of foundation models is used here to describe specific contexts in which foundation models or technologies based on foundation models may be inappropriately used and lead to negative consequences. This includes uses that can have both civilian and military applications, or contribute to the handling, manipulation, or application of biological materials, organisms, or technologies. |  19
Lock-in and opacity risks
Challenges to benchmarking:
(Awais et al., 2023)
Difficult to ensure explainability:
(Piñeiro-Martin et al., 2023; Thieme et al., 2023; Triguero et al., 2023;
Wasserman Rozen et al., 2023)
Lack of replicability and transparency:
(Farina and Lavazza, 2023; Hosseini and Horbach, 2023; Kang et al., 2023b;
Khowaja et al., 2023; Liao and Vaughan, 2023; O’Neill and Connor, 2023;
Piñeiro-Martin et al., 2023; Yan et al., 2023; Zhou et al., 2023a; Lu et al.,
2023)
Low technological readiness:
(Yan et al., 2023)
Reliance on proprietary software:
(Liao and Vaughan, 2023; Liesenfeld et al., 2023) |  Risks and harms associated with foundation models that are difficult to understand, replicate, or modify. These risks and harms arise from issues including reliance on proprietary software, challenges to benchmarking, lack of transparency, difficulty to ensure explainaibility, and low technological readiness. |  15
Overdependency in human-computer interaction
(Connor and O’Neill, 2023; Kirk et al., 2023; Liao and Vaughan, 2023; Ma et al., 2023; Mhlanga, 2023; Piñeiro-Martin et al., 2023; Solaiman et al., 2023; Thieme et al., 2023; Weidinger et al., 2021, 2022; Peris et al., 2023) |  Overreliance of humans interacting with computer systems, interfaces, and technologies built on foundation models for various aspects of their lives and potentially negatively impacting their well-being, safety, decision-making abilities, or interpersonal relationships. |  11
Data risks and harms
Data extractivism:
(Huang and Siddarth, 2023)
Data quality:
(Huang and Siddarth, 2023; Kandpal et al., 2022)
Datasets containing toxic data:
(Birhane et al., 2023; Fan et al., 2023; O’Neill and Connor, 2023; Piñeiro-
Martin et al., 2023; Wang et al., 2024)
Degradation of the digital commons:
(Chan, 2023; Huang and Siddarth, 2023) |  A range of risks and harms associated with the collection, storage, processing, and use of data within the data pipeline. These risks extend beyond the cybersecurity risks and harms described above and includes broader issues of ethical, social, and economic implications of data stewardship and management. They include issues of poor data quality, datasets containing toxic data, data extractivism, and the degradation of the digital commons. |  8
Value misalignment
(Huang and Siddarth, 2023; Kasirzadeh and Gabriel, 2023) |  Misalignment of the output content of conversational agents with the norms and values of the human interacting with such agent. |  2
Extreme or catastrophic risks and harms
Extreme risks:
(Shevlane et al., 2023)
Catastrophic risks:
(Barrett et al., 2023) |  Speculative far-reaching or irreversible adverse impacts of foundation models at societal scale that extend beyond immediate impacts. |  2