Imperial/TP/2023/KS/01

Localised Gravity and Resolved Braneworlds

Rahim Leung 111<EMAIL_ADDRESS>and K.S. Stelle 222
<EMAIL_ADDRESS>
The Blackett Laboratory, Imperial College London
Prince Consort Road, London, SW7 2AZ

Dedicated to the Memory of Stanley Deser

###### Abstract

Deriving an effective massless field theory for fluctuations about a
braneworld spacetime requires analysis of the transverse-space-wavefunction’s
second-order differential equation. There can be two strikingly different
types of effective theory. For a supersymmetric braneworld, one involves a
technically consistent embedding of a supergravity theory on the worldvolume;
the other can produce, in certain situations, a genuine localisation of
gravity near the worldvolume but not via a technically consistent embedding.
So, in the latter situation, the theory’s dynamics remains higher-dimensional
but there can still be a lower-dimensional effective-theory interpretation of
the dynamics at low worldvolume momenta / large worldvolume distances.

This paper examines the conditions for such a gravity localisation to be
possible. Localising gravity about braneworld spacetimes requires finding
solutions to transverse-space self-adjoint Sturm-Liouville problems admitting
a normalisable zero mode in the noncompact transverse space. This in turn
requires analysis of Sturm-Liouville problems with radial singular endpoints
following a formalism originating in the work of Hermann Weyl. Examples of
such gravity-localising braneworld systems are found and analysed in this
formalism with underlying “skeleton” braneworlds of Salam-Sezgin, resolved
D3-brane and Randall-Sundrum II types.

###### Contents

  1. 1 Introduction: Brane dynamics taxonomy
  2. 2 Resolved D3-branes on Eguchi-Hanson space: consistently embedded worldvolume supergravity versus genuine localisations
  3. 3 Localised worldvolume gravitons on resolved branes
  4. 4 Some Sturm-Liouville theory
  5. 5 Examples
    1. 5.1 Example 1: Randall-Sundrum
    2. 5.2 Example 2: Salam-Sezgin
    3. 5.3 Example 3: D3-branes on a resolved conifold over $S^{5}/\mathbb{Z}_{3}$
    4. 5.4 Example 4: D3-branes on resolved conifolds over $Y^{p,q}$
    5. 5.5 Example 5: D3-branes on a resolved cone over $T^{1,1}/\mathbb{Z}_{2}$
    6. 5.6 Example 6 (no localisation): NS5-branes on Taub-NUT
  6. 6 Conclusion and a Conjecture
  7. Appendices
  8. A M-theory and Type IIA supergravity conventions
  9. B Type IIB supergravity conventions
  10. C Bucher rules for Abelian T-duality
  11. D Type IIB embedding of the Salam-Sezgin theory

##  1 Introduction: Brane dynamics taxonomy

That superstring theories are about more than just perturbative strings has
become clear over the past two decades. The wide array of brane solutions of
the corresponding effective supergravity theories fills out a solitonic sector
which also carries representations of the associated duality symmetries in the
originating and reduced lower spacetime dimensions. Those dualities also are
the key links between the variants of perturbatively quantisable superstring
theories. The descent through effective theory spacetime dimensions has mostly
been considered in the context of dimensional compactification of the extra
dimensions transverse to a given lower dimensional spacetime.

Compactification of the transverse dimensions is not the only possibility,
however. The array of supergravity brane solution worldvolumes presents
another kind of locale in which to situate lower-dimensional physics. There is
a basic difference, however: unless they are “stacked” [1] so as to generate
compact dimensions transverse to their worldvolumes, brane transverse spaces
are generically noncompact. Dimensional reduction on noncompact spaces in turn
generically runs into a problem identified by Hull and Warner [2], that the
transverse wave-equation spectrum is generically a continuum running down to
zero eigenvalue, leading to a vanishing Newton constant for coupling the lower
dimensional gravity to other fields. Generically does not mean universally,
however, as was shown by Randall and Sundrum [3] in a construction based on
sliced and reflected AdS subspaces.

Another situation where noncompact transverse spaces can avoid the problem of
a vanishing Newton constant is where Newton’s constant is not fundamentally
meaningful, owing to the existence (at least at the classical level in
effective theory) of a “trombone” symmetry [4] such as that in pure vacuum
Einstein theory. Pure Einstein theory without external field-theory sources
has, up to bothersome issues about GR delta-function singularities [5], black
hole solutions such as the Schwarzschild solution. The black-hole orbits of
pure GR’s trombone symmetry are parametrised by the mass of the black hole.
This is also the situation for pure-gravity worldvolume dynamics obtained by
replacing the flat brane worldvolume metric by a Ricci-flat metric [6],
producing a consistent truncation of the higher dimensional originating theory
down to pure GR on the worldvolume. Moreover, for an underlying “skeleton”
brane with unbroken supersymmetry, this sort of consistent lower-dimensional
theory embedding can extend to a full supergravity theory corresponding to the
skeleton unbroken supersymmetry [7]. In the terminology of Ref. [8], such a
consistent-truncation effective-theory origin involves Type I boundary
conditions for the transverse-space eigenfunctions as one approaches the
skeleton-brane worldvolume.

For transverse space differential equations of second order in derivatives,
however, there will be a variety of boundary condition choices as one
approaches a brane worldvolume. Type III boundary conditions, when available,
can produce a different type of reduced dimension effective theory – not
involving a consistent truncation, but with a genuine dynamical localisation
of gravity and companion fields into the region near the brane worldvolume. An
initial example [9] of such Type III localisation involves the lift to $D=10$
Type IIA supergravity (or further up to $D=11$ M-Theory) of the $D=6$ Salam-
Sezgin vacuum (called the SS-CGP solution in the following) [10, 9], which
involves a hyperbolic ${\cal H}^{(2,2)}$ transverse-space metric. That
transverse-space geometry allows for a dynamical localisation of gravity near
the lower-dimensional worldvolume owing to a special feature of the
transverse-space wave equation. When transformed into a Schrödinger equation
form, it has a Pöschl-Teller potential for which the spectrum has a single
$L^{2}$ normalisable bound state separated from the edge of the continuous
spectrum by a mass gap. It is the existence of the $L^{2}$ normalisable
transverse state that gives rise to gravity localisation near the lower-
dimensional worldvolume. The required boundary conditions on the transverse
spectrum is of Robin form for this Type III system, distinct from the
Dirichlet-skeleton conditions of a Type I reduction.111Distinctions between
boundary condition choices are also related to distinctions between possible
source terms at transverse-space endpoints. Consideration of source terms have
been treated in Ref. [9] and also more recently in Ref. [11].

In this paper, we begin in Section 2 with an initial example that allows for
such a choice between either a Type I consistent supergravity embedding or a
Type III localisation, involving a resolved D3 brane with an Eguchi-Hanson
transverse geometry. This is related, via a sequence of dualities, to the SS-
CGP skeleton of Ref. [9].

This is followed in Section 3 with a review of the formalism for localised
graviton fluctuations and the transverse-space wave equation about a
“skeleton” quiescent brane, particularly for resolved branes.

In Section 4, we set the stage for a general discussion of localisation
possibilities based on Sturm-Liouville theory for systems with singular
endpoints, originating in the work of Hermann Weyl [12] and fully developed in
the work of Anton Zettl and collaborators, as presented in Ref. [13]. The
heart of the matter is a characterisation of the available boundary conditions
as one approaches the lower-dimensional worldvolume and at infinity.

We then apply this analysis in Section 5 to five examples where boundary
conditions giving gravity localisation are successfully found and also one
example where localisation is not found. We end with a conjecture on the
relevance of a two-dimensional subspace of the transverse space and the
attendant logarithmic structure of the normalisable transverse zero mode.

After the Conclusion in Section 6, we give four technical Appendices.

##  2 Resolved D3-branes on Eguchi-Hanson space: consistently embedded
worldvolume supergravity versus genuine localisations

We’ll consider here, a particular system of resolved D3-branes by
transgression, the general formalism of which is discussed in Ref. [14]. Here,
the RR and NSNS three-form fluxes are present and depend on the six-
dimensional space transverse to the worldvolume of the branes. Such a
configuration is also given the name of fractional D3-branes, and are
holographically dual to $\mathcal{N}=1$ super Yang-Mills in four dimensions
[15, 16, 17, 18, 19]. Our conventions for M-theory, Type IIA and Type IIB
supergravities are given in Appendices A and B.

The resolved D3-branes that we consider are geometrically a warped product
$\mathbb{R}^{1,3}\times_{W}(M_{4}\times T^{2})$, where $M_{4}$ is the Eguchi-
Hanson space. To obtain such a solution, we will follow the recipe of [14] and
propose the ansatz222In the language of [14], we have $\hat{G}_{(3)}=L_{(3)}$,
where $L_{(3)}$ is a complex self-dual three-form satisfying
${\ast}_{6}L_{(3)}=iL_{(3)}$, where ${\ast}_{6}$ is the Hodge dual with
respect to the Ricci-flat metric on $M_{4}\times T^{2}$.

$\begin{split}&d\hat{s}^{2}=H^{-1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H^{1/2}(ds^{2}_{EH}+dzd\overline{z})\,,\quad\hat{\Phi}=0\,,\quad\hat{C}_{0}=0\,,\\\
&\hat{G}_{(3)}=\mathcal{F}_{(2)}\wedge
d\overline{z}\,,\quad\hat{F}_{(5)}=H^{-2}\operatorname{vol}_{4}\wedge
dH-\frac{i}{2}{\ast}_{EH}dH\wedge dz\wedge d\overline{z}\,.\end{split}$ (2.1)

Here, $\operatorname{vol}_{4}$ is the volume form on $\mathbb{R}^{1,3}$, $z$
is the complex coordinate on $T^{2}$, the Ricci-flat metric on the Eguchi-
Hanson space is

$ds^{2}_{EH}=\cosh
2\rho\left(d\rho^{2}+\frac{1}{4}\tanh^{2}2\rho\,(d\chi+\cos\theta\,d\varphi)^{2}+\frac{1}{4}(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\right)\,,$
(2.2)

with $\rho\geq 0$, $\chi\in[0,2\pi)$, $\theta\in[0,\pi]$ and
$\varphi\in[0,2\pi)$; ${\ast}_{EH}$ is the Hodge dual with respect to
$ds^{2}_{EH}$, $H$ is a function on the Eguchi-Hanson space, and
$\mathcal{F}_{(2)}$ is the unique (up to normalisation), self-dual harmonic
two-form on the Eguchi-Hanson space given by

$\mathcal{F}_{(2)}=d\mathcal{A}_{(1)}\,,\quad\mathcal{A}_{(1)}=\frac{1}{2}\operatorname{sech}2\rho\,(d\chi+\cos\theta\,d\varphi)\,.$
(2.3)

All of the equations of motion are satisfied provided that $H$ is a solution
to the Poisson equation

$d{\ast}_{EH}dH=-\mathcal{F}_{(2)}\wedge{\ast}_{EH}\mathcal{F}_{(2)}\,.$ (2.4)

Assuming spherical symmetry, $H=H(\rho)$, the explicit solution is given by

$H=\operatorname{sech}2\rho-k\log\tanh\rho\,,$ (2.5)

where the logarithmic term is the homogeneous solution of the Poisson equation
(2.4). For $H$ to be positive definite for all $\rho$, we will take $k\geq 0$.
When $k=0$, the solution (2.1) is smooth for all values of $\rho$, and
interpolates between the Ricci-flat vacuum $\mathbb{R}^{1,3}\times M_{4}\times
T^{2}$ at small $\rho$ and the warped
$\mathbb{R}^{1,3}\times\mathbb{R}\text{P}^{3}\times\mathbb{H}^{3}$ solution at
large $\rho$, where the $\mathbb{R}\text{P}^{3}$ is the constant $\rho$ slice
of the Eguchi-Hanson space. When $k>0$, the large $\rho$ asymptotic structure
remains the same, but there is a curvature singularity at $\rho=0$.

Our solution (2.1) is a cousin of the Type IIA solution of [10] which
describes the ten-dimensional embedding of the $\mathbb{R}^{1,3}\times S^{2}$
Salam-Sezgin vacuum with an NS5-brane inclusion, where the NS5-brane has a
tension proportional to the parameter $k$. This solution, which we will call
the CGP-SS solution, is

$\begin{split}&d\tilde{s}^{2}=H^{-1/4}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}+(d\psi+\mathcal{A}_{(1)})^{2}\right)+H^{3/4}ds^{2}_{EH}\,,\\\
&e^{2\tilde{\phi}}=\operatorname{sech}2\rho\,,\quad\tilde{B}_{(2)}=\frac{1}{4}(1+k)\cos\theta\,d\chi\wedge
d\varphi-\mathcal{A}_{(1)}\wedge d\psi\,,\end{split}$ (2.6)

where $H$ is given by (2.5), the Salam-Sezgin gauge parameter $\overline{g}$
is set to $1/\sqrt{2}$ for convenience, and we use tildes to distinguish the
Type IIA fields from the Type IIB ones.

Although the geometric structure of our resolved D3 solution and the CGP-SS
solution are different, they are related by a series of dualities. In terms of
branes, this duality map is

$\text{NS5}\underbrace{\to}_{\text{M-theory/IIA}}\text{D4}\underbrace{\to}_{\text{T-duality}}\text{D3}\,.$
(2.7)

To construct this explicitly, we first lift the CGP-SS solution to M-theory on
a circle with coordinate $\zeta$, then reduce back to Type IIA on the
worldvolume circle $y$. The resulting Type IIA solution describes a resolved
D4-brane, with

$\begin{split}&d\tilde{s}^{2}_{str}=H^{-1/2}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\left(d\psi+\mathcal{A}_{(1)}\right)^{2}\right)+H^{1/2}\left(ds^{2}_{EH}+d\zeta^{2}\right)\,,\\\
&e^{2\tilde{\phi}}=H^{-1/2}\,,\quad\tilde{F}_{(4)}=(2(1+k)\operatorname{vol}(\mathbb{R}\text{P}^{3})-\mathcal{F}_{(2)}\wedge
d\psi)\wedge d\zeta\,,\end{split}$ (2.8)

where

$\operatorname{vol}(\mathbb{R}\text{P}^{3})=\frac{1}{8}\sin\theta\,d\chi\wedge
d\theta\wedge d\varphi$ (2.9)

is the volume form of the asymptotic $\mathbb{R}\text{P}^{3}$ at infinity of
the Eguchi-Hanson space. We have expressed the solution in string frame,
$d\tilde{s}^{2}_{str}=e^{\tilde{\phi}/2}d\tilde{s}^{2}_{Ein}$, which is more
convenient for T-dualisation. Applying the Bucher rules for Abelian T-duality
on the circle $\psi$, which are detailed in Appendix C, we obtain the Type IIB
solution

$\begin{split}&d\hat{s}^{2}=H^{-1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H^{1/2}\left(ds^{2}_{EH}+d\zeta^{2}+d\xi^{2}\right)\,,\quad\hat{\Phi}=0\,,\quad\hat{C}_{0}=0\,,\\\
&\hat{H}_{(3)}=-\mathcal{F}_{(2)}\wedge
d\xi\,,\quad\hat{F}_{(3)}=\mathcal{F}_{(2)}\wedge d\zeta\,,\\\
&\hat{F}_{(5)}=2(k+\tanh^{2}2\rho)(1+\hat{\ast})\operatorname{vol}(\mathbb{R}\text{P}^{3})\wedge
d\zeta\wedge d\xi\,,\end{split}$ (2.10)

where $\xi$ is the T-dual coordinate of $\psi$. Identifying the complex
coordinate $z=\zeta+i\xi$, we find that (2.10) is exactly our resolved
D3-brane solution (2.1). Since the original CGP-SS solution preserves eight
rigid supercharges with a Killing spinor that is independent of the T-dualised
coordinate $\psi$ [9], our resolved D3-brane solution (2.1) also preserves
eight rigid supercharges.

A straightforward extension of this is the construction of the Type IIB T-dual
of the Type IIA embedding of the full six-dimensional Salam-Sezgin theory.
This is done in Appendix D. The result is an embedding of Type IIB
supergravity on a four-dimensional Kähler manifold that is a product
$M_{2}\times T^{2}$, where the metric on $M_{2}$ is the induced metric on the
Eguchi-Hanson space at constant values on the two-sphere. Properties of this
four-manifold, including a construction of its Kähler structure, are explored
in Appendix D.

The fact that (2.1) preserves eight rigid supercharges suggests that we can
embed the four-dimensional $\mathcal{N}=2$ supergravity into its worldvolume,
just as for the usual half-supersymmetric D3-branes. The bosonic sector of the
$\mathcal{N}=2$ theory is the Einstein-Maxwell theory with Lagrangian

$\mathcal{L}_{4}=R{\ast}_{4}1-\frac{1}{2}F_{(2)}\wedge{\ast}_{4}F_{(2)}\,.$
(2.11)

Using similar arguments to [7], one may propose the consistent-truncation
ansatz

$\begin{split}&d\hat{s}^{2}=H^{-1/2}g_{\mu\nu}dx^{\mu}dx^{\nu}+H^{1/2}(ds^{2}_{EH}+dzd\overline{z})\,,\quad\hat{\Phi}=0\,,\quad\hat{C}_{0}=0\,,\\\
&\hat{G}_{(3)}=\mathcal{F}_{(2)}\wedge
d\overline{z}+\frac{1}{2}(F_{(2)}+i{\ast}_{4}F_{(2)})\wedge dz\,,\quad\\\
&\hat{F}_{(5)}=H^{-2}\operatorname{vol}_{4}\wedge
dH-\frac{i}{2}{\ast}_{EH}dH\wedge dz\wedge d\overline{z}\,,\end{split}$ (2.12)

where $\operatorname{vol}_{4}$ is the volume form associated to the four-
dimensional metric $g_{\mu\nu}$, and ${\ast}_{4}$ is the Hodge dual with
respect to $g_{\mu\nu}$. We find that the Type IIB equations of motion and
Bianchi identities are satisfied provided that $(g_{\mu\nu},F_{(2)})$ satisfy
the equations of motion and the Bianchi identity of $\mathcal{N}=2$
supergravity in four dimensions. We can use this to obtain an embedding of the
$\mathcal{N}=2$ supergravity on the worldvolume of the CGP-SS solution by
reversing the duality map (2.7).

In the language of [8], the embedding (2.12) is a Type I embedding of Type IIB
supergravity to the $\mathcal{N}=2$ supergravity in four dimensions. But there
is another possibility. One striking property of the CGP-SS solution is that,
although the Eguchi-Hanson space is non-compact, transverse-traceless graviton
perturbations on the NS5-brane worldvolume having a non-trivial dependence on
the Eguchi-Hanson radial coordinate $\rho$ may instead be genuinely localised
[9, 8]. In the language of Ref. [8], this gives a Type III embedding. As we
will now show, this is also a property of our resolved D3-brane solution
(2.1). Consider a transverse-traceless perturbation of the Minkowski
worldvolume that depends both on the worldvolume coordinates as well as the
Eguchi-Hanson radius $\rho$,

$\eta_{\mu\nu}\mapsto\eta_{\mu\nu}+\mathcal{H}_{\mu\nu}(x,\rho)\,,\quad\partial^{\mu}\mathcal{H}_{\mu\nu}=0\,,\quad\eta^{\mu\nu}\mathcal{H}_{\mu\nu}=0\,.$
(2.13)

From the general results of [20], the equations of motion to first-order in
perturbations are satisfied provided that

$(\Box_{4}+H^{-1}\Box_{EH})\mathcal{H}_{\mu\nu}(x,\rho)=0\,,\quad
H=\operatorname{sech}2\rho-k\log\tanh\rho\,,$ (2.14)

where $\Box_{4}$ and $\Box_{EH}$ are the respectively the Laplacians on
$\mathbb{R}^{1,3}$ and Eguchi-Hanson space. For functions that only depend on
$\rho$,

$\Box_{EH}f(\rho)=\frac{1}{\sinh 2\rho\cosh 2\rho}\frac{d}{d\rho}\left(\sinh
2\rho\frac{d}{d\rho}f(\rho)\right)\,.$ (2.15)

The graviton spectrum can be identified by performing a decomposition of
$\mathcal{H}_{\mu\nu}$ in terms of the complete set of eigenfunctions of the
operator $H^{-1}\Box_{EH}$,

$\mathcal{H}_{\mu\nu}(x,\rho)=h^{(\lambda)}_{\mu\nu}(x)\xi_{(\lambda)}(\rho)\,,\quad
H^{-1}\Box_{EH}\xi_{(\lambda)}=-\lambda^{2}\xi_{(\lambda)}\,,$ (2.16)

where $\lambda\in\mathbb{R}$, and there is a sum/integral over the $\lambda$
index. The mode corresponding to $\lambda=0$ is a massless graviton, and the
modes with $\lambda\neq 0$ are massive with mass $|\lambda|$. Localisation is
achieved if the zero eigenfunction $\xi_{0}(\rho)$ is square-integrable with
respect to the induced measure $\mu=H\mu_{EH}$, where $\mu_{EH}=\sinh
2\rho\cosh 2\rho$ is the $\rho$-dependent part of the measure on the Eguchi-
Hanson space. If there is a mass gap separating the zero mode from the massive
modes, corrections to the effective four-dimensional gravitational potential
are of Yukawa form, otherwise they are polynomial. The localisation of gravity
on the brane worldvolume thus depends solely on the spectrum of the operator
$H^{-1}\Box_{EH}$. Fortunately, this is the same operator as that considered
in [9, 8]. By imposing the Robin boundary condition at $\rho=0$

$((\sinh
2\rho)\log\tanh\rho\,\partial_{\rho}-2)\xi_{(\lambda)}(\rho)\big{\rvert}_{\rho=0}=0\,,$
(2.17)

and requiring normalisability, in Refs [9, 8] it was shown that the spectrum
of the operator consists of a single zero mode bound state

$\xi_{(0)}(\rho)=\frac{1}{\pi}\left(\frac{24}{2+3k}\right)^{1/2}\log\tanh\rho\,,$
(2.18)

where the normalisation constant is defined so that

$\int_{0}^{\infty}d\rho\,\mu\,\xi_{(0)}^{2}=1\,,$ (2.19)

and a continuum of massive mode scattering states with eigenvalues
$\lambda^{2}>1+k$. This gives a mass gap of $\delta=1+k$.

In the Type III perspective of the effective field theory for the graviton
modes, it is inconsistent to truncate away the massive gravitons. This is
because the massive gravitons couple to the massless graviton at cubic order
in the action with coefficients proportional to the overlap integrals of
$\xi_{(0)}^{2}\xi_{(\lambda)}$ or
$\xi_{(0)}\xi_{(\lambda)}\xi_{(\lambda^{\prime})}$. Neither of these integrals
vanish, and the same situation occurs also at higher orders in the action.
However, due to the non-zero mass gap $\delta$, the effective theory obtained
by integrating out the massive modes is a valid theory up to the energy scale
defined by $\delta$. Such effective theories of massless modes supported by
non-constant zero eigenfunctions were studied in [21, 22]. A particular
feature of these theories is that a delayed, covert form of spontaneous
symmetry breaking can occur, revealing itself only at fourth order in the
action.333Symmetry breaking at fourth order in fields is characteristic of
situations where field redefinitions are incapable of restoring a symmetry
that is apparent at lower order [23]. The interaction coefficients at fourth
order and higher do not to have the values one expects for a generally-
covariant theory in an order-by-order field expansion [24, 25]. A quick way to
see this is to note that the $n^{\text{th}}$-order terms in the effective
action involving just the massless mode are proportional to the integral

$I_{n}=\int_{0}^{\infty}d\rho\,\mu\,\xi_{(0)}^{n}\,.$ (2.20)

In the perturbative formulation of gravity, the fourth-order coefficient is
the square of the third-order coefficient. However, an explicit calculation
shows that $I_{4}\neq(I_{3})^{2}$. This does not mean that the gauge symmetry
is broken. As shown in [21], the gauge symmetry is preserved due to the
existence of a field with a Stueckelberg-like transformation that only appears
when we consider the full, ten-dimensional gravity perturbation
$\hat{g}_{MN}\mapsto\hat{g}_{MN}+\hat{H}_{MN}$, and not just the worldvolume,
transverse-traceless part. This Stueckelberg-like field was shown in [22] to
be a phantom mode – it only contributes to the energy of the effective field
theory at interacting level.

##  3 Localised worldvolume gravitons on resolved branes

The metric of a magnetic brane resolved by transgression in $d$ dimensions
that is sourced by a $(p+1)$-form potential has the form [14]

$ds^{2}_{d}=H^{-\frac{d_{e}}{d-2}}ds^{2}(\mathbb{R}^{1,d_{m}-1})+H^{\frac{d_{m}}{d-2}}ds^{2}(B)\,,$
(3.1)

where $d_{e}=p+1$, $d_{m}=d-p-3$, and $B$ is a non-compact,
$(p+3)$-dimensional, Ricci-flat manifold. The function $H$ satisfies

$\Delta_{B}H=-\frac{1}{2k!}L^{2}_{(k)}\,,$ (3.2)

where $\Delta_{B}$ is the Laplacian on $B$, and $L_{(k)}$ is a harmonic
$k$-form on $B$. Compared to the Salam-Sezgin solution of Type IIA
supergravity, which can be interpreted as a resolved NS5-brane with $B$ being
Eguchi-Hanson space, we find that the metric in (3.1) lacks a fibration of one
of the worldvolume coordinates over $B$ (see [9]). This does not affect our
discussion here because we can always consider the same solution reduced along
the fibre direction. For the Salam-Sezgin solution, this is equivalent to
considering a reduction along the fibre coordinate to 9 dimensions, in which
the resolved NS5-brane becomes a resolved 4-brane. Since the fibre direction
is external to $B$, reducing on it does not introduce any curvature
singularities.

What we are interested in is the behaviour of worldvolume gravitational waves
on this background. As demonstrated in [20], it is consistent to consider
perturbations of the form

$\eta_{\mu\nu}\mapsto\eta_{\mu\nu}+h_{\mu\nu}(x,y)\,,$ (3.3)

where $h_{\mu\nu}$ is transverse and traceless, and $y$ are the coordinates on
$B$. The perturbation $h_{\mu\nu}$ satisfies the wave equation

$\Box_{d}h_{\mu\nu}=0\,,$ (3.4)

where $\Box_{d}$ is the scalar Laplacian associated to the metric (3.1). A
little algebra shows that (3.4) can be rewritten as

$\left(\partial^{2}+H^{-1}\Delta_{B}\right)h_{\mu\nu}=0\,,$ (3.5)

where $\partial^{2}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$. The lower-
dimensional physics is then controlled by the spectrum of the operator
$H^{-1}\Delta_{B}$. In particular, we can expand $h_{\mu\nu}$ into eigenmodes
of $H^{-1}\Delta_{B}$,

$h_{\mu\nu}(x,y)=h^{(m)}_{\mu\nu}(x)\xi_{(m)}(y)\,,$ (3.6)

where

$\Delta_{B}\xi_{(m)}=-m^{2}H\xi_{(m)}\,,$ (3.7)

and

$(\partial^{2}-m^{2})h^{(m)}_{\mu\nu}=0\,.$ (3.8)

From this, we observe that in order for there to be an isolated, massless
graviton on the worldvolume, the spectrum of $H^{-1}\Delta_{B}$ must contain a
normalisable zero mode $\xi_{(0)}(y)$. To determine the precise measure with
which to normalise $\xi_{(0)}(y)$, we can look at the $h^{2}$ terms in the
perturbative action. This is most easily done by changing to a frame where the
metric is a direct product,

$ds^{2}_{d}=H^{-\frac{d_{e}}{d-2}}d\tilde{s}^{2}_{d}\,,\quad
d\tilde{s}^{2}_{d}=(\eta_{\mu\nu}+h_{\mu\nu})dx^{\mu}dx^{\nu}+Hds^{2}(B)\,.$
(3.9)

The Einstein-Hilbert term under this change of frame is

$\sqrt{-g}R=H^{-d_{e}/2}\left(\sqrt{-\tilde{g}}\tilde{R}+\cdots\right)\,.$
(3.10)

From this, we can read off the measure for integration over $B$ to be

$\mu(y)=H^{-d_{e}/2}H^{(d_{e}+2)/2}\sqrt{g_{B}}=H\sqrt{g_{B}}\,,$ (3.11)

where $g_{B}$ is the metric on $B$. Although we have derived this measure by
analysing the perturbative action, it is easy to see, from a differential
equation point of view, that this is indeed the correct measure. The defining
equation for the eigenmodes is (3.7). Assuming appropriate boundary conditions
on the eigenfunctions to ensure the self-adjointness of $\Delta_{B}$, (3.7)
takes the form of a Sturm-Liouville equation on $B$ with weight function $H$.
The Sturm-Liouville theorem then states that the measure with respect to which
the eigenmodes are orthonormalised is the product of the weight function and
the natural measure on $B$, which indeed agrees with (3.11).

In the following section, we will review some mathematical theory concerning
differential equations of the type (3.7), and apply these results to several
supergravity solutions, and will show how they can realise worldvolume gravity
localisation. We will also consider an example of a supergravity solution
where worldvolume gravity localisation cannot be realised.

##  4 Some Sturm-Liouville theory

Hermann Weyl began the analysis of Sturm-Liouville systems with singular
endpoints in connection with the geometric construction of concentric circles
in the plane, each one contained within the preceding one [12]. He introduced
the terminology of limit circles (LC) and limit points (LP) to describe the
nature of the singularity at an endpoint. The analysis for Sturm-Liouville
problems is much more general, however and was developed in particular by A.
Zettl and collaborators [13].

All of the examples that we will consider here consist of transverse
geometries that have one distinguished non-compact direction. Let $z$ be this
non-compact coordinate normalised such that $z\in[0,\infty)$. Since there is
only one non-compact direction, we may integrate over the transverse-space
compact directions so that the remaining coordinate dependence of the
eigenfunctions $\xi_{(m)}$ is only on the coordinate $z$. One may also view
this as restricting to a standard consistent truncation for the compact
transverse directions. Then the eigenvalue equation (3.7) becomes a Sturm-
Liouville problem (SLP),

$\mathcal{L}\xi_{(m)}\coloneqq-(p\,\xi_{(m)}^{\prime})^{\prime}=m^{2}\mu\,\xi_{(m)}\,,\quad
z\in[0,\infty)\,,$ (4.1)

where $p(z)>0$, $\mu(z)>0$ is the measure, and the primes denote $z$
derivatives. The combination $p\,\xi_{(m)}^{\prime}$ is called the quasi-
derivative of $\xi_{(m)}$ with respect to the SLP (4.1), or just the quasi-
derivative for short. As opposed to the general canonical form of an SLP,
there is no undifferentiated term on the LHS of (4.1) proportional to
$\xi_{(m)}$. This is because the differential operator in (3.7) is the
Laplacian, which has a $d(\cdot d)$ structure. For completeness, we also will
extend the analysis to the case where $m$ can be imaginary in the following
section, so that the SLP becomes

$\mathcal{L}\xi_{(\lambda)}\coloneqq-(p\,\xi_{(\lambda)}^{\prime})^{\prime}=\lambda\mu\,\xi_{(\lambda)}\,,\quad\lambda\in\mathbb{R}\,,\quad
z\in[0,\infty)\,.$ (4.2)

In the following examples, the endpoint $z=\infty$ will be singular while
$z=0$ will either be regular or singular. This makes the problem of defining
boundary conditions somewhat intricate because the solutions to (4.2) and
their quasi-derivatives are, in general, undefined at the singular points. A
consequence of this is that a particular self-adjoint realisation
$\mathcal{L}^{s}$ of the SL operator $\mathcal{L}$ needs to be constructed.
Only when $\mathcal{L}$ is made self-adjoint by $\mathcal{L}^{s}$ does one
enjoy desirable spectral properties such as the reality of eigenvalues and the
orthogonality of eigenfunctions with different eigenvalues.

There are, in general, many self-adjoint realisations of $\mathcal{L}$. The
differences between them boil down to the boundary conditions imposed on the
endpoints, which are in turn dependent on the nature of the endpoints. We then
need to understand the different types of endpoint as well as how they
influence the construction of self-adjoint realisations of $\mathcal{L}$. This
will be the focus of this subsection, following the presentation given in Ref.
[13].

Before diving into the different types of singular endpoints, we note that the
self-adjoint domain of $\mathcal{L}^{s}$, for any type of singular endpoint,
is a subset of the maximal domain

$D_{\text{max}}\coloneqq\\{\xi:[0,\infty)\to\mathbb{R}:\xi,p\xi^{\prime}\in\Omega^{0}((0,\infty))\,,\xi,\mu^{-1}\mathcal{L}\xi\in
L^{2}([0,\infty),\mu)\\}\,,$ (4.3)

which is characterised by boundary conditions involving linear combinations of
the Lagrange sesquilinear form

$[f,g]\coloneqq f(pg^{\prime})-g(pf^{\prime})$ (4.4)

defined at one or both endpoints. The space $L^{2}([0,\infty),\mu)$ of square-
integrable functions with respect to the measure $\mu$ is the Hilbert space
$\mathcal{H}$,

$\mathcal{H}=L^{2}([0,\infty),\mu)\,.$ (4.5)

We note that the final condition in the definition of the maximal domain,
$\mu^{-1}\mathcal{L}\xi\in\mathcal{H}$, is implied by the penultimate
condition, $\xi\in\mathcal{H}$, for all $\xi$ solving (4.2).

There are many classifications of singular endpoints. Classifying whether a
singular endpoint is a regular singular point or an irregular singular point,
for example, is useful for performing a Frobenius analysis. For our purposes,
however, it is more useful to classify the singular endpoints as either limit-
circle (LC) or limit-point (LP). An endpoint is LC if all solutions to (4.2)
are square-integrable with respect to the measure $\mu$ in a neighbourhood of
the endpoint. If a singular endpoint is not LC, it is LP. From this
definition, the LC/LP classification of an endpoint appears to depend on the
eigenvalue $\lambda$. This, however, is not the case due to the following
theorem:444See Theorem 7.2.2 in Ref. [13].

######  Theorem 4.1.

If all solutions of (4.2) for a particular $\lambda\in\mathbb{R}$ are in
$L^{2}(I,\mu)$ for some interval $I\subset[0,\infty)$, then this holds for
every $\lambda$.

Consequently, we can classify whether an endpoint is LC or LP by looking at
the solutions to (4.2) for a particular value of $\lambda$. The simplest
solutions are the zero modes, corresponding to $\lambda=0$. Since (4.2) does
not have a term proportional to $\xi_{(\lambda)}$ on the LHS, the zero modes
can be obtained by directly integrating the quasi-derivative,

$\xi_{(0)}(z)=a_{(0)}+b_{(0)}\int^{z}\frac{dt}{p(t)}\coloneqq\xi^{c}_{(0)}+\xi^{nc}_{(0)}(z)\,,$
(4.6)

where $a_{(0)}$ and $b_{(0)}$ are constants, and the superscripts $c$ and $nc$
stand for constant and non-constant. The condition for the endpoint $z=0$ to
be LC is then

$\int_{0}^{b}dz\,\mu(z)<\infty\,,\quad\text{and}\quad\int_{0}^{b}dz\,\mu(z)\xi^{nc}_{(0)}(z)^{2}<\infty\,,$
(4.7)

where $b$ is a non-zero constant. Similar conditions are made at $z=\infty$.
For the examples below that admit worldvolume gravity localisation, the point
$z=0$ is either regular or LC, while the point $z=\infty$ is LP. In such
cases555See Theorem 10.4.5 in Zettl [13]., for any pair of non-zero real
numbers $(A_{1},A_{2})$,

$D^{s}(\omega,A_{1},A_{2})\coloneqq\\{\xi\in
D_{\text{max}}:A_{1}[\xi,u_{(\omega)}](0)+A_{2}[\xi,v_{(\omega)}](0)=0\\}$
(4.8)

is a self-adjoint domain, where $u_{(\omega)}$ and $v_{(\omega)}$ are the
linearly independent solutions to (4.2) with $\lambda=\omega$. For $\omega=0$,
these are $\xi^{c}_{(0)}$ and $\xi^{nc}_{(0)}$. In other words, the self-
adjoint extension $\mathcal{L}^{s}$ of $\mathcal{L}$ with domain $D$ is
constructed by imposing the boundary condition

$A_{1}[\xi_{(\lambda)},u_{(\omega)}](0)+A_{2}[\xi_{(\lambda)},v_{(\omega)}](0)=0\,,\quad(A_{1},A_{2})\in\mathbb{R}^{2}\setminus\\{(0,0)\\}\,.$
(4.9)

Importantly, there are no special conditions other than $L^{2}$ normalisation
required at the LP point, $z=\infty$. All of the required information is
encoded in the above boundary condition at the LC point $z=0$.

With the self-adjoint realisation $\mathcal{L}^{s}$ of $\mathcal{L}$
constructed with the domain (4.8), we can now study its spectrum. Since
$\mathcal{L}^{s}$ is self-adjoint, the spectrum $\sigma(\mathcal{L}^{s})$ is a
subset of $\mathbb{R}$, and is the union of two disjoint sets

$\sigma(\mathcal{L}^{s})=\sigma_{d}(\mathcal{L}^{s})\cup\sigma_{\text{ess}}(\mathcal{L}^{s})\,,$
(4.10)

where $\sigma_{d}(\mathcal{L}^{s})$ is the discrete spectrum and
$\sigma_{\text{ess}}(\mathcal{L}^{s})$ is the essential spectrum. The discrete
spectrum consists of all isolated eigenvalues of finite multiplicity. and
their associated eigenfunctions are square-integrable. In particular,
$\xi_{(0)}$ is square-integrable. On the other hand, the essential spectrum
consists of real numbers $\lambda$ such that the solutions to (4.2) are not
square-integrable. In the language of quantum mechanics, the discrete spectrum
corresponds to bound states, while the essential spectrum corresponds to a
continuum of scattering states.

In the cases that we will be considering, where $z=0$ is either regular or LC
and $z=\infty$ is LP, the essential spectrum does not depend on the particular
boundary condition (4.9), or in other words, on the particular self-adjoint
realisation of $\mathcal{L}$, but the discrete spectrum does.666See
Proposition 10.4.4 (2) in Zettl [13]. We will thus abbreviate the notation for
the essential spectrum to $\sigma_{\text{ess}}$. Let

$\sigma_{0}\coloneqq\inf\sigma_{\text{ess}}$ (4.11)

be the starting point of the essential spectrum. The possible spectra depend
on the following three possibilities of $\sigma_{0}$:

1. $\sigma_{0}=-\infty$. In this case, $\sigma_{\text{ess}}$ may be the whole real line, or it may consist of disjoint closed intervals separated by gaps. Given an eigenvalue $\lambda\in\sigma_{\text{ess}}$, its corresponding eigenfunction has an infinite number of zeroes in $[0,\infty)$. A function with an infinite number of zeroes is called oscillatory, whereas a function with a finite number of zeroes is called non-oscillatory.

2. $\sigma_{0}=\infty$. This is the case where $\sigma_{\text{ess}}$ is empty, so the spectrum of every self-adjoint realisation is discrete, $\sigma(\mathcal{L}^{s})=\sigma_{d}(\mathcal{L}^{s})$, and there are only bound states.

3. $\sigma_{0}\in(-\infty,\infty)$. Here, the point $\sigma_{0}$ is called the oscillation number. For $\lambda<\sigma_{0}$, each nontrivial solution has no zeroes or a finite number of zeroes. If $\lambda>\sigma_{0}$, then each nontrivial solution has an infinite number of zeroes. If $\lambda=\sigma_{0}$, then either of the two cases above can occur.

In the following, we will employ the Sturm-Liouville analysis reviewed here to
two known examples (Randall-Sundrum and Salam-Sezgin), and a few new examples
of brane gravity localisation. For all of these examples, the self-adjoint
domain will be chosen with respect to the zero eigenfunction. Recall that a
bound-state solution to (4.2) is an eigenfunction if it is in the Hilbert
space $\mathcal{H}$, i.e. if it is square-integrable. Since $z$ is a non-
compact coordinate, the zero eigenfunction is not necessarily constant. In any
case, the boundary condition at $z=0$ is

$[\xi_{(\lambda)},\xi_{(0)}](0)=0\,,$ (4.12)

where $\xi_{(0)}$ is the zero eigenfunction. We will denote the self-adjoint
realisation of $\mathcal{L}$ with the domain defined by (4.12) by
$\mathcal{L}^{0}$. Note that $\xi_{(0)}$ itself automatically obeys this
boundary condition. The solutions to (4.2) obeying the boundary condition
(4.12) are orthogonal to $\xi_{(0)}$. In particular, the square-integrable
bound states will satisfy

$\int_{0}^{\infty}dz\,\mu\,\xi_{(\lambda)}\xi_{(\omega)}=\delta_{\lambda\omega}\,,\quad\lambda,\omega\in\sigma_{d}(\mathcal{L}^{0})\,,$
(4.13)

while the scattering states obey

$\int_{0}^{\infty}dz\,\mu\,\xi_{(\lambda^{\prime})}\xi_{(\omega^{\prime})}=\delta(\lambda^{\prime}-\omega^{\prime})\,,\quad\lambda^{\prime},\omega^{\prime}\in\sigma_{\text{ess}}\,.$
(4.14)

Consequently, we can use (4.13) as an easy way to see which eigenvalues are
allowed in $\sigma_{d}(\mathcal{L}^{0})$. Any solution to (4.2) with
$\lambda\notin\sigma_{\text{ess}}$ that is not square-integrable is thus not
part of the spectrum. In some cases, it may be difficult to compute the
integral (4.13) explicitly. However, for $\omega\neq\lambda$ and $\omega\neq
0$, a little algebra shows that

$0=\int_{0}^{\infty}dz\,\mu\,\xi_{(\lambda)}\xi_{(\omega)}=\frac{\lambda}{\omega}\int_{0}^{\infty}dz\,\mu\,\xi_{(\lambda)}\xi_{(\omega)}-\frac{1}{\omega}[\xi_{(\lambda)},\xi_{(\omega)}]\Big{|}_{0}^{\infty}\,,$
(4.15)

which is equivalent to

$[\xi_{(\lambda)},\xi_{(\omega)}]\Big{|}_{0}^{\infty}=0\,.$ (4.16)

Since $\xi_{(\lambda)}$ and $\xi_{(\omega)}$ obey (4.12), the condition at
$z=0$ is satisfied trivially. Therefore, the condition for orthonormality is
then

$[\xi_{(\lambda)},\xi_{(\omega)}](\infty)=0\,.$ (4.17)

for all pairs $\xi_{(\lambda)}$ and $\xi_{(\omega)}$, $\omega\neq\lambda$.
This shows that only the asymptotic behaviour of the solutions at infinity is
required to check whether they belong in the spectrum.

In the following, the supergravity solution for Example 1 is from [3], Example
2 is from [9], Examples 3-5 are from [26], and Example 6 is from [14].

##  5 Examples

###  5.1 Example 1: Randall-Sundrum

Although the Randall-Sundrum (RSII) geometry is not that of a resolved brane,
but rather that of an orbifolded $\hbox{AdS}_{5}$ sourced by a codimension 1
brane, we can still understand the brane gravity localisation in this case
using the above framework. The relevant SLP is777We have shifted the $z$
coordinate of [3] by $1/k$ so that it starts at 0.

$\xi_{(\lambda)}^{\prime\prime}-\frac{3}{(z+1/k)}\xi_{(\lambda)}=-\lambda\xi_{(\lambda)}\,,$
(5.1)

where $k>0$ is a constant, $\lambda\in\mathbb{R}$, and $z\in[0,\infty)$. The
case of physical interest is when $\lambda=m^{2}>0$, but we will first perform
an analysis for all real $\lambda$. Equation (5.1) can be written in the form
of (4.2) with $p(z)=(z+1/k)^{-3}$ and $\mu(z)=p(z)$. The endpoint $z=0$ is
regular, but $z=\infty$ is singular, and is LP. To see that it is LP, consider
the zero mode solutions to (5.1),

$\xi_{(0)}(z)=a_{(0)}+b_{(0)}(z+1/k)^{4}=\xi^{c}_{(0)}+\xi^{nc}_{(0)}(z)\,.$
(5.2)

We have $\mu\,\xi^{c}_{(0)}\propto(z+1/k)^{-3}$ and
$\mu\,\xi^{nc}_{(0)}\propto(z+1/k)$. Thus, $\xi^{c}_{(0)}$ is square-
integrable in the neighbourhood of $z=\infty$, but $\xi^{nc}_{(0)}$ is not.
Hence, $z=\infty$ is LP. This also shows that the zero eigenfunction is the
constant mode

$\xi_{(0)}(z)=a_{(0)}\,.$ (5.3)

The general solution to (5.1) is given by

$\xi_{(\lambda)}(z)=(z+1/k)^{2}\left(a_{(\lambda)}J_{2}\left(\sqrt{\lambda}(z+1/k)\right)+b_{(\lambda)}Y_{2}\left(\sqrt{\lambda}(z+1/k)\right)\right)\,,$
(5.4)

where $a_{(\lambda)}$ and $b_{(\lambda)}$ are constants, and $J$ and $Y$ are
Bessel functions of the first and second kind respectively. These functions
are oscillatory (scattering) for all $\lambda>0$, so the essential spectrum is
$\sigma_{\text{ess}}=(0,\infty)$, and the oscillation number is

$\sigma_{0}=\inf\sigma_{\text{ess}}=0\,.$ (5.5)

To find the discrete spectrum, we need to impose the boundary condition (4.12)
at $z=0$. Since the zero mode is constant, this is equivalent to the Neumann
condition

$\xi_{(\lambda)}^{\prime}(0)=0\,.$ (5.6)

For $\lambda=m^{2}>0$, we have

$b_{(\lambda)}=-\frac{J_{1}(\sqrt{\lambda}/{k})}{Y_{1}(\sqrt{\lambda}/{k})}a_{(\lambda)}=-\frac{J_{1}(m/{k})}{Y_{1}(m/{k})}a_{(\lambda)}\,.$
(5.7)

For $\lambda<0$, the solutions obeying (5.6) are not square-integrable, so
they are not part of the discrete spectrum. Thus, we find that the orthonormal
solutions to (5.1) in the self-adjoint domain defined by (5.6) are given by

$\xi_{(0)}=a_{(0)}\,,\quad\xi_{(m)}=a_{(m)}(z+1/k)^{2}\left(J_{2}\left(m(z+1/k)\right)-\frac{J_{1}(m/{k})}{Y_{1}(m/{k})}Y_{2}\left(m(z+1/k)\right)\right)\,,$
(5.8)

with $\lambda=m^{2}$. The coefficients $a_{(0)}$ and $a_{(m)}$ are determined
by the normalisations

$\int_{0}^{\infty}dz\,\mu\,\xi^{2}_{(0)}=1\,,\quad\int_{0}^{\infty}dz\,\mu\,\xi_{(m)}\xi_{(n)}=\delta(\lambda(m)-\lambda(n))=\delta(m^{2}-n^{2})=\frac{\delta(m-n)}{2n}\,.$
(5.9)

For the zero eigenfunction, we have

$1=\int_{0}^{\infty}dz\,\mu\,\xi^{2}_{(0)}=\frac{a_{(0)}^{2}k^{2}}{2}\,.$
(5.10)

###  5.2 Example 2: Salam-Sezgin

The Salam-Sezgin $\mathbb{R}^{1,3}\times S^{2}$ vacuum lifted to Type IIA
supergravity has the form of a completely resolved NS5-brane on Eguchi-Hanson
space. The metric on the Eguchi-Hanson space is given by (2.2). By relabelling
$z=\rho$, the relevant SLP is

$\xi^{\prime\prime}_{(\lambda)}+2\coth
2z\,\xi^{\prime}_{(\lambda)}=-\lambda\,\xi_{(\lambda)}\,,$ (5.11)

which can be put into the form (4.2) with $p(z)=\mu(z)=\sinh 2z$. The endpoint
$z=0$ is LC, whereas $z=\infty$ is LP. To see this, note that the zero modes
are given by

$\xi_{(0)}(z)=a_{(0)}+b_{(0)}\log\tanh z=\xi^{c}_{(0)}+\xi^{nc}_{(0)}(z)\,.$
(5.12)

These are both square-integrable with respect to the measure $\mu=\sinh 2z$ in
a neighbourhood of $z=0$, but $\xi^{nc}_{(0)}$ is singular at $z=0$, so $z=0$
is LC. The constant mode $\xi^{c}_{(0)}$ is clearly not square-integrable in a
neighbourhood of $z=\infty$, so $z=\infty$ is LP. To deduce the essential
spectrum, we consider the asymptotic form of (5.11) near $z=\infty$. This is

$\xi^{\prime\prime}_{(\lambda)}+2\xi^{\prime}_{(\lambda)}=-\lambda\,\xi_{(\lambda)}\,,$
(5.13)

which has solutions

$\xi_{(\lambda)}=e^{-\rho}\left(a_{(\lambda)}\cos(\sqrt{\lambda-1}z)+b_{(\lambda)}\sin(\sqrt{\lambda-1}z)\right)\,.$
(5.14)

These are oscillatory for $\lambda>1$, so the essential spectrum is
$\sigma_{\text{ess}}=(1,\infty)$, and the oscillation number is

$\sigma_{0}=\inf\sigma_{\text{ess}}=1\,.$ (5.15)

To deduce the discrete spectrum, we impose the boundary condition (4.12) at
$z=0$,

$[\xi_{(\lambda)},\xi_{(0)}](0)=0\,,$ (5.16)

where the square-integrable zero eigenfunction $\xi_{(0)}$ is given by

$\xi_{(0)}(z)=b_{(0)}\log\tanh z\,.$ (5.17)

Explicitly, the boundary condition is

$\lim_{z\to 0^{+}}\left(\sinh 2z\,\log\tanh
z\,\xi^{\prime}_{(\lambda)}(z)-2\xi_{(\lambda)}(z)\right)=0\,,$ (5.18)

where we used the fact that $\sinh 2z\,\xi^{\prime}_{(0)}(z)=2b_{(0)}$. To see
which bound states other than the zero eigenfunction are allowed, we note that
the general solution to (5.11) for arbitrary $\lambda$ is

$\xi_{(\lambda)}(z)=a_{(\lambda)}\mathcal{Q}_{-\frac{1}{2}-\frac{\sqrt{1-\lambda}}{2}}(\cosh
2z)+b_{(\lambda)}\mathcal{Q}_{-\frac{1}{2}+\frac{\sqrt{1-\lambda}}{2}}(\cosh
2z)\,,$ (5.19)

where $\mathcal{Q}$ is the Legendre function of the second kind. Since
$\lambda<1$ for the (possible) bound states, the orders of the Legendre
functions are real, and we define
$\nu_{1}=-\tfrac{1}{2}-\tfrac{\sqrt{1-\lambda}}{2}$ and
$\nu_{2}=-\tfrac{1}{2}+\tfrac{\sqrt{1-\lambda}}{2}$. The boundary condition at
$z=0$ then sets

$b_{(\lambda)}=-\frac{H(\nu_{1})}{H(\nu_{2})}a_{(\lambda)}\,,$ (5.20)

where $H(n)\coloneqq\int_{0}^{1}dt\,(1-t^{n})/(1-t)$ is Euler’s representation
of the harmonic number $H(n)$.888This relation is also true for the scattering
states. Although the boundary condition at $z=0$ can be satisfied, solutions
with $\lambda<1$ and $\lambda\neq 0$ are not normalisable. Thus, they are not
in the Hilbert space $\mathcal{H}$, and are therefore not part of the
spectrum. So, we find that the orthonormal solutions to (5.11) in the self-
adjoint domain defined by (5.18) are given by

$\begin{split}&\xi_{(0)}=b_{(0)}\log\tanh z\,,\quad\\\
&\xi_{(m)}=a_{(m)}\left(\mathcal{Q}_{-\frac{1}{2}-\frac{i\sqrt{m^{2}-1}}{2}}(\cosh
2z)-\frac{H(\nu_{1})}{H(\nu_{2})}\mathcal{Q}_{-\frac{1}{2}+\frac{i\sqrt{m^{2}-1}}{2}}(\cosh
2z)\right)\,,\quad m>1\,,\end{split}$ (5.21)

with $\lambda=m^{2}$, $\nu_{1}=-\tfrac{1}{2}-\tfrac{i\sqrt{m^{2}-1}}{2}$, and
$\nu_{2}=\nu_{1}^{*}$. The coefficients $b_{(0)}$ and $a_{(m)}$ are determined
by the normalisations

$\int_{0}^{\infty}dz\,\mu\,\xi^{2}_{(0)}=1\,,\quad\int_{0}^{\infty}dz\,\mu\,\xi_{(m)}\xi_{(n)}=\frac{\delta(m-n)}{2n}\,.$
(5.22)

For the zero eigenfunction, we have

$1=\int_{0}^{\infty}dz\,\mu\,\xi^{2}_{(0)}=\frac{b_{(0)}^{2}\pi^{2}}{12}\,.$
(5.23)

###  5.3 Example 3: D3-branes on a resolved conifold over
$S^{5}/\mathbb{Z}_{3}$

The transverse metric is given by

$\begin{split}&ds^{2}(B)=f^{-1}dr^{2}+\frac{r^{2}}{9}f\left(d\psi-3\sin^{2}\theta\sigma_{3}\right)^{2}+r^{2}ds^{2}(\mathbb{C}\text{P}^{2})\,,\quad
f=1-\frac{b^{6}}{r^{6}}\,,\\\
&ds^{2}(\mathbb{C}\text{P}^{2})=d\theta^{2}+\sin^{2}\theta\left(\sigma^{2}_{1}+\sigma_{2}^{2}+\cos^{2}\theta\,\sigma_{3}^{2}\right)\,,\end{split}$
(5.24)

where $r\geq b$, and $\sigma_{i}$ are the left-invariant 1-forms of $SU(2)$
satisfying

$d\sigma_{i}=2\epsilon_{ijk}\sigma_{j}\wedge\sigma_{k}\,.$ (5.25)

The corresponding function $H$ is given by

$H=\frac{1}{4b^{6}r^{4}}\,.$ (5.26)

To understand the geometry as $r\to b^{+}$, consider the coordinate
transformation

$r=b(\cosh\rho)^{1/3}\,.$ (5.27)

Then, the metric is

$ds^{2}(B)=\frac{b^{2}}{9}(\cosh\rho)^{2/3}\left(d\rho^{2}+(\tanh\rho)^{2}\left(d\psi-3\sin^{2}\theta\sigma_{3}\right)^{2}+9ds^{2}(\mathbb{C}\text{P}^{2})\right)\,.$
(5.28)

Near $\rho=0$, this is

$ds^{2}(B)_{\rho\to
0^{+}}=\frac{b^{2}}{9}\left(d\rho^{2}+\rho^{2}\left(d\psi-3\sin^{2}\theta\sigma_{3}\right)^{2}+9ds^{2}(\mathbb{C}\text{P}^{2})\right)\,,$
(5.29)

so $B$ is asymptotically $\mathbb{R}^{2}\times\mathbb{C}\text{P}^{2}$ provided
$\psi\in[0,2\pi)$. As such, we expect the zero mode to exhibit a logarithmic
behaviour near the origin.

The variables $r$ and $\rho$ are useful in understanding the geometry of the
transverse space. However, to analyse the eigenvalue equation (3.7), it is
more useful to consider the following coordinate redefinition

$r=b(z+1)^{1/2}\,,$ (5.30)

where $z\in[0,\infty)$. The SLP then reads

$z(z^{2}+3z+3)\xi^{\prime\prime}_{(\lambda)}+3(z+1)^{2}\xi^{\prime}_{(\lambda)}=-\frac{\lambda}{16b^{8}}\xi_{(\lambda)}\,.$
(5.31)

This can be put into the canonical form (4.2) with $p(z)=z(z^{2}+3z+3)$ and
$\mu(z)=1$. For the rest of this section, we will absorb the factor of
$16b^{8}$ into $\lambda$. To classify whether the endpoints $z=0$ and
$z=\infty$ are LC or LP, we note that the zero modes are

$\xi_{(0)}(z)=a_{(0)}+b_{(0)}\left(2\sqrt{3}\tan^{-1}\left(\frac{\sqrt{3}}{3+2z}\right)+\log\left(\frac{z^{2}}{z^{2}+3z+3}\right)\right)=\xi^{c}_{(0)}+\xi^{nc}_{(0)}(z)\,.$
(5.32)

Both $\xi^{c}_{(0)}$ and $\xi^{nc}_{(0)}$ are square-integrable near $z=0$,
but $\xi^{nc}_{(0)}$ is logarithmically divergent at this point, so $z=0$ is
LC. Near $z=\infty$, $\xi^{c}_{(0)}$ is not square-integrable, so $z=\infty$
is LP. To determine the essential spectrum, consider the variable redefinition
$z=1/y$. In the $y$ coordinate, the eigenvalue equation becomes

$y(1+3y(1+y))\xi^{\prime\prime}_{(\lambda)}+(3y^{2}-1)\xi^{\prime}_{(\lambda)}=-\lambda\xi_{(\lambda)}\,.$
(5.33)

In a finite interval near $y=0$, this becomes

$\xi^{\prime\prime}_{(\lambda)}-\frac{1}{y}\xi^{\prime}_{(\lambda)}=-\frac{\lambda}{y}\xi_{(\lambda)}\,,$
(5.34)

which has solutions

$\xi_{(\lambda)}=y\left(a_{(\lambda)}J_{2}\left(2\sqrt{\lambda
y}\right)+b_{(\lambda)}Y_{2}\left(2\sqrt{\lambda y}\right)\right)\,.$ (5.35)

As in the Randall-Sundrum case, the Bessel functions $J_{2}$ and $Y_{2}$ are
oscillatory for all $\lambda>0$. This means that the oscillation number is

$\sigma_{0}=\inf\sigma_{\text{ess}}=0\,,$ (5.36)

and there is no mass gap.

We also note that in order for the zero mode (5.32) to be square-integrable,
we require $a_{(0)}=0$. The zero eigenfunction, written back in the $z$
coordinate, is then

$\xi_{(0)}=b_{(0)}\left(2\sqrt{3}\tan^{-1}\left(\frac{\sqrt{3}}{3+2z}\right)+\log\left(\frac{z^{2}}{z^{2}+3z+3}\right)\right)\,,$
(5.37)

with

$\begin{split}\int_{0}^{\infty}dz\,\xi^{2}_{(0)}&=\frac{b_{(0)}^{2}}{3}\left(\psi^{(1)}\left(\frac{1}{6}\right)+\psi^{(1)}\left(\frac{1}{3}\right)-\psi^{(1)}\left(\frac{2}{3}\right)-\psi^{(1)}\left(\frac{5}{6}\right)-2\pi^{2}\right)\\\
&\approx 7.5\times b_{(0)}^{2}\,,\end{split}$ (5.38)

where $\psi^{(1)}(x)$ is the first derivative of the digamma function,
$\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$, and we recall that the measure
$\mu(z)=1$.

The function $H$ given in (5.26) is analogous to the one in (2.5) with $k=0$.
It is the function describing a fully resolved background with no singularity.
We can similarly add to $H$ a term corresponding to the homogeneous solution
to the Poisson equation that it satisfies, i.e. a term proportional to
$\xi_{(0)}$. This addition corresponds to an inclusion of a brane. Since
$\xi_{(0)}$ is negative definite for all $z$, we write

$H=\frac{1}{4b^{10}(z+1)^{2}}-k\left(2\sqrt{3}\tan^{-1}\left(\frac{\sqrt{3}}{3+2z}\right)+\log\left(\frac{z^{2}}{z^{2}+3z+3}\right)\right)\,,\quad
k\geq 0\,.$ (5.39)

With this new $H$, the endpoints $z=0$ and $z=\infty$ of the SLP are still LC
and LP respectively, and near $y=1/z=0$, the SLP becomes

$\xi^{\prime\prime}_{(\lambda)}-\frac{1}{y}\xi^{\prime}_{(\lambda)}=-\frac{\lambda(1+12b^{10}k)}{y}\xi_{(\lambda)}\,.$
(5.40)

The solutions to this SLP are scattering for all $k\geq 0$, so even with the
inclusion of the brane, this system does not have a mass gap.

###  5.4 Example 4: D3-branes on resolved conifolds over $Y^{p,q}$

The metric on the transverse space is a resolved cone over the Sasaki-Einstein
spaces $Y^{p,q}$ constructed in [27],

$ds^{2}(B)=f^{-1}dr^{2}+\frac{1}{9}fr^{2}(d\psi+\mathcal{A}_{(1)})^{2}+r^{2}ds^{2}_{4}\,,$
(5.41)

where

$f=1-\frac{b^{6}}{r^{6}}\,,\quad\mathcal{A}_{(1)}=-\cos\theta\,d\varphi+y(d\beta+\cos\theta\,d\varphi)$
(5.42)

and the Kähler-Einstein base is

$ds^{2}_{4}=\frac{1}{6}(1-y)(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})+\frac{dy^{2}}{w(y)q(y)}+\frac{1}{36}w(y)q(y)(d\beta+\cos\theta\,d\varphi)^{2}\,,$
(5.43)

with

$w(y)=\frac{2(a-y^{2})}{1-y}\,,\quad q(y)=\frac{a-2y^{2}+2y^{3}}{a-y^{2}}\,.$
(5.44)

The coordinate ranges are $r\geq b$, $\theta\in[0,\pi]$, $\varphi\in[0,2\pi)$,
and $y\in[y_{1},y_{2}]$ with

$a=\frac{1}{2}-\frac{p^{2}-3q^{2}}{4p^{3}}\sqrt{4p^{2}-3q^{2}}\,,\quad
y_{1}=\frac{1}{4p}\left(2p-3q-\sqrt{4p^{2}-3q^{2}}\right)\,,\quad
y_{2}=y_{1}+\frac{3q}{2p}\,.$ (5.45)

The short-distance geometry is $\mathbb{R}^{2}\times KE_{4}$, where $KE_{4}$
is the Kähler-Einstein base, and the requisite function $H$ is the same as for
the previous example,

$H=\frac{1}{4b^{6}r^{4}}\,.$ (5.46)

For functions that only depend on $r$, the Laplacian on the space (5.41) is
the same as that on (5.24). Thus, the relevant eigenvalue problem is just
(5.31) again, and the same analysis from the previous section goes through.

###  5.5 Example 5: D3-branes on a resolved cone over $T^{1,1}/\mathbb{Z}_{2}$

The transverse metric is

$ds^{2}(B)=f^{-1}dr^{2}+\frac{r^{2}}{36}f(\sigma_{3}-\Sigma_{3})^{2}+\frac{1}{12}R_{1}^{2}(\sigma_{1}^{2}+\sigma_{2}^{2})+\frac{1}{12}R_{2}^{2}(\Sigma_{1}^{2}+\Sigma_{2}^{2})\,,$
(5.47)

where $\sigma_{i}$ and $\Sigma_{i}$ are left-invariant 1-forms of $SU(2)\times
SU(2)$, $R^{2}_{i}=r^{2}+l_{i}^{2}$, $r\geq 0$ and

$f=\frac{2r^{4}+3(l_{1}^{2}+l_{2}^{2})r^{2}+6l_{1}^{2}l_{2}^{2}}{R_{1}^{2}R_{2}^{2}}\,.$
(5.48)

Near $r=0$, the geometry becomes $\mathbb{R}^{2}\times S^{2}\times S^{2}$.
Let’s focus on the case $l_{1}=l_{2}=l$ with $R_{1}=R_{2}=R$, and with the
normalisable, harmonic 2-form

$L_{(2)}=u_{0}\theta^{0}\wedge\theta^{5}+u_{1}\theta^{1}\wedge\theta^{2}+u_{2}\theta^{3}\wedge\theta^{4}\,,$
(5.49)

where

$u_{0}=\frac{2}{R^{6}}\,,\quad u_{1}=\frac{1}{R^{6}}\,,\quad
u_{2}=-\frac{1}{R^{6}}\,,$ (5.50)

and where the vielbeins are

$\begin{split}&\theta^{0}=f^{-1/2}dz\,,\quad\theta^{1}=\frac{R}{\sqrt{12}}\sigma_{1}\,,\quad\theta^{2}=\frac{R}{\sqrt{12}}\sigma_{2}\,,\\\
&\theta^{3}=\frac{R}{\sqrt{12}}\Sigma_{1}\,,\quad\theta^{4}=\frac{R}{\sqrt{12}}\Sigma_{2}\,,\quad\theta^{5}=\frac{r}{6}f^{1/2}(\sigma_{3}-\Sigma_{3})\,.\end{split}$
(5.51)

The requisite function $H$ that is sourced by (5.49) is given by

$H=\frac{1}{8l^{6}R^{4}}\,,$ (5.52)

and the relevant eigenvalue equation is

$\xi^{\prime\prime}_{(\lambda)}+\frac{5r^{4}+9l^{2}r^{2}+3l^{4}}{r(r^{4}+3l^{2}r^{2}+3l^{4})}\xi^{\prime}_{(\lambda)}=-\frac{\lambda}{16l^{6}}\frac{1}{r^{4}+3l^{2}r^{2}+3l^{4}}\xi_{(\lambda)}\,.$
(5.53)

Absorbing the factor of $16l^{6}$ into $\lambda$, this can be put into the
canonical form of (4.2) with $p(r)=r(r^{4}+3l^{2}r^{2}+3l^{4})$, and
$\mu(r)=r$. In fact, by performing the coordinate redefinition

$r=l\sqrt{z}\,,\quad z\in[0,\infty)\,,$ (5.54)

we find that (5.53) becomes again (5.31). Thus, the same analysis for (5.31)
holds in this case as well.

###  5.6 Example 6 (no localisation): NS5-branes on Taub-NUT

The metric on the Taub-NUT space is given by

$ds^{2}(B)=\left(\frac{r+b}{r-b}\right)dr^{2}+4b^{2}\left(\frac{r-b}{r+b}\right)(d\psi+\cos\theta\,d\varphi)^{2}+(r^{2}-b^{2})(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2})\,,$
(5.55)

where $r\geq b$, $\psi\in[0,4\pi)$, $\theta\in[0,\pi]$ and
$\varphi\in[0,2\pi)$. Near $r=b$, the space is asymptotically
$\mathbb{R}^{4}$, whereas near $r=\infty$, it becomes $\mathbb{R}^{3}\times
S^{1}$. From [14], the requisite function $H$ is

$H=\frac{1}{4b(r+b)}\,.$ (5.56)

Defining $z=r-b\in[0,\infty)$, the relevant SLP is then

$\xi^{\prime\prime}_{(\lambda)}+\frac{2}{z}\xi^{\prime}_{(\lambda)}=-\frac{\lambda}{4bz}\xi_{(\lambda)}\,,$
(5.57)

which can be put into the form of (4.2) with $p(z)=z^{2}$ and $\mu(z)=z$. The
zero modes are given by999The radial coordinate of the $\mathbb{R}^{4}$
asymptote of the Taub-NUT space is $u=\sqrt{z}$, hence the $1/z$ structure of
the zero modes.

$\xi_{(0)}(z)=a_{(0)}+\frac{b_{(0)}}{z}=\xi^{c}_{(0)}+\xi^{nc}_{(0)}(z)\,.$
(5.58)

$\xi^{c}_{(0)}$ is square-integrable near $z=0$, but $\xi^{nc}_{(0)}$ is not,
so $z=0$ is LP. Similarly, $\xi^{nc}_{(0)}$ is square-integrable near
$z=\infty$, but $\xi^{c}_{(0)}$ is not, so $z=\infty$ is also LP. Since both
endpoints are LP, it is clear that there exists no zero eigenfunction. As
such, a resolved NS5-brane on the Taub-NUT space does not admit worldvolume
gravity localisation.

##  6 Conclusion and a Conjecture

The analysis of Sturm-Liouville problems with singular endpoints in the LC-LP
language of limit circles and limit points clearly identifies the asymptotic
properties of brane transverse wave equations that allow gravity localisation
in the region near a brane worldvolume. As explored in Ref. [8], gravity
localisation is characterised by a calculable Newton constant and a clear
transition between higher-dimensional near-field behaviour and lower
worldvolume-dimensional far-field behavior as the spatial worldvolume
separation between gravitating sources increases. Gravity localisation can
take place with or without a mass gap – the key point is the existence of a
normalisable transverse wavefunction zero mode. The corresponding difference
in the effective theories is between Yukawa versus power-law corrections to
the long-distance lower-dimensional behaviour, as has been clear since the
time of the RSII construction [3].

So far, this investigation has been carried out just in a classical or
semiclassical supergravity context and a next level should be to see how much
of this structure is realised at the quantum/superstring level.

From the localising examples found, we observe that only the transverse spaces
that are asymptotically $\mathbb{R}^{2}\times\\{\text{compact}\\}$ at small
distances101010The origin of the $\mathbb{R}^{2}$ is the brane worldvolume.
admit a normalisable zero mode (except for RSII, which does not have a
resolved brane structure). The pattern is that the measure $\mu(z)$ is always
a power of $z$ near $z=0$ where the zero eigenfunction is logarithmically
divergent. It is the logarithmic behaviour of the eigenfunction that allows
for its normalisability, even though the transverse space is non-compact. As
such, we conjecture that only resolved branes with transverse spaces that
asymptote to $\mathbb{R}^{2}\times\\{\text{compact}\\}$ at small distances
allow for worldvolume gravity localisation.

## Acknowledgments

We are grateful to Costas Bachas and Chris Pope for helpful discussions. The
work of KSS was supported in part by the STFC under Consolidated Grants
ST/T000791/1 and ST/X000575/1.

## Appendices

##  Appendix A M-theory and Type IIA supergravity conventions

The bosonic sector of M-theory consists of a metric $\hat{g}_{MN}$ and a four-
form flux $\hat{F}_{(4)}$. The Bianchi identity for the four-form is

$d\hat{F}_{(4)}=0\,,$ (A.1)

and the equations of motion are

$\displaystyle
d\hat{\ast}\hat{F}_{(4)}-\frac{1}{2}\hat{F}_{(4)}\wedge\hat{F}_{(4)}=0\,,$
(A.2)
$\displaystyle\hat{R}_{MN}=\frac{1}{12}\left(\hat{F}_{MPQR}\hat{F}_{N}^{\phantom{N}PQR}-\frac{1}{12}(\hat{F}_{(4)})^{2}\hat{g}_{MN}\right)\,.$
(A.3)

The dynamics of Type IIA supergravity are encoded in M-theory about
backgrounds where the eleven-dimensional spacetime has a $U(1)$ isometry. We
write

$\begin{split}&d\hat{s}^{2}_{11}=e^{-\tilde{\Phi}/6}d\tilde{s}^{2}_{10}+e^{4\tilde{\Phi}/3}(d\theta+\tilde{A}_{(1)})^{2}\,,\\\
&\hat{F}_{(4)}=\tilde{F}_{(4)}+\tilde{H}_{(3)}\wedge(d\theta+\tilde{A}_{(1)})\,,\end{split}$
(A.4)

where the ten-dimensional Type IIA fields are distinguished by a tilde and are
independent of $\theta$. The metric $\tilde{g}$ is the Einstein frame metric
of the Type IIA theory. The Bianchi identity of $\hat{F}_{(4)}$ yields

$d\tilde{H}_{(3)}=0\,,\quad
d\tilde{F}_{(4)}-\tilde{H}_{(3)}\wedge\tilde{F}_{(2)}=0\,,$ (A.5)

where $\tilde{F}_{(2)}=d\tilde{A}_{(1)}$. These can be integrated by
introducing the NSNS two-form potential $\tilde{B}_{(2)}$ and the RR three-
form potential $\tilde{A}_{(3)}$,

$\tilde{H}_{(3)}=d\tilde{B}_{(2)}\,,\quad\tilde{F}_{(4)}=d\tilde{A}_{(3)}-\tilde{H}_{(3)}\wedge\tilde{A}_{(1)}\,.$
(A.6)

##  Appendix B Type IIB supergravity conventions

Our conventions are taken from [28]. The bosonic sector of Type IIB
supergravity contains the RR forms $\hat{F}_{(1)}$, $\hat{F}_{(3)}$,
$\hat{F}_{(5)}$, the NSNS three-form $\hat{H}_{(3)}$, the dilaton $\hat{\Phi}$
and the metric, which will be taken to be the Einstein frame metric. The
Bianchi identities are

$\displaystyle d\hat{F}_{(5)}+\hat{F}_{(3)}\wedge\hat{H}_{(3)}=0\,,$ (B.1)
$\displaystyle d\hat{F}_{(3)}+\hat{F}_{(1)}\wedge\hat{H}_{(3)}=0\,,$ (B.2)
$\displaystyle d\hat{H}_{(3)}=0\,,$ (B.3) $\displaystyle d\hat{F}_{(1)}=0\,,$
(B.4)

which can be solved by introducing potentials:

$\begin{split}&\hat{F}_{(1)}=d\hat{C}_{0}\,,\quad\hat{H}_{(3)}=d\hat{B}_{(2)}\,,\\\
&\hat{F}_{(3)}=d\hat{C}_{(2)}-\hat{C}_{0}d\hat{B}_{(2)}\,,\quad\hat{F}_{(5)}=d\hat{C}_{(4)}-\hat{C}_{(2)}\wedge\hat{H}_{(3)}\,.\end{split}$
(B.5)

The equations of motion read

$\displaystyle\hat{\ast}\hat{F}_{(5)}=\hat{F}_{(5)}\,,$ (B.6) $\displaystyle
d(e^{\hat{\Phi}}\hat{\ast}\hat{F}_{(3)})-\hat{F}_{(5)}\wedge\hat{H}_{(3)}=0\,,$
(B.7) $\displaystyle
d(e^{-\hat{\Phi}}\hat{\ast}\hat{H}_{(3)})-e^{\hat{\Phi}}\hat{F}_{(1)}\wedge\hat{\ast}\hat{F}_{(3)}-\hat{F}_{(3)}\wedge\hat{F}_{(5)}=0\,,$
(B.8) $\displaystyle
d(e^{2\hat{\Phi}}\hat{\ast}\hat{F}_{(1)})+e^{\hat{\Phi}}\hat{H}_{(3)}\wedge\hat{\ast}\hat{F}_{(3)}=0\,,$
(B.9) $\displaystyle
d\hat{\ast}d\hat{\Phi}-e^{2\hat{\Phi}}\hat{F}_{(1)}\wedge\hat{\ast}\hat{F}_{(1)}+\frac{1}{2}e^{-\hat{\Phi}}\hat{H}_{(3)}\wedge\hat{\ast}\hat{H}_{(3)}-\frac{1}{2}e^{\hat{\Phi}}\hat{F}_{(3)}\wedge\hat{\ast}\hat{F}_{(3)}=0\,,$
(B.10)
$\displaystyle\hat{R}_{MN}=\frac{1}{2}\hat{\nabla}_{M}\hat{\Phi}\hat{\nabla}_{N}\hat{\Phi}+\frac{1}{2}e^{2\hat{\Phi}}\hat{\nabla}_{M}C_{0}\hat{\nabla}_{N}C_{0}+\frac{1}{96}\hat{F}_{MP_{1}P_{2}P_{3}P_{4}}\hat{F}_{N}^{\phantom{M}P_{1}P_{2}P_{3}P_{4}}$
$\displaystyle\quad\qquad+\frac{1}{4}e^{-\hat{\Phi}}\left(\hat{H}_{MP_{1}P_{2}}\hat{H}_{N}^{\phantom{N}P_{1}P_{2}}-\frac{1}{12}(\hat{H}_{(3)})^{2}\hat{g}_{MN}\right)$
$\displaystyle\quad\qquad+\frac{1}{4}e^{\hat{\Phi}}\left(\hat{F}_{MP_{1}P_{2}}\hat{F}_{N}^{\phantom{N}P_{1}P_{2}}-\frac{1}{12}(\hat{F}_{(3)})^{2}\hat{g}_{MN}\right)\,.$
(B.11)

These admit a trombone symmetry, where

$\begin{split}&\hat{g}_{MN}\mapsto
k^{2}\hat{g}_{MN}\,,\quad\hat{\Phi}\mapsto\hat{\Phi}\,,\quad\hat{C}_{0}\mapsto\hat{C}_{0}\,,\\\
&\hat{F}_{(3)}\mapsto k^{2}\hat{F}_{(3)}\,,\quad\hat{H}_{(3)}\mapsto
k^{2}\hat{H}_{(3)}\,,\quad\hat{F}_{(5)}\mapsto
k^{4}\hat{F}_{(5)}\,,\end{split}$ (B.12)

for a constant $k$.

In a configuration where $\hat{\Phi}=0$ and $\hat{F}_{(1)}=0$, it is
convenient to package the three-form fluxes into a complex three-form

$\hat{G}_{(3)}=\hat{F}_{(3)}+i\hat{H}_{(3)}\,,$ (B.13)

whose Bianchi identity and equation of motion are

$d\hat{G}_{(3)}=0\,,$ (B.14)

and

$d\hat{\ast}\hat{G}_{(3)}-i\hat{G}_{(3)}\wedge\hat{F}_{(5)}=0\,.$ (B.15)

The Bianchi identity for the five-form then reads

$d\hat{F}_{(5)}+\frac{i}{2}\hat{G}_{(3)}\wedge\overline{\hat{G}}_{(3)}=0\,,$
(B.16)

and the consistency conditions for setting $\hat{\Phi}=0$ and
$\hat{F}_{(1)}=0$ are contained in the single condition

$\hat{G}_{(3)}\wedge\hat{\ast}\hat{G}_{(3)}=0\,.$ (B.17)

With this condition, the Einstein equations simplify to

$\hat{R}_{MN}=\frac{1}{96}\hat{F}_{MP_{1}P_{2}P_{3}P_{4}}\hat{F}_{N}^{\phantom{M}P_{1}P_{2}P_{3}P_{4}}+\frac{1}{8}\left(\hat{G}_{MP_{1}P_{2}}\overline{\hat{G}}_{N}^{\phantom{N}P_{1}P_{2}}+\overline{\hat{G}}_{MP_{1}P_{2}}\hat{G}_{N}^{\phantom{N}P_{1}P_{2}}\right)\,.$
(B.18)

##  Appendix C Bucher rules for Abelian T-duality

We provide here a brief review of the Buscher rules for Abelian T-duality,
following the presentation in Ref. [29]. The rules for the NSNS sector were
derived in [30, 31], and those for the RR sector were derived in [32].

Suppose we have a (massive) Type IIA background with a $U(1)$ isometry
generated by $\partial/\partial\psi$. The most general solution compatible
with this symmetry is given by

$\begin{split}&d\tilde{s}^{2}_{str}=ds^{2}_{9}+e^{2A}(d\psi+A_{(1)})^{2}\,,\\\
&\tilde{F}_{(0)}=m\,,\\\ &\tilde{B}_{(2)}=L_{(2)}+B_{(1)}\wedge d\psi\,,\\\
&\tilde{F}_{(2)}=G_{(2)}+G_{(1)}\wedge(d\psi+A_{(1)})\,,\\\
&\tilde{F}_{(4)}=G_{(4)}+G_{(3)}\wedge(d\psi+A_{(1)})\,,\end{split}$ (C.1)

where $A$, $ds^{2}_{9},L_{(2)},B_{(1)}$, $G_{(i)}$, and the dilaton
$\tilde{\phi}$ are independent of $\psi$, where $m$ is the Romans mass, and
the metric is in string frame. The assumption that the NSNS 2-form
$\tilde{B}_{(2)}$ is independent of $\psi$ can be relaxed slightly by noting
that only its field strength $\tilde{H}_{(3)}$ appears in the equations of
motion even though $\tilde{B}_{(2)}$ itself appears in the string sigma model.
As long as $\tilde{H}_{(3)}$ is of the form
$L_{(3)}+H_{(2)}\wedge(d\psi+A_{(1)})$, one can T-dualise along $\psi$, since
$\partial/\partial\psi$ will remain a Killing vector. There are
generalisations required in the Buscher rules for such cases, but these will
not be necessary here, as such solutions are not our main focus.

T-dualising along $\psi$ gives a Type IIB solution,

$\begin{split}&d\hat{s}^{2}_{str}=ds^{2}_{9}+e^{-2A}(d\xi+B_{(1)})^{2}\,,\\\
&e^{\hat{\Phi}}=e^{\phi-A}\,,\\\ &\hat{F}_{(1)}=-G_{(1)}+m(d\xi+B_{(1)})\,,\\\
&\hat{B}_{(2)}=L_{(2)}+A_{(1)}\wedge(d\xi+B_{(1)})\,,\\\
&\hat{F}_{(3)}=-G_{(3)}+G_{(2)}\wedge(d\xi+B_{(1)})\,,\\\
&\hat{F}_{(5)}=(1+\hat{\ast})G_{(4)}\wedge(d\xi+B_{(1)})\,,\end{split}$ (C.2)

where $\xi$ is the dual coordinate of $\psi$, the metric is in string frame,
and we set $\alpha^{\prime}=1$.

##  Appendix D Type IIB embedding of the Salam-Sezgin theory

The Salam-Sezgin theory is an $\mathcal{N}=2$ six-dimensional supergravity
whose bosonic sector contains a metric, a two-form gauge potential, a vector,
and a scalar [33]. The Lagrangian for the bosonic sector is

$\mathcal{L}_{6}=\overline{R}\overline{\ast}_{6}1-\frac{1}{4}d\overline{\phi}\wedge\overline{\ast}_{6}d\overline{\phi}-\frac{1}{2}e^{-\overline{\phi}/2}\overline{F}_{(2)}\wedge\overline{\ast}_{6}\overline{F}_{(2)}-\frac{1}{2}e^{\overline{\phi}}\overline{H}_{(3)}\wedge\overline{\ast}\overline{H}_{(3)}-8\overline{g}^{2}e^{-\overline{\phi}/2}\overline{\ast}_{6}1\,,$
(D.1)

where $\overline{F}_{(2)}=d\overline{A}_{(1)}$,
$\overline{H}_{(3)}=d\overline{B}_{(2)}-\tfrac{1}{2}\overline{A}_{(1)}\wedge\overline{F}_{(2)}$,
and we put bars over six-dimensional fields. In [10], this theory was embedded
into the NSNS sector of Type IIA supergravity with

$\begin{split}&d\tilde{s}^{2}=(\cosh
2\rho)^{1/4}\left(e^{-\overline{\phi}/4}d\overline{s}^{2}_{6}+e^{\overline{\phi}/4}dy^{2}+\frac{1}{2\overline{g}^{2}}e^{\overline{\phi}/4}\left[d\rho^{2}+\frac{1}{4}\tanh^{2}2\rho\left(d\chi-2\overline{g}\overline{A}_{(1)}\right)^{2}\right.\right.\\\
&\left.\left.\qquad\qquad\qquad\qquad+\frac{1}{4}\left(d\psi+\operatorname{sech}2\rho\,(d\chi-2\overline{g}\overline{A}_{(1)})\right)^{2}\right]\right)\,,\\\
&\tilde{H}_{(3)}=\overline{H}_{(3)}-\frac{\sinh
2\rho}{4\overline{g}^{2}\cosh^{2}2\rho}d\rho\wedge
d\psi\wedge(d\chi-2\overline{g}\overline{A}_{(1)})\\\
&\qquad\quad+\frac{1}{4\overline{g}\cosh
2\rho}\overline{F}_{(2)}\wedge\left(d\psi+\cosh
2\rho\,(d\chi-2\overline{g}\overline{A}_{(1)})\right)\,,\\\
&e^{\tilde{\phi}}=(\cosh 2\rho)^{-1/2}e^{-\overline{\phi}/2}\,,\end{split}$
(D.2)

where the metric is in Einstein frame. We will now lift this into M-theory on
a circle parametrised by $\zeta$ and then reduce the solution back to Type IIA
on the circle parametrised by $y$. The resulting solution in string frame is

$\displaystyle d\tilde{s}^{2}_{str}=(\cosh
2\rho)^{1/2}\left(d\overline{s}^{2}_{6}+e^{\overline{\phi}/2}dy^{2}+\frac{1}{2\overline{g}^{2}}e^{\overline{\phi}/2}\left[d\rho^{2}+\frac{1}{4}\tanh^{2}2\rho\left(d\chi-2\overline{g}\overline{A}_{(1)}\right)^{2}+\left(d\psi+\mathcal{A}_{(1)}\right)^{2}\right]\right)$
$\displaystyle\qquad\quad+(\cosh
2\rho)^{-1/2}e^{-\overline{\phi}/2}d\zeta^{2}\,,$
$\displaystyle\tilde{F}_{(4)}=\left(\overline{H}_{(3)}+\frac{1}{4\overline{g}}\tanh^{2}2\rho\,\overline{F}_{(2)}\wedge(d\chi-2\overline{g}\overline{A}_{(1)})\right)\wedge
d\zeta$ (D.3) $\displaystyle\qquad\quad+\left(\frac{1}{\overline{g}^{2}}\tanh
2\rho\,d\rho\wedge
d\zeta\wedge\mathcal{A}_{(1)}-\frac{1}{2\overline{g}}\operatorname{sech}2\rho\,\overline{F}_{(2)}\wedge
d\zeta\right)\wedge(d\psi+\mathcal{A}_{(1)})\,,$ $\displaystyle
e^{\tilde{\phi}}=(\cosh 2\rho)^{1/4}e^{\overline{\phi}/4}\,.$

where we have defined

$\mathcal{A}_{(1)}=\frac{1}{2}\operatorname{sech}2\rho\,(d\chi-2\overline{g}\overline{A}_{(1)})\,,$
(D.4)

and rescaled $\psi\mapsto\psi/2$. Applying the Bucher rules presented in
Appendix C on the coordinate $\psi$, we obtain the Type IIB solution

$\begin{split}&d\hat{s}^{2}_{str}=(\cosh
2\rho)^{1/2}\left(d\overline{s}^{2}_{6}+\frac{1}{2\overline{g}^{2}}e^{\overline{\phi}/2}\left[d\rho^{2}+\frac{1}{4}\tanh^{2}2\rho\left(d\chi-2\overline{g}\overline{A}_{(1)}\right)^{2}\right]\right)\\\
&\qquad\quad+(\cosh
2\rho)^{-1/2}e^{-\overline{\phi}/2}\left(d\zeta^{2}+2\overline{g}^{2}d\xi^{2}\right)\,,\\\
&e^{\hat{\Phi}}=\sqrt{2}\overline{g}\,,\quad\hat{C}_{0}=0\,,\\\
&\hat{H}_{(3)}=-d\mathcal{A}_{(1)}\wedge
d\xi\,,\quad\hat{F}_{(3)}=\frac{1}{2\overline{g}^{2}}d\mathcal{A}_{(1)}\wedge
d\zeta\,,\\\
&\hat{F}_{(5)}=(1+\hat{\ast})\left[\left(\overline{H}_{(3)}+\frac{1}{4\overline{g}}\tanh^{2}2\rho\,\overline{F}_{(2)}\wedge(d\chi-2\overline{g}\overline{A}_{(1)})\right)\wedge
d\zeta\wedge d\xi\right]\,,\end{split}$ (D.5)

where we have flipped the signs $(\zeta,\xi)\mapsto(-\zeta,-\xi)$. We can
freely shift the dilaton vacuum value to zero by an $SL(2,\mathbb{R})$
transformation with matrix $\Lambda=\operatorname{diag}(p,1/p)$, where
$p=2^{1/4}\overline{g}^{1/2}$, which also rescales $\hat{F}_{(3)}$ and
$\hat{H}_{(3)}$ by $p$ and $1/p$ respectively.

We can interpret (D.5) as a truncation of the Type IIB theory on a four-
manifold $M_{2}\times T^{2}$, where $T^{2}$ is the two-torus parametrised by
$(\zeta,\xi)$, and $M_{2}$ is a non-compact two-manifold with metric

$ds^{2}_{2}=\cosh 2\rho\,d\rho^{2}+\frac{1}{4}\frac{\sinh^{2}2\rho}{\cosh
2\rho}d\chi^{2}\,.$ (D.6)

From (2.2), we recognise this two-dimensional metric to be the induced metric
on Eguchi-Hanson space where the two-sphere coordinates are held constant;
$ds^{2}_{2}=ds^{2}_{EH}\rvert_{(\theta,\varphi)=(\theta_{0},\varphi_{0})}$. As
$\rho\to 0$, the metric becomes

$ds^{2}_{2}\big{\rvert}_{\rho\to 0}=d\rho^{2}+\rho^{2}d\chi^{2}\,.$ (D.7)

The coordinate $\chi\in[0,2\pi)$, so $M_{2}$ is $\mathbb{R}^{2}$ in this
limit. For the asymptote at infinity, it is useful to consider a change of
coordinates $r^{2}=\cosh 2\rho$, in which the metric becomes

$ds^{2}_{2}=\left(1-\frac{1}{r^{4}}\right)^{-1}dr^{2}+\frac{r^{2}}{4}\left(1-\frac{1}{r^{4}}\right)d\chi^{2}\,.$
(D.8)

From this, we find that as $r\to\infty$,

$ds^{2}_{2}\big{\rvert}_{r\to\infty}=dr^{2}+\frac{r^{2}}{4}d\chi^{2}\,,$ (D.9)

so $M_{2}$ is $\mathbb{R}^{2}$ with a conical deficit angle $\pi$ in this
limit, which is simply the upper half-plane $\mathbb{R}\times\mathbb{R}^{+}$.
The Euler characteristic of $M_{2}$ can be computed using the Gauss-Bonnet
theorem,

$\chi(M_{2})=\frac{1}{2\pi}\left(\int_{M_{2}}K\operatorname{vol}(M_{2})+\int_{\partial
M_{2}}\kappa\,ds\right)=1\,.$ (D.10)

where $K=R/2$ is the Gaussian curvature, $\partial M_{2}$ is the boundary of
$M_{2}$ at infinity with line element $ds$, and $\kappa$ is the geodesic
curvature at infinity. Note however that the Euler characteristic of the four-
manifold $M_{2}\times T^{2}$ is

$\chi(M_{2}\times T^{2})=\chi(M_{2})\chi(T^{2})=0\,.$ (D.11)

The two-manifold $M_{2}$ is Kähler. It is easiest to see this by performing
the change of coordinates $u^{4}=r^{4}-1$, after which the metric reads

$ds^{2}_{2}=\left(1+\frac{1}{u^{4}}\right)^{-1/2}\left(du^{2}+\frac{u^{2}}{4}d\chi^{2}\right)\,.$
(D.12)

Defining the complex coordinate $w=ue^{i\chi/2}$, we can rewrite the metric as

$ds^{2}_{2}=\left(1+\frac{1}{|w|^{4}}\right)^{-1/2}dwd\overline{w}\,,$ (D.13)

from which we deduce the Kähler potential

$K=(1+|w|^{4})^{1/2}-\tanh^{-1}\left((1+|w|^{4})^{1/2}\right)\,.$ (D.14)

Since the two-torus is also Kähler, the four-manifold $M_{2}\times T^{2}$ is
Kähler.

## References

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