extremality (4.6) and angular momenta of the black holes as

$E=R\left(L_{u}+L_{v}-\frac{2p}{\alpha^{\prime}}+2\sqrt{4L_{u}L_{v}+\frac{p^{2}}{\alpha^{\prime
2}}}\right)$ (4.43)

where we used the fact that $g_{6}^{2}=k/p$, implying that
$\alpha_{1}=\alpha_{5}$. This highly resembles the formula in [45], up to the
factor of $2$, which reflects the different scaling of $\alpha_{5}$. The
momentum is simply

$P=\frac{n}{R}=\frac{Rvr_{0}^{2}\sinh 2\alpha_{n}}{2g_{s}^{2}\alpha^{\prime
2}}=R(L_{u}-L_{v})$ (4.44)

We will now use this parametrisation of the black string solutions to work out
the asymptotic symmetries of this spacetime, in order to facilitate the
comparison with the corresponding formulae in the ALD case.

###  4.4 Asymptotic symmetries of the self-dual black holes

We would now like to find the asymptotic symmetries of the self-dual black
holes within this truncation. Since we do not know, _a priori_ , which
boundary conditions to impose, we will be following the same strategy that we
used in [45] to find the asymptotic symmetries of the ALD background, namely
to require that the asymptotic symmetry generators have zero symplectic
product with the metric and scalar field perturbations that simply change the
mass and angular momentum parameters of the black holes - namely, the
parameters $L_{u,v}$ in the solutions above - as the latter should clearly be
allowed perturbations of the spacetime.

To find the asymptotic symmetries, we first choose a gauge that fixes the
radial dependence of the diffeomorphism. We will thus be working in radial
gauge

$\displaystyle(\mathcal{L}_{\xi}g)_{\rho\mu}$ $\displaystyle=0$ (4.45)

The resulting gauge-fixed diffeomorphism takes the form

$\displaystyle\xi$
$\displaystyle=\frac{\rho^{5}}{(\rho^{4}+\rho^{2}\beta/\alpha^{\prime}+L_{u}L_{v})^{2}}F_{\rho}(U,V)\partial_{\rho}+\bigg{(}F_{U}(U,V)+\frac{(\alpha^{\prime}g_{6})^{2}(\rho^{2}\partial_{v}F_{\rho}-L_{v}\partial_{u}F_{\rho})}{\rho^{4}-L_{u}L_{v}}\bigg{)}\partial_{U}+$
$\displaystyle\;\;\;\;+\,\bigg{(}F_{V}(U,V)+\frac{(\alpha^{\prime}g_{6})^{2}(\rho^{2}\partial_{u}F_{\rho}-L_{u}\partial_{v}F_{\rho})}{\rho^{4}-L_{u}L_{v}}\bigg{)}\partial_{V}$
(4.46)

where we introduced the notation $\beta=\sqrt{p^{2}+4\alpha^{\prime
2}L_{u}L_{v}}$ . This is very similar to the corresponding radial
diffeomorphism in the ALD spacetime, except for the fall-off of the radial
component.

The symplectic form receives contributions from both the metric and the scalar

$\omega=\frac{1}{16\pi G_{3}}(\omega_{g}+\omega_{sc})$ (4.47)

For an action of the form (4.34), the general expressions for these
contributions, for two arbitrary variations $\delta_{1},\delta_{2}$ of the
fields are [80]

$\omega^{\mu}_{g}=\frac{1}{2}\bigg{(}(2\nabla_{\lambda}h_{1\rho}^{\mu}-\nabla^{\mu}h_{1\lambda\rho})h_{2}^{\lambda\rho}-\nabla_{\rho}h_{1}h_{2}^{\rho\mu}+h_{1}(\nabla_{\rho}h_{2}^{\rho\mu}-\nabla^{\mu}h_{2})-(1\leftrightarrow
2)\bigg{)}$ (4.48)
$\omega^{\mu}_{sc}=\frac{\kappa}{2}\bigg{(}-\delta_{1}(\nabla^{\mu}F)\delta_{2}F-\frac{1}{2}h_{1}\nabla^{\mu}F\delta_{2}F-(1\leftrightarrow
2)\bigg{)}$ (4.49)

where $\kappa/2$ is the coefficient of the scalar kinetic term in the action
($\kappa=24$ in our case) and we use the notation
$h^{i}_{\mu\nu}=\delta_{i}g_{\mu\nu}$ and
$h_{i}=g^{\mu\nu}\delta_{i}g_{\mu\nu}$. The falloff conditions on the
symplectic form that ensure it is conserved and normalisable are, in our
coordinates

$\displaystyle\omega_{UV}=o(\rho^{0})\hskip 28.45274pt\omega_{\rho U,\rho
V}=o(\rho^{-1})$ (4.50)

where $\omega_{\mu\nu}=\epsilon_{\mu\nu\alpha}\omega^{\alpha}$ and the
notation $o(\rho^{a})$ signifies that the falloff needs to be faster than
$\rho^{a}$. We evaluate the symplectic form on a perturbation that simply
changes the black hole parameters

$\delta_{1}=\delta L_{u}\frac{\partial}{\partial L_{u}}+\delta
L_{v}\frac{\partial}{\partial L_{v}}$ (4.51)

and one that corresponds to the action of a diffeomorphism of the form (4.46),
$\delta_{2}=\mathcal{L}_{\xi}$.

Requiring that the coefficients of $\delta L_{u},\delta L_{v}$ vanish
independently, the asymptotic falloff condition on $\omega_{UV}$ translates
into the following two constraints

$\displaystyle\partial_{U}F_{U}+\partial_{V}F_{V}$
$\displaystyle=\frac{\beta}{2\alpha^{\prime}L_{v}}\partial_{V}F_{U}=\frac{\beta}{2\alpha^{\prime}L_{u}}\partial_{U}F_{V}$
(4.52)

These equations, up to a factor of 2 difference, are the same as in the ALD
background. The solutions to the equations are given in terms of two arbitrary
functions $f,\bar{f}$

$\displaystyle
F_{U}(U,V)=f(u)+\frac{\beta-\tilde{\beta}}{4\alpha^{\prime}L_{u}}\bar{f}(v)\hskip
28.45274ptF_{V}(U,V)=\bar{f}(v)+\frac{\beta-\tilde{\beta}}{4\alpha^{\prime}L_{v}}f(u)$
(4.53)

of the following field-dependent coordinates:

$\displaystyle u=U+\frac{4\alpha^{\prime}L_{v}}{\beta+\tilde{\beta}}\,V\hskip
28.45274ptv=V+\frac{4\alpha^{\prime}L_{u}}{\beta+\tilde{\beta}}\,U$ (4.54)

where we have introduced the notation

$\displaystyle\tilde{\beta}=\sqrt{p^{2}-12\alpha^{\prime 2}L_{u}L_{v}}$ (4.55)

The field-dependent coordinates (4.54) reduce to $U$ and, respectively, $V$ in
the $L_{u},L_{v}\rightarrow 0$ or $\alpha^{\prime}\rightarrow 0$ limit, while
$F_{U,V}$ become purely holomorphic functions of these fixed coordinates.
Another interesting fact to note is that the expression for $\tilde{\beta}$
breaks down at an energy or, correspondingly, horizon radius that _precisely_
coincides with the maximum value of the radius of the small black strings we
discussed in section 4.1

$(L_{u}L_{v})_{*}=\frac{(\tilde{r}_{0}^{*})^{4}}{16}=\frac{p^{2}(\hat{r}_{0}^{*})^{4}}{16\alpha^{\prime
2}}=\frac{p^{2}}{12\alpha^{\prime 2}}$ (4.56)

where $\hat{r}_{0}^{*}=\sqrt[4]{\frac{4}{3}}$ is the value of the rescaled
radius that corresponds to $T_{max}$ for $\varkappa=1$.

The $\omega_{\rho U,\rho V}$ components of the symplectic form contain terms
that behave have both as $\rho$ and $\frac{1}{\rho}$ at infinity. However,
these terms are total derivatives, in the sense that
$\omega_{\rho\sigma}=\partial_{\sigma}(...)$, which vanishes upon integration
on the $\sigma$ circle. The radial function $F_{\rho}$ is not fixed by these
constraints but, just like in the ALD background, it is fixed by the
symplectic form evaluated on variations generated by two different
diffeomorphisms of the form above, which yield the constraint

$\displaystyle
L_{u}\partial^{2}_{V}F_{\rho}+L_{v}\partial^{2}_{U}F_{\rho}=\frac{\beta}{2\;\alpha^{\prime}}\partial_{U}\partial_{V}F_{\rho}$
(4.57)

The solution to this equation takes the general form

$\displaystyle F_{\rho}(U,V)$ $\displaystyle=f_{\rho}(u)+\bar{f}_{\rho}(v)$
(4.58)

for two new functions of the field-dependent coordinates.

We would now like to compute the charges associated to these asymptotic
diffeomorphisms and show they are non-vanishing. Since our background depends
on constant parameters, the only nontrivial charges will correspond to the
energy and momentum, already computed in (4.43), (4.44). To obtain non-trivial
charges associated with the general functions $f,\bar{f}$ in (4.53), we first
perturb the background using an allowed diffeomorphism, denoted $\eta$ and
parametrized by $h(u),\bar{h}(v),h_{\rho}(u),\bar{h}_{\rho}(v)$, and then
compute the charge difference associated to another asymptotic diffeomorphism,
$\xi$, parametrized by $f(u),\bar{f}(v),f_{\rho}(u),\bar{f}_{\rho}(v)$. The
result takes the form

$\displaystyle\delta_{\eta}Q_{\xi}$
$\displaystyle=\frac{g_{6}^{2}\alpha^{\prime 2}}{4\pi
G_{3}}\frac{\tilde{\beta}}{\beta+\tilde{\beta}}\oint
d\sigma\bigg{[}r_{u}\bigg{(}2L_{u}f(u)h^{\prime}(u)+g_{6}^{2}\alpha^{\prime
2}(f(u)h^{\prime\prime}_{\rho}(u)-h(u)f^{\prime\prime}_{\rho}(u))\bigg{)}-$
$\displaystyle-
r_{v}\bigg{(}2L_{v}\bar{f}(v)\bar{h}^{\prime}(v)+g_{6}^{2}\alpha^{\prime
2}(\bar{f}(v)\bar{h}^{\prime\prime}_{\rho}(v)-\bar{h}(v)\bar{f}^{\prime\prime}_{\rho}(v))\bigg{)}+$
$\displaystyle+\frac{1}{2}\bigg{(}r_{u}f^{\prime}_{\rho}(u)\bar{h}^{\prime}(v)+r_{v}f_{\rho}(u)\bar{h}^{\prime\prime}(v)-r_{v}\bar{f}^{\prime}_{\rho}(v)h^{\prime}(u)-r_{u}\bar{f}_{\rho}(v)h^{\prime\prime}(u)\bigg{)}\bigg{]}$
(4.59)

where $r_{u,v}$ are the normalised radii of the field-dependent coordinates
(4.54)

$\displaystyle
r_{u}:=\partial_{\sigma}u=\frac{4\alpha^{\prime}L_{v}+\beta+\tilde{\beta}}{\beta+\tilde{\beta}}\hskip
56.9055ptr_{v}:=\partial_{\sigma}v=\frac{4\alpha^{\prime}L_{u}+\beta+\tilde{\beta}}{\beta+\tilde{\beta}}$
(4.60)

It is clear that the last line vanishes after integration by parts. We are
left with the following result

$\displaystyle\delta_{\eta}Q_{\xi}$
$\displaystyle=\frac{g_{6}^{2}\alpha^{\prime 2}}{4\pi
G_{3}}\frac{\tilde{\beta}}{\beta+\tilde{\beta}}\oint
d\sigma\;r_{u}\bigg{(}2L_{u}f(u)h^{\prime}(u)+g_{6}^{2}\alpha^{\prime
2}(f(u)h^{\prime\prime}_{\rho}(u)-h(u)f^{\prime\prime}_{\rho}(u))\bigg{)}+RM$
(4.61)

where ‘RM’ stands for the expression obtained by replacing $f,h$ etc. by
$\bar{f},\bar{h}$ and $u\rightarrow v$. This result is extremely similar with
the analogous expression in the ALD spacetime. Setting, for simplicity, the
radial piece of the diffeomorphism to zero, the difference simply amounts to
replacing $\tilde{\beta}$ by $p$ (the same holds at the level of the field-
dependent coordinates), which becomes negligible at low energies.

To conclude, it is rather remarkable that i) the asymptotic symmetries that we
have obtained in this simple manner are so similar to the symmetries of
single-trace $T\bar{T}$ (identified with the asymptotic symmetries of the ALD
spacetime) and ii) they appear to know about the termodynamic stability of the
backgrounds that they depend on, in the sense that the symmetry generators
become imaginary precisely when one reaches the large, unstable black string
region. It would be very interesting to better understand this connection
between asymptotically flat backgrounds and $T\bar{T}$.

##  5 Discussion

In this article, we have argued that turning on a particular $21$-dimensional
subset of the $22$ maximally supersymmetric single-trace irrelevant
deformations of the D1-D5 CFT for a _finite_ amount leads to a UV-complete
theory decoupled from gravity, which is in general a deformation of little
string theory, compactified on K3. Our argument was largely based on the
supergravity description of these deformations, which are characterised by the
degeneration of the flat asymptotia to linear dilaton-like ones in specific
ways that we could identify with known decoupling limits of string theory. By
studying the behaviour of entropy in these theories, inferred from that of the
dual black holes, we also argued that the resulting two-dimensional theories
should admit a $T\bar{T}$ \- like effective description. Finally, we put forth
evidence that, for low enough energies, the asymptotically flat black strings,
too, admit a similar description, as suggested by the resemblance between the
asymptotic symmetry generators we found for flat space and the field-dependent
coordinate transformations characteristic of $T\bar{T}$.

If this global picture is correct, there are many interesting future
directions to explore.

One such direction is to gain a better technical understanding of the general
ALD-like spacetimes we discussed and their associated holographic dictionary.
One may, for example, study the asymptotic symmetries of the general such
spacetimes and check whether they, too, display $T\bar{T}$ \- like features.
Another interesting question is whether the behaviour of the entropy and
symmetries changes if several $\lambda_{\bar{\imath}}$ deformations are turned
on at the same time. One may also try to understand the precise map between
supergravity modes and their dual operators, by adapting the holographic
renormalisation technique to this setup (see [81] for some early work); the
careful linearised analysis of the fields in a consistent truncation of the
standard ALD background, performed in [45], already showed several unusual
features. It is also interesting to understand how the matching of
supersymmetry multiplets works in these cases; for example, whether fields in
the same multiplet acquire the same momentum-dependence in their radial fall-
offs.

All these analyses would presumably provide important clues for what the
properties of the dual theories should be and, in particular, which
observables should be expected to be universal. In addition, this could prove
to be a useful warm-up exercise for flat space holography, with which it
shares many features [82], but without some of the puzzles related to
decoupling and the black hole instability.

Another possibility is to approach the same problem from the field-theory
perspective, using conformal perturbation theory and/or supersymmetry. One may
hope, for example, that for the case of the $SO(21)$ \- singlet irrelevant
deformation, its mixing with other operators may be constrained enough to
yield the deformation solvable. The conformal perturbation theory calculation
may be performed at a generic point in moduli space, or at the orbifold point.
In the latter case, it may be possible to understand the exact single-trace
$T\bar{T}$ deformation in terms of the dual worldsheet theory and, to the
extent this field progresses [83], study the other types of deformations by
turning on RR fields in the bulk.

Still from the boundary perspective, it would be very interesting to further
explore the properties of non-commutative open string theories and, in
particular, understand the assumptions that underlie the particular energy
dependence of the five-dimensional NCOS decoupling parameters, which is
ultimately responsible for the observed Hagedorn behaviour in the supergravity
dual. It would of course be interesting if this could be further extended to
general OD$p$ theories.

Finally, it would be interesting to see if the perspective of section 4 can
shed light on more general black holes, such as Reissner-Nordström or Kerr
ones.

### Acknowledgements

We are grateful to Iosif Bena, Nikolay Bobev, Soumangsu Chakraborty, Ruben
Minasian and especially Kyriakos Papadodimas for insightful conversations. The
work of NK is supported by the ERC Consolidator Grant 772408-Stringlandscape.
The work of SG is supported by the ERC Starting Grant 853507. SG and NK also
thank CERN and EPFL, where part of this work was carried out, for their
hospitality.

##  Appendix A Details of the six-dimensional supersymmetric solutions

In this appendix, we fill in many of the details of the supersymmetric
solutions discussed in section 2, including conventions and the explicit form
of the Killing spinors.

###  A.1 Supersymmetry conventions

In six dimensions, the R-symmetry group is USp$(2)$ for $\mathcal{N}=(1,0)$
supergravity and USp$(4)$ for $\mathcal{N}=(2,0)$. These groups are isomorphic
to spin groups [59]

$\displaystyle\text{USp}(2)\cong SU(2)\cong Spin(3)$
$\displaystyle\text{USp}(4)\cong Spin(5)$ (A.1)

and we can construct their generators in the standard way using gamma matrices
in three and, respectively, five Euclidean dimensions. In $\mathcal{N}=(1,0)$
supergravity, the spinors are USp$(2)$ doublets of six-dimensional Weyl
spinors satisfying the symplectic Majorana condition; in $\mathcal{N}=(2,0)$,
they are quadruplets of six-dimensional Weyl spinors that transform under
USp$(4)$ and satisfy the corresponding symplectic Majorana condition.

In the following, $\mu,\nu$ will denote curved spacetime indices, and $a,b$
will be six-dimensional tangent space indices. In $\mathcal{N}=(1,0)$
theories, the $USp(2)$ structure is trivial, as all bosonic quantities are
singlets under it, and all fermionic ones are doublets; we will consequently
drop all associated indices. In $\mathcal{N}=(2,0)$ theories, given the
correspondence between the fundamental and, respectively, traceless
antisymmetric representation of $USp(4)$ and the spinor and, respectively,
vector representation of $Spin(5)$, we will be using $Spin(5)$ spinor indices
$A,B=\\{1,\ldots,4\\}$ on the fermions, and vector indices $i,j=\\{1,\ldots
5\\}$ on the bosons. Meanwhile, the indices
$\bar{\imath},\bar{\jmath}=\\{1,\ldots Q\\}$ transform in the vector
representation of $SO(Q)$. We will be using the notation $\gamma^{a}$ for the
six-dimensional Lorentzian gamma matrices, and $\Gamma^{i}$ for the five-
dimensional Euclidean gamma matrices. The chirality matrix in six dimensions
is defined as

$\displaystyle\gamma_{7}$ $\displaystyle=-\gamma^{0}...\gamma^{5}$ (A.2)

and the chirality projectors are given by
$P_{\pm}=\frac{I_{7}\pm\gamma_{7}}{2}$. The left and right spinors satisfy
$P_{+}\varepsilon_{+}=\varepsilon_{+}$ and
$P_{-}\varepsilon_{-}=\varepsilon_{-}$, respectively.

In $\mathcal{N}=(2,0)$ theories, the supersymmetry variations are given by

$\displaystyle\delta\psi_{\mu A}$
$\displaystyle=\nabla_{\mu}\varepsilon_{A}+\frac{1}{4}F_{\mu
ab}^{+i}\gamma^{a}\gamma^{b}(\Gamma^{i})_{A}^{\;B}\varepsilon_{B}$ (A.3)
$\displaystyle\delta\chi^{\bar{\imath}}_{A}$
$\displaystyle=\frac{1}{\sqrt{2}}\gamma^{\mu}P_{\mu}^{i\bar{\imath}}(\Gamma^{i})_{A}^{\;B}\varepsilon_{B}-\frac{1}{12}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}F^{-\bar{\imath}}_{\mu\nu\rho}\varepsilon_{A}$
(A.4)

where the covariant derivative contains both the spacetime spin connection,
$\omega_{\mu}{}^{ab}$, and the $SO(P)$ one, $Q_{\mu}^{ij}$

$\nabla_{\mu}\varepsilon_{A}=\bigg{(}\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{\;ab}\gamma_{a}\gamma_{b}\bigg{)}\varepsilon_{A}-\frac{1}{4}Q_{\mu
ij}(\Gamma^{i}\Gamma^{j})_{A}^{\;B}\varepsilon_{B}$ (A.5)

whose construction in terms of the decomposition of the Lie algebra element
$dg\,g^{-1}$ was explained in section 2 and is given by

$Q^{ij}=(dgg^{-1})^{ij}=-X^{i\Lambda}dX^{j}_{\Lambda}\;,\;\;\;\;\;\;Q^{\bar{\imath}\bar{\jmath}}=(dgg^{-1})^{\bar{\imath}\bar{\jmath}}=X^{\bar{\imath}\Lambda}dX^{\bar{\jmath}}_{\Lambda}$
(A.6)

We have also included the $SO(Q)$ connection, for completeness. In
$\mathcal{N}=(1,0)$ theories, there is only one self-dual form, and so the
index $i$ takes a single value. Consequently, $Q=0$, case in which the
equations simplify to (2.16).

###  A.2 Details of the computation of the $(1,0)$ solutions

In the $\mathcal{N}=(1,0)$ case, we can replace the doublet of spinors
satisfying the Majorana condition by a single unconstrained spinor. Let us now
explicitly solve the Killing spinor equations (2.16) in the $(1,0)$ theory.
Our derivation closely follows that of [61, 60] for $\mathcal{N}=(2,0)$. We
choose (Hopf) coordinates on the sphere, denoted $\\{\psi,\theta,\varphi\\}$

$\displaystyle d\Omega^{2}_{3}$
$\displaystyle=\frac{1}{4}(d\theta^{2}+d\psi^{2}+d\varphi^{2}+2\cos\theta
d\varphi d\psi)$ (A.7)

The orientation on $S^{3}$ is

$\displaystyle\omega_{S^{3}}=\frac{\sin\theta}{8}d\psi\wedge d\theta\wedge
d\varphi$ (A.8)

and use the convention $\epsilon_{r\sigma t\psi\theta\varphi}=+\sqrt{g}$ when
computing the Hodge dual. The vielbein associated to the metric (2.17) is
given by

$e^{0}=e^{U}dt\;,\hskip 8.5359pte^{1}=e^{U}d\sigma\;,\hskip
8.5359pte^{2}=e^{-U}dr\;,\hskip
8.5359pte^{3}=\frac{r\,e^{-U}}{2}(d\psi+\cos\theta d\varphi)$ (A.9)
$e^{4}=\frac{r\,e^{-U}}{2}d\theta\;,\hskip
8.5359pte^{5}=\frac{r\,e^{-U}}{2}\sin\theta d\varphi$ (A.10)

and the nonzero components of the spin connection
$\omega_{\mu}^{\;ab}=e^{\;a}_{\nu}\nabla_{\mu}e^{\;\nu b}$ are

$\displaystyle\omega_{02}$ $\displaystyle=e^{U}U^{\prime}e^{0}\;,\hskip
14.22636pt\omega_{12}=-e^{U}U^{\prime}e^{1}\;,\hskip
14.22636pt\omega_{23}=\frac{e^{U}(1-rU^{\prime})}{r}e^{3}\;,\hskip
14.22636pt\omega_{24}=\frac{e^{U}(1-rU^{\prime})}{r}e^{4}$ (A.11)
$\displaystyle\omega_{25}$
$\displaystyle=\frac{e^{U}(1-rU^{\prime})}{r}e^{5}\;,\hskip
14.22636pt\omega_{34}=\frac{e^{U}}{r}e^{5}\;,\hskip
14.22636pt\omega_{35}=-\frac{e^{U}}{r}e^{4}\;,\hskip
14.22636pt\omega_{45}=\frac{e^{U}}{r}(-e^{3}+2\cot\theta e^{5})$ (A.12)

where $U^{\prime}=\partial_{r}U$. In terms of the vielbeine, our Ansatz (2.19)
for the 3-forms is given by

$\displaystyle H^{+}$ $\displaystyle=\frac{Ze^{3U}}{r^{3}}\big{(}e^{3}\wedge
e^{4}\wedge e^{5}-e^{0}\wedge e^{1}\wedge e^{2}\big{)}$ (A.13) $\displaystyle
H^{-\bar{\imath}}$
$\displaystyle=\frac{Z^{\bar{\imath}}e^{3U}}{r^{3}}\big{(}e^{3}\wedge
e^{4}\wedge e^{5}+e^{0}\wedge e^{1}\wedge e^{2}\big{)}$ (A.14)

Plugging this into the tensorino equations (A.4) and using the chirality of
the $6d$ spinor they can be reduced to

$\displaystyle\frac{1}{\sqrt{2}}\gamma^{2}e_{2}^{r}P^{\bar{\imath}}_{r}\varepsilon-\frac{Z^{\bar{\imath}}e^{3U}}{r^{3}}\gamma^{0}\gamma^{1}\gamma^{2}\varepsilon=0$
(A.15)

As explained in the main text, for $Z_{\bar{\imath}}\neq 0$ this leads to the
projection

$\displaystyle\gamma^{0}\gamma^{1}\varepsilon=\varepsilon$ (A.16)

as well as the equation (2.37) for $P_{r}^{\bar{\imath}}$.

Let us now turn to the gravitino equations. Writing them explicitly in
components, they read

$\displaystyle
e^{-U}\partial_{t}\varepsilon+\frac{1}{2}e^{U}U^{\prime}\gamma^{0}\gamma^{2}\varepsilon-\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{1}\gamma^{2}\varepsilon=0$
(A.17) $\displaystyle
e^{-U}\partial_{\sigma}\varepsilon+\frac{1}{2}e^{U}U^{\prime}\gamma^{2}\gamma^{1}\varepsilon-\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{2}\gamma^{0}\varepsilon=0$
(A.18) $\displaystyle
e^{U}\partial_{r}\varepsilon-\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{0}\gamma^{1}\varepsilon=0$
(A.19)
$\displaystyle\frac{2e^{U}}{r}\partial_{\psi}\varepsilon+\frac{1}{2}\bigg{(}\frac{e^{U}(1-rU^{\prime})}{r}\gamma^{2}\gamma^{3}-\frac{e^{U}}{r}\gamma^{4}\gamma^{5}\bigg{)}\varepsilon+\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{4}\gamma^{5}\varepsilon=0$
(A.20)
$\displaystyle\frac{2e^{U}}{r}\partial_{\theta}\varepsilon+\frac{1}{2}\bigg{(}\frac{e^{U}(1-rU^{\prime})}{r}\gamma^{2}\gamma^{4}-\frac{e^{U}}{r}\gamma^{3}\gamma^{5}\bigg{)}\varepsilon+\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{5}\gamma^{3}\varepsilon=0$
(A.21)
$\displaystyle\frac{2e^{U}}{r\sin\theta}(\partial_{\varphi}-\cos\theta\partial_{\psi})\varepsilon+\frac{1}{2}\bigg{(}\frac{e^{U}(1-rU^{\prime})}{r}\gamma^{2}\gamma^{5}+\frac{e^{U}}{r}\gamma^{3}\gamma^{4}+\frac{e^{U}2\cot\theta}{r}\gamma^{4}\gamma^{5}\bigg{)}\varepsilon+\frac{1}{2}\frac{Ze^{3U}}{r^{3}}\gamma^{3}\gamma^{4}\varepsilon=0$
(A.22)

Using the projection (A.16) inside these equations, one can easily factorize
the $r$ dependence. Requiring that the $t$ (or $\sigma$) equation does not
impose any additional projection on $\varepsilon$ sets 171717More generally,
one easily sees $e^{2U}(U^{\prime}-Ze^{2U}/r^{3})$ is $r$ independent, as
should be.

$U^{\prime}=\frac{Ze^{2U}}{r^{3}}$ (A.23)

which in particular implies that, in this coordinate system,
$\partial_{t}\varepsilon=\partial_{\sigma}\varepsilon=0$. The radial and
$S^{3}$ equations can then be trivially solved to give

$\displaystyle\varepsilon$
$\displaystyle=e^{\frac{U(r)}{2}}e^{-\frac{\theta}{2}\gamma^{2}\gamma^{4}}e^{-\frac{\varphi}{2}\gamma^{2}\gamma^{3}}\varepsilon_{0}$
(A.24)

with $\varepsilon_{0}$ coordinate-independent and satisfying

$\displaystyle\gamma^{0}...\gamma^{5}\varepsilon_{0}=-\varepsilon_{0}\hskip
28.45274pt\gamma^{0}\gamma^{1}\varepsilon_{0}=\varepsilon_{0}$ (A.25)

which follow from the corresponding projections on $\varepsilon$. The metric
and scalar fields $X^{\Lambda}$ obey (2.20), (2.23); how to solve these
equations is explained in the main text.

###  A.3 Details on the computation of the $(2,0)$ solutions

In the $\mathcal{N}=(2,0)$ case, there are four Killing spinors,
$\varepsilon_{A}$, related via a symplectic Majorana condition. We will
package them into a four-index object, on which the $\Gamma^{i}$ matrices act.
The formulae for the vielbeine, spin connection and field strengths are the
same as before (A.9), (A.11) and (A.13). The gravitino equations, written
explicitly in components are

$\displaystyle
2e^{-2U}\partial_{t}\varepsilon+\gamma^{0}\gamma^{2}\left(U^{\prime}-\frac{Z^{i}e^{2U}}{r^{3}}\Gamma^{i}\gamma^{0}\gamma^{1}\right)\varepsilon=0$
(A.26)
$\displaystyle\partial_{\sigma}\varepsilon-\gamma^{0}\gamma^{1}\partial_{t}\varepsilon=0$
(A.27) $\displaystyle
2D_{r}\varepsilon-\frac{Z^{i}e^{2U}}{r^{3}}\Gamma^{i}\gamma^{0}\gamma^{1}\varepsilon=0$
(A.28) $\displaystyle
4\partial_{\psi}\varepsilon+(1-rU^{\prime})\gamma^{2}\gamma^{3}\varepsilon-\left(1-r\frac{Z^{i}e^{2U}}{r^{3}}\Gamma^{i}\right)\gamma^{4}\gamma^{5}\varepsilon=0$
(A.29) $\displaystyle
4\partial_{\theta}\varepsilon+(1-rU^{\prime})\gamma^{2}\gamma^{4}\varepsilon-(1+r\frac{Z^{i}e^{2U}}{r^{3}}\Gamma^{i})\gamma^{3}\gamma^{5}\varepsilon=0$
(A.30) $\displaystyle
4\partial_{\varphi}\varepsilon+\big{(}\cos\theta+\sin\theta\gamma^{3}\gamma^{5}\big{)}\bigg{(}(1-rU^{\prime})\gamma^{2}\gamma^{3}+(1+r\frac{Z^{i}e^{2U}}{r^{3}}\Gamma^{i})\gamma^{4}\gamma^{5}\bigg{)}\varepsilon=0$
(A.31)

Looking for time-independent solutions, we see from the first two equations
that they also need to be $\sigma$-independent and satisfy

$\displaystyle U^{\prime}=\pm\frac{|Z^{+}|e^{2U}}{r^{3}}\hskip
56.9055pt\hat{Z}^{i}\Gamma^{i}\gamma^{0}\gamma^{1}\varepsilon=\pm\varepsilon$
(A.32)

We focus on plus sign. Plugging the projection into the above equations and
using six-dimensional chirality, one recovers exactly the same sphere
equations as in the $(1,0)$ case, obviously with the same solution. After
using the projection, the $r$ equation reads

$\displaystyle
2D_{r}\varepsilon=2\bigg{(}\partial_{r}\varepsilon-\frac{1}{4}Q_{rij}\Gamma^{ij}\varepsilon\bigg{)}=\frac{|Z^{+}|\hat{Z}^{i}e^{2U}}{r^{3}}\Gamma^{i}\gamma^{0}\gamma^{1}\varepsilon=U^{\prime}\varepsilon$
(A.33)

which fixes the $r$ dependence and yields the same solution as before, if we
set $Q^{ij}=0$.

The solution for the Killing spinors is of the form

$\displaystyle\varepsilon(r,t,\sigma,S^{3})$
$\displaystyle=e^{-\frac{\theta}{2}\gamma^{2}\gamma^{4}}e^{-\frac{\varphi}{2}\gamma^{2}\gamma^{3}}\varepsilon_{0}(r)$
(A.34)

where $\varepsilon_{0}(r)$ is a $6d$ Majorana-Weyl spinor that depends on $r$
only. It needs to satisfy:

$\displaystyle D_{r}\varepsilon_{0}=\frac{U^{\prime}}{2}\varepsilon_{0}$
(A.35)

and the projection since the $\gamma^{0}\gamma^{1}$ commute with the
exponentials from $S^{3}$:

$\displaystyle\hat{Z}^{i}\Gamma^{i}\varepsilon_{0}=\gamma^{0}\gamma^{1}\varepsilon_{0}$
(A.36)

We can see that the covariant derivative on the projection gives:

$\displaystyle
D_{r}(\hat{Z}^{i}\Gamma^{i})\varepsilon_{0}+\hat{Z}^{i}\Gamma^{i}\frac{U^{\prime}}{2}\varepsilon_{0}=\frac{U^{\prime}}{2}\gamma^{0}\gamma^{1}\varepsilon_{0}=\frac{U^{\prime}}{2}\hat{Z}^{i}\Gamma^{i}\varepsilon_{0}$
(A.37)

which requires for consistency

$\displaystyle D_{r}\hat{Z}^{i}=0$ (A.38)

The tensorini equations, after using the projection
$\hat{Z}^{i}\Gamma^{i}\varepsilon=\gamma^{0}\gamma^{1}\varepsilon$ give

$\displaystyle\bigg{(}\frac{1}{\sqrt{2}}P_{r}^{i\bar{\imath}}-\frac{Z^{\bar{\imath}}e^{2U}}{r^{3}}\hat{Z}^{i}\bigg{)}\Gamma^{i}\varepsilon_{0}=0$
(A.39)

which implies that we have the following equations for the function $U(r)$

$\displaystyle U^{\prime}=\frac{|Z^{+}|e^{2U}}{r^{3}}\;,\hskip
28.45274pt\frac{1}{\sqrt{2}}P_{r}^{i{\bar{\imath}}}=\frac{Z^{\bar{\imath}}e^{2U}}{r^{3}}\hat{Z}^{i}$
(A.40)

In the second equation, after plugging in the definition of $P_{r}$ and
$Z_{\bar{\imath}}$, we obtain

$\displaystyle-\frac{1}{2}X^{\bar{\imath}}_{\Lambda}\partial_{r}X^{i\Lambda}=\frac{q^{\Lambda}e^{2U}}{r^{3}}\hat{Z}^{i}X^{\bar{\imath}}_{\Lambda}$
(A.41)

We multiply by $X^{\bar{\imath}}_{\Sigma}$, sum over $\bar{\imath}$ and use
$X^{\bar{\imath}}_{\Lambda}X^{\bar{\imath}}_{\Sigma}=X^{j}_{\Lambda}X^{j}_{\Sigma}-\eta_{\Lambda\Sigma}$:

$\displaystyle\partial_{r}X^{i}_{\Sigma}-X^{j}_{\Lambda}X^{j}_{\Sigma}\partial_{r}X^{i\Lambda}=2\frac{e^{2U}}{r^{3}}\hat{Z}^{i}(X^{j}_{\Sigma}Z^{j}-q_{\Sigma})$
(A.42)

Then, using the $SO(5)$ connection
$Q_{\mu}^{ij}=-X^{i\Sigma}\partial_{\mu}X^{j}_{\Sigma}$ and
$X^{i}_{\Lambda}X^{j\Lambda}=\delta^{ij}$, we recognize on the left-hand side
the covariant derivative

$\displaystyle D_{r}X^{i}_{\Lambda}$
$\displaystyle=\partial_{r}X^{i}_{\Lambda}-Q^{ij}_{r}X^{j}_{\Lambda}=\partial_{r}X^{i}_{\Lambda}+X^{i\Sigma}X^{j}_{\Lambda}\partial_{r}X^{j}_{\Sigma}$
(A.43)

Hence, we can write

$\displaystyle
D_{r}X^{i}_{\Lambda}=\frac{2e^{2U}}{r^{3}}\hat{Z}^{i}(X^{j}_{\Lambda}Z^{j}-q_{\Lambda})$
(A.44)

Introducing $\tau=\frac{1}{r^{2}}$, we obtain the equations

$\displaystyle\partial_{\tau}(e^{-2U})=|Z^{+}|\hskip
28.45274ptD_{\tau}X^{i}_{\Lambda}=e^{2U}\hat{Z}^{i}(q_{\Lambda}-Z^{j}X^{j}_{\Lambda})$
(A.45)

which imply that

$\displaystyle D_{\tau}(e^{-2U}X^{i}_{\Lambda})$
$\displaystyle=\hat{Z}^{i}(q_{\Lambda}-Z^{j}X^{j}_{\Lambda})+|Z^{+}|X^{i}_{\Lambda}=\hat{Z}^{i}q_{\Lambda}+|Z^{+}|(X^{i}_{\Lambda}-\hat{Z}^{i}\hat{Z}^{j}X^{j}_{\Lambda})$
(A.46)

Multiplying by $\hat{Z}^{i}$ and summing over $i$ kills the second term on the
left-hand side and, since $D_{r}\hat{Z}^{i}=0$, we can write

$\displaystyle D_{\tau}(e^{-2U}\hat{Z}^{i}X^{i}_{\Lambda})=q_{\Lambda}$ (A.47)

where we used the normalization $\hat{Z}^{i}\hat{Z}^{i}=1$. Since the quantity
above is a $SO(5)$ scalar, we can turn the covariant derivative into a partial
derivative and since $q_{\Lambda}$ is constant, we can integrate:

$\displaystyle e^{-2U}\hat{Z}^{i}X^{i}_{\Lambda}=\tau q_{\Lambda}+c_{\Lambda}$
(A.48)

We can now solve for the parallel component
$X^{i}_{\Lambda}\propto\hat{Z}^{i}$, and obtain

$\displaystyle X^{i}_{\Lambda}$ $\displaystyle=\hat{Z}^{i}(\tau
q_{\Lambda}+c_{\Lambda})e^{2U}+\bar{g}^{i}_{\Lambda}$ (A.49)

where we denoted by $\bar{g}^{i}_{\Lambda}$ the perpendicular part. From here
we can extract the function entering the metric to be given by

$\displaystyle
e^{-4U}=\big{(}q_{\Lambda}\tau+c_{\Lambda}\big{)}\big{(}q^{\Lambda}\tau+c^{\Lambda}\big{)}$
(A.50)

The remaining details of the solution are given in the main text.

##  Appendix B Attractor solutions within a consistent truncation

In this appendix, we explicitly work out the map between solutions to ten-
dimensional supergravity whose bosonic part fits within a consistent
truncation to six dimensions and the six-dimensional language of section 2.

###  B.1 A consistent truncation with $SO(2,2)$ symmetry

We will be using a bosonic consistent truncation of ten-dimensional
supergravity that preserves an $SO(2,2)$ subgroup of the original global
$SO(5,21)$ symmetry group of this low-energy theory, which is given in181818To
match the standard conventions, we have flipped the sign of $\chi_{1}$ with
respect to this reference.  [57]

$\displaystyle\mathcal{L}$ $\displaystyle=\frac{1}{8\pi^{3}\alpha^{\prime
2}}\bigg{(}R-\frac{1}{2}(\partial\phi_{1})^{2}-\frac{1}{2}(\partial\phi_{2})^{2}-\frac{1}{2}e^{2\phi_{1}}(\partial\chi_{1})^{2}-\frac{1}{2}e^{2\phi_{2}}(\partial\chi_{2})^{2}-\frac{e^{-\phi_{1}-\phi_{2}}}{12}H_{3}^{2}-$
$\displaystyle-\frac{e^{\phi_{1}-\phi_{2}}}{12}(F_{3}-\chi_{1}H_{3})^{2}+\chi_{2}H_{3}\wedge
F_{3}\bigg{)}$ (B.1)

where $\phi_{1}+\phi_{2}\equiv 2\phi$, $\phi_{1}-\phi_{2}\equiv-2\rho$ is the
logarithm of the string frame volume of the K3, $\chi_{1}$ is the $10d$ axion
and $\chi_{2}$, the part of $C^{(4)}$ that is proportional to the volume of
K3. As discussed at length in [57] the global $SO(2,2)\sim
SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ acts on the scalar combinations

$\displaystyle\tau_{1}=\chi_{1}+ie^{-\phi_{1}}\hskip
28.45274pt\tau_{2}=\chi_{2}+ie^{-\phi_{2}}$ (B.2)

via fractional linear transformations

$\displaystyle\tau\rightarrow\frac{a\tau+b}{c\tau+d}\;,\;\;\;ad-bc=1\hskip
28.45274pta,b,c,d\in\mathbb{Z}$ (B.3)

where $\tau\in\\{\tau_{1},\tau_{2}\\}$. Note that the transformation of
$\tau_{1}$ is noting by type IIB S-duality.

We would now like to put this action in the general form described in section
2.

#### The $SO(2,2)$ vielbein

As explained in the main text, the vielbein can be found from the matrix
$\mathcal{N}=g^{T}g$, where $\mathcal{N}$ itself is constructed from the gauge
field kinetic ($\mathcal{G}$) and axionic ($\mathcal{B}$) terms. The gauge
kinetic matrix is

$\mathcal{G}=\left(\begin{array}[]{cc}e^{-\phi_{1}-\phi_{2}}+\chi_{1}^{2}e^{\phi_{1}-\phi_{2}}&-\chi_{1}e^{\phi_{1}-\phi_{2}}\\\
-\chi_{1}e^{\phi_{1}-\phi_{2}}&e^{\phi_{1}-\phi_{2}}\end{array}\right)\;,\;\;\;\;\;\;\mathcal{B}=\left(\begin{array}[]{cc}0&\chi_{2}\\\
-\chi_{2}&0\end{array}\right)$ (B.4)

From this, one can easily build the matrix $\mathcal{N}_{\Lambda\Sigma}$ using
its definition (2.9). Writing $\mathcal{N}=g^{T}g$, it is easy to see that, up
to an $SO(2)$ rotation, the ’vielbein’ satisfies

$C^{T}g\,C=\left(\begin{array}[]{cc}\mathcal{E}&0\\\
0&(\mathcal{E}^{T})^{-1}\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-\mathcal{B}&1\end{array}\right)\;,\;\;\;\;\;\mathcal{E}^{T}\mathcal{E}=\mathcal{G}$
(B.5)

for the matrix $C$ defined in (2.14). A convenient choice for the matrix
$\mathcal{E}$ is, letting $\phi_{1}+\phi_{2}=2\phi$ and
$\phi_{1}-\phi_{2}=-2\rho$

$\mathcal{E}=\left(\begin{array}[]{cc}e^{-\phi}&0\\\
-\chi_{1}e^{-\rho}&e^{-\rho}\end{array}\right)$ (B.6)

which leads to the following choice of parametrisation of the vielbein in
terms of the physical fields

$C^{T}g\,C=\left(\begin{array}[]{cccc}e^{-\phi}&0&0&0\\\
-\chi_{1}e^{-\rho}&e^{-\rho}&0&0\\\
\chi_{1}\chi_{2}e^{\phi}&-\chi_{2}e^{\phi}&e^{\phi}&\chi_{1}e^{\phi}\\\
\chi_{2}e^{\rho}&0&0&e^{\rho}\end{array}\right)$ (B.7)

Note this precisely coincides with the upper $4\times 4$ block of (2.15), up
to a transposition of the second and third row, as well as a global $O(2,2)$
sign flip. On this restricted subspace, $\Omega_{4}$ is exactly the same as
$C$. Performing the multiplications by the $C$’s, we find

$g=\frac{1}{2}\left(\begin{array}[]{cccc}e^{\phi}+e^{-\phi}+\chi_{1}\chi_{2}e^{\phi}&e^{\phi}(\chi_{1}-\chi_{2})&e^{\phi}-e^{-\phi}-\chi_{1}\chi_{2}e^{\phi}&e^{\phi}(\chi_{1}+\chi_{2})\\\
\chi_{2}e^{\rho}-\chi_{1}e^{-\rho}&e^{\rho}+e^{-\rho}&\chi_{1}e^{-\rho}-\chi_{2}e^{\rho}&e^{\rho}-e^{-\rho}\\\
e^{\phi}-e^{-\phi}+\chi_{1}\chi_{2}e^{\phi}&e^{\phi}(\chi_{1}-\chi_{2})&e^{\phi}+e^{-\phi}-\chi_{1}\chi_{2}e^{\phi}&e^{\phi}(\chi_{1}+\chi_{2})\\\
\chi_{1}e^{-\rho}+\chi_{2}e^{\rho}&e^{\rho}-e^{-\rho}&-\chi_{1}e^{-\rho}-\chi_{2}e^{\rho}&e^{\rho}+e^{-\rho}\end{array}\right)$
(B.8)

Let us check that the scalar kinetic terms in (B.1) are reproduced using the
general construction reviewed in section 2. For this, we first compute

$\displaystyle\partial_{\mu}gg^{-1}$ $\displaystyle=$ $\displaystyle
C\left(\begin{array}[]{cc}\partial_{\mu}\mathcal{E}&0\\\
-\partial_{\mu}[(\mathcal{E}^{-1})^{T}\mathcal{B}]&\partial_{\mu}(\mathcal{E}^{T})^{-1}\end{array}\right)\left(\begin{array}[]{cc}\mathcal{E}^{-1}&0\\\
\mathcal{B}\mathcal{E}^{-1}&\mathcal{E}^{T}\end{array}\right)C^{T}=$ (B.16)
$\displaystyle=\;\frac{1}{2}\left(\begin{array}[]{cc}\partial_{\mu}\mathcal{E}\mathcal{E}^{-1}+\partial_{\mu}(\mathcal{E}^{-1})^{T}\mathcal{E}^{T}-(\mathcal{E}^{-1})^{T}\partial_{\mu}\mathcal{B}\mathcal{E}^{-1}&(\mathcal{E}^{-1})^{T}\partial_{\mu}\mathcal{B}\mathcal{E}^{-1}-\partial_{\mu}\mathcal{E}\mathcal{E}^{-1}+\partial_{\mu}(\mathcal{E}^{-1})^{T}\mathcal{E}^{T}\\\\[5.69054pt]
-\partial_{\mu}\mathcal{E}\mathcal{E}^{-1}+\partial_{\mu}(\mathcal{E}^{-1})^{T}\mathcal{E}^{T}-(\mathcal{E}^{-1})^{T}\partial_{\mu}\mathcal{B}\mathcal{E}^{-1}&\partial_{\mu}\mathcal{E}\mathcal{E}^{-1}+\partial_{\mu}(\mathcal{E}^{-1})^{T}\mathcal{E}^{T}+(\mathcal{E}^{-1})^{T}\partial_{\mu}\mathcal{B}\mathcal{E}^{-1}\end{array}\right)$

The elements in the upper and, respectively, lower block represent
$Q_{\mu}^{ij}$ and $Q^{\bar{\imath}\bar{\jmath}}_{\mu}$ in the gauge (B.7). To
find the scalar kinetic term, we simply remove these blocks, and compute the
trace of the remaining matrix squared. Using

$\partial_{\mu}\mathcal{E}\mathcal{E}^{-1}=\left(\begin{array}[]{cc}-\partial_{\mu}\phi&0\\\
-\partial_{\mu}\chi_{1}\,e^{\phi-\rho}&-\partial_{\mu}\rho\end{array}\right)=-(\partial_{\mu}(\mathcal{E}^{-1})^{T}\mathcal{E}^{T})^{T}$
(B.17)

$(\mathcal{E}^{-1})^{T}\partial_{\mu}\mathcal{B}\mathcal{E}^{-1}=\left(\begin{array}[]{cc}0&e^{\phi+\rho}\partial_{\mu}\chi_{2}\\\
-e^{\phi+\rho}\partial_{\mu}\chi_{2}&0\end{array}\right)$ (B.18)

one may easily check the Lagrangian agrees with the scalar part of (B.1).

#### The basis of three-form fields

Let us now write down explicitly the the three-form field strengths
$F^{\Lambda}$ that satisfy the modified self-duality condition (2.8). The
general procedure for constructing them is outlined in [51]. Stating with a
lagrangian $\mathcal{L}(F)$ for the three-form fields that takes the schematic
form

$\mathcal{L}=\mathcal{G}\,F\wedge\star F+\mathcal{B}\,F\wedge F$ (B.19)

one forms the linear combinations

$\displaystyle
F^{\Lambda}\equiv\frac{1}{\sqrt{2}}\;C\left(\begin{array}[]{c}F\\\
\partial_{F}\mathcal{L}\end{array}\right)=\frac{1}{2}\begin{pmatrix}F+\partial_{F}\mathcal{L}\\\
-F+\partial_{F}\mathcal{L}\end{pmatrix}$ (B.22)

where, for a Lagrangian of the general form we assume,

$\displaystyle\partial_{F}\mathcal{L}$ $\displaystyle=\mathcal{G}\star
F+\mathcal{B}F$ (B.23)

Using the relation $C^{T}\eta\,C=\sigma_{1}\otimes I_{2}$, it is easy to check
that the forms $F^{\Lambda}$ so defined satisfy (2.8), with $\mathcal{N}$
given in (2.9). The charges $q^{\Lambda}$ associated to the sphere integrals
of $F^{\Lambda}$ read, in terms of the usual electric and magnetic charges

$q^{\Lambda}=\eta^{\Lambda\Sigma}q_{\Sigma}=\frac{1}{2}\left(\begin{array}[]{c}q_{el}+q_{mag}\\\
q_{el}-q_{mag}\end{array}\right)=\frac{1}{\sqrt{2}}\;C\left(\begin{array}[]{c}q_{mag}\\\
q_{el}\end{array}\right)$ (B.24)

To sum up, the 3-forms that satisfy (2.8), are the combinations $(\pm
F+\partial_{F}\mathcal{L})/2$. For the particular case of the $SO(2,2)$
truncation, these 3-forms are:

$\displaystyle\frac{1}{2}\bigg{[}\pm
H_{3}+(e^{-\phi_{1}-\phi_{2}}{}^{*}H_{3}-\chi_{1}e^{\phi_{1}-\phi_{2}}{}^{*}(F_{3}-\chi_{1}H_{3})-\chi_{2}F_{3})\bigg{]}$
(B.25) $\displaystyle\frac{1}{2}\bigg{[}\pm
F_{3}+(e^{\phi_{1}-\phi_{2}}{}^{*}(F_{3}-\chi_{1}H_{3})+\chi_{2}H_{3})\bigg{]}$
(B.26)

where the combination $F_{3}-\chi_{1}H_{3}$ is the modified field strength. We
denote the charges associated to these forms by $(q_{F1}\pm q_{NS5})/2$ and,
respectively, $(q_{D1}\pm q_{D5})/2$.

#### A useful gauge rotation

While the gauge (B.7) allows for a simple relation between the vielbein and
the physical fields, the gauge $Q^{ij}_{\mu}=0$ is much more convenient for
the study of the black string attractor solutions. To reach it, we note the
gauge connection reads

$Q^{ij}_{\mu}=i\sigma_{2}(\partial_{\mu}\chi_{1}e^{\phi_{1}}-\partial_{\mu}\chi_{2}{e^{\phi}_{2}})\equiv-i\sigma_{2}\partial_{\mu}\alpha$
(B.27)

This can be set to zero via a gauge transformation that satisfies
$h^{-1}\partial_{\mu}h=-Q_{\mu}$, with solution $h=e^{i\alpha\sigma_{2}}$. One
can also set
$Q_{\mu}^{\bar{\imath}\bar{\jmath}}=i\sigma_{2}(\partial_{\mu}\chi_{1}e^{\phi_{1}}+\partial_{\mu}\chi_{2}e^{\phi_{2}})$
to zero via a similar gauge transformation, $e^{i\sigma_{2}\beta}$. Note that
the constant piece of $\alpha,\beta$ is not fixed by the vanishing of
$Q^{ij}$, and can be used to further rotate $\hat{Z}^{i}$ by a constant angle.

These gauge transformations act on the group element as

$g\rightarrow\left(\begin{array}[]{cc}R_{\alpha}&0\\\
0&R_{\beta}\end{array}\right)g\;,\;\;\;\;\;R_{\alpha}=\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{array}\right)$ (B.28)

Choosing $\alpha$ to be given by (B.27), one can rotate the corresponding
gauge connection to zero.

#### The central charge

It is not hard to see that

$C^{T}Z=C^{T}g\,C\left(\begin{array}[]{c}q_{mag}\\\
q_{el}\end{array}\right)=\left(\begin{array}[]{c}q_{NS5}\,e^{-\phi}\\\
e^{-\rho}(q_{D5}-\chi_{1}q_{NS5})\\\
e^{\phi}(q_{F1}+\chi_{1}\,q_{D1}-\chi_{2}\,q_{D5}+\chi_{1}\chi_{2}\,q_{NS5})\\\
e^{\rho}(q_{D1}+\chi_{2}\,q_{NS5})\end{array}\right)$ (B.29)

We immediately find

$\displaystyle|Z_{+}|^{2}$
$\displaystyle=\big{[}q_{NS5}\,e^{-\phi}+e^{\phi}(q_{F1}+\chi_{1}q_{D1}-\chi_{2}q_{D5}+\chi_{1}\chi_{2}\,q_{NS5})\big{]}^{2}+$
$\displaystyle+\big{[}e^{-\rho}(q_{D5}-\chi_{1}q_{NS5})+e^{\rho}(q_{D1}+\chi_{2}q_{NS5})\big{]}^{2}$
(B.30)

The matter central charge $|Z_{-}|^{2}$ is given by the same expression with
relative minus signs, and one can check that $|Z_{+}|^{2}-|Z_{-}|^{2}$ is
moduli-independent, as expected.

###  B.2 Putting various string solutions in six-dimensional form

In this appendix, we exemplify how we put the string solutions of section 3 in
the form above.

#### The pure NS5-F1 solution

For this solution, written in the main text, the axions are zero, and so is
$F_{3}$. The vielbein (B.8) splits as (up to a transposition of the second and
the third row)

$g=\left(\begin{array}[]{cc}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)+\left(\begin{array}[]{cc}\cosh\rho&\sinh\rho\\\
\sinh\rho&\cosh\rho\end{array}\right)\otimes\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)$ (B.31)

with the only non-trivial dynamics affecting the six-dimensional dilaton

$e^{2\phi}=g_{6}^{2}\frac{f_{5}}{f_{1}}\;,\;\;\;\;\;\;\;e^{-2\rho}=v=const.$
(B.32)

Note that only $H_{3}$ contributes to the forms $F^{\Lambda}$. Numbering the
$\Lambda$ index as $\Lambda=\\{1,2,\bar{1},\bar{2}\\}$, where $1,\bar{1}$
corespond to the combinations in (B.25), and $2,\bar{2}$ to those in (B.26),
the non-zero field strengths that satisfy the modified (anti)-self duality
conditions (2.8) are

$F^{1}=\frac{1}{4}(H_{3}+e^{-2\phi}\star
H_{3})\;,\;\;\;\;\;F^{\bar{1}}=\frac{1}{4}(-H_{3}+e^{-2\phi}\star H_{3})$
(B.33)

We immediately note that the charges of these combinations are integer
quantized, being given by $(p\pm k)/2$. The magnetic charge of $F^{1,\bar{1}}$
equals plus or minus its electric charge, as expected.

The combinations that appear in the supersymmetry variations are

$F^{+}=X^{1}_{\Lambda}F^{\Lambda}=\frac{e^{-\phi}}{4}(H_{3}+\star
H_{3})\;,\;\;\;\;\;F^{-}=X^{\bar{1}}_{\Lambda}F^{\Lambda}=\frac{e^{-\phi}}{4}(-H_{3}+\star
H_{3})$ (B.34)

which are, indeed, properly (anti)-self-dual. The explicit expresion for the
quantities in parantheses is

$H_{3}\pm\star
H_{3}=2\alpha^{\prime}\bigg{(}k\pm\frac{g_{s}^{2}pf_{5}}{vf_{1}}\bigg{)}(\omega_{S^{3}}\pm\star\omega_{S^{3}})$
(B.35)

Note that the anti-self-dual part of $H_{3}$ vanishes in the near-horizon
limit.

Let us now check this solution matches indeed the general one, (2.17). The
Einstein frame metric is

$ds^{2}=\frac{1}{g_{6}\sqrt{f_{1}f_{5}}}(-dt^{2}+d\sigma^{2})+\frac{\sqrt{f_{1}f_{5}}}{g_{6}}(dr^{2}+r^{2}d\Omega_{3}^{2})$
(B.36)

with $g_{6}=g_{s}/\sqrt{v}$. From this, we read off that

$H^{\Lambda}H_{\Lambda}=g_{6}^{2}f_{1}f_{5}$ (B.37)

and that we need to rescale $r$ by a factor of $g_{6}$: $r=g_{6}\tilde{r}$.
Note from this that $c_{\Lambda}c^{\Lambda}=g_{6}^{2}$, which indeed
degenerates in the decoupling limit.

The harmonic functions are given by identifying the various rows of (B.31)
with our general solution (2.43), leading to

$H^{1}=\cosh\phi\,\sqrt{H_{\Lambda}H^{\Lambda}}=\frac{1}{2}(f_{1}+g_{6}^{2}f_{5})=\frac{\alpha^{\prime}(p+k)}{2}+\frac{1}{2}(1+g_{6}^{2})$
(B.38)
$H^{\bar{1}}=-\sinh\phi\,\sqrt{H_{\Lambda}H^{\Lambda}}=\frac{1}{2}(f_{1}-g_{6}^{2}f_{5})=\frac{\alpha^{\prime}(p-k)}{2\tilde{r}^{2}}+\frac{1}{2}(1-g_{6}^{2})$
(B.39)

The modulus $v$ simply enters the D1-D5 CFT moduli, via

$\bar{g}_{2\Lambda}=\left(0,\cosh v,0\,\sinh v\right)$ (B.40)

which is obviously orthogonal to the charge and the $c_{\Lambda}$.

#### The pure D1-D5 solution

This entirely parallels the previous discussion, we just concentrate on $\rho$
instead of $\phi$ and switch $H_{3}\leftrightarrow F_{3}$. The solution for
the scalars is now

$e^{-2\rho}=v\frac{f_{1}}{f_{5}}\;,\;\;\;\;\;e^{2\phi}=g_{6}^{2}=const.$
(B.41)

where $v,g_{6}$ are now the corresponding string frame K3 volume and six-
dimensional string coupling in this frame. Note that S-duality exchanges
$\phi$ with $\rho$ and $g_{6}^{2}$ with $v^{-1}$, so the solution above is the
same as (B.32), with these identifications.

To find the map to our $H_{\Lambda}$ we put the metric (3.21) in $6d$ Einstein
frame. One should be careful that S-duality yields an overall factor of
$g_{s}$ (which we ommitted in (3.21)), and thus the overall factor we obtain
is $g_{s}/g_{6}=\sqrt{v}$. We thus find the identification

$H_{\Lambda}H^{\Lambda}=\frac{f_{1}f_{5}}{v}\;,\;\;\;\;\tilde{r}=r\sqrt{v}$
(B.42)

The non-zero harmonic functions then are

$H^{2,\bar{2}}=\frac{1}{2}(e^{-\rho}\pm
e^{\rho})\sqrt{H_{\Lambda}H^{\Lambda}}=\frac{1}{2}(f_{1}\pm\frac{f_{5}}{v})=\frac{g_{s}\alpha^{\prime}(p\pm
k)}{2\tilde{r}^{2}}+\frac{1}{2}(1\pm 1/v)$ (B.43)

Now, the horizon moduli are given by $g_{6}$, which maps as expected to $v$ in
the NS case.

#### The D1-D5 solution with axionic moduli

It is also interesting to understand how to turn on more general moduli in
this solution. We will be working within the $SO(2,2)$ truncation described
above, and choose the D1-D5 frame for concreteness. As discussed above, in
this frame the volume of the K3 is an attracted scalar, while the $6d$ dilaton
is unfixed and constant throughout. We can trivially turn on two constant
modes for $\chi_{1,2}$ via large gauge transformations; they are denoted as
$s_{0},s_{4}$. These shifts in the axions induce an F1 charge, as the charges
of the solution are

$q_{NS5}=0\;,\;\;\;\;q_{D5}=k\;,\;\;\;\;\;q_{F1}=ks_{4}-ps_{0}\;,\;\;\;\;\;q_{D1}=p$
(B.44)

Note, however, that in the full string theory this charge is quantized.

At the level of the $6d$ fields, the scalars shift predictably

$e^{\phi}=g_{6}\;,\;\;\;\;\;e^{\rho}=\sqrt{\frac{f_{5}}{vf_{1}}}\;,\;\;\;\;\;\chi_{1}=s_{0}\;,\;\;\;\;\;\chi_{2}=s_{4}$
(B.45)

and the scalar matrix is given by (B.8) with these replacements.

$\displaystyle
g=\begin{pmatrix}\frac{g_{6}^{2}(s_{0}s_{4}+1)+1}{2g_{6}}&\frac{1}{2}g_{6}(s_{0}-s_{4})&\frac{g_{6}^{2}(1-s_{0}s_{4})-1}{2g_{6}}&\frac{1}{2}g_{6}(s_{0}+s_{4})\\\
\frac{f_{5}s_{4}-f_{1}s_{0}v}{2\sqrt{vf_{1}f_{5}}}&\frac{f_{5}+vf_{1}}{2\sqrt{vf_{1}f_{5}}}&\frac{f_{1}s_{0}v-f_{5}s_{4}}{2\sqrt{vf_{1}f_{5}}}&\frac{f_{5}-vf_{1}}{2\sqrt{vf_{1}f_{5}}}\\\
\frac{g_{6}^{2}(s_{0}s_{4}+1)-1}{2g_{6}}&\frac{1}{2}g_{6}(s_{0}-s_{4})&\frac{g_{6}^{2}(1-s_{0}s_{4})+1}{2g_{6}}&\frac{1}{2}g_{6}(s_{0}+s_{4})\\\
\frac{f_{5}s_{4}+f_{1}s_{0}v}{2\sqrt{vf_{1}f_{5}}}&\frac{f_{5}-vf_{1}}{2\sqrt{vf_{1}f_{5}}}&-\frac{f_{1}s_{0}v+f_{5}s_{4}}{2\sqrt{vf_{1}f_{5}}}&\frac{f_{5}+vf_{1}}{2\sqrt{vf_{1}f_{5}}}\end{pmatrix}$
(B.46)

#### The D1-D5 frame TsT solution

We generate the (non)-extremal such solution by performing a TsT on the
general/extremal D1-D5 one. In the following, we will be giving the
expressions for the non-extremal solutions, generated from the seed

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{\sqrt{f_{1}f_{5}}}\left[d\sigma^{2}-dt^{2}+\frac{r_{0}^{2}}{r^{2}}(\cosh\alpha_{n}dt+\sinh\alpha_{n}d\sigma)^{2}\right]+\sqrt{f_{1}f_{5}}\left(\frac{dr^{2}}{f}+r^{2}d\Omega_{3}^{2}\right)+\sqrt{\frac{f_{1}}{f_{5}}}ds^{2}_{K3}$
(B.47)

where the function $f$ is the blackening factor, $f=1-r_{0}^{2}/r^{2}$ and
$f_{1,5}=1+r_{1,5}^{2}/r^{2}$, with $r_{1},r_{5}$ defined in (4.15). The
dilaton and the 3-form field have the same expressions as in the extremal
case, in terms of the non-extremal functions $f_{1},f_{5}$. After the TsT
(followed by two trivial axionic shifts, which will be useful later), we
obtain:

$\displaystyle ds^{2}$
$\displaystyle=\frac{\sqrt{f_{1}f_{5}}}{f_{1}f_{5}-\gamma^{2}f}\big{(}d\sigma^{2}-fdt^{2}\big{)}+\sqrt{f_{1}f_{5}}\big{(}\frac{dr^{2}}{f}+r^{2}d\Omega^{2}_{3}\big{)}+\sqrt{\frac{f_{1}}{f_{5}}}\,ds^{2}_{K3}$
(B.48) $\displaystyle B$ $\displaystyle=\frac{\gamma
f}{f_{1}f_{5}-\gamma^{2}f}dt\wedge d\sigma$ (B.49) $\displaystyle e^{\Phi}$
$\displaystyle=g_{s}\frac{f_{1}}{\sqrt{f_{1}f_{5}-\gamma^{2}f}}$ (B.50)
$\displaystyle C_{0}$
$\displaystyle=\frac{Q_{1}}{g_{s}r_{1}^{2}}\frac{\gamma}{f_{1}}+s_{0}$ (B.51)
$\displaystyle C_{2}$
$\displaystyle=-\frac{2Q_{5}}{g_{s}}\frac{1}{8}\cos\theta d\psi\wedge
d\phi+\frac{Q_{1}f_{5}-s_{0}g_{s}r_{1}^{2}\gamma
f}{g_{s}r_{1}^{2}(f_{1}f_{5}-\gamma^{2}f)}dt\wedge d\sigma$ (B.52)
$\displaystyle C_{4}$
$\displaystyle=\bigg{(}-\frac{v}{g_{s}}\frac{Q_{5}}{r_{5}^{2}}\frac{\gamma}{f_{5}}+s_{4}\bigg{)}\omega_{K3}-\frac{\gamma
fQ_{5}\cos\theta}{4g_{s}(f_{1}f_{5}-f\gamma^{2})}dt\wedge d\sigma\wedge
d\psi\wedge d\phi$ (B.53)

We also introduced the notations
$Q_{1}=\frac{\alpha^{\prime}g_{s}p}{v},Q_{5}=\alpha^{\prime}g_{s}k$. To obtain
the extremal solution, we simply send $r_{0}\rightarrow 0$, keeping
$r_{1}^{2}=Q_{1},r_{5}^{2}=Q_{5}$ fixed. Note this sets $f=1$.

The six-dimensional fields can be readily extracted from the above. We
particularize to the extremal case, which is the only one we need to match.
The scalars are

$e^{2\phi}=g_{6}^{2}\frac{f_{1}f_{5}}{f_{1}f_{5}-\gamma^{2}}\;,\;\;\;\;\;e^{2\rho}=\frac{f_{5}}{vf_{1}}\;,\;\;\;\;\chi_{1}=\frac{\gamma}{g_{s}f_{1}}+s_{0}\;,\;\;\;\;\chi_{2}=-\frac{v\gamma}{g_{s}f_{5}}+s_{4}$
(B.54)

and the Einstein frame metric

$ds^{2}_{6}=\frac{1}{\sqrt{\frac{1}{v}(f_{1}f_{5}-\gamma^{2})}}\big{(}d\sigma^{2}-dt^{2}\big{)}+\sqrt{\frac{1}{v}(f_{1}f_{5}-\gamma^{2})}\bigg{(}dr^{2}+r^{2}d\Omega_{3}^{2}\bigg{)}$
(B.55)

where we have rescaled $r$ as in (B.42), and then dropped the tilde. The six-
dimensional charges are obtained by integrating the field strengths (B.25),
(B.26), evaluated on this solution, on $S^{3}$

$q^{\Lambda}={\color[rgb]{0,0.88,0}}\frac{1}{2}\\{ks_{4}-ps_{0},k+p,ks_{4}-ps_{0},p-k\\}$
(B.56)

Using the $6d$ solution and the above charges, we can compute the central
charge vector. Its modulus is

$|Z_{+}|=\frac{f_{5}p+f_{1}kv}{2\sqrt{v}\sqrt{f_{1}f_{5}-\gamma^{2}}}$ (B.57)

and the following rotation aligns it with the previous direction

$\left(\begin{array}[]{cc}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}Z_{1}\\\
Z_{2}\end{array}\right)=\left(\begin{array}[]{c}0\\\
|Z_{+}|\end{array}\right)\;,\;\;\;\;\;\sin\alpha=\frac{\gamma}{\sqrt{f_{1}f_{5}}}$
(B.58)

Note the rotation angle vanishes at the horizon, while approaching $\pi/2$ at
infinity for the decoupled NCOS solutions with $\gamma=1$.

Upon this rotation, the coset solution takes the standard form (2.43), and we
read off

$\displaystyle\bar{g}_{i\Lambda}=\bigg{\\{}\frac{g_{6}^{2}(s_{0}s_{4}+1)+1}{2g_{6}},\;\frac{1}{2}g_{6}(s_{0}-s_{4}),\;\frac{g_{6}^{2}(1-s_{0}s_{4})-1}{2g_{6}},\;\frac{1}{2}g_{6}(s_{0}+s_{4})\bigg{\\}}$
(B.59)

exactly as in the undeformed D1-D5 solution, and

$\displaystyle H_{\Lambda}=\bigg{\\{}$
$\displaystyle\frac{-f_{1}g_{6}vs_{0}+g_{6}f_{5}s_{4}+\gamma\sqrt{v}(g_{6}^{2}(s_{0}s_{4}+1)-1)}{2g_{6}v},\;\frac{vf_{1}+f_{5}+\gamma\sqrt{v}g_{6}(s_{0}-s_{4})}{2v},\;$
$\displaystyle\frac{f_{1}g_{6}vs_{0}-f_{5}g_{6}s_{4}+\gamma\sqrt{v}(g_{6}^{2}(1-s_{0}s_{4})+1)}{2g_{6}v},\;\frac{-f_{1}v+f_{5}+\gamma\sqrt{v}g_{s}(s_{0}+s_{4})}{2v}\bigg{\\}}$
(B.60)

#### The S-dual (NS frame) TsT solution

The non-extremal solution obtained by S-dualising (B.48) is given by

$\displaystyle ds^{2}$
$\displaystyle=\sqrt{\frac{f_{1}f_{5}-\gamma^{2}f+\bigg{(}\frac{Q_{1}\gamma}{r_{1}^{2}}+f_{1}s_{0}\bigg{)}^{2}}{f_{1}f_{5}}}\bigg{[}\frac{f_{5}}{f_{1}f_{5}-\gamma^{2}f}(d\sigma^{2}-fdt^{2})+f_{5}\bigg{(}\frac{dr^{2}}{f}+r^{2}d\Omega^{2}_{3}\bigg{)}+ds^{2}_{K3}\bigg{]}$
$\displaystyle e^{\Phi}$
$\displaystyle=g_{s}\frac{f_{1}f_{5}-\gamma^{2}f+(f_{1}g_{s}s_{0}+\frac{Q_{1}\gamma}{r_{1}^{2}})^{2}}{f_{1}\sqrt{f_{1}f_{5}-\gamma^{2}f}}$
(B.61) $\displaystyle C_{0}$
$\displaystyle=\frac{1}{g_{s}}\frac{-f_{1}(\frac{Q_{1}\gamma}{r_{1}^{2}}+f_{1}g_{s}s_{0})}{f_{1}f_{5}-\gamma^{2}f+(f_{1}g_{s}s_{0}+\frac{Q_{1}\gamma}{r_{1}^{2}})^{2}}$
(B.62) $\displaystyle B$ $\displaystyle=-2Q_{5}\frac{1}{8}\cos\theta
d\psi\wedge d\phi+\frac{Q_{1}f_{5}-s_{0}g_{s}r_{1}^{2}\gamma
f}{r_{1}^{2}(f_{1}f_{5}-\gamma^{2}f)}dt\wedge d\sigma$ (B.63) $\displaystyle
C_{2}$ $\displaystyle=-\frac{\gamma f}{g_{s}(f_{1}f_{5}-\gamma^{2}f)}dt\wedge
d\sigma$ (B.64)

where $s_{0}$ is the previously introduced arbitrary shift of the S-dual
axion. The type IIB four-form potential can be obtained from the one in the
D1-D5 frame by $\tilde{C}_{4}=C_{4}-B\wedge C_{2}$ and its part on $K3$ is not
changed.

In order for the dilaton to not diverge at infinity, we need to fix
$s_{0}=-\frac{\gamma}{g_{s}}\frac{Q_{1}}{r_{1}^{2}}$. Charge quantizationthen
fixes $s_{4}$ in terms of $s_{0}$ as explained in section 3.

The six-dimensional fields read (particularising again to the extremal
solution)

$e^{-2\rho}=\frac{v}{g_{s}^{2}}\frac{f_{5}+\gamma^{2}(f_{1}-2)}{f_{5}}\;,\;\;\;\;\;e^{2\phi}=\frac{1}{v}\frac{f_{5}(f_{5}+\gamma^{2}(f_{1}-2))}{f_{1}f_{5}-\gamma^{2}}$
(B.65)

and axions

$\chi_{1}=\frac{1}{g_{s}}\frac{\gamma(f_{1}-1)}{f_{5}+\gamma^{2}(f_{1}-2)}\;,\;\;\;\;\;\;\chi_{2}=--\frac{v}{g_{s}}\frac{Q_{5}}{r_{5}^{2}}\frac{\gamma}{f_{5}}+s_{4}$
(B.66)

There is again an $SO(2)$ rotation that brings the solution to the standard
form.

##  Appendix C Energy of the non-extremal NCOS and OD1 backgrounds

In this appendix we compute the energy of the backgrounds discussed in section
3, which fit into the six-dimensional truncation (B.1). We work in the
covariant phase space formalism, for which we summarize the necessary formulae
below, following [85, 84].

###  C.1 Useful formulae from the covariant phase space formalism

Let us consider an asymptotic Killing vector $\xi$ and a path in phase space
between two solutions of the eom associated to a Lagrangian. The infinitesimal
variation in phase space of the charge associated to $\xi$ is given by:

$\displaystyle\delta\mathcal{Q}_{\xi}$
$\displaystyle=\frac{1}{2!4!}\frac{1}{8\pi
G_{6}}\int_{S_{4}}\epsilon_{\alpha\beta\gamma\sigma\mu\nu}k^{\mu\nu}_{\xi}dx^{\alpha}\wedge
dx^{\beta}\wedge dx^{\gamma}\wedge dx^{\sigma}$ (C.1)

where $S_{4}$ is a compact four-dimensional spacelike surface and
$\epsilon_{\alpha\beta\gamma\sigma\mu\nu}k^{\mu\nu}_{\xi}$ are the components
of a four-form that receives contributions from the various fields of the
theory. The formula is written for six-dimensional theories, but can be
trivially generalized to any number of dimensions. For the particular case
$\xi=\partial_{t}$, one obtains, after integration on the path in phase space,
the energy difference between the two solutions.

In the case of a Lagrangian of the form:

$\displaystyle
R-\frac{1}{2}f_{ab}(\phi)\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}-V(\phi)-\frac{1}{12}k_{ij}(\phi)F^{i}_{\mu\nu\rho}F^{j\mu\nu\rho}+\frac{1}{3!3!}h_{ij}(\phi)\epsilon^{\mu\nu\rho\sigma\alpha\beta}F^{i}_{\mu\nu\rho}F^{j}_{\sigma\alpha\beta}$
(C.2)

the contributions come from the metric, the scalars and the 3-forms, for which
we can split them again into kinetic and topological:

$\displaystyle k^{\mu\nu}_{\xi}$
$\displaystyle=k^{\mu\nu}_{\xi,g}+k^{\mu\nu}_{\xi,sc}+k^{\mu\nu}_{\xi,kin}+k^{\mu\nu}_{\xi,top}$
(C.3)

The explicit expressions are written below, in the standard notation
$h_{\mu\nu}=\delta g_{\mu\nu},h=g^{\mu\nu}\delta g_{\mu\nu}$ and $\delta
f^{\alpha_{1}...\alpha_{n}}=g^{\alpha_{1}\beta_{1}}...g^{\alpha_{n}\beta_{n}}\delta
f_{\beta_{1}...\beta_{n}}$:

$\displaystyle k^{\mu\nu}_{\xi,g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\xi^{\mu}\nabla_{\alpha}h^{\alpha\nu}-\xi^{\mu}\nabla^{\nu}h+\xi_{\alpha}\nabla^{\nu}h^{\alpha\mu}+\frac{1}{2}h\nabla^{\nu}\xi^{\mu}-\frac{1}{2}h^{\nu\alpha}\nabla_{\alpha}\xi^{\mu}+\frac{1}{2}h^{\nu\alpha}\nabla^{\mu}\xi_{\alpha}-(\mu\leftrightarrow\nu)\bigg{)}$
$\displaystyle k^{\mu\nu}_{\xi,sc}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\xi^{\nu}f_{ab}(\phi)\nabla^{\mu}\phi^{a}\delta\phi^{b}-(\mu\leftrightarrow\nu)\bigg{)}$
(C.4) $\displaystyle k^{\mu\nu}_{\xi,kin}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigg{[}-\partial_{\phi^{a}}k_{ij}(\phi)F^{j\mu\nu\lambda}\xi^{\alpha}C_{\alpha\lambda}\delta\phi^{a}+k_{ij}(\phi)\bigg{(}-\delta
F^{j\mu\nu\lambda}+2h^{\mu\sigma}F^{j\;\;\nu\lambda}_{\;\sigma}+h^{\lambda\sigma}F^{j\mu\nu}_{\;\;\sigma}-$
$\displaystyle-\frac{1}{2}hF^{j\mu\nu\lambda}\bigg{)}\xi^{\alpha}C^{i}_{\alpha\lambda}-k_{ij}(\phi)F^{j\mu\nu\lambda}\xi^{\alpha}\delta
C^{i}_{\alpha\lambda}-k_{ij}(\phi)\xi^{\mu}F^{j\nu\alpha\lambda}\delta
C^{i}_{\alpha\lambda}-(\mu\leftrightarrow\nu)\bigg{]}$ $\displaystyle
k^{\mu\nu}_{\xi,top}$ $\displaystyle=$
$\displaystyle\frac{1}{24}\bigg{[}\epsilon^{\mu\nu\alpha\beta\gamma\sigma}\bigg{(}(\partial_{\phi^{a}}h_{ij}(\phi))F^{j}_{\beta\gamma\sigma}\xi^{\rho}C^{i}_{\alpha\rho}\delta\phi^{a}+h_{ij}(\phi)\delta
F^{j}_{\beta\gamma\sigma}\xi^{\rho}C^{i}_{\alpha\rho}+h_{ij}(\phi)F^{j}_{\beta\gamma\sigma}\xi^{\rho}\delta
C^{i}_{\alpha\rho}\bigg{)}-$ (C.5)
$\displaystyle-2h_{ij}(\phi)\xi^{\nu}\epsilon^{\mu\eta\alpha\beta\gamma\sigma}F^{j}_{\beta\gamma\sigma}\delta
C^{i}_{\eta\alpha}-(\mu\leftrightarrow\nu)\bigg{]}$

where $F^{i}=dC^{i}$ and we ignored the terms in [84] that are automatically 0
for $\xi$ isometry.

###  C.2 Details of the computation

The Lagrangian B.1 is of the form C.2, so we can apply the formulae above in
order to compute the energy of the backgrounds discussed in section 3. For our
particular case, $\xi=\partial_{t}$ and the integration is on $S^{1}\times
S^{3}$, so we are interested in the $tr$ component of the expressions written
above.

Since the metric only depends on $r_{0}$, the variation in phase space that we
need to consider in order to compute the metric contribution to the energy is
simply $\delta=\delta r_{0}\partial_{r_{0}}$. The result, after the trivial
integration in phase space, is:

$\displaystyle E_{g}=\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}\bigg{(}\frac{r_{0}^{2}}{2}+\frac{r_{0}^{2}+r_{1}^{2}+r_{5}^{2}}{1-\gamma^{2}}\bigg{)}$
(C.6)

For $\gamma=0$, this is the energy, before subtracting the extremal
contribution, for the D1-D5 background at zero momentum.

For the scalars $\chi_{1,2}$, we allow for a variation of their asymptotic
values $\delta=\delta r_{0}\partial_{r_{0}}+\delta
a_{0}\partial_{a_{0}}+\delta a_{4}\partial_{a_{4}}$, where
$a_{0}=s_{0}-\frac{q_{D1}\gamma}{r_{1}^{2}}$ and
$a_{4}=s_{4}+\frac{q_{D5}\gamma}{r_{5}^{2}}$. We obtain:

$\displaystyle E_{sc}=\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}\frac{\gamma}{1-\gamma^{2}}(q_{D5}s_{4}-q_{D1}s_{0})$ (C.7)

For the contribution for the 3-forms, we first introduce by hand a constant
piece $s_{b}dt\wedge d\sigma$ in the B-field and allow again for a variation
of its asymptotic value, which we will denote by $b_{1}$, namely
$b_{1}=s_{b}-\frac{\gamma}{1-\gamma^{2}}$. We obtain that we need to set
$\delta b_{1}=0$, namely to take it to be parameter independent, in order for
the result to be integrable in phase space. With this restriction, we obtain:

$\displaystyle E_{form}$ $\displaystyle=-\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}b_{1}\big{(}\gamma(r_{0}^{2}+r_{1}^{2}+r_{5}^{2})-q_{D1}s_{0}\big{)}$
(C.8) $\displaystyle E_{top}$ $\displaystyle=-\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}b_{1}q_{D5}s_{4}$ (C.9)

Adding the four contributions, we find

$E=\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}\left[\frac{r_{0}^{2}}{2}+\frac{r_{0}^{2}+r_{1}^{2}+r_{5}^{2}}{1-\gamma^{2}}(1-\gamma
b_{1})+\left(\frac{\gamma}{1-\gamma^{2}}-b_{1}\right)(q_{D5}s_{4}-q_{D1}s_{0})\right]$
(C.10)

From here it follows that by setting $b_{1}=\frac{\gamma}{1-\gamma^{2}}$ i.e.
not doing anything to the B-field obtained by TsT, the dependence on the
asymptotic values of the scalars $a_{0},a_{4}$ drops out and we obtain:

$\displaystyle E|_{b_{1}=\frac{\gamma}{1-\gamma^{2}}}$
$\displaystyle=\frac{Rv}{\alpha^{\prime
2}g_{s}^{2}}\bigg{(}\frac{r_{0}^{2}}{2}+r_{0}^{2}+r_{1}^{2}+r_{5}^{2}\bigg{)}$
(C.11)

where we can see that also the $\gamma$ dependence drops out.

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