U-Folds From Geodesics in Moduli Space

D. Astesianoa, D. Ruggeri b and M. Trigianteb 111<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
a Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavík, Iceland,

bDepartment of Applied Science and Technology, Politecnico di Torino,

C.so Duca degli Abruzzi, 24, I-10129 Torino, Italy

Abstract

We exploit the presence of moduli fields in the ${\rm
AdS}_{3}\times{S}^{3}\times CY_{2}$, where $CY_{2}=T^{4}$ or $K3$, solution to
Type IIB superstring theory, to construct a U-fold solution with geometry
${\rm AdS}_{2}\times S^{1}\times{\rm S}^{3}\times CY_{2}$. This is achieved by
giving a non-trivial dependence of the moduli fields in ${\rm SO}(4,n)/{\rm
SO}(4)\times{\rm SO}(n)$ ($n=4$ for $CY_{2}=T^{4}$ and $n=20$ for $CY_{2}=K3$
), on the coordinate $\eta$ of a compact direction $S^{1}$ along the boundary
of ${\rm AdS}_{3}$, so that these scalars, as functions of $\eta$, describe a
geodesic on the corresponding moduli space. The back-reaction of these
evolving scalars on spacetime amounts to a splitting of ${\rm AdS}_{3}$ into
${\rm AdS}_{2}\times S^{1}$ with a non-trivial monodromy along $S^{1}$ defined
by the geodesic. Choosing the monodromy matrix in ${\rm
SO}(4,n;\,\mathbb{Z})$, this supergravity solution is conjectured to be a
consistent superstring background. We generalize this construction starting
from an ungauged theory in $D=2d$, $d$ odd, describing scalar fields non-
minimally coupled to $(d-1)$-forms and featuring solutions with topology ${\rm
AdS}_{d}\times S^{d}$, and moduli scalar fields. We show, in this general
setting, that giving the moduli fields a geodesic dependence on the $\eta$
coordinate of an $S^{1}$ at the boundary of ${\rm AdS}_{d}$ is sufficient to
split this space into ${\rm AdS}_{d-1}\times S^{1}$, with a monodromy along
$S^{1}$ defined by the starting and ending points of the geodesic. This
mechanism seems to be at work in the known J-fold solutions in $D=10$ Type IIB
theory and hints towards the existence of similar solutions in the Type IIB
theory compactified on $CY_{2}$. We argue that the holographic dual theory on
these backgrounds is a 1+0 CFT on an interface in the 1+1 theory at the
boundary of the original ${\rm AdS}_{3}$.

x

###### Contents

  1. 1 Introduction
  2. 2 The General Construction of the ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ U-Fold
    1. 2.1 Boundary conditions on $S^{1}$
    2. 2.2 Examples in Type IIB superstring theory
  3. 3 The Bosonic Sector of the $\mathcal{N}=(2,2),\,D=6$ Theory
    1. 3.1 An Explicit Parametrization of the Scalar Manifold
    2. 3.2 ${\rm AdS}_{3}\times{S}^{3}$ Solution of Type IIB Theory on $T^{4}$
    3. 3.3 ${\rm AdS}_{2}\times S^{1}\times{S}^{3}$ U-Folds
  4. 4 Final remarks and discussion of the solution
  5. A Type IIB Supergravity on $T^{4}$: Geometric Characterization of the Scalar and Tensor Fields.
  6. B The Type IIB Static Black String Solutions
    1. B.1 Attractor mechanism for extremal solutions
  7. C  String solutions in the double commuting $\frac{{\rm SL}(2,\mathbb{R})}{{\rm SO}(2)}$ truncation
    1. C.1 D1-D5 Solution
    2. C.2 D1-D5 solution with non-trivial moduli
  8. D Extremization of the effective scalar potential

##  1 Introduction

Solutions of superstring theory of $D=11$ supergravity with geometry ${\rm
AdS}_{d}\times M_{{\rm int.}}$ have been the focus of intense study because of
their relevance to the ${\rm AdS/CFT}$ correspondence. Type IIB superstring
theory, in particular, features a number of backgrounds of the form ${\rm
AdS}_{d}\times S^{d}\times M_{10-2d}$ which are characterized by moduli
fields, i.e. scalar fields which can be assigned an arbitrary constant value
throughout spacetime without affecting its geometry. These fields are dual to
exactly marginal operators in the dual CFT. The simplest example is ${\rm
AdS}_{5}\times S^{5}$ on which the holographic duality was originally
conjectured, and whose moduli fields are the Type IIB axion and dilaton fields
which correspond to the complexified coupling constant in the dual
$\mathcal{N}=4$ SYM theory. Other examples of such backgrounds, or variants
thereof, are222For a review of the backgrounds ${\rm AdS}_{3}\times
S^{3}\times CY_{2}$, with $CY_{2}=T^{4}$ or $K3$, and of their holographic
dual descriptions, see [1, 2, 3]. In the present work, we shall not
concentrate on the dual CFT side of these backgrounds.

  * i)

${\rm AdS}_{3}\times S^{3}\times T^{4}$ describing the near-horizon geometry
of a D1-D5 or of an F1-NS5 system, in which the 5-branes are wrapped around
the 4-torus $T^{4}$;

  * ii)

${\rm AdS}_{3}\times S^{3}\times K3$ describing the near-horizon geometry of a
D1-D5 or of an F1-NS5 system, in which the 5-branes are wrapped around the K3
manifold;

  * iii)

${\rm AdS}_{5}\times S^{5}/\mathbb{Z}_{k}$ describing the near-horizon
geometry of a stack of $D3$ branes at the apex of an orbifold
$\mathbb{C}^{3}/\mathbb{Z}_{k}$ with $k>1$ [4, 5].

The backgrounds $i)$ can be described within Type IIB superstring theory
compactified on $T^{4}$ to $D=6$, which is described by a maximal six-
dimensional supergravity featuring the global symmetry group ${\rm SO}(5,5)$
at the classical level. The classical moduli space of the ${\rm AdS}_{3}\times
S^{3}$ solution of the six-dimensional maximal supergravity is ${\rm
SO}(4,5)/{\rm SO}(4)\times{\rm SO}(5)$, ${\rm SO}(4,5)$ being the stabilizer
in ${\rm SO}(5,5)$ of the brane charges.
The background $ii)$ is a solution to Type IIB superstring theory compactified
on K3, which is described, at the classical level, by a half-maximal six-
dimensional supergravity and the classical moduli space is ${\rm
SO}(4,21)/{\rm SO}(4)\times{\rm SO}(21)$.
Finally, in the last case $iii)$, the classical moduli space is ${\rm
SU}(1,k)/{\rm U}(k)$ and acts on the $k$ complexified coupling constants of
the dual necklace quiver theory [6, 7], .
All these moduli occur in the $d$-dimensional gauged supergravity featuring
the ${\rm AdS}_{d}$ factor ($d=3$ and 5 in the above examples) as vacuum
solution, as flat directions of the scalar potential.
Assuming a dependence of these scalar fields on the spacetime coordinates will
in general affect the geometry of the background in some non-trivial way. In
[8, 9, 5, 10] the authors have investigated spacetime-dependent configurations
of the moduli fields in the Wick-rotated moduli space within the Euclidean
version of the theory. The resulting solutions were described by geodesics in
the (pseudo-Riemannian) moduli space, which were classified according to their
being lightlike, spacelike, and timelike, and an interpretation of the
corresponding backgrounds, in the dual CFT, was worked out.

Here we shall work in a suitable, ungauged Lorentzian theory in $D=2d$, $d$
odd, exhibiting a ${\rm AdS}_{d}\times S^{d}$ background with moduli fields.
We show how to construct a ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ geometry
from the ${\rm AdS}_{d}\times S^{d}$ one by giving a suitable spatial
dependence to the scalar moduli.

The construction proceeds along the following steps. We first compactify one
direction in the boundary of ${\rm AdS}_{d}$. Let us denote by $\eta$ the
coordinate of the corresponding $S^{1}$. We then assume a suitable subset
$\varphi^{a}$ of the moduli fields to depend on $\eta$ only and to describe,
in their evolution, a geodesic in the moduli space as $\eta$ varies in its
defining interval of values $\eta\in[0,T]$, $T$ being the length of $S^{1}$.
As a consequence of the back-reaction of the evolving scalar fields on
spacetime, the ${\rm AdS}_{d}\times S^{d}$ background is transformed into an
${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ geometry, with a non-trivial
monodromy along $S^{1}$. The two solutions cannot be continuously deformed
into one another. If we denote by $\mathcal{M}_{0}=G_{0}/H_{0}$ the moduli
space spanned by $\varphi^{a}$, and by $O$ and $P$ the starting and ending
points of the geodesic, parameterized by $\varphi^{a}(0)$ and
$\varphi^{a}(T)$, respectively, the $G_{0}$-element $g$ connecting $P$ to $O$,
defines the monodromy: $P=g\,O$. If we assume that $G_{0}(\mathbb{Z})$ is a
symmetry of the underlying superstring theory reduced to ${\rm AdS}_{d}\times
S^{d}$, then by choosing $g\in G_{0}(\mathbb{Z})$, the backgrounds defined by
the initial and final points of the geodesic would be identified from the
string perspective and the corresponding supergravity solution could be
conjectured to be a consistent superstring background. The component of the
metric in the $S^{1}$ direction is proportional to the squared "velocity"
along the geodesic. Therefore the ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ is
not continuously connected to ${\rm AdS}_{d}\times S^{d}$ as this velocity is
set to zero and the moduli fields to constant values.

A prototype of this background is the non-supersymmetric J-fold [11], with
geometry ${\rm AdS}_{4}\times S^{1}\times S^{5}$, characterized by an axio-
dilaton field evolving along a geodesic in their moduli space ${\rm
SL}(2,\mathbb{R})/{\rm SO}(2)$. In the corresponding J-fold description, the
initial and final points of the geodesic are connected by a monodromy
$g=J_{n}\in{\rm SL}(2,\mathbb{Z})_{{\rm IIB}}$, $J_{n}$ being a hyperbolic
element of the Type IIB global symmetry group. However, the mechanism analyzed
here, of the construction of ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ U-folds
by giving the moduli fields a geodesic dependence on the coordinate of
$S^{1}$, seems to be at work in the J-fold solutions studied in [12, 13, 11,
14, 15, 16, 17, 18, 19, 20, 21, 22] with geometry ${\rm AdS}_{4}\times
S^{1}\times\tilde{S}^{5}$, $\tilde{S}^{5}$ being a deformed 5-sphere. We shall
further comment on this in the final Discussion section.

In this paper, we shall focus on solutions of the form ${\rm AdS}_{2}\times
S^{1}\times S^{3}$ within Type IIB theory compactified on a $CY_{2}$-manifold,
which can either be $T^{4}$ or $K3$. Such backgrounds feature a richer class
of monodromies. In the former case, $CY_{2}=T^{4}$, the solutions have an
$O(4,4;\mathbb{Z})$-monodromy, in the latter, $CY_{2}=K3$, the monodromy is
chosen in $O(4,20;\mathbb{Z})$. These discrete groups are contained inside the
so-called U-duality group which is conjectured to encode all string dualities
and to be an exact symmetry of the underlying, though yet unknown, unifying
quantum theory of gravity [23]. We shall work extensively on the
$CY_{2}=T^{4}$ case and construct explicit U-fold solutions within the
effective $\mathcal{N}=(2,2)$ six-dimensional supergravity.

The paper is organized as follows. In section 2 we describe the general
construction of such ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ backgrounds. We
start from a ${\rm AdS}_{d}\times S^{d}$ solution of an ungauged model in
$D=2d$, $d$ odd, which describes scalar fields, spanning the symmetric target
space of sigma model, and non-minimally coupled to a set of $(d-1)$-forms. We
discuss, in subsection 2.1, the issue of the boundary conditions along $S^{1}$
in relation to the geodesic motion of the moduli fields. In subsection 2.1.1
we also introduce, on such backgrounds, the so-called $\chi$-deformations
originally studied, in the context $D=10$ J-fold solutions, in [15, 16, 20,
22]. To give concrete examples of these solutions, we focus on the Type IIB
theory compactified on a 4-torus. We start reviewing, in section 3, the
bosonic sector of the resulting ungauged maximal 6-dimensional supergravity.
This is the model where we construct explicit U-fold solutions. Then, in
section 3.2 we review the known AdS${}_{3}\times S^{3}$ solutions and their
string interpretation. Starting from these backgrounds, in section 3.3 we
build the U-Fold for a certain choice of the monodromy matrix.

In Appendix A we review the geometric characterization of the Type IIB string
origin of the scalar and tensor fields in the maximal six-dimensional
supergravity. In Appendix B we review the general construction of a black-
string solution with ${\rm SO}(4)$-symmetry and the attractor mechanism at
work for the extremal ones. In Appendix C we review the construction of the
D1-D5 solution within the maximal six-dimensional theory, also in the presence
of certain, radially evolving, moduli fields. In Appendix D we give the
explicit form of the effective potential in the presence of D1-D5 charges, as
a function of the string 0-modes, and discuss its extremization.

##  2 The General Construction of the ${\rm AdS}_{d-1}\times S^{1}\times
S^{d}$ U-Fold

In this section, we show how to build a new class of solutions with topology
${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ (with dimension $D=2d$) by giving
the moduli a suitable non-trivial profile. For the sake of concreteness, we
take $d$ odd since we have in mind the type IIB backgrounds with $d=3$ and
$5$.
We start considering a model, in $D$-dimensions, describing $n$ non-chiral
$p$-forms $C^{I}_{(p)}$ ($I=1,\dots n$), with $p=d-1$, coupled to gravity and
to a number of scalar fields $\phi^{s}$, $s=1,\dots n_{s}$. The latter are
described by a sigma-model with homogeneous symmetric, Riemannian target
space:

${M}_{{\rm scal.}}=\frac{G}{H}\,.$

We assume the isometry group $G$ of the scalar manifold to admit a pseudo-
orthogonal representation $\mathscr{R}_{p}$ in terms of ${\rm
O}(n,n)$-transformations. Our analysis will be extended to models in which the
number $n$ of self-dual and $m$ anti-self-dual $d$-form field strengths of the
$p$-form fields are different. In this case, we require $G$ to admit a
representation in terms of ${\rm O}(n,m)$-transformations.

In an extended $D$-dimensional supergravity model, this geometric feature is
built-in. As is the case in ungauged $D$-dimensional supergravity, the scalar
fields are non-minimally coupled to the tensor ones, that is they couple to
their field strengths $H^{I}_{(d)}\equiv dC_{(p)}^{I}$ of the latter in their
kinetic terms. The general form of the bosonic action we are considering is
333Throughout the paper we adopt the mostly plus notation for the metric.
Moreover we define  $\displaystyle\sigma_{(p)}\cdot\omega_{(d)}$
$\displaystyle\equiv\sigma_{\hat{\mu}_{1}\dots\hat{\mu}_{d}}\,\omega^{\hat{\mu}_{1}\dots\hat{\mu}_{d}}/d!\,,$
$\displaystyle({}^{*}\omega)_{\hat{\mu}_{1}\dots\hat{\mu}_{D-p}}$
$\displaystyle\equiv\frac{e_{D}}{p!}\,\epsilon_{\hat{\mu}_{1}\dots\hat{\mu}_{D-p}\hat{\nu}_{1}\dots\hat{\nu}_{p}}\,\omega^{\hat{\nu}_{1}\dots\hat{\nu}_{p}}\,,$
(2.1) where $\epsilon_{0,\dots D-1}=1$ and
$\star^{2}\omega_{(p)}=(-1)^{(D-p)p+1}\,\omega_{(p)}$.

$\displaystyle\mathcal{L}_{2d}=e_{2d}\left[R-\frac{1}{2}\,\partial_{\hat{\mu}}\phi^{r}\partial^{\hat{\mu}}\phi^{s}\mathcal{G}_{rs}(\phi)-\frac{1}{2}\,\mathcal{I}(\phi)_{IJ}H^{I}\cdot
H^{J}-\frac{1}{2}\,\mathcal{R}(\phi)_{IJ}{H}^{I}\cdot{}^{*}H^{J}\right],$
(2.2)

where $\mathcal{G}_{rs}(\phi)>0$ is the target space metric of the scalar
sigma model, and $e_{D}=e_{2d}\equiv\sqrt{|{\rm
det}(g_{\hat{\mu}\hat{\nu}})|}$. The matrices $\mathcal{I}_{IJ}(\phi)>0$ and
$\mathcal{R}_{IJ}(\phi)=-\mathcal{R}_{JI}(\phi)$ are functions of the scalar
fields and describe the non-minimal coupling between them and the tensor field
strengths $H^{I}$. We define the dual field strengths as

$\displaystyle
G_{I\,\hat{\mu}_{1}\dots\hat{\mu}_{d}}=\epsilon_{\hat{\mu}_{1}\dots\hat{\mu}_{d}\hat{\nu}_{1}..\hat{\nu}_{d}}\frac{\partial\mathcal{L}_{2d}}{\partial
H^{I}_{\,\hat{\nu}_{1}\dots\hat{\nu}_{d}}},$ (2.3)

which, in matrix form and suppressing the spacetime indices, reads:

$\displaystyle G=-\mathcal{I}\,{}^{\star}H-\mathcal{R}\,H\,.$ (2.4)

In what follows we treat the field strength and the dual field strength on
equal footing and define this column vector

$\displaystyle\mathbb{H}=(\mathbb{H}^{M})=\begin{pmatrix}H^{I}\\\ G_{I}\\\
\end{pmatrix}.$ (2.5)

From this the following _twisted self-duality condition_ follows

$\displaystyle{}^{\star}\mathbb{H}=-\Omega\,\mathcal{M}\,\mathbb{H}\,,$ (2.6)

where

$\displaystyle\Omega=\begin{pmatrix}0&\mathbb{I}\\\
\mathbb{I}&0\end{pmatrix}\,\,,\,\,\,\mathcal{M}(\phi)\equiv\left(\begin{matrix}\mathcal{I}-\mathcal{R}\,\mathcal{I}^{-1}\,\mathcal{R}&-\mathcal{R}\mathcal{I}^{-1}\cr\mathcal{I}^{-1}\mathcal{R}&\mathcal{I}^{-1}\end{matrix}\right)\,,$
(2.7)

$\Omega$ is the $O(n,n)$-invariant matrix, $\Omega^{2}=\mathbb{I}$, and
$\mathcal{M}$ is a scalar-dependent pseudo-orthogonal, positive-definite
symmetric matrix: $\mathcal{M}\,\Omega\,\mathcal{M}=\Omega$. The Maxwell
equations and the Bianchi equations read,

$\displaystyle d\mathbb{H}=0\,.$ (2.8)

The Einstein and scalar field equations of motion for this model are

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\,$
$\displaystyle=\,T^{(s)}_{\mu\nu}+T^{(H)}_{\mu\nu}\,,$ $\displaystyle
D_{\hat{\mu}}(\partial^{\hat{\mu}}\phi^{s})=\nabla_{\hat{\mu}}(\partial^{\hat{\mu}}\phi^{s})+\tilde{\Gamma}^{s}_{uv}\partial_{\hat{\mu}}\phi^{u}\partial^{\hat{\mu}}\phi^{v}$
$\displaystyle=\frac{1}{4}\,\mathcal{G}^{st}\,\mathbb{H}^{T}\,\partial_{t}\mathcal{M}\,\,\mathbb{H}\,,$
(2.9)

where the energy-momentum tensors are defined as:

$\displaystyle T^{(s)}_{\hat{\mu}\hat{\nu}}$
$\displaystyle\equiv\frac{1}{2}\mathcal{G}_{rs}\left(\partial_{\hat{\mu}}\phi^{r}\partial_{\hat{\nu}}\phi^{s}-\frac{1}{2}g_{\hat{\mu}\hat{\nu}}\partial_{\hat{\rho}}\phi^{r}\partial^{\hat{\rho}}\phi^{s}\right)\,,$
$\displaystyle T^{(H)}_{\hat{\mu}\hat{\nu}}$
$\displaystyle\equiv\frac{1}{4\,p!}\,\mathbb{H}^{T}_{\hat{\mu}\hat{\mu}_{1}\dots\hat{\mu}_{p}}\,\mathcal{M}(\phi)\,\mathbb{H}_{\hat{\nu}}{}^{\hat{\mu}_{1}\dots\hat{\mu}_{p}}\,$
(2.10)

and $\tilde{\Gamma}^{s}_{uv}(\phi)$ denotes the Levi-Civita connection on the
scalar manifold. The scalar kinetic term can also be written in the following
form

$\displaystyle\frac{{\tt
k}}{8}\text{Tr}\left(\mathcal{M}^{-1}\partial\mathcal{M}\mathcal{M}^{-1}\partial\mathcal{M}\right)=\frac{1}{2}\mathcal{G}_{st}\partial_{\hat{\mu}}\phi^{s}\partial^{\hat{\mu}}\phi^{t}\,,$
(2.11)

where the constant ${\tt k}$ depends of the matrix representation of
$\mathcal{M}$.

It is straightforward to prove, using the above expression for $T^{(H)}$, that
the trace of this tensor vanishes

$\displaystyle
T^{(H)}{}_{\hat{\mu}}{}^{\hat{\mu}}=\frac{1}{8}\mathbb{H}^{T}_{\hat{\mu}}\mathcal{M}\mathbb{H}^{\hat{\mu}}=\frac{1}{8}\mathbb{H}^{T}\mathcal{M}\wedge\star\mathbb{H}=-\frac{1}{8}\mathbb{H}^{T}\wedge\mathcal{M}\Omega\mathcal{M}\mathbb{H}=-\frac{1}{8}\mathbb{H}^{T}\wedge\Omega\mathbb{H}=0\,.$
(2.12)

The global symmetry of the above equations in supergravity is the invariance
under the two-fold action of the group $G$ on the scalar fields as the
isometry group of the scalar manifold, and on the field strengths $\mathbb{H}$
through a pseudo-orthogonal representation $\mathscr{R}_{p}$ (we suppress all
pseudo-orthogonal indices):444For the sake of simplicity, we use the short-
hand notation $A^{-T}\equiv(A^{-1})^{T}$.

$\displaystyle g\in G$
$\displaystyle\,\,\stackrel{{\scriptstyle\mathscr{R}_{p}}}{{\longrightarrow}}\,\,\mathscr{R}_{p}[g]\in{\rm
O}(n,n)\,,$ $\displaystyle\mathbb{H}$
$\displaystyle\rightarrow\,\,\mathbb{H}^{\prime}=\mathscr{R}_{p}[g]\,\mathbb{H}\,,$
$\displaystyle\mathcal{M}(\phi)$
$\displaystyle\rightarrow\,\,\mathcal{M}(\phi^{\prime})=\mathscr{R}_{p}[g]^{-T}\,\mathcal{M}(\phi)\,\mathscr{R}_{p}[g]^{-1}\,.$
(2.13)

In extended supergravities the matrix $\mathcal{M}(\phi)$ obeys the above
transformation property since it is expressed in terms of the coset
representative $\mathbb{L}(\phi^{s})$ of ${M}_{{\rm scal.}}$ in the
representation $\mathscr{R}_{p}$ as follows:555Here we describe the scalar
manifold as a left-coset space.

$\mathcal{M}(\phi)=\mathscr{R}_{p}[\mathbb{L}(\phi)]^{-T}\,\mathscr{R}_{p}[\mathbb{L}(\phi)]^{-1}\,.$
(2.14)

The transformation property (2.13) of $\mathcal{M}(\phi)$ then follows from
the action of an isometry element $g\in G$ on $\phi^{s}$:

$g\,\mathbb{L}(\phi)=\mathbb{L}(\phi^{\prime})\,h(g,\phi)\,\,,\,\,\,h(g,\phi)\in
H\,$ (2.15)

and the fact that $\mathscr{R}_{p}[h(g,\phi)]\in{\rm O}(n)\times{\rm O}(n)$.
We now consider solitonic solutions described only by the scalar and the
tensor fields, with spacetime geometry ${\rm M}_{d}\times_{w}S^{d}$, where we
will choose ${\rm M}_{d}={\rm AdS}_{d-1}\times S^{1}$. The metric will read

$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}+g_{ij}\,d\upxi^{i}\,d\upxi^{j}=v_{1}^{2}\,ds^{2}_{{\rm
AdS}_{d-1}}+v_{3}^{2}\,d\eta^{2}+v_{2}^{2}\,ds^{2}_{S^{d}}\,,$ (2.16)

where $\mu=0,\dots,d-1$ and $i=1,\dots,d$, $ds^{2}_{{\rm AdS}_{d-1}}$ is the
metric of an AdSd-1 space of radius 1 and $,ds^{2}_{S^{d}}$ is the metric of a
$S^{d}$-sphere of radius 1. The coefficients $v_{1},\,v_{2},\,v_{3}$ are
constant and we have split the $M_{d}$ coordinates
$(x^{\mu})=(x^{\alpha},\eta)$, where $\alpha=0\,\dots,d-2$ and $x^{d-1}=\eta$.
We shall also define:

$\tilde{e}_{d}=\sqrt{|{\rm det}(g_{\mu\nu})|}\,\,,\,\,\,{e}_{d}=\sqrt{{\rm
det}(g_{ij})}\,.$

The radius $v_{2}$ of the $d$-sphere will also be denoted by $L$.

The general ansatz for the tensor field strengths reads:

$\displaystyle\mathbb{H}=-\Omega\,\mathcal{M}\,\Gamma\,\boldsymbol{\epsilon}_{\text{M}_{d}}+\Gamma\,\boldsymbol{\epsilon}_{\text{S}^{d}},$
(2.17)

where $\boldsymbol{\epsilon}_{{\rm M}}$,
$\boldsymbol{\epsilon}_{\text{S}^{d}}$ denote the volume densities of the
manifold ${\rm M}_{d}$ and $\text{S}^{d}$:

$\boldsymbol{\epsilon}_{{\rm
M}}\equiv\frac{\tilde{e}_{d}}{d!\,L^{d}}\,\epsilon_{\mu_{1}\dots\mu_{d}}\,dx^{\mu_{1}}\wedge\dots\,dx^{\mu_{d}}\,,\,\,\,\boldsymbol{\epsilon}_{\text{S}^{d}}\equiv\frac{{e}_{d}}{d!\,L^{d}}\,\epsilon_{i_{1}\dots
i_{d}}\,d\upxi^{i_{1}}\wedge\dots\,d\upxi^{i_{d}}\,,$

where we recall that $L$ is the radius $v_{2}$ of the $d$-sphere. We have also
denoted by $\Gamma=(\Gamma^{M})$ the pseudo-orthogonal vector of charges:

$\Gamma^{M}\equiv\frac{1}{{\rm
S}_{\text{S}^{d}}}\,\int_{\text{S}^{d}}\mathbb{H}^{M}\,,$ (2.18)

where $\text{S}_{\text{S}^{d}}$ is the surface area of a $d$-sphere of radius
1. This ansatz is covariant under (2.13) provided the charge vector is
transformed accordingly:

$g\in
G\,\,,\,\,\,\,\Gamma\rightarrow\,\,\Gamma^{\prime}=\mathscr{R}_{p}[g]\,\Gamma\,.$
(2.19)

Quantum effects would restrict $\Gamma^{M}$ to belong to an even, unimodular
charge lattice $\boldsymbol{\Gamma}^{n,n}$ and this would, in turn, break the
global symmetry group $G$ to a discrete subgroup
$G(\mathbb{Z})\sim\mathscr{R}_{p}[G]\cap{\rm O}(n,n;\mathbb{Z})$ preserving
the lattice $\boldsymbol{\Gamma}^{n,n}$. In the effective supergravity
description of superstring/M-theories, this group $G(\mathbb{Z})$ was
conjectured to encode the known string-dualities [23]. Field configurations
connected by $G(\mathbb{Z})$ should be identified from the string theory point
of view. The values of the $D$-dimensional scalar fields defining inequivalent
string backgrounds should then span the manifold:

$G(\mathbb{Z})\backslash G/H\,.$

In chiral models, in which the number $n$ of self-dual and the number $m$ of
anti-self-dual $d$-form field strengths in $D$-dimensions are different, as is
the case of Type IIB superstring theory compactified in $K3$ to $D=6$, the
duality action of the global symmetry group $G$ on the $d$-form field
strengths is implemented by transformations in ${\rm O}(n,m)$:
$\mathscr{R}_{p}[G]\subset{\rm O}(n,m)$. Quantum corrections break ${\rm
O}(n,m)$ to ${\rm O}(n,m;\,\mathbb{Z})$ leaving the even, unimodular charge
lattice $\boldsymbol{\Gamma}^{n,m}$ invariant. The global symmetry group is
consequently broken to $G(\mathbb{Z})$. Our construction also applies to this
more general setting.

In the remainder of this paragraph, we shall focus on the classical
description and thus consider the global symmetry group $G$ to be continuous.

As for the space-time dependence of the scalar fields, we need first to define
the solution’s moduli. We define the _little group_ , or _stabilizer_ , of the
charge vector $\Gamma$ the maximal subgroup $G_{\ell}$ of $G$ such that:

$\forall g\in G_{\ell}\,\,:\,\,\,\,\mathscr{R}_{p}[g]\,\Gamma=\Gamma\,.$
(2.20)

If $H_{\ell}$ denotes the maximal compact subgroup of $G_{\ell}$, the moduli
fields $\varphi^{u}$, $u=1,\dots,n_{\ell}$, are defined as the subset of
scalar fields $\phi^{s}$ spanning the submanifold:

${M}_{\ell}\equiv\frac{G_{\ell}}{H_{\ell}}\,,$ (2.21)

of ${M}_{{\rm scal.}}$ at the origin. We split the scalar fields accordingly:

$\\{\phi^{s}\\}=(\varphi^{u},\,\phi^{k})\,\,,\,\,\,u=1,\dots,n_{\ell}\,\,,\,\,\,k=1,\dots,n_{s}-n_{\ell}\,,$

by defining the coset representative of ${M}_{{\rm scal.}}$ as [24]:

$\mathbb{L}(\phi^{s})=\mathbb{L}_{\ell}(\varphi^{u})\,\hat{\mathbb{L}}(\phi^{k})\,\,,\,\,\,\mathbb{L}_{\ell}(\varphi^{u})\in\frac{G_{\ell}}{H_{\ell}}\,,$
(2.22)

where the scalars $\hat{\phi}\equiv(\phi^{k})$ are defined as follows.
Consider the Cartan decomposition of the Lie algebra $\mathbb{G}$ of $G$ into
the Lie algebra $\mathbb{H}$ generating $H$ and the coset space $\mathbb{K}$
isomorphic to the tangent space to ${M}_{{\rm scal.}}$ at the origin:

$\mathbb{G}=\mathbb{H}\oplus\mathbb{K}\,.$ (2.23)

The latter space $\mathbb{K}$ supports a representation of $H$ under its
adjoint action. This representation is completely reducible under the adjoint
action of $H_{\ell}\subset H$ and $\mathbb{K}$ splits accordingly in
$H_{\ell}$-invariant subspaces

$\mathbb{K}=\mathbb{K}_{\ell}\oplus\hat{\mathbb{K}}\,,$ (2.24)

where $\mathbb{K}_{\ell}$ is the coset space of $G_{\ell}/H_{\ell}$, while
$\hat{\mathbb{K}}$ supports a representation of $H_{\ell}$. We define

$\hat{\mathbb{L}}(\phi^{k})\in e^{\hat{\mathbb{K}}}\,.$ (2.25)

We note that the scalar fields $\varphi^{u}$ span a symmetric submanifold
$G_{\ell}/H_{\ell}$ of ${M}_{{\rm scal.}}$ only at $\phi^{k}=0$. For fixed
$\hat{\phi}=(\phi^{k})\neq(0)$, $\varphi^{u}$ span a space which is not a
symmetric submanifold $G_{\ell}/H_{\ell}$ of ${M}_{{\rm scal.}}$ since this
space does not support a transitive action of $G_{\ell}$. Indeed, using the
short-hand notation $\phi\equiv(\phi^{s})$ and
$\varphi_{\ell}\equiv(\varphi^{u})$, the action of an element $g_{\ell}\in
G_{\ell}$ on these scalars also affects $\hat{\phi}$:

$g_{\ell}\,\mathbb{L}(\phi)=g_{\ell}\,\mathbb{L}_{\ell}(\varphi_{\ell})\,\hat{\mathbb{L}}(\hat{\phi})=\mathbb{L}_{\ell}(\varphi^{\prime}_{\ell})\,\hat{\mathbb{L}}(\hat{\phi}^{\prime})\,h_{\ell}\,,$
(2.26)

where $h_{\ell}\in H_{\ell}$ is the compensating transformation in:

$g_{\ell}\,\mathbb{L}_{\ell}(\varphi_{\ell})=\mathbb{L}_{\ell}(\varphi^{\prime}_{\ell})\,h_{\ell}\,,$

and

$\hat{\mathbb{L}}(\hat{\phi}^{\prime})\equiv
h_{\ell}\,\hat{\mathbb{L}}(\hat{\phi})\,h_{\ell}^{-1}\,.$

Since, for our analysis, we are interested in a symmetric submanifold $M_{0}$
of ${M}_{{\rm scal.}}$ at fixed, non-vanishing values
$\hat{\phi}_{*}\equiv(\phi^{k}_{*})$ of $\hat{\phi}=(\phi^{k})$ (to be
identified with the extremum of the scalar potential) we will have to restrict
the moduli fields to a symmetric submanifold $G_{0}/H_{0}$ of
$G_{\ell}/H_{\ell}$ ($G_{0}\subset G_{\ell},\,H_{0}\subset H_{\ell}$)
characterized by the property that $[G_{0},\,\mathbb{L}(\hat{\phi}_{*})]=0$.
To this end, we first reduce $\hat{\phi}_{*}$ to its simplest form (_normal
form_), by acting on it using $H_{\ell}$ and then define $G_{0}$ to be a non-
compact, semisimple subgroup of $G_{\ell}$ commuting with
$\mathbb{L}(\hat{\phi}_{*})$. The manifold $M_{0}=G_{0}/H_{0}$,$H_{0}$ being
the maximal compact subgroup of $G_{0}$, is now a symmetric submanifold of
${M}_{{\rm scal.}}$ at $\hat{\phi}=\hat{\phi}_{*}$ and, as such, it is
_totally geodesic_.666A totally geodesic submanifold $M_{0}$ of a Riemannian
manifold ${M}_{{\rm scal.}}$ is characterized by the property that a geodesic
in ${M}_{{\rm scal.}}$ originating in a point of $M_{0}$ and initially tangent
to $M_{0}$, lies entirely in the submanifold itself.  Let us denote a point in
it by $\varphi_{0}\equiv(\varphi^{a})$, $a=1,\dots,\,n_{m}$, where
$n_{m}<n_{\ell}$ is the dimension of $M_{0}$.

Using the ansatz (2.17) and the property of $\mathcal{M}$ of being pseudo-
orthogonal symmetric, we can write

$\mathbb{H}^{T}\frac{\partial}{\partial\phi^{s}}\mathcal{M}\,\mathbb{H}=4\,\frac{\partial}{\partial\phi^{s}}V(\phi,\Gamma)\,L^{-2d}\,,$
(2.27)

where:

$V(\phi,\Gamma)\equiv\frac{1}{2}\,\Gamma^{T}\mathcal{M}(\phi)\Gamma\,.$ (2.28)

The scalar field equations, on the ansatz, read:

$\displaystyle
D_{\hat{\mu}}(\partial^{\hat{\mu}}\phi^{s})=\nabla_{\hat{\mu}}(\partial^{\hat{\mu}}\phi^{s})+\tilde{\Gamma}^{s}_{uv}\partial_{\hat{\mu}}\phi^{u}\partial^{\hat{\mu}}\phi^{v}$
$\displaystyle=\mathcal{G}^{st}\,\partial_{t}V\,L^{-2d}\,.$ (2.29)

Notice that, in the chosen parametrization defined by the coset representative
of the form (2.22), the scalar potential is independent of the moduli fields
$\varphi^{u}$. Indeed, by using eqs. (2.14) and (2.20):

$V(\varphi^{s},\Gamma^{M})=V(\varphi^{u},\,\phi^{k},\Gamma^{M})=V(\phi^{k},\,\Gamma^{M})\,.$

Let us define $\hat{\phi}_{*}\equiv(\phi^{k}_{*})$ to be the value of the non-
moduli fields $\phi^{k}$ which extremize the scalar potential $V$:

$\left.\frac{\partial
V}{\partial\phi^{k}}\right|_{\phi^{k}=\phi^{k}_{*}}=0\,\,\Rightarrow\,\,\phi^{k}_{*}=\phi^{k}_{*}(\Gamma)\,.$
(2.30)

The fixed values $\hat{\phi}_{*}=(\phi^{k}_{*})$ only depend on the quantized
charges and are defined modulo the action of a $H_{\ell}$ transformation. In
light of our previous discussion, we fix the action of $H_{\ell}$ to bring
$\hat{\phi}_{*}$ to its normal form and then restrict the moduli fields to the
coordinates $\varphi^{a}$, $a=1,\dots,n_{m}$, of the symmetric submanifold
$M_{0}=G_{0}/H_{0}$. The value of the scalar potential at the minimum is
denoted by $V_{*}$ and only depends on the quantized charges $\Gamma^{M}$:

$V_{*}(\Gamma)=V(\phi^{k}_{*},\Gamma)>0\,.$

We complete our ansatz by choosing

$\phi^{k}(x)\equiv\phi^{k}_{*}\,\,,\,\,\,\,\varphi^{a}(x)=\varphi^{a}(\eta)\,,$
(2.31)

where $\eta$ is the coordinate parametrizing $S^{1}$ in ${\rm M}_{d}={\rm
AdS}_{d-1}\times S^{1}$. All other scalar fields are set to zero. The scalar
field equation reduces to the geodesic equation for $\varphi^{a}(\eta)$ on
${M}_{0}$:

$\ddot{\varphi^{a}}+\tilde{\Gamma}^{a}{}_{bc}\,\dot{\varphi}^{b}\,\dot{\varphi}^{c}=g_{\eta\eta}\,\mathcal{G}^{ak}\,\left.\frac{\partial
V}{\partial\phi^{k}}\right|_{\phi^{k}=\phi^{k}_{*}}\,L^{-2d}=0\,,$ (2.32)

where we have used the fact that $V$ is independent of $\varphi^{a}$ and have
defined $\dot{\varphi}\equiv d\varphi/d\eta$. Having assumed the non-moduli
fields $\phi^{k}$ to be constant and equal to the values that minimize $V$,
all the derivatives of $\phi^{k}$ on the solution vanish and the corresponding
field equations read:

$g^{\eta\eta}\,\tilde{\Gamma}^{k}{}_{bc}\,\dot{\varphi}^{b}\,\dot{\varphi}^{c}=L^{-2d}\,\mathcal{G}^{kk^{\prime}}\,\left.\frac{\partial
V}{\partial\phi^{k^{\prime}}}\right|_{\phi^{k}=\phi^{k}_{*}}\,L^{-2d}=0\,,$

which is satisfied provided $\tilde{\Gamma}^{k}{}_{bc}=0$ for
$\phi^{k}=\phi^{k}_{*}$ and $\varphi^{a}$ generic. This condition follows from
the property of the moduli space $G_{0}/H_{0}$ of being a totally geodesic
submanifold of $G/H$.777Notice indeed that the coset space
$\mathbb{K}_{0}\subset\mathbb{K}$ of $G_{0}/H_{0}$ is also contained in the
tangent space
$\mathbb{K}_{*}\equiv\hat{\mathbb{L}}(\phi_{*}^{k})\,\mathbb{K}\,\hat{\mathbb{L}}(\phi_{*}^{k})^{-1}$
to $M_{{\rm scal.}}$ at the point $\varphi^{u}=0,\,\phi^{k}=\phi^{k}_{*}$
since
$\mathbb{K}_{0}=\hat{\mathbb{L}}(\phi_{*}^{k})\,\mathbb{K}_{0}\,\hat{\mathbb{L}}(\phi_{*}^{k})^{-1}\subset\mathbb{K}_{*}$.
Being $M_{0}$ symmetric, it follows that
$[[\mathbb{K}_{0},\,\mathbb{K}_{0}],\,\mathbb{K}_{0}]\subset\mathbb{K}_{0}$,
namely $\mathbb{K}_{0}$ is a _Lie triple system_ and thus $M_{0}$ is totally
geodesic [25].888By the same token, also the field equations for the moduli
fields $\varphi^{u}$, $u\neq a$, which are set to zero, are satisfied since
$\tilde{\Gamma}^{u}{}_{bc}=0$, $u\neq a$.
The moduli fields $\varphi^{a}(\eta)$, therefore, describe a geodesic in
$M_{0}$. Let us denote by $\kappa$ the corresponding line element (or
"velocity") along it:

$\kappa^{2}=\frac{1}{2}\,\mathcal{G}_{ab}\,\dot{\varphi}^{a}\,\dot{\varphi}^{b}\,.$
(2.33)

The reader can verify that the tensor-field equations $d\mathbb{H}^{M}=0$ are
satisfied by the ansatz.

As far as the Einstein equations are concerned, the only non-vanishing
components of the Riemann and Ricci tensors are:

$\displaystyle R_{\alpha\beta\gamma\delta}$
$\displaystyle=-v_{1}^{-2}\,(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})\,\,\Rightarrow,\,\,\,R_{\alpha\beta}=R_{\alpha\gamma\beta}{}^{\gamma}=-\frac{(d-2)}{v_{1}^{2}}\,g_{\alpha\beta}\,,$
$\displaystyle R_{ijkl}$
$\displaystyle=v_{2}^{-2}\,(g_{ik}g_{jl}-g_{il}g_{jk})\,\,\Rightarrow,\,\,\,R_{ij}=R_{ikj}{}^{k}=\frac{(d-1)}{v_{2}^{2}}\,g_{ij}\,.$
(2.34)

The Einstein equation can be conveniently recast in the form:

$R_{\hat{\mu}\hat{\nu}}=\,\frac{1}{2}\mathcal{G}_{rs}\,\partial_{\hat{\mu}}\phi^{r}\partial_{\hat{\nu}}\phi^{s}+T^{(H)}_{\hat{\mu}\hat{\nu}}\,.$
(2.35)

The components of $T^{(H)}$ are:

$\displaystyle T^{(H)}_{\alpha\beta}$
$\displaystyle=-\frac{1}{2}\,V_{*}\,L^{-2d}\,g_{\alpha\beta}\,,$
$\displaystyle T^{(H)}_{\eta\eta}$
$\displaystyle=-\frac{1}{2}\,V_{*}\,v_{3}^{2}\,L^{-2d}\,,$ $\displaystyle
T^{(H)}_{ij}$ $\displaystyle=\frac{1}{2}\,V_{*}\,L^{-2d}\,g_{ij}\,.$ (2.36)

Taking into account the contribution to the energy-momentum tensor coming from
the geodesic motion of the moduli fields $\varphi^{a}$, the Einstein equations
imply the following relations:

$v_{1}^{2}=\frac{2(d-2)\,L^{2d}}{V_{*}}\,\,,\,\,\,v_{2}^{2}=L^{2}=\frac{2(d-1)\,L^{2d}}{V_{*}}\,,\,\,\,v_{3}^{2}=\frac{2\,\kappa^{2}\,L^{2d}}{V_{*}}\,.$
(2.37)

From the second equation, it follows that:

$L^{2(d-1)}=\frac{V_{*}}{2(d-1)}\,,$

and, in terms of $L$, the radii $v_{1}$ of ${\rm AdS}_{d-1}$ and $v_{3}$ of
$S^{1}$ read

$v_{1}=\sqrt{\frac{d-2}{d-1}}\,L\,,\,\,\,v_{3}=\,\frac{\kappa}{\sqrt{d-1}}\,L\,.$

We notice that the radius $v_{3}$ of $S^{1}$ is proportional to the "velocity"
along the geodesic, that is its constant unit measure $\kappa$.
This solution is to be compared to the known one with geometry ${\rm
AdS}_{d}\times S^{d}$ in which

$\phi^{k}=\phi^{k}_{*}(\Gamma)={\rm const.}\,\,,\,\,\,\varphi^{u}={\rm
const.}\,.$

In that case, the ansatz for the metric reads

$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}+g_{ij}\,d\upxi^{i}\,d\upxi^{j}=v_{1}^{2}\,ds^{2}_{{\rm
AdS}_{d}}+v_{2}^{2}\,ds^{2}_{S^{d}}\,,$ (2.38)

and that for the tensor field strengths has the same general form (2.17). For
this solution, we find:

$v_{1}^{2}=v_{2}^{2}=L^{2}=\frac{2(d-1)\,L^{2d}}{V_{*}}\,\,\Rightarrow\,\,\,v_{1}=L=\left(\frac{V_{*}}{2(d-1)}\right)^{\frac{1}{2(d-1)}}\,.$
(2.39)

Notice that the ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ discussed above and
the known ${\rm AdS}_{d}\times S^{d}$ solution are not continuously connected.
Indeed setting $\kappa=0$ amounts in the former to shrinking $S^{1}$ to zero
radius. We may say that the ${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ solution
may be obtained from the ${\rm AdS}_{d}\times S^{d}$ by compactifying one
direction on the boundary of ${\rm AdS}_{d}$ and giving the moduli fields
$\varphi^{a}$ a geodesic dependence on the coordinate of the corresponding
$S^{1}$. This procedure amounts to performing a Scherk-Schwarz reduction on
$S^{1}$ with a twist matrix described by a _hyperbolic element_ of $G_{0}$, as
we will show below. It is the back-reaction of the evolving scalar fields on
space-time that deforms ${\rm AdS}_{d}$ into ${\rm AdS}_{d-1}\times S^{1}$.

It is interesting to note that this construction can be generalized. We can
take $k$-directions along the boundary of AdSd and compactify them, allowing
the moduli to depend on the coordinates $\eta_{1},\dots,\eta_{k}$ along these
$k$ directions:

$\varphi^{a}=\varphi^{a}(\boldsymbol{\eta})=\varphi^{a}(\eta_{1},\dots,\eta_{k})\,,$

where $\boldsymbol{\eta}\equiv(\eta_{1},\dots,\eta_{k})$, while the non-moduli
fields $\phi^{k}$ are still fixed at their attractor values $\phi^{k}_{*}$. We
assume $d-k>1$. Let us further assume that there is a symmetric submanifold of
$G_{\ell}/H_{\ell}$ consisting of the product of $k$-factors:

$M_{0}=\frac{G_{0}}{H_{0}}=\frac{G_{1}}{H_{1}}\times\frac{G_{2}}{H_{2}}\times\dots\times\frac{G_{k}}{H_{k}}\,,$
(2.40)

where the elements of $G_{0}=G_{1}\times\dots\times G_{k}$ are required to
commute with $\hat{\mathbb{L}}(\hat{\phi}_{*})$. Then we take
$\varphi^{a}(\eta_{1},\dots,\eta_{k})$ to describe $k$ geodesics, one within
each factor $\frac{G_{\tt j}}{H_{\tt j}}$, ${\tt j}=1,\dots,k$, so that:

$\frac{1}{2}\,\mathcal{G}_{ab}(\varphi(\boldsymbol{\eta}))\partial_{\eta_{\tt
i}}\varphi^{a}\partial_{\eta_{\tt j}}\varphi^{b}=\delta_{\tt
ij}\kappa^{2}_{\tt i}\,.$

The back-reaction of the moduli on the background yields a spacetime with
geometry:

$\displaystyle{\rm AdS}_{d-k}\times\underbrace{S^{1}\times\dots\times
S^{1}}_{k}\times S^{d}\,.$ (2.41)

Writing the metric in the form

$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}+g_{ij}\,d\upxi^{i}\,d\upxi^{j}=v_{1}^{2}\,ds^{2}_{{\rm
AdS}_{d-k}}+\sum_{\tt j=1}^{k}\,v_{2+\tt j}^{2}\,d\eta_{\tt
j}^{2}+v_{2}^{2}\,ds^{2}_{S^{d}}\,,$ (2.42)

where $\eta_{1},\dots,\eta_{k}$ parameterize the $k$ circles, we find, from
the Einstein equation:

$v_{1}^{2}=\frac{2(d-k-1)\,L^{2d}}{V_{*}}\,\,,\,\,\,v_{2}^{2}=L^{2}=\frac{2(d-1)\,L^{2d}}{V_{*}}\,,\,\,\,v_{\tt
j+2}^{2}=\frac{2\,\kappa_{\tt j}^{2}\,L^{2d}}{V_{*}}\,,\,\,{\tt j}=1,\dots
k\,.$ (2.43)

Having assumed $d-k>1$, $v_{1}$ is non-vanishing. The above equations imply

$L^{2(d-1)}=\frac{V_{*}}{2(d-1)}\,,\,\,v_{1}=\sqrt{\frac{d-1-k}{d-1}}\,L\,,\,\,\,v_{\tt
j+2}=\,\frac{\kappa_{\tt j}}{\sqrt{d-1}}\,L\,.$

In this section, we proved that the general ansatz describing the new
solutions satisfies the field equations. In this, no role is played by the
global properties of the solutions themselves. In particular, we could have
also considered our backgrounds in the form

$\displaystyle{\rm
AdS}_{d-k}\times\underbrace{\mathbb{R}\times\dots\times\mathbb{R}}_{k}\times
S^{d}\,.$ (2.44)

For $d=5$ and $k=1$ the background is a singular instance of Janus solutions
[26, 27, 28, 29] in Type IIB superstring theory.

However, since the geodesics on the moduli space $G_{0}/H_{0}$ are non-
compact, if the affine parameter $\eta_{j}$ varies in $\mathbb{R}$, the
corresponding geodesic would stretch to the boundary of the manifold and the
solution would not be regular. Considering the product of $k$ circles $S^{1}$,
we need to specify, in a consistent way, the corresponding boundary conditions
of the bosonic fields.999Alternatively, we could consider a product of $k$
finite intervals (${I}^{k}$) or of $k$ circles. We shall expand on this issue,
within superstring theory, in the next subsection.

Let us conclude this section by noting that if we started from the background
${\rm EAdS}_{d}\times S^{d}$ in the Euclidean version of the model, the moduli
space would be a pseudo-Riemannian, Wick-rotated version of the one in the
Lorentzian theory. As such it can also describe time-like geodesics for which

$\frac{1}{2}\,\mathcal{G}_{ab}\,\dot{\varphi}^{a}\,\dot{\varphi}^{b}=-\kappa^{2}<0\,.$
(2.45)

These geodesics would allow for a decomposition of the $d$-sphere yielding
geometries of the form:

$\displaystyle{\rm EAdS}_{d}\times\underbrace{S^{1}\times\dots\times
S^{1}}_{k}\times S^{d-k}\,.$ (2.46)

###  2.1 Boundary conditions on $S^{1}$

Consider the solution with spacetime of the form ${\rm AdS}_{d-1}\times
S^{1}\times S^{d}$ within superstring theory. Let us denote by $T$ the length
of $S^{1}$ so that $\eta\in[0,\,T)$. Since the dependence of the moduli fields
$\varphi^{a}(\eta)$ on $\eta$ describes a geodesic on $M_{0}$ which connects
two distinct points as we move around the circle $S^{1}$, consistency of the
background as a string theory solution requires the _monodromy_ matrix
connecting the initial and final points to be a string duality, namely to
belong to $G_{0}(\mathbb{Z})$. More precisely, suppose $\varphi^{a}(\eta)$
describe a geodesic on $M_{0}$ connecting an initial point, which we can
choose to be the origin of the moduli space $\varphi^{a}(0)=0$, to a final
point in $M_{0}$ with coordinate $\varphi^{a}(T)$. The coset representative
along the geodesic defines an $\eta$-dependent "twist" matrix in
$G_{0}/H_{0}$:

$\mathcal{A}(\eta)\equiv\mathbb{L}_{0}(\varphi^{a}(\eta))\,.$ (2.47)

The monodromy matrix reads:

$\mathfrak{M}\equiv\mathcal{A}(T)\,\mathcal{A}(0)^{-1}\,,$ (2.48)

and maps the coset representative in $M_{{\rm scal.}}$ at $\eta=0$ to that at
$\eta=T$:

$\mathfrak{M}\,\mathbb{L}(\varphi^{a}(0),\,\hat{\phi}_{*})=\mathbb{L}(\varphi^{a}(T),\,\hat{\phi}_{*})\,.$
(2.49)

This matrix is defined modulo a compensating transformation in the isotropy
group of $\varphi^{a}(0)$ in $M_{0}$:

$\mathfrak{M}\,\sim\,\mathfrak{M}\,h_{0}\,\,,\,\,\,h_{0}\in\mathbb{L}_{0}(\varphi^{a}(0))\,H_{0}\,\mathbb{L}_{0}(\varphi^{a}(0))^{-1}\,.$
(2.50)

We require one representative of the equivalence class of $\mathfrak{M}$ to be
an integer matrix:

$\mathfrak{M}\in G_{0}(\mathbb{Z})\,.$ (2.51)

A conjugation of $\mathfrak{M}$ by an element of $G_{0}(\mathbb{Z})$ can be
reabsorbed by a redefinition of the scalar fields $\varphi^{a}$ in the
background so that it is unphysical. The monodromy matrix is therefore defined
by conjugacy classes of hyperbolic elements in $G_{0}(\mathbb{Z})$.

The geodesic on $M_{0}$ is uniquely defined, by the initial point
$\varphi_{0}(0)\equiv(\varphi^{a}(0))$ and by a velocity vector $\mathbb{Q}\in
T_{\varphi_{0}(0)}(M_{0})$. The latter can be described by a matrix of the
form

$\mathbb{Q}=\mathbb{L}_{0}(\varphi_{0}(0))\,\mathbb{Q}_{0}\,\mathbb{L}_{0}(\varphi_{0}(0))^{-1}\,,$

where $\mathbb{Q}_{0}$, being an element fo the coset space of $M_{0}$, in a
suitable basis of the chosen real matrix representation, is a symmetric
matrix: $\mathbb{Q}_{0}=\mathbb{Q}_{0}^{T}$. The moduli along the
corresponding geodesic $\varphi_{0}(\eta)\equiv(\varphi^{a}(\eta))$ are
solutions to the matrix equation:

$\mathcal{M}_{0}(\varphi_{0}(\eta))\equiv\mathbb{L}_{0}(\varphi_{0}(\eta))\mathbb{L}_{0}(\varphi_{0}(\eta))^{T}=\mathcal{M}_{0}(\varphi_{0}(0))\,e^{\mathbb{Q}^{T}\,\eta}=\mathbb{L}_{0}(\varphi_{0}(0))\,e^{\mathbb{Q}_{0}\,\eta}\,\mathbb{L}_{0}(\varphi_{0}(0))^{T}\,.$
(2.52)

In terms of $\mathbb{Q}_{0}$ the "velocity" $\kappa$ along the geodesic reads:

$\kappa^{2}=\frac{{\tt k}}{8}\,{\rm Tr}(\mathbb{Q}_{0}^{2})\,.$ (2.53)

We can restrict ourselves to geodesics originating in the origin of $M_{0}$,
$\varphi^{a}(0)=0$, where
$\mathbb{L}_{0}(\varphi_{0}(0))=\mathbb{L}_{0}(0)={\bf 1}$. Having chosen
$\mathfrak{M}\in G_{0}(\mathbb{Z})$, then $T$ and $\mathbb{Q}_{0}$ have to be
fixed so that:

$\mathcal{M}_{0}(\varphi_{0}(T))=\mathfrak{M}\,\mathcal{M}_{0}(\varphi_{0}(0))\,\mathfrak{M}^{T}=\mathfrak{M}\,\mathfrak{M}^{T}=e^{\mathbb{Q}_{0}\,T}\,.$
(2.54)

In the more general solution (2.41), we have a monodromy matrix
$\mathfrak{M}_{\tt j}$ associated with each of the $k$ 1-cycles. Consistency
as a string background requires, for each of these matrices:

$\mathfrak{M}_{\tt j}\in G_{\tt j}(\mathbb{Z})\subset
G(\mathbb{Z})\,\,,\,\,\,\,{\tt j}=1,\dots,k\,.$ (2.55)

The solutions described here are instances of $U$-folds [30]. They feature
non-contractible 1-cycles along which there is a monodromy matrix is a string
duality in $G(\mathbb{Z})$. In general, this duality is a combination of $S$
and $T$-dualities and it is referred to as $U$-duality.

####  2.1.1 Deformations of the global geometry of the solution

Let us discuss a deformation of the global geometry of the solution effected
by introducing suitable metric moduli. Since $d$ is an odd number, it is
convenient to choose the following parametrization for $S^{d}$

$\\{\upxi^{i}\\}=\\{\mu_{I},\,\upphi_{I}\\}\,,\,\,\,i=1,\dots,d\,,\,\,\,I=1,\dots,{\tt
p}=\frac{d+1}{2}\,.$ (2.56)

where $\mu_{I}\geq 0,\,\upphi_{I}\in[0,\,2\pi)$, in which its metric reads:

$ds^{2}_{S^{d}}=L^{2}\sum_{I=1}^{\tt
p}(d\mu_{I}^{2}+\mu_{I}^{2}\,d\upphi_{I}^{2})\,.$ (2.57)

The coordinates $\mu^{I}$ are subject to the constraint: $\sum_{I=1}^{\tt
p}\mu_{I}^{2}=1\,.$ Let us now perform, on the solution discussed in the
previous sections, the following local change of variables:

$\upphi_{I}\,\rightarrow\,\upphi_{I}^{\prime}=\upphi_{I}+\chi_{I}\,\eta\,.$

The resulting solution is locally, though not globally equivalent to the
original one, as discussed in [15, 16, 20, 22]. The parameters $\chi_{I}$ are
metric deformation of the manifold $S^{1}\times S^{d}$ which can be understood
as follows. Consider the _local_ torus

$T^{{\tt p}+1}=S^{1}_{\upphi_{1}}\times S^{1}_{\upphi_{2}}\times\dots\times
S^{1}_{\eta}\,,$

where $S^{1}_{\upphi_{I}}$ are the local circles parametrized by $\upphi_{I}$
and $S^{1}_{\eta}$ is the circle parametrized by $\eta$. For $\chi_{I}=0$,
$T^{{\tt p}+1}$ can be described as the quotient

$T^{{\tt p}+1}=\mathbb{R}^{{\tt p}+1}/\Lambda_{{\rm cubic}}\,,$

where $\Lambda_{{\rm cubic}}$ is a cubic lattice generated by an orthonormal
basis of vectors ${\bf u}_{I},\,{\bf u}$. The position vector ${\bf x}$ of a
point in $T^{{\tt p}+1}$ is defined modulo a vector in the lattice (summation
over the repeated index $I$ is understood):

${\bf x}=\frac{\upphi_{I}}{2\pi}\,{\bf u}_{I}+\frac{\eta}{T}\,{\bf u}\sim{\bf
x}+n_{I}\,{\bf u}_{I}+n\,{\bf u}\,\,;\,\,\,n_{I},\,n\in\mathbb{Z}\,,$ (2.58)

which implies $\upphi_{I}\sim\upphi_{I}+2\pi$ and $\eta\sim\eta+T$.

After the deformation, that is for $\chi_{I}\neq 0$, the local torus is
described as:

$T^{{\tt p}+1}=\mathbb{R}^{{\tt p}+1}/\Lambda[\chi_{I}]\,,$

where $\Lambda[\chi_{I}]$ is a lattice generated by the non-orthogonal basis
$\\{\tilde{{\bf u}}_{I},\,\tilde{\bf u}\\}$, where

$\tilde{{\bf u}}_{I}\equiv{{\bf u}}_{I}\,\,,\,\,\,\,\tilde{\bf
u}\equiv\frac{\chi_{I}T}{2\pi}\,{\bf u}_{I}+{\bf u}\,.$

Now the position vector of a point on the local torus is subject to the
identification:

${\bf x}=\frac{\upphi_{I}}{2\pi}\,{\bf u}_{I}+\frac{\eta}{T}\,\tilde{{\bf
u}}\sim{\bf x}+n_{I}\,{\bf u}_{I}+n\,\tilde{{\bf
u}}\,\,,\,\,\,n_{i},\,n\in\mathbb{Z}\,,$ (2.59)

which implies the same identifications on $\upphi_{i},\,\eta$ given earlier.
Defining instead new coordinates $\upphi^{\prime}_{I},\,\eta^{\prime}$ with
respect to the old orthonormal basis

${\bf x}=\frac{\upphi^{\prime}_{I}}{2\pi}\,{\bf
u}_{I}+\frac{\eta^{\prime}}{T}\,{\bf u}\,,$

we have the following relation:

$\upphi_{I}^{\prime}=\upphi_{I}+\chi_{I}\,\eta\,,\,\,\eta^{\prime}=\eta\,.$

While in the coordinates $\\{\upphi_{I},\,\eta\\}$ the parameters $\chi_{I}$
appear in the metric, in the coordinates $\\{\upphi^{\prime}_{I},\,\eta\\}$,
they only appear in _twisted_ identifications:

$\upphi^{\prime}_{I}\sim\upphi^{\prime}_{I}+2\pi\,\,,\,\,\,\eta\sim\eta+T\,\wedge\,\,\upphi^{\prime}_{I}\sim\upphi^{\prime}_{I}+\chi_{I}\,T\,.$
(2.60)

The deformations $\chi_{I}$ break the isometry group ${\rm SO}(d+1)$ of
$S^{d}$ to its maximal torus ${\rm U}(1)^{\tt p}$ acting on $\upphi_{I}$ as
shift-isometries.

This global deformation was originally studied in [15, 16, 20, 22] in more
sophisticated variants of the U-folds described here, for $d=5$, and the
parameters $\chi_{I}$ corresponded to exactly marginal deformations of the
dual SCFT at the boundary.

From the above characterization, it follows that $\chi_{I}$ are periodic
since:

$\chi_{I}\rightarrow\chi_{I}\pm\frac{2\pi}{T}\,\,\Rightarrow\,\,\,\tilde{{\bf
u}}_{I}\,\rightarrow\,\tilde{{\bf u}}_{I}\,\,,\,\,\,\tilde{{\bf
u}}\,\rightarrow\,\tilde{{\bf u}}\pm\tilde{{\bf
u}}_{I}\in\Lambda[\chi_{I}]\,,$ (2.61)

so that $\chi_{I}\sim\chi_{I}\pm\frac{2\pi}{T}$.

###  2.2 Examples in Type IIB superstring theory

##### Type IIB in $D=10$.

The first setting where our construction can be applied is Type IIB theory in
ten dimensions [31]. The global symmetry group of the classical low-energy
supergravity description is $G={\rm SL}(2,\mathbb{R})_{{\rm IIB}}$ within
which only $G(\mathbb{Z})={\rm SL}(2,\mathbb{Z})_{{\rm IIB}}$ is a symmetry of
the superstring theory. It is known that this theory admits maximally
supersymmetric background of the form ${\rm AdS}_{5}\times S^{5}$ in which the
3-form field strengths vanish and the self-dual five-form field strength
$\hat{F}_{(5)}={}^{*}\hat{F}_{(5)}$ has a non-vanishing flux, related to the
radii of the ${\rm AdS}_{5}$ and $S^{5}$. The construction discussed earlier
applies with $d=5,\,p=4$. In this case the representation
$\mathscr{R}_{p}=\mathscr{R}_{4}={\bf 1}$, namely it is the singlet
representation since the electric and magnetic 5-form field strengths coincide
$G=F=\mathbb{H}=\hat{F}_{(5)}$, $\mathcal{M}(\phi)=1$. Moreover, $G_{0}=G$ and
the scalar manifold

$M_{{\rm scal.}}=\frac{G}{H}=\frac{{\rm SL}(2,\mathbb{R})_{{\rm IIB}}}{{\rm
SO}(2)}\,,$

coincides with the moduli space $M_{0}=G_{0}/H_{0}$ and describes the axion
$C_{(0)}$ and the dilaton $\phi$. In our previous notation we identify
$\varphi^{a}=(\phi,\,C_{(0)})$ and the (left) coset representative in the
${\bf 2}$ of ${\rm SL}(2,\mathbb{R})$ is
$\mathbb{L}=\mathbb{L}_{0}(\varphi^{a})=L_{(2)}(\phi,\,C_{(0)})$, where
$L_{(2)}$ is defined in eq. (3.6) below.101010 In this representation, the
action from the left of a matrix $\left(\begin{matrix}a&b\cr
c&d\end{matrix}\right)$, with $ad-bc=1$, amounts to the following
transformation on the complex axion-dilaton field $\rho=C_{(0)}+i\,e^{-\phi}$:
$\rho\rightarrow\rho^{\prime}=(d\rho+c)/(b\rho+a)$.  The background with
geometry ${\rm AdS}_{4}\times S^{1}\times S^{5}$ was constructed in [11]. The
axion and the dilaton describe a geodesic on $M_{{\rm scal.}}$. This
background is non-supersymmetric and unstable. The solution in [11] can be
generalized by considering a geodesic in the moduli space originating in a
generic point $\phi(0),\,C_{(0)}(0)$ with "velocity" $\kappa$:

$e^{\phi(\eta)}=\cosh(\sqrt{2}\kappa\,\eta)\,e^{\phi(0)}\,\,,\,\,C_{(0)}(\eta)=e^{-\phi(0)}\tanh\left(\sqrt{2}\kappa\,\eta\right)+C_{(0)}(0)\,.$
(2.62)

This solution, however, does not satisfy the appropriate boundary conditions
for it to be a possible solution to Type IIB superstring theory: the monodromy
matrix as $\eta\rightarrow\eta+T$ must be in ${\rm SL}(2,\mathbb{Z})_{{\rm
IIB}}$. Choosing, for instance, 111111In the definition of $J_{\tt n}$,
$\mathcal{S}\equiv\left(\begin{matrix}0&-1\cr 1&0\end{matrix}\right)$ and
$\mathcal{T}\equiv\left(\begin{matrix}1&0\cr 1&1\end{matrix}\right)$. $J_{\tt
n}$ is an elliptic element of ${\rm SL}(2,\mathbb{Z})$ for ${\tt n}=0,1$,
parabolic for ${\tt n}=2$ and hyperbolic for ${\tt n}>2$. It was shown that
hyperbolic elements of ${\rm SL}(2,\mathbb{Z})$, modulo conjugations, can be
either brought to the standard form $J_{\tt n}$, ${\tt n}>2$, or to coincide
with sporadic monodromies, see [30].

$\mathfrak{M}=J_{\tt n}\equiv-\mathcal{S}\,\mathcal{T}^{\tt
n}=\left(\begin{matrix}{\tt n}&1\cr-1&0\end{matrix}\right)\,\,,\,\,{\tt
n}\in\mathbb{Z}\,,\,\,{\tt n}>0\,,$ (2.63)

and $C_{(0)}=\phi(0)=0$, solving eq. (2.54), we find the following solution:

$\displaystyle e^{\phi(\eta)}$ $\displaystyle=\frac{{\tt
n}\sinh\left(\sqrt{2}\kappa\eta\right)}{\sqrt{{\tt
n}^{2}+4}}+\cosh\left(\sqrt{2}\kappa\eta\right)\,\,,\,\,C_{(0)}(\eta)=-\frac{2\sinh\left(\sqrt{2}\kappa\eta\right)}{\sqrt{{\tt
n}^{2}+4}\cosh\left(\sqrt{2}\kappa\eta\right)+{\tt
n}\sinh\left(\sqrt{2}\kappa\eta\right)}\,,$ $\displaystyle T$
$\displaystyle=\frac{1}{\sqrt{2}\kappa}\cosh^{-1}\left(\frac{{\tt
n}^{2}}{2}+1\right)\,.$ (2.64)

Defining the complex field $\rho=C_{(0)}+i\,e^{-\phi}$ one can verify that

$\rho(T)=-\frac{1}{\rho(0)+{\tt n}}\,,$ (2.65)

which is the effect of the transformation $J_{\tt
n}=-\mathcal{S}\,\mathcal{T}^{\tt n}$ on $\rho(0)$. Notice that, on this
solution

$\mathfrak{M}=\mathcal{A}(T)\,\mathcal{A}(0)^{-1}\,h_{0}=L_{(2)}(\phi(T),\,C_{(0)}(T))L_{(2)}(\phi(0),\,C_{(0)}(0))^{-1}\,h_{0}\,,$

where the compensating transformation $h_{0}$ reads:

$h_{0}=\frac{1}{\sqrt{1+{\tt n}^{2}}}{}\left(\begin{matrix}{\tt n}&1\cr-1&{\tt
n}\end{matrix}\right)\in{\rm SO}(2)\,.$ (2.66)

We could start from the background of the form [4, 6, 7]:

${\rm AdS}_{5}\times S^{5}/\mathbb{Z}_{k}\,,$ (2.67)

which preserves 16 supercharges. The classical supergravity description
features the following moduli space

$M_{{\rm scal.}}=\frac{{\rm SU}(1,k)}{{\rm U}(1)\times{\rm SU}(k)}\,.$ (2.68)

It describes $k$ complex scalars dual to the complexified coupling constants
$\tau_{i}$ of the dual $\mathcal{N}=2$ supersymmetric $A_{k-1}$ quiver gauge
theory. Also in this case $\mathscr{R}_{p}=\mathscr{R}_{4}={\bf 1}$,
$G=F=\mathbb{H}=\hat{F}_{(5)}$, $\mathcal{M}(\phi)=1$ and $G_{0}=G={\rm
SU}(1,k)$. Applying our construction to this situation would require the
monodromy matrix to belong to the quantum duality which generalizes the ${\rm
SL}(2,\mathbb{Z})$ symmetry for the $k=1$ case to $k>1$. For a discussion on
the definition of this group and its relation to the classical symmetry ${\rm
SU}(1,k)$ of the supergravity moduli space see, for instance, [32, 6].

##### Type IIB on $T^{4}$ or $K3$

The six-dimensional theory resulting from the compactification of Type IIB
superstring theory on a 4-torus $T^{4}$ is described, in its low-energy limit,
by the maximal $\mathcal{N}=(2,2)$ six-dimensional supergravity [33]. This
theory features a solution with spacetime ${\rm AdS}_{3}\times S^{3}$,
describing the near-horizon geometry of a system of D1-D5 branes with two
common Neumann directions (at the boundary of ${\rm AdS}_{3}$) and the five-
branes wrapping the 4-torus. Alternatively, in an S-dual picture, the same
background describes the near-horizon geometry of an F1-NS5 system.

The global symmetry group $G$ of the classical six-dimensional supergravity is
${\rm Spin}(5,5)$, double cover of ${\rm SO}(5,5)$. Quantum effects break this
group to $G(\mathbb{Z})={\rm Spin}(5,5;\,\mathbb{Z})$, which is conjectured to
encode superstring U-dualities. This group acts on the charge lattice
$\boldsymbol{\Gamma}^{5,5}$. The bosonic sector of the theory describes $n=5$
2-forms, $25$ scalar fields in the coset (in the classical theory)

$M_{{\rm scal.}}=\frac{G}{H}=\frac{{\rm Spin}(5,5)}{[({\rm Spin}(5)\times{\rm
Spin}(5))/\mathbb{Z}_{2}]}\,,$

and 16 vector fields in the ${\bf 16}$ of ${\rm Spin}(5,5)$. The ten self-dual
and anti-self-dual components of the 3-form field strengths
$\mathbb{H}^{M}_{\hat{\mu}\hat{\nu}\hat{\rho}}$, $M=1\dots,10$, transform,
under the action of the classical global symmetry group ${\rm SO}(5,5)$ in the
representation $\mathscr{R}_{2}={\bf 10}$. On the bosonic backgrounds under
consideration, the vector fields vanish. Since we shall be focusing on the
bosonic sector only, the global symmetry group $G$ will be described as ${\rm
SO}(5,5)$. The charge vector $\Gamma^{M}$ of the D1-D5 system has little group
$G_{\ell}={\rm SO}(4,5)\subset G$ in the classical supergravity. The coset
space $\mathbb{K}$ of $M_{{\rm scal.}}$, under the adjoint action of
$H_{\ell}={\rm SO}(4)\times{\rm SO}(5)$ splits into
$\mathbb{K}_{\ell}\oplus\hat{\mathbb{K}}$ in the representation
${\bf(4,5)}\oplus{\bf(1,5)}$, see eq. (2.24). The non-moduli scalar fields
$\hat{\phi}^{k}$ are chosen to be parameters of $\hat{\mathbb{K}}$ in the
${\bf(1,5)}$ of $H_{\ell}$. As discussed earlier, we fix ${\rm SO}(5)$ to
rotate $\hat{\phi}^{k}$ to their simplest form. This corresponds to the
parameter of a Cartan generator in the coset space. Next, we consider a group
$G_{0}$ commuting with the corresponding element
$\hat{\mathbb{L}}(\hat{\phi}^{k})$. Our choice for $G_{0}$ needs not to be
maximal. The maximal choice of $G_{0}$ is ${\rm SO}(4,4)$ and thus of the
moduli space $M_{0}=G_{0}/H_{0}={\rm SO}(4,4)/{\rm SO}(4)\times{\rm SO}(4)$.

The D5-branes of the D1-D5 system wrap the whole $T^{4}$ and the D1, D5
charges $d_{1},\,d_{5}$ belong to a $\boldsymbol{\Gamma}^{1,1}$ sublattice of
$\boldsymbol{\Gamma}^{5,5}$, which thus decomposes as [34]

$\boldsymbol{\Gamma}^{5,5}=\boldsymbol{\Gamma}^{1,1}\oplus\boldsymbol{\Gamma}^{4,4}\,.$

The subgroup $G_{0}$ acts on $\boldsymbol{\Gamma}^{4,4}$ leaving
$\boldsymbol{\Gamma}^{1,1}$ invariant.

Choosing one direction along ${\rm AdS}_{3}$ to be compact and fixing the
dependence $\varphi^{a}(\eta)$ of the moduli fields $\varphi^{a}$ in the space
$G_{0}/H_{0}$ on the compact boundary coordinate $\eta$, to describe a
geodesic in $G_{0}/H_{0}$, the back-reaction of these moduli deforms ${\rm
AdS}_{3}\times S^{3}$ into ${\rm AdS}_{2}\times S^{1}\times S^{3}$ as
discussed earlier. Consistency as a string background requires the monodromy
matrix $\mathfrak{M}$, connecting the starting and ending points of the
geodesic, to be a _hyperbolic_ element of ${\rm SO}(4,4;\,\mathbb{Z})$:

$\mathfrak{M}\equiv\mathbb{L}_{0}(\varphi^{a}(T))\,\mathbb{L}_{0}(\varphi^{a}(0))^{-1}\in{\rm
SO}(4,4;\,\mathbb{Z})\,.$ (2.69)

We could also choose the geodesic describing the moduli, to lie in a non-
maximal symmetric subspace of $M_{{\rm scal.}}$ such as:

$M_{0}=\frac{G_{0}}{H_{0}}=\left(\frac{{\rm SL}(2,\mathbb{R})}{{\rm
SO}(2)}\right)^{4}\subset\frac{{\rm SO}(4,4)}{{\rm SO}(4)\times{\rm
SO}(4)}\subset M_{{\rm scal.}}\,.$

In this particular case, $\mathfrak{M}$ has the general form:

$\mathfrak{M}=\mathfrak{M}_{1}\,\mathfrak{M}_{2}\,\mathfrak{M}_{3}\,\mathfrak{M}_{4}\,,$
(2.70)

where $\mathfrak{M}_{\tt j}$, ${\tt j}=1,\dots,4$, is an element of the ${\rm
SL}(2,\mathbb{Z})$ subgroup of the corresponding ${\rm SL}(2,\mathbb{R})$
factor in $G_{0}$. We shall expand on this class of U-fold solutions in the
maximal six-dimensional theory in the next section.

Considering Type IIB superstring theory compactified on $K3$, this is a six-
dimensional $(2,0)$ ungauged supergravity [1]. The group $G$ is ${O}(5,21)$
and the charge lattice is $\boldsymbol{\Gamma}^{5,21}$. Considering a
D1-D5-brane system in which the D5 branes wrap the whole $K3$, the little
group $G_{\ell}$ of corresponding charge vector $\Gamma^{M}$ is ${\rm
O}(4,21)$, yielding 84 moduli fields in $G_{\ell}/H_{\ell}={\rm SO}(4,21)/{\rm
SO}(4)\times{\rm SO}(21)$. According to our construction, we do not consider
geodesics within this manifold, but rather within a smaller one
$M_{0}=G_{0}/H_{0}$. The maximal choice of $G_{0}$ is ${O}(4,20)$. The D1, D5
charges $d_{1},\,d_{5}$ belong to a $\boldsymbol{\Gamma}^{1,1}$ sublattice of
$\boldsymbol{\Gamma}^{5,21}$, which thus decomposes as [34]

$\boldsymbol{\Gamma}^{5,21}=\boldsymbol{\Gamma}^{1,1}\oplus\boldsymbol{\Gamma}^{4,20}\,.$

The subgroup $G_{0}$ acts on $\boldsymbol{\Gamma}^{4,20}$ leaving
$\boldsymbol{\Gamma}^{1,1}$ invariant. The only non-vanishing moduli fields
$\varphi^{a}$ in the solution are required to describe a geodesic in
$G_{0}/H_{0}$. The monodromy matrix is then chosen as follows:

$\mathfrak{M}\in G_{0}(\mathbb{Z})\,\subset\,{\rm O}(4,20;\,\mathbb{Z})\,.$
(2.71)

##  3 The Bosonic Sector of the $\mathcal{N}=(2,2),\,D=6$ Theory

In this section, we shall describe an instance of $U$-fold of the form ${\rm
AdS}_{2}\times S^{1}\times S^{3}$, obtained applying the construction
discussed in section 2 to the maximal six-dimensional model obtained from Type
IIB supergravity upon compactification on a 4-torus and dualization of all
forms are dualized to lower-order ones. As pointed out above, this model, in
its classical limit, features a scalar manifold of the form:

$\mathcal{M}_{{\rm scal}}=\frac{{\rm SO}(5,5)}{{\rm SO}(5)\times{\rm
SO}(5)}\,.$ (3.1)

Let us start reviewing the general mathematical description of the model. We
recall that the scalar fields originating from toroidal dimensional reduction
to $D=6$ consist in the moduli of the internal metric $G_{ij}$, the axion-
dilaton field $\rho=C_{(0)}+i\,e^{-\phi}$, the scalars
$B^{\upalpha}_{ij}=(C_{ij},\,-B_{ij})$, $\upalpha=1,2$, originating from the
ten-dimensional 2-forms $C_{(2)},\,B_{(2)}$ and the scalar
$C_{ijkl}=c\,\epsilon_{ijkl}$ from the 4-form. As mentioned earlier, the five
2-form fields and their duals transform, under the global symmetry group ${\rm
SO}(5,5)$, in its fundamental representation $\mathscr{R}_{(2)}={\bf 10}$.
They are $B^{\upalpha}_{\mu\nu}=(C_{\mu\nu},\,-B_{\mu\nu})$, their duals
$\tilde{B}_{\upalpha\,\mu\nu}=(\tilde{C}_{\mu\nu},\,-\tilde{B}_{\mu\nu})$ and
the six $C_{ij\mu\nu}$. The identification of the $D=6$ scalar fields and
2-forms with the above Type IIB fields is effected by decomposing the adjoint
and the fundamental representations of ${\rm SO}(5,5)$ with respect to the
subgroup ${\rm SL}(4,\mathbb{R})\times{\rm SO}(1,1)\times{\rm
SL}(2,\mathbb{R})_{{\rm IIB}}$, where ${\rm SL}(4,\mathbb{R})\times{\rm
SO}(1,1)$ is the group of acting transitively on the metric moduli of the
torus:

$\displaystyle{\rm Adj}({\rm SO}(5,5))$ $\displaystyle\rightarrow\,{\rm
Adj}({\rm GL}(4,\mathbb{R}))+{\rm Adj}({\rm SL}(2,\mathbb{R})_{{\rm
IIB}})+{\bf(6,2)}_{+1}+{\bf(1,1)}_{+2}+$
$\displaystyle+{\bf(6^{\prime},2)}_{-1}+{\bf(1,1)}_{-2}\,,$ $\displaystyle{\bf
10}$
$\displaystyle\rightarrow\,{\bf(1,2)}_{-1}+{\bf(6,1)}_{0}+{\bf(1,2)}_{+1}\,,$
(3.2)

where the grading refers to the ${\rm SO}(1,1)$ factor. The scalars
$B^{\upalpha}_{ij}$ parametrize the nilpotent generators in the
${\bf(6,2)}_{+1}$, $c$ the highest-grading generator ${\bf(1,1)}_{+2}$. As for
the 2-forms, $B^{\upalpha}_{\mu\nu}$ and $\tilde{B}_{\upalpha\,\mu\nu}$ are
defined by the ${\bf(1,2)}_{-1}$ and ${\bf(1,2)}_{+1}$ representations in the
branching of the ${\bf 10}$, while $C_{ij\mu\nu}$ transform in the
${\bf(6,1)}_{0}$.

Besides these fields, the bosonic sector also consists of 16 vector fields
which, however, will not play a role in our analysis.

Below we describe the subsector of the maximal D=6 theory describing the
graviton, the scalar fields and the 2-forms. We shall collectively denote by
$C_{(2)}^{I}$, $I=1,\dots,5$ the tensor fields and by $\phi^{s}$ the 25 scalar
fields. The corresponding Lagrangian (2.2)) reads

$\displaystyle\mathcal{L}_{6}=e_{6}R_{6}-\frac{e_{6}}{12}\left(\mathcal{I}_{IJ}H^{I}_{\,\,\hat{\mu}\hat{\nu}\hat{\rho}}H^{J\,\hat{\mu}\hat{\nu}\hat{\rho}}+\mathcal{R}_{IJ}H^{I}_{\,\,\hat{\mu}\hat{\nu}\hat{\rho}}{}^{\star}H^{J\hat{\mu}\hat{\nu}\hat{\rho}}\right)-e_{6}\frac{{\rm
Tr}}{8}\left(\mathcal{M}^{-1}\partial\mathcal{M}\mathcal{M}^{-1}\partial\mathcal{M}\right),$
(3.3)

where $\hat{\mu}=0,...,5$ and I,J=$1,...,n$. The matrices
$\mathcal{I}(\phi)=(\mathcal{I}(\phi)_{IJ})=\mathcal{I}(\phi)^{T}>0$ and
$\mathcal{R}(\phi)=(\mathcal{R}(\phi)_{IJ})=-\mathcal{R}(\phi)^{T}$ were
introduced earlier in section 2, together with the symmetric, positive-
definite ${\rm O}(5,5)$-matrix $(\mathcal{M})=\mathcal{M}_{MN}$, where
$M,N=1,...,10$, see eq. (2.7).

###  3.1 An Explicit Parametrization of the Scalar Manifold

In this section, we review the explicit description of the scalar manifold
$M_{{\rm scal.}}$ in terms of the so-called _solvable_ or _Borel_ one [35, 36]
parametrization, in which the Type IIB origin of the 25 scalar fields is
manifest. A detailed account of this parametrization, for this model, is also
given in [37]. Here we use, for the description of the Type IIB theory in
$D=10$, the conventions defined in Appendix B of [11].

The basis of the $\mathscr{R}_{(2)}={\bf 10}$ is chosen as follows:121212In
this representation the coefficient ${\tt k}$ in (2.11) is equal to 1.

$V^{M}=(V^{\alpha},\,V_{ij},\,V_{\alpha})\,,$ (3.4)

where the components $V^{\alpha},\,V_{ij},\,V_{\alpha}$ transform in the
${\bf(1,2)}_{-1}$, ${\bf(6,1)}_{0}$ and ${\bf(1,2)}_{+1}$, respectively. In
this basis, the ${\rm SO}(5,5)$ invariant matrix in the fundamental reads:

$\Omega_{MN}\equiv\left(\begin{matrix}{\bf 0}&{\bf
0}&\delta^{\alpha}_{\beta}\cr{\bf 0}&\epsilon_{ij\,kl}&{\bf
0}\cr\delta^{\alpha}_{\beta}&{\bf 0}&{\bf 0}\end{matrix}\right)\,.$ (3.5)

In the solvable parametrization of the scalar manifold the scalar fields are
defined as parameters of the Borel subalgebra of $\mathfrak{so}(5,5)$.

We start by discussing the axion-dilaton system and then describe the rest of
the scalar fields.

##### Axion-dilaton system.

The dilaton $\phi$ and the axion field $C_{(0)}$, in the ten-dimensional
theory, span the coset ${\rm SL}(2,\mathbb{R})_{{\rm IIB}}/{\rm SO}(2)$. The
corresponding coset representative is chosen of the following form:

$L_{(2)}=(L_{\upalpha}{}^{\upbeta})=\left(\begin{array}[]{cc}e^{\phi/2}&0\\\
C_{(0)}\,e^{\phi/2}&e^{-\phi/2}\\\ \end{array}\right)\,.$ (3.6)

This allows us to define the matrix

$\mathbf{m}_{\upalpha\upbeta}\equiv(L_{(2)}L_{(2)}^{T})_{\upalpha\upbeta}=\frac{1}{{\rm
Im}(\rho)}\left(\begin{matrix}1&{\rm Re}(\rho)\cr{\rm
Re}(\rho)&|\rho|^{2}\end{matrix}\right)\,.$ (3.7)

##### The ${\rm SO}(5,5)/{\rm SO}(5)\times{\rm SO}(5)$ manifold.

As mentioned above, the scalars $B^{\upalpha}_{ij}$ and $c$ parametrize the
generators $t_{\upalpha}^{ij},\,t$ of the Borel subalgebra of
$\mathfrak{so}(5,5)$ in the representations ${\bf(6,2)}_{+1}$ and
${\bf(1,1)}_{+2}$ of ${\rm GL}(4,\,\mathbb{R})$, respectively. In the basis
(3.4), these generators read:

$\displaystyle\mathbb{B}$
$\displaystyle=\frac{1}{2}\,B^{\upalpha}_{ij}\,t_{\upalpha}^{ij}=\left(\begin{matrix}{\bf
0}&{\bf 0}&{\bf 0}\cr B_{\upalpha ij}&{\bf 0}&{\bf 0}\cr{\bf
0}&-B_{\upalpha}{}^{ij}&{\bf
0}\end{matrix}\right)\,,\,\,\,\,\,\mathbb{C}=c\,t=\left(\begin{matrix}{\bf
0}&{\bf 0}&{\bf 0}\cr{\bf 0}&{\bf 0}&{\bf 0}\cr
c\,\epsilon_{\upalpha\upbeta}&{\bf 0}&{\bf 0}\end{matrix}\right)\,,$ (3.8)

where

$B_{\upalpha\,ij}\equiv\,\epsilon_{\upalpha\upbeta}\,B^{\beta}_{ij}=(B_{ij},\,C_{ij})\,,\,\,\,\,B_{\upalpha}{}^{ij}\equiv\frac{1}{2}\,\epsilon^{ijkl}\,B_{\upalpha\,kl}\,.$
(3.9)

Next we write the coset representatives $\mathbb{L}_{(2)},\,\mathbb{L}_{(4)}$
of ${\rm SL}(2,\mathbb{R})_{{\rm IIB}}/{\rm SO}(2)$ and ${\rm
GL}(4,\mathbb{R})/{\rm SO}(4)$ in the same representation:

$\displaystyle\mathbb{L}_{(2)}$
$\displaystyle=\left(\begin{matrix}L_{(2)}^{-T}&{\bf 0}&{\bf 0}\cr{\bf 0}&{\bf
1}&{\bf 0}\cr{\bf 0}&{\bf 0}&L_{(2)}\end{matrix}\right)\,,$
$\displaystyle\mathbb{L}_{(4)}$
$\displaystyle=\left(\begin{matrix}G^{-\frac{1}{4}}\,{\bf 1}&{\bf 0}&{\bf
0}\cr{\bf 0}&2\,G^{-\frac{1}{4}}\,E_{[i}{}^{k}E_{j]}{}^{k}&{\bf 0}\cr{\bf
0}&{\bf 0}&G^{\frac{1}{4}}\,{\bf 1}\end{matrix}\right)\,,$ (3.10)

where $G\equiv{\rm det}(G_{ij})$, the metric $G_{ij}$ being in the $D=10$
Einstein frame and $E_{i}{}^{k}$ being the vielbein matrix:
$G_{ij}=E_{i}{}^{k}E_{j}{}^{k}$ (summation over $k$ is understood). The
$\frac{{\rm SO}(5,5)}{{\rm SO}(5)\times{\rm SO}(5)}$ coset representative is
then written as follows:

$\mathbb{L}^{M}{}_{N}=e^{\mathbb{C}}\cdot
e^{\mathbb{B}}\cdot\mathbb{L}_{(4)}\cdot\mathbb{L}_{(2)}\,.$ (3.11)

We can now compute the components of $\mathcal{M}=\mathbb{L}\,\mathbb{L}^{T}$:

$\displaystyle\mathcal{M}^{\upalpha\upbeta}$
$\displaystyle=G^{-\frac{1}{2}}\,\mathbf{m}^{\upalpha\upbeta}\,\,,$
$\displaystyle\mathcal{M}_{ij}{}^{\upbeta}$
$\displaystyle=G^{-\frac{1}{2}}\,\mathbf{m}^{\upalpha\upbeta}\,B_{\upalpha\,ij}\,\,,$
$\displaystyle\mathcal{M}_{\upalpha}{}^{\upbeta}$
$\displaystyle=G^{-\frac{1}{2}}\,c_{\upalpha\upsigma}\,\mathbf{m}^{\upsigma\upbeta}\,\,,$
$\displaystyle\mathcal{M}_{ij,\,kl}$
$\displaystyle=G^{-\frac{1}{2}}\,\left(2\,G_{i[k}G_{l]j}+\mathbf{m}^{\upalpha\upbeta}\,B_{\upalpha\,ij}\,B_{\upbeta,kl}\right)\,,$
$\displaystyle\mathcal{M}_{\upalpha\,ij}$
$\displaystyle=G^{-\frac{1}{2}}\,\left(c_{\upalpha\upbeta}\,\mathbf{m}^{\upbeta\upsigma}\,B_{\upsigma\,ij}-B_{\upalpha}{}^{kl}\,G_{ki}G_{lj}\right)\,,$
$\displaystyle\mathcal{M}_{\upalpha\upbeta}$
$\displaystyle=G^{-\frac{1}{2}}\,\left(c_{\upalpha\upsigma}c_{\upbeta\upgamma}\,\mathbf{m}^{\upsigma\upgamma}+\frac{1}{2}\,B_{\upalpha}{}^{ij}B_{\upbeta}{}^{kl}\,G_{ik}G_{jl}+G\,\mathbf{m}_{\upalpha\upbeta}\right)\,,$
(3.12)

where:

$c_{\upalpha\upbeta}\equiv
c\,\epsilon_{\upalpha\upbeta}-\frac{1}{4}\,B_{\upalpha\,ij}B_{\upbeta}{}^{ij}\,.$
(3.13)

We refer to the Appendix A for more details about the solvable parametrization
of the scalar manifold as well as the description of the tensor fields in
terms of the $\mathfrak{so}(5,5)$ weights of the ${\bf 10}$.

###  3.2 ${\rm AdS}_{3}\times{S}^{3}$ Solution of Type IIB Theory on $T^{4}$

In the appendix C we present a general discussion of the static black-string
solutions in the presence of non-trivial moduli. A known instance is the D1-D5
system in which the D1 and D5-branes have only two common Neumann directions,
defining the worldsheet of the black string in $D=6$, and the D5-branes wrap
the whole $T^{4}$. This solution is of particular importance in string theory
since it provided the setting for the first, successful black hole entropy
calculation through string microstate counting [38].

The charge vector reads

$\Gamma^{M}=(d_{5},0,0,0,0,0,0,0,d_{1},0)\,,$ (3.14)

where $d_{5},\,d_{1}$ are taken to be both positive and denote the D5 and the
D1-brane charges, respectively. Through a (charge-dependent) change of basis,
see Appendix D, the above vector can be brought to the form:

$\Gamma^{\prime M}=(\sqrt{2|d_{1}d_{5}|},0,0,0,0,0,0,0,0,0)\,.$ (3.15)

In this new basis, the invariant pseudo-orthogonal metric reads:

$\Omega^{\prime}={\rm diag}(+1,+1,+1,+1,+1,-1,-1,-1,-1,-1)\,.$ (3.16)

The above form of $\Gamma^{\prime}$ makes its little group $G_{\ell}={\rm
SO}(4,5)$ manifest: it is the subgroup of $G$ acting non-trivially only on the
entries $2,3,\dots,10$. The matrix representation of $G_{\ell}$ in the
original basis (3.14) is $d_{1},\,d_{5}$-dependent. As explained in sections 2
and 2.2, the coset space $\mathbb{K}$ of $M_{{\rm scal.}}$ splits into
$\mathbb{K}_{\ell}$ and $\hat{\mathbb{K}}$ supporting the representations
${\bf(4,5)}$ and ${\bf(1,5)}$ of $H_{\ell}={\rm SO}(4)\times{\rm SO}(5)$. The
non-moduli fields $\phi^{k}$ parametrize the five-dimensional
$\hat{\mathbb{K}}$ and can be brought to a normal form $\phi^{k}=(g,0,0,0,0)$,
through the action of ${\rm SO}(5)$, in which the only non-vanishing scalar
$g$ parametrizes the Cartan generator:

$\mathfrak{h}={\rm diag}(1,0,0,0,0,0,0,0,-1,0)\,.$ (3.17)

The scalar field $g$ is identified as follows:
$e^{g}=e^{\phi}\,G^{\frac{1}{2}}$, where $G\equiv{\rm det}(G_{ij})$, see
Appendix A.

The effective potential $V$ for the D1-D5 system, and its extremization, is
discussed in Appendix D. The maximal subgroup of ${\rm SO}(5,5)$ commuting
with $\mathfrak{h}$ is ${\rm SO}(4,4)$, which represents the maximal choice of
$G_{0}\subset G_{\ell}$. The maximal moduli space $M_{0}$ is:

$M_{0}=\frac{{\rm SO}(4,4)}{{\rm SO}(4)\times{\rm SO}(4)}\,,$

parametrized by $C_{ij}$ and $\tilde{G}_{ij}=e^{-\frac{\phi}{2}}\,G_{ij}$, see
Appendix A. Had we chosen the system of F1 and NS5 branes dual to the D1-D5
one, the maximal moduli space $M_{0}$ would be spanned by
$B_{ij},\,G^{(s)}_{ij}\equiv e^{\frac{\phi}{2}}\,G_{ij}$, where $G^{(s)}_{ij}$
is the torus metric in the string frame. The ${\rm O}(4,4;\,\mathbb{Z})$
global symmetry group would, in this case, encode the T-dualities along
directions of $T^{4}$, and transformations in ${\rm O}(4,4;\,\mathbb{Z})/{\rm
SO}(4,4;\,\mathbb{Z})$ would map the Type IIB and Type IIA descriptions of the
same F1-NS5 solution.

Coming back to the D1-D5 system, when we only switch on the moduli fields
$C_{ij}$, $\tilde{G}_{ij}$ and the field $g$, the matrix $\mathcal{M}$ is the
product of two commuting symmetric matrices:

$\mathcal{M}(\varphi^{a},\,g)=\mathcal{M}_{0}(\varphi^{a})\,\hat{\mathcal{M}}(g)\,,$
(3.18)

where

$\displaystyle\hat{\mathcal{M}}(g)$
$\displaystyle=\text{diag}(e^{g},1,1,1,1,1,1,1,1,e^{-g},1)\,,$ (3.19)

and
$\Gamma^{T}\,\mathcal{M}(\varphi^{a},g)\Gamma=\Gamma^{T}\,\hat{\mathcal{M}}(g)\Gamma$,
so that the effective potential reads:

$V=\frac{1}{2}\left(d_{1}^{2}\,e^{-g}+d_{5}^{2}\,e^{g}\right)\,.$ (3.20)

By further fixing the moduli fields $C_{ij}=0,\,\tilde{G}_{ij}=\delta_{ij}$,
we have $g=2\phi$ and the solution takes the known form (see appendix C.1)

$\displaystyle ds^{2}=$
$\displaystyle(Z_{1}Z_{5})^{-\frac{1}{2}}(-dt^{2}+dx^{2})+(Z_{1}Z_{5})^{\frac{1}{2}}(dx^{i}dx^{i})\,,$
(3.21) $\displaystyle e^{g}$
$\displaystyle=e^{2\phi}=\log\left(\frac{Z_{1}}{Z_{5}}\right)\,,$
$\displaystyle dx^{i}dx^{i}=$ $\displaystyle
dr^{2}+r^{2}d\Omega^{2}_{3},\qquad Z_{1}=1+\frac{d_{1}}{2r^{2}},\qquad
Z_{5}=1+\frac{d_{5}}{2r^{2}}.$ (3.22)

To avoid the factor $1/2$ in the harmonic functions, it can be convenient to
define rescaled charges: $d_{1}=2Q_{1},\,d_{5}=2Q_{5}$. The charges $d_{1}$
and $d_{5}$ are defined in eq.(D.1). At the horizon the $Z$ functions become

$\displaystyle Z_{1}=\frac{d_{1}}{2r^{2}}=\frac{Q_{1}}{r^{2}},\qquad
Z_{5}=\frac{d_{5}}{2r^{2}}=\frac{Q_{5}}{r^{2}}.$ (3.23)

This background is AdS${}_{3}\times S^{3}$, which is the throat of the D1-D5
system. The non-moduli scalars are attracted towards configurations that
extremizes the potential. We give a general discussion in appendix D. In this
particular case, in which we set $C_{ij}=0,\,\tilde{G}_{ij}=\delta_{ij}$, the
scalar field $g=2\phi$ is fixed at the horizon, through the attractor
mechanism, to the value

$\displaystyle
g=2\phi=\log\left(\frac{d_{1}}{d_{5}}\right)=\log\left(\frac{Q_{1}}{Q_{5}}\right)\,.$
(3.24)

At the minimum the effective potential has the value

$V_{*}=d_{1}d_{5}\,.$ (3.25)

The near-horizon metric is conveniently written in the following form:

$\displaystyle
ds^{2}=\frac{r^{2}}{L^{2}}(-dt^{2}+dx^{2})+\frac{L^{2}}{r^{2}}dr^{2}+L^{2}\left[d\psi^{2}+\sin^{2}\psi\left(d\theta^{2}+\sin^{2}\theta
d\omega^{2}\right)\right],$ (3.26)

where $L^{2}=\frac{\sqrt{d_{1}d_{5}}}{2}$, as obtained in eqs.(2.39), (3.25).

###  3.3 ${\rm AdS}_{2}\times S^{1}\times{S}^{3}$ U-Folds

In this section, we apply the general construction of section 2 to the
construction of $U-$fold solutions with geometry AdS${}_{2}\times S^{1}\times
S^{3}$ in Type IIB theory on $T^{4}$. As discussed in section 2, we only
switch on the field $e^{g}=e^{\phi}\,G^{\frac{1}{2}}$ and the moduli fields
$\varphi^{a}$ spanning either the manifold ${\rm SO}(4,4)/[{\rm
SO}(4)\times{\rm SO}(4)]$, parametrized by
$\tilde{G}_{ij}=e^{-\frac{\phi}{2}}G_{ij},\,C_{ij}$, or a symmetric
submanifold $M_{0}=G_{0}/H_{0}$ of it. We start with a metric of the form

$\displaystyle
ds^{2}=-k_{1}\frac{r^{2}}{L^{2}}dt^{2}+k_{1}\frac{L^{2}}{r^{2}}dr^{2}+k_{2}d\eta^{2}+L^{2}\left[d\psi^{2}+\sin^{2}\psi\left(d\theta^{2}+\sin^{2}\theta
d\omega^{2}\right)\right],$ (3.27)

where $k_{1},\,k_{2}$ are positive constants and we compactify the $\eta$
direction: $\eta\in[0,T]$. Just as in the D1-D5 solution, we turn on the
charges $d_{1},\,d_{5}$. The ansatz for the three form field strength is again

$\displaystyle\mathbb{H}=-\gamma\Omega\mathcal{M}\Gamma dt\wedge d\eta\wedge
dr+\zeta(\theta,\psi)\Gamma d\psi\wedge d\theta\wedge d\omega,$ (3.28)

which is duality-covariant. On the 3-form field we need to impose the twisted
self-duality condition, the Maxwell equations, and the Bianchi equations,
which read respectively

$\displaystyle\star\mathbb{H}=-\Omega\,\mathcal{M}\mathbb{H},\quad
d\mathbb{H}=0\,.$ (3.29)

These are all solved if we pick

$\displaystyle\gamma=\frac{k_{1}\sqrt{k_{2}}}{L^{3}},\quad\zeta(\theta,\psi)=\sin\theta\sin^{2}\psi.$
(3.30)

The moduli $\varphi^{a}$ in $M_{0}$ are switched on along a geodesic arc:

$\varphi^{a}=\varphi^{a}(\eta)\,,$ (3.31)

with "velocity" $\kappa$ and affine parameter $\eta$, so that

$\displaystyle\frac{1}{2}\mathcal{G}_{ab}\partial_{\eta}\varphi^{a}\partial_{\eta}\varphi^{b}=\kappa^{2}.$
(3.32)

Thanks to the ansatz for the field strength $\mathbb{H}$ (3.28), the equation
of motion for the scalars has the same form as eq.(2.9), where
$V(\phi,\Gamma)=V(g,\Gamma)$ is given in eq. (3.20). The scalar equation of
motion is solved if we extremize the potential as

$\displaystyle
g=\log\left(e^{\phi}\,G^{\frac{1}{2}}\right)=\log\left(\frac{d_{1}}{d_{5}}\right)={\rm
const.},$ (3.33)

since we forced the moduli fields $\varphi^{a}$ to evolve in $\eta$ along a
geodesic.
The last equation to check is the trace-reversed Einstein equation. The
computation of the energy-momentum part which involves the three form
$\mathbb{H}$ gives

$\displaystyle T^{(H)}_{\hat{\mu}\hat{\nu}}=\frac{d_{1}d_{5}}{2L^{4}}\,{\rm
diag}\left(\frac{k_{1}r^{2}}{L^{4}},-\frac{k_{2}}{L^{2}},-\frac{k_{1}}{r^{2}},1,\sin^{2}(\psi),\sin^{2}(\theta)\sin^{2}(\psi)\right)\,,$
(3.34)

where the second element on the diagonal is the $(\eta,\eta)$ component of the
tensor and has to be canceled, in the Einstein equation, by the only non-
vanishing component of the scalar energy-momentum tensor, which equals the
left-hand side of eq.(3.32), namely $\kappa^{2}$. The Einstein equation then
completely fixes the coefficients as follows

$\displaystyle k_{1}=\frac{1}{2},\quad
k_{2}=\frac{\kappa^{2}\,\sqrt{d_{1}d_{5}}}{4},\quad
L^{2}=\frac{\sqrt{d_{1}d_{5}}}{2}.$ (3.35)

To give an instance of a geodesic curve described by string 0-modes, we can
restrict the moduli fields $\varphi^{a}$ to the submanifold

$M_{0}=\frac{G_{0}}{H_{0}}=\left(\frac{{\rm SL}(2,\mathbb{R})}{{\rm
SO}(2)}\right)^{2}\,,$ (3.36)

which is obtained by restricting the moduli
$\tilde{G}_{ij}=e^{-\frac{\phi}{2}}\,G_{ij},\,C_{ij}$ to the following non-
vanishing fields:

$e^{-\phi_{12}}\equiv\tilde{G}_{11}=\tilde{G}_{22}\,,\,\,e^{-\phi_{34}}\equiv\tilde{G}_{33}=\tilde{G}_{44}\,,\,\,\,C_{12},\,C_{34}\,.$
(3.37)

The metric of the scalar manifold, restricted to $g$,
$\phi_{12},\,\phi_{34},\,C_{12},\,C_{34}$ reads:

$d\tilde{s}^{2}=\frac{dg^{2}}{2}+d\phi_{12}^{2}+d\phi_{34}^{2}+e^{2\phi_{12}}\,dC_{12}^{2}+e^{2\phi_{34}}\,dC_{34}^{2}\,.$
(3.38)

The geodesic can be chosen as the product of two geodesics spanned by
$\phi_{12},\,C_{12}$ and by $\phi_{34},\,C_{34}$, respectively, unfolding in
the two factors of $M_{0}$. The monodromy matrix $\mathfrak{M}$ will be the
product $\mathfrak{M}=\mathfrak{M}_{1}\,\mathfrak{M}_{2}$, of two elements
$\mathfrak{M}_{1},\,\mathfrak{M}_{2}$ in the two factors ${\rm
SL}(2,\mathbb{Z})$ in $G_{0}(\mathbb{Z})={\rm SL}(2,\mathbb{Z})^{2}$.
Choosing, for instance:

$\mathfrak{M}_{1}=J_{{\tt n}_{1}}\,,\,\,\,\,\mathfrak{M}_{2}=J_{{\tt
n}_{2}}\,,$

the geodesic reads:

$\displaystyle e^{\phi_{12}(\eta)}$ $\displaystyle=\frac{\sqrt{{\tt
n}_{1}^{2}+4}\cosh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{1}^{2}+2\right)\right)}{T}\right)+{\tt
n}_{1}\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{1}^{2}+2\right)\right)}{T}\right)}{\sqrt{{\tt n}_{1}^{2}+4}}\,,$
$\displaystyle C_{12}(\eta)$
$\displaystyle=-\frac{2\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{1}^{2}+2\right)\right)}{T}\right)}{\sqrt{{\tt
n}_{1}^{2}+4}\cosh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{1}^{2}+2\right)\right)}{T}\right)+{\tt
n}_{1}\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{1}^{2}+2\right)\right)}{T}\right)}\,,$ $\displaystyle e^{\phi_{34}(\eta)}$
$\displaystyle=\frac{\sqrt{{\tt
n}_{2}^{2}+4}\cosh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{2}^{2}+2\right)\right)}{T}\right)+{\tt
n}_{2}\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{2}^{2}+2\right)\right)}{T}\right)}{\sqrt{{\tt n}_{2}^{2}+4}}\,,$
$\displaystyle C_{34}(\eta)$
$\displaystyle=-\frac{2\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{2}^{2}+2\right)\right)}{T}\right)}{\sqrt{{\tt
n}_{2}^{2}+4}\cosh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{2}^{2}+2\right)\right)}{T}\right)+{\tt
n}_{2}\sinh\left(\frac{\eta\cosh^{-1}\left(\frac{1}{2}\left({\tt
n}_{2}^{2}+2\right)\right)}{T}\right)}\,,$

which satisfy the equations explicitly presented in appendix C, after the
appropriate coordinate transformations to bring the target metric (3.38) in
the form (C.1). It is straightforward to verify that, as
$\eta\rightarrow\eta+T$ the complexified fields $\rho_{12}\equiv
C_{12}+i\,e^{-\phi_{12}},\,\rho_{34}\equiv C_{34}+i\,e^{-\phi_{34}}$ transform
under the action of the monodromies $J_{{\tt n}_{1}}$, $J_{{\tt n}_{2}}$,
respectively:

$\rho_{12}\rightarrow-\frac{1}{\rho_{12}+{\tt
n}_{1}}\,\,,\,\,\,\rho_{34}\rightarrow-\frac{1}{\rho_{34}+{\tt n}_{2}}\,.$

In this case, we have:

$\displaystyle\kappa^{2}=\frac{1}{2T^{2}}\left(\cosh^{-1}\left(\frac{{\tt
n}_{1}^{2}}{2}+1\right)+\cosh^{-1}\left(\frac{{\tt
n}_{2}^{2}}{2}+1\right)\right)\,.$ (3.39)

Along the same lines, we can construct the ${\rm AdS}_{2}\times S^{1}\times
S^{3}$ T-fold starting from the near horizon geometry of an F1-NS5 system with
charges

$\Gamma^{M}=(0,n_{5},0,0,0,0,0,0,0,f_{1})\,.$

This solution is obtained by applying an S-duality $\mathcal{S}\in{\rm
SL}(2,\mathbb{Z})_{{\rm IIB}}$ to the D1-D5 U-fold. As pointed out earlier,
the maximal choice ${\rm SO}(4,4)/{\rm SO}(4)\times{\rm SO}(4)$ for the moduli
space $M_{0}$ is spanned by $B_{ij},\,G^{(s)}_{ij}\equiv
e^{\frac{\phi}{2}}\,G_{ij}$, where $G^{(s)}_{ij}$ is the torus metric in the
string frame, while

$\hat{\mathcal{M}}={\rm diag}(1,e^{g},1,1,1,1,1,1,1,e^{-g})\,,$ (3.40)

with $e^{g}=e^{-2\phi_{6}}\equiv e^{-\phi}\,G^{\frac{1}{2}}$, see Appendix A.
Since now the monodromy matrix is an element of ${\rm O}(4,4;\mathbb{Z})$,
which is now the T-duality group associated with the internal 4-torus, the
resulting U-fold is a T-fold.

We can choose a monodromy matrix $\mathfrak{M}\in{\rm O}(4,4;\mathbb{Z})/{\rm
SO}(4,4;\mathbb{Z})$, which is not connected continuously to the identity.
This can be achieved by describing $S^{1}$ through two open patches: one
consisting of $S^{1}$ minus the point $\eta=0$, the other obtained by
subtracting to $S^{1}$ the point $\eta=T/2$. Within each patch the evolution
of the scalar fields is geodesic. However, in the interval $(0,T/2)$ the
moduli fields are described in both patches by131313Recall that
$\mathbb{L}_{0}$ is in the identity sector of the isometry group.
$\mathbb{L}_{0}(\varphi^{a}(\eta))$ while in $(T/2,T)$ they are described in
the first patch by $\mathbb{L}_{0}(\varphi^{a}(\eta))$ and in the second one
by $\mathscr{O}\cdot\mathbb{L}_{0}(\varphi^{a}(\eta))$, where
$\mathscr{O}\in{\rm O}(4,4;\mathbb{Z})/{\rm SO}(4,4;\mathbb{Z})$,
$\mathscr{O}^{2}={\bf 1}$. The twist matrix has the following property:
$\mathcal{A}(0)=\mathbb{L}_{0}(\varphi^{a}(0))$,
$\mathcal{A}(T)=\mathscr{O}\cdot\mathbb{L}_{0}(\varphi^{a}(T))$, so that
$\mathfrak{M}$, given by eq. (2.48), is in ${\rm O}(4,4;\mathbb{Z})/{\rm
SO}(4,4;\mathbb{Z})$. This T-fold solution with NS-NS charges is therefore
described in one patch within the Type IIB theory and in the other one within
the Type IIA one, the transition function being a pseudo-orthogonal matrix
with a negative determinant.

In the next subsection, we review the action of a duality transformation on
the general, duality covariant, ansatz discussed in section 2.

####  3.3.1 Duality

Let us discuss here the action of the duality symmetry on a generic solution.
Consider a duality transformation

$\mathcal{O}=(\mathcal{O}^{M}{}_{N})\in{\rm
SO}(5,5)\,\,\Leftrightarrow\,\,\mathcal{O}\,\Omega\,\mathcal{O}^{T}=\Omega\,.$
(3.41)

Its action on the scalar fields is described by the transformation property of
the matrix $\mathcal{M}(\phi)$:

$\phi\equiv(\phi^{s})\,\stackrel{{\scriptstyle\mathcal{O}}}{{\longrightarrow}}\,\,\phi^{\prime}\equiv(\phi^{\prime
s})\,\,\Leftrightarrow\,\,\,\,\mathcal{O}^{-T}\mathcal{M}(\phi)\mathcal{O}^{-1}=\mathcal{M}(\phi^{\prime})\,,$
(3.42)

while the corresponding action on $\mathbb{H}\equiv(\mathbb{H}^{M})$ reads:

$\displaystyle\mathbb{H}^{\prime}=\mathcal{O}\,\mathbb{H}=-\Omega\mathcal{O}^{-T}\mathcal{M}(\phi)\mathcal{O}^{-1}\mathcal{O}\Gamma\boldsymbol{\epsilon}_{(3)}+\mathcal{O}\Gamma\tilde{\boldsymbol{\epsilon}}_{(3)}=-\Omega\mathcal{M}(\phi^{\prime})\Gamma^{\prime}\boldsymbol{\epsilon}_{(3)}+\Theta\Gamma^{\prime}\tilde{\boldsymbol{\epsilon}}_{(3)}\,.$
(3.43)

A particular instance of duality symmetry, which we shall use in the sequel,
is $S$-duality, implemented, in the basis we are using, by the following
matrix

$\displaystyle\Theta$
$\displaystyle=\left(\begin{matrix}{\theta^{\alpha}_{\,\,\beta}}&{\bf 0}&{\bf
0}\cr{\bf 0}&2\delta_{ij}^{\,\,kl}&{\bf 0}\cr{\bf 0}&{\bf
0}&{\tilde{\theta}_{\alpha}^{\,\,\beta}}\end{matrix}\right),$ (3.44)

where

$\displaystyle\theta^{\alpha}_{\,\,\beta}=\left(\begin{matrix}0&1\cr-1&0\end{matrix}\right),\quad\tilde{\theta}_{\alpha}^{\,\,\beta}=\left(\begin{matrix}0&1\cr-1&0\end{matrix}\right),$
(3.45)

such that
$\tilde{\theta}_{\alpha}^{\,\,\beta}=(\theta^{-1})^{\beta}_{\,\,\alpha}$.
Besides being pseudo-orthogonal, $\Theta$ is orthogonal as well:
$\Theta\,\Theta^{T}={\bf 1}.$ From eq. (3.42) we deduce the following
transformation rule for the scalar fields:

$\displaystyle B^{\prime}_{\alpha}$
$\displaystyle=\tilde{\theta}_{\alpha}^{\,\,\beta}B_{\beta}=\left(\begin{matrix}0&1\cr-1&0\end{matrix}\right)\left(\begin{matrix}B_{1}\cr
B_{2}\end{matrix}\right)=\left(\begin{matrix}B_{2}\cr-
B_{1}\end{matrix}\right),$ (3.46) $\displaystyle
c^{\prime}\epsilon_{\alpha\sigma}$
$\displaystyle=c(\theta^{-1})^{\beta}_{\,\,\alpha}\epsilon_{\beta\gamma}\tilde{\theta}_{\sigma}^{\,\,\gamma}=c\epsilon_{\alpha\sigma},$
(3.47) $\displaystyle\rho^{\prime}$ $\displaystyle=-\frac{1}{\rho},$ (3.48)

all other fields being invariant.

##  4 Final remarks and discussion of the solution

In this work, we discuss the general construction of a U-fold solution with
${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ geometry and a monodromy along
$S^{1}$ in the duality group, within theories admitting a ${\rm AdS}_{d}\times
S^{d}$ background with moduli fields. More in general we build U-fold
backgrounds of the form

$\displaystyle{\rm AdS}_{d-k}\times\underbrace{S^{1}\times\dots\times
S^{1}}_{k}\times S^{d}\,,$ (4.1)

with $d-k>1$.

This was achieved by introducing an appropriate spatial dependence of the
scalar moduli along a compact direction which, in the original ${\rm
AdS}_{d}\times S^{d}$ solution, is in the boundary of ${\rm AdS}_{d}$. The
effect of the back-reaction of the evolving moduli on the geometry is to split
${\rm AdS}_{d}$ into ${\rm AdS}_{d-1}\times S^{1}$. The geodesic arc connects
two distinct points in the classical moduli space. Consequently, the
consistency of this background in superstring theory requires the initial and
ending points to be identified in the string moduli space.

Since the dependence of the fields, in the U-fold solution, on the
$S^{1}$-coordinate $\eta$ is factorized in the action of a twist matrix
$\mathcal{A}(\eta)\in G_{0}$ on the fields, it is natural to expect this
background to be a solution of the $D=2d$-dimensional model compactified along
$S^{1}$ á la Cremmer-Scherk-Schwarz [39] to $(2d-1)$ dimensions, with a twist
matrix $\mathcal{A}(\eta)$ in the global symmetry group $G_{0}$. Indeed the
only fields in the solution transforming non-tivially under $G_{0}$ are the
moduli fields $\varphi^{a}$. We extensively discussed, as a concrete
application of our general construction, Type IIB superstring theory
compactified on $T^{4}$ to six dimensions, and utilized the moduli fields
present in the ${\rm AdS}_{3}\times{S}^{3}\times T^{4}$ solution to create a
U-fold with geometry ${\rm AdS}_{2}\times S^{1}\times{S}^{3}\times T^{4}$.
This construction involves introducing a non-trivial geodesic dependence of
the moduli fields in ${\rm SO}(4,4)/{\rm SO}(4)\times{\rm SO}(4)$, or in a
symmetric submanifold $M_{0}$ thereof, on a compact direction $S^{1}$ along
the boundary of ${\rm AdS}_{3}$.

Cremmer-Scherk-Schwarz reduction of maximal $\mathcal{N}=(2,2)$ six-
dimensional supergravity to five dimensions admit Minkowski five-dimensional
vacua only if the twist matrix is compact, that is in ${\rm Spin}(5)\times{\rm
Spin}(5)$. In this case the reduction was originally studied in [40], [41]
and, more recently, in [37, 42] in relation to the study of black hole
solutions. Our setting, however, is substantially different since it relies on
the moduli fields describing a geodesic in $M_{0}=G_{0}/H_{0}$, which requires
the twist matrix $\mathcal{A}(\eta)\in G_{0}/H_{0}$ to be intrinsically non-
compact. This is necessary for the evolving moduli to back-react on the
geometry through a non-vanishing component of the energy-momentum tensor:
$T^{(S)}_{\eta\eta}\neq 0$.

The general mechanism analyzed here in detail, underlying the construction of
${\rm AdS}_{d-1}\times S^{1}\times S^{d}$ U-folds, seems to be at work in the
more sophisticated class of J-fold solutions with geometry ${\rm
AdS}_{4}\times S^{1}\times\tilde{S}^{5}$, which have recently attracted
considerable interest [12, 13, 11, 14, 15, 16, 17, 18, 43, 19, 20, 21, 22].
All these solutions correspond to ${\rm AdS}_{4}$ vacua of maximal four-
dimensional supergravity with gauge group $\mathcal{G}=[{\rm SO}(6)\times{\rm
SO}(1,1)]\ltimes T^{12}$. Aside from the non-supersymmetric J-fold with ${\rm
SO}(6)$ symmetry mentioned earlier and found in [11], which fits the general
class discussed in section 2, in all other solutions the the axion-dilaton
system features a dependence on the coordinates $\upxi^{i}$ of the deformed
5-sphere $\tilde{S}^{5}$ and on $\eta$. The former is sourced by 3-form field
strengths while the latter describes a geodesic arc in the moduli space, as
noticed in [17]. In the present work, we have highlighted this common feature
of all these solutions and generalized it to the case $d\neq 5$, showing that
the geodesic dependence of the moduli fields on the $\eta$ coordinate of an
$S^{1}$ at the boundary of ${\rm AdS}_{d}$ is sufficient to split this space
into ${\rm AdS}_{d-1}\times S^{1}$, where the ratio of the warp factors
multiplying the metric ${\rm AdS}_{d-1}$ and $d\eta^{2}$ is a suitable
constant.

An important issue to address is the supersymmetry and stability of the ${\rm
AdS}_{2}\times S^{1}\times S^{3}$ solution discussed in this work. Secondly,
it would be of considerable interest to construct variants of these solutions,
with geometry ${\rm AdS}_{2}\times S^{1}\times\tilde{S}^{3}$, where
$\tilde{S}^{3}$ is a deformed 3-sphere, in which the moduli fields
$\varphi^{a}$, aside from the geodesic dependence on $\eta$, also feature a
dependence on the coordinates of $\tilde{S}^{3}$ sourced by 2-form field-
strengths. These new solutions would be the counterpart in $D=6$ of the
$\mathcal{N}=1,2$ and $4$ J-folds in $D=10$. In analogy with the latter
solutions, we may argue that these ${\rm AdS}_{2}\times
S^{1}\times\tilde{S}^{3}$ U-folds should be dual to the IR limit of a $1+1$
SCFT compactified along one spatial direction, with an interface localized
along this direction and an ${\rm O}(4,4;\,\mathbb{Z})$ monodromy acting on
the exactly marginal operators dual to $\varphi^{a}$, when crossing the
interface. Although we do not expect the solutions ${\rm AdS}_{2}\times
S^{1}\times S^{3}$ to be supersymmetric, we expect their variants ${\rm
AdS}_{2}\times S^{1}\times\tilde{S}^{3}$ to preserve some supersymmetry, in
analogy with the $D=10$ J-fold counterparts. It would also be interesting to
relate generalizations of our backgrounds to a compactified version of the
Janus solutions found in [44] which were also put in correspondence with an
interface CFT.

Similar to the Janus solutions, the approach outlined in this paper is
relevant for the supersymmetric wormhole solutions recently found in [45]. The
existence of moduli, or scalar fields becoming moduli near the wormhole’s
neck, might be a crucial element in constructing such wormhole solutions.

We leave these projects to a future investigation.

Acknowledgements

MT is grateful to Emanuel Malek, Tomas Ortin, Henning Samtleben and Thomas Van
Riet for fruitful discussions. DA is grateful to Friðrik Freyr Gautason for
useful discussions. DA is supported by the Icelandic Research Fund under grant
228952-052.

##  Appendix A Type IIB Supergravity on $T^{4}$: Geometric Characterization of
the Scalar and Tensor Fields.

In the solvable parametrization of the scalar manifold, the ten-dimensional
dilaton $\phi$ and the radial moduli $\sigma_{i}$, $i=1,\dots,4$, of $T^{4}$,
parametrize the Cartan subalgebra of $\mathfrak{so}(5,5)$ while the remaining
scalar fields are parameters of the shift generators $E_{\boldsymbol{\alpha}}$
corresponding to the positive roots $\boldsymbol{\alpha}$. The latter are
conveniently expressed in terms of the simple roots which, in a suitable
orthonormal basis $\boldsymbol{\epsilon}_{i},\,\boldsymbol{\epsilon}_{5}$, are
chosen to have the following form:

$\displaystyle\boldsymbol{\alpha}_{1}$
$\displaystyle=\boldsymbol{\epsilon}_{1}-\boldsymbol{\epsilon}_{2}\,,\,\,\boldsymbol{\alpha}_{2}=\boldsymbol{\epsilon}_{2}-\boldsymbol{\epsilon}_{3}\,,\,\,\boldsymbol{\alpha}_{3}=\boldsymbol{\epsilon}_{3}-\boldsymbol{\epsilon}_{4}\,,\,\,\boldsymbol{\alpha}_{4}=\boldsymbol{\epsilon}_{3}+\boldsymbol{\epsilon}_{4}\,,\,\,\boldsymbol{\alpha}_{5}={\bf
a}=-\frac{1}{2}\left(\sum_{i=1}^{4}\boldsymbol{\epsilon}_{i}\right)+\boldsymbol{\epsilon}_{5}\,.$
(A.1)

The generators $t_{\upalpha}{}^{ij},\,t$ of the solvable Lie algebra
generating the coset, which enter the definition (3.11) of the coset
representative correspond to the following shift-generators
$E_{\boldsymbol{\alpha}}$:

$\displaystyle t_{\upalpha=1}{}^{ij}=E_{{\bf
a}+\boldsymbol{\epsilon}_{i}+\boldsymbol{\epsilon}_{j}}\,,\,\,\,t_{\upalpha=2}{}^{ij}=E_{\boldsymbol{\epsilon}_{i}+\boldsymbol{\epsilon}_{j}}\,,\,\,\,t=E_{{\bf
a}+\boldsymbol{\epsilon}_{1}+\boldsymbol{\epsilon}_{2}+\boldsymbol{\epsilon}_{3}+\boldsymbol{\epsilon}_{4}}\,.$
(A.2)

The six positive roots $\boldsymbol{\epsilon}_{i}-\boldsymbol{\epsilon}_{j}$,
$i<j$, are the positive roots of the ${\rm GL}(4,\mathbb{R})$ group of the
4-torus and enter the definition of the vielbein matrix ${\bf
E}\equiv(E_{i}{}^{k})$. The latter can be written as

${\bf E}=\hat{\bf E}\cdot{\bf R}\,,$

where

$\hat{\bf
E}=\prod_{i<j}e^{\gamma_{i}{}^{j}\,E_{\boldsymbol{\epsilon}_{i}-\boldsymbol{\epsilon}_{j}}}\,\,,\,\,\,{\bf
R}\equiv(\delta^{i}_{j}e^{\sigma_{j}})\,,$ (A.3)

and the generators $E_{\boldsymbol{\epsilon}_{i}-\boldsymbol{\epsilon}_{j}}$
are meant, in the above formula, as $4\times 4$ nilpotent matrices in the
${\bf 4}$ of ${\rm GL}(4,\mathbb{R})$. The dependence of the coset
representative on the dilatonic scalars $\phi,\,\sigma_{i}$ is through the
matrix

$\mathbb{L}_{{\rm
dil.}}\equiv\exp\left(\sum_{i=1}^{4}\sigma_{i}\,\left(H_{\boldsymbol{\epsilon}_{i}}+\frac{1}{2}\,H_{\boldsymbol{\epsilon}_{5}}\right)-\frac{\phi}{2}\,H_{{\bf
a}}\right)\,.$ (A.4)

The dependence of the kinetic term of the non-dilatonic scalar
($\gamma_{i}{}^{j},\,B^{\upalpha}{}_{ij},\,c_{4}$) on the dilatonic ones, in
the Lagrangian, is through the characteristic exponential $e^{-2\,{\bf
h}\cdot\boldsymbol{\alpha}}$, where

${\bf
h}\equiv\left(\sum_{i=1}^{4}\sigma_{i}\,\left({\boldsymbol{\epsilon}_{i}}+\frac{1}{2}\,{\boldsymbol{\epsilon}_{5}}\right)-\frac{\phi}{2}\,{{\bf
a}}\right)\,,$

and $\boldsymbol{\alpha}$ is the positive root associated with the scalar.

Similarly to the non-dilatonic scalars we can associate each of the ten 3-form
field-strength $\mathbb{H}^{M}$ with weights ${\bf w}^{M}$ of the ${\bf 10}$
of $\mathfrak{so}(5,5)$ which, the chosen conventions, read:

${\bf w}^{M}\,=\,({\bf w}^{\upalpha},{\bf w}_{ij},\,{\bf w}_{\upalpha})\,,$
(A.5)

where:

$\displaystyle{\bf w}^{\upalpha}$ $\displaystyle=\left({\bf
w}_{0},\,-\boldsymbol{\epsilon}_{5}\right)\,,\,\,{\bf w}_{ij}={\bf
w}_{0}+\boldsymbol{\epsilon}_{i}+\boldsymbol{\epsilon}_{j}\,,\,\,\,{\bf
w}_{\upalpha}=-{\bf w}^{\upalpha}\,,\,\,\,\,\,\,{\bf
w}_{0}\equiv-\frac{1}{2}\sum_{i=1}^{4}\boldsymbol{\epsilon}_{i}\,.$ (A.6)

The dependence of the kinetic term of $\mathbb{H}^{M}$ on the dilatonic
scalars in the Lagrangian is through the characteristic exponential:
$e^{-2\,{\bf h}\cdot{\bf w}}$.

The $d_{5},\,d_{1}$ charges are associated with the weights $\pm{\bf w}_{0}$.
The Cartan generator $\mathfrak{h}$ in (3.17) is defined by the component of
${\bf h}$ along ${\bf W}_{0}$ and is spanned by the scalar $g$:

$e^{g}=e^{-{\bf h}\cdot{\bf w}_{0}}=g_{s}\,G^{\frac{1}{2}}\,.$

The roots of ${\rm SO}(5,5)$ which are orthogonal to ${\bf w}_{0}$ define an
${\rm SO}(4,4)$ algebra representing the maximal choice of $G_{0}$, subgroup
of $G$ which commutes with $\mathfrak{h}$. The positive roots of ${\rm
SO}(4,4)$ and the corresponding non-dilatonic scalars are:

$\displaystyle{\bf
a}+\boldsymbol{\epsilon}_{i}+\boldsymbol{\epsilon}_{j}\,\,\leftrightarrow\,\,\,\,C_{ij}\,,$
$\displaystyle\boldsymbol{\epsilon}_{i}-\boldsymbol{\epsilon}_{j}\,\,,\,(i<j)\,\,\,\leftrightarrow\,\,\,\,\gamma_{i}{}^{j}\,,$

As for the dilatonic scalar fields, these are the components of ${\bf h}$
along the orthonormal basis ${\bf v}_{i}\equiv\frac{{\bf
a}}{2}+\boldsymbol{\epsilon}_{i}$, orthogonal to ${\bf w}_{0}$:

$\tilde{\sigma}_{i}={\bf h}\cdot{\bf v}_{i}=\sigma_{i}-\frac{\phi}{4}\,.$
(A.7)

The scalars $\gamma_{i}{}^{j},\,\tilde{\sigma}_{i}$ define the components of
the matrix $\tilde{G}_{ij}\equiv e^{-\frac{\phi}{2}}\,G_{ij}$.

The NS-NS charges $n_{5},\,f_{1}$, associated with the NS 5-brane and the
fundamental string, respectively, correspond to the weights
$\mp\boldsymbol{\epsilon}_{5}$. The scalar $g$ parametrizing $\mathfrak{h}$
and entering eq. (3.40) is now given by

$e^{g}=e^{{\bf
h}\cdot\boldsymbol{\epsilon}_{5}}=g_{s}^{-1}\,G^{\frac{1}{2}}=e^{-2\phi_{6}}\,,$

where the six-dimensional dilaton is defined as follows: $e^{\phi_{6}}\equiv
e^{\phi}\,{\rm det}\left(G^{(s)}_{ij}\right)^{-\frac{1}{4}}$ and
$G^{(s)}_{ij}\equiv e^{\frac{\phi}{2}}\,G_{ij}$ is the torus metric in the
string frame. The dilatonic scalar fields in the ${\rm SO}(4,4)/{\rm
SO}(4)\times{\rm SO}(4)$ coset, are given by the components of ${\bf h}$ along
the orthonormal basis

${\bf v}_{i}\equiv\boldsymbol{\epsilon}_{i}\,,\,\,\,i=1,\dots,4\,.$

They are:

$\sigma_{i}^{(s)}\equiv{\bf h}\cdot{\bf v}_{i}=\sigma_{i}+\frac{\phi}{4}\,,$

and are the radial moduli in the string frame. On the other hand, the positive
roots which are orthogonal to $\boldsymbol{\epsilon}_{5}$ are

$\displaystyle\boldsymbol{\epsilon}_{i}+\boldsymbol{\epsilon}_{j}\,\,\leftrightarrow\,\,\,\,B_{ij}\,,$
$\displaystyle\boldsymbol{\epsilon}_{i}-\boldsymbol{\epsilon}_{j}\,\,,\,(i<j)\,\,\,\leftrightarrow\,\,\,\,\gamma_{i}{}^{j}\,.$

This proves that the manifold ${\rm SO}(4,4)/{\rm SO}(4)\times{\rm SO}(4)$ is
spanned by the moduli $G^{(s)}_{ij},\,B_{ij}$.

##  Appendix B The Type IIB Static Black String Solutions

Let us study the most general ansatz for a static black string solution with
${\rm SO}(4)$ symmetry. The corresponding metric has the following general
form

$\displaystyle
ds^{2}=A(r)^{2}(-dt^{2}+dx^{2})+A(r)^{-2}dr^{2}+B(r)^{2}\left[d\psi^{2}+\sin^{2}\psi\left(d\theta^{2}+\sin^{2}\theta
d\omega^{2}\right)\right],$ (B.1)

where $r,\psi,\theta,\omega$ are the polar coordinates of the four-dimensional
space transverse to the string world volume and the last three angles
parameterize a 3-sphere $S^{3}$. The coordinates $x^{\hat{\mu}}$,
$\hat{\mu}=0,\dots,6$ naturally split into

$x^{\mu}=\\{t,x,r\\}\,\,,\,\,\,x^{\tilde{\mu}}=\\{\psi,\theta,\omega\\}\,,$

where $\mu=0,1,2$ and $\tilde{\mu}=3,4,5$.

The Ricci tensor is

$\displaystyle R_{tt}=$ $\displaystyle-
R_{xx}=A^{2}\left(AA^{\prime\prime}+3\frac{AA^{\prime}B^{\prime}}{B}+2A^{\prime
2}\right),$ $\displaystyle R_{rr}=$
$\displaystyle\frac{1}{A^{2}B}\left[-3A\left(A^{\prime}B^{\prime}+AB^{\prime\prime}\right)-2B\left(AA^{\prime\prime}+A^{\prime
2}\right)\right],$ $\displaystyle R_{\psi\psi}=$
$\displaystyle\frac{R_{\theta\theta}}{\sin^{2}\psi}=\frac{R_{\omega\omega}}{\sin^{2}\psi\sin^{2}\theta}=-3ABA^{\prime}B^{\prime}-A^{2}\left[(BB^{\prime\prime}+2B^{\prime
2})\right].$

We emphasize that this ansatz described the most general static black-string
solution coupled to any number of scalar fields. The constant charge vector
$\Gamma\equiv(\Gamma^{M})$ is defined as follows:

$\Gamma^{M}=\frac{1}{2\pi^{2}}\,\int_{S^{3}}\,\mathbb{H}^{M}\,,$

and characterizes the solution. Let us consider the already introduced (2.17)
ansatz for the tensor field

$\displaystyle\mathbb{H}=-\xi(r)\Omega\mathcal{M}\Gamma dt\wedge dx\wedge
dr+\chi(\theta,\psi)\Gamma d\psi\wedge d\theta\wedge d\omega,$ (B.2)

and this solves the twisted self-duality condition, the Maxwell equations, and
the Bianchi equations if

$\displaystyle\xi(r)=\frac{A}{B^{3}},\qquad\chi(\theta,\psi)=\sin^{2}\psi\sin\theta.$
(B.3)

With this choice, the energy-momentum tensor reads

$\displaystyle T_{tt}=-T_{xx}=-A^{4}T_{rr}=\frac{A^{2}}{2B^{6}}V,$ (B.4)
$\displaystyle T_{\psi\psi}=\frac{V}{2B^{4}}.$ (B.5)

On the inverse trace of Einstein’s equation

$\displaystyle
R_{\hat{\mu}\hat{\nu}}=T^{(H)}_{\hat{\mu}\hat{\nu}}+\frac{1}{2}\mathcal{G}_{ts}\partial_{\hat{\mu}}\phi^{t}\partial_{\hat{\nu}}\phi^{s},$
(B.6)

we impose the consistency requirement $\phi=\phi(r)$ and the equations
explicitly read

$\displaystyle R^{t}_{t}=R^{x}_{x}=-\frac{V}{2B^{6}},$ (B.7) $\displaystyle
R^{r}_{r}=-\frac{V}{2B^{6}}+\frac{1}{2}\mathcal{G}_{ts}\partial_{r}\phi^{t}\partial^{r}\phi^{s},$
(B.8) $\displaystyle
R^{\psi}_{\psi}=R^{\theta}_{\theta}=R^{\omega}_{\omega}=\frac{V}{2B^{6}}.$
(B.9)

These equations force some consistency relations such that

$\displaystyle R^{t}_{t}=R^{x}_{x},\qquad
R^{\psi}_{\psi}=R^{\theta}_{\theta}=R^{\omega}_{\omega},$ (B.10)

which are already verified, but there is another condition that we still have
to impose, i.e.

$\displaystyle
R^{t}_{t}+R^{\psi}_{\psi}=0\rightarrow\frac{1}{2}\left(A^{2}B^{2}\right)^{\prime\prime}+(AB)^{\prime
2}-2=0.$ (B.11)

Posing $AB=\pm\sqrt{u}$ this condition becomes

$\displaystyle\frac{d^{2}u}{dr^{2}}+\frac{1}{2u}\left(\frac{du}{dr}\right)^{2}-4=0.$
(B.12)

To solve this we define

$\displaystyle\Omega(u)=\frac{du}{dr},\qquad\dot{\Omega}=\frac{d}{du}\Omega,$
(B.13)

and we get the Bernoulli equation

$\displaystyle\Omega\dot{\Omega}+\frac{\Omega^{2}}{2u}-4=0,$ (B.14)

with solutions

$\displaystyle\Omega=\pm\sqrt{\frac{a+4u^{2}}{u}}.$ (B.15)

We choose the "+" sign and, from eq. $(\ref{COND})$, we have

$\displaystyle
r-c=\int\frac{du}{\sqrt{\frac{a+4u^{2}}{u}}}=\frac{2u\sqrt{\frac{4u^{2}}{a}+1}}{3\sqrt{\frac{a}{u}+4u}}{}_{2}\text{F}_{1}\left(\frac{1}{2},\frac{3}{4},\frac{7}{4},\frac{-4u^{2}}{a}\right),\qquad
u=A(r)^{2}B(r)^{2}.$ (B.16)

The extremal solutions are recovered with $a\rightarrow 0$. In fact, this
gives

$\displaystyle(r-c)^{2}=A(r)^{2}B(r)^{2},$ (B.17)

from which we can recognize the structure of the double coincident horizon at
$r=c$.
At radial infinity, $r$ and $u$ go to $+\infty$, while, near the horizon
$r\sim c$ and $u\sim 0$. In these parameterizations, however, the inner and
outer horizons do not appear. Now with a little abuse of notation we regard
$A$ and $B$ as functions of $u$ and we trade $B$ with $u$ in the metric:

$\displaystyle
ds^{2}=A(u)^{2}\left(-dt^{2}+dx^{2}\right)+u\,A(u)^{-2}\left(\frac{du^{2}}{a+4u^{2}}+d\Omega^{2}_{S^{3}}\right).$
(B.18)

The affine parameter $\tau$ can be defined by the differential equation

$\displaystyle\frac{d\tau}{du}=\frac{1}{u\sqrt{a+4u^{2}}},$ (B.19)

which can be solved as follows:

$u(\tau)=-\frac{\sqrt{a}}{2\sinh(\sqrt{a}\tau)}\,.$ (B.20)

Notice that we have chosen the sign of $u(\tau)$ so that, since $u\geq 0$,
$\tau$ be non-positive. At radial infinity ($u\rightarrow\infty$)
$\tau\rightarrow 0^{-}$and near the horizon ($u\rightarrow 0$)
$\tau\rightarrow-\infty$.
In terms of the affine parameter and defining $A(u)=e^{U(\tau)}$ the non-
extremal metric is

$\displaystyle
ds^{2}=e^{2U}(-dt^{2}+dx^{2})+\frac{1}{2}\frac{\sqrt{a}}{\sinh(\sqrt{a}\tau)}e^{-2U(\tau)}\left(\frac{1}{4}\frac{a}{\sinh^{2}(\sqrt{a}\tau)}d\tau^{2}+d\Omega^{2}_{S^{3}}\right).$
(B.21)

Regularity requires $a>0$. Einstein’s equations and the scalar fields
equations are

$\displaystyle\ddot{U}=$ $\displaystyle e^{4U}\frac{V}{2},$ (B.22)
$\displaystyle\ddot{\phi}^{s}+\tilde{\Gamma}^{s}_{tu}\dot{\phi}^{t}\dot{\phi}^{u}=$
$\displaystyle e^{4U}\mathcal{G}^{st}\partial_{s}V,$ (B.23)
$\displaystyle\frac{3a}{8}=$
$\displaystyle\dot{U}^{2}-e^{4U}\frac{V}{4}+\frac{1}{8}\mathcal{G}_{ts}\dot{\phi}^{t}\dot{\phi}^{s},$
(B.24)

where the dots are the derivatives with respect to the affine parameter
$\tau$. The first two equations can be deduced from the effective action

$\displaystyle S_{\text{eff}}=\int
d\tau\,\left(\dot{U}^{2}+\frac{1}{8}G_{su}\dot{\phi}^{s}\dot{\phi}^{u}+e^{4U}\frac{V}{4}\right),$
(B.25)

while the third one can be interpreted as a Hamiltonian constraint, with
energy $\frac{3a}{8}$.
The near horizon limit can be obtained by taking $\tau\rightarrow-\infty$ and
the metric reads

$\displaystyle
ds^{2}=e^{2U}(-dt^{2}+dx^{2})+\sqrt{a}e^{\sqrt{a}\tau}e^{-2U(\tau)}\left(ae^{-2\sqrt{a}\tau}d\tau^{2}+d\Omega^{2}_{S^{3}}\right).$
(B.26)

To have a finite horizon area, we must require the following behavior

$\displaystyle
e^{-2U}\sim\left(\frac{A_{H}}{2\pi^{2}}\right)^{2/3}\frac{1}{\sqrt{a}e^{\sqrt{a}\tau}},\text{as}\quad\tau\rightarrow\infty.$
(B.27)

The two horizons can be described by changing the radial variable into a new
one $\rho$ defined through the relation:

$\sinh^{2}(\sqrt{a}\tau)=\frac{a}{(\rho-\rho_{0})^{2}-a}=\frac{a}{4u^{2}}\,,$
(B.28)

which can be solved in the two variables as follows:

$\rho=\rho_{0}-\sqrt{a}\,\coth(\sqrt{a}\tau)\,\,,\,\,\,\tau=\frac{1}{2\sqrt{a}}\,\log\left(\frac{\rho-\rho_{+}}{\rho-\rho_{-}}\right)\,,$
(B.29)

where we have denoted by $\rho_{\pm}$ the radial location of the two horizons
defined as:

$\rho_{\pm}\equiv\rho_{0}\pm\sqrt{a}\,.$ (B.30)

Therefore, the metric near the horizon reads

$\displaystyle
ds^{2}=\left(\frac{A_{H}}{2\pi^{2}}\right)^{-2/3}\sqrt{a}e^{\sqrt{a}\tau}(-dt^{2}+dx^{2})+\left(\frac{A_{H}}{2\pi^{2}}\right)^{2/3}\left(ae^{2\sqrt{a}\tau}d\tau^{2}+d\Omega^{2}_{\text{S}^{3}}\right).$
(B.31)

The existence of a timelike killing vector $\xi=\partial_{t}$, guarantees the
existence of the Komar mass for this class of solutions, which is given by

$\displaystyle M=\frac{c^{2}}{8\pi
G}\int_{S^{3}_{\infty}}\sqrt{g}|_{\theta,\phi,\omega}\epsilon_{\theta\phi\omega\mu\nu}\nabla^{\mu}\xi^{\nu}=\frac{\pi
c^{2}}{4G}\lim_{\tau\rightarrow 0^{-}}\dot{U}.$ (B.32)

This allows us to fix the following conditions at infinity

$\displaystyle U(0)=0,\quad\dot{U}(0)=M\frac{4G}{\pi c^{2}}.$ (B.33)

###  B.1 Attractor mechanism for extremal solutions

If $a=0$ we have

$\displaystyle\tau=$
$\displaystyle-\frac{1}{2u}\,\,,\,\,\,\,\rho-\rho_{0}=-\frac{1}{\tau}=2\,r^{2}\,,$
(B.34)

and the two horizons coincide $\rho_{\pm}=\rho_{0}$. The metric reads

$\displaystyle
ds^{2}=e^{2U}\left(-dt^{2}+dx^{2}\right)+\frac{1}{(-2\tau)}\,e^{-2U}\left(\frac{d\tau^{2}}{4\tau^{2}}+d\Omega^{2}_{S^{3}}\right).$
(B.35)

The horizon is at again at $u\rightarrow 0$ and $\tau\rightarrow-\infty$ and
the near horizon behaviour of the function $e^{-U}$ is now given by

$\displaystyle
e^{-2U}=\left(\frac{A_{H}}{2\pi^{2}}\right)^{2/3}(-2\tau),\text{as}\quad\tau\rightarrow\infty.$
(B.36)

Since the area of $S_{3}$ is $A_{H}=2\pi^{2}L^{3}$, we can also replace
$A_{H}$ with the radius of the three-sphere $L$. The coordinate which defines
the proper spatial distance from the horizon is

$\displaystyle
d\zeta=\frac{e^{-U}}{\sqrt{(-8\tau^{3})}}\,d\tau=-\frac{L}{2}d\log(-\tau).$
(B.37)

Thanks to the regularity condition in eq.(B.36) and fixing a not-useful
constant we find

$\displaystyle\zeta=-\frac{L}{2}\log(-\tau)+{\rm const.},$ (B.38)

while the position of the horizon is now at $\zeta\rightarrow-\infty$. Again,
we require the scalar fields to be regular at the horizon

$\displaystyle\lim_{\zeta\rightarrow-\infty}\phi^{s}(\zeta)=\phi^{s}_{*}<\infty.$
(B.39)

If the functions $\phi^{s}$ are uniformly continuous this request necessarily
implies that the derivatives

$\displaystyle\lim_{\zeta\rightarrow-\infty}\frac{d^{k}}{d\zeta^{k}}\phi^{s}=0.$
(B.40)

This implies

$\displaystyle\lim_{\tau\rightarrow-\infty}\tau\frac{d}{d\tau}\phi^{s}=\lim_{\tau\rightarrow-\infty}\tau^{2}\frac{d^{2}}{d\tau^{2}}\phi^{s}=...=0.$
(B.41)

The scalar field equation (B.23) can be rewritten as

$\displaystyle\tau^{2}\partial^{2}_{\tau}\phi^{s}+\tilde{\Gamma}^{s}_{uv}\left(\tau\partial_{\tau}\phi^{u}\right)\left(\tau\partial_{\tau}\phi^{v}\right)=\mathcal{G}^{st}\partial_{t}V\,L^{-6}.$
(B.42)

Taking the limit at the near horizon and applying the rule for regularity
(B.41) we get the _attractor mechanism_ 141414The attractor mechanism was
first found for $D=4$ asymptotically-flat, extremal black holes in [46, 47].
equation

$\displaystyle\lim_{\phi^{s}\rightarrow\phi^{s}_{*}}\partial_{t}V=0.$ (B.43)

The scalar fields which $V$ depends on are attracted towards fixed value
extremizing the potential, while the others are called flat directions.
Therefore, the attractor mechanism forces the effective potential $V$ to have
an extremum at the horizon, namely for $\phi^{s}=\phi^{s}_{*}$:

$\displaystyle\Gamma^{T}\partial_{s}\mathcal{M}\Gamma|_{\text{horizon}}=0\,.$
(B.44)

We can also find the explicit critical value of the black hole potential at
the horizon. To do this, we note that the function $U$ near the horizon is
explicitly given by

$\displaystyle U=-\frac{1}{2}\log\left({L^{2}(-2\tau)}\right).$ (B.45)

If we insert this result inside eq.(B.22)-(B.24) we get that the value of the
black hole potential $V$ at the horizon is fixed to be

$\displaystyle V_{\text{horizon}}=V_{*}=4\,L^{4}.$ (B.46)

Restoring the coordinate $r$, and setting $c=0$, the metric at the horizon
takes the form

$\displaystyle
ds^{2}=\frac{r^{2}}{r^{2}_{H}}(-dt^{2}+dx^{2})+\frac{L^{2}}{r^{2}}dr^{2}+L^{2}d\Omega_{S^{3}},$
(B.47)

which describes an ${\rm AdS}_{3}\times S^{3}$ geometry.

##  Appendix C  String solutions in the double commuting $\frac{{\rm
SL}(2,\mathbb{R})}{{\rm SO}(2)}$ truncation

We can now write the solutions of the equations (B.22)-(B.24) by minimizing
the potential $V$, as discussed in appendix D. After this minimization, to
solve the equations we just need the moduli to move along geodesics on the
target space. Here we discuss the case of two commuting ${\rm
SL}(2,\mathbb{R})$ of the scalar manifold, for which the potential and the
metric reads

$\displaystyle d\tilde{s}=$ $\displaystyle
d\phi^{2}+d\psi^{2}+d\varphi^{2}+e^{\varphi-\sqrt{2}\psi+\phi}(dC_{12})^{2}+e^{\varphi+\sqrt{2}\psi+\phi}(dC_{34})^{2},$
(C.1) $\displaystyle V=$
$\displaystyle\frac{e^{\varphi}}{2}\left(d_{1}^{2}e^{-\phi}+d_{5}^{2}e^{\phi-2\varphi}\right),$
(C.2)

where $B^{1}_{12}=C_{12}$ and $B^{1}_{34}=C_{34}$. From the perspective of the
10D, the metric in Einstein’s frame is

$\displaystyle
ds^{2}_{10}=e^{2\alpha\varphi}ds^{2}_{6}+e^{2\beta\varphi}\left(e^{2\gamma\psi}d\theta_{1}^{2}+e^{2\gamma\psi}d\theta_{2}^{2}+e^{-2\gamma\psi}d\theta_{3}^{2}+e^{-2\gamma\psi}d\theta_{4}^{2}\right)\,,$
(C.3) $\displaystyle C_{2}=\tilde{C}_{2}+C_{12}d\theta_{1}\wedge
d\theta_{2}+C_{34}d\theta_{3}\wedge d\theta_{4}\,,$ (C.4)

where $\alpha=1/4=-\beta$ and $\gamma^{2}=1/8$.
In this case the einstein equations are, directly from eq.(B.22)-(B.24)

$\displaystyle\ddot{U}(\tau)$
$\displaystyle-\frac{1}{4}e^{\varphi(\tau)+4U(\tau)}\left(d_{1}^{2}e^{-\phi(\tau)}+d_{5}^{2}e^{\phi(\tau)-2\varphi(\tau)}\right)=0,$
$\displaystyle\ddot{\phi}(\tau)$
$\displaystyle+\frac{1}{2}\,e^{\varphi(\tau)+4U(\tau)}\left(-d_{5}^{2}e^{\phi(\tau)-2\varphi(\tau)}+d_{1}^{2}e^{\phi(\tau)}\right)-\frac{1}{2}e^{\varphi(\tau)-\sqrt{2}\psi(\tau)+\phi(\tau)}(\dot{C}_{12}(\tau)^{2}+e^{2\sqrt{2}\psi(\tau)}\dot{C}_{34}(\tau)^{2})=0,$
$\displaystyle\ddot{\varphi}(\tau)$
$\displaystyle+\frac{1}{2}\,e^{4U(\tau)}\left(d_{5}^{2}e^{\phi(\tau)-\varphi(\tau)}-d_{1}^{2}e^{+\varphi(\tau)-\phi(\tau)}\right)-\frac{1}{2}e^{\varphi(\tau)-\sqrt{2}\psi(\tau)+\phi(\tau)}(\dot{C}_{12}(\tau)^{2}+e^{2\sqrt{2}\psi(\tau)}\dot{C}_{34}(\tau)^{2})=0,$
$\displaystyle\ddot{\psi}(\tau)$
$\displaystyle+\frac{1}{\sqrt{2}}e^{\varphi(\tau)-\sqrt{2}\psi(\tau)+\phi(\tau)}(\dot{C}_{12}(\tau)^{2}-e^{2\sqrt{2}\psi(\tau)}\dot{C}_{34}(\tau)^{2})=0,$
$\displaystyle\ddot{C}_{12}(\tau)$
$\displaystyle+\dot{C}_{12}(\tau)\left(\dot{\varphi}(\tau)-\sqrt{2}\dot{\psi}(\tau)+\dot{\phi}(\tau)\right)=0,$
$\displaystyle\ddot{C}_{34}(\tau)$
$\displaystyle+\dot{C}_{34}(\tau)\left(\dot{\varphi}(\tau)+\sqrt{2}\dot{\psi}(\tau)+\dot{\phi}(\tau)\right)=0.$
(C.5)

In order to separate the moduli from the other scalars we introduce the
function

$\displaystyle\varphi=\frac{h(\tau)-g(\tau)}{2},\quad\phi=\frac{h(\tau)+g(\tau)}{2},$
(C.6)

therefore the system totally splits into two different sets of equations. This
is due to the diagonal form of the metric, see eq.(C.1), in this particular
truncation. The first equations are for the scalars which are not modules

$\displaystyle\ddot{U}(\tau)$
$\displaystyle-\frac{1}{4}e^{4U(\tau)-g(\tau)}\left(d_{1}^{2}+d_{5}^{2}e^{2g(\tau)}\right)=0,$
(C.7) $\displaystyle\ddot{g}(\tau)$
$\displaystyle+e^{4U(\tau)-g(\tau)}\left(d_{1}^{2}-d_{5}^{2}e^{2g(\tau)}\right)=0,$
(C.8)

and the second system is for the moduli

$\displaystyle\ddot{h}(\tau)$
$\displaystyle-e^{h(\tau)-\sqrt{2}\psi(\tau)}\left(\dot{C}_{12}(\tau)^{2}+e^{2\sqrt{2}\psi(\tau)}\dot{C}_{34}(\tau)^{2}\right)=0,$
(C.9) $\displaystyle\ddot{\psi}(\tau)$
$\displaystyle+\frac{1}{\sqrt{2}}e^{h(\tau)-\sqrt{2}\psi(\tau)}\left(\dot{C}_{12}(\tau)^{2}-e^{2\sqrt{2}\psi(\tau)}\dot{C}_{34}(\tau)^{2}\right)=0,$
(C.10) $\displaystyle\ddot{C}_{12}(\tau)$
$\displaystyle+\dot{C}_{12}(\tau)\left(\dot{h}(\tau)-\sqrt{2}\dot{\psi}(\tau)\right)=0,$
(C.11) $\displaystyle\ddot{C}_{34}(\tau)$
$\displaystyle+\dot{C}_{34}(\tau)\left(\dot{h}(\tau)+\sqrt{2}\dot{\psi}(\tau)\right)=0.$
(C.12)

###  C.1 D1-D5 Solution

When the moduli are equal to zero, to solve the first system we can define the
superpotential to be

$\displaystyle W=e^{-\frac{g(\tau)}{2}}\left(d_{1}+d_{5}e^{g(\tau)}\right).$
(C.13)

Thanks to this definition we can prove that the system can be cast into a
first-order system

$\displaystyle\dot{g}(\tau)-2\,e^{2U(\tau)}\partial_{g(\tau)}W$
$\displaystyle=0,$ (C.14) $\displaystyle\dot{U}(\tau)-\frac{We^{2U(\tau)}}{4}$
$\displaystyle=0.$ (C.15)

Now it is easy to see that the solution is given by

$\displaystyle U(\tau)$
$\displaystyle=-\frac{1}{4}\log[{(1-d_{1}\tau)(1-d_{5}\tau)}],$ (C.16)
$\displaystyle-\varphi(\tau)$
$\displaystyle=\phi(\tau)=\frac{1}{2}\log\left[{\frac{1-d_{1}\tau}{1-d_{5}\tau}}\right],$
(C.17) $\displaystyle\psi(\tau)$ $\displaystyle=c_{12}(\tau)=c_{34}(\tau)=0,$
(C.18)

for which the metric reads

$\displaystyle
ds^{2}=\frac{(-dt^{2}+dx^{2})}{\sqrt{(-d_{1}\tau+1)(-d_{5}\tau+1)}}+\frac{{\sqrt{(-d_{1}\tau+1)(-d_{5}\tau+1)}}}{(-2\tau)}\left(\frac{d\tau^{2}}{4\tau^{2}}+d\Omega^{2}_{S^{3}}\right).$
(C.19)

We can now define the usual radial coordinate $\tau=(-2r^{2})^{-1}$ and we end
up with the usual form of the D1-D5 system

$\displaystyle ds^{2}=$
$\displaystyle(Z_{1}Z_{5})^{-\frac{1}{2}}(-dt^{2}+dx^{2})+(Z_{1}Z_{5})^{\frac{1}{2}}(dx^{i}dx^{i}),$
(C.20) $\displaystyle dx^{i}dx^{i}=$ $\displaystyle
dr^{2}+r^{2}d\Omega^{2}_{3},\qquad Z_{1}=1+\frac{Q_{1}}{r^{2}},\qquad
Z_{5}=1+\frac{Q_{5}}{r^{2}},$ (C.21)

where $Q_{1}=d_{1}/2$ and $Q_{5}=d_{5}/2$. At the horizon the $Z$ functions
become

$\displaystyle Z_{1}=\frac{Q_{1}}{r^{2}},\qquad Z_{5}=\frac{Q_{5}}{r^{2}},$
(C.22)

and the physical distance from the horizon is

$\displaystyle\rho=\log\left(\frac{r}{\ell}\right).$ (C.23)

This background is ${\rm AdS}_{3}\times S^{3}$, which is the throat of the
D1-D5 system. Now, the scalars are attracted towards configurations that
extremizes this potential, called critical points

$\displaystyle\partial_{\phi}V=0\rightarrow V=V_{*}=d_{1}\,d_{5}.$ (C.24)

Since we are in the near-horizon region where this critical point is reached
by the scalar fields, we have $V=V_{*}$ and the energy-momentum tensor for the
scalars is zero $T^{(S)}_{\hat{\mu}\hat{\nu}}=0$. The near-horizon metric is
conveniently written in the following form:

$\displaystyle
ds^{2}=\frac{r^{2}}{L^{2}}(-dt^{2}+dx^{2})+\frac{L^{2}}{r^{2}}dr^{2}+L^{2}\left[d\psi^{2}+\sin^{2}\psi\left(d\theta^{2}+\sin^{2}\theta
d\omega^{2}\right)\right],$ (C.25)

where $L^{2}=\frac{\sqrt{d_{1}d_{5}}}{2}$. Einstein’s equations boil down to

$\displaystyle
G_{\mu}^{\nu}=-\frac{1}{2}V_{0}\delta_{\mu}^{\nu}=T^{(H)\nu}_{\,\,\mu},\quad
G_{\bar{\mu}}^{\bar{\nu}}=\frac{1}{2}V_{0}\delta_{\bar{\mu}}^{\bar{\nu}}=T^{(H)\bar{\nu}}_{\,\,\bar{\mu}},$
(C.26)

solved by the condition $L^{2}=\sqrt{Q_{1}Q_{5}}$.

###  C.2 D1-D5 solution with non-trivial moduli

The equations for the moduli (C.9)-(C.12) are consistent with all of them to
be zero, this was the matter of the last section C.1 which describes the D1-D5
system. In this subsection, we consider a solution in which some of the moduli
fields are allowed to evolve in the radial variable $\tau$. The resulting
solution, however, is not related to the general construction considered in
this work, in which the evolution of the moduli is on one of the boundary
coordinates instead.
We can still keep part of the solution we wrote before

$\displaystyle U(\tau)$
$\displaystyle=-\frac{1}{4}\log[{(1-d_{1}\tau)(1-d_{5}\tau)}],$ (C.27)
$\displaystyle g(\tau)$
$\displaystyle=\log\left[{\frac{1-d_{1}\tau}{1-d_{5}\tau}}\right],$ (C.28)

while the moduli remain to be fixed now. If we now introduce

$\displaystyle s(\tau)=h(\tau)+\sqrt{2}\psi(\tau),\quad
t(\tau)=h(\tau)-\sqrt{2}\psi(\tau),$ (C.29)

the equations decouple into two independent systems

$\displaystyle\ddot{t}(\tau)$
$\displaystyle-2e^{t(\tau)}\dot{c}_{12}(\tau)^{2}=0,\quad\ddot{c}_{12}(\tau)+\dot{c}_{12}(\tau)\dot{t}(\tau)=0,$
(C.30) $\displaystyle\ddot{s}(\tau)$
$\displaystyle-2e^{s(\tau)}\dot{c}_{34}(\tau)^{2}=0,\quad\ddot{c}_{34}(\tau)+\dot{c}_{34}(\tau)\dot{s}(\tau)=0.$
(C.31)

We can directly integrate $c_{12}$ and $c_{34}$ as

$\displaystyle c_{12}(\tau)=c_{1}\int e^{-t(\tau)}d\tau,\quad
c_{34}(\tau)=c_{3}\int e^{-s(\tau)}d\tau,$ (C.32)

while the remaining equations are

$\displaystyle\quad\ddot{t}(\tau)=2c_{1}^{2}e^{-t(\tau)},\quad\ddot{s}(\tau)=2c_{3}^{2}e^{-s(\tau)},$
(C.33)

where $c_{1}$ and $c_{3}$ are constants. The whole system can now be
integrated and the solution reads

$\displaystyle t$
$\displaystyle=\log\left[-2c_{1}^{2}(\cosh(\tau)-1)\right],\quad
c_{12}=\frac{\coth\left(\frac{\sqrt{6a}}{2}\tau\right)}{2c_{1}},$
$\displaystyle s$
$\displaystyle=\log\left[-2c_{3}^{2}(\cosh(\tau)-1)\right],\quad
c_{34}=\frac{\coth\left(\frac{\sqrt{6a}}{2}\tau\right)}{2c_{3}}.$

In terms of the field appearing in the truncation given by (C.1), the solution
is

$\displaystyle\varphi$
$\displaystyle=\frac{1}{4}\left[-2\log\left(\frac{1-d_{1}\tau}{1-d_{5}\tau}\right)+\log\left(4c_{1}{}^{2}c_{3}{}^{2}(\cosh(\sqrt{6a}\tau)-1)^{2}\right)\right],$
$\displaystyle\phi$
$\displaystyle=\frac{1}{4}\left[2\log\left(\frac{1-d_{1}\tau}{1-d_{5}\tau}\right)+\log\left(4c_{1}{}^{2}c_{3}{}^{2}(\cosh(\sqrt{6a}\tau)-1)^{2}\right)\right],$
$\displaystyle\psi$
$\displaystyle=\frac{\log\left(-\frac{c_{3}}{c_{1}}\right)}{\sqrt{2}},\quad
c_{12}=\frac{\coth\left(\frac{\sqrt{6a}}{2}\tau\right)}{2c_{1}},\quad
c_{34}=\frac{\coth\left(\frac{\sqrt{6a}}{2}\tau\right)}{2c_{3}}.$

This solution is non-extremal. We can directly build other solutions using the
S-duality transformation given by eq.(3.46)-(3.48).

##  Appendix D Extremization of the effective scalar potential

In this section, we wish to discuss the extremization of $V(\phi,\Gamma)$. To
this end, we first need to explicitly construct $\mathcal{M}$ in terms of the
scalar fields by using the solvable parametrization of the coset manifold.
Using the explicit dependence of $\mathcal{M}_{MN}$ on the dimensionally
reduced Type IIB fields we can derive the moduli space of the ${\rm
AdS}_{3}\times{\rm S}^{3}$ background. We can derive the expression for the
potential $V$ by first writing the charge vector $\Gamma^{M}$ in components:

$\Gamma^{M}=\left(\begin{array}[]{ccc}n{}^{\alpha}&D_{ij}&n_{\alpha}\\\
\end{array}\right)\,,$ (D.1)

where $n^{\alpha}=(d_{5},n_{5})$, $n_{\alpha}=(d_{1},f_{1})$, being
$d_{1},\,d_{5}$ the D1, D5 charges and $f_{1},n_{5}$ the charges of the
fundamental string and of the NS 5-brane, respectively, while $D_{ij}$ are the
D3-brane charges. We then restrict ourselves to the D1-D5 charges and obtain:

$\displaystyle 2\,V\,$
$\displaystyle=\,\Gamma^{M}\mathcal{M}_{MN}\Gamma^{N}=\frac{G^{-\frac{1}{2}}}{g_{s}}\left(d_{1}-\frac{1}{8}\epsilon^{ijkl}B_{ij}B_{kl}d_{5}\right)^{2}+$
(D.2)
$\displaystyle+g_{s}G^{-\frac{1}{2}}\left[\left(d_{1}-\frac{1}{8}\epsilon^{ijkl}B_{ij}B_{kl}d_{5}\right)C_{(0)}-\left(c-\frac{1}{8}\epsilon^{ijkl}B_{ij}C_{kl}\right)d_{5}\right]^{2}+$
$\displaystyle+g_{s}G^{\frac{1}{2}}\left(\frac{1}{2g_{s}}B_{ij}B_{kl}G^{ik}G^{jl}+1\right)d_{5}^{2}\,.$

The mass formula for the D1D5 system given in [48] is obtained by adding an
invariant, constant shift-term to the potential $V$, that is
$\Gamma^{M}\Omega_{MN}\Gamma^{N}/2$. The minimizing conditions over the moduli
space are the following:

$\displaystyle c-\frac{1}{8}\epsilon^{ijkl}B_{ij}C_{kl}$
$\displaystyle=g_{s}G^{\frac{1}{2}}C_{(0)}\,,$ (D.3) $\displaystyle
g_{s}G^{\frac{1}{2}}+\frac{1}{8}\epsilon^{ijkl}B_{ij}B_{kl}$
$\displaystyle=\frac{d_{1}}{d_{5}}\,,$ (D.4) $\displaystyle
G^{\frac{1}{2}}B_{ij}G^{ik}G^{jl}$
$\displaystyle=\frac{1}{2}\,B_{ij}\epsilon^{ijkl}\,.$ (D.5)

With these conditions the potential at the minimum is

$V_{*}=d_{1}d_{5}\,,$ (D.6)

which, after the mentioned shift, becomes exactly double. The initial space
for the scalar manifold is ${\rm SO}(5,5)/\left({\rm SO}(5)\times{\rm
SO}(5)\right)$; due to the attractor mechanism, $5$ of these scalars are fixed
at the minimum of the potential. These conditions are explicitly given in
(D.3), (D.4) and (D.5) .
Then, the scalar manifold reduces to ${\rm SO}(4,5)/\left({\rm
SO}(4)\times{\rm SO}(5)\right)$ as required by CFT duality (see Cecotti) . To
see this, let us show that ${\rm SO}(4,5)$ is the little group of the charge
vector $\Gamma^{M}$ when only the D1-D5 charges are switched on:

$\Gamma^{M}=(d_{5},0,0,0,0,0,0,0,d_{1},0)\,.$ (D.7)

Supersymmetry requires $d_{1}\,d_{5}>0$. Let us perform the following duality
transformation:

$\Gamma^{M}\,\rightarrow\,\,\,\Gamma^{\prime
M}=\mathcal{O}^{M}{}_{N}\,\Gamma^{N}\,,$ (D.8)

where

$\mathcal{O}^{M}{}_{N}={\rm
diag}\left(\sqrt{\frac{d_{1}}{d_{5}}},1,1,1,1,1,1,1,\sqrt{\frac{d_{5}}{d_{1}}},1\right)\in{\rm
SO}(5,5)\,.$ (D.9)

The new charge vector reads:

$\Gamma^{\prime M}=\sqrt{d_{1}d_{5}}(1,0,0,0,0,0,0,0,1,0)\,.$ (D.10)

Changing the basis of the representation space by a Cayley transformation

$\mathcal{C}^{M}{}_{N}=\frac{1}{\sqrt{2}}\,\left(\begin{matrix}{\bf 1}&{\bf
1}\cr{\bf 1}&-{\bf 1}\end{matrix}\right)\,,$

the invariant matrix becomes:

$\Omega\rightarrow\mathcal{C}^{t}\,\Omega\,\mathcal{C}={\rm
diag}\left(1,1,1,1,1,-1,-1,-1,-1,-1\right)$ (D.11)

and the charge vector acquires the following form:

$\Gamma\rightarrow\mathcal{C}\,\Gamma=\sqrt{2\,d_{1}d_{5}}\,{\rm
diag}\left(1,0,0,0,0,0,0,0,0,0\right)\,.$ (D.12)

From the above form of the charge vector, it is straightforward to identify
the ${\rm SO}(4,5)$ subgroup of ${\rm SO}(5,5)$ which leaves it invariant. It
consists of those matrices which have a trivial action of the first entry of
the vector.
In the non-BPS case in which $d_{1}d_{5}<0$ (e.g. $D1$-anti-$D5$ system), the
same transformation yields the vector:

$\mathcal{C}\,\Gamma=\sqrt{2\,|d_{1}d_{5}|}\,{\rm
diag}\left(0,0,0,0,0,0,0,0,1,0\right)\,.$ (D.13)

The stabilizer being still ${\rm SO}(4,5)$. This orbit, as opposed to the BPS
one, is characterized by the invariant property $\Gamma^{t}\Omega\Gamma<0$.

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