$\alpha^{\prime}$-corrections and their double formulation

Eric Lescano

Instituto de Astronomía y Física del Espacio, Universidad de Buenos aires
(UBA-CONICET)
Ciudad Universitaria, Pabellón IAFE, 1428 Buenos Aires, Argentina

<EMAIL_ADDRESS>
Abstract

The present notes are based on three lectures, each ninety minutes long,
prepared for the international school “Integrability, dualities and
deformations”, hosted by Universidade de Santiago de Compostela, from August
23 to August 27, 2021. These lectures, aimed at graduate students, require
only a basic knowledge of string theory. The main goal is to introduce
$\alpha^{\prime}$-corrections to the gravitational sector of different
formulations of closed string theory and to reformulate them using novel
techniques based on double field theory.

###### Contents

  1. 0 Introduction
    1. 0.1 Student background and string bibliography
    2. 0.2 Outline
  2. 1 Lecture 1: Closed string theory and supergravity
    1. 1.1 Part A: low-energy effective formulations
      1. 1.1.1 Action principles
      2. 1.1.2 Symmetry transformations in bosonic/type II supergravity
        1. 1.1.3 Symmetry transformations in heterotic supergravity
        2. 1.1.4 Digression: vielbein formalism
    2. 1.2 Part B: $\alpha^{\prime}$-corrections
      1. 1.2.1 The Metsaev-Tseytlin approach
      2. 1.2.2 The Bergshoeff-de Roo approach
    3. 1.3 Exercises
  3. 2 Lecture 2: Double field theory and higher-derivative terms
    1. 2.1 Part A: zeroth-order formulation
      1. 2.1.1 Heterotic/gauged double field theory
      2. 2.1.2 Parametrization
    2. 2.2 Part B: the generalized Green-Schwarz mechanism
      1. 2.2.1 Heterotic DFT
      2. 2.2.2 Bi-parametric extension
      3. 2.2.3 Comments about HSZ theory
    3. 2.3 Exercises
  4. 3 Lecture 3: Extended space and higher-order formulations
    1. 3.1 Part A: a systematic procedure to first-order formulations
      1. 3.1.1 Proposal
      2. 3.1.2 Extended symmetries and gauge fixing
      3. 3.1.3 Transformation rules for the $O(D,D)$ field content
      4. 3.1.4 First-order identification
      5. 3.1.5 Fluxes and action principle
    2. 3.2 Part B: extension to higher-order formulations
      1. 3.2.1 The generalized Bergshoeff-de Roo identification
      2. 3.2.2 Current research
    3. 3.3 Exercises
  5. 4 Summary
  6. 5 Acknowledgments

##  0 Introduction

In the last decades there has been a huge progress made in the study of the
low energy limit of string theory. The corresponding field theories,
collectively known as supergravity theories, describe particle dynamics and
any string structure is completely suppressed. Many related programs, however,
incorporate higher-order corrections beyond this low-energy limit and take
some string features into account. Some current examples are: tests of duality
conjectures [1], microstate counting of black hole entropy [2], moduli
stabilization [3], the swampland program [4], among others.

Historically, $\alpha^{\prime}$-corrections have been computed through
scattering amplitudes [5] [6] [7] and beta-function computations [8]. There
are different string formulations and the order of the first non-trivial
$\alpha^{\prime}$-correction depends strongly on the theory. For instance,
bosonic and heterotic string theories both contain four-derivative corrections
in their leading-order Lagrangians ($\alpha^{\prime}$ first-order correction)
while type-II superstrings contain eight-derivative terms as their first
corrections (corrections $\sim\alpha^{\prime 3}$)111In these lectures we use
the convention that $2i$-derivative terms in the string theory Lagrangian are
of order $(\alpha^{\prime})^{i-1}$.. All these higher-derivative
supergravities incorporate string effects to their dynamics.

Other methods used to compute higher-order corrections are based on symmetry
arguments: some terms of the effective string Lagrangian can be unambiguously
fixed by means of symmetry transformations. For example, the classical
Einstein-Hilbert Lagrangian is given by an object that transforms as a scalar
under infinitesimal diffeomorphisms, the Ricci scalar. Therefore, a possible
four-derivative correction to the Einstein-Hilbert action could be a $(\rm
Riem)^{2}$ term, that also transforms as a scalar under infinitesimal
diffeomorphisms. In [9] $\alpha^{\prime}$-corrections were studied using this
method. The authors present a relation between gauge fields and the
gravitational sector that allows to construct four-derivative corrections in a
simple way.

In these lectures we will review both [7] and [9] approaches, and we will
generalize the construction of the latter in a double field theory (DFT)
context [10]. DFT is a novel framework to construct T-duality invariant
theories, T-duality being an exact symmetry of string theory. Since DFT is
invariant under the same symmetries as string theory, it is a promising
framework to compute higher-derivative corrections. The idea of these lectures
is to give a self-contained introduction to duality invariant approaches
related to $\alpha^{\prime}$-corrections and to present the state of the art
of this topic.

###  0.1 Student background and string bibliography

These lectures give an introduction to the study of
$\alpha^{\prime}$-corrections to the effective action of closed string theory.
Our intention is to keep it self-contained, even though some knowledge on non-
linear sigma models and their effective actions [11] is strongly recommended
to get the most out of the course. A very pedagogical introduction to these
topics can be found in [12]. For people new to string theory, we recommend
Barton Zwiebach’s book [13].

###  0.2 Outline

The plan of the course is as follows:

  * •

Lecture 1: Closed string theory and supergravity

In part A we present the supergravity limit of closed string theory. We pay
special attention to the NS-NS Lagrangian of type II superstring theory, since
it is common to other closed string formulations. We analyze the symmetries of
this Lagrangian, and the additional symmetries of the heterotic supergravity.
We finish with a digression about the vielbein formalism.

In part B we present two approaches to study the four-derivative terms in
closed string theory. We first present the Metsaev-Tseytlin approach [7]. The
corresponding effective action is constructed taking into account three- and
four-point scattering amplitudes for the massless states of the string. Then
we present the Bergshoeff-de Roo approach [9], which is mostly based on
supersymmetry considerations. We review an interesting relation between gauge
and Lorentz vectors that allows to construct the gravitational
$\alpha^{\prime}$-corrections. We shall follow [14], where the equivalence of
these approaches was also discussed.

  * •

Lecture 2: Double field theory and higher-derivative terms

In part A we give a self-contained review of double field theory [10]. We
present the construction of the theory in four steps: We start by introducing
the double geometry, describing its symmetries, discussing the fundamental
fields and, finally, introducing the action principle. Then, we study the
heterotic/gauged double field theory, and analyze the generalization of the
previous steps for this case. We discuss the parametrizations needed in order
to connect with the bosonic part of the heterotic supergravity.

In part B we introduce the generalized Green-Schwarz mechanism at the DFT
level [15]. This mechanism consists of a suitable modification of the Lorentz
transformation of the generalized frame [14]. We start by analyzing the
heterotic DFT case and then we will generalize the ansatz for a bi-parametric
case. We show that the parametrization of this deformation reproduces the
Bergshoeff-de Roo approach after a suitable field redefinition of the
vielbein. We finish this part with some comments about HSZ theory [16]. This
theory is not a string theory but it can be understood as a higher-derivative
deformation of DFT. However the four-derivative correction of its supergravity
Lagrangian is closely related to heterotic string theory while its six-
derivative correction is related to bosonic string theory.

  * •

Lecture 3: Extended space and higher-order formulations

This lecture is focused on introducing a systematic procedure to finding
higher-derivative corrections at the DFT level [24]. Our starting point is an
$O(D,D+K)$ invariant theory (“gauged DFT”). By splitting the fields, metrics
and parameters in terms of $O(D,D)$ degrees of freedom, we obtain a DFT
formulation with an extra $O(D,D)$ vector that plays the role of gauge
connection at the DFT level. We then identify this degree of freedom with a
projection of the generalized flux. This method allows to construct the
generalized Green-Schwarz mechanism and the four-derivative action principle
from geometric first principles. This identification between the $O(D,D)$
vector and the derivatives of the generalized frame works up to first-order.

In part B we discuss about the extension of the previous method to higher-
order. We present the generalized Bergshoeff-de Roo identification, which is
an exact identification which allows us to construct symmetry transformations
and action principles up to an arbitrary order. At the end, we present some
current research lines around this generalization.

After each lecture a problem sheet is included. Those exercises with a * mark
require a little bit more computations than the others.

##  1 Lecture 1: Closed string theory and supergravity

String theory is a proposal to describe nature in a unified approach, where
gravity and gauge symmetries coexist. The idea of string theory is very
simple: to replace point-like particles with string excitations. Strings can
be open or closed and their length are given by
$l_{s}=\sqrt{\alpha^{\prime}}$. As we learned from undergraduate courses, a
string can oscillate in different modes, and the higher the mode, the higher
the energy that is required. The lightest physical modes of the closed
(super-)strings spectrum are massless and, in this sense, string theory
predicts the existence of a finite number of massless particles and an
infinite number of massive particles ($M^{2}\sim\frac{1}{\alpha^{\prime}}$).
From a phenomenological point of view it is common to focus only in the
massless states, integrating out the massive ones. This procedure gives rise
to a particular higher-derivative structure that corrects the low energy limit
of the effective theory. The study of these $\alpha^{\prime}$-corrections is
the main topic of these lectures.

An interesting aspect of string theory is that the starting point is not a
4-dimensional theory. The D-dimensional target space which contains our
strings has an arbitrary dimension from a classical point of view, but a fixed
and $D>4$ dimension if we want a consistent quantum theory [11]. This is one
of the most exciting aspects of string theory and the reason is due to anomaly
cancellation: the classical worldsheet formulation of the string is invariant
under local Weyl transformations, and cannot be preserved in any dimension.

In these lectures we are interested in three different formulations of closed
string theories: the bosonic string with critical dimension $D=26$, the type
II superstrings with critical dimension $D=10$, and the heterotic superstring
also defined on a 10-dimensional target space. The superstring formulations
are related to each other by string dualities, but each formulation receives
different kinds of corrections. In order to avoid confusion, we will use the
letter $D$ for the critical dimension of each string and we always consider
$\frac{\sqrt{\alpha^{\prime}}}{R}<<1$ with R the curvature of the target
space. Thus, we are able to study the physics of the massless states of the
different formulations by considering an $\alpha^{\prime}$ expansion.

###  1.1 Part A: low-energy effective formulations

####  1.1.1 Action principles

The low-energy effective action depends on the string formulation under
consideration [1]. For example, the massless spectrum for bosonic string
theory consists of a metric tensor $g_{\mu\nu}$, a 2-form or Kalb-Ramond
field, $b_{\mu\nu}$, and a scalar field called the dilaton, $\phi$, where
$\mu,\nu=0,\dots,D-1$ and in this case $D=26$. The action principle for this
setup (in string frame) is given by

$\displaystyle S=\int
d^{D}x\sqrt{-g}e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\mathcal{O}(\alpha^{\prime}))\,,$
(1.1)

where $R$ is the Ricci scalar and

$\displaystyle
H_{\mu\nu\rho}=\partial_{\mu}b_{\nu\rho}+\partial_{\nu}b_{\rho\mu}+\partial_{\rho}b_{\mu\nu}\,,$
(1.2)

is a 3-form known as the $b$-field curvature. As the reader can easily verify,
each term of the action principle (1.1) contains two derivatives. The first-
order correction (or the $\alpha^{\prime}$-correction) to this effective
action are given by four-derivative terms, while the second-order corrections
are given by 6-derivative terms and so on. Since the order of the correction
can be easily identified through the number of derivatives we adopt the
convention $\alpha^{\prime}=1$ to present a light notation. These
$\alpha^{\prime}$-corrections were historically computed from scattering
amplitudes222The beta-function computation is a different way to obtain
corrections. See [8]. as we will see in part B. Before moving to another
formulation it is interesting to observe that the low-energy effective bosonic
action consists of an Einstein-Hilbert action coupled to a 2-form and a scalar
field.

As we have mentioned in the first part of this lecture, we are also interested
in the type-II superstring formulation. There exist two different consistent
frameworks, type IIA and type IIB. Both of them contain bosons and fermions in
their spectrum and this is the main difference with respect to the bosonic
string that only contains bosons. Schematically, the type-II effective
Lagrangian can be written as,

$\displaystyle L_{\rm{type-II}}=L_{\rm NS-NS}+L_{\rm R-R}+L_{\rm NS-R}+L_{\rm
R-NS}\,.$ (1.3)

The first term, $L_{\rm NS-NS}$ , encodes the dynamics of the following
massless fields: a metric tensor, a 2-form and a dilaton. Morever, the two-
derivative effective action principle for this part of the spectrum is exactly
given by (1.1), but considering $D=10$. The second term of (1.3), $L_{\rm
R-R}$, describes the dynamics of a 1-form $C_{1}$ and a 3-form $C_{3}$ for
type IIA superstring and a zero-form $C_{0}$, a two form $C_{2}$ and a four-
form $C_{4}$ for type IIB superstring. Both $L_{\rm NS-NS}$ and $L_{\rm R-R}$
are bosonic Lagrangians, meaning that they depend on bosonic quantitites. On
the other hand, both $L_{\rm NS-R}$ and $L_{\rm R-NS}$ depend on fermionic
degrees of freedom, related to the bosonic spectrum through ${\cal N}=2$
supersymmetry. This symmetry interchanges bosons and fermions. The critical
dimension of this formulation is $D=10$. The Lagrangian (1.3) is our first
example of a supergravity Lagrangian, since it contains both a dynamical
metric tensor and fermionic degrees of freedom related by supersymmetry.
Indeed, the low energy limit of al superstring theory formulations is given by
a supergravity. Moreover, supergravity theories in arbitrary dimensions can be
studied independently of string theory. It turns out that the first
$\alpha^{\prime}$-corrections for the type II superstrings consist of eight-
derivative terms, so neither $\alpha^{\prime}$ nor $\alpha^{\prime 2}$
corrections are present in this theory.

Last but not least, we consider heterotic string theory [17]. This string is a
hybrid formulation. To construct it we consider a left-right decomposition for
the closed string oscillation. This is analogous to a travelling wave
decomposition, where left and right are the propagation directions of a wave
on the closed string. The heterotic string contains a $D=26$ bosonic string
formulation on its left part, compatified on a $16$-dimensional torus (for
dimensional consistency), and a $D=10$ superstring formulation on its right
side. The 10-dimensional bosonic and massless degrees of freedom are a metric
tensor, a 2-form, a dilaton and a non-abelian gauge field $A_{\mu i}$. The
index $i$ is a gauge index that runs from $1,\dots,n$ with $n$ the dimension
of the gauge group, $E_{8}\times E_{8}$ or $SO(32)$, depending on the
heterotic formulation.

This superstring theory is invariant under ${\cal N}=1$ supersymmetry so it
contains fermionic degrees of freedom besides the former fields. The effective
action is given by a heterotic supergravity,

$\displaystyle S_{\rm{het}}=\int
d^{D}x\sqrt{-g}e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\bar{H}_{\mu\nu\rho}\bar{H}^{\mu\nu\rho}-\frac{1}{4}F_{\mu\nu}{}^{i}F^{\mu\nu}{}_{i}+L_{f}+\mathcal{O}(\alpha^{\prime}))\,,$
(1.4)

where $L_{f}$ are two-derivative fermionic terms and333We use the convention
$T_{[\mu\nu]}=\frac{1}{2}(T_{\mu\nu}-T_{\nu\mu})$ and
$T_{(\mu\nu)}=\frac{1}{2}(T_{\mu\nu}+T_{\nu\mu})$.

$\displaystyle
F_{\mu\nu}{}^{i}=2\partial_{[\mu}A_{\nu]}{}^{i}-f^{i}{}_{jk}A_{\mu}{}^{j}A_{\nu}{}^{k}\,,$
(1.5)

is the gauge field curvature. In the previous expression gauge indices are
contracted using a constant Cartan-Killing metric $\kappa_{ij}$ with inverse
$\kappa^{ij}$. The first higher-derivative correction to this action principle
is a correction of order $\alpha^{\prime}$. We will discuss the form of these
terms in the next section. In order to keep track of the $\alpha^{\prime}$
terms, we assume that the structure constants $f_{ijk}$ count as a derivative.
So, the first non-trivial $\alpha^{\prime}$-corrections are terms with
$f^{m}\partial^{n}$ such that $n+m=4$, and so on. Therefore we still think in
(1.4) as a zeroth-order Lagrangian.

An important difference with respect to the previous formulations is that the
curvature of the $b$-field contains extra terms [18],

$\bar{H}_{\mu\nu\rho}=3\left(\partial_{[\mu}b_{\nu\rho]}-C_{\mu\nu\rho}^{(g)}\right)\,,$
(1.6)

where $C_{\mu\nu\rho}^{(g)}$ is a so-called Chern-Simons 3-form, defined as

$C_{\mu\nu\rho}^{(g)}=A^{i}_{[\mu}\partial_{\nu}A_{\rho]i}-\frac{1}{3}f_{ijk}A_{\mu}^{i}A_{\nu}^{j}A_{\rho}^{k}\,.$
(1.7)

The necessity for this additional contribution will be clear when we inspect
the symmetry rules of this formulation.

####  1.1.2 Symmetry transformations in bosonic/type II supergravity

We start by discussing the symmetry transformations of the universal sector,

$\displaystyle S=\int
d^{D}x\sqrt{-g}e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho})\,.$
(1.8)

This action is common to all the effective close string formulations, and it
is invariant under:

  * •

Infinitesimal diffeomorphisms: When acting on an arbitrary vector $v^{\mu}$ of
weight $\omega$ this transformation takes the form

$\displaystyle\delta_{\xi}v^{\mu}=L_{\xi}v^{\mu}=\xi^{\nu}(\partial_{\nu}v^{\mu})-(\partial_{\nu}\xi^{\mu})v^{\nu}+\omega(\partial_{\nu}\xi^{\nu})v^{\mu}\,,$
(1.9)

where $\xi^{\mu}$ is an arbitrary parameter and $L_{\xi}$ is the Lie
derivative. The fundamental fields transform as 444We use the convention that
contracted indices are not part of the (anti-)symmetrizers.

$\displaystyle\delta_{\xi}g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
L_{\xi}g_{\mu\nu}=\xi^{\rho}(\partial_{\rho}g_{\mu\nu})+2(\partial_{(\mu}\xi^{\rho})g_{\rho\nu)}\,,$
(1.10) $\displaystyle\delta_{\xi}b_{\mu\nu}$ $\displaystyle=$ $\displaystyle
L_{\xi}b_{\mu\nu}=\xi^{\rho}(\partial_{\rho}b_{\mu\nu})+2(\partial_{[\mu}\xi^{\rho})b_{\rho\nu]}\,,$
(1.11) $\displaystyle\delta_{\xi}\phi$ $\displaystyle=$ $\displaystyle
L_{\xi}\phi=\xi^{\rho}(\partial_{\rho}\phi)\,,$ (1.12)

and the closure of these transformations acting on a generic (1,1) tensor
$T_{\mu}{}^{\nu}$, i.e.,

$\displaystyle\Big{[}\delta_{\xi_{1}},\delta_{\xi_{2}}\Big{]}T_{\mu}{}^{\nu}=\delta_{\xi_{21}}T_{\mu}{}^{\nu}$
(1.13)

is given by the Lie bracket

$\displaystyle\xi^{\mu}_{12}(x)=\xi^{\rho}_{1}\frac{\partial\xi^{\mu}_{2}}{\partial
x^{\rho}}-(1\leftrightarrow 2)\,.$ (1.14)

At this point we observe that the transformation of the derivative of a
vector,
$\delta_{\xi}(\partial_{\mu}v^{\nu})=\partial_{\mu}(\delta_{\xi}v^{\nu})$,
does not transform as a (1,1) tensor. Consequently, we define a covariant
derivative in the usual way,

$\displaystyle\nabla_{\rho}T_{\mu}{}^{\nu}=\partial_{\rho}T_{\mu}{}^{\nu}-\Gamma_{\rho\mu}^{\sigma}T_{\sigma}{}^{\nu}+\Gamma_{\rho\sigma}^{\nu}T_{\mu}{}^{\sigma}\,,$
(1.15)

where $\Gamma_{\mu\nu}^{\rho}$ is the Christoffel symbol,
$\Gamma_{\mu\nu}^{\rho}=\frac{1}{2}g^{\rho\sigma}(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu})$.

Each term of the Lagrangian

$L=e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda})$
(1.16)

transforms as a scalar (with $\omega=0$),

$\delta_{\xi}L=\xi^{\mu}\partial_{\mu}L\,,$ (1.17)

while the square root of the determinant transforms as

$\delta_{\xi}(\sqrt{-g})=\partial_{\mu}(\sqrt{-g}\xi^{\mu})\,.$ (1.18)

Consequently, the universal action (1.8) is invariant up to total derivatives.

  * •

Abelian gauge transformations: This transformation only acts on the $b$-field
due to its 2-form nature,

$\delta_{\zeta}b_{\mu\nu}=2\partial_{[\mu}\zeta_{\nu]}\,,$ (1.19)

where $\zeta$ is an arbitrary parameter. Since

$\displaystyle\delta_{\zeta}H_{\mu\nu\rho}=3\partial_{[\mu}(\delta
b_{\nu\rho]})=6\partial_{[\mu}(\partial_{\nu}\zeta_{\rho]})=0\,,$ (1.20)

the Lagrangian (1.16) is gauge invariant.

####  1.1.3 Symmetry transformations in heterotic supergravity

The low-energy heterotic effective Lagrangian,

$L_{\rm{het}}=e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\bar{H}_{\mu\nu\lambda}\bar{H}^{\mu\nu\lambda}-\frac{1}{4}F_{\mu\nu}{}^{i}F^{\mu\nu}{}_{i})\,,$
(1.21)

is invariant under the symmetries previously described since $A_{\mu i}$
transforms as a 1-form under infinitesimal diffeomorphisms, i.e.,

$\displaystyle\delta_{\xi}A_{\mu i}=L_{\xi}A_{\mu
i}=\xi^{\rho}(\partial_{\rho}A_{\mu i})+(\partial_{\mu}\xi^{\rho})A_{\rho
i}\,,$ (1.22)

and is invariant under abelian gauge transformations. The Lagrangian (1.21) is
also invariant under non-abelian gauge transformations.

  * •

Non-abelian gauge transformations: The non-abelian gauge transformations act
on $A_{\mu i}$ and $b_{\mu\nu}$ in the following way,

$\displaystyle\delta_{\lambda}A_{\mu}^{i}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda^{i}+f^{i}{}_{jk}\lambda^{j}A_{\mu}^{k}\,,$
(1.23) $\displaystyle\delta_{\lambda}b_{\mu\nu}$ $\displaystyle=$
$\displaystyle-(\partial_{[\mu}\lambda^{i})A_{\nu]i}\,,$ (1.24)

where $\lambda^{i}$ is an arbitrary parameter. Considering an arbitrary gauge
vector $v^{i}$, the partial derivative of this vector is not covariant and we
need to define a covariant derivative as

$\displaystyle\nabla_{\mu}v^{i}=\partial_{\mu}v^{i}-f^{i}{}_{jk}A_{\mu}{}^{j}v^{k}\,.$
(1.25)

Here we use the same notation $\nabla$ for the gauge covariant derivative as
in (1.15), so our convention is that $\nabla$ covariantizes the derivative of
an object with respect to all the symmetries that the object transforms under.
On the other hand, $F_{\mu\nu i}$ transforms covariantly under non-abelian
gauge transformations

$\displaystyle\delta_{\lambda}F_{\mu\nu
i}=f_{ijk}\lambda^{j}F_{\mu\nu}{}^{k}\,,$ (1.26)

unlike $A_{\mu i}$ whose transformation is not covariant due to the presence
of the $\partial_{\mu}\lambda^{i}$ term. Moreover, we need
$\delta\bar{H}_{\mu\nu\rho}=0$ to ensure the non-abelian gauge invariance of
the effective action. Now it is clear the Chern-Simons terms in (1.6) were
need as they transform as

$\displaystyle\delta_{\lambda}C^{(g)}_{\mu\nu\rho}=\partial_{[\mu}(A_{\nu}{}^{i}\partial_{\rho]}\lambda_{i})\,.$
(1.27)

The previous transformation exactly cancels the
$3\partial_{[\mu}(\delta_{\lambda}b_{\nu\rho]})$ contribution such that (1.4)
is non-abelian gauge invariant.

####  1.1.4 Digression: vielbein formalism

All supersymmetric invariant theories, such as type II or heterotic string
theory, must be written in the vielbein formalism of general relativity,
instead of the metric formalism, in order to couple the fermions.
Consequently, we consider flat vectors $v^{a}$ defined on the tangent space of
each point of the target space, where the indices $a,b=0,\dots,D-1$ are known
as “flat indices”. Flat vectors transform under Lorentz transformations
according to

$\displaystyle\delta_{\Lambda}v^{a}=v^{b}\Lambda_{b}{}^{a}$ (1.28)

where all the contractions are made with a constant flat (inverse) metric
$\eta^{ab}$. The Lorentz parameter satisfies $\Lambda_{ab}=-\Lambda_{ba}$.
Since we are abandoning the metric formulation, we need to consider a new
fundamental field $e_{\mu}{}^{a}$, the vielbein, whose inverse is given by
$e^{\mu}{}_{a}$. This object satisfies,

$\displaystyle e_{\mu}{}^{a}\eta_{ab}e_{\nu}{}^{b}=g_{\mu\nu}\,,$ (1.29)

and transforms covariantly under infinitesimal diffeomorphisms and Lorentz
transformations,

$\displaystyle\delta_{\xi,\Lambda}e_{\mu}{}^{a}$ $\displaystyle=$
$\displaystyle\xi^{\nu}\partial_{\nu}e_{\mu}{}^{a}+\partial_{\mu}\xi^{\nu}e_{\nu}{}^{a}+e_{\mu}{}^{b}\Lambda_{b}{}^{a}\,,$
(1.30) $\displaystyle\delta_{\xi,\Lambda}e^{\mu}{}_{a}$ $\displaystyle=$
$\displaystyle\xi^{\nu}\partial_{\nu}e^{\mu}{}_{a}-\partial_{\nu}\xi^{\mu}e^{\nu}{}_{a}+e^{\mu}{}_{b}\Lambda^{b}{}_{a}\,.$
(1.31)

The other fundamental fields, $b_{\mu\nu},A_{\mu i}$ and $\phi$ are Lorentz
invariant. Analogously to what happens with infinitesimal diffeomorphisms or
non-abelian gauge transformations, the transformation of the partial
derivative of a flat vector does not match with the transformation of a flat
tensor. Considering a generic flat vector $v^{a}$, we define a flat covariant
derivative as

$\displaystyle\nabla_{\mu}v^{a}=\partial_{\mu}v^{a}-w_{\mu}{}^{a}{}_{b}v^{b}\,,$
(1.32)

where $w_{\mu ab}$ is the spin connection. Imposing

$\displaystyle\nabla_{\mu}e_{\nu}{}^{a}=\partial_{\mu}e_{\nu}{}^{a}-\Gamma_{\mu\nu}^{\rho}e_{\rho}{}^{a}-w_{\mu}{}^{a}{}_{b}e_{\nu}{}^{b}=0\,,$
(1.33)

we can fully determine the spin connection in terms of the vielbein, $w_{\mu
bc}=w_{\mu bc}(e)$. The 2-form version of the Riemann tensor can be written in
terms of the vielbein as

$\displaystyle R_{\mu\nu ab}=-2\partial_{[\mu}w_{\nu]ab}+2w_{[\mu
a}{}^{c}w_{\nu]cb}\,,$ (1.34)

and the previous object satisfies $R_{\mu\nu ab}e^{\rho
a}e_{\sigma}{}^{b}=R^{\rho}{}_{\sigma\mu\nu}$ where

$\displaystyle
R^{\rho}{}_{\sigma\mu\nu}=2\partial_{[\mu}\Gamma^{\rho}_{\nu]\sigma}+2\Gamma^{\rho}_{[\mu\tau}\Gamma_{\nu]\sigma}^{\tau}\,$
(1.35)

is the Riemann tensor in terms of the metric. Traces of the Riemann tensor
give the Ricci tensor and scalar, respectively

$\displaystyle R_{\mu\nu}=R^{\rho}{}_{\mu\rho\nu}\,,\quad
R=R_{\mu\nu}g^{\mu\nu}\ .$ (1.36)

###  1.2 Part B: $\alpha^{\prime}$-corrections

In this part we will study two possible ways of presenting the gravitational
$\alpha^{\prime}$-corrections. These contributions are four-derivative terms
that correct the low energy effective Lagrangian of bosonic/heterotic string
theory. On the one hand, we will study the Metsaev-Tseytlin approach, which is
obtained considering scattering amplitudes. Then, we will study the
Bergshoeff-de Roo approch, related to the former using field redefinitions.

####  1.2.1 The Metsaev-Tseytlin approach

The four-derivative corrections to the universal action principle of the
different formulations of closed string theory were historically computed
considering three- and four-point scattering amplitudes for the massless
states [5] [6] [7]. This method is based on the study of the different string
interactions through the S-matrix, in order to construct an effective
Lagrangian that be able to reproduce that interactions. The effective action,
originally computed by Metsaev and Tseytlin, takes the form

$\displaystyle S_{MT}=\int d^{D}x\sqrt{-g}e^{-2\phi}(L^{(0)}+L^{(1)}_{MT})\,,$
(1.37)

where $L^{(0)}$ is given by (1.1) and

$\displaystyle L^{(1)}_{\rm MT}$ $\displaystyle=$
$\displaystyle-\frac{a+b}{8}\Big{[}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-\frac{1}{2}H^{\mu\nu\rho}H_{\mu\sigma\lambda}R_{\nu\rho}{}^{\sigma\lambda}+\frac{1}{24}H^{4}-\frac{1}{8}H^{2}_{\mu\nu}H^{2\mu\nu}\Big{]}$
(1.38) $\displaystyle+\frac{a-b}{4}H^{\mu\nu\rho}C_{\mu\nu\rho}\,$

corresponds to the first-order $\alpha^{\prime}$-correction with $a$ and $b$
units of $\alpha^{\prime}$. In the previous expression we have introduced the
following quantitites,

$\displaystyle C_{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle
w_{[\mu}{}^{ab}\partial_{\nu}w_{\rho]ab}+\frac{2}{3}w_{[\mu}{}^{ab}w_{\nu
b}{}^{c}w_{\rho]ca}\,,$ (1.39) $\displaystyle H^{2}_{\mu\nu}$ $\displaystyle=$
$\displaystyle H_{\mu}{}^{\rho\sigma}H_{\nu\rho\sigma}\,,$ (1.40)
$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle
H_{\mu\nu\rho}H^{\mu\nu\rho}\,.$ (1.41)

The bi-parametric form of the Lagrangian (1.38) was introduced in [14]. As we
will see, this compact form allows comparisons between different formulations.
In order to find $\alpha^{\prime}$-corrections we need to impose

$\displaystyle(a,b)=\begin{cases}(-1,-1)&\mbox{bosonic string \, , }\\\
(-1,0)&\mbox{heterotic string \, ,}\\\ (0,0)&\mbox{type II strings \,
.}\end{cases}$ (1.42)

At this point it is important to recall that we are showing only the
gravitational corrections to the universal NS-NS sector which contains only a
metric tensor, a 2-form and a dilaton which means that, for instance, we are
turning off the non-abelian gauge sector of the heterotic supergravity.

The full action (1.37) requires a modification of the Lorentz transformation
of the $b$-field at order $\alpha^{\prime}$ 555A zeroth-order Lorentz
transformation has no derivatives, so a first-order transformation must
contain two derivatives.,

$\displaystyle\delta_{\Lambda}b_{\mu\nu}=\frac{1}{2}(a-b)\partial_{[\mu}\Lambda^{ab}w_{\nu]ab}\,.$
(1.43)

This expression is the so-called Green-Schwarz transformation [18] and is a
cornerstone of string theory phenomenology. We can absorb the second line of
(1.38) by deforming the curvature of the $b$-field once again

$\displaystyle\bar{H}_{\mu\nu\rho}=3\Big{(}\partial_{[\mu}b_{\nu\rho]}+\frac{1}{2}(b-a)C_{\mu\nu\rho}\Big{)}\,.$
(1.44)

As the reader can easily check, the Green-Schwarz transformation and the
3-form deformation are present only in the heterotic case. Moreover, the new
terms in the curvature of the $b$-field guarantees that $\bar{H}_{\mu\nu\rho}$
transforms trivially under Lorentz transformations,

$\displaystyle\delta_{\Lambda}\bar{H}_{\mu\nu\rho}=0$ (1.45)

and then $L_{\rm MT}$ is Lorentz invariant when four-derivative corrections
are taking into account.

####  1.2.2 The Bergshoeff-de Roo approach

Another historical way to compute higher-order derivative corrections is to
appeal to symmetry arguments. For instance, if we focus on the bosonic terms
in the heterotic supergravity Lagrangian,

$L_{het}=e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\bar{H}_{\mu\nu\lambda}\bar{H}^{\mu\nu\lambda}-\frac{1}{4}F_{\mu\nu}{}^{i}F^{\mu\nu}{}_{i})\,,$
(1.46)

each term of this Lagrangian is invariant under infinitesimal diffeomorphisms,
abelian/non abelian gauge transformations and Lorentz transformations. In this
sense, the coefficients $\Big{\\{}1,4,-\frac{1}{12},-\frac{1}{4}\Big{\\}}$ are
not determined by these symmetries. However, ${\cal N}=1$ supersymmetric
invariance unequivocally determines all the coefficients. In other words, if
we change one of these coefficients by hand the full action is not ${\cal
N}=1$ supersymmetric invariant anymore.

Using this kind of argument, Bergshoeff and de Roo obtained the following
corrections [9],

$\displaystyle S_{\rm BdR}=\int
d^{D}x\sqrt{-g}e^{-2\phi}(\tilde{L}_{0}+L_{1})$ (1.47)

where

$\displaystyle
L_{1}=-\frac{a}{8}R^{(-)}_{\mu\nu}{}^{ab}R^{(-)\mu\nu}{}_{ab}-\frac{b}{8}R^{(+)}_{\mu\nu}{}^{ab}R^{(+)\mu\nu}{}_{ab}$
(1.48)

and the $(\pm)$ notation means that we need to add a torsion term in the spin
connection $w_{\mu ab}\rightarrow w^{(\pm)}_{\mu ab}=w_{\mu
ab}\pm\frac{1}{2}H_{\mu\nu\rho}e^{\nu}{}_{a}e^{\rho}{}_{b}$. The zeroth-order
Lagrangian is given by

$\displaystyle\tilde{L}_{0}=R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\tilde{H}^{2}\,,$
(1.49)

where
$\tilde{H}_{\mu\nu\rho}=H_{\mu\nu\rho}-\frac{3}{2}aC^{(-)}_{\mu\nu\rho}+\frac{3}{2}bC^{(+)}_{\mu\nu\rho}\,$
and $C_{\mu\nu\rho}$ was defined in (1.39). In order to reproduce the
different corrections for each string we need to use (1.42) again. This
compact form of the Bergshoeff-de Roo action was introduced in [14].

The action principle (1.47) is invariant under Lorentz transformations if we
impose the following Green-Schwarz mechanism,

$\displaystyle\delta_{\Lambda}b_{\mu\nu}=\frac{1}{2}(a-b)\partial_{[\mu}\Lambda^{ab}w_{\nu]ab}-\frac{1}{4}(a+b)\partial_{[\mu}\Lambda^{ab}H_{\nu]ab}\,.$
(1.50)

To prove the equivalence between $S_{\rm MT}$ and $S_{\rm BdR}$ we need to
consider the following non-covariant field redefinition for the $b$-field,

$\displaystyle
b^{\rm{MT}}_{\mu\nu}=b^{\rm{BdR}}_{\mu\nu}+\frac{1}{4}(a+b)H_{[\mu}{}^{ab}w_{\nu]ab}\,,$
(1.51)

and extra covariant field redefinitions [14]. Since both approaches are
equivalent, let us now fix the parameters $(a,b)=(-1,0)$ in (1.47) in order to
review an interesting relation found by Bergshoeff-de Roo. The action
principle is given by

$\displaystyle\int
d^{D}x\sqrt{-g}e^{-2\phi}(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\tilde{H}_{\mu\nu\lambda}\tilde{H}^{\mu\nu\lambda}+\frac{1}{8}R^{(-)}_{\mu\nu}{}^{ab}R^{(-)\mu\nu}{}_{ab})$
(1.52)

where
$\tilde{H}_{\mu\nu\rho}=3\left(\partial_{[\mu}b_{\nu\rho]}+\frac{1}{2}C^{(-)}_{\mu\nu\rho}\right)\,$
and the only higher-derivative correction to the symmetry transformations is
the Green-Schwarz mechanism,

$\displaystyle\delta_{\Lambda}b_{\mu\nu}=-\frac{1}{2}\partial_{[\mu}\Lambda^{ab}w^{(-)}_{\nu]ab}\,.$
(1.53)

The key observation here is that the structure of the action and
transformations resembles the form of a Yang-Mills theory coupled to the NS-NS
sector. In fact, the Lagrangian (1.52) is rather similar to (1.21), and the
Green-Schwarz transformation (1.53) is the analogous to (1.24). Moreover, in
both Lagrangians the 3-form is deformed by Chern-Simons terms. Therefore, here
we review an alternative way for constructing the four-derivative
gravitational corrections in the heterotic supergravity. Let us define the
gauge generators $(t_{i})^{ab}$ such that

$\displaystyle(t_{i})^{ab}(t^{i})_{cd}$ $\displaystyle=$
$\displaystyle-\delta_{[c}^{a}\delta_{d]}^{b}\,,$ (1.54)
$\displaystyle(t^{i})_{ab}(t_{j})^{ab}$ $\displaystyle=$
$\displaystyle-\delta_{j}^{i}\,,$ (1.55) $\displaystyle
f_{i}{}^{kl}(t^{i})_{ab}$ $\displaystyle=$ $\displaystyle
2\sqrt{2}(t^{k})_{[a|c}(t^{l})^{c}{}_{b]}\,.$ (1.56)

The gauge generators are antisymmetric in their flat indices,
$(t_{i})^{ab}=-(t_{i})^{ba}$ and, thanks to them, each generic gauge
vector/parameter can be written in the following way,

$\displaystyle v^{i}$ $\displaystyle=$ $\displaystyle-v^{ab}(t^{i})_{ab}\,,$
(1.57) $\displaystyle\lambda^{i}$ $\displaystyle=$
$\displaystyle-\lambda^{ab}(t^{i})_{ab}\,.$ (1.58)

If we apply the previous relations to $\delta_{\lambda}b_{\mu\nu}$ in (1.24)
we find

$\displaystyle\delta_{\lambda}b_{\mu\nu}=\partial_{[\mu}\lambda_{ab}A_{\nu]}{}^{ab}\,.$
(1.59)

Now we identify

$\displaystyle\lambda_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\Lambda_{ab}\,,$ (1.60) $\displaystyle
A_{\nu}{}^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}w^{(-)}_{\nu}{}^{ab}\,,$ (1.61)

obtaining the Lorentz Green-Schwarz transformation from the gauge Green-
Schwarz transformation
$\delta_{\lambda}b_{\mu\nu}\rightarrow\delta_{\Lambda}b_{\mu\nu}$. Moreover if
we apply (1.56) at the level of the action considering the identification
(1.61), it is straightforward to find

$\displaystyle-\frac{1}{4}F_{\mu\nu}{}^{i}F^{\mu\nu}{}_{i}$
$\displaystyle\rightarrow$
$\displaystyle\frac{1}{8}R^{(-)}_{\mu\nu}{}^{ab}R^{(-)\mu\nu}{}_{ab}$ (1.62)
$\displaystyle 3(\partial_{[\mu}b_{\nu\rho]}-C^{(g)}_{\mu\nu\rho})$
$\displaystyle\rightarrow$ $\displaystyle
3(\partial_{[\mu}b_{\nu\rho]}+\frac{1}{2}C^{(-)}_{\mu\nu\rho})\,.$ (1.63)

The assignment (1.62) indicates that we can identify,

$\displaystyle
F_{\mu\nu}{}^{ab}=-\frac{1}{\sqrt{2}}R^{(-)}_{\mu\nu}{}^{ab}\,.$ (1.64)

In [9], the authors use the previous identification to construct a higher-
derivative heterotic supergravity. Nowadays, an active topic of investigation
is to follow a similar program by considering DFT. The main goal of the next
lectures is to show that it is possible to perform identifications such as
(1.60) and (1.61) in a DFT framework to find a higher-derivative DFT
formulation related to the Bergshoeff-de Roo approach.

###  1.3 Exercises

  1. 1.

Show that the curvature for the $b$-field can be written as
$H_{\mu\nu\rho}=\nabla_{[\mu}b_{\nu\rho]}$ in bosonic supegravity. Then show
its Bianchi identity $\nabla_{[\mu}H_{\nu\rho\sigma]}=0$.

  2. 2.

Consider a generic vector $v^{\mu}$ and show that the closure
$\Big{[}\delta_{\xi_{1}},\delta_{\xi_{2}}\Big{]}v^{\mu}=\delta_{\xi_{21}}v^{\mu}\,$
holds if $\xi_{12}=-\xi_{21}$ is given by (1.14).

  3. 3.

Using the Jacobi identity for $f_{ijk}$, i.e.,
$f_{il}{}^{m}f_{jk}{}^{l}+f_{jl}{}^{m}f_{ki}{}^{l}+f_{kl}{}^{m}f_{ij}{}^{l}=0\,,$
show that $F_{\mu\nu i}$ transforms covariantly under non-abelian gauge
transformations.

  4. 4.

Using
$f^{i}{}_{jk}f_{i}{}^{mn}=2\alpha^{\prime}\delta_{[j}^{m}\delta_{k]}^{n}$
compute $\delta_{\lambda}C^{(g)}_{\mu\nu\rho}$.

  5. 5.

Consider a generic gauge vector $v^{i}$. Show
$\big{[}\nabla_{\mu},\nabla_{\nu}\big{]}v^{i}=f_{jk}{}^{i}v^{j}F_{\mu\nu}{}^{k}$.

  6. 6.

Show that
$\nabla_{[\mu}C^{(g)}_{\nu\rho\sigma]}=\frac{1}{4}F_{[\mu\nu}{}^{i}F_{\rho\sigma]i}$.
Then use this result to obtain the Bianchi identity for
$\bar{H}_{\mu\nu\rho}$.

  7. 7.

Consider a generic flat vector $v^{a}$ and compute
$\delta_{\Lambda}(\partial_{\mu}v^{a})$ and
$\delta_{\Lambda}(\nabla_{\mu}v^{a})$. Compare both expressions to obtain
$\delta_{\Lambda}w_{\mu ab}$.

  8. 8.

Use the vielbein compatibility to obtain
$w_{abc}=-e^{\mu}{}_{[a}{}e^{\nu}{}_{b]}{}\partial_{\mu}e_{\nu
c}+e^{\mu}{}_{[a}{}e^{\nu}{}_{c]}{}\partial_{\mu}e_{\nu
b}+e^{\mu}{}_{[b}{}e^{\nu}{}_{c]}{}\partial_{\mu}e_{\nu a}$ .

  9. 9.

Show that $\delta_{\Lambda}\bar{H}_{\mu\nu\rho}=0$ in the Metsaev-Tseytlin
approach and $\delta_{\Lambda}\tilde{H}_{\mu\nu\rho}=0$ in the Bergshoeff-de
Roo approach.

  10. 10.

Compute $\delta_{\Lambda}b^{MT}_{\mu\nu}$ from (1.51) to prove that the
b-redefinition holds.

  11. 11.

Compute the following decomposition $R^{(\pm)}_{\mu\nu ab}=R_{\mu\nu
ab}\mp\nabla_{[\mu}H_{\nu]ab}+\frac{1}{2}H_{[\mu a}{}^{c}H_{\nu]cb}$ where
$H_{\mu ab}=H_{\mu\nu\rho}e^{\nu}{}_{a}e^{\rho}{}_{b}$.

  12. 12.

Show the following identities:

$\displaystyle(\nabla_{\mu}H_{\nu\rho\sigma})H^{\mu\rho}{}_{\tau}H^{\nu\sigma\tau}=0\,,\quad(\nabla_{\mu}H^{\nu\rho\sigma})R^{\mu}{}_{\nu\rho\sigma}=0\,.$
(1.65)

  13. 13.

Show that the assignments (1.62) and (1.63) hold.

##  2 Lecture 2: Double field theory and higher-derivative terms

As we mentioned in the previous lecture, the different formulations of string
theory are defined on a $D$-dimensional manifold. From a phenomenological
point of view, this $D$-dimensional target space is often divided into an
external non-compact spacetime $M_{D-N}$ and an internal compact space
$M_{N}$,

$\displaystyle M_{D}=M_{D-N}\times M_{N}\,,$ (2.1)

where the simplest case is to consider an internal toroidal manifold. In this
scenario, the resulting theory is invariant under different symmetries. One of
these symmetries is T-duality or, more precisely, $O(N,N)$ invariance. This
symmetry group appears after compactifying the universal NS-NS sector on a
N-dimensional torus and it is an exact symmetry of string theory [25].

The main idea of double field theory (DFT) [10] [20] [21] is to rewrite a
supergravity theory as a T-duality invariant theory before compactification.
In this sense, DFT represents an $O(D,D)$ invariant theory, where $D$ is the
dimension of the target space of the embedded supergravity666For complementary
material about DFT see Dr. Gianluca Inverso’s lecture notes and the following
DFT reviews [22].. All the DFT fields and parameters are $O(D,D)$ multiplets
or, in other words, covariant objects. DFT coordinates lie in the fundamental
representation of $O(D,D)$ whose dimension is $2D$. Then
$X^{M}=(\tilde{x}_{\mu},x^{\mu})$ are coordinates of a double space. Here
$x^{\mu}$ are the coordinates of the embedded supergravity, and
$\tilde{x}_{\mu}$ are $D$ extra coordinates that we need for consistency. The
dual coordinates are taken away considering that fields and parameters depend
only on $x^{\mu}$ and therefore $\tilde{\partial}=0$. This is, in fact, the
simplest solution to the “strong constraint”,

$\partial_{M}(\partial^{M}\star)=(\partial_{M}\star)(\partial^{M}\star)=0\,,$
(2.2)

where $\star$ means a product of arbitrary DFT fields/parameters. Contractions
here are given by the $O(D,D)$ invariant metric, $\eta_{MN}$.

###  2.1 Part A: zeroth-order formulation

The DFT construction can be performed following four steps,

  1. 1.

Double geometry: We consider a double geometry with coordinates $X^{M}$ with
$M=0,\dots,2D-1$. We equip a group invariant metric $\eta_{MN}$ and its
inverse $\eta^{MN}$. These metrics are 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 $V^{A}$, with $A$ a flat index,
$A=0,\dots,2D-1$. 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,

$\displaystyle P_{AB}=\frac{1}{2}\left(\eta_{AB}-{\cal H}_{AB}\right)\,,\ \ \
\ \overline{P}_{AB}=\frac{1}{2}\left(\eta_{AB}+{\cal H}_{AB}\right)\ \,,$
(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)
$\displaystyle{P}_{{AB}}{\overline{P}}^{B}{}_{C}={\overline{P}}_{{AB}}{P}^{B}{}_{C}=0\,,$
$\displaystyle\quad{\overline{P}}_{{AB}}+{P}_{{AB}}=\eta_{{AB}}\,.$

Thus, $P$ and $\bar{P}$ project onto complementary orthogonal subspaces and we
can write any arbitrary vector as

$\displaystyle
V^{A}=P^{A}{}_{B}V^{B}+\overline{P}^{A}{}_{B}V^{B}=V^{\underline{A}}+V^{\overline{A}}\,.$
(2.5)

The notation underline and overline means that we use the projectors to
lower/raise the indices, instaed of using an invariant metric.

  2. 2.

Symmetries: DFT is a T-duality invariant formulation. Moreover, all the fields
and parameters are written in representations of the duality group and,
consequently, duality invariance is always guaranteed. Infinitesimal $O(D,D)$
transformations acting on an arbitrary double vector reads

$\displaystyle\delta_{h}V^{M}=V^{N}h_{N}{}^{M}\,,$ (2.6)

where $h\in o(D,D)$ 777We use the lower-case because $h$ belongs to the
associated algebra. is an arbitrary parameter.

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.,

$\displaystyle\delta_{\hat{\xi}}V^{M}={\mathcal{L}}_{\hat{\xi}}V^{M}=\hat{\xi}^{N}\partial_{N}V^{M}+(\partial^{M}\hat{\xi}_{P}-\partial_{P}\hat{\xi}^{M})V^{P}+\omega\partial_{N}\hat{\xi}^{N}V^{M}\,.$
(2.7)

In the previous expression we consider a generic parameter $\hat{\xi}^{M}$ and
a density weight factor $\omega$. The generalized Lie derivative differs from
the ordinary one since we need to ensure ${\cal L}_{\hat{\xi}}\eta_{MN}=0$.
The closure of the generalized diffeomorphism transformations on an arbitrary
generalized tensor $V^{M}{}_{N}$,

$\displaystyle\Big{[}\delta_{\hat{\xi}_{1}},\delta_{\hat{\xi}_{2}}\Big{]}V^{M}{}_{N}=\delta_{\hat{\xi}_{21}}V^{M}{}_{N}\,,$
(2.8)

is provided by the C-bracket,

$\displaystyle\hat{\xi}^{M}_{12}=\hat{\xi}^{P}_{1}\frac{\partial\hat{\xi}^{M}_{2}}{\partial
X^{P}}-\frac{1}{2}\hat{\xi}^{P}_{1}\frac{\partial\hat{\xi}_{2P}}{\partial
X_{M}}-(1\leftrightarrow 2)\,.$ (2.9)

An important comment here is that the closure of the generalized
diffeomorphisms requires the strong constraint (2.2). Moreover, and from an
algebraic point of view, DFT contains a non-trivial Jacobiator and therefore
only satisfies the Jacobi identity ‘up to homotopy’ [23].

On the other hand, the partial derivative of a generic double vector does not
transform as a tensor. Therefore we define a covariant derivative as follows

$\displaystyle D_{M}V^{N}=\partial_{M}V^{N}-\Gamma_{M}{}^{N}{}_{P}V^{P}\,.$
(2.10)

The compatibility $D_{M}\eta_{NP}=0$ implies $\Gamma_{MNP}=-\Gamma_{MPN}$. It
is usual to impose

$\displaystyle{\cal T}_{MNP}=3\Gamma_{[MNP]}=0\,,$ (2.11)

since ${\cal T}_{MNP}$ plays the role of generalized torsion. A curious aspect
of the double geometry is that there are not enough compatibility conditions
to fully determine $\Gamma_{MNP}$ [26], unlike in general relativity.
Consequently, the generalized Riemann tensor cannot be fully determined but,
nevertheless, a generalized Ricci tensor ${\cal R}_{MN}$ can be constructed as
well as a generalized Ricci scalar ${\cal R}$, and these are fully determined
in terms of the fundamental fields of DFT.

Another symmetry of DFT is the double Lorentz transformation that acts in the
following way,

$\displaystyle\delta_{\Lambda}V^{A}=V^{B}\Lambda_{B}{}^{A}\,,$ (2.12)

on an arbitrary flat vector $V^{A}$. Demanding $\delta_{\Lambda}\eta_{AB}=0$
we have $\Lambda_{AB}=-\Lambda_{BA}$. Moreover, using the
decomposition/notation (2.5), the condition $\delta_{\Lambda}{\cal H}_{AB}=0$
implies

$\displaystyle\Lambda_{\underline{A}\overline{B}}=\Lambda_{\overline{A}\underline{B}}=0\,.$
(2.13)

The partial derivative of an arbitrary flat vector $V^{A}$ does not transform
as a tensor and therefore we introduce a flat covariant derivative,

$\displaystyle D_{M}V^{A}=\partial_{M}V^{A}-W_{M}{}^{A}{}_{B}V^{B}\,,$ (2.14)

where $W_{MAB}=-W_{MBA}$ is the generalized spin connection. Imposing
compatibility with the generalized frame we find that $W_{MAB}$ is not fully
determined.

  3. 3.

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 satisfies

$\displaystyle E_{MA}{\cal H}^{AB}E_{NB}$ $\displaystyle=$ $\displaystyle{\cal
H}_{MN}\,,$ (2.15) $\displaystyle E_{MA}\eta^{AB}E_{NB}$ $\displaystyle=$
$\displaystyle\eta_{MN}\,.$ (2.16)

DFT can be written in terms of $d$ and ${\cal H}_{MN}$, the latter known as
the generalized metric. In this case, the double Lorentz invariance is
implicit. The generalized metric is an $O(D,D)$ element, i.e.

$\displaystyle{\cal H}_{MP}\eta^{PQ}{\cal H}_{QN}=\eta_{MN}\,,$ (2.17)

and with the help of this dynamical metric and $\eta_{MN}$ one can define
curved projectors,

$\displaystyle P_{MN}=\frac{1}{2}\left(\eta_{MN}-{\cal H}_{MN}\right)\,,\ \ \
\ \ \overline{P}_{MN}=\frac{1}{2}\left(\eta_{MN}+{\cal H}_{MN}\right)\ .$
(2.18)

The previous projectors satisfy

$\displaystyle{\overline{P}}_{{MQ}}{\overline{P}}^{Q}{}_{N}={\overline{P}}_{{MN}}\,,$
$\displaystyle\quad{P}_{{MQ}}{P}^{Q}{}_{N}={P}_{{MN}},$ (2.19)
$\displaystyle{P}_{{MQ}}{\overline{P}}^{Q}{}_{N}={\overline{P}}_{{MQ}}{P}^{Q}{}_{N}=0\,,$
$\displaystyle\quad{\overline{P}}_{{MN}}+{P}_{{MN}}=\eta_{{MN}}\,,$

similar to the flat projectors. The main difference here is that $P_{AB}$ and
$\overline{P}_{AB}$ are constants, while the curved ones are not for an
arbitrary double background.

The fundamental fields transform with respect to generalized diffeomorphisms
and double Lorentz transformations as follows,

$\displaystyle\delta_{\hat{\xi},\Lambda}E_{MA}$ $\displaystyle=$
$\displaystyle{\cal L}_{\hat{\xi}}E_{MA}+E_{MB}\Lambda^{B}{}_{A}\,,$ (2.20)
$\displaystyle\delta_{\hat{\xi}}d$ $\displaystyle=$
$\displaystyle\hat{\xi}^{N}\partial_{N}d-\frac{1}{2}\partial_{M}\hat{\xi}^{M}\,.$
(2.21)

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 $\omega=1$. On the other hand,
both $E_{MA}$ and ${\cal H}_{MN}$ satisfy compatibility conditions

$\displaystyle D_{M}{\cal H}_{NP}$ $\displaystyle=$ $\displaystyle 0\,,\quad
D_{M}{E}_{NP}=0\,.$ (2.22)

The generalized fluxes are defined as

$\displaystyle F_{ABC}=3E_{[A}E^{M}{}_{B}E_{MC]}\,,$ (2.23)

with $E_{A}=\sqrt{2}E^{M}{}_{A}\partial_{M}$. It is possible to determine the
totally antisymmetric part of the spin connection,
$W_{ABC}\equiv\sqrt{2}W_{MBC}E^{M}{}_{A}$, in the following way

$\displaystyle F_{ABC}=-3W_{[ABC]}\,.$ (2.24)

Finally, it is convenient to define

$\displaystyle F_{\underline{M}\overline{A}\overline{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}E_{M}{}^{\underline{C}}F_{\underline{C}\overline{A}\overline{B}}\,,$
(2.25) $\displaystyle F_{\overline{M}\underline{A}\underline{B}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}E_{M}{}^{\overline{C}}F_{\overline{C}\underline{A}\underline{B}}\,,$
(2.26)

for later use. The transformation rule of the previous objects is

$\displaystyle\delta_{\hat{\xi},\Lambda}F_{\underline{M}\overline{A}\overline{B}}$
$\displaystyle=$ $\displaystyle{\cal
L}_{\hat{\xi}}F_{\underline{M}\overline{A}\overline{B}}+\partial_{\underline{M}}\Lambda_{\overline{A}\overline{B}}+2F_{\underline{M}\overline{C}[\overline{B}}\Lambda^{\overline{C}}{}_{\overline{A}]}\,,$
(2.27)
$\displaystyle\delta_{\hat{\xi},\Lambda}F_{\overline{M}\underline{A}\underline{B}}$
$\displaystyle=$ $\displaystyle{\cal
L}_{\hat{\xi}}F_{\overline{M}\underline{A}\underline{B}}+\partial_{\overline{M}}\Lambda_{\underline{A}\underline{B}}+2F_{\overline{M}\underline{C}[\underline{B}}\Lambda^{\underline{C}}{}_{\underline{A}]}\,.$
(2.28)

  4. 4.

Action principle: The action of DFT is given by

$\displaystyle\int d^{2D}Xe^{-2d}{\cal R}(E,d)\,,$ (2.29)

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 [27],

$\displaystyle{\cal R}(d,E_{MA})$ $\displaystyle=$ $\displaystyle
2E_{\underline{A}}F^{\underline{A}}+F_{\underline{A}}F^{\underline{A}}-\frac{1}{6}{F}_{\underline{ABC}}F^{\underline{ABC}}-\frac{1}{2}{F}_{\overline{A}\underline{BC}}F^{\overline{A}\underline{BC}}\,,$
(2.30)

where $F_{A}=\sqrt{2}\partial^{M}E_{MA}-2E_{A}d\,.$

The present formulation is enough to recover the action principle (1.1) and
its symmetry rules, as we will see. For this reason, many extensions of DFT
have been studied. Frameworks related to the canonical formulation of DFT can
be found in [29], the inclusion of the R-R sector can be found in [30], while
supersymmetric extensions were worked out in [31]. Now we are interested in
obtaining extra non-abelian terms from DFT. Therefore, in the following part,
we generalize the construction presented here.

####  2.1.1 Heterotic/gauged double field theory

The duality group of the heterotic/gauged formulation of DFT [32] is
$O(D,D+n)$ with $n$ the dimension of the gauge group. Since this group differs
from $O(D,D)$, the construction described in the previous part requires the
following substantial modifications:

  1. 1.

Double geometry: We need to use a double geometry with coordinates $X^{\cal
M}=(\tilde{x}_{\mu},x^{\mu},x^{i})$ with ${\cal M}=0,\dots,2D-1+n$ and
$i=1,\dots,n$. On each point of the extended double space we consider an
extended double tangent space, so we are able to define flat vectors $V^{\cal
A}$, with ${\cal A}$ a flat index, ${\cal A}=0,\dots,2D-1+n$.

  2. 2.

Symmetries: The generalized diffeomorphisms now contain an extra term that
depends on generalized structure constants $f_{\cal MNP}$,

$\displaystyle\delta_{\hat{\xi}}V^{\cal M}={\mathcal{L}}_{\hat{\xi}}V^{\cal
M}=\hat{\xi}^{\cal N}\partial_{\cal N}V^{\cal M}+(\partial^{\cal
M}\hat{\xi}_{\cal P}-\partial_{\cal P}\hat{\xi}^{\cal M})V^{\cal P}+f^{\cal
M}{}_{\cal NP}\hat{\xi}^{\cal N}V^{\cal P}+\omega\partial_{\cal
N}\hat{\xi}^{\cal N}V^{\cal M}\,,$ (2.31)

where $V^{\cal M}$ is an arbitrary double vector. The structure constants are
fully antisymmetric and satisfy a Jacobi rule [33],

$\displaystyle f_{{\cal MNP}}=f_{[{\cal MNP}]}\,,\qquad f_{[\cal MN}{}^{\cal
R}f_{{\cal P}]{\cal R}}{}^{\cal Q}=0\,.$ (2.32)

The closure of the transformations is given by a deformed bracket

$\displaystyle[\hat{\xi}_{1},\hat{\xi}_{2}]^{\cal
M}_{(C_{f})}=2\hat{\xi}^{\cal P}_{[1}\partial_{\cal P}\hat{\xi}_{2]}^{\cal
M}-\hat{\xi}_{[1}^{\cal N}\partial^{\cal M}\hat{\xi}_{2]\cal N}+f_{{\cal
PQ}}{}^{\cal M}\hat{\xi}_{1}^{\cal P}\hat{\xi}_{2}^{\cal Q}\,,$ (2.33)

which reduces to the C-bracket when the structure constants vanish.
Interesting enough, the closure requires the strong constraint (2.2) plus an
extra constraint,

$\displaystyle f_{\cal MNP}\partial^{\cal M}\star=0\,.$ (2.34)

We solve the new constraint considering that the generalized structure
constants are non-vanishing only when ${\cal M,N,P}=i,j,k$, i.e. $f_{\cal
MNP}=f_{ijk}$, and $\partial_{i}=0$.

The covariant derivative (2.10) remains unchanged, but $\Gamma_{\cal MNP}$
receives an extra term in its transformation rule

$\displaystyle\delta_{\hat{\xi}}\Gamma_{\cal MNP}=\delta^{\rm
ungauged}_{\hat{\xi}}\Gamma_{\cal MNP}+f_{\cal NQP}\partial_{\cal
M}\hat{\xi}^{\cal Q}\,,$ (2.35)

where the explicit form of the first term corresponds to an exercise (see
Problem Sheet 2.3). On the other hand, the double Lorentz symmetry remains
unchanged.

  3. 3.

Fundamental fields: The fundamental fields of heterotic/gauged DFT are a
generalized frame $E_{\cal M}{}^{\cal A}(X)$ and a generalized dilaton $d(X)$.
The only modification at this point is related to the construction of the
generalized fluxes $F_{\cal ABC}$, that now contain an extra term

$\displaystyle F_{\cal ABC}=3E_{[\cal A}E^{\cal M}{}_{\cal B}E_{\cal
MC]}+\sqrt{2}f_{\cal MNP}E^{\cal M}{}_{\cal A}E^{\cal N}{}_{\cal B}E^{\cal
P}{}_{\cal C}\,.$ (2.36)

Similarly to the heterotic supergravity case we assume that $f_{\cal MNP}$ has
the same units as derivatives.

  4. 4.

Action principle: The action of heterotic/gauged DFT is given by

$\displaystyle\int d^{2D+n}Xe^{-2d}{\cal R}(E,d)\,,$ (2.37)

where ${\cal R}(E,d)$ is a two derivative scalar under (2.31) and it is
invariant under Lorentz transformations. This action principle now contains
$f_{\cal MNP}$ contributions provided by the generalized fluxes (2.36) and,
schematically, ${\cal R}$ has the same form as (2.30) but promoting the
indices ${A}\rightarrow{\cal A}$.

####  2.1.2 Parametrization

Here we present the parametrization of the different parameters, metrics and
fields of heterotic/gauged DFT. The parametrization of the ungauged DFT is a
particular case of this one. We follow the same structure of the previous
part, step by step.

  1. 1.

Double geometry: The parametrization of the coordinates is $X^{\cal
M}=(\tilde{x}_{\mu},x^{\mu},x^{i})$. Since we solve the strong constraint
using $\tilde{\partial}^{\mu}=\partial_{i}=0$, the parameters and fields of
the resulting gauged supergravity do not depend on extra coordinates. The
parametrization of the invariant metric is

${\eta}_{{\cal
MN}}=\left(\begin{matrix}\eta^{\mu\nu}&\eta^{\mu}{}_{\nu}&\eta^{\mu}{}_{i}\\\
\eta_{\mu}{}^{\nu}&\eta_{\mu\nu}&\eta_{\mu i}\\\
\eta_{i}{}^{\nu}&\eta_{i\nu}&\eta_{ij}\end{matrix}\right)=\left(\begin{matrix}0&\delta^{\mu}{}_{\nu}&0\\\
\delta_{\mu}{}^{\nu}&0&0\\\ 0&0&\kappa_{ij}\end{matrix}\right)\ ,$ (2.38)

with $\mu,\nu=0,\dots,D-1$, $i,j=1,\dots,n$ and $\kappa_{ij}$ the Killing
metric of the gauge group. The flat indices can be split in the following way
${\cal A}=(\underline{a},\overline{a},\overline{i})$ where
$\underline{a}=1,\dots,D$, $\bar{a}=1,\dots,D$ and $\overline{i}$ runs from
$1,\dots,n$. The latter is the flat version of the gauge index. The Lorentz
invariant metrics are parametrized as

$\eta_{\cal AB}=\left(\begin{matrix}-\eta_{\underline{a}\underline{b}}&0&0\\\
0&\eta_{\bar{a}\bar{b}}&0\\\ 0&0&\kappa_{\bar{i}\bar{j}}\end{matrix}\right)\
,\quad{\cal H}_{\cal
AB}=\left(\begin{matrix}\eta_{\underline{a}\underline{b}}&0&0\\\
0&\eta_{\overline{a}\overline{b}}&0\\\
0&0&\kappa_{\bar{i}\bar{j}}\end{matrix}\right)\ ,$ (2.39)

where $\eta_{\underline{a}\underline{b}}$ and
$\eta_{\overline{a}\overline{b}}$ can be identified with a flat and constant
metric $\eta_{ab}$

$\displaystyle\eta_{\bar{a}\bar{b}}\delta_{ab}^{\bar{a}\bar{b}}=\eta_{\underline{a}\underline{b}}\delta_{ab}^{\underline{a}\underline{b}}=\eta_{ab}\,,$
(2.40)

and $\kappa_{\bar{i}\bar{j}}$ is the flat version of the Killing metric.
Defining the following constant object $e_{i}{}^{\bar{i}}$ we have a relation
between the Cartan-Killing metric and its flat version,

$\displaystyle\kappa_{ij}=e_{i}{}^{\bar{i}}\kappa_{\bar{i}\bar{j}}e_{j}{}^{\bar{j}}\,.$
(2.41)

The same holds for the inverses of these metrics, defining $e^{i}{}_{\bar{i}}$
as the inverse of $e_{i}{}^{\bar{i}}$.

Before analyzing the symmetry rules, we include in this part the
parametrization of the generalized frame,

$E^{\cal M}{}_{\cal
A}=\left(\begin{matrix}{E}_{\mu\underline{a}}&{E}^{\mu}{}_{\underline{a}}&E^{i}{}_{\underline{a}}\\\
E_{\mu\overline{a}}&E^{\mu}{}_{\overline{a}}&E^{i}{}_{\overline{a}}\\\
E_{\mu\overline{i}}&E^{\mu}{}_{\overline{i}}&E^{i}{}_{\overline{i}}\end{matrix}\right)\
=\
\frac{1}{\sqrt{2}}\left(\begin{matrix}-{e}_{\mu\underline{a}}-C_{\rho\mu}{e}^{\rho}{}_{\underline{a}}&{e}^{\mu}{}_{\underline{a}}&-A_{\rho}{}^{i}{e}^{\rho}{}_{\underline{a}}\,,\\\
e_{\mu\overline{a}}-C_{\rho\mu}{}e^{\rho}{}_{\overline{a}}&e^{\mu}{}_{\overline{a}}&-A_{\rho}{}^{i}e^{\rho}{}_{\overline{a}}\\\
\sqrt{2}A_{\mu
i}e^{i}{}_{\overline{i}}&0&\sqrt{2}e^{i}{}_{\overline{i}}\end{matrix}\right)\,,$
(2.42)

where ${e}_{\mu\underline{a}}$ and $e_{\mu\overline{a}}$ are a pair of
vielbeins for the same metric tensor $g_{\mu\nu}$, i.e.,

$\displaystyle
e_{\mu}{}^{\overline{a}}\eta_{\overline{a}\overline{b}}e_{\nu}{}^{\overline{b}}$
$\displaystyle=$ $\displaystyle g_{\mu\nu}\,,$ (2.43) $\displaystyle
e_{\mu}{}^{\underline{a}}\eta_{\underline{a}\underline{b}}e_{\nu}{}^{\underline{b}}$
$\displaystyle=$ $\displaystyle g_{\mu\nu}\,.$ (2.44)

We identify each vielbein (and their inverses) considering the following gauge
fixing of the double Lorentz transformations to a single copy of Lorentz
transformations,

$\displaystyle{e}_{\mu\underline{a}}\delta_{a}^{\underline{a}}$
$\displaystyle=$
$\displaystyle{e}_{\mu\overline{a}}\delta_{a}^{\overline{a}}=e_{\mu a}\,,$
(2.45) $\displaystyle{e}^{\mu}{}_{\underline{a}}\delta_{a}^{\underline{a}}$
$\displaystyle=$
$\displaystyle{e}^{\mu}{}_{\overline{a}}\delta_{a}^{\overline{a}}=e^{\mu}{}_{a}\,.$
(2.46)

Finally in (2.42) we use the notation
$C_{\mu\nu}=b_{\mu\nu}+\frac{1}{2}A_{\mu}{}^{i}A_{\nu i}$.

  2. 2.

Symmetries: The parametrization of the symmetry parameters is given by,

$\displaystyle\hat{\xi}^{\cal M}$ $\displaystyle=$
$\displaystyle(\zeta_{\mu},\xi^{\mu},\lambda^{i})$ (2.47)
$\displaystyle\Lambda_{\overline{a}\overline{b}}$ $\displaystyle=$
$\displaystyle\Lambda_{ab}\delta_{\overline{a}\overline{b}}^{ab}\,,\quad\Lambda_{\underline{a}\underline{b}}=-\Lambda_{ab}\delta_{\underline{a}\underline{b}}^{ab}$
(2.48) $\displaystyle\Lambda_{\overline{a}\overline{i}}$ $\displaystyle=$
$\displaystyle
0\,,\quad\Lambda_{\overline{i}\overline{j}}=f_{ijk}\lambda^{j}e^{k}{}_{\bar{i}}e^{i}{}_{\overline{j}}{}\,.$
(2.49)

The components $\Lambda_{\overline{a}\overline{i}}$ and
$\Lambda_{\overline{i}\overline{j}}$ are fixed to ensure $\delta
E^{\mu}{}_{\overline{i}}=0$ and $\delta E^{i}{}_{\overline{i}}=0$ as required
by (2.42). From (2.47) we can understand how the generalized diffeomorphisms
encode ordinary diffeomorphisms plus gauge transformations, while double
Lorentz transformation encode the ordinary Lorentz transformations. For
instance, from $\delta E^{\mu\overline{a}}$ we obtain

$\delta E^{\mu}{}_{\overline{a}}\delta^{\overline{a}}_{a}=\delta
e^{\mu}{}_{a}=\xi^{\rho}\partial_{\rho}e^{\mu}{}_{a}-\partial_{\rho}\xi^{\mu}e^{\rho}{}_{a}+e^{\mu}{}_{b}\Lambda^{b}{}_{a}\,,$
(2.50)

in agreement with (1.31).

  3. 3.

Fundamental fields: The parametrization of the generalized frame was given in
(2.42), while the parametrization of the generalized metric,

$\displaystyle{\cal H}_{\cal
MN}=\left(\begin{matrix}g^{\mu\nu}&-g^{\mu\rho}C_{\rho\nu}&-g^{\mu\rho}A_{\rho
i}\\\
-C_{\rho\mu}g^{\nu\rho}&g_{\mu\nu}+C_{\rho\mu}C_{\sigma\nu}g^{\rho\sigma}+A_{\mu}{}^{i}\kappa_{ij}A_{\nu}{}^{j}&C_{\rho\mu}g^{\rho\sigma}A_{\sigma
i}+A_{\mu}{}^{j}\kappa_{ji}\\\ -g^{\nu\rho}A_{\rho
i}&C_{\rho\nu}g^{\rho\sigma}A_{\sigma
i}+A_{\nu}{}^{j}\kappa_{ij}&\kappa_{ij}+A_{\rho i}g^{\rho\sigma}A_{\sigma
j}\end{matrix}\right)\ ,$ (2.51)

can be easily obtained from the $E_{\cal M}{}^{\cal A}{\cal H}_{\cal
AB}E_{\cal N}{}^{\cal B}={\cal H}_{\cal MN}$. This parametrization agrees with
the results of [28].

The generalized dilaton is given by

$\displaystyle e^{-2d}=\sqrt{-g}e^{-2\phi}\,.$ (2.52)

Using the parametrization of the generalized frame and dilaton, it is
straightforward to compute the parametrization of the different projections of
the generalized fluxes [19],

$\displaystyle F_{\overline{a}\underline{bc}}$ $\displaystyle=$
$\displaystyle-w^{(+)}_{abc}\delta_{\overline{a}\underline{bc}}^{abc}\,,$
(2.53) $\displaystyle F_{\underline{a}\overline{bc}}$ $\displaystyle=$
$\displaystyle w^{(-)}_{abc}\delta_{\underline{a}\overline{bc}}^{abc}\,,$
(2.54) $\displaystyle F_{\overline{abc}}$ $\displaystyle=$ $\displaystyle
3\left(w_{[{abc}]}-\frac{1}{6}\bar{H}_{{abc}}\right)\delta_{\overline{abc}}^{abc}\,,$
(2.55) $\displaystyle F_{\underline{abc}}$ $\displaystyle=$
$\displaystyle-3\left(w_{[abc]}+\frac{1}{6}\bar{H}_{abc}\right)\delta_{\underline{abc}}^{abc}\,,$
(2.56) $\displaystyle F_{\overline{i}\underline{ab}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}e^{\mu}{}_{a}e^{\nu}{}_{b}e{}_{i\overline{i}}F^{i}_{\mu\nu}\delta_{\underline{ab}}^{ab}\,,$
(2.57) $\displaystyle F_{\underline{a}\overline{i}\overline{j}}$
$\displaystyle=$
$\displaystyle-e^{i}{}_{\overline{i}}e^{j}{}_{\overline{j}}e^{\mu}{}_{a}A_{\mu}{}^{k}f_{ijk}\delta_{\underline{a}}^{a}\,,$
(2.58) $\displaystyle F_{\overline{ijk}}$ $\displaystyle=$
$\displaystyle\sqrt{2}e^{i}{}_{\overline{i}}e^{j}{}_{\overline{j}}e^{k}{}_{\overline{k}}f_{ijk}\,,$
(2.59) $\displaystyle F_{\underline{a}}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}e_{a}^{\mu}+e_{a}^{\mu}e_{b}^{\nu}\partial_{\mu}e^{b}_{\nu}-2e^{\mu}{}_{a}\partial_{\mu}\phi\right)\delta_{\underline{a}}^{a}\,,$
(2.60)

where

$\displaystyle\bar{H}_{abc}$ $\displaystyle=$ $\displaystyle
3e^{\mu}{}_{a}e^{\nu}{}_{b}e^{\rho}{}_{c}\left(\partial_{[\mu}b_{\nu\rho]}-A_{[\mu}^{i}\partial_{\nu}A_{\rho]i}+\frac{1}{3}f_{ijk}A_{\mu}{}^{i}A_{\nu}{}^{j}A_{\rho}{}^{k}\right)\,.$
(2.61)

  4. 4.

Action principle: The action of heterotic/gauged DFT reduces to (1.4) after
parametrization. For example, the last term of the heterotic/gauged DFT action
principle,
$-\frac{1}{2}F_{\bar{i}\underline{b}\underline{c}}F^{\bar{i}\underline{b}\underline{c}}$,
reproduces the term $-\frac{1}{4}F_{\mu\nu}{}^{i}F^{\mu\nu}{}_{i}$ when the
component $F_{\overline{i}\underline{b}\underline{c}}$ is replaced by (2.57).

###  2.2 Part B: the generalized Green-Schwarz mechanism

####  2.2.1 Heterotic DFT

In the first lecture we showed that the $b$-field transforms non-trivially
under Lorentz transformations when four-derivative terms are considered in the
heterotic supergravity action principle (we turn off the b-parameter),

$\displaystyle\delta_{\Lambda}b_{\mu\nu}=\frac{a}{2}\partial_{[\mu}\Lambda^{ab}w^{(-)}_{\nu]ab}\,,$
(2.62)

where in the last expression we chose the Bergshoeef-de Roo approach (1.53)
keeping $a$ undetermined. Our goal now is to deform the transformation of the
generalized frame in order to construct a higher-derivative DFT formulation
mimicking this behavior 888We turn off the gauged part, for simplicity. Once
the DFT is deformed, we can promote $O(D,D)\rightarrow O(D,D+n)$.. The idea is
to consider a generalized Green-Schwarz mechanism at the DFT level,

$\displaystyle\delta_{\Lambda}E_{M}{}^{\overline{A}}$ $\displaystyle=$
$\displaystyle
E_{M}{}^{\overline{B}}\Lambda_{\overline{B}}{}^{\overline{A}}+\frac{a}{2}T_{1M}{}^{\overline{A}}\,,$
(2.63) $\displaystyle\delta_{\Lambda}E_{M}{}^{\underline{A}}$ $\displaystyle=$
$\displaystyle
E_{M}{}^{\underline{B}}\Lambda_{\underline{B}}{}^{\underline{A}}+\frac{a}{2}T_{2M}{}^{\underline{A}}\,,$
(2.64)

where the natural proposal is

$\displaystyle
T_{1M}{}^{\overline{A}}=\partial_{P}\Lambda^{\overline{B}\overline{C}}F_{\underline{M}\overline{B}\overline{C}}E^{P\overline{A}}\,,$
(2.65)

since $F_{\underline{M}\overline{B}\overline{C}}$ is related to $w^{(-)}_{\mu
bc}$. Before testing this ansatz, let us show that the curved index of a
generic first-order correction to the generalized frame transformation must
satisfy

$\displaystyle T_{1M\overline{A}}$ $\displaystyle=$ $\displaystyle
T_{1\underline{M}\overline{A}}\,,$ (2.66) $\displaystyle T_{2M\underline{A}}$
$\displaystyle=$ $\displaystyle T_{2\overline{M}\underline{A}}\,.$ (2.67)

Any first order-transformation $\delta^{(1)}_{\Lambda}$ has to ensure the
following compatibility condition,

$\displaystyle\delta^{(1)}_{\Lambda}(E_{M}{}^{\overline{A}}E^{M\underline{B}})$
$\displaystyle=$ $\displaystyle 0\,,$ (2.68)

and then (2.66) and (2.67) are mandatory. Moreover, both relations are
compatible with

$\displaystyle\delta^{(1)}_{\Lambda}\eta_{AB}$ $\displaystyle=$ $\displaystyle
0\,,$ (2.69) $\displaystyle\delta^{(1)}_{\Lambda}{\cal H}_{AB}$
$\displaystyle=$ $\displaystyle 0\,.$ (2.70)

Finally, from $\delta^{(1)}_{\Lambda}\eta_{MN}$ we have

$\displaystyle\delta^{(1)}_{\Lambda}(E_{M}{}^{\overline{A}}E_{N\overline{A}}+E_{M}{}^{\underline{A}}E_{N\underline{A}})=0\,,$
(2.71)

and, consequently, $T_{2\overline{M}\underline{A}}$ must be constructed from
$T_{1\underline{M}\overline{A}}$ according to

$\displaystyle
T_{2(M}{}^{\underline{A}}E_{N)\underline{A}}=-T_{1(M}{}^{\overline{A}}E_{N)\overline{A}}\,.$
(2.72)

Then,

$\displaystyle
T_{2M}{}^{\underline{A}}=-\partial_{\overline{M}}\Lambda^{\overline{A}\overline{B}}F_{\underline{N}\overline{A}\overline{B}}E^{N\underline{A}}\,,$
(2.73)

and the generalized Green-Schwarz mechanism can be written as 999Convention:
$[\overline{N}\,\underline{M}]=\frac{1}{2}\overline{N}\,\underline{M}-\frac{1}{2}\overline{M}\,\underline{N}$
(we exchange indices but projections remain unchanged). [14] [15]

$\displaystyle\delta_{\Lambda}E_{M}{}^{A}=E_{M}{}^{B}\Lambda_{B}{}^{A}+a\partial_{[\overline{N}}\Lambda^{\overline{B}\overline{C}}F_{\underline{M}]\overline{B}\overline{C}}E^{NA}\,.$
(2.74)

On the other hand the generalized dilaton does not have a first order
transformation. Now we can test the ansatz in order to see if the Green-
Schwarz mechanism for the $b$-field is inherited. For simplicity we will study
ungauged DFT, so the duality group is $O(D,D)$. However, a crucial question
is: how do we know that the fields that we use to parametrize the generalized
frame in (2.42), $\Big{\\{}e,b\Big{\\}}$, are the same fields as the ones in
the Bergshoeff-de Roo approach, $\Big{\\{}e_{\rm BdR},b_{\rm BdR}\Big{\\}}$?
We have seen, for example, that the previous set differs from that used in the
Metsaev-Tseytlin approach, $\Big{\\{}e_{\rm MT},b_{\rm MT}\Big{\\}}$ , by a
$b$-field redefinition, as we showed in (1.51). Consequently, we do not know
if our set of fields agrees with other approaches. Moreover, the set of fields
that we are using to parametrize the DFT fields are covariant under Buscher
transformations [34], so it is very likely that field redefinitions will be
required.

The safest way to go ahead is to use an arbitrary set of fields,
$\Big{\\{}\tilde{e},\tilde{b}\Big{\\}}$, and to consider a new parametrization
for the generalized frame,

$E^{M}{}_{A}=\frac{1}{\sqrt{2}}\left(\begin{matrix}-{\tilde{e}}_{\mu\underline{a}}-\tilde{b}_{\rho\mu}{\tilde{e}}^{\rho}{}_{\underline{a}}&{\tilde{e}}^{\mu}{}_{\underline{a}}\,,\\\
\tilde{e}_{\mu\overline{a}}-\tilde{b}_{\rho\mu}{}\tilde{e}^{\rho}{}_{\overline{a}}&\tilde{e}^{\mu}{}_{\overline{a}}\end{matrix}\right)\,.$
(2.75)

If we want to obtain the transformation rule $\delta\tilde{e}^{\mu\bar{a}}$,
we need to inspect $\delta E^{\mu\bar{a}}$,

$\displaystyle\delta
E^{\mu\bar{a}}=E^{\mu\bar{b}}\lambda_{\bar{b}}{}^{\bar{a}}+\frac{a}{2}\bar{P}_{P}{}^{Q}\partial_{Q}\Lambda^{\overline{b}\overline{c}}(\frac{1}{\sqrt{2}}E_{M}{}^{\underline{a}}F_{\underline{a}\overline{b}\overline{c}})E^{P\overline{a}}\,.$
(2.76)

Parametrizing the previous expression we get,

$\displaystyle\delta\tilde{e}^{\mu a}=\tilde{e}^{\mu
b}\Lambda_{b}{}^{a}-\frac{a}{4}\tilde{\partial}^{a}\Lambda^{bc}\tilde{w}^{(-)}_{dbc}\tilde{e}^{\mu
d}\,,$ (2.77)

where $\tilde{\partial}_{a}=\tilde{e}^{\mu}{}_{a}\partial_{\mu}$ and
$\tilde{w}_{abc}=w_{abc}(\tilde{e})$. Let us observe that due to the
generalized Green-Schwarz mechanism, the inverse vielbein
$\tilde{e}^{\mu}{}_{a}$ obtains an undesired transformation as well as
$\tilde{e}_{\mu a}$. The latter can be easily computed considering

$\displaystyle\delta_{\Lambda}(\tilde{e}^{\mu
a}\tilde{e}_{\mu}{}^{b})=\delta^{(1)}_{\Lambda}(\eta^{ab})=0\,,$ (2.78)

and we get

$\displaystyle\delta\tilde{e}_{\mu a}=\tilde{e}_{\mu
b}\Lambda^{b}{}_{a}+\frac{a}{4}\partial_{\mu}\Lambda^{bc}\tilde{w}^{(-)}_{abc}\,.$
(2.79)

Since we want to match with the set of fields used by Bergshoeff-de Roo we
need to propose the following field redefinition for the vielbein,

$\displaystyle e_{\rm BdR}^{\mu a}$ $\displaystyle=$
$\displaystyle\tilde{e}^{\mu a}+\frac{a}{8}\tilde{e}^{\mu
b}\tilde{w}^{(-)}_{b}{}^{cd}\tilde{w}^{(-)}_{\nu cd}\tilde{e}^{\nu a}\,,$
(2.80) $\displaystyle e^{\rm BdR}_{\mu a}$ $\displaystyle=$
$\displaystyle\tilde{e}_{\mu
a}-\frac{a}{8}\tilde{w}^{(-)}_{a}{}^{cd}\tilde{w}^{(-)}_{\mu cd}\,,$ (2.81)

and similarly for the Lorentz parameter,

$\displaystyle\Lambda^{\rm
BdR}_{ab}=\Lambda_{ab}-\frac{a}{4}\tilde{w}^{(-)}_{[a}{}^{cd}\tilde{\partial}_{b]}\Lambda_{cd}\,.$
(2.82)

Inspecting the transformation rule of the $E_{\mu\bar{a}}$ it is
straightforward to obtain

$\displaystyle\delta\tilde{b}_{\mu\nu}=\frac{a}{2}\partial_{[\mu}\Lambda^{ab}\tilde{w}^{(-)}_{\nu]ab}$
(2.83)

which is exactly the transformation of the Bergshoeff-de Roo approach
presented in the previous lecture101010The way to match with the fields
presented in the original publication [9] is very well discussed in the
appendix of [19]. when we turn off the b-parameter. Some readers may be
worried about the $\tilde{w}_{\nu ab}$ in (2.83) but the difference between
${w}_{\nu ab}$ and $\tilde{w}_{\nu ab}$ can be neglected at this order since
it introduces $\alpha^{\prime 2}$ contributions to (2.83). Then
$\tilde{b}_{\mu\nu}=b^{\rm BdR}_{\mu\nu}$ and no more field redefinitions are
required. This generalized Green-Schwarz mechanism agrees with [35]. In the
next part we will generalize this method to include the bi-parametric case.

####  2.2.2 Bi-parametric extension

In the previous part we have given some arguments to propose a generalized
Green-Schwarz transformation at the heterotic DFT level. We mainly follow
[14], where the following bi-parametric proposal was analyzed in detail,

$\displaystyle\delta_{\Lambda}E_{M}{}^{A}=E_{M}{}^{B}\Lambda_{B}{}^{A}+a\partial_{[\overline{P}}\Lambda^{\overline{B}\overline{C}}F_{\underline{M}]\overline{B}\overline{C}}E^{PA}+b\partial_{[\underline{P}}\Lambda^{\underline{B}\underline{C}}F_{\overline{M}]\underline{B}\underline{C}}E^{PA}\,.$
(2.84)

Here the parameters are related to (super)gravity formulations when

$\displaystyle(a,b)=\begin{cases}(-1,-1)&\mbox{bosonic DFT \, , }\\\
(-1,0)&\mbox{heterotic DFT \, ,}\\\ (0,0)&\mbox{type II DFT \, .}\end{cases}$
(2.85)

It is important to clarify that the previous parameters allow us to access the
ungauged bosonic supergravity Lagrangians and their gravitational
corrrections. A higher-derivative study related to $Z_{2}$-transformations in
DFT was done in [36], preceding [14]. In [36] the combination (-1,-1) was
referred to as DFT(+) and the combination (-1,1) as DFT(-). We leave some
comments related to this theory for the next part 111111Nowadays some papers
use the name HSZ for Hohm, Siegel and Zwiebach after the authors of [16] for
the combination (-1,1), which is a very particular higher-derivative extension
of DFT..

Let’s see what field redefinitions are required in the bi-parametric case.
Parametrizing $\delta_{\Lambda}E^{\mu a}$ we have,

$\displaystyle\delta\tilde{e}^{\mu a}$ $\displaystyle=$
$\displaystyle\tilde{e}^{\mu
b}\Lambda_{b}{}^{a}-\frac{a}{4}\tilde{\partial}^{a}\Lambda^{bc}\tilde{w}^{(-)}_{dbc}\tilde{e}^{\mu
d}+\frac{b}{4}\tilde{\partial}^{\mu}\Lambda^{bc}\tilde{w}^{(+)}_{\nu
bc}\tilde{e}^{\nu a}\,.$ (2.86)

Therefore, the field redefinition for the vielbein now is 121212Once the
vielbein redefinition is imposed, then we must propose a dilaton redefinition
according to
$e^{-2d}=\sqrt{-\tilde{g}}e^{-2\tilde{\phi}}=\sqrt{-g}e^{-2\phi}$.,

$\displaystyle e_{\rm BdR}^{\mu a}$ $\displaystyle=$
$\displaystyle\tilde{e}^{\mu a}+\frac{a}{8}\tilde{e}^{\mu
b}\tilde{w}^{(-)}_{b}{}^{cd}\tilde{w}^{(-)}_{\nu cd}\tilde{e}^{\nu
a}-\frac{b}{8}\tilde{e}^{\mu b}\tilde{w}^{(+)}_{b}{}^{cd}\tilde{w}^{(+)}_{\nu
cd}\tilde{e}^{\nu a}\,,$ (2.87) $\displaystyle e^{\rm BdR}_{\mu a}$
$\displaystyle=$ $\displaystyle\tilde{e}_{\mu
a}-\frac{a}{8}\tilde{w}^{(-)}_{a}{}^{cd}\tilde{w}^{(-)}_{\mu
cd}+\frac{b}{8}\tilde{w}^{(+)}_{a}{}^{cd}\tilde{w}^{(+)}_{\mu cd}\,,$ (2.88)

while the Lorentz parameter field also needs a redefinition,

$\displaystyle\Lambda^{\rm
BdR}_{ab}=\Lambda_{ab}-\frac{a}{4}\tilde{w}^{(-)}_{[a}{}^{cd}\tilde{\partial}_{b]}\Lambda_{cd}+\frac{b}{4}\tilde{w}^{(+)}_{[a}{}^{cd}\tilde{\partial}_{b]}\Lambda_{cd}.$
(2.89)

Finally the $b$-field transformation is

$\displaystyle\delta\tilde{b}_{\mu\nu}=\frac{a}{2}\partial_{[\mu}\Lambda^{ab}\tilde{w}^{(-)}_{\nu]ab}-\frac{b}{2}\partial_{[\mu}\Lambda^{ab}\tilde{w}^{(+)}_{\nu]ab}\,.$
(2.90)

Up to this point we have deformed the double Lorentz transformations in a
consistent way. Moreover, it is straightforward to verify that the two-
parameter deformation closes as follows

$\displaystyle[\delta_{\Lambda_{1}},\delta_{\Lambda_{2}}]E_{MA}=\delta_{\Lambda_{21},\xi_{21}}E_{MA}\,.$
(2.91)

The previous computation is part of the Problem Sheet 2.3. However we still
need a recipe to construct the four-derivative action at the DFT level,

$\displaystyle S_{DFT}=\int d^{2D}Xe^{-2d}({\cal R}+a{\cal R}^{(-)}+b{\cal
R}^{(+)})\,.$ (2.92)

Since $\delta^{(1)}_{\Lambda}{\cal R}\neq 0$, then ${\cal R}^{(\pm)}\neq 0$ to
ensure the invariance of the action. We leave this point to discuss in the
third lecture.

####  2.2.3 Comments about HSZ theory

All $(a,b)$ combinations different from $(-1,-1)$, $(-1,0)$,$(0,0)$ produce
$O(D,D)$ invariant theories that are not string theories. One example is HSZ
theory, where $(a,b)=(-1,1)$, originally constructed in [16] and explored in
[37] [38] [39]. This theory can be written in the generalized metric
formulation, where Lorentz invariance is implicit. In this case the
generalized diffeomorphims are deformed as,

$\displaystyle\delta_{\hat{\xi}}{\cal H}_{MN}=$ $\displaystyle{\cal
L}_{\hat{\xi}}{\cal H}_{MN}-\frac{1}{2}\partial_{(M}{\cal
H}^{PQ}\partial_{P}K_{Q|N)}+\frac{1}{2}{\cal H}_{(M}{}^{P}{\cal
H}_{N)}{}^{Q}\partial_{P}{\cal H}^{RS}\partial_{R}K_{SQ}$ (2.93)
$\displaystyle-\partial_{P}{\cal H}_{Q(M}\partial_{N)}K^{QP}+{\cal
H}_{(M}{}^{P}{\cal H}_{N)}{}^{Q}\partial_{R}{\cal
H}_{SP}\partial_{Q}K^{SR}\,,$

where $K_{MN}=2\partial_{[M}\hat{\xi}_{N]}$. On the other hand, the
generalized dilaton does not have a first order transformation.

In HSZ the C-bracket is consistently deformed,

$\left[\delta_{\hat{\xi}_{1}}\,,\,\delta_{\hat{\xi}_{2}}\right]=\delta_{[\hat{\xi}_{2},\hat{\xi}_{1}]_{C^{\prime}}}\
,$ (2.94)

where

$\left[\hat{\xi}_{1},\hat{\xi}_{2}\right]_{C^{\prime}}^{M}=2\hat{\xi}^{P}_{[1}\partial_{P}\hat{\xi}_{2]}^{M}-\hat{\xi}_{[1}^{N}\partial^{M}\hat{\xi}_{2]N}+\partial_{P}\hat{\xi}_{[1}^{Q}\partial^{M}\partial_{Q}\hat{\xi}_{2]}^{P}\
.$ (2.95)

The four-derivative action principle of the HSZ theory at the supergravity
level is given by,

$\displaystyle S=\int
d^{D}x\sqrt{-g}e^{-2\phi}\left(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\widehat{H}_{\mu\nu\rho}\widehat{H}^{\mu\nu\rho}\right)\
,$ (2.96)

where

$\widehat{H}_{\mu\nu\rho}=3\,\partial_{[\mu}b_{\nu\rho]}+3\Omega_{\mu\nu\rho}\
,\ \ \ \
\Omega_{\mu\nu\rho}=\Gamma_{[\mu|\sigma|}^{\delta}\partial_{\nu}\Gamma_{\rho]\delta}^{\sigma}+\frac{2}{3}\Gamma_{[\mu|\sigma|}^{\delta}\Gamma_{\nu|\lambda|}^{\sigma}\Gamma_{\rho]\delta}^{\lambda}\
.$ (2.97)

In this formulation, the Chern-Simons terms deform the 3-form through
Christoffel connections and the $b$-field transformation is corrected
according to

$\delta^{(1)}_{\xi}b_{\mu\nu}=\partial_{[\mu}\partial_{|\rho|}\xi^{\sigma}\Gamma_{\nu]\sigma}^{\rho}\
.$ (2.98)

The Green-Schwarz mechanism appears as a correction to the diffeomorphism
transformation for this field. Both the dilaton and the metric tensor do not
receive corrections to their transformations.

While HSZ theory does not correspond to a string theory, it could be an
important theory to understand the $\alpha^{\prime}$ structure of string
theory for several reasons:

  1. 1.

The full higher-derivative Lagrangian (as a deformed DFT) is known to all
orders131313Stricktly speaking, we cannot say that this is an
$\alpha^{\prime}$ expansion, since this is not a string theory.. It was
computed in [39] in terms of a double metric ${\cal M}_{MN}$ (which is not an
$O(D,D)$ element) and a generalized dilaton $d$. It contains terms up to six
derivatives.

  2. 2.

The odd higher-derivative terms of HSZ theory (in powers of $\alpha^{\prime}$,
i.e., 4-derivative terms, 8-derivative terms, and so on), at the supergravity
level, have the same structure of the $\alpha^{\prime}$-corrections with odd
$b$-field for heterotic supergravity. For this reason HSZ does not have $(\rm
Riem)^{2}$ terms, but it has Chern-Simons contributions.

  3. 3.

The even higher-derivative terms of this theory (in powers of
$\alpha^{\prime}$, i.e., 6-derivative terms, 10-derivative terms, and so on),
at the supergravity level, have the same structure of the
$\alpha^{\prime}$-corrections with even $b$-field for bosonic supergravity (up
to an overall sign). For this reason HSZ has $(\rm Riem)^{3}$ terms. The
following action reproduces the on-shell cubic amplitudes of HSZ theory [38],

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{-g}e^{-2\phi}(R+4\nabla_{\mu}\phi\nabla^{\mu}\phi-\frac{1}{12}\widehat{H}_{\mu\nu\rho}\widehat{H}^{\mu\nu\rho}$
(2.99)
$\displaystyle-\frac{1}{48}\,R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\rho\sigma}R_{\rho\sigma}{}^{\mu\nu})\,.$

The coefficient of the Riemann-cubed term is minus the coefficient of the same
term in bosonic string theory and no such term appears in the heterotic
string. It was shown in [40] that interactions that contribute to three-point
amplitudes with three external gravitons lead to causality violations which
require the existence of an infinite tower of new particles with spin higher
than two. Such violations are avoided when the full structure of string theory
is taken into account [41]. In bosonic string theory there is also a non-zero
Gauss-Bonnet Riemann-cubed term $G_{3}$

$G_{3}=R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\rho\sigma}R_{\rho\sigma}{}^{\mu\nu}-2\,R^{\mu\nu\alpha\beta}R_{\nu\lambda\beta\gamma}R^{\lambda}{}_{\mu}{}^{\gamma}{}_{\alpha}\
,$ (2.100)

but its presence can only be seen from four-point amplitudes [42]. These terms
(up to a sign) are part of the HSZ Lagrangian, as shown in [39].

###  2.3 Exercises

  1. 1.

Consider a generic double vector $V^{M}$ and show that
$\Big{[}\delta_{\hat{\xi}_{1}},\delta_{\hat{\xi}_{2}}\Big{]}V^{M}=\delta_{\hat{\xi}_{21}}V^{M}$
holds (consider both ungauged/gauged DFT).

  2. 2.

Decide if the following projections are determined or undetermined:
$\Gamma_{\underline{M}\underline{N}\underline{P}}$,
$\Gamma_{\underline{M}\underline{N}\overline{P}}$,
$\Gamma_{\underline{M}\overline{N}\overline{P}}$,
$\Gamma_{\overline{M}\overline{N}\overline{P}}$, and write the determined
projections in terms of ${\cal H}_{MN}$.

  3. 3.

Demanding $\int d^{2D}Xe^{-2d}V\nabla_{M}V^{M}=-\int
d^{2D}Xe^{-2d}V^{M}\nabla_{M}V$ show that the trace of the connection is
$\Gamma_{MN}{}^{M}=-2\partial_{N}d$. Then show that $W_{AB}{}^{A}=-F_{B}$.

  4. 4.

Find $\delta_{\hat{\xi}}\Gamma_{MNP}$. Then show that $\Gamma_{[MNP]}$
transforms as a generalized tensor.

  5. 5.

Show that (2.53)-(2.60) hold.

  6. 6.

Using the parametrization (2.42), show that $\delta E^{\mu}{}_{\bar{i}}=\delta
E^{i}{}_{\bar{i}}=0$ implies (2.49).

  7. 7.

Show that $\delta E^{\mu}{}_{\underline{a}}\delta_{a}^{\underline{a}}$ also
reproduces (2.50).

  8. 8.

Compute the transformation rule for $b_{\mu\nu}$ and $A_{\mu i}$ coming from
heterotic/gauged DFT.

  9. 9.

* Show that the ungauged DFT action reduces to (1.1) after parametrization.

  10. 10.

Show that the redefinitions (2.81) trivialize the transformation of the
Bergshoeff-de Roo vielbein up to a Lorentz transformation.

  11. 11.

Compute $\delta\tilde{b}_{\mu\nu}$ from $\delta E_{\mu a}$ in order to obtain
(2.90).

  12. 12.

Compute $\Lambda_{21}$ and $\hat{\xi}_{21}$ to show that (2.91) holds.

  13. 13.

* Show that $\delta_{\Lambda}{\cal R}\neq 0$ when we deformed the generalized frame double Lorentz transformation. Impose $b=0$ (heterotic DFT) to simplify the computation, and then generalize the result to the bi-parametric case.

  14. 14.

Compute the closure of the deformed generalized diffeomorphisms acting on
${\cal H}_{MN}$ in HSZ theory.

  15. 15.

Show that $\delta_{\hat{\xi}}({\cal H}_{MP}\eta^{PQ}{\cal H}_{QN})=0$ in HSZ
theory.

##  3 Lecture 3: Extended space and higher-order formulations

###  3.1 Part A: a systematic procedure to first-order formulations

In the previous lecture we showed the form of a suitable generalization of the
Green-Schwarz mechanism at the DFT level. Our starting point was an ansatz for
the deformations of the double Lorentz transformations acting on the
generalized frame, and we proceeded very carefully to satisfy all the DFT
constraints. These transformations imply new four-derivative terms in the DFT
Lagrangian, but we still do not have a method to compute them. For this
reason, now we study a systematic procedure to reproduce both the symmetries
and the higher-derivative corrections of the DFT Lagrangian. This procedure
was originally presented in [24] together with a generalization to higher-
order corrections in heterotic DFT. We will discuss this point and the state
of the art of this topic in the last part.

####  3.1.1 Proposal

As we have seen in (1.2), it is possible to construct gravitational
$\alpha^{\prime}$-corrections considering a map between gauge and Lorentz
symmetries, provided by gauge generators. This relation was first observed by
Bergshoeff and de Roo [9], and the assignments were $A_{\mu
i}\rightarrow\frac{1}{\sqrt{2}}w^{(-)}_{\mu ab}$ and
$\lambda^{i}\rightarrow-\frac{1}{\sqrt{2}}\Lambda^{ab}$. Now we want to
generalized these identifications at the DFT level, but we deal with the
following problem: the $O(D,D+K)$ formulation of gauged DFT does not have a
generalized gauge field or parameter to perform the identifications. Here we
focus on an arbitrary gauge group of dimension $K$.

A possible solution to the previous problem is to break the $O(D,D+K)$
representations in terms of $O(D,D)$ representations [43]. This procedure
allows us to obtain a generalized gauged field/parameter at the DFT level. We
start by considering a generalized frame for gauged DFT ${\cal E}_{\cal
M}{}^{\cal A}$, where ${\cal M}=1,\dots,2D-1+K$ and ${\cal A}=1,\dots,2D+K$.
Keeping the notation of Lecture 2, we can split the extended indices as ${\cal
M}=(M,\alpha)$ and ${\cal A}=(A,\overline{\alpha})$, where $\alpha$ and
$\overline{\alpha}$ are curved/flat gauge indices that run from $1,\dots,K\,,$
respectively. The $O(D,D+K)$ frame can be parametrized in terms of $O(D,D)$
multiplets as

${\cal E}_{\cal M}{}^{\cal A}=\left(\begin{matrix}{\cal E}_{M}{}^{A}&{\cal
E}_{M}{}^{\bar{\alpha}}\\\ {\cal E}_{\alpha}{}^{A}&{\cal
E}_{\alpha}{}^{\bar{\alpha}}\end{matrix}\right)=\left(\begin{matrix}({\chi}^{\frac{1}{2}})_{M}{}^{N}E_{N}{}^{A}&-A_{M}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}\\\
{A}^{M}{}_{\alpha}E_{M}{}^{A}&({\Box}^{\frac{1}{2}})_{\alpha}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}\end{matrix}\right)\
,$ (3.1)

where $e_{\alpha}{}^{\overline{\alpha}}$ (and its inverse
$e^{\alpha}{}_{\overline{\alpha}}$) are constant objects that are related to
the Cartan-Killing metric $\kappa_{\alpha\beta}$ and its flat version
$\kappa_{\overline{\alpha}\overline{\beta}}$ as

$\displaystyle
e_{\alpha}{}^{\overline{\alpha}}\kappa_{\overline{\alpha}\overline{\beta}}e_{\beta}{}^{\overline{\beta}}$
$\displaystyle=$ $\displaystyle\kappa_{\alpha\beta}\,,$ (3.2) $\displaystyle
e^{\alpha}{}_{\overline{\alpha}}\kappa_{\alpha\beta}e^{\beta}{}_{\overline{\beta}}$
$\displaystyle=$ $\displaystyle\kappa_{\overline{\alpha}\overline{\beta}}\,.$
(3.3)

In (3.1) we defined the following objects,

$\displaystyle{\chi}_{MN}=\eta_{MN}-A_{M}{}^{\alpha}A_{N\alpha}$
$\displaystyle\longrightarrow$
$\displaystyle{(\chi^{\frac{1}{2}}})_{MN}=\eta_{MN}-\frac{1}{2}A_{M}{}^{\alpha}A_{N\alpha}+{\cal
O}(A^{4})\,,$ (3.4)
$\displaystyle{\Box}_{\alpha\beta}=\kappa_{\alpha\beta}-A_{M\alpha}A^{M}{}_{\beta}$
$\displaystyle\longrightarrow$
$\displaystyle{(\Box^{\frac{1}{2}})}_{\alpha\beta}=\kappa_{\alpha\beta}-\frac{1}{2}A_{M\alpha}A^{M}{}_{\beta}+{\cal
O}(A^{4})\,,$ (3.5)

and we impose

$\displaystyle E^{M\overline{A}}A_{M\alpha}=0\,,$ (3.6)

or equivalently

$\displaystyle A_{M\alpha}=A_{\underline{M}\alpha},$ (3.7)

in order to eliminate extra degrees of freedom. From the previous condition we
need ${\cal E}_{\alpha}{}^{\overline{A}}=0$, which will require a suitable
gauge fixing. On the other hand, the projectors ${\cal P}_{\cal MN}$,
$\overline{\cal P}_{\cal MN}$ and ${\cal P}_{\cal AB}$, $\overline{\cal
P}_{\cal AB}$ can be decomposed as

$P_{\cal MN}=\left(\begin{matrix}{P}_{MN}&0&\ \ 0\\\ 0&0&\ \ 0\\\ 0&0&\ \
0\end{matrix}\right)\,,\quad\overline{\cal P}_{\cal
MN}=\left(\begin{matrix}0&0&0\\\ 0&{\overline{P}}_{MN}&0\\\
0&0&{\kappa}_{\alpha\beta}\end{matrix}\right)\,,$ (3.8) $P_{\cal
AB}=\left(\begin{matrix}{P}_{AB}&0&\ \ 0\\\ 0&0&\ \ 0\\\ 0&0&\ \
0\end{matrix}\right)\,,\quad\overline{\cal P}_{\cal
AB}=\left(\begin{matrix}0&0&0\\\ 0&{\overline{P}}_{AB}&0\\\
0&0&{\kappa}_{\overline{\alpha}\overline{\beta}}\end{matrix}\right)\,.$ (3.9)

From an $O(D,D)$ perspective, $\Big{\\{}E_{MA},A_{M\alpha},d\Big{\\}}$ are now
the fundamental fields of the gauged DFT. Similarly to the field decomposition
(3.1), the generalized diffeomorphim transformations given by $\hat{\xi}_{\cal
M}$, and the double Lorentz transformations given by $\Gamma_{\cal A\cal B}$,
must be decomposed in a consistent way 141414Notice that we have changed the
notation for the double Lorentz parameter $\Lambda\rightarrow\Gamma$.. We
dedicate the next part for doing so.

####  3.1.2 Extended symmetries and gauge fixing

The symmetry transformations of the $O(D,D+K)$ fundamental fields are

$\displaystyle\delta{\cal E}_{\cal MA}$ $\displaystyle=$
$\displaystyle\hat{\xi}^{\cal N}\partial_{\cal N}{\cal E}_{\cal
MA}+(\partial_{\cal M}\hat{\xi}^{\cal P}-\partial^{\cal P}\hat{\xi}_{\cal
M}){\cal E}_{\cal PA}+f_{\cal MNP}\hat{\xi}^{\cal N}{\cal E}^{\cal P}{}_{\cal
A}+{\cal E}_{\cal MB}\Gamma^{\cal B}{}_{\cal A}\,,$ (3.10)
$\displaystyle\delta d$ $\displaystyle=$ $\displaystyle\hat{\xi}^{\cal
N}\partial_{\cal N}d-\frac{1}{2}\partial_{\cal M}\hat{\xi}^{\cal M}\,,$ (3.11)

where $f_{\cal MNP}$ only takes values when ${\cal MNP}=\alpha\beta\gamma$.
The $\partial_{\cal M}$ derivative is split according to $\partial_{\cal
M}=(\partial_{M},0)$, while the generalized diffeomorphism parameter as

$\displaystyle\hat{\xi}^{\cal M}$ $\displaystyle=$
$\displaystyle(\hat{\xi}^{M},\lambda^{\alpha})\,.$ (3.12)

Using the previous decomposition, $\delta d$ is what we expect. On the other
hand, the double Lorentz parameter, ${\Gamma_{\cal AB}}$, requires a gauge
fixing to ensure $\delta{\cal E}_{\alpha}{}^{\overline{A}}=0$ and $\delta
e_{i}{}^{\overline{i}}=0$. From the former we find,

$\displaystyle\delta{\cal
E}_{\alpha}{}^{\overline{A}}=-\partial^{P}\lambda_{\alpha}{\cal
E}_{P}{}^{\overline{A}}+{\cal
E}_{\alpha}{}^{\overline{\beta}}\Gamma_{\overline{\beta}}{}^{\overline{A}}=0\,.$
(3.13)

Notice that ${\cal E}_{M\overline{A}}=E_{M\overline{A}}$, and then,

$\displaystyle\Gamma_{\overline{\beta}}{}^{\overline{A}}=(\Box^{-1/2})^{\alpha}{}_{\beta}e^{\beta}{}_{\overline{\beta}}(\partial^{P}\lambda_{\alpha})E_{P}{}^{\overline{A}}\,.$
(3.14)

Similarly to the previous computation, we can now demand $\delta{\cal
E}_{\alpha}{}^{\overline{\alpha}}=\delta(\Box^{\frac{1}{2}})_{\alpha}{}^{\beta}e_{\beta}{}^{\overline{\alpha}}$,
which is the condition that we need to impose $\delta
e_{\alpha}{}^{\overline{\alpha}}=0$. This condition is satisfied if

$\displaystyle\Gamma_{\overline{\beta}\overline{\alpha}}=(\Box^{-1/2})^{\alpha}{}_{\beta}e^{\beta}{}_{[\overline{\beta}}\Big{(}-\hat{\xi}_{P}\partial^{P}{\cal
E}_{\alpha\overline{\alpha}]}+\partial^{P}\lambda_{\alpha}{\cal
E}_{P\overline{\alpha}]}-f_{\alpha\delta\gamma}\lambda^{\delta}{\cal
E}^{\gamma}{}_{\overline{\alpha}]}+\delta_{\Lambda}({\Box^{\frac{1}{2}}})_{\alpha\delta}e^{\delta}{}_{\overline{\alpha}]}\Big{)}\,.$
(3.15)

Then, equations (3.14) and (3.15) are the gauge fixing conditions that we need
to write a GDFT in terms of fundamental fields which are in representations of
$O(D,D)$. In the next part we will explore the transformation rules for these
fields.

####  3.1.3 Transformation rules for the $O(D,D)$ field content

We start by defining the following field

$\displaystyle
C_{M\alpha}=-A_{M}{}^{\beta}(\Box^{-\frac{1}{2}})_{\beta\alpha}$ (3.16)

which is constrained by $E_{M}{}^{\overline{A}}C_{M\alpha}=0$. We also define

$\displaystyle\Delta_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\kappa_{\alpha\beta}+C_{M\alpha}C^{M}{}_{\beta}$ (3.17)
$\displaystyle\Theta_{MN}$ $\displaystyle=$
$\displaystyle\eta_{MN}+C_{M}{}^{\alpha}C_{N\alpha}\,.$ (3.18)

Now we inspect the transformation of $\delta{\cal
E}_{M\overline{A}}=\delta{E}_{M\overline{A}}$. In terms of the $C_{M\alpha}$
field we have,

$\displaystyle\delta{E}_{M\overline{A}}={\cal
L}_{\hat{\xi}}E_{M\overline{A}}+E_{M}{}^{\overline{B}}\Lambda_{\overline{B}\overline{A}}+C_{M}{}^{\gamma}\partial^{P}\lambda_{\gamma}E_{P\overline{A}}\,,$
(3.19)

where we have identified

$\displaystyle\Gamma_{\overline{A}\overline{B}}=\Lambda_{\overline{A}\overline{B}}\,.$
(3.20)

At this point we observe that the transformation rule for $\delta
E_{M\overline{A}}$ now contains an extra gauge transformation. Our idea is to
construct gauge generators that allow us to identify

$\displaystyle
C_{M}{}^{\gamma}\partial_{P}\lambda_{\gamma}E^{P\overline{A}}\rightarrow\frac{a}{2}F_{\underline{M}}{}^{\overline{B}\overline{C}}\partial_{P}\Lambda_{\overline{B}\overline{C}}E^{P\overline{A}}\,,$
(3.21)

similarly to the Bergshoeff-de Roo identification. Let’s keep this idea on
mind but, before defining the gauge generators at the DFT level, we need to
inspect the other generalized frame projection $\delta E_{M\underline{A}}$,

$\displaystyle\delta{\cal
E}_{M\underline{A}}=\delta(\chi^{\frac{1}{2}})_{M}{}^{N}E_{N}{}^{\underline{A}}+(\chi^{\frac{1}{2}})_{M}{}^{N}\delta
E_{N}{}^{\underline{A}}\,.$ (3.22)

From the previous expresion we find,

$\displaystyle\delta E_{N\underline{A}}={\cal
L}_{\hat{\xi}}E_{N\underline{A}}+E_{N}{}^{\underline{B}}\Gamma_{\underline{B}\underline{A}}-(\Theta^{\frac{1}{2}})^{M}{}_{N}\Big{(}\partial_{M}\lambda^{\alpha}C^{R}{}_{\beta}(\Delta^{\frac{1}{2}})^{\beta}{}_{\alpha}+\delta_{\Lambda}(\Theta^{-\frac{1}{2}})_{M}{}^{R}\Big{)}E_{R\underline{A}}\,,$
(3.23)

which need a parameter redefinition since we want (3.23) to have a generalized
Green-Schwarz form. Then, we must define

$\displaystyle
Q_{N}{}^{R}=(\Theta^{\frac{1}{2}})^{M}{}_{N}\Big{(}\partial_{M}\lambda^{\alpha}C^{R}{}_{\beta}(\Delta^{\frac{1}{2}})^{\beta}{}_{\alpha}+\delta_{\Lambda}(\Theta^{-\frac{1}{2}})_{M}{}^{R}\Big{)}E_{R\underline{A}}\,,$
(3.24)

and
$S_{MN}=\overline{P}_{M}{}^{P}\partial_{P}\lambda^{\alpha}C_{N\alpha}-Q_{MN}$,
in order to identify

$\displaystyle\Gamma_{\underline{A}\underline{B}}=\Lambda_{\underline{A}\underline{B}}-E^{M}{}_{\underline{A}}S_{MN}E^{N}{}_{\underline{B}}\,.$
(3.25)

In terms of the previous parameter, the transformation of this component takes
its expected form,

$\displaystyle\delta E_{N\underline{A}}={\cal
L}_{\hat{\xi}}E_{N\underline{A}}+E_{N}{}^{\underline{B}}\Lambda_{\underline{B}\underline{A}}-\partial_{\overline{N}}\lambda^{\alpha}C^{R}{}_{\alpha}E_{R\underline{A}}\,.$
(3.26)

The only remaining transformation is $\delta C_{M\alpha}$. There exists
several ways to obtain it. For instance, if we focus on $\delta{\cal
E}_{M}{}^{\overline{\alpha}}$ we can obtain the transformation of the
$C$-field (or $A$-field) according to

$\displaystyle\delta{\cal
E}_{M}{}^{\overline{\alpha}}=\delta(-A_{M}{}^{\alpha})e_{\alpha}{}^{\overline{\alpha}}=\delta(C_{M}{}^{\gamma}(\Box^{\frac{1}{2}})_{\gamma}{}^{\beta})e_{\beta}{}^{\overline{\alpha}}\,.$
(3.27)

From the previous expression we find,

$\displaystyle\delta C_{M}{}^{\gamma}={\cal
L}_{\hat{\xi}}C_{M}{}^{\gamma}+\partial_{M}\lambda^{\gamma}-(\Box^{-1})^{\gamma\alpha}\partial_{\overline{M}}\lambda_{\alpha}+C_{M}{}^{\delta}(\Box^{\frac{1}{2}})_{\delta}{}^{\epsilon}e_{\epsilon}{}^{\overline{\beta}}\Gamma_{\overline{\beta}}{}^{\overline{\alpha}}e^{\beta}{}_{\overline{\alpha}}(\Box^{-\frac{1}{2}})^{\gamma}{}_{\beta}\,.$
(3.28)

Moreover, using (3.15) one gets

$\displaystyle\delta C_{M}{}^{\gamma}={\cal
L}_{\hat{\xi}}C_{M}{}^{\gamma}+\partial_{M}\lambda^{\gamma}-(\Box^{-1})^{\gamma\alpha}\partial_{\overline{M}}\lambda_{\alpha}+C_{M}{}^{\alpha}\partial^{P}\lambda_{\alpha}C_{P}{}^{\gamma}-C_{M}{}^{\alpha}f_{\alpha\beta}{}^{\gamma}\lambda^{\beta}\,.$
(3.29)

So far we have computed the transformation rules for all the fundamental
fields of this formulation of gauged DFT. In the next part we will introduce
the gauge generator $(t_{\alpha})^{\overline{A}\overline{B}}$ to obtain
(3.21).

####  3.1.4 First-order identification

We start by taking the leading order contribution from the C-transformation
(3.29),

$\displaystyle\delta C_{M}{}^{\gamma}={\cal
L}_{\hat{\xi}}C_{M}{}^{\gamma}+\partial_{\underline{M}}\lambda^{\gamma}-C_{M}{}^{\alpha}f_{\alpha\beta}{}^{\gamma}\lambda^{\beta}\,.$
(3.30)

Indeed we are expecting that the second term of this transformation turns into
a non-covariant double Lorentz transformation and the third one into a
covariant double Lorentz transformation, as in (2.27). If this happens, we
could identify the $C$-field with $F_{\underline{M}\overline{B}\overline{C}}$.

We define the gauge generators $(t_{\alpha})^{\overline{A}\overline{B}}$ such
that,

$\displaystyle(t^{\alpha})_{\overline{A}\overline{B}}(t_{\beta})^{\overline{A}\overline{B}}$
$\displaystyle=$ $\displaystyle X_{R}\delta_{\beta}^{\alpha}\,,$ (3.31)
$\displaystyle(t^{\alpha})_{\overline{A}\overline{B}}(t_{\alpha})^{\overline{C}\overline{D}}$
$\displaystyle=$ $\displaystyle
X_{R}\delta_{\overline{A}\overline{B}}^{\overline{C}\overline{D}}\,,$ (3.32)

and for generic gauge/Lorentz vectors and parameters we have

$\displaystyle V_{\overline{B}\overline{C}}$ $\displaystyle=$ $\displaystyle-
V_{\alpha}(t^{\alpha})_{\overline{B}\overline{C}}\,,$ (3.33)
$\displaystyle\lambda_{\overline{B}\overline{C}}$ $\displaystyle=$
$\displaystyle-\lambda_{\alpha}(t^{\alpha})_{\overline{B}\overline{C}}\,.$
(3.34)

The previous expressions mean that we are identifying the gauge symmetry of
GDFT with the right sector of the double Lorentz transformations. Once these
identifications are imposed, the GDFT turn into a DFT with higher-derivative
corrections.

The relation between the structure constants and the gauge generators is

$\displaystyle
f_{\alpha\beta}{}^{\gamma}(t_{\gamma})^{\overline{A}\overline{B}}=2(t_{[\alpha})^{[\overline{A}\overline{C}}(t_{\beta]})_{\overline{C}}{}^{\overline{B}]}\,.$
(3.35)

The map (3.32) can be applied on (3.30),

$\displaystyle\delta C_{M\overline{B}\overline{C}}={\cal
L}_{\hat{\xi}}C_{M\overline{B}\overline{C}}+\partial_{\underline{M}}\lambda_{\overline{B}\overline{C}}+C_{M}{}^{\alpha}f_{\alpha\beta\gamma}\lambda^{\beta}(t^{\gamma})_{\overline{B}\overline{C}}\,,$
(3.36)

and using (3.35) one gets

$\displaystyle\delta C_{M\overline{B}\overline{C}}={\cal
L}_{\hat{\xi}}C_{M\overline{B}\overline{C}}+\partial_{\underline{M}}\lambda_{\overline{B}\overline{C}}+2C_{M[\overline{B}|\overline{D}}\lambda^{\overline{D}}{}_{\overline{C}]}\,.$
(3.37)

Finally the identification is

$\displaystyle C_{M\overline{B}\overline{C}}$ $\displaystyle=$ $\displaystyle
F_{\underline{M}\overline{B}\overline{C}}\,,$ (3.38)
$\displaystyle\lambda_{\overline{B}\overline{C}}$ $\displaystyle=$
$\displaystyle\Lambda_{\overline{B}\overline{C}}\,,$ (3.39)

as expected. The correction to the transformation of the projections of the
generalized frame now takes the form of a generalized Green-Schwarz mechanism,

$\displaystyle\delta_{\Lambda}E_{M\overline{A}}$ $\displaystyle=$
$\displaystyle
E_{M\overline{B}}\Lambda^{\overline{B}}{}_{\overline{A}}+\frac{1}{X_{R}}F_{\underline{M}}{}^{\overline{B}\overline{C}}\partial^{P}\Lambda_{\overline{B}\overline{C}}E_{P\overline{A}}\,,$
(3.40) $\displaystyle\delta_{\Lambda}E_{M\underline{A}}$ $\displaystyle=$
$\displaystyle
E_{M\underline{B}}\Lambda^{\underline{B}}{}_{\underline{A}}-\frac{1}{X_{R}}\partial_{\overline{M}}\Lambda^{\overline{B}\overline{C}}F^{\underline{R}}{}_{\overline{B}\overline{C}}E_{R\underline{A}}\,,$
(3.41)

and therefore $\frac{1}{X_{R}}=\frac{a}{2}$. Up to this point, all our effort
has produced the same corrections that we have proposed in the previous
lecture to deform the Lorentz transformations. However, the identifications
(3.38) and (3.39) can also be applied on the $O(D,D+K)$ action principle in
order to construct the $4$-derivative corrections to the heterotic DFT
Lagrangian. This method can be performed to obtain the corrections associated
to the b-parameter, so we have found a systematic procedure to construct the
bi-parametric first-order corrections to the DFT action. We show the form of
all these corrections in the next part.

####  3.1.5 Fluxes and action principle

In order to construct the higher-derivative action principle we need some
components of the fluxes $\Big{\\{}{\cal F}_{\cal A},{\cal F}_{\cal
ABC}\Big{\\}}$. The $O(D,D+K)$ action is given by,

$\displaystyle\int d^{2D+K}Xe^{-2d}{\cal R}({\cal E},d)\,,$ (3.42)

where ${\cal R}({\cal E},d)$ is,

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle 2{\cal
E}_{\cal\underline{A}}{\cal F}^{\cal\underline{A}}+{\cal
F}_{\cal\underline{A}}{\cal F}^{\cal\underline{A}}-\frac{1}{6}{\cal
F}_{\underline{\cal ABC}}{\cal F}^{\cal\underline{ABC}}-\frac{1}{2}{\cal
F}_{\overline{\cal A}\underline{\cal BC}}{\cal F}^{\overline{\cal
A}\underline{\cal BC}}\,.$ (3.43)

It turns out that the expression for the generalized fluxes is

$\displaystyle{\cal F}_{\cal A}$ $\displaystyle=$
$\displaystyle\sqrt{2}\partial^{\cal M}{\cal E}_{\cal MA}-2{\cal E}_{\cal
A}d\,,$ (3.44) $\displaystyle{\cal F}_{\cal ABC}$ $\displaystyle=$
$\displaystyle 3{\cal E}_{[\cal A}{\cal E}^{\cal M}{}_{\cal B}{\cal E}_{\cal
MC]}+\sqrt{2}f_{\cal MNP}{\cal E}^{\cal M}{}_{\cal A}{\cal E}^{\cal N}{}_{\cal
B}{\cal E}^{\cal P}{}_{\cal C}\,.$ (3.45)

Let us start inspecting the single index flux. The identification between
$A_{M\alpha}$ and the generalized flux is

$\displaystyle A_{M\alpha}=-C_{M\alpha}\rightarrow-
F_{\underline{M}\overline{B}\overline{C}}$ (3.46)

and the first order contribution to the generalized frame with underline index
is

$\displaystyle{\cal
E}_{M\underline{A}}=E_{M\underline{A}}-\frac{1}{2}A_{M}{}^{\alpha}A_{N\alpha}E^{N}{}_{\underline{A}}\,,$
(3.47)

according to (3.4). Using the previous expressions, the underline component of
the single-index flux is

$\displaystyle{\cal
F}_{\underline{A}}=\sqrt{2}\partial^{M}(E_{M\underline{A}}-\frac{1}{2}A_{M}{}^{\alpha}A^{N}{}_{\alpha}E_{N\underline{A}})-2\sqrt{2}(E_{M\underline{A}}-\frac{1}{2}A_{M}{}^{\alpha}A^{N}{}_{\alpha}E_{N\underline{A}})\partial^{M}d\,.$
(3.48)

We can rewrite (3.48) as

$\displaystyle{\cal
F}_{\underline{A}}=F_{\underline{A}}-\frac{1}{4}F^{\underline{B}}A_{\underline{B}}{}^{\alpha}A_{\underline{A}\alpha}-\frac{1}{4}E^{\underline{B}}(A_{\underline{B}}{}^{\alpha}A_{\underline{A}\alpha})\,$
(3.49)

using the notation
$A_{\underline{B}\alpha}=\sqrt{2}A_{M\alpha}E^{M}{}_{\underline{B}}$. Now we
impose the identification,

$\displaystyle
A_{\underline{B}}{}^{\alpha}A_{\underline{A}\alpha}=\frac{1}{X_{R}}A_{\underline{B}}{}^{\overline{C}\overline{D}}A_{\underline{A}\overline{C}\overline{D}}=\frac{a}{2}F_{\underline{B}}{}^{\overline{C}\overline{D}}F_{\underline{A}\overline{C}\overline{D}}$
(3.50)

to finally obtain

$\displaystyle{\cal
F}_{\underline{A}}=F_{\underline{A}}-\frac{a}{8}F^{\underline{B}}F_{\underline{B}}{}^{\overline{C}\overline{D}}F_{\underline{A}\overline{C}\overline{D}}-\frac{a}{8}E^{\underline{B}}(F_{\underline{B}}{}^{\overline{C}\overline{D}}F_{\underline{A}\overline{C}\overline{D}})\,.$
(3.51)

The same can be done for the different projections of ${\cal F}_{\cal ABC}$,

$\displaystyle{\cal F}_{\underline{ABC}}$ $\displaystyle=$ $\displaystyle
F_{\underline{ABC}}+\frac{3a}{4}\left(E_{[\underline{A}}{F}^{\overline{CD}}{}_{\underline{B}}-\frac{1}{2}F_{\underline{D}[\underline{AB}}F^{\underline{D}\overline{CD}}-\frac{2}{3}F^{\overline{C}}{}_{\overline{E}[\underline{A}}F_{\underline{B}}{}^{\overline{ED}}\right)F_{\underline{C}]\overline{CD}}\,,$
(3.52) $\displaystyle{\cal F}_{\overline{A}\underline{BC}}$ $\displaystyle=$
$\displaystyle
F_{\overline{A}\underline{BC}}+\frac{a}{4}\left(E_{\overline{A}}{F}^{\overline{CD}}{}_{[\underline{B}}+{F}^{\underline{E}\overline{CD}}{F}_{\overline{A}\underline{E}[\underline{B}}\right)F_{\underline{C}]\overline{CD}}\,,$
(3.53) $\displaystyle{\cal F}_{\underline{A}\overline{BC}}$ $\displaystyle=$
$\displaystyle{F}_{\underline{A}\overline{BC}}-\frac{a}{8}F_{\underline{D}\overline{EF}}F^{\overline{EF}}{}_{\underline{A}}F^{\underline{D}}{}_{\overline{BC}}\,.$
(3.54)

If we now analyze the four-derivative contributions to the action (3.43) we
find,

$\displaystyle 2{\cal E}^{\underline{A}}{\cal F}_{\underline{A}}$
$\displaystyle=$ $\displaystyle 2{\cal
E}^{\underline{A}}{F}_{\underline{A}}+2{E}^{\underline{A}}{\cal
F}_{\underline{A}}$ $\displaystyle{\cal F}_{\underline{A}}{\cal
F}^{\underline{A}}$ $\displaystyle=$ $\displaystyle 2{F}_{\underline{A}}{\cal
F}^{\underline{A}}$ $\displaystyle-\frac{1}{6}{\cal
F}_{\underline{A}\underline{B}\underline{C}}{\cal
F}^{\underline{A}\underline{B}\underline{C}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}{\cal
F}_{\underline{A}\underline{B}\underline{C}}{F}^{\underline{A}\underline{B}\underline{C}}$
$\displaystyle-\frac{1}{2}{\cal F}_{\overline{\cal
A}\underline{B}\underline{C}}{\cal F}^{\overline{\cal
A}\underline{B}\underline{C}}$ $\displaystyle=$
$\displaystyle-{F}_{\overline{A}\underline{B}\underline{C}}{\cal
F}^{\overline{A}\underline{B}\underline{C}}-\frac{1}{2}{\cal
F}_{\overline{\alpha}\underline{B}\underline{C}}{\cal
F}^{\overline{\alpha}\underline{B}\underline{C}}\,.$ (3.55)

The only flux projection that we have not computed yet is

$\displaystyle{\cal
F}_{\overline{\alpha}\underline{B}\underline{C}}=-\frac{1}{\sqrt{2}}A^{\underline{D}}{}_{\alpha}e^{\alpha}{}_{\overline{\alpha}}F_{\underline{D}\underline{B}\underline{C}}+\frac{1}{2}f_{\alpha\beta\gamma}e^{\alpha}{}_{\overline{\alpha}}A_{\underline{B}}{}^{\beta}A_{\underline{C}}{}^{\gamma}\,,$
(3.56)

and replacing all the flux projections in (3.55) we get,

$\displaystyle{\cal R}({\cal E},d)={\cal R}(E,d)+a{\cal R}^{(-)}$ (3.57)

where

$\displaystyle{\cal R}^{(-)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left[(E_{\underline{A}}E_{\underline{B}}{F}^{\underline{B}}{}_{\overline{CD}}){F}^{\underline{A}\overline{CD}}+(E_{\underline{A}}E_{\underline{B}}{F}^{\underline{A}}{}_{\overline{CD}}){F}^{\underline{B}\overline{CD}}+2(E_{\underline{A}}{F}_{\underline{B}}{}^{\overline{CD}}){F}^{\underline{A}}{}_{\overline{CD}}{F}^{\underline{B}}\right.\,$
$\displaystyle+(E_{\underline{A}}{F}^{\underline{A}\overline{CD}})(E_{\underline{B}}{F}^{\underline{B}}{}_{\overline{CD}})+(E_{\underline{A}}{F}_{\underline{B}}{}^{\overline{CD}})(E^{\underline{A}}{F}^{\underline{B}}{}_{\overline{CD}})+2(E_{\underline{A}}F_{\underline{B}}){F}^{\underline{B}}{}_{\overline{CD}}{F}^{\underline{A}\overline{CD}}\,$
$\displaystyle+(E_{\overline{A}}F_{\underline{B}\overline{CD}})F_{\underline{C}}{}^{\overline{CD}}F^{\overline{A}\underline{BC}}-(E_{\underline{A}}F_{\underline{B}\overline{CD}})F_{\underline{C}}{}^{\overline{CD}}{F}^{\underline{ABC}}+2(E_{\underline{A}}{F}^{\underline{A}}{}_{\overline{CD}}){F}_{\underline{B}}{}^{\overline{CD}}{F}^{\underline{B}}\,$
$\displaystyle-4(E_{\underline{A}}{F}_{\underline{B}}{}^{\overline{CD}}){F}^{\underline{A}}{}_{\overline{CE}}{F}^{\underline{B}\overline{E}}{}_{\overline{D}}+\frac{4}{3}{F}^{\overline{E}}{}_{{\underline{A}}\overline{C}}{F}_{{\underline{B}}\overline{ED}}F_{\underline{C}}{}^{\overline{CD}}{F}^{\underline{ABC}}+{F}^{\underline{B}}{}_{\overline{CD}}{F}_{\underline{A}}{}^{\overline{CD}}F_{\underline{B}}{F}^{\underline{A}}\,$
$\displaystyle\left.+{F}_{\underline{A}}{}^{\overline{CE}}{F}_{\underline{B}\overline{E}\overline{D}}{F}^{\underline{A}}{}_{\overline{CG}}{F}^{\underline{B}\overline{G}\overline{D}}-{F}_{\underline{B}}{}^{\overline{CE}}{F}_{\underline{A}\overline{E}\overline{D}}{F}^{\underline{A}}{}_{\overline{CG}}{F}^{\underline{B}\overline{G}\overline{D}}-F_{\overline{A}\underline{BD}}F^{\underline{D}}{}_{\overline{CD}}F_{\underline{C}}{}^{\overline{CD}}F^{\overline{A}\underline{BC}}\right]\,.$

The systematic procedure used to construct ${\cal R}^{(-)}$ can be easily
adapted to construct ${\cal R}^{(+)}$. The strategy that one should follow is
to start from an $O(D+K,D)$ invariant theory and to identify

$\displaystyle C_{M\alpha}=C_{\overline{M}\alpha}$ (3.59)

with $F_{\overline{M}\underline{B}\underline{C}}$ considering gauge generators
that allow the index identification
$\alpha\rightarrow[\underline{B}\underline{C}]$. This other correction to
${\cal R}(E,d)$, that we will call it ${\cal R}^{(+)}$, has the form (3.1.5)
but exchanging the projections of the different fields. Then, the action
principle

$\displaystyle S=\int d^{2D}Xe^{-2d}({\cal R}(E,d)+a{\cal R}^{(-)}+b{\cal
R}^{(+)})\,,$ (3.60)

is the explicit form of (2.92). The invariance of this action under double
Lorentz transformations was studied in [14] considering an hybrid formalism
that mixes the generalized flux formalism with the generalized metric
formalism. The equivalence between (3.60) and the hybrid formulation was given
in [44].

###  3.2 Part B: extension to higher-order formulations

In the previous part we used a systematic procedure to deform both the
symmetries and the action of DFT. The double Lorentz transformations are
deformed by a generalized Green-Schwarz mechanism that was constructed
considering an $O(D,D+K)$ (or $O(D+K,D)$) invariant theory and we split its
fundamental fields in terms of $O(D,D)$ multiplets. Using this method, we
obtained a gauged DFT with generalized gauge fields $C_{M\alpha}$ which we
identified with a suitable projection of the generalized fluxes. However, this
procedure cannot be extended to the next order using (3.38) since the gauge
field does not transform as a flux projection. This problem was solved in
[24]. We explain some details of the resolution in the following part.

####  3.2.1 The generalized Bergshoeff-de Roo identification

In [24] the authors proposed to identify the gauge group of an $O(D,D+K)$
theory with $O(1,D+K-1)_{R}$. At first sight, this would seem impossible since
the dimension of both groups does not match for finite $K$, i.e.,

$K=\frac{(D+K)(D+K-1)}{2}\ ,$ (3.61)

However, the identification works when $K\to\infty$ 151515This method is
heuristic. Its mathematical foundation deserve further study, as the authors
commented in [24].. First we need to introduce the map

$V_{\overline{\cal A}}{}^{\overline{\cal
B}}=-\,V_{\alpha}\,(t^{\alpha})_{\overline{\cal A}}{}^{\overline{\cal B}}\ ,$
(3.62)

where $\left(t_{\alpha}\right)_{\overline{\cal A}}{}^{\overline{\cal B}}$
denote the generators of the gauge algebra. We impose
${A}_{\overline{M}\alpha}=0$ and
$e_{\alpha}{}^{\overline{\alpha}}=\rm{const}.$, which require non vanishing
$\Gamma_{\overline{A\alpha}}$ and $\Gamma_{\overline{\alpha\beta}}$
parameters. The only degree of freedom to identify is

$\displaystyle{\cal
E}_{\alpha\underline{A}}=E^{M}{}_{\underline{A}}{A}_{M\alpha}=\frac{1}{\sqrt{2}}{A}_{\underline{A}\alpha}\,,$
(3.63)

which transforms as

$\delta{A}_{\underline{A}\alpha}=\hat{\xi}^{P}\partial_{P}A_{\underline{A}\alpha}-{\cal
E}_{\underline{A}}\lambda_{\alpha}+f_{\alpha\beta}{}^{\gamma}\lambda^{\beta}{A}_{\underline{A}\gamma}+{A}_{\underline{B}\alpha}\Gamma^{\underline{B}}{}_{\underline{A}}\
.$ (3.64)

Using the map (3.62) and the relation between the structure constants and the
generators

$\displaystyle f_{\alpha\beta}{}^{\gamma}(t_{\gamma})^{\overline{\cal
A}\overline{\cal B}}=2(t_{[\alpha})^{\overline{\cal A}\overline{\cal
C}}(t_{\beta]})_{\overline{\cal C}}{}^{\overline{\cal B}}\,,$ (3.65)

(3.64) can be written as

$\displaystyle\delta{A}_{\underline{A}\overline{\cal BC}}={\cal
L}_{\hat{\xi}}{A}_{\underline{A}\overline{\cal BC}}-{\cal
E}_{\underline{A}}\lambda_{\overline{\cal
BC}}+2{A}_{\underline{A}\overline{\cal D}[\overline{\cal
C}}\,\lambda^{\overline{\cal D}}{}_{\overline{\cal
B}]}+{A}_{\underline{D}\overline{\cal
BC}}\,\Gamma^{\underline{D}}{}_{\underline{A}}\ .$ (3.66)

This is precisely the way in which the following projection of the extended
fluxes transforms

$\displaystyle\delta{\cal F}_{\underline{A}\overline{\cal BC}}={\cal
L}_{\hat{\xi}}{\cal F}_{\underline{A}\overline{\cal BC}}+{\cal
E}_{\underline{A}}\Gamma_{\overline{\cal BC}}+2{\cal
F}_{\underline{A}\overline{\cal D}[\overline{\cal C}}\Gamma^{\overline{\cal
D}}{}_{\overline{\cal B}]}+{\cal F}_{\underline{D}\overline{\cal
BC}}\Gamma^{\underline{D}}{}_{\underline{A}}\ .$ (3.67)

The resemblance between (3.66) and (3.67) turns into an exact identification
provided by

$\displaystyle\lambda_{\overline{\cal AB}}$ $\displaystyle=$
$\displaystyle-\,\lambda_{\alpha}\,(t^{\alpha})_{\overline{\cal AB}}\ =\
\Gamma_{\overline{\cal AB}}\ ,$ (3.68)
$\displaystyle{A}_{\underline{A}\overline{\cal BC}}$ $\displaystyle=$
$\displaystyle-\,{\cal E}_{\alpha\underline{A}}\,(t^{\alpha})_{\overline{\cal
BC}}\ =\ -{\cal F}_{\underline{A}\overline{\cal BC}}\ .$ (3.69)

In [24] these identifications were named as the generalized Bergshoeff-de Roo
identification. Interesting enough, this recursive procedure is useful to
construct higher-order contributions, where $X_{R}$ acts as a regulator. The
method can be implemented in an iterative way, following the rule

$\displaystyle{A}_{\underline{A}\cal\overline{B}\overline{C}}\rightarrow-{\cal
F}_{\underline{A}{\cal\overline{B}\overline{C}}}({A}_{\underline{A}\cal\overline{B}\overline{C}}\rightarrow-{\cal
F}_{\underline{A}{\cal\overline{B}\overline{C}}}({A}_{\underline{A}\cal\overline{B}\overline{C}}\rightarrow-{\cal
F}_{\underline{A}{\cal\overline{B}\overline{C}}}(\dots)))\,,$ (3.70)

and converges up to second-order (and possible to all orders.) The state of
the art related to this procedure is discussed in the next part.

####  3.2.2 Current research

The generalized Bergshoeff-de Roo identification discussed in the last part of
these lectures contains a supersymmetric extension that we have not reviewed.
In [19] the authors showed how this extension works to first-order in
$\alpha^{\prime}$. Therefore, an explicit relation between the four-derivative
heterotic supergravity and a higher-derivative ${\cal N}=1$ supersymmetric DFT
was achieved.. Moreover, since the identification (3.69) is exact, the
procedure can be performed to obtain corrections to any desired order.
However, it is likely that higher-order corrections do not reproduce the full
string theory Lagrangian. The reason of the previous statement is related to
quartic Riemann interactions, which contain terms proportional to the
transcendental coefficient $\zeta(3)$ in heterotic string theory [5] [6].
These kind of contributions presumably require additional deformations to the
generalized Bergshoeff-de Roo identification. Other possibility is the
existence of a new invariant in DFT, with eight derivatives. This means that
our starting point could be a higher derivative GDFT, and then the iterative
procedure introduce new corrections. Nevertheless, it was recently shown in
[45] that such an object cannot be constructed in a T-duality invariant
approach before compactification, so this interesting problem is still open.

In [46], the extension to include second-order corrections beyond heterotic
DFT/supergravity was achieved. The idea of the authors was to start with an
extended duality group $O(D+p,D+q)$ and then perform an $O(D,D)$
decomposition. This method is an extension of the generalized Bergshoeff-de
Roo identification that captures the bi-parametric freedom, but the quartic
Riemann interaction problem is still present. On the other hand, higher-
derivative formulations have been recently used at the level of perturbative
DFT to inspect double copy relations from a duality covariant approach [47],
in non-abelian and Poisson-Lie T-duality [48] and in closely related
integrable deformations of the string sigma model [49]. The study of higher-
dimensional black holes and entropy was also studied considering higher-
derivative deformations [50]. Finally, serveral works restrict the study of
the higher-derivative corrections to backgrounds with commuting isometries
[51].

###  3.3 Exercises

  1. 1.

Compute the leading order and the next-to-leading order contributions to
$\Gamma_{\overline{\alpha}\overline{B}}$ and
$\Gamma_{\overline{a}\overline{b}}$.

  2. 2.

* Show that $\Gamma_{(\overline{a}\overline{b})}=0$, and therefore expression (3.15) holds.

  3. 3.

Prove the following identities:
$\Delta_{\alpha\beta}=(\Box^{-1})_{\alpha\beta}$ and
$\chi_{MN}=(\Theta^{-1})_{MN}$ .

  4. 4.

Starting from $\delta{\cal E}_{M\underline{A}}$, show that (3.26) holds.

  5. 5.

* Show that $S_{(MN)}=0$ and $S_{MN}=S_{\underline{M}\underline{N}}$.

  6. 6.

Starting from $\delta{\cal E}_{M\overline{\alpha}}$, show that (3.29) holds.

  7. 7.

Compute the next-to-leading order correction to (3.30). Conclude that the
identification (3.38) is not valid to next order.

  8. 8.

Compute the flux components (3.52)-(3.54).

  9. 9.

Compute ${\cal R}^{(-)}$ and show that (3.1.5) holds.

  10. 10.

Show that there is no solution to (3.61) for finite $K>1$.

  11. 11.

* Show that the generalized Bergshoeff-de Roo identification is compatible to first-order finding $X_{R}$ in terms of the $a$ parameter.

##  4 Summary

In these lectures we have reviewed some aspects of the low-energy limit of
string theory when four-derivative contributions are considered. We focus on
the gravitational corrections for the different formulations of closed string
theory. We have studied the Metsaev-Tseytlin approach, related to scatering
amplitude computations, and the Bergshoeff-de Roo approach, which can be
obtained considering field redefinitions in the Metsaev-Tseytlin approach. In
the Bergshoeff-de Roo approach, a relation between non-abelian gauge
symmetries and Lorentz tranformations was analyzed for the heterotic
supergravity. It turns out that the non-abelian gauge transformations can be
written as higher-derivative Lorentz transformations using gauge generators
that relate gauge/Lorentz indices. The Green-Schwarz mechanism corrects only
the $b$-field Lorentz transformation, since no other field different from the
gauge field $A_{\mu i}$, transforms under non-abelian gauge transformations.
Interesting enough, the gravitational four-derivative terms in the heterotic
supergravity action can be reproduced using this identification.

The generalization of the previous relation to the DFT level allowed us to
propose a generalized Green-Schwarz mechanism to deform the double Lorentz
symmetry for the generalized frame using a pair of parameters $(a,b)$. All
choices reproduce T-duality invariant theories and only a few cases reproduce
string supergravities. Among the non-string theories we discuss about the HSZ
theory, which is a deformation of DFT that can be expressed in the generalized
metric formulation. In this case, the generalized diffeomorphism
transformations contain higher-derivative deformations, and the C-bracket is
consistently deformed. At the supergravity Lagrangian level, the higher-
derivative corrections of HSZ theory are closely related to heterotic string
theory for odd powers of $\alpha^{\prime}$ and to bosonic string theory for
even powers of $\alpha^{\prime}$.

Finally, in the last lecture, we introduce a systematic procedure to construct
deformations in DFT. The idea is to consider an $O(D,D+K)$ invariant theory
and to write the fields, parameters and metrics in terms of $O(D,D)$ degrees
of freedom. This procedure requires a suitable gauge fixing. The fundamental
fields, from an $O(D,D)$ perspective, include a gauge field $C_{M\alpha}$ that
can be identified with a projection of the generalized fluxes. Using this
method, to first order, one can construct deformations for both the symmetries
and the action principle in a systematic way. The generalization of this
procedure to higher-orders (the generalized Bergshoeff-de Roo identification)
was partially discussed together with some current research.

##  5 Acknowledgments

The author is very grateful to W. Baron, D. Marques, C. Nunez and A. Rodriguez
for collaboration in some works covered by these lectures. I warmly thank R.
Borsato, E. Malek and D. Marques for enlightening comments which help me to
organize the content of each lecture. I also indebted with S. Iguri, N. Miron-
Granese and A. Rodriguez for helping me to improve the presentation of the
first version of these notes, and with R. Borsato and E.Malek for helping me
to improve the present version. Finally, I would like to thank Riccardo
Borsato, Saskia Demulder, Sibylle Driezen, Fedor Levkovich-Maslyuk and Emanuel
Malek for all the support and help during the school. Support by CONICET is
also gratefully acknowledged.

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