UPR-1308-T MI-TH-212

Conformal Symmetries for Extremal Black Holes with General Asymptotic Scalars
in STU Supergravity

M. Cvetič1,2, C.N. Pope3,4 and A. Saha3

$\,{}^{1}$Department of Physics and Astronomy, and Department of Mathematics,
University of Pennsylvania, Philadelphia, PA 19104, USA

$\,{}^{2}$Center for Applied Mathematics and Theoretical Physics,
University of Maribor, SI2000 Maribor, Slovenia

$\,{}^{3}$George P. & Cynthia Woods Mitchell Institute for Fundamental Physics
and Astronomy,
Texas A&M University, College Station, TX 77843, USA

$\,{}^{4}$DAMTP, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road, Cambridge CB3 OWA, UK

###### Abstract

We present a construction of the most general BPS black holes of STU
supergravity (${\cal N}=2$ supersymmetric $D=4$ supergravity coupled to three
vector super-multiplets) with arbitrary asymptotic values of the scalar
fields. These solutions are obtained by acting with a subset of of the global
symmetry generators on STU BPS black holes with zero values of the asymptotic
scalars, both in the U-duality and the heterotic frame. The solutions are
parameterized by fourteen parameters: four electric and four magnetic charges,
and the asymptotic values of the six scalar fields. We also present BPS black
hole solutions of a consistently truncated STU supergravity, which are
parameterized by two electric and two magnetic charges and two scalar fields.
These latter solutions are significantly simplified, and are very suitable for
further explicit studies. We also explore a conformal inversion symmetry of
the Couch-Torrence type, which maps any member of the fourteen-parameter
family of BPS black holes to another member of the family. Furthermore, these
solutions are expected to be valuable in the studies of various swampland
conjectures in the moduli space of string compactifications.

###### Contents

  1. 1 Introduction
  2. 2 The STU Supergravity Theory
    1. 2.1 STU supergravity from six dimensions
    2. 2.2 Heterotic formulation of STU supergravity
    3. 2.3 The U-duality formulation of STU supergravity
    4. 2.4 Mapping between the heterotic and the U-duality formulations
    5. 2.5 STU supergravity in the $3+1$ formulation
  3. 3 General Extremal BPS Static Black Holes In The U-Duality Formulation
    1. 3.1 The general 8-charge static BPS metric
    2. 3.2 Scalar fields in the U-duality formulation
    3. 3.3 Gauge fields for the 8-charge BPS black holes
  4. 4 General 8-Charge Static BPS Black Holes In The Heterotic Formulation
    1. 4.1 8-charge static BPS black hole metric
    2. 4.2 Scalar fields in the heterotic formulation
    3. 4.3 Gauge fields for the 8-charge BPS black holes
  5. 5 Truncation to Pairwise-Equal Charges
  6. 6 Symmetry Transformations and Asymptotic Scalars
  7. 7 Concluding Remarks and Outlook
  8. A Kaluza-Klein $T^{2}$ Reduction From Six Dimensions

##  1 Introduction

Many intriguing non-perturbative aspects of string theory and M-theory have
been brought to light by studying the black hole and higher $p$-brane
solutions (see, for example, [1, 2]). Especially important in this context are
the supersymmetric BPS solutions, which are expected to be protected in the
face of stringy corrections to the leading-order effective action. Thus by
studying the BPS solutions in the low-energy supergravity limit one can expect
to gain insights that could remain relevant in the full theory.

The full four-dimensional supergravity theories resulting from the dimensional
reduction of the ten-dimensional heterotic or type II superstring are quite
complicated, with many field strengths and scalar fields in their bosonic
sectors. For many purposes, however, it suffices to focus on the black hole
solutions residing within a truncation of the theories to the so-called STU
supergravity, which comprises ${\cal N}=2$ supergravity coupled to three
vector supermultiplets. Solutions within the STU theory can be rotated using
the global symmetries of the full heterotic or type II theories to “fill out”
solution sets of the larger theories.

There are different ways to present the STU supergravity, which are related by
dualisations of one or more of the four gauge field strengths. These different
presentations of the theory arise naturally in different contexts. For
example, one of the duality complexions, which we refer to as the heterotic
formulation, is the one that arises naturally when one performs a toroidal
reduction from the ten-dimensional heterotic theory and then makes a
consistent truncation to the STU supergravity. A different duality complexion,
which we refer to as the U-duality formulation, arises if one makes a
consistent reduction of eleven-dimensional supergravity on the 7-sphere,
truncates further to the subsector of fields that are invariant under the
$U(1)$ maximal torus of the $SO(8)$ isometry group of the 7-sphere, and then
turns off the gauge coupling by sending the radius of the sphere to infinity.
This formulation of STU supergravity is related to the heterotic formulation
by a dualisation of two out of the four field strengths. There is also an
intermediate duality complexion which corresponds to dualising any one of the
four fields in the U-duality formulation. We refer to this as the 3+1
formulation of the theory.

As a preliminary to subsequent calculations, one of the purposes of this paper
is to obtain fully explicit relations between the fields in the three above-
mentioned formulations of the STU supergravity theory. These will then be
utilised later in the paper, when we construct explicit expressions for the
most general static BPS black hole solutions in the STU supergravity theory.
The most general such solutions are characterised by a total of eight charges,
corresponding to four electric and four magnetic charges carried by the four
gauge field strengths. One can use the compact $U(1)^{3}$ subgroup of the
$SL(2,{\mathbb{R}})^{3}$ global symmetry of the STU theory to rotate these
solutions to ones involving only 5 independent charges, but for our purposes
it is more useful to present the solutions in the more symmetrical 8-charge
characterisation. In fact, we obtain this symmetrical form by using the
$U(1)^{3}$ symmetry in the opposite direction, to go from a 5-charge to an
8-charge expression. This procedure was originally partially implemented in
[3]; for later purposes we also wish to generalise the solutions by allowing
for arbitrary asymptotic values of the six scalar fields of the STU
supergravity. This can be done by acting with the remaining six generators of
the coset $SL(2,{\mathbb{R}})^{3}/U(1)^{3}$ within the global symmetry group.

One of our reasons for constructing the most general static BPS black hole
solutions explicitly is to study in detail a phenomenon that was first noticed
by Couch and Torrence in 1984 in the case of the extremal Reissner-Nordström
black hole [4]. This solution, which is a special case [5] of the static BPS
black holes of STU supergravity in which all four electric charges are set
equal and the magnetic charges are all set to zero, exhibits a conformal
inversion symmetry as follows. Writing the extreme Reissner-Nordström metric
as

$ds^{2}=-\Big{(}1+{\frac{Q}{r}}\Big{)}^{-2}\,dt^{2}+\Big{(}1+{\frac{Q}{r}}\Big{)}^{2}\,(dr^{2}+r^{2}\,d\Omega_{2}^{2})\,,$
(1.1)

where $d\Omega_{2}^{2}$ is the metric on the unit 2-sphere, the horizon is
located at $r=0$. Defined an inverted coordinate $\hat{r}$ and a conformally-
related metric $d\hat{s}^{2}$ by

$\hat{r}={\frac{Q^{2}}{r}}\,,\qquad
d\hat{s}^{2}={\frac{Q^{2}}{r^{2}}}\,ds^{2}\,,$ (1.2)

one finds that metric $d\hat{s}^{2}$ takes exactly the same form as the
original metric $ds^{2}$, written now in terms of the inverted coordinate:

$d\hat{s}^{2}=-\Big{(}1+{\frac{Q}{\hat{r}}}\Big{)}^{-2}\,dt^{2}+\Big{(}1+{\frac{Q}{\hat{r}}}\Big{)}^{2}\,(d\hat{r}^{2}+\hat{r}^{2}\,d\Omega_{2}^{2})\,.$
(1.3)

The Couch-Torrence symmetry therefore maps the near-horizon region to the
asymptotic region near infinity, and vice versa. This symmetry has been
employed in more recent times to related conserved quantities on the horizon
to conserved quantities at infinity. (See, for example, [6, 7, 8].)

Recently, generalisations of the Couch-Torrence symmetry have been studied for
the static black hole solutions of STU supergravity. In [8] it was shown that
the set of solutions carrying four electric charges and no magnetic charges
maps into itself under a similar conformal inversion. The mapping is no longer
a symmetry, in that a black hole with charges $(Q_{1},Q_{2},Q_{3},Q_{4})$ maps
into a different member of the 4-charge family, with different charges
$(\hat{Q}_{1},\hat{Q}_{2},\hat{Q}_{3},\hat{Q}_{4})$, related by
$\hat{Q}_{i}=Q_{i}^{-1}\,(Q_{1}\,Q_{2}\,Q_{3}\,Q_{4})^{1/2}$ [8]. More
recently, we studied the problem for the general 8-charge static BPS black
holes [9]. Our focus was on the metrics where the asymptotic values of the
scalar fields were taken to vanish. We established that any such 8-charge
black hole will certainly be related to other members of this family under a
conformal inversion, although it does not seem to be possible any longer to
present an explicit mapping of the charges in the way that could be done for
the 4-charge solutions.

A slightly different approach was developed in [10], which could be applied
not only to the static BPS black holes of STU supergravity, but more generally
in Einstein-Maxwell-scalar theories of type $E_{7}$ [10]. An important aspect
of this approach is that it is applied in the context of the general class of
static BPS black holes including the parameters associated with the asymptotic
values of the scalar fields. We exhibit this description of the conformal
inversion mapping in detail for the STU supergravity case, where it allows an
explicit mapping of the general 14-parameter family of solutions (comprising 8
charges plus 6 asymptotic scalar values). At the price of the conformal
inversion always entailing a mapping from one set of asymptotic scalar values
to a different set of values, the approach allows one to give fully explicit
transformations under the inversion.

The organisation of this paper is as follows. In section 2, we discuss how the
four-dimensional STU supergravity can be obtained by Kaluza-Klein reduction
from higher dimensions, where the internal space is a torus. In particular, it
suffices for our purposes to start from the pure ${\cal N}=2$ non-chiral
supergravity in six dimensions (which itself can be obtained from a toroidal
reduction of supergravity in ten dimensions), and then perform a reduction on
a 2-torus to four dimensions. In addition to discussing the resulting
formulation of STU supergravity, known sometimes as the heterotic formulation,
we also carry out explicitly the dualisation of two of the four gauge fields.
This results in the formulation known as the U-duality basis. This is the form
in which the theory would arise if one began with the gauged ${\cal N}=8$
supergravity coming from the reduction of eleven-dimensional supergravity on
$S^{7}$, truncated this to ${\cal N}=2$ supergravity plus three vector
supermultiplets, and then took the ungauged limit. We also discuss the STU
supergravity in a third duality complex, which we refer to as the 3+1
formulation, where just one of the field strengths of the heterotic
formulation is dualised. We shall make use of all three of these formulations
of STU supergravity, and the explicit relations between them, in the remainder
of the paper.

In section 3, we consider the general extremal BPS static black hole solutions
of STU supergravity in the U-duality formulation. These solutions carry 8
charges in total, comprising electric and magnetic charges for each of the
four gauge fields. These are really equivalent to solutions with just 5
independent charges, since the $U(1)^{3}$ subgroup of the global
$SL(2,{\mathbb{R}})^{3}$ symmetry group of the STU theory allows 3 of the 8
charges to be transformed away (while maintaining the vanishing of the
asymptotic values of the scalar fields at infinity). In fact the 8-charge
solutions were constructed in [3], in the heterotic formulation, by starting
from a 5-charge “seed solution” and then acting with the $U(1)^{3}$
transformations. The analogous process was employed recently in [9] to
construct the 8-charge solution in the U-duality formulation of STU
supergravity. The new feature that we implement now in section 3 is to allow
also for arbitrary asymptotic values of the six scalar fields in the 8-charge
black hole solutions.111Partial results including the scalars were obtained in
[3]. This can be achieved by employing the action of the remaining six
symmetries in the coset $SL(2,{\mathbb{R}})^{3}/U(1)^{3}$.

In section 4, we carry out the same steps of generating the general static BPS
black hole solutions in the heterotic formulation. Again, we first “fill out”
the 5-charge seed solution to the full set of 8 charges by acting with the
$U(1)^{3}$ subgroup of the global symmetry group. We highlight similarities
and also differences with the description in the U-duality formulation.

In section 5 we describe the consistent truncation of the STU supergravity to
a supersymmetric theory whose bosonic sector comprises the metric, two gauge
fields and two scalar fields (a dilaton and an axion). The theory is
considerably simpler to work with than the full STU supergravity, and as we
show, the general static BPS black hole solutions, characterised by four
charges and asymptotic values for the two scalar fields, are very much
simpler.

In section 6, we consider the static BPS black holes in the 3+1 formulation,
using the description of the theory in terms of the Kähler geometry of the
scalar manifold. We then use this to investigate the behaviour of the black
hole metrics under an inversion of the radial coordinate, which, together with
a conformal rescaling, maps the horizon to infinity and vice versa. In this
description of the conformal inversion, which was discussed previously in
[10], one can show how any given member of the general 14-parameter family of
static BPS black holes (characterised by the 4 electric charges, 4 magnetic
charges and 6 asymptotic scalar values) is mapped by the conformal inversion
to another member of the 14-parameter family. In this description, unlike one
considered previously in [8, 9], the asymptotic values of the scalar fields
are always different in the original and the conformally-inverted metrics.

Section 7 contains a summary of our conclusions, and also further discussion.
Some details of the Kaluza-Klein reduction from six to four dimensions are
relegated to appendix A.

##  2 The STU Supergravity Theory

In this section, we shall provide explicit expressions for the bosonic sector
of four-dimensional ungauged STU supergravity in the three different
formulations we shall be using in this paper, and the explicit relations
between them. Specifically, we shall consider what may be called the heterotic
formulation, which is the way the theory arises if one starts from ten-
dimensional supergravity and toroidally reduces on $T^{6}$, together with
appropriate truncations. The truncated theory has ${\cal N}=2$ supersymmetry,
and comprises ${\cal N}=2$ supergravity coupled to three vector multiplets.
The theory was presented in this form in [11]. We shall also consider the
formulation of STU supergravity that one would obtain by reducing eleven-
dimensional supergravity on $S^{7}$, performing an appropriate truncation from
${\cal N}=8$ to ${\cal N}=2$ supersymmetry, and also turning off the gauge
coupling constant in the four-dimensional theory. This formulation of STU
supergravity is sometimes referred to as the $U$-duality invariant
formulation. The two formulations are related in four dimensions by performing
an appropriate dualisation of two of the four gauge fields. We shall also
consider STU supergravity in an intermediate formulation which we refer to as
the 3+1 formulation. In this case, instead of dualising two field strengths in
the heterotic formulation only one is dualised.

###  2.1 STU supergravity from six dimensions

In the heterotic formulation one can in fact conveniently describe the STU
theory by first reducing from ten dimensions to six dimensions on $T^{4}$ and
truncating to pure ${\cal N}=2$ non-chiral supergravity. The STU theory is
then obtained by reducing this on $T^{2}$ with no further truncation. (Except,
of course, the usual Kaluza-Klein truncation in which only the singlets under
the $U(1)^{2}$ isometry of the $T^{2}$ are retained.) The bosonic sector of
the non-chiral supergravity in six-dimensional supergravity is described by
the Lagrangian

$\displaystyle{\cal L}_{6}=\hat{R}\,{\hat{*}\hbox to0.0pt{1\hss}\mkern
4.0mu{\rm l}}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}{\hat{*}d\hat{\phi}}\wedge d\hat{\phi}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}e^{-\sqrt{2}\,\hat{\phi}}\,{\hat{*}\hat{H}_{{\scriptscriptstyle(3)}}}\wedge\hat{H}_{{\scriptscriptstyle(3)}}\,,$
(2.1)

where $\hat{H}_{{\scriptscriptstyle(3)}}=d\hat{B}$. The reduction down to four
dimensions is described in detail in appendix A. After dualising the 2-form
potential for the field $H_{{\scriptscriptstyle(3)}}$ to an axion $\chi_{1}$,
the resulting Lagrangian for the bosonic sector of STU supergravity is given
by (A):

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle R\,{*\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\sum_{i=1,3}({*d\varphi_{i}}\wedge
d\varphi_{i}+e^{2\varphi_{i}}\,{*d\chi_{i}}\wedge
d\chi_{i})-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}{*d{{\tilde{\varphi}}}_{2}}\wedge
d{{\tilde{\varphi}}}_{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{2{{\tilde{\varphi}}}_{2}}\,{*d{{\widetilde{\chi}}}_{2}}\wedge
d{{\widetilde{\chi}}}_{2}$ $\displaystyle-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}e^{-\varphi_{1}}\,\Big{[}e^{{{\tilde{\varphi}}}_{2}-\varphi_{3}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{1}\wedge{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{1}+e^{-{{\tilde{\varphi}}}_{2}+\varphi_{3}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{2}\wedge{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}}_{2}$
$\displaystyle\qquad\qquad+e^{-{{\tilde{\varphi}}}_{2}-\varphi_{3}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}^{3}\wedge{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}^{3}+e^{{{\tilde{\varphi}}}_{2}+\varphi_{3}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}^{4}\wedge{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}^{4}\Big{]}+\chi_{1}\,({{\widetilde{F}}}_{1}\wedge{{\widetilde{F}}}_{2}+F^{3}\wedge
F^{4})\,,$

where ${{\widetilde{F}}}_{1}=d{{\widetilde{A}}}_{1}$,
${{\widetilde{F}}}_{2}=d{{\widetilde{A}}}_{2}$, $F^{3}=dA^{3}$ and
$F^{4}=dA^{4}$ are the “raw” field strengths, and

$\displaystyle{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}}_{1}$
$\displaystyle=$
$\displaystyle{{\widetilde{F}}}_{1}-{{\widetilde{\chi}}}_{2}\,F^{3}\,,\qquad{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}={{\widetilde{F}}}_{2}-\chi_{3}\,F^{3}\,,$ $\displaystyle{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{3}$ $\displaystyle=$
$\displaystyle F^{3}\,,\qquad{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}^{4}=F^{4}+\chi_{3}\,{{\widetilde{F}}}_{1}+{{\widetilde{\chi}}}_{2}\,{{\widetilde{F}}}_{2}-\chi_{2}\,\chi_{3}\,F^{3}\,,$
(2.3)

with overbars, denote the “dressed” field strengths that appear in the kinetic
terms in (2.1). The gauge fields numbered 1 and 2 are denoted with tildes
here; later on, we shall dualise these fields in order to obtain the STU
supergravity theory in the formalism in which the $SL(2,{\mathbb{R}})^{3}$
symmetry is manifest.

###  2.2 Heterotic formulation of STU supergravity

In this formulation the raw field strengths
$({{\widetilde{F}}}_{1},{{\widetilde{F}}}_{2},F^{3},F^{4})$ are organised into
the column vector

$\boldsymbol{{{\cal F}}}\equiv\begin{pmatrix}{{\cal F}}^{1}\cr{{\cal
F}}^{2}\cr{{\cal F}}^{3}\cr{{\cal
F}}^{4}\end{pmatrix}=\begin{pmatrix}F^{3}\cr{{\widetilde{F}}}_{1}\cr
F^{4}\cr{{\widetilde{F}}}_{2}\end{pmatrix}\,,$ (2.4)

and the $(\varphi_{1},\chi_{1})$ dilaton/axion pair are redefined as

$\Phi={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\varphi_{1}\,,\qquad\Psi=\chi_{1}\,.$ (2.5)

The Lagrangian (2.1) can then be written as

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle R\ {*\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 8}}}{{\hbox{Tr}}}({*dM}\wedge LdML)-2{*d\Phi}\wedge
d\Phi-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{4\Phi}\,{*d\Psi}\wedge d\Psi$ (2.6)
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{-2\Phi}\,(LML)_{ij}\,{*{{\cal F}}^{i}}\wedge{{\cal
F}}^{j}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\Psi\,L_{ij}\,{{\cal F}}^{i}\wedge{{\cal F}}^{j}\,.$

where

$L=\sigma_{1}\otimes{{\mathbb{I}}}_{2}=\begin{pmatrix}0&{{\mathbb{I}}}_{2}\cr{{\mathbb{I}}}_{2}&0\end{pmatrix}\,.$
(2.7)

The scalar matrix $M$ can be read off by comparing the kinetic terms for the
gauge fields ${{\cal F}}^{i}$ in (2.6) with the kinetic terms in (2.1). It is
straightforward to see that $M$ can then be written as

$M=\begin{pmatrix}{{\mathbb{G}}}^{-1}&-{{\mathbb{G}}}^{-1}\,{{\mathbb{B}}}\cr{{\mathbb{B}}}\,{{\mathbb{G}}}^{-1}&{{\mathbb{G}}}-{{\mathbb{B}}}\,{{\mathbb{G}}}^{-1}\,{{\mathbb{B}}}\end{pmatrix}\,,$
(2.8)

where

${{\mathbb{G}}}=e^{-\varphi_{3}}\,\begin{pmatrix}e^{-{{\tilde{\varphi}}}_{2}}+{{\widetilde{\chi}}}_{2}^{2}\,e^{{{\tilde{\varphi}}}_{2}}&-{{\widetilde{\chi}}}_{2}\,e^{{{\tilde{\varphi}}}_{2}}\cr-{{\widetilde{\chi}}}_{2}\,e^{{{\tilde{\varphi}}}_{2}}&e^{\tilde{\varphi}_{2}}\end{pmatrix}\,,\qquad{{\mathbb{B}}}=\begin{pmatrix}0&-\chi_{3}\cr\chi_{3}&0\end{pmatrix}\,.$
(2.9)

One can recognise ${{\mathbb{G}}}$ as being associated with the internal
metric on the 2-torus in the Kaluza-Klein reduction (A.1), ( i.e.
$ds_{2}^{2}={{\mathbb{G}}}_{ij}\,dz^{i}\,dz^{j}$ is the metric enclosed in
square brackets in (A.1), in the 2-torus directions), and ${{\mathbb{B}}}$ as
being associated with the internal component $A_{{{\scriptscriptstyle(0)}}12}$
of the 2-form potential in (A.2).

The four-dimensional Lagrangian (2.6) is in the form that was obtained in [3].
It is invariant under the $O(2,2)$ transformations (the $T$-duality of the
2-torus compactification):

$M\longrightarrow\Omega M\Omega^{T}\,,\qquad{{\cal
A}}^{i}\longrightarrow\Omega^{i}{}_{j}\,{{\cal A}}^{j}\,,$ (2.10)

where $g_{\mu\nu}$ and $S$ are inert, and ${{\cal A}}^{i}$ are the potentials
for the field strengths ${{\cal F}}^{i}$; that is, ${{\cal F}}^{i}=d{{\cal
A}}^{i}$. The transformation matrix $\Omega\in O(2,2)$ preserves the
$O(2,2)$-invariant matrix $L$:

$\Omega^{T}L\Omega=L\ ,\ \ \ \ \
L=\begin{pmatrix}0&\mathbb{I}_{2}\cr\mathbb{I}_{2}&0\end{pmatrix}\,.$ (2.11)

$L$ is in fact the metric tensor in the $(2,2)$-signature flat space on which
the $O(2,2)$ transformations $\Omega$ act. $L$ has components $L_{ij}$ with
$L_{13}=L_{31}=L_{24}=L_{42}=0$ and the remainder being zero. The inverse
metric $L^{-1}$ has components $L^{ij}$, and for these too
$L^{13}=L^{31}=L^{24}=L^{42}=0$ with the remainder being zero. Note that one
also has

$L\,M\,L=M^{-1}\,.$ (2.12)

Note that $M$ has components $M^{ij}$ and its inverse $M^{-1}$ has components
$(M^{-1})_{ij}=L_{ik}\,L_{j\ell}\,M^{k\ell}$.

The equations of motion and Bianchi identities are in addition invariant under
the $SL(2,{\mathbb{R}})$ transformations (electromagnetic $S$-duality):

$S\longrightarrow{{aS+b}\over{cS+d}}\,,\qquad{\cal F}^{i}\to(c\Psi+d)\,{{\cal
F}}^{i}+c\,e^{-2\Phi}\,(ML)_{ij}\,{*{{\cal F}}^{j}}\,,$ (2.13)

where $g_{\mu\nu}$ and the scalar matrix $M$ are inert.

The equations of motion for the electromagnetic fields that follow from (2.6)
are

$d{\cal G}_{i}=0\,,\qquad\hbox{where}\qquad{\cal
G}_{i}\equiv-e^{-2\Phi}\,(LML)_{ij}\,{*{{\cal F}}^{j}}+\Psi\,L_{ij}\,{{\cal
F}}^{j}\,,$ (2.14)

which implies that the canonical electric charges $\vec{\alpha}$ will be given
by222We have changed the overall sign of the definition of the dual field
strength ${\cal G}_{i}$ in (2.14), relative to the one in [3]. This is for
consistency with our conventions for the Hodge dualisation of differential
forms (which is made explicit in eqn (3.65), and our conventions in the rest
of the paper.

$\alpha_{i}={\frac{1}{4\pi}}\,\int{\cal G}_{i}\,.$ (2.15)

The magnetic charges $\vec{p}$ are given by

$p^{i}={\frac{1}{4\pi}}\,\int{{\cal F}}^{i}\,.$ (2.16)

It is convenient for later purposes to define also magnetic charges
$\vec{\beta}$ with a lowered $O(2,2)$ index:

$\beta_{i}=L_{ij}\,p^{j}\,.$ (2.17)

These will be referred to later as the “canonical” magnetic charges.333In
toroidally compactified heterotic string theory the canonical electric charges
${\vec{\alpha}}$ are quantised and span an even self-dual lattice, subject to
the constraint: ${\vec{\alpha}}^{T}L{\vec{\alpha}}=-2,0,2,\cdots$. For BPS-
saturated configurations one further requires
${\vec{\alpha}}^{T}L{\vec{\alpha}}>0$. By S-duality the same conditions are
satisfied for quantised magnetic charges ${\vec{\beta}}$, along with the the
condition that when ${\vec{\alpha}}\propto{\vec{\beta}}$ with magnetic and
electric charge vector components being co-prime integers [12]. If the
asymptotic values of the scalars are taken to be zero, one has
$S_{\infty}={\rm i}$ and $M_{\infty}=\mathbb{I}_{4}$.

###  2.3 The U-duality formulation of STU supergravity

We arrive at this formulation by dualising the gauge fields
${{\widetilde{A}}}_{1}$ and ${{\widetilde{A}}}_{2}$ appearing in the
Lagrangian (2.1). To do this, we employ the standard procedure of introducing
dual potentials $A^{1}$ and $A^{2}$ as Lagrange multipliers, and adding the
terms $A^{1}\wedge d{{\widetilde{F}}}_{1}+A^{2}\wedge d{{\widetilde{F}}}_{2}$
to the Lagrangian (A). Up to total derivatives, this is equivalent to adding

${\cal
L}_{LM}=F^{1}\wedge{{\widetilde{F}}}_{1}+F^{2}\wedge{{\widetilde{F}}}_{2}\,,$
(2.18)

where $F^{1}=dA^{1}$ and $F^{2}=dA^{2}$ are the raw dualised field strengths.
Varying the total Lagrangian with respect to ${{\widetilde{F}}}_{1}$ and
${{\widetilde{F}}}_{2}$, now treated as independent fields, shows that
${{\widetilde{F}}}_{1}$ and ${{\widetilde{F}}}_{2}$ satisfy

$\displaystyle F^{1}+\chi_{1}\,{{\widetilde{F}}}_{2}$ $\displaystyle=$
$\displaystyle
e^{-\varphi_{1}+{{\tilde{\varphi}}}_{2}}\,(e^{-\varphi_{3}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{1}+\chi_{3}\,e^{\varphi_{3}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{4}})\,,$
$\displaystyle F^{2}+\chi_{1}\,{{\widetilde{F}}}_{1}$ $\displaystyle=$
$\displaystyle
e^{-\varphi_{1}+\varphi_{3}}\,(e^{-{{\tilde{\varphi}}}_{2}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{2}+{{\widetilde{\chi}}}_{2}\,e^{{{\tilde{\varphi}}}_{2}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{4}})\,,$ (2.19)

and hence, using (2.3),

$\displaystyle F^{1}+\chi_{1}\,{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}+\chi_{1}\,\chi_{3}\,{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{3}$ $\displaystyle=$
$\displaystyle
e^{-\varphi_{1}+{{\tilde{\varphi}}}_{2}}\,(e^{-\varphi_{3}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{1}+\chi_{3}\,e^{\varphi_{3}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{4}})\,,$
$\displaystyle F^{2}+\chi_{1}\,{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{1}+\chi_{1}\,{{\widetilde{\chi}}}_{2}\,{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{3}$ $\displaystyle=$
$\displaystyle
e^{-\varphi_{1}+\varphi_{3}}\,(e^{-{{\tilde{\varphi}}}_{2}}\,{*{{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}}_{2}+{{\widetilde{\chi}}}_{2}\,e^{{{\tilde{\varphi}}}_{2}}\,{*{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}^{4}})\,,$ (2.20)

These two equations, together with their Hodge duals, can be solved
algebraically for ${{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}}_{1}$ and
${{\widetilde{{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}$, with the results expressed, using (2.3) again, in terms of
$(F^{1},F^{2},F^{3},F^{4})$ and $(*F^{1},*F^{2},*F^{3},*F^{4})$. Since the
expressions are a little unwieldy, we shall not present them here.

Having solved for ${{\widetilde{{\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}}}}_{1}$ and
${{\widetilde{{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}$ these may be substituted back into the Lagrangian (2.1) plus
(2.18), to give the theory in the dualised form, with the gauge fields
$(A^{1},A^{2},A^{3},A^{4})$. After one final step, in which the
${{\tilde{\varphi}}}_{2}$ and ${{\widetilde{\chi}}}_{2}$ scalars are subjected
to the discrete involutive $SL(2,{\mathbb{R}})$ transformation

$\tilde{\tau}_{2}=-{\frac{1}{\tau_{2}}}\,,\qquad\hbox{where}\quad\tilde{\tau}_{2}={{\widetilde{\chi}}}_{2}+i\,e^{-{{\tilde{\varphi}}}_{2}}\,,\qquad\tau_{2}=\chi_{2}+i\,e^{-\varphi_{2}}\,,$
(2.21)

one finds that the Lagrangian is precisely equal to the one described in
appendix B of [14] and appendix A of [9]. (The dualisation to go from the
heterotic to the U-duality formulation has also been discussed in [13].) The
kinetic terms involving the field strengths given, as in eqn (A.2) of [9], by

${\cal L}_{F}=-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}f^{R}_{AB}\,{*F}^{A}\wedge F^{B}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}f^{I}_{AB}\,F^{A}\wedge F^{B}\,,$ (2.22)

where $f^{R}_{AB}$ and $f^{I}_{AB}$ are the real and imaginary parts of the
scalar matrix given in eqn (A.9) of [9]. This is written in the formalism and
notation described in Freedman and Van Proeyen [15], in which, defining

$G_{A}=-f^{R}_{AB}\,{*F^{B}}-f^{I}_{AB}\,F^{B}\,,$ (2.23)

the field equations $dG_{A}=0$ and Bianchi identities $dF^{A}=0$ are invariant
under the transformations

$\displaystyle\begin{pmatrix}{\bf F}\cr{\bf
G}\end{pmatrix}\longrightarrow{\cal S}\,\begin{pmatrix}{\bf F}\cr{\bf
G}\end{pmatrix}\,,\qquad{\cal S}=\begin{pmatrix}A&B\cr C&D\end{pmatrix}\,,$
(2.24)

where the constant $4\times 4$ matrices $A$, $B$, $C$ and $D$ obey

$A^{T}C-C^{T}A=0\,,\qquad B^{T}D-D^{T}B=0\,,\qquad
A^{T}D-C^{T}B={{\mathbb{I}}}_{4}\,,$ (2.25)

and we have defined ${\bf F}=(F^{1},F^{2},F^{3},F^{4})^{T}$ and ${\bf
G}=(G_{1},G_{2},G_{3},G_{4})^{T}$. The equations (2.25) are precisely the
conditions for the matrix ${\cal S}$ to be an element of $Sp(8,{\mathbb{R}})$,
obeying [15]

$S^{T}\Omega
S=\Omega\,,\qquad\hbox{where}\qquad\Omega=\begin{pmatrix}0&{{\mathbb{I}}}_{4}\cr-{{\mathbb{I}}}_{4}&0\end{pmatrix}\,.$
(2.26)

The scalar field Lagrangian

${\cal L}_{\rm scal}=-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\sum_{i=1}^{3}({*d\varphi_{i}}\wedge
d\varphi_{i}+e^{2\varphi_{i}}\,{*d\chi_{i}}\wedge d\chi_{i})$ (2.27)

is invariant under $SL(2,{\mathbb{R}})^{3}$, which is a subgroup of
$Sp(8,{\mathbb{R}})$, and so in this formulation the STU supergravity theory
has a manifest $SL(2,{\mathbb{R}})^{3}$ symmetry, at the level of the
equations of motion. The explicit forms of the $A$, $B$, $C$ and $D$ matrices
corresponding to the $SL(2,{\mathbb{R}})^{3}$ subgroup of $Sp(8,{\mathbb{R}})$
are given in [9].

The conserved electric and magnetic charges in the U-duality formalism are
given by

$Q_{A}={\frac{1}{4\pi}}\,\int G_{A}\,,\qquad P^{A}={\frac{1}{4\pi}}\,\int
F^{A}\,.$ (2.28)

If we define the charge vectors ${\bf P}=(P^{1},P^{2},P^{3},P^{4})^{T}$ and
${\bf Q}=(Q_{1},Q_{2},Q_{3},Q_{4})$ then the action of
$SL(2,{\mathbb{R}})^{3}$ on the charges takes the same form as for the fields
and their duals in (2.24), namely

$\displaystyle\begin{pmatrix}{\bf P}\cr{\bf
Q}\end{pmatrix}\longrightarrow{\cal S}\,\begin{pmatrix}{\bf P}\cr{\bf
Q}\end{pmatrix}\,,$ (2.29)

where again the $4\times 4$ matrices $A$, $B$, $C$ and $D$ that form ${\cal
S}$ are restricted to the $SL(2,{\mathbb{R}})^{3}$ subgroup of
$Sp(8,{\mathbb{R}})$, as given in [9].

There is in fact a simpler way to present the action of the
$SL(2,{\mathbb{R}})^{3}$ global symmetries on the charges, by introducing the
charge tensor $\gamma_{aa^{\prime}a^{\prime\prime}}$, where $a$, $a^{\prime}$
and $a^{\prime\prime}$ are doublet indices of the three $SL(2,{\mathbb{R}})$
factors in the symmetry group (see, for example, [11]). If we make the
assignments

$\displaystyle\gamma_{000}$ $\displaystyle=$
$\displaystyle-P^{1}\,,\qquad\gamma_{011}=P^{2}\,,\qquad\gamma_{101}=P^{3}\,,\qquad\gamma_{110}=P^{4}\,,$
$\displaystyle\gamma_{111}$ $\displaystyle=$ $\displaystyle-
Q_{1}\,,\qquad\gamma_{100}=Q_{2}\,,\qquad\gamma_{010}=Q_{3}\,,\qquad\gamma_{001}=Q_{4}\,,$
(2.30)

then the $SL(2,{\mathbb{R}})^{3}$ transformation of the charges is given by

$\gamma_{aa^{\prime}a^{\prime\prime}}\longrightarrow(S_{1})_{a}{}^{b}\,(S_{2})_{a^{\prime}}{}^{b^{\prime}}\,(S_{3})_{a^{\prime\prime}}{}^{b^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(2.31)

where

$S_{i}=\begin{pmatrix}a_{i}&b_{i}\cr c_{i}&d_{i}\end{pmatrix}\,,\qquad
a_{i}\,d_{i}-b_{i}\,c_{i}=1\quad\hbox{for each}\ i\,.$ (2.32)

This gives exactly the same $SL(2,{\mathbb{R}})^{3}$ transformation as in
appendix A of [9]. It furthermore demonstrates that
$\gamma_{aa^{\prime}a^{\prime\prime}}$ is covariant with respect to
$SL(2,{\mathbb{R}})^{3}$ transformations.

One can, of course, conveniently write the $SL(2,{\mathbb{R}})^{3}$
transformation (2.24) in the analogous way, by defining the field strength
tensor

$\displaystyle\Phi_{000}$ $\displaystyle=$
$\displaystyle-F^{1}\,,\qquad\Phi_{011}=F^{2}\,,\qquad\Phi_{101}=F^{3}\,,\qquad\Phi_{110}=F^{4}\,,$
$\displaystyle\Phi_{111}$ $\displaystyle=$ $\displaystyle-
G_{1}\,,\qquad\Phi_{100}=G_{2}\,,\qquad\Phi_{010}=G_{3}\,,\qquad\Phi_{001}=G_{4}\,,$
(2.33)

which transforms according to

$\Phi_{aa^{\prime}a^{\prime\prime}}\longrightarrow(S_{1})_{a}{}^{b}\,(S_{2})_{a^{\prime}}{}^{b^{\prime}}\,(S_{3})_{a^{\prime\prime}}{}^{b^{\prime\prime}}\,\Phi_{bb^{\prime}b^{\prime\prime}}\,,$
(2.34)

Furthermore, if the dilaton/axion pairs of scalar fields
$(\varphi_{1},\chi_{1})$, $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$ are assembled into the matrices

$M_{i}=\begin{pmatrix}e^{\varphi_{i}}&-\chi_{i}\,e^{\varphi_{i}}\cr-\chi_{i}\,e^{\varphi_{i}}&e^{-\varphi_{i}}+\chi_{i}^{2}\,e^{\varphi_{i}}\end{pmatrix}\,,\qquad\hbox{for}\qquad
i=1,2,3\,,$ (2.35)

whose components are $(M_{1})^{ab}$, $(M_{2})^{a^{\prime}b^{\prime}}$ and
$(M_{3})^{a^{\prime\prime}b^{\prime\prime}}$ respectively, then the scalars
transform under the $SL(2,{\mathbb{R}})$ factors according to

$M_{i}\longrightarrow M_{i}^{\prime}=(S_{i}^{T})^{-1}\,M_{i}\,S_{i}^{-1}\,,$
(2.36)

(with the $i$’th scalars transforming only under the $i$’th
$SL(2,{\mathbb{R}})$ group).

###  2.4 Mapping between the heterotic and the U-duality formulations

As we saw in section 2.2, the scalar fields in the heterotic formulation
comprise the dilaton/axion pairs $(\Phi,\Psi)$, $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$. These come from the Kaluza-Klein reduction from six
dimensions, as described in appendix A, with
$\Phi={\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\varphi_{1}$ and
$\Psi=\chi_{1}$. The pairs $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$ are packaged into the $O(2,2)/U(1)^{2}$ scalar coset
matrix $M$ given by (2.8) and (2.9), while $(\Phi,\Psi)$ parameterise the
$SL(2,{\mathbb{R}})/U(1)$ scalar coset.

In the U-duality formulation the scalars comprise the same set
$(\varphi_{1},\chi_{1})$, $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$, except that now the $(\varphi_{2},\chi_{2})$ pair
are subjected to the involution $\tau_{2}\rightarrow-1/\tau_{2}$, where
$\tau_{2}=\chi_{2}+ie^{-\varphi_{2}}$, as given in (2.21). Thus, in total, the
mapping of fields from those of the heterotic formulation and to those of the
U-duality formulation is as follows:

$\displaystyle\Phi$ $\displaystyle\longrightarrow$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\varphi_{1}\,,\qquad\Psi\longrightarrow\chi_{1}\,,\qquad\varphi_{3}\longrightarrow\varphi_{3}\,,\qquad\chi_{3}\longrightarrow\chi_{3}\,,$
$\displaystyle e^{{{\tilde{\varphi}}}_{2}}$ $\displaystyle\longrightarrow$
$\displaystyle(1+\chi_{2}^{2}\,e^{2\varphi_{2}})\,e^{-\varphi_{2}}\,,\qquad{{\widetilde{\chi}}}_{2}\longrightarrow-\chi_{2}\,e^{2\varphi_{2}}\,(1+\chi_{2}^{2}\,e^{2\varphi_{2}})^{-1}\,.$
(2.37)

The relation between the electric and magnetic charges in the two formulations
follows from the relations between the field strengths, which we discussed
earlier. The dual field strengths ${\cal G}_{i}$ in the heterotic formulation,
given by (2.14), can be seen, after making use of the equations (2.20) to
express the fields $\widetilde{F}_{1}$ and $\widetilde{F}_{2}$ in (2.4) in
terms of the dual fields $F^{1}$ and $F^{2}$, to be related to the fields
$F^{A}$ and dual fields $G_{A}$ of the U-duality formulation by

${\cal G}_{1}=G_{3}\Big{|}_{\tau_{2}\rightarrow-1/\tau_{2}}\,,\qquad{\cal
G}_{2}=-F^{1}\,,\qquad{\cal
G}_{3}=G_{4}\Big{|}_{\tau_{2}\rightarrow-1/\tau_{2}}\,,\qquad{\cal
G}_{4}=-F^{2}\,,$ (2.38)

where in addition to the involution of $(\varphi_{2},\chi_{2})$ indicated
here, we also make the replacements
$\Phi\rightarrow{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\varphi_{1}$ and $\Psi\rightarrow\chi_{1}$, as given in (2.37). Similarly,
one finds that the fields ${{\cal F}}^{i}$ of the heterotic formulation are
related to $F^{A}$ and $G_{A}$ by

${{\cal F}}^{1}=F^{3}\,,\qquad{{\cal
F}}^{2}=G_{1}\Big{|}_{\tau_{2}\rightarrow-1/\tau_{2}}\,,\qquad{{\cal
F}}^{3}=F^{4}\,,\qquad{{\cal
F}}^{4}=G_{2}\Big{|}_{\tau_{2}\rightarrow-1/\tau_{2}}\,.$ (2.39)

From the definitions (2.15) and (2.16) for the electric and magnetic charges
$\alpha_{i}$ and $p^{i}$ in the heterotic formulation, and the definitions
(2.28) for the electric and magnetic charges $Q_{A}$ and $P^{A}$ in the
U-duality formulation we therefore have

${{\vec{\alpha}\,}}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(Q_{3},-P^{1},Q_{4},-P^{2})$
(2.40)

and

$\vec{p}=(p^{1},p^{2},p^{3},p^{4})=(P^{3},Q_{1},P^{4},Q_{2})\,.$ (2.41)

It is also useful to record the mapping from the magnetic charges with lowered
index in the heterotic formulation, $\beta_{i}=L_{ij}\,p^{j}$, for which we
therefore have

${{\vec{\beta}\,}}=(\beta_{1},\beta_{2},\beta_{3},\beta_{4})=(P^{4},Q_{2},P^{3},Q_{1})\,.$
(2.42)

###  2.5 STU supergravity in the $3+1$ formulation

There is a third choice of duality frame for STU supergravity that is useful
for some purposes. In this frame, which we refer to as the “$3+1$ formalism,”
the starting point is again the STU supergravity Lagrangian given in eqns
(2.1) and (2.3). We then dualise just the ${{\widetilde{F}}}_{2}$ field. in
this formalism, therefore, the simple 4-charge BPS solution would carry 3
electric charges and 1 magnetic (or vice versa).

To perform the dualisation of ${{\widetilde{F}}}_{2}$, we add a Lagrange
multiplier term ${\cal L}_{LM}=F^{2}\wedge{{\widetilde{F}}}_{2}$, where
$F^{2}=dA^{2}$ is the dualised field, and then solve the algebraic equation of
motion obtained by varying $F^{2}$. This allows us to solve for the dressed
field ${{\widetilde{{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}={{\widetilde{F}}}_{2}-\chi_{3}\,F^{3}$, finding

${{\widetilde{{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu}}}}_{2}=-{\frac{1}{(1+{{\widetilde{\chi}}}_{2}^{2}\,e^{2{{\tilde{\varphi}}}_{2}})}}\,\Big{[}e^{\varphi_{1}+{{\tilde{\varphi}}}_{2}+\varphi_{3}}\,({*F}^{2}+\chi_{1}\,{*{{\widetilde{F}}}}_{1})+{{\widetilde{\chi}}}_{2}\,e^{2{{\tilde{\varphi}}}_{2}}\,(F^{4}+\chi_{3}\,{{\widetilde{F}}}_{1})\Big{]}\,.$
(2.43)

Substituting back into the original Lagrangian plus the Lagrange multiplier
term, we obtain the bosonic STU supergravity Lagrangian written in the $3+1$
formulation. It is convenient to define the new $(\varphi_{2},\chi_{2})$
scalars related to $({{\tilde{\varphi}}}_{2},{{\widetilde{\chi}}}_{2})$ by the
involution $\tilde{\tau}_{2}=-1/\tau_{2}$, exactly as in (2.37), and the three
“dressed” field strengths:

$\hat{F}^{2}=F^{2}+\chi_{1}\,{{\widetilde{F}}}_{1}\,,\qquad\hat{F}^{3}=F^{3}+\chi_{2}\,{{\widetilde{F}}}_{1}\,,\qquad\hat{F}^{4}=F^{4}+\chi_{3}\,{{\widetilde{F}}}_{1}\,.$
(2.44)

In terms of these, the $3+1$ STU Lagrangian takes the form444A dualisation to
obtain the STU theory in the 3+1 formulation can also be found in [16].

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle R\,{*\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\sum_{i=1}^{3}({*d\varphi_{i}}\wedge
d\varphi_{i}+e^{2\varphi_{i}}\,{*d\chi_{i}}\wedge
d\chi_{i})-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\,{*{{\widetilde{F}}}}_{1}\wedge{{\widetilde{F}}}_{1}$
(2.45) $\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{\varphi_{1}-\varphi_{2}-\varphi_{3}}\,{*\hat{F}}^{2}\wedge\hat{F}^{2}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}e^{-\varphi_{1}+\varphi_{2}-\varphi_{3}}\,{*\hat{F}}^{3}\wedge\hat{F}^{3}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}e^{-\varphi_{1}-\varphi_{2}+\varphi_{3}}\,{*\hat{F}}^{4}\wedge\hat{F}^{4}$
$\displaystyle+\chi_{1}\,F^{3}\wedge F^{4}+\chi_{2}\,F^{2}\wedge
F^{4}+\chi_{3}\,F^{2}\wedge F^{3}$
$\displaystyle+(\chi_{2}\,\chi_{3}\,F^{2}+\chi_{1}\,\chi_{3}\,F^{3}+\chi_{1}\,\chi_{2}\,F^{4})\wedge{{\widetilde{F}}}_{1}+\chi_{1}\,\chi_{2}\,\chi_{3}\,{{\widetilde{F}}}_{1}\wedge{{\widetilde{F}}}_{1}\,.$

Note that ${{\widetilde{F}}}_{1}$, $F^{2}$, $F^{3}$ and $F^{4}$ are the bare
field strengths:

${{\widetilde{F}}}_{1}=d{{\widetilde{A}}}_{1}\,,\qquad F^{2}=dA^{2}\,,\qquad
F^{3}=dA^{3}\,,\qquad F^{4}=dA^{4}\,.$ (2.46)

Note also that the scalar fields here are exactly the same as the ones in the
U-duality formulation. In this $3+1$ formulation there is a permutation
symmetry among the sets of fields

$(\varphi_{1},\chi_{1},F^{2})\,,\qquad(\varphi_{2},\chi_{2},F^{3})\,,\qquad(\varphi_{3},\chi_{3},F^{4})\,.$
(2.47)

We shall return to a discussion of the 3+1 formulation of STU supergravity in
section 6.

##  3 General Extremal BPS Static Black Holes In The U-Duality Formulation

The most general extremal BPS static black hole solution in STU supergravity
will carry 8 independent charges, since each of the four field strengths can
carry an electric charge and a magnetic charge. Since the theory has an
$SL(2,{\mathbb{R}})^{3}$ global symmetry, this can be used in order to map one
solution into another solution that is equivalent under the group action. The
3-parameter compact subgroup $U(1)^{3}$ leaves the asymptotic values of the
six scalar fields unchanged, and this means that it suffices to consider an
$8-3=5$ parameter “seed” solution, with only 5 independent charges, in order
to fill out the full 8-parameter family by acting with $U(1)^{3}$. If ones
starts with a 5-charge seed solution in which the dilatonic and axionic
scalars vanish asymptotically at infinity, then the 8-charge solutions
obtained in this way will all have vanishing asymptotic scalars. One can also
then choose to fill out the solution set further by then acting with the
remaining 6-parameter coset $SL(2,{\mathbb{R}})^{3}/U(1)^{3}$ transformations,
thereby giving arbitrary asymptotic values to the six scalar fields.

In the subsections below, we shall present the general 8-charge static BPS
black hole solutions in the U-duality formulation, both for vanishing values
of the asymptotic scalar fields and for arbitrary values of the asymptotic
scalar fields.

###  3.1 The general 8-charge static BPS metric

The general 8-charge solution for the case of vanishing asymptotic scalars was
constructed in this formulation in [9]. The starting point was the solution
with 5 independent charges that was constructed in [3], after translating it
into the U-duality formulation. The five independent charges could be taken to
be555In [9] the magnetic charges were actually taken to be ${\bf
P}=(0,0,p,-p)$, but we are making the equally valid choice in (3.1) here, for
consistency with the choice (4.1) in the heterotic formulation, given the
mapping (2.41).

${\bf Q}=({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1},{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2},{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{3},{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})\,,\qquad{\bf P}=({{\bar{p}}},-{{\bar{p}}},0,0)\,.$ (3.1)

(We are placing bars on the charges in the 5-charge seed solution.) For this
solution, the metric is given by [3]

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-{\frac{r^{2}}{\sqrt{V}}}\,dt^{2}+\frac{\sqrt{V}}{{r^{2}}}\,(dr^{2}+r^{2}d\Omega^{2})\,,$
(3.2) $\displaystyle V$ $\displaystyle=$ $\displaystyle
r^{4}+\alpha\,r^{3}+\beta\,r^{2}+\gamma\,r+\Delta\,,$ (3.3)

where

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\sum_{i}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{i}\,,\qquad\beta=\sum_{i<j}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{i}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{j}-{{\bar{p}}}^{2}\,,\qquad\gamma=\sum_{i<j<k}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{i}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{j}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{k}-{{\bar{p}}}^{2}\,({{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1}+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})\,,$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\prod_{i}{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{i}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}{{\bar{p}}}^{2}\,({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1}+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})^{2}\,.$ (3.4)

This was filled out to give the general 8-charge solution by acting with the
$U(1)^{3}$ subgroup of the $SL(2,{\mathbb{R}})^{3}$ global symmetry of the STU
supergravity in [9]. It was shown that the constants $\alpha$, $\beta$,
$\gamma$ and $\Delta$ are then given by

$\displaystyle\alpha^{2}$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{A}Q_{A}\Big{)}^{2}+\Big{(}\sum_{A}P^{A}\Big{)}^{2}\,,$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sum_{A<B}(Q_{A}Q_{B}+P^{A}P^{B})\,,$
$\displaystyle\alpha\gamma$ $\displaystyle=$ $\displaystyle
4\Delta+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\beta^{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\sum_{A<B}\Big{(}(P^{A}P^{B})^{2}+(Q_{A}Q_{B})^{2}+P^{AB}\,Q_{AB}\Big{)}-3\prod_{A}P^{A}-3\prod_{A}Q_{A}\,,$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\prod_{A}Q_{A}+\prod_{A}P^{A}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\sum_{A<B}P^{A}Q_{A}P^{B}Q_{B}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}\sum_{i}(P^{A})^{2}\,(Q_{A})^{2}\,,$ (3.5)

where $P^{AB}\equiv P^{A}P^{B}+(\prod_{C}P^{C})/(P^{A}P^{B})$ and
$Q_{AB}\equiv Q_{A}Q_{B}+(\prod_{C}Q_{C})/(Q_{A}Q_{B})$ (so
$P^{12}=P^{1}P^{2}+P^{3}P^{4}$, etc.). Stated equivalently, the expressions
for $\alpha^{2}$, $\beta$, $\alpha\gamma$ and $\Delta$ given in (3.5) are the
unique polynomials of degrees 2, 2, 4 and 4 in the charges, respectively, that
are invariant under $U(1)^{3}$ and that reduce to those following from (3.4)
under the 5-charge specialisation (3.1). Note that $\Delta$ is actually
invariant under the entire $SL(2,{\mathbb{R}})^{3}$ symmetry group; it is the
usual quartic charge invariant of STU supergravity.

The solution described above has 8 independent charges, with the scalar fields
$(\varphi_{i},\chi_{i})$ going to zero at infinity. In order to generalise the
solution to include arbitrary values for the six asymptotic scalars, we need
only act with the symmetry generators in the six-dimensional coset
$SL(2,{\mathbb{R}})^{3}/U(1)^{3}$. This is most easily implemented by using
the charge-tensor formulation discussed at the end of section 2.3. We proceed
in two stages, the first being to re-express the quantities $\alpha$, $\beta$,
$\gamma$ and $\Delta$ in (3.5) in terms of the charge tensor
$\gamma_{aa^{\prime}a^{\prime\prime}}$. To do this, we first introduce the
$SL(2,{\mathbb{R}})$-invariant antisymmetric tensors ${\epsilon}^{ab}$,
${\epsilon}^{a^{\prime}b^{\prime}}$ and
${\epsilon}^{a^{\prime\prime}b^{\prime\prime}}$, with ${\epsilon}^{01}=$, etc.
These obey

$(S_{1})_{a}{}^{c}\,(S_{1})_{b}{}^{d}\,{\epsilon}^{ab}={\epsilon}^{cd}\,,\qquad\hbox{etc.}$
(3.6)

We also note that since $\alpha$, $\beta$ and $\gamma$ are invariant only
under the $U(1)^{3}$ subgroup but not the full $(SL2,{\mathbb{R}})^{3}$, we
may in addition employ the Kronecker deltas $\delta^{ab}$,
$\delta^{a^{\prime}b^{\prime}}$ and
$\delta^{a^{\prime\prime}b^{\prime\prime}}$ in their construction, since
$\delta$ is invariant under the $U(1)$ subgroup of $SL(2,{\mathbb{R}})$. It is
straightforward to see that we may now rewrite $\alpha$, $\beta$, $\gamma$ and
$\Delta$ in (3.5) as follows:

$\displaystyle\alpha^{2}$ $\displaystyle=$
$\displaystyle\delta^{ab}\,\delta^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}+2\beta\,,$
(3.7) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}(\delta^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}+{\epsilon}^{ab}\,\delta^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}+{\epsilon}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}})\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(3.8) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle{\frac{1}{\alpha}}\,\Big{[}\Delta+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\beta^{2}-Y\Big{]}\,,$ (3.9) $\displaystyle\Delta$
$\displaystyle=$ $\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}{\epsilon}^{ac}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,{\epsilon}^{bd}\,{\epsilon}^{c^{\prime}d^{\prime}}\,{\epsilon}^{c^{\prime\prime}d^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,\gamma_{cc^{\prime}c^{\prime\prime}}\,\gamma_{dd^{\prime}d^{\prime\prime}}\,,$
(3.10)

where

$\displaystyle Y$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}\Big{(}\delta^{ac}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\delta^{bd}\,{\epsilon}^{c^{\prime}d^{\prime}}\,{\epsilon}^{c^{\prime\prime}d^{\prime\prime}}+\delta^{a^{\prime}c^{\prime}}\,{\epsilon}^{ab}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\delta^{b^{\prime}d^{\prime}}\,{\epsilon}^{cd}\,{\epsilon}^{c^{\prime\prime}d^{\prime\prime}}$
(3.11)
$\displaystyle\quad+\delta^{a^{\prime\prime}c^{\prime\prime}}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{ab}\,\delta^{b^{\prime\prime}d^{\prime\prime}}\,{\epsilon}^{c^{\prime}d^{\prime}}\,{\epsilon}^{cd}\Big{)}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,\gamma_{cc^{\prime}c^{\prime\prime}}\,\gamma_{dd^{\prime}d^{\prime\prime}}\,.$

Note that, as expected, $\Delta$ in (3.10) is written without the use of the
Kronecker deltas, since it is invariant under the entire
$SL(2,{\mathbb{R}})^{3}$ symmetry.

The final step is to introduce the non-zero asymptotic values for the scalar
fields. As stated above, this could be accomplished by acting on the
expressions (3.7) – (3.10) with the coset symmetries
$SL(2,{\mathbb{R}})^{3}/U(1)^{3}$. In fact a simpler way, with the added
advantage that it will directly express $\alpha$, $\beta$, $\gamma$ and
$\Delta$ in terms of the asymptotic values of the scalars
$(\varphi_{i},\chi_{i})$, is to note that the only thing that will cause the
expressions to change under the action of $SL(2,{\mathbb{R}})^{3}/U(1)^{3}$ is
the fact that the Kronecker deltas are used in the construction of $\alpha$,
$\beta$ and $\gamma$. If we replace all the Kronecker deltas in the
expressions for $\alpha$, $\beta$ and $\gamma$ by the corresponding scalar
matrices (2.35), evaluated at infinity,

$\displaystyle\delta^{ab}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{1}^{ab}\,,\qquad\delta^{a^{\prime}b^{\prime}}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}b^{\prime}}\,,\qquad\delta^{a^{\prime\prime}b^{\prime\prime}}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}b^{\prime\prime}}\,,$ (3.12)

where the bars on the $\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{i}$ indicate that the scalars in (2.35) are evaluated at infinity, then
the expressions for these quantities will be invariant under the full
$SL(2,{\mathbb{R}})^{3}$ symmetry. Thus we see that we shall now have

$\displaystyle\alpha^{2}$ $\displaystyle=$ $\displaystyle\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{1}^{ab}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}b^{\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}+2\beta\,,$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}(\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{1}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}+{\epsilon}^{ab}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}+{\epsilon}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}b^{\prime\prime}})\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle{\frac{1}{\alpha}}\,\Big{[}\Delta+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\beta^{2}-Y\Big{]}\,,$ (3.13)

where now

$\displaystyle Y$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 8}}}\Big{(}\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{1}^{ac}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{1}^{bd}\,{\epsilon}^{c^{\prime}d^{\prime}}\,{\epsilon}^{c^{\prime\prime}d^{\prime\prime}}+\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}c^{\prime}}\,{\epsilon}^{ab}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{b^{\prime}d^{\prime}}\,{\epsilon}^{cd}\,{\epsilon}^{c^{\prime\prime}d^{\prime\prime}}$
(3.14) $\displaystyle\quad+\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}c^{\prime\prime}}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{ab}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{b^{\prime\prime}d^{\prime\prime}}\,{\epsilon}^{c^{\prime}d^{\prime}}\,{\epsilon}^{cd}\Big{)}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,\gamma_{cc^{\prime}c^{\prime\prime}}\,\gamma_{dd^{\prime}d^{\prime\prime}}\,.$

$\Delta$ is unchanged, and is still given by (3.10).

###  3.2 Scalar fields in the U-duality formulation

The dilatons and axions in the 5-charge static black solutions in the
heterotic formulation can be read off from equations (40), (41) and (42) of
[3]666The labelling of the torus coordinates is opposite in [3] to the
labelling we are using in this paper, with our coordinates $(z^{1},z^{2})$ in
appendix A being equal to $(y^{2},y^{1})$ in [3]. This means that the torus
metric and 2-form components $G_{11}$, $G_{22}$, $G_{12}$ and $B_{12}$ in eqns
(42) in [3] should be interpreted as our ${{\mathbb{G}}}_{22}$,
${{\mathbb{G}}}_{11}$, ${{\mathbb{G}}}_{12}$ and $-{{\mathbb{B}}}_{12}$
respectively (see eqns (2.8) and (2.9))., together with eqn (2.9). Mapping to
the charges of U-duality formulation using (2.41) and (2.40), we have

$\displaystyle e^{2\Phi}$ $\displaystyle=$ $\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})}{\sqrt{V}}}\,,\qquad\Psi=-{\frac{{{\bar{p}}}\,({{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1}-{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})}{2(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})}}\,,$ (3.15)
$\displaystyle e^{{{\tilde{\varphi}}}_{2}}$ $\displaystyle=$
$\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}{\sqrt{V}}}\,,\qquad{{\widetilde{\chi}}}_{2}={\frac{{{\bar{p}}}\,[r+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1}+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})]}{(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}}\,,$ (3.16) $\displaystyle e^{\varphi_{3}}$ $\displaystyle=$
$\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}{\sqrt{V}}}\,,\qquad\chi_{3}=-{\frac{{{\bar{p}}}\,[r+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1}+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})]}{(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1})(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}}\,.$ (3.17)

After mapping into the scalar variables of the U-duality formulation, using
eqn (2.37), we therefore have

$\displaystyle e^{\varphi_{1}}$ $\displaystyle=$
$\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})}{\sqrt{V}}}\,,\qquad\chi_{1}=-{\frac{{{\bar{p}}}\,({{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1}-{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})}{2(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{2})}}\,,$ (3.18)
$\displaystyle e^{\varphi_{2}}$ $\displaystyle=$
$\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{3})}{\sqrt{V}}}\,,\qquad\chi_{2}=-{\frac{{{\bar{p}}}\,[r+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1}+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})]}{(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1})(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{3})}}\,,$ (3.19) $\displaystyle e^{\varphi_{3}}$ $\displaystyle=$
$\displaystyle{\frac{(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1})(r+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}{\sqrt{V}}}\,,\qquad\chi_{3}=-{\frac{{{\bar{p}}}\,[r+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}({{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1}+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})]}{(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{1})(r+{{\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{4})}}\,.$ (3.20)

To find the expressions for the scalar fields in the general 8-charge
solution, we follow the same strategy that we used previously for the metric.
Namely, we take the expressions above for the scalar fields in the 5-charge
seed solution, and then fill these out to 8-charge solutions by acting with
the $U(1)^{3}\in SL(2,{\mathbb{R}})^{3}$ global symmetry transformations. A
new feature that arises here is that the scalar fields, unlike the metric,
themselves transform under the global symmetries, and so these transformations
must be included also in the calculation. To be precise, the dilaton/axion
pair $(\varphi_{1},\chi_{1})$ transforms under $SL(2,{\mathbb{R}})_{1}$ but is
inert under $SL(2,{\mathbb{R}})_{2}$ and $SL(2,{\mathbb{R}})_{3}$. Analogous
statements apply to $(\varphi_{2},\chi_{2})$ and to $(\varphi_{3},\chi_{3})$.

As in our construction of the metric functions for the 8-charge solution, we
find it convenient to first obtain the scalar solutions for the case where the
asymptotic values of the dilatons and axions are all zero. A very simple
replacement at the final stage of the calculation allows the introduction of
arbitrary values for the asymptotic scalars.

The process of promoting the 5-charge seed solution to a full 8-charge
solution proceeds in the same we that we described in [9]. Because we have
made a small adjustment in the conventions in this paper, in order to allow a
direct mapping to equivalent the results in the heterotic formulation, we need
first to record the explicit results for the $U(1)^{3}$ rotations that yield
the 8-charge solution. Specifically, it was convenient in this paper to change
the 5-charge configuration in the U-duality formalism from the one specified
in eqn (B.4) of [9] to the one specified in eqn (3.1) of this paper. (That is,
${\bf P}=(p,-p,0,0)$ rather than ${\bf P}=(0,0,p,-p)$.) This modifies the
expressions that were found in [9] when solving for the 5 charges and the
three $U(1)$ angles $\theta_{i}$ in terms of the 8 generic charges $Q_{i}$ and
$P^{i}$. Thus we solve the 8 equations contained in

$\bar{\gamma}_{aa^{\prime}a^{\prime\prime}}=(U_{1})_{a}{}^{b}\,(U_{2})_{a^{\prime}}{}^{b^{\prime}}\,(U_{3})_{a^{\prime\prime}}{}^{b^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(3.21)

where $\bar{\gamma}_{aa^{\prime}a^{\prime\prime}}$ is the charge tensor for
the 5-charge seed configuration in (3.1);
$\gamma_{aa^{\prime}a^{\prime\prime}}$ is the charge tensor of the generic
8-charge configuration, and $U_{i}$ denote the three $U(1)$ matrices with
$\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ as parameters:

$U_{i}=\begin{pmatrix}\cos\theta_{i}&\sin\theta_{i}\cr-\sin\theta_{i}&\cos\theta_{i}\end{pmatrix}\,.$
(3.22)

(See eqn (2.30) for a definition of $\gamma_{aa^{\prime}a^{\prime\prime}}$.)
Eqn (3.21) thus yields expressions for the three angles $\theta_{i}$ and the 5
charges of the seed solution, expressed in terms of the 8 arbitrary charges of
the generic 8-charge solution. We find

$\displaystyle\tan\theta_{+}$ $\displaystyle=$
$\displaystyle{\frac{(P^{1}+P^{2})-(Q_{1}+Q_{2})\tan\theta_{1}}{(Q_{3}+Q_{4})+(P^{3}+P^{4})\tan\theta_{1}}}\,,\qquad\tan\theta_{-}={\frac{(P^{3}-P^{4})+(Q_{3}-Q_{4})\tan\theta_{1}}{(Q_{1}-Q_{2})-(P^{1}-P^{2})\tan\theta_{1}}}\,,$
(3.23) $\displaystyle\tan 2\theta_{1}$ $\displaystyle=$
$\displaystyle{\frac{2(P^{3}+P^{4})(Q_{3}+Q_{4})-2(P^{1}+P^{2})(Q_{1}+Q_{2})}{(P^{1}+P^{2}+P^{3}+P^{4})(P^{1}+P^{2}-P^{3}-P^{4})-(Q_{1}+Q_{2}+Q_{3}+Q_{4})(Q_{1}+Q_{2}-Q_{3}-Q_{4})}}\,,$

where $\theta_{\pm}=\theta_{2}\pm\theta_{3}$, and

$\displaystyle\bar{p}$ $\displaystyle=$
$\displaystyle-\bar{\gamma}_{000}\,,\quad\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu_{1}=-\bar{\gamma}_{111}\,,\quad\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu_{2}=\bar{\gamma}_{100}\,,\quad\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu_{3}=\bar{\gamma}_{010}\,,\quad\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu_{4}=\bar{\gamma}_{001}\,,$ (3.24)

where we have placed bars on the 5 seed charges $\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu_{i}$ and $\bar{p}$.

To begin, let us consider the dilaton/axion pair $(\varphi_{1},\chi_{1})$. The
first step is to promote the right-hand sides of eqns (3.18) to 8-charge
expressions, by using eqns (3.24) together with (3.2). We shall write the
expressions in (3.18) as

$e^{-\varphi_{1}}={\frac{\sqrt{V(r)}}{D(r)}}\,\qquad\chi_{1}={\frac{\delta}{D(r)}}\,,$
(3.25)

where

$\displaystyle D(r)$ $\displaystyle=$ $\displaystyle
r^{2}+\alpha_{D}\,r+\beta_{D}\,,$ (3.26) $\displaystyle\alpha_{D}^{2}$
$\displaystyle=$ $\displaystyle({{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1}+{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2})^{2}\,,\qquad\beta_{D}={{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu}}_{1}\,{{\mkern
1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern
1.5mu}}_{2}\,,\qquad\delta={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\bar{p}\,(\bar{Q}_{1}-\bar{Q}_{2})\,.$ (3.27)

We first promote these 5-charge expressions to 7-charge expressions, by acting
with the $U(1)_{2}$ and $U(1)_{3}$ rotations, while keeping $\theta_{1}=0$.
From (3.2), this latter requirement means the 7-charge restriction implies the
$Q_{A}$ and $P^{A}$ must obey

$(P^{1}+P^{2})(Q_{1}+Q_{2})-(P^{3}+P^{4})(Q_{3}+Q_{4})=0\,.$ (3.28)

This leads straightforwardly to the 7-charge augmentations

$\displaystyle{\alpha_{D}^{2}}_{{\scriptscriptstyle(7)}}$ $\displaystyle=$
$\displaystyle(Q_{1}+Q_{2})^{2}+(P^{3}+P^{4})^{2}\,,\qquad{\beta_{D}}_{{\scriptscriptstyle(7)}}=Q_{1}\,Q_{2}+P^{3}\,P^{4}\,,$
$\displaystyle\delta_{{\scriptscriptstyle(7)}}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}[(P^{1}-P^{2})(Q_{1}-Q_{2})-(P^{3}-P^{4})(Q_{3}-Q_{4})]\,.$ (3.29)

The augmentation from 7 to 8 charges, relaxing the constraint (3.28), is
achieved by further rotating the quantities in (3.29) under the remaining
$U(1)_{1}$ transformation (now with $\theta_{2}$ and $\theta_{3}$ set to
zero). It is helpful at this point to establish a general result for the
$U(1)_{1}$ rotation of a general 2-index symmetric tensor $W_{ab}$; this obeys
the transformation

$\displaystyle
W_{ab}\longrightarrow\widetilde{W}_{ab}=(S_{1})_{a}{}^{c}\,(S_{1})_{b}{}^{d}\,W_{cd}\,,$
(3.30)

where $S_{1}$ is the $U(1)_{1}\in SL(2,{\mathbb{R}})_{1}$ matrix

$S_{1}=\begin{pmatrix}\cos\theta_{1}&\sin\theta_{1}\cr-\sin\theta_{1}&\cos\theta_{1}\end{pmatrix}\,.$
(3.31)

If we define the $U(1)_{2}\times U(1)_{3}$ -invariant symmetric tensor

$Z_{ab}=({\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}-\delta^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}})\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(3.32)

where the charge tensor $\gamma_{aa^{\prime}a^{\prime\prime}}$ is defined in
(2.30), then the expression for $\tan 2\theta_{1}$ in eqn (3.2) can be written
in the compact form

$\tan 2\theta_{1}={\frac{2Z_{01}}{Z_{00}-Z_{11}}}\,.$ (3.33)

From this we find

$\cos 2\theta_{1}={\frac{Z_{00}-Z_{11}}{\Xi}}\,,\qquad\sin
2\theta_{1}={\frac{2Z_{01}}{\Xi}}\,,$ (3.34)

where $\Xi$ is the $U(1)^{3}$-invariant quantity defined by

$\Xi^{2}=Z_{ab}\,Z_{cd}\,(\delta^{ac}\,\delta^{bd}-{\epsilon}^{ac}\,{\epsilon}^{bd})\,.$
(3.35)

It then follows straightforwardly from (3.30) that

$\displaystyle\widetilde{W}_{00}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\Big{[}\delta^{ab}\,W_{ab}+{\frac{G(W)}{\Xi}}\Big{]}\,,\qquad\widetilde{W}_{11}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\Big{[}\delta^{ab}\,W_{ab}-{\frac{G(W)}{\Xi}}\Big{]}\,,$
$\displaystyle\widetilde{W}_{01}$ $\displaystyle=$
$\displaystyle{\frac{1}{\Xi}}\,Z_{ab}\,W_{cd}\,\delta^{ac}\,{\epsilon}^{bd}\,,$
(3.36)

where

$G(W)\equiv
W_{ab}\,Z_{cd}\,(\delta^{ac}\,\delta^{bd}-{\epsilon}^{ac}\,{\epsilon}^{bd})\,.$
(3.37)

(Note from (3.35) that $G(Z)=\Xi^{2}$.)

To proceed with the augmentation of the 7-charge expressions (3.29), we note
that if we define the two $U(1)_{2}\times U(1)_{3}$ -invariant tensors

$X_{ab}={\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,\qquad
Y_{ab}=\delta^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(3.38)

where the charge tensor $\gamma_{aa^{\prime}a^{\prime\prime}}$ is defined in
eqn (2.30), then the 7-charge expressions in (3.29) can be written as

${\alpha_{D}^{2}}_{{\scriptscriptstyle(7)}}=-\widetilde{Z}_{11}\,,\qquad{\beta_{D}}_{{\scriptscriptstyle(7)}}=-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\widetilde{X}_{11}\,,\qquad\delta_{{\scriptscriptstyle(7)}}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}(\widetilde{X}_{01}+\widetilde{Y}_{01})\,.$ (3.39)

Using the $U(1)_{1}$ transformations (3.36),777The transformation (3.36) is
viewed here as the inverse, taking us from the general 8-charge (untilded)
configuration to the 7-charge (tilded) configuration. we therefore obtain the
8-charge expressions

$\displaystyle\alpha_{D}^{2}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}(\delta^{ab}\,Z_{ab}-\Xi)\,,$ $\displaystyle\beta_{D}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}\Big{[}\delta^{ab}\,X_{ab}-{\frac{G(X)}{\Xi}}\Big{]}\,,$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle-{\frac{1}{4\Xi}}\,(X_{ab}+Y_{ab})\,Z_{cd}\,\delta^{ac}\,{\epsilon}^{bd}\,,$
(3.40)

and $G(X)$ is given by substituting $W_{ab}=X_{ab}$ in (3.37). These
expressions are all manifestly invariant under $U(1)^{3}$. It is useful to
note that $\alpha_{D}$ can be written in terms of $\alpha$ (given in (3.7))
and $\Xi$ as

$\alpha_{D}={\frac{\alpha^{2}+\Xi}{2\alpha}}\,.$ (3.41)

It is also worth noting that if we define the $U(1)^{3}$ invariants

$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle-\delta^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
$\displaystyle Z_{2}$ $\displaystyle=$
$\displaystyle-{\epsilon}^{ab}\,\delta^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
$\displaystyle Z_{3}$ $\displaystyle=$
$\displaystyle-{\epsilon}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
$\displaystyle Z_{0}$ $\displaystyle=$
$\displaystyle\delta^{ab}\,\delta^{a^{\prime}b^{\prime}}\,\delta^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,,$
(3.42)

then $\Xi^{2}$, defined in (3.35), can be written in the factorised form

$\Xi^{2}=(Z_{0}+Z_{1}+Z_{2}+Z_{3})(Z_{0}+Z_{1}-Z_{2}-Z_{3})\,.$ (3.43)

It remains to implement the $U(1)^{3}$ transformations that augment the 5
charges to 8 charges on the scalar fields. Since we are looking specifically
at $\varphi_{1}$ and $\chi_{1}$, which transform only under $U(1)_{1}$, this
means that we just have to make the replacement

$\chi_{1}+i\,e^{-\varphi_{1}}\longrightarrow{\frac{(\chi_{1}+i\,e^{-\varphi_{1}})\,\cos\theta_{1}+\sin\theta_{1}}{\cos\theta_{1}-(\chi_{1}+i\,e^{-\varphi_{1}})\,\sin\theta_{1}}}\,.$
(3.44)

In other words, the scalars $\varphi_{1}$ and $\chi_{1}$ will be given by

$e^{\varphi_{1}}={\frac{{\cal
D}\,D(r)}{\sqrt{V(r)}}}\,,\qquad\chi_{1}={\frac{{\cal C}}{{\cal D}}}\,,$
(3.45)

where

$\displaystyle{\cal C}$ $\displaystyle=$
$\displaystyle{\frac{\delta}{D(r)}}\,\cos
2\theta_{1}-{\frac{1}{2}}\Big{[}{\frac{V(r)+\delta^{2}}{D(r)^{2}}}-1\Big{]}\,\sin
2\theta_{1}\,,$ $\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{[}1+{\frac{V(r)+\delta^{2}}{D(r)^{2}}}\Big{]}-{\frac{\delta}{D(r)}}\,\sin
2\theta_{1}+{\frac{1}{2}}\Big{[}1-{\frac{V(r)+\delta^{2}}{D(r)^{2}}}\Big{]}\,\cos
2\theta_{1}\,,$ (3.46)

where $V(r)$ in (3.2) is calculated using the 8-charge expressions
(3.7)–(3.10), the quantities $\delta$, $\alpha_{D}$ and $\beta_{D}$ appearing
in $D(r)$ are given by the 8-charge expressions in (3.40), and $\cos
2\theta_{1}$ and $\sin 2\theta_{1}$ are given by (3.34).

The expressions for $\varphi_{1}$ and $\chi_{1}$ are in fact considerably
simpler than is immediately evident from (3.45) and (3.46). This is because
the quadratic function $D(r)$ is actually a divisor of the quartic function
$V(r)+\delta^{2}$ that appears in (3.46). In fact we can show that

$V(r)+\delta^{2}=D(r)\,\widetilde{D}(r)\,,\qquad\widetilde{D}(r)=r^{2}+\tilde{\alpha}_{D}\,r+\tilde{\beta}_{D}\,,$
(3.47)

where the coefficients in the quadratic function $\widetilde{D}(r)$ are given
by

$\tilde{\alpha}_{D}^{2}=-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}(\delta^{ab}\,Z_{ab}+\Xi)\,,\qquad\tilde{\beta}_{D}=-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}\Big{[}\delta^{ab}\,X_{ab}+{\frac{G(X)}{\Xi}}\Big{]}\,.$
(3.48)

(Note that $\tilde{\alpha}_{D}$ and $\tilde{\beta}_{D}$ are closely related to
the coefficients $\alpha_{D}$ and $\beta_{D}$ in the function
$D(r)=r^{2}+\alpha_{D}\,r+\beta_{D}$ given in (3.40).) Thus we see that
$\varphi_{1}$ and $\chi_{1}$ are given by

$e^{\varphi_{1}}={\frac{r^{2}+b_{1}\,r+b_{0}}{\sqrt{V(r)}}}\,,\qquad\chi_{1}={\frac{c_{1}\,r+c_{0}}{r^{2}+b_{1}\,r+b_{0}}}\,,$
(3.49)

where

$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle{\frac{\alpha_{D}+\tilde{\alpha}_{D}}{2}}+{\frac{(\alpha_{D}-\tilde{\alpha}_{D})(Z_{00}-Z_{11})}{2\Xi}}\,,$
$\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle{\frac{\beta_{D}+\tilde{\beta}_{D}}{2}}-{\frac{2\delta\,Z_{01}}{\Xi}}+{\frac{(\beta_{D}-\tilde{\beta}_{D})(Z_{00}-Z_{11})}{2\Xi}}\,,$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle{\frac{(\alpha_{D}-\tilde{\alpha}_{D})\,Z_{01}}{\Xi}}\,,\qquad
c_{0}={\frac{\delta\,(Z_{00}-Z_{11})+(\beta_{D}-\tilde{\beta}_{D})\,Z_{01}}{\Xi}}\,,$
(3.50)

Using our previous expressions for the various quantities appearing in (3.50),
we find that in terms of the eight charges of the general static BPS black
holes,

$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle{\frac{1}{\alpha}}\,\Big{[}(P^{3}+P^{4})\,\sum_{A}P^{A}+(Q_{1}+Q_{2})\,\sum_{A}Q_{A}\Big{]}\,,$
$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle
P^{3}\,P^{4}+Q_{1}\,Q_{2}\,,$ $\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle{\frac{1}{\alpha}}\,\Big{[}(P^{1}+P^{2})(Q_{1}+Q_{2})-(P^{3}+P^{4})(Q_{3}+Q_{4})\Big{]}\,,$
$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}(P^{1}\,Q_{1}+P^{2}\,Q_{2}-P^{3}\,Q_{3}-P^{4}\,Q_{4})\,,$ (3.51)

where as usual $\alpha=\sqrt{(\sum_{A}P^{A})^{2}+(\sum_{A}Q_{A})^{2}}$.

Finally, it remains to introduce non-vanishing asymptotic values for the
scalar fields. As before when we discussed the metric, this is accomplished by
making $SL(2,{\mathbb{R}})^{3}/U(1)^{3}$ coset transformations on the charges,
and now also on the scalar fields. Thus we introduce an asymptotic coset
vielbein $\mkern 1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{i}$ for each of the $SL(2,{\mathbb{R}})_{i}/U(1)_{i}$ coset factors,
with

$\mkern 1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{i}=\begin{pmatrix}e^{{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\bar{\varphi}_{i}}&-\bar{\chi}_{i}\,e^{{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\bar{\varphi}_{i}}\cr 0&e^{-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\bar{\varphi}_{i}}\end{pmatrix}\,,$ (3.52)

where $(\bar{\varphi}_{i},\bar{\chi}_{i})$ denotes the asymptotic values of
the scalar fields. The corresponding transformation of the charge tensor is
therefore

$\gamma_{aa^{\prime}a^{\prime\prime}}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu^{b}_{1\,a}\,\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu^{b^{\prime}}_{2\,a^{\prime}}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu^{b^{\prime\prime}}_{3\,a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,.$
(3.53)

The effect of this is that all $U(1)^{3}$ invariants, such as $\alpha$,
$\beta$, $\gamma$, $\alpha_{D}$, $\beta_{D}$, $\delta$, $\tilde{\alpha}_{D}$,
$\tilde{\beta}_{D}$ will receive modification, namely that each Kronecker
delta in the expressions (3.7)–(3.9), (3.32), (3.42), (3.50), etc., will be
replaced by the corresponding asymptotic scalar matrix

$\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{i}=\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern 1.5mu_{i}^{T}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern 1.5mu_{i}\,,$ (3.54)

as in eqn (3.12). Additionally, the components of the tensor $Z_{ab}$
appearing in the expressions (3.50) will be transformed by making the
replacement

$Z_{ab}\longrightarrow\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu_{1}^{c}{}_{a}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{1}^{d}{}_{b}\,Z_{cd}\,.$ (3.55)

As already observed, the quantity $\Delta$ receives no modification because
its construction in (3.10) uses only the $SL(2,{\mathbb{R}})$-invariant
epsilon tensors, and no Kronecker deltas.

It is worth recording that even when the asymptotic scalars are non-zero, the
quantity $\Xi$, defined now by (3.35) with all Kronecker deltas replaced by
the asymptotic scalar $M$ matrices as in (3.12), is again simply factorisable,
as

$\Xi=\alpha\,\tilde{\alpha}\,,$ (3.56)

where $\alpha$ is given in (3.13) and $\tilde{\alpha}$ is the closely related
quantity given by

$\displaystyle\tilde{\alpha}^{2}$ $\displaystyle=$ $\displaystyle\Big{(}\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{1}^{ab}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}b^{\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}b^{\prime\prime}}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}-\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{1}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}$
(3.57) $\displaystyle\ \ +{\epsilon}^{ab}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{2}^{a^{\prime}b^{\prime}}\,{\epsilon}^{a^{\prime\prime}b^{\prime\prime}}+{\epsilon}^{ab}\,{\epsilon}^{a^{\prime}b^{\prime}}\,\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{3}^{a^{\prime\prime}b^{\prime\prime}}\Big{)}\,\gamma_{aa^{\prime}a^{\prime\prime}}\,\gamma_{bb^{\prime}b^{\prime\prime}}\,.$

In other words $\Xi^{2}$, which is a quartic multinomial in the charges with
coefficients that are multinomial in the $\bar{\chi}_{i}$ and
$e^{\bar{\varphi}_{i}}$ asymptotic scalar values, factorises as the product of
the two quadratic multinomials $\alpha^{2}$ and $\tilde{\alpha}^{2}$.

It is also useful to note that the quantities $\alpha_{D}$ and
$\tilde{\alpha}_{D}$, given by (3.40) and (3.48) after the replacements
(3.12), are related to $\alpha$ and $\tilde{\alpha}$ by

$\alpha_{D}+\tilde{\alpha}_{D}=\alpha\,,\qquad\alpha_{D}-\tilde{\alpha}_{D}=\tilde{\alpha}\,.$
(3.58)

Thus after introducing the asymptotic scalar values the coefficients $b_{1}$
and $c_{1}$ in eqns (3.50) are particularly simple, and are given by

$b_{1}={\frac{1}{2\alpha}}\,\Big{[}\alpha^{2}+\widetilde{Z}_{00}-\widetilde{Z}_{11}\Big{]}\,,\qquad
c_{1}={\frac{1}{\alpha}}\,\widetilde{Z}_{01}\,,$ (3.59)

where $\widetilde{Z}_{ab}=\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu_{1}^{c}{}_{a}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{1}^{d}{}_{b}\,Z_{cd}$ and all Kronecker deltas involved in the
construction of $\alpha$ and $Z_{ab}$ are as usual replaced by $M$ matrices
according to (3.12).

The action of the coset transformations on the scalar fields themselves will
be given by sending

$M_{i}\longrightarrow\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu_{i}^{T}\,M_{i}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{i}\,,\qquad\hbox{for}\quad i=1,2,3\,,$ (3.60)

where the $M_{i}$ scalar matrices are given in (2.35). These transformations
amount to

$e^{-\varphi_{i}}\longrightarrow
e^{-\varphi_{i}-\bar{\varphi}_{i}}\,,\qquad\chi_{i}\longrightarrow\bar{\chi}_{i}+\chi_{i}\,e^{-\bar{\varphi}_{i}}\,.$
(3.61)

The $U(1)$ rotation angle $\theta_{1}$ will be modified by the coset
transformations, as dictated by substituting the
$\gamma_{aa^{\prime}a^{\prime\prime}}$ transformations (3.53) into the
expression (3.32) for $Z_{ab}$, and then substituting these transformed
$Z_{ab}$ components into (3.33). However, since we have already re-expressed
the expressions in (3.45) and (3.46) for $\varphi_{1}$ and $\chi_{1}$ in the
simpler forms given by (3.49) and (3.50), we no longer need to implement the
coset transformation on $\theta_{1}$ explicitly.

The procedure we have described above provides specifically the expressions
for the dilaton/axion pair $(\varphi_{1},\chi_{1})$ in the general 8-charge
static BPS black hole solutions. One could repeat the discussion, starting
from the same 5-charge seed solution, to augment the expressions (3.19) and
(3.20) for the $(\varphi_{2},\chi_{2})$ and $(\varphi_{3},\chi_{3})$
dilaton/axion pairs. In fact a simpler way of arriving at the same result is
to exploit the triality symmetry of the U-duality formulation of STU
supergravity. The precise statement of this triality can be seen from the
definition of the $SL(2,{\mathbb{R}})^{3}$ charge tensor
$\gamma_{aa^{\prime}a^{\prime\prime}}$ in (2.30). The first
$SL(2,{\mathbb{R}})$ index, $a$, is associated with the
$(\varphi_{1},\chi_{1})$ pair, and the charges $(P^{2},Q_{2})$; the
$a^{\prime}$ index with $(\varphi_{2},\chi_{2})$ and the charges
$(P^{3},Q_{3})$, and the $a^{\prime\prime}$ index with
$(\varphi_{3},\chi_{3})$ and the charges $(P^{4},Q_{4})$. Thus from the
expressions we have obtained for $(\varphi_{1},\chi_{1})$, we just have to
permute the labellings according to this triality correspondence, in order to
obtain the expressions for $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$.

Acting with the ${\mathbb{Z}}_{2}\in\hbox{triality}$ symmetry

$(\varphi_{1},\chi_{1};P^{2},Q_{2})\longleftrightarrow(\varphi_{2},\chi_{2};P^{3},Q_{3})$
(3.62)

on the results obtained above for the the $\varphi_{1}$ and $\chi_{1}$ scalars
will give the expressions for $\varphi_{2}$ and $\chi_{2}$. Acting instead
with the ${\mathbb{Z}}_{2}\in\hbox{triality}$ symmetry

$(\varphi_{1},\chi_{1};P^{2},Q_{2})\longleftrightarrow(\varphi_{3},\chi_{3};P^{4},Q_{4})$
(3.63)

will give the expressions for $\varphi_{3}$ and $\chi_{3}$.

If we specialise for simplicity to the case with vanishing asymptotic values
for the scalar fields, acting with the triality transformations (3.62) or
(3.63) on the constants in (3.51) will map (3.49) into expressions for
$\varphi_{2}$ and $\chi_{2}$, or $\varphi_{3}$ and $\chi_{3}$, respectively.

As a check on the calculations in this section, we can take the expressions of
the $\varphi_{i}$ and $\chi_{i}$ scalars and make the 5-charge specialisation
given by eqn (3.1). Doing this, it is straightforward to see that we do indeed
recover the expressions given in eqns (3.18), (3.19) and (3.20). Expressed
conversely, this shows that if we were instead to act with the U-duality
transformations in order to elevate the 5-charge expressions for
$(\varphi_{2},\chi_{2})$ and $(\varphi_{3},\chi_{3})$ to general 8-charge
expressions, we would indeed obtain the results that follow by making the
triality transformations on the general 8-charge expressions for
$(\varphi_{1},\chi_{1})$ that we have constructed.

###  3.3 Gauge fields for the 8-charge BPS black holes

Having obtained the expressions for the metric and the scalar fields, the form
of the gauge fields in the general 8-charge static BPS black hole solutions
follow straightforwardly from the gauge fields equations of motion.

In the U-duality formulation, we find

$F^{A}=P^{A}\,\sin\theta\,d\theta\wedge
d\varphi+{\frac{1}{\sqrt{V}}}\,\big{(}(f^{R})^{-1}\big{)}^{AB}\,(Q_{B}+f^{I}_{BC}\,P^{C})\,dt\wedge
dr\,,$ (3.64)

which is consistent with the definitions of the electric and magnetic charges
in (2.28).

Note that our conventions for Hodge dualisation are such that in the metric
(3.2) we have

$*(\sin\theta\,d\theta\wedge d\varphi)={\frac{1}{\sqrt{V}}}\,dt\wedge
dr\,,\qquad*(dt\wedge dr)=-\sqrt{V}\,\sin\theta\,d\theta\wedge d\varphi\,.$
(3.65)

##  4 General 8-Charge Static BPS Black Holes In The Heterotic Formulation

###  4.1 8-charge static BPS black hole metric

The starting point for writing the general 8-charge static BPS black holes
with arbitrary asymptotic values for the scalar fields is again the metric
(3.2). The charges in the seed solution with five independent charges and with
vanishing asymptotic scalars take the form

$\vec{\alpha}_{0}=(q_{1},-{\bar{q}},q_{3},{\bar{q}})\,,\qquad\vec{p}_{0}=(0,p^{2},0,p^{4})$
(4.1)

in the heterotic basis [3]. (The superscripts 2 and 4 refer to the indexed
labelling of charges $p^{i}$. See section 2.2 for the notation.) The quantity
${\bar{q}}$ parameterises the introduction of the fifth, independent, charge)
The coefficients $\alpha$, $\beta$, $\gamma$ and $\Delta$ appearing in the
metric function $V(r)$ are given by [3]

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
q_{1}+q_{3}+p^{2}+p^{4}\,,$ $\displaystyle\beta$ $\displaystyle=$
$\displaystyle
q_{1}\,q_{3}-{\bar{q}}^{2}+p^{2}p^{4}+(q_{1}+q_{3})(p^{2}+p^{4})\,,$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle(q_{1}q_{3}-{\bar{q}}^{2})(p^{2}+p^{4})+p^{2}p^{4}\,(q_{1}+q_{3})\,,$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle(q_{1}q_{3})(p^{2}p^{4})-\tfrac{1}{4}{\bar{q}}^{2}\,(p^{2}+p^{4})^{2}\,.$
(4.2)

(Here ${\bar{q}}^{2}$ means just the square of ${\bar{q}}$.)

The general solutions with 8 charges and non-vanishing asymptotic scalars can
be filled out by acting with the global $O(2,2)\times SL(2,{\mathbb{R}})$
symmetry of the heterotic formulation, where $O(2,2)$ is the T-duality
symmetry from the 2-torus and $SL(2,{\mathbb{R}})$ is the electric/magnetic
S-duality symmetry. We proceed in three stages; first, acting with the
$U(1)^{2}$ subgroup of $O(2,2)$ to augment the 5-charge solution to 7 charges;
then acting with the $U(1)$ subgroup of $SL(2,{\mathbb{R}})$ to augment
further to 8 charges; and finally, acting with the remaining $O(2,2)/U(1)^{2}$
and $SL(2,{\mathbb{R}})/U(1)$ cosets in order to introduce the non-vanishing
asymptotic scalars.

Acting with $U(1)^{2}\in O(2,2)$:

The 5-charge starting point can be filled out to 7 independent charges by
acting with the $U(1)^{2}$ compact subgroup of $O(2,2)$. Since $\alpha$,
$\beta$ and $\gamma$ should be invariant under $U(1)^{2}$ but not under the
remaining $O(2,2)/U(1)^{2}$ coset action (that is, when the asymptotic scalars
vanish), the 2 additional charges can be introduced by writing $\alpha$,
$\beta$ and $\gamma$ as $U(1)^{2}$-invariant expressions in $\vec{\alpha}$ and
$\vec{p}$ that reduce to (4.2) under the 5-charge specialisation (4.1). In
fact, rather than using the magnetic charges $\vec{p}$ with components $p^{i}$
we shall instead find it convenient to lower the index and work with the
magnetic charges $\vec{\beta}$, which have the components
$\beta_{i}=L_{ij}\,p^{j}$, as defined in (2.17). The available “building
blocks” are the matrix $L$ defined in (2.11), which is fully $O(2,2)$
invariant, and the $4\times 4$ identity matrix ${{\mathbb{I}}}_{4}$, which is
invariant only under the $U(1)^{2}$ subgroup.888Actually things are a bit more
complicated. There are two further matrices that are invariant under the
$U(1)^{2}$ subgroup, namely $K_{1}$ and $K_{2}$ given by
$K_{1}=\begin{pmatrix}i\sigma_{2}&0\cr 0&i\sigma_{2}\end{pmatrix}\,,\qquad
K_{2}=\begin{pmatrix}0&i\sigma_{2}\cr i\sigma_{2}&0\end{pmatrix}\,,$ (4.3)
where $\sigma_{2}$ is the usual Pauli matrix. Note that $K_{1}$ and $K_{2}$
are antisymmetric. It turns out that by using $K=K_{1}+K_{2}$, one can write
the quantity $\Sigma^{2}$ that appears later in (4.15) as a manifest perfect
square. It then gets dressed up with asymptotic scalars once these are turned
on. See later in this section for a discussion of this point. We can also
construct the more general expression for $\Delta$ along similar lines. Since
$\Delta$ is in fact invariant under the full $O(2,2)$ symmetry its expression
will not involve the use of ${{\mathbb{I}}}_{4}$.

Defining the matrices

$\nu_{\pm}=I_{4}\pm L^{-1}\,,$ (4.4)

one can straightforwardly establish that $\alpha$, $\beta$, $\gamma$ and
$\Delta$ may be written for the 7-charge solutions in $U(1)^{2}$-invariant
terms as follows:

$\displaystyle\alpha^{2}$ $\displaystyle=$
$\displaystyle{{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}+2\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu\,,$ (4.5)
$\displaystyle 2\beta$ $\displaystyle=$
$\displaystyle{{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}}+2\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu\,,$ (4.6)
$\displaystyle 2\alpha\gamma$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})\mkern
1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern 1.5mu$ (4.7)
$\displaystyle+({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})+({{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}})\,,$
$\displaystyle 4\Delta$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})-({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})^{2}\,,$
(4.8)

where

$\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu^{2}=({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})\,.$
(4.9)

These expressions reduce to those in (4.2) under the 5-charge specialisation
(4.1). Note also that in the original 5-charge configuration (4.1), the
charges satisfy the constraint

${{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}=0\,,$ (4.10)

and that this continues to be true after filling out to 7 charges, as we have
done by acting with $U(1)^{2}$.

Acting with $U(1)\in SL(2,{\mathbb{R}})$:

The constraint (4.10) is removed once the 8th and final charge is introduced.
This is achieved by acting with the $U(1)$ subgroup of the remaining
$SL(2,{\mathbb{R}})$ S-duality symmetry. Its action on ${{\vec{\alpha}\,}}$
and ${{\vec{\beta}\,}}$ is

${{\vec{\alpha}\,}}\to{{\vec{\alpha}\,}}\,\cos\psi-{{\vec{\beta}\,}}\,\sin\psi\,,\qquad{{\vec{\beta}\,}}\to{{\vec{\alpha}\,}}\,\sin\psi+{{\vec{\beta}\,}}\,\cos\psi\,,$
(4.11)

where

$\tan
2\psi=-\frac{2({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})}{({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}})-({{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})}\,.$
(4.12)

Thus we have

$\cos
2\psi={\frac{1}{\Xi}}\,({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})\,,\qquad\sin
2\psi=-{\frac{2}{\Xi}}\,{{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}\,,$
(4.13)

where

$\displaystyle\Xi^{2}$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})^{2}-4\Sigma^{2}\,,$
(4.14) $\displaystyle\Sigma^{2}$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})-({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})^{2}\,.$
(4.15)

We now apply the transformation (4.11), with $\cos 2\psi$ and $\sin 2\psi$
given by (4.13), to the expressions (4.5)–(4.8) in order to add in the 8th
charge. It is useful to note that for any symmetric $4\times 4$ matrix $X$ we
shall therefore have

$\displaystyle{{\vec{\alpha}\,}}^{T}X\,{{\vec{\alpha}\,}}$
$\displaystyle\longrightarrow$ $\displaystyle{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}({{\vec{\alpha}\,}}^{T}X\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}X\,{{\vec{\beta}\,}})+{\frac{G(X)}{2\Xi}}\,,$
$\displaystyle{{\vec{\beta}\,}}^{T}X\,{{\vec{\beta}\,}}$
$\displaystyle\longrightarrow$ $\displaystyle{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}({{\vec{\alpha}\,}}^{T}X\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}X\,{{\vec{\beta}\,}})-{\frac{G(X)}{2\Xi}}\,,$
(4.16) $\displaystyle{{\vec{\alpha}\,}}^{T}X\,{{\vec{\beta}\,}}$
$\displaystyle\longrightarrow$
$\displaystyle{\frac{1}{\Xi}}\,\Big{[}({{\vec{\alpha}\,}}^{T}X\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})-({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}X\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}X\,{{\vec{\beta}\,}})\Big{]}\,,$

where

$G(X)=({{\vec{\alpha}\,}}^{T}X\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}X\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})+4({{\vec{\alpha}\,}}^{T}X\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})\,.$
(4.17)

Note that $G(\nu_{+})=\Xi^{2}$.

Applying these results in (4.5)–(4.8), we therefore find that the general
8-charge expressions of $\alpha$, $\beta$, $\gamma$ and $\Delta$ are given by

$\displaystyle\alpha^{2}$ $\displaystyle=$
$\displaystyle{{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}+2\Sigma\,,$
(4.18) $\displaystyle 2\beta$ $\displaystyle=$
$\displaystyle{{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}}+2\Sigma\,,$
(4.19) $\displaystyle 2\alpha\gamma$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})\,\Sigma-2({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})$
(4.20)
$\displaystyle+({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})+({{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}})\,,$
$\displaystyle 4\Delta$ $\displaystyle=$
$\displaystyle({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}})({{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})-({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})^{2}\,.$
(4.21)

We may now observe that these 8-charge expressions may be written in a more
compact notation by introducing the $SL(2,{\mathbb{R}})$-valued 8-charge
vector ${{\vec{v}\,}}^{a}$, where

${{\vec{v}\,}}^{1}={{\vec{\alpha}\,}}\,,\qquad{{\vec{v}\,}}^{2}={{\vec{\beta}\,}}\,.$
(4.22)

We also introduce the $SL(2,{\mathbb{R}})$-invariant antisymmetric tensor
$\varepsilon_{ab}$, with $\varepsilon_{12}=1$, and the Kronecker delta
$\delta_{ab}$, which is invariant only under the $U(1)$ subgroup of
$SL(2,{\mathbb{R}})$. Using these, the quantities $\alpha$, $\beta$, $\gamma$
and $\Delta$ in (4.18)–(4.21) may be rewritten as

$\displaystyle\alpha^{2}$ $\displaystyle=$
$\displaystyle\delta_{ab}\,{{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b}+2\Sigma\,,$
(4.23) $\displaystyle 2\beta$ $\displaystyle=$
$\displaystyle\delta_{ab}\,{{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b}+2\Sigma\,,$
(4.24) $\displaystyle 2\alpha\gamma$ $\displaystyle=$
$\displaystyle\delta_{ab}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})\,\Sigma+\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\nu_{+}\,{{\vec{v}\,}}^{d})\,,$
(4.25) $\displaystyle 8\Delta$ $\displaystyle=$
$\displaystyle\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}L^{-1}\,{{\vec{v}\,}}^{d})\,,$
(4.26)

where

$\Sigma^{2}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\nu_{+}\,{{\vec{v}\,}}^{d})\,.$
(4.27)

Introducing the asymptotic scalars:

This final step is achieved by acting with the cosets $O(2,2)/U(1)^{2}$ and
$SL(2,{\mathbb{R}})/U(1)$, appropriately parameterised in terms of the
asymptotic values of the scalar fields. This can be done by using a vielbein
formulation for both the $O(2,2)/U(1)^{2}$ scalar coset matrix and the
$SL(2,{\mathbb{R}})/U(1)$ scalar coset matrix.

For $O(2,2)/U(1)^{2}$, the scalar matrix $M$ given in (2.8) can be written as

$M={{\cal V}}^{T}\,{{\cal V}}\,,\qquad{{\cal
V}}=\begin{pmatrix}E^{-1}&-E^{-1}\,{{\mathbb{B}}}\cr 0&E^{T}\end{pmatrix}\,,$
(4.28)

where $E$ is the zweibein for the internal 2-torus metric ${{\mathbb{G}}}$.
Note that ${{\cal V}}^{T}L^{-1}\,{{\cal V}}=L^{-1}$. In terms of indices we
have

$M^{ij}=\delta^{k\ell}\,{{\cal V}}_{k}{}^{i}\,{{\cal V}}_{\ell}{}^{j}\,,\qquad
L^{ij}=L^{k\ell}\,{{\cal V}}_{k}{}^{i}\,{{\cal V}}_{\ell}{}^{j}\,.$ (4.29)

We shall then make the $O(2,2)/U(1)^{2}$ transformation $\boldsymbol{{{\cal
F}}}\rightarrow(\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu^{T})^{-1}\,\boldsymbol{{{\cal F}}}$ on the gauge
fields, which implies the transformations

${{\vec{\alpha}\,}}\longrightarrow\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern
1.5mu\,{{\vec{\alpha}\,}}\,,\qquad{{\vec{\beta}\,}}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu\,{{\vec{\beta}\,}}\,,$ (4.30)

on the charges, where $\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu$ denotes the asymptotic value of the scalar
vielbein ${{\cal V}}$. Since ${{\cal V}}^{T}L^{-1}\,{{\cal V}}=L$ and ${{\cal
V}}^{T}\,{{\cal V}}=M$, this means that the only change in the expressions
(4.18)–(4.27) will be that the matrix $I_{4}$ in $\nu_{+}=I_{4}+L^{-1}$ will
change to $\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu$, and
so

$\nu_{+}\longrightarrow\mu_{+}\,,\qquad\mu_{+}=\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu+L^{-1}\,.$ (4.31)

In a similar way, the scalars $\Phi$ and $\Psi$ enter the
$SL(2,{\mathbb{R}})/U(1)$ scalar coset in the form

${\cal N}={\rm e}^{2\Phi}\begin{pmatrix}1&-\Psi\cr-\Psi&\Psi^{2}+{\rm
e}^{-4\Phi}\end{pmatrix}\,,$ (4.32)

and this can be written in terms of a vielbein $U={{\cal U}}$ as

${{\cal N}}={{\cal U}}^{T}\,{{\cal U}}\,,\qquad{{\cal
U}}=\begin{pmatrix}e^{\Phi}&-\Psi\,e^{\Phi}\cr 0&e^{-\Phi}\end{pmatrix}\,.$
(4.33)

The components ${{\cal U}}^{a}{}_{b}$ of this matrix obey the
$SL(2,{\mathbb{R}})$ invariance condition

${{\cal U}}^{c}{}_{a}\,{{\cal
U}}^{d}{}_{b}\,\varepsilon_{cd}=\varepsilon_{ab}\,,$ (4.34)

while, contracted instead with $\delta_{cd}$, we have

${{\cal U}}^{c}{}_{a}\,{{\cal U}}^{d}{}_{b}\,\delta_{cd}={{\cal N}}_{ab}\,,$
(4.35)

where ${{\cal N}}_{ab}$ are the components of the $SL(2,{\mathbb{R}})/U(1)$
scalar matrix (4.32). Thus if we transform the electric and magnetic charges
under the $SL(2,{\mathbb{R}})/U(1)$ transformation

${{\vec{v}\,}}^{a}\longrightarrow\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
U}}\mkern-1.5mu}\mkern 1.5mu^{a}{}_{b}\,{{\vec{v}\,}}^{b}\,,$ (4.36)

where $\mkern 1.5mu\overline{\mkern-1.5mu{{\cal U}}\mkern-1.5mu}\mkern
1.5mu^{a}{}_{b}$ is the scalar vielbein (4.33) with the scalars set equal to
their asymptotic values, then the only change in the expressions (4.23)–(4.26)
and (4.27) will be that the Kronecker delta $\delta_{ab}$ will be replaced by
the corresponding $SL(2,{\mathbb{R}})/U(1)$ scalar matrix $\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern 1.5mu_{ab}$, where
the scalars in ${{\cal N}}_{ab}$ defined in (4.32) are set equal to their
asymptotic values.

Pulling together the threads of the previous discussion, if we act with
$O(2,2)/U(1)^{2}$ and $SL(2,{\mathbb{R}})/U(1)$ coset elements to introduce
asymptotic values for all the scalar fields, the final expressions for the
quantities $\alpha$, $\beta$, $\gamma$ and $\Delta$ given previously in
(4.23)–(4.26) will be given by

$\displaystyle\alpha^{2}$ $\displaystyle=$ $\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern
1.5mu_{ab}\,{{\vec{v}\,}}^{a\,T}\mu_{+}\,{{\vec{v}\,}}^{b}+\Sigma\,,$ (4.37)
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern
1.5mu_{ab}\,{{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\Sigma\,,$ (4.38) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle{\frac{1}{2\alpha}}\,\Big{[}\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern
1.5mu_{ab}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})\,\Sigma+\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\mu_{+}\,{{\vec{v}\,}}^{d})\Big{]}\,,$
(4.39) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}L^{-1}\,{{\vec{v}\,}}^{d})\,,$
(4.40)

where

$\Sigma^{2}={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\varepsilon_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}\mu_{+}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\mu_{+}\,{{\vec{v}\,}}^{d})\,.$
(4.41)

As was first observed in [3], the quartic invariant $\Delta$ does not depend
on the values of the asymptotic scalars.

Having obtained these general expressions in the heterotic formulation for the
coefficients $\alpha$, $\beta$, $\gamma$ and $\Delta$ for static BPS black
holes with eight independent charges and arbitrary asymptotic values for the
six scalar fields, it is instructive to compare them with the analogous
expressions in the U-duality formulation, which we obtained in (3.13) and
(3.10). After making use of the mappings (2.42) and (2.40) between the charges
${{\vec{\alpha}\,}}$ and ${{\vec{\beta}\,}}$ of the heterotic formulation and
the charges $(P^{i},Q_{i})$ of the U-duality formulation, and also the mapping
(2.37) between the scalar fields in the two formulations, it is
straightforward to verify that the two sets of expressions for $\alpha$,
$\beta$, $\gamma$ and $\Delta$ agree.

There is, however, one respect in which the two sets of expressions for the
constants $\alpha$, $\beta$ and $\gamma$ ostensibly differ in the two
formulations. In the U-duality formulation, the
$SL(2,{\mathbb{R}})^{3}$-invariant expressions in (3.13) have the feature that
$\alpha^{2}$, $\beta$ and $\alpha\gamma$ are manifestly polynomial in the
eight charges. By contrast, in the heterotic formulation the corresponding
$O(2,2)\times SL(2,{\mathbb{R}})$-invariant expressions (4.37)–(4.39) for
$\alpha^{2}$, $\beta$ and $\alpha\gamma$ are not manifestly polynomial in the
eight charges, because they are written using $\Sigma$, which is defined via
the expression (4.41) for $\Sigma^{2}$. In fact one finds that $\Sigma^{2}$
defined in (4.41) turns out to be a perfect square when one evaluates it, and
so $\Sigma$ is indeed a quadratic polynomial in the charges. However, one
cannot write $\Sigma$ itself as a manifestly $O(2,2)\times
SL(2,{\mathbb{R}})$-invariant quadratic polynomial in the charges and
asymptotic scalars. The explanation for this phenomenon turns out to be
related to the observation made earlier, in footnote 8, namely that there
exists a three-dimensional vector space of matrices that are invariant under
the $U(1)\times U(1)$ subgroup of $O(2,2)$ while being non-invariant under the
full $O(2,2)$ group.

As we proceeded through the steps described above, we first promoted the
5-charge expressions for $\alpha$, $\beta$, $\gamma$ and $\Delta$ to 7-charge
expressions. At that stage, we could have used the $U(1)^{2}$-invariant matrix
(4.3)

$K=K_{1}+K_{2}=\begin{pmatrix}0&1&0&1\cr-1&0&-1&0\cr
0&1&0&1\cr-1&0&-1&0\end{pmatrix}$ (4.42)

in order to write $\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu$ in (4.9) directly, unsquared, as

$\mkern 1.5mu\overline{\mkern-1.5mu\Sigma\mkern-1.5mu}\mkern
1.5mu=-{{\vec{\alpha}\,}}^{T}K\,{{\vec{\beta}\,}}\,,$ (4.43)

as may readily be verified. This 7-charge expression can then be augmented to
an 8-charge expression by making the replacement (4.11), with $\psi$ given by
(4.12). Because the matrix $K$ is antisymmetric, this replacement leaves
(4.43) unchanged, and so in the 8-charge expressions for $\alpha$, $\beta$,
$\gamma$ and $\Delta$ in (4.18)–(4.21), the expression for $\Sigma$ following
from (4.15) can instead be replaced directly by the unsquared expression

$\Sigma=-{{\vec{\alpha}\,}}^{T}K\,{{\vec{\beta}\,}}\,.$ (4.44)

Finally, when the asymptotic scalars are introduced by acting with the
$O(2,2)/U(1)^{2}$ and $SL(2,{\mathbb{R}})/U(1)$ coset matrices (4.28) and
(4.33) as we did previously, the expression (4.44) for $\Sigma$ will become

$\Sigma={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\epsilon_{ab}\,{{\vec{v}\,}}^{a\,T}\mkern
1.5mu\overline{\mkern-1.5muP\mkern-1.5mu}\mkern 1.5mu\,{{\vec{v}\,}}^{b}\,,$
(4.45)

where

$\mkern 1.5mu\overline{\mkern-1.5muP\mkern-1.5mu}\mkern 1.5mu=\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern 1.5mu^{T}K\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern 1.5mu\,.$ (4.46)

Thus we have

$\Sigma=-K^{k\ell}\,\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu_{k}{}^{i}\,\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{\ell}{}^{j}\,\alpha_{i}\,\beta_{j}=-(\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{1}{}^{i}+\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern 1.5mu_{3}{}^{i})(\mkern
1.5mu\overline{\mkern-1.5mu{{\cal V}}\mkern-1.5mu}\mkern
1.5mu_{2}{}^{j}+\mkern 1.5mu\overline{\mkern-1.5mu{{\cal
V}}\mkern-1.5mu}\mkern
1.5mu_{4}{}^{j})(\alpha_{i}\,\beta_{j}-\alpha_{j}\,\beta_{i})\,.$ (4.47)

###  4.2 Scalar fields in the heterotic formulation

The derivation of the expressions for the scalar fields in the heterotic
formulation proceeds in an analogous fashion. Here, we shall just present the
results for the $(\Phi,\Psi)$ dilaton/axion pair, associated with the
S-duality in the heterotic formalism. They are given in the 5-charge special
case (4.1) by

$e^{-2\Phi}=\frac{V(r)^{\frac{1}{2}}}{D(r)}\,,\qquad\Psi=\frac{\delta}{D(r)}\,,$
(4.48)

where

$D(r)=r^{2}+\alpha_{D}\,r+\beta_{D}\,,$ (4.49)

with

$\alpha_{D}=p^{2}+p^{4}\,,\qquad\beta_{D}=p^{2}\,p^{4}\,,\qquad\delta={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}q\,(p^{2}-p^{4})\,.$ (4.50)

As in the corresponding derivation in the U-duality formulation, we shall
assume the asymptotic values of the scalar fields are all zero until the final
stage in the calculation.

The augmentation to a 7-charge solution is accomplished by acting with the
$O(2)\times O(2)$ subgroup of the $O(2,2)$ T-duality, and this results in
expressions for $\alpha_{D}^{2}$, $\beta_{D}$ and $\delta$ that are
$O(2)\times O(2)$ invariant:

$\displaystyle{\alpha_{D}^{2}}_{{\scriptscriptstyle(7)}}$ $\displaystyle=$
$\displaystyle(p^{1}+p^{3})^{2}+(p^{2}+p^{4})^{2}={{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}\,,$
$\displaystyle{\beta_{D}}_{{\scriptscriptstyle(7)}}$ $\displaystyle=$
$\displaystyle p^{1}\,p^{3}+p^{2}\,p^{4}={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}}\,,$
$\displaystyle\delta_{{\scriptscriptstyle(7)}}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}[(p^{1}-p^{3})(q_{1}-q_{3})+(p^{2}-p^{4})(q_{2}-q_{4})]={\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}{{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}}\,.$ (4.51)

These quantities are further augmented to 8-charge expressions by acting with
the remaining $U(1)\in SL(2,{\mathbb{R}})$ symmetry, with angle $\psi$ given
by (4.12). Thus the right-hand-most expressions in (4.51) are replaced
according to the rules (4.16), leading to the 8-charge expressions

$\displaystyle\alpha_{D}^{2}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{(}{{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}}-\Xi\Big{)}\,,$
$\displaystyle\beta_{D}$ $\displaystyle=$
$\displaystyle{\frac{1}{4}}\Big{(}{{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}+{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}}-{\frac{\sigma}{\Xi}}\Big{)}\,,$
(4.52) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle{\frac{1}{2\Xi}}\Big{[}({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})-({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})\Big{]}\,,$

where $\Xi$ is given by (4.14) and (4.15), and

$\sigma=G(L^{-1})=({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\alpha}\,}}-{{\vec{\beta}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})+4({{\vec{\alpha}\,}}^{T}L^{-1}\,{{\vec{\beta}\,}})({{\vec{\alpha}\,}}^{T}\nu_{+}\,{{\vec{\beta}\,}})\,.$
(4.53)

These expressions can be written more compactly in terms of the 8-component
charge vector ${{\vec{v}\,}}^{a}$ defined in (4.22), giving

$\displaystyle\alpha_{D}^{2}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{(}\delta_{ab}\,{{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b}-\Xi\Big{)}\,,$
$\displaystyle\beta_{D}$ $\displaystyle=$
$\displaystyle{\frac{1}{4}}\Big{(}\delta_{ab}\,{{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b}-{\frac{\sigma}{\Xi}}\Big{)}\,,$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle-{\frac{1}{2\Xi}}\,\delta_{ac}\,\varepsilon_{bd}\,({{\vec{v}\,}}^{a\,T}L^{-1}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\nu_{+}\,{{\vec{v}\,}}^{d})\,,$
(4.54)

with $\Xi$ and $\sigma$ given by

$\displaystyle\Xi^{2}$ $\displaystyle=$
$\displaystyle(\delta_{ab}\,{{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b})^{2}-4\Sigma^{2}\,,$
$\displaystyle=$
$\displaystyle(\delta_{ab}\,\delta_{cd}-2\varepsilon_{ac}\,\varepsilon_{bd})\,({{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}\nu_{+}\,{{\vec{v}\,}}^{d})\,,$
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle(\delta_{ab}\,\delta_{cd}-2\varepsilon_{ac}\,\varepsilon_{bd})({{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b})({{\vec{v}\,}}^{c\,T}L^{-1}\,{{\vec{v}\,}}^{d})\,.$
(4.55)

Thus $\alpha_{D}^{2}$, $\beta_{D}$ and $\delta$ are all written for the
general 8-charge configuration, in forms that are manifestly invariant under
the $U(1)^{3}$ subgroup of the full $O(2,2)\times SL(2,{\mathbb{R}})$ duality
group.

As in the analogous earlier discussion in the U-duality formulation, here the
dilaton/axion pair $(\Phi,\Psi)$ transforms under the $U(1)$ subgroup of the
$SL(2,{\mathbb{R}})$ S-duality group, and so to obtain the expressions for
$\Phi$ and $\Psi$ after the augmentation from the 5-charge seed solution to
the general 8-charge solution, we should transform these too, according to
(2.13) with $a=d=\cos\psi$ and $b=-c=\sin\psi$, where the $U(1)$ angle is
given by (4.12). Thus, analogously to (3.45) the dilaton and axion will then
be given by

$e^{-2\Phi}={\frac{\sqrt{V(r)}}{{\cal D}\,D(r)}}\,,\qquad\Psi={\frac{{\cal
C}}{{\cal D}}}\,,$ (4.56)

where

$\displaystyle{\cal C}$ $\displaystyle=$
$\displaystyle{\frac{\delta}{D(r)}}\,\cos
2\psi-{\frac{1}{2}}\Big{[}{\frac{V(r)+\delta^{2}}{D(r)^{2}}}-1\Big{]}\,\sin
2\psi\,,$ $\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{[}1+{\frac{V(r)+\delta^{2}}{D(r)^{2}}}\Big{]}-{\frac{\delta}{D(r)}}\,\sin
2\psi+{\frac{1}{2}}\Big{[}1-{\frac{V(r)+\delta^{2}}{D(r)^{2}}}\Big{]}\,\cos
2\psi\,.$ (4.57)

The function $V(r)$ is given by the expression in (3.2) with the coefficients
$\alpha$, $\beta$, $\gamma$ and $\Delta$ given by (4.23)–(4.26); the function
$D(r)$ is given by (3.26) with the coefficients given by (4.54); the
coefficient $\delta$ is given also in (4.54); and the expressions for $\cos
2\psi$ and $\sin 2\psi$ are given, as in (4.13), by

$\cos
2\psi={\frac{1}{\Xi}}\,(\delta_{a1}\,\delta_{b1}-\delta_{a2}\,\delta_{b2})\,{{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b}\,,\qquad\sin
2\psi={\frac{2}{\Xi}}\,\delta_{a1}\,\delta_{b2}\,{{\vec{v}\,}}^{a\,T}\nu_{+}\,{{\vec{v}\,}}^{b}\,.$
(4.58)

(Of course these last expressions should not be invariant under the $U(1)$
subgroup of the $SL(2,{\mathbb{R}})$ S-duality, since $\psi$ is the $U(1)$
angle of the final rotation that introduced the 8th charge.)

Finally, the process of turning on non-vanishing asymptotic values for the
scalar fields can be accomplished by means of transformations under the
$[O(2,2)\times SL(2,{\mathbb{R}})]/U(1)^{3}$ coset, as in section 4.1. This
means that in all the expressions above one makes the replacements

$\nu_{+}\longrightarrow\mu_{+}\,,\qquad\delta_{ab}\longrightarrow\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern 1.5mu_{ab}\,,$ (4.59)

where $\mu_{+}$ is defined in eqn (4.31) and $\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern 1.5mu_{ab}$ is
obtained by setting $\Phi$ and $\Psi$ to their asymptotic values $\mkern
1.5mu\overline{\mkern-1.5mu\Phi\mkern-1.5mu}\mkern 1.5mu$ and $\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu$ in (4.32). Note that
$\delta_{a1}$ and $\delta_{a2}$ in (4.58) will be replaced by $\mkern
1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern 1.5mu_{a1}$ and
$\mkern 1.5mu\overline{\mkern-1.5mu{{\cal N}}\mkern-1.5mu}\mkern 1.5mu_{a2}$
also. Finally, the dilaton/axion pair $(\Phi,\Psi)$ will transform under the
$SL(2,{\mathbb{R}})/U(1)$ coset also, undergoing the replacements

$e^{-2\Phi}\longrightarrow e^{-2\Phi-2\mkern
1.5mu\overline{\mkern-1.5mu\Phi\mkern-1.5mu}\mkern
1.5mu}\,,\qquad\Psi\longrightarrow\mkern
1.5mu\overline{\mkern-1.5mu\Psi\mkern-1.5mu}\mkern 1.5mu+\Psi\,e^{-2\mkern
1.5mu\overline{\mkern-1.5mu\Phi\mkern-1.5mu}\mkern 1.5mu}\,.$ (4.60)

###  4.3 Gauge fields for the 8-charge BPS black holes

Having obtained the expressions for the metric and the scalar fields, the form
of the gauge fields in the general 8-charge static BPS black hole solutions
follow straightforwardly from the gauge fields equations of motion.

In the heterotic formulation we find

${{\cal F}}^{i}=p^{i}\,\sin\theta\,d\theta\wedge
d\varphi+{\frac{e^{2\Phi}}{\sqrt{V}}}\,M^{ij}\,(\alpha_{j}-\Psi\,\beta_{j})\,dt\wedge
dr\,.$ (4.61)

As may be verified, these expressions are consistent with the definitions of
the electric and magnetic charges in eqns (2.15) and (2.16).

##  5 Truncation to Pairwise-Equal Charges

The STU supergravity theory admits a consistent truncation in which the four
gauge fields are set equal in pairs, and at the same time two of the
dilaton/axion pairs are set to zero. Because of the triality symmetry in the
U-duality formulation, the three possible ways of equating pairs of gauge
fields are equivalent. We shall choose the pairing

$A^{1}=A^{2}\,,\qquad A^{3}=A^{4}$ (5.1)

in the theory described in section 2.3. It is straightforward to see that this
truncation is consistent provided that at the same time one sets

$\varphi_{2}=\varphi_{3}=0\,,\qquad\chi_{2}=\chi_{3}=0\,.$ (5.2)

The entire bosonic Lagrangian then reduces to

$\displaystyle{\cal L}_{U}$ $\displaystyle=$ $\displaystyle R\,{*\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}{*d\varphi_{1}}\wedge
d\varphi_{1}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{2\varphi_{1}}\,{*d\chi_{1}}\wedge d\chi_{1}$
$\displaystyle-{\frac{e^{2\varphi_{1}}}{1+\chi_{1}^{2}\,e^{2\varphi_{1}}}}\,\Big{[}e^{-\varphi_{1}}\,{*F^{1}}\wedge
F^{1}+\chi_{1}\,F^{1}\wedge F^{1}\Big{]}-e^{-\varphi_{1}}\,{*F^{3}}\wedge
F^{3}+\chi_{1}\,F^{3}\wedge F^{3}\,.$

Note that this Lagrangian is invariant under the discrete symmetry

$A^{1}\longrightarrow A^{3}\,,\qquad A^{3}\longrightarrow
A^{1}\,,\qquad\tau_{1}\longrightarrow-{\frac{1}{\tau_{1}}}\,,$ (5.4)

where as usual $\tau_{1}=\chi_{1}+{{\rm i\,}}e^{-\varphi_{1}}$, that is,

$e^{-\varphi_{1}}\longleftrightarrow{\frac{e^{\varphi_{1}}}{(1+\chi_{1}^{2}\,e^{2\varphi_{1}})}}\,,\qquad\chi_{1}\longleftrightarrow-{\frac{\chi_{1}\,e^{2\varphi_{1}}}{(1+\chi_{1}^{2}\,e^{2\varphi_{1}})}}\,.$
(5.5)

In the heterotic formulation, as can be seen from (2.38) and (2.39), the
corresponding truncation amounts to setting

${{\cal A}}^{1}={{\cal A}}^{3}\,,\qquad{{\cal A}}^{2}={{\cal A}}^{4}\,,$ (5.6)

together, again, with

$\varphi_{2}=\varphi_{3}=0\,,\qquad\chi_{2}=\chi_{3}=0\,.$ (5.7)

The bosonic Lagrangian in the heterotic formulation, described in section 2.2,
therefore reduces to

$\displaystyle{\cal L}_{H}$ $\displaystyle=$ $\displaystyle R\,{*\hbox
to0.0pt{1\hss}\mkern 4.0mu{\rm l}}-2{*d\Phi}\wedge
d\Phi-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}e^{4\Phi}\,{*d\Psi}\wedge d\Psi$ (5.8)
$\displaystyle-e^{-2\Phi}\,({*{{\cal F}}^{1}}\wedge{{\cal F}}^{1}+{*{{\cal
F}}^{2}}\wedge{{\cal F}}^{2})+\Psi\,({{\cal F}}^{1}\wedge{{\cal F}}^{1}+{{\cal
F}}^{2}\wedge{{\cal F}}^{2})\,.$

The general static BPS black hole solutions in the truncated theory are
obtained by setting the charges associated with the corresponding pairs of
equated fields to be equal also. Thus in the U-duality formulation we set

$P^{1}=P^{2}\,,\qquad Q_{1}=Q_{2}\,,\qquad P^{3}=P^{4}\,,\qquad
Q_{3}=Q_{3}\,.$ (5.9)

In the heterotic formulation, this corresponds to setting

$p^{1}=p^{3}\,,\qquad q_{1}=q_{3}\,,\qquad p^{2}=p^{4}\,,\qquad
q_{2}=q_{4}\,.$ (5.10)

Not only is the STU theory greatly simplified in the pairwise-equal truncation
of the fields, as we saw above, but also the form of the black hole solutions
becomes considerably simpler. In particular, the quartic metric function
$V(r)$ (3.2) now becomes a perfect square, The metric function $V$ in eqn
(3.2) also becomes a perfect square:

$V(r)=\Big{(}r^{2}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\alpha\,r+\Delta^{1/2}\Big{)}^{2}\,,$ (5.11)

with $\alpha$ and $\Delta$ now given by

$\alpha=2\sqrt{(P^{1}+P^{3})^{2}+(Q_{1}+Q_{3})^{2}}\,,\qquad\Delta=(P^{1}\,P^{3}+Q_{1}\,Q_{3})^{2}\,.$
(5.12)

(For simplicity, we first consider the case where the asymptotic values of the
scalar fields are taken to be zero here.)

The scalar fields $(\varphi_{1},\chi_{1})$ themselves can be read off from
eqns (3.49) and (3.50), together with the triality-related expressions for
$(\varphi_{2},\chi_{2})$ and $(\varphi_{3},\chi_{3})$ as detailed in (3.62)
and (3.63), after making the pairwise-equal specialisation (5.9). These latter
expressions reproduce the vanishing of the $(\varphi_{2},\chi_{2})$ and
$(\varphi_{3},\chi_{3})$ as in (5.7), and the former give (3.49)

$\displaystyle e^{\varphi_{1}}$ $\displaystyle=$
$\displaystyle{\frac{r^{2}+b_{1}\,r+b_{0}}{r^{2}+d_{1}\,r+d_{0}}}\,,\qquad\chi_{1}={\frac{c_{1}\,r+c_{0}}{r^{2}+b_{1}\,r+b_{0}}}\,,$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle{\frac{4}{\alpha}}\,\Big{[}P^{3}\,(P^{1}+P^{3})+Q_{1}\,(Q_{1}+Q_{3})\Big{]}\,,\qquad
b_{0}=(P^{3})^{2}+(Q_{1})^{2}\,,$ $\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle{\frac{4}{\alpha}}\,(P^{1}\,Q_{1}-P^{3}\,Q_{3})\,,\qquad
c_{0}=P^{1}\,Q_{1}-P^{3}\,Q_{3}\,,$ $\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}\alpha\,,\qquad d_{0}=P^{1}\,P^{3}+Q_{1}\,Q_{3}\,.$ (5.13)

It is worth noting that the asymmetry of the expressions for $\varphi_{1}$ and
$\chi_{1}$ with respect to exchanging $(P^{1},Q_{1})$ and $(P^{3},Q_{3})$,
because of the asymmetry of $b_{1}$ and $b_{0}$ given in (5.13), is in fact
precisely consistent with the exchange symmetry of the pairwise-equal
truncation of the STU theory. This symmetry is given in (5.4). One may verify
that the scalar fields $\varphi_{1}$ and $\chi_{1}$ given in (5.13) have the
property that

${\frac{e^{\varphi_{1}}}{(1+\chi_{1}^{2}\,e^{2\varphi_{1}})}}={\frac{r^{2}+d_{1}\,r+d_{0}}{r^{2}+e_{1}\,r+e_{0}}}\,,\qquad{\frac{\chi_{1}\,e^{2\varphi_{1}}}{(1+\chi_{1}^{2}\,e^{2\varphi_{1}})}}=-{\frac{c_{1}\,r+c_{0}}{r^{2}+e_{1}\,r+e_{0}}}\,,$
(5.14)

where

$e_{1}={\frac{4}{\alpha}}\,\Big{[}P^{1}\,(P^{1}+P^{3})+Q_{3}\,(Q_{1}+Q_{3})\Big{]}\,,\qquad
e_{0}=(P^{1})^{2}+(Q_{3})^{2}\,,$ (5.15)

and so indeed the pairwise-equal solution is compatible with the exchange
symmetry given by (5.5).

If the asymptotic values of the scalar fields are taken to be non-vanishing,
$\varphi_{1}\rightarrow\bar{\varphi}_{1}$ and
$\chi_{1}\rightarrow\bar{\chi}_{1}$, it is straightforward to check using the
results in section 3.2 that the constants $b_{1}$, $b_{0}$, $c_{1}$ and
$c_{0}$ in (5.13) are replaced by

$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle{\frac{4}{\alpha}}\,\Big{\\{}e^{-\bar{\varphi}_{1}}\,[(P^{3})^{2}+Q_{1}^{2}]+P^{1}\,P^{3}+Q_{1}\,Q_{3}+\bar{\chi}_{1}\,(P^{1}\,Q_{1}-P^{3}\,Q_{3})$
$\displaystyle\qquad+\bar{\chi}_{1}\,e^{\bar{\varphi}_{1}}\,\Big{[}P^{1}\,Q_{1}-P^{3}\,Q_{3}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\bar{\chi}_{1}\,[(P^{1})^{2}+(P^{3})^{2}+Q_{1}^{2}+Q_{3}^{2}]\Big{]}\Big{\\}}\,,$
$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle
e^{-\bar{\varphi}_{1}}\,[(P^{3})^{2}+Q_{1}^{2}]+\bar{\chi}_{1}\,(P^{1}\,Q_{1}-P^{3}\,Q_{3})$
$\displaystyle\qquad+\bar{\chi}_{1}\,e^{\bar{\varphi}_{1}}\,\Big{[}P^{1}\,Q_{1}-P^{3}\,Q_{3}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}\bar{\chi}_{1}\,[(P^{1})^{2}+(P^{3})^{2}+Q_{1}^{2}+Q_{3}^{2}]\Big{]}\,,$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle{\frac{4}{\alpha}}\,\Big{[}P^{1}\,Q_{1}-P^{3}\,Q_{3}+\bar{\chi}_{1}\,e^{\bar{\varphi}_{1}}\,[(P^{1})^{2}+Q_{3}^{2}]\Big{]}\,,$
$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle
P^{1}\,Q^{1}-P^{3}\,Q_{3}+\bar{\chi}_{1}\,e^{\bar{\varphi}_{1}}\,[(P^{1})^{2}+Q_{3}^{2}]\,.$
(5.16)

Note that $b_{1}={\frac{4}{\alpha}}\,(b_{0}+P^{1}\,P^{3}+Q_{1}\,Q_{3})$, and
$c_{1}={\frac{4}{\alpha}}\,c_{0}$. The quantity $\alpha$ is now given by

$\displaystyle\alpha^{2}$ $\displaystyle=$ $\displaystyle
4e^{\bar{\varphi}_{1}}\,\Big{[}(P^{1}+\bar{\chi}_{1}\,Q_{1})^{2}+(Q_{3}-\bar{\chi}_{1}\,P^{3})^{2}\Big{]}+8(P^{1}\,P^{3}+Q_{1}\,Q_{3})$
(5.17) $\displaystyle+4e^{-\bar{\varphi}_{1}}\,[(P^{3})^{2}+Q_{1}^{2}]\,.$

Finally, the scalar fields $\varphi_{1}$ and $\chi_{1}$ themselves should be
transformed, as in eqn (3.61).

##  6 Symmetry Transformations and Asymptotic Scalars

In this section, we shall address a conformal transformation for BPS black
holes, studied in [10], in the context of STU theory. In order to study the
conformal transformation, we need to go outside the approach of the rest of
this paper, and adopt the pre-potential formalism studied in [17, 18]. In
[18], the authors discuss various stationary solutions of $D=4,\,{\cal N}=2$
supergravity, among which the one with pre-potential

$F(X^{I})=-{\frac{X^{1}X^{2}X^{3}}{X^{0}}}$ (6.1)

is of interest to us. The resulting theory, as we shall see later, corresponds
to STU supergravity in the 3+1 formulation that we discussed in section 2.5.

Let us consider a Kähler manifold with Kähler potential given by

$K=-\log[\,{{\rm i\,}}({\bar{X}^{I}}W_{I}-X^{I}{\bar{W}_{I}})]$ (6.2)

where $X^{I}$ and $W_{I}$999$W_{I}$ is denoted as $F_{I}$ in most of the
literature dealing with supergravity and special geometry, however we choose
not to use the notation $F_{I}$ to avoid confusion with field strength. are
related to the “holomorphic section” of the underlying Kähler manifold via the
usual definition

$\begin{pmatrix}X^{I}\\\
W_{I}\end{pmatrix}=e^{-{\frac{K}{2}}}\begin{pmatrix}L^{I}\\\
M_{I}\end{pmatrix}$ (6.3)

where $\begin{pmatrix}L^{I}\\\ M_{I}\end{pmatrix}$ constitute the “holomorphic
section”. The four quantities, $W_{I}$, are evaluated from the pre-potential
mentioned in (6.1) using the definition:
$W_{I}={\frac{{\partial}F}{{\partial}X^{I}}}$. Physical scalars
$z^{A},\,A=1,2,3$, which parameterise the Kähler manifold, are given in terms
of $X^{I}$ functions as follows

$z^{A}={\frac{X^{A}}{X^{0}}},\quad A=1,2,3.$ (6.4)

It can be shown that the metric function of the 8-charge static BPS black hole
should be related to the Kähler potential (since the metric has to be duality
invariant, it should be related to a similar duality invariant quantity in
special geometry), and $X^{I}$, $W_{I}$ and their complex conjugates should be
given by harmonic functions. These are made to ensure that the bosonic
solutions are supersymmetric (i.e. solutions which render the supersymmetry
variations $\delta\psi_{\mu}$ and $\delta\lambda^{A}$ equal to zero). We refer
to [18] for detailed results.

$\displaystyle e^{-2U}=e^{-K}=\,{{\rm
i\,}}(\bar{X}^{I}W_{I}-X^{I}\bar{W}_{I})\,,$ $\displaystyle{{\rm
i\,}}(X^{I}-{\bar{X}^{I}})={{\widetilde{H}}}^{I},\quad\text{where}\quad{{\widetilde{H}}}^{I}={{\tilde{h}}}^{I}+{\frac{p^{I}}{r}},$
$\displaystyle{{\rm
i\,}}(W_{I}-{\bar{W}_{I}})={{H}}_{I},\quad\text{where}\quad{{H}}_{I}={{h}}_{I}+{\frac{q_{I}}{r}}.$
(6.5)

Here ${{\tilde{h}}}^{I}$ and ${{h}}_{I}$ are integration constants, while
$q_{I}$ and $p^{I}$ are the electric and magnetic charges.Using (6.4) and
(6.5), the metric and scalar fields are expressed in terms of the harmonic
functions below,

$\displaystyle z^{A}$ $\displaystyle=$
$\displaystyle{\frac{\left(2{{\widetilde{H}}}^{A}{{H}}_{A}-{{\widetilde{H}}}^{I}{{H}}_{I}\right)-{{\rm
i\,}}e^{-2U}}{\left(d_{ABC}{{\widetilde{H}}}^{B}{{\widetilde{H}}}^{C}+2{{\widetilde{H}}}^{0}{{H}}_{A}\right)}}\,,\qquad
A=1,2,3\,,$ $\displaystyle e^{-4U}$ $\displaystyle=$
$\displaystyle{{\frac{1}{r^{4}}}}V(r)$ (6.6) $\displaystyle=$
$\displaystyle-({{\widetilde{H}}}^{I}{{H}}_{I})^{2}+(d_{ABC}{{\widetilde{H}}}^{B}{{\widetilde{H}}}^{C}d^{ADE}{{H}}_{D}{{H}}_{E})$
$\displaystyle+\,4\,{{\widetilde{H}}}^{0}{{H}}_{1}{{H}}_{2}{{H}}_{3}-4\,{{H}}_{0}\,{{\widetilde{H}}}^{1}{{\widetilde{H}}}^{2}{{\widetilde{H}}}^{3}\,.$

$d_{ABC}$ is the “symmetrised $\epsilon$ tensor,” $d_{ABC}=|\epsilon_{ABC}|$,
so

$d_{123}=1=d_{213}=\text{(any permutation of 1,2,3 indices)}\,.$ (6.7)

Towards the end of this section we shall relate the charges, scalar fields and
the metric mentioned in the above equation to those in the U-duality
formulation, via the 3+1 formulation discussed in section 2.5.

Not all the eight constants ${{h}}_{I}$ and ${{\tilde{h}}}^{I}$ are
independent; they are constrained by two conditions. Since the metric is
asymptotically flat, we have the constraint

$\displaystyle 1$ $\displaystyle=$
$\displaystyle-({{\tilde{h}}}^{I}{{h}}_{I})^{2}+(d_{ABC}\,{{\tilde{h}}}^{B}{{\tilde{h}}}^{C}d^{ADE}\,{{h}}_{D}{{h}}_{E})$
(6.8)
$\displaystyle+\,4\,{{\tilde{h}}}^{0}{{h}}_{1}{{h}}_{2}{{h}}_{3}-4\,{{h}}_{0}\,{{\tilde{h}}}^{1}{{\tilde{h}}}^{2}{{\tilde{h}}}^{3}$

The other constraint is [17]

${{\tilde{h}}}^{I}\,q_{I}-{{h}}_{I}\,p^{I}=0\,,$ (6.9)

which arises from a more fundamental requirement of special geometry:

$\displaystyle\langle V,{\cal D}_{A}V\rangle=0,\quad\text{where}\quad
V=\begin{pmatrix}L^{I}\\\ M_{I}\end{pmatrix},\qquad{\cal