${\cal N}=(2,2)$ AdS3 from D3-branes wrapped on Riemann surfaces

Christopher<EMAIL_ADDRESS>Niall T.
<EMAIL_ADDRESS>Achilleas
<EMAIL_ADDRESS>
$a$: Institute for Theoretical Physics, Utrecht University Princetonplein 5,
3584 CC Utrecht, The Netherlands

$b$: Department of Physics, University of Oviedo, Avda. Federico Garcia Lorca
s/n, 33007 Oviedo, Spain

$c$: Department of Nuclear and Particle Physics, Faculty of Physics, National
and Kapodistrian University of Athens, Athens 15784, Greece

$d$: Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université
PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France

###### Contents

  1. 1 Introduction
  2. 2 $\mathcal{N}=(2,2)$ AdS3 solutions
    1. 2.1 Embedding into the $(0,2)$ classification
    2. 2.2 Canonical form of solutions with $\mathbb{T}^{3}$ fibration
  3. 3 D3-branes wrapping a constant curvature Riemann surface
    1. 3.1 Twisted D3-branes
    2. 3.2 A simple ansatz for constant curvature Riemann surfaces
    3. 3.3 Analysis of solutions
      1. 3.3.1 $\mathbb{T}^{2}$ case
      2. 3.3.2 S2 case
      3. 3.3.3 H2 case
      4. 3.3.4 Flux quantisation
      5. 3.3.5 Central charge
    4. 3.4 A more general solution
      1. 3.4.1 $\mathbb{T}^{2}$ case
      2. 3.4.2 S2 case
      3. 3.4.3 H2 case
      4. 3.4.4 Toric data and three-cycles
      5. 3.4.5 Flux quantisation and observables
  4. 4 D3-branes wrapping a topological disk
    1. 4.1 Summary of Boido:2021szx
    2. 4.2 An ${\cal N}=(2,2)$ tuning
    3. 4.3 Flux quantisation and central charge
    4. 4.4 Discussion
  5. A Reducing the conditions
    1. A.1 Reducing the conditions
    2. A.2 Simplifying the conditions
  6. B Embedding the topological disk solution into section 2.2

##  1 Introduction

Since the seminal work of Maldacena:2000mw , studying the low energy dynamics
of branes wrapped on compact manifolds has been a successful research
direction. From the field-theoretic point of view one may generate families of
lower-dimensional field theories from a higher-dimensional parent theory
living on the worldvolume of the branes in flat spacetime. By placing the
branes on a non-trivially curved manifold and requiring the preservation of
some supersymmetry, one must perform a twist of the field theory.
Geometrically, this is realised by studying the embedding of a calibrated
cycle, on which the brane is wrapped, into a special holonomy manifold. The
different ways of embedding the compactification manifold give rise to the
different families of lower-dimensional field theories.

For a large enough number of branes, and small enough curvatures, one may use
the holographic correspondence. Since the original work of Maldacena:2000mw ,
which studied the holographic duals of 4d $\mathcal{N}=4$ super-Yang–Mills
(SYM) and the 6d $\mathcal{N}=(2,0)$ theory compactified on a Riemann surface,
there have been many generalisations. The theories of class $\mathcal{S}$
study the twisted compactification of the 6d $(2,0)$ SCFT living on M5-branes
on a punctured Riemann surface Gaiotto:2009gz  preserving $\mathcal{N}=2$
supersymmetry in four dimensions. These were later extended to theories
preserving $\mathcal{N}=1$ supersymmetry in four dimensions in Bah:2015fwa ;
Bah:2011je ; Bah:2012dg ; Bah:2013qya  and the holographic dual of
Argyres–Douglas theories in Bah:2021mzw ; Bah:2021hei . Theories of class
$\mathcal{R}$ study the compactification of the 6d $(2,0)$ $A_{N-1}$ theory on
hyperbolic three-manifolds, Bobev:2019zmz . Whilst the theories in Bah:2018lyv
arise from compactifying the D4-D8 bound state on a Riemann surface. In
addition, the work of Bobev:2019ore  studies various compactifications of
branes on Riemann surfaces with punctures using gauged supergravity, and
Ferrero:2020laf ; Ferrero:2021wvk ; Boido:2021szx ; Hosseini:2021fge  consider
field theories wrapped on spindles.

In this work we will restrict to studying the holographic dual of a stack of
D3-branes compactified on a Riemann surface with a twist preserving
$\mathcal{N}=(2,2)$ supersymmetry and flowing to a two-dimensional SCFT. The
UV theory is 4d $\mathcal{N}=4$ SYM which flows to the 2d SCFT after being
placed on the Riemann surface. Via the AdS/CFT correspondence the two
theories, the UV and IR, admit AdS duals. For the UV theory this is of course
AdS${}_{5}\times\text{S}^{5}$ whilst for the IR one obtains the AdS3 solutions
which form the content of this work. The full flow solution is a five-
dimensional asymptotically AdS5 black string with AdS${}_{3}\times\Sigma_{2}$
near-horizon geometry.444Static black string solutions in 5d STU supergravity
have been found in Azzola:2018sld  for example.

The microscopic origin of the Bekenstein–Hawking entropy of black objects is
one of the fundamental problems in theoretical physics. Certain thermodynamic
quantities of black objects may be computed in gravity in either the
asymptotic or near-horizon limit without loss of information. One such
quantity is the Bekenstein–Hawking entropy. The fact that one only needs the
near-horizon to compute the Bekenstein–Hawking entropy and not the full black
string solution is a considerable simplification to the problem. One can then
identify the microstates of the black string as the SCFT dual to the AdS3
near-horizon theory. In this paper we will take this latter approach and
interest ourselves in the near-horizon of black strings preserving
$\mathcal{N}=(2,2)$ supersymmetry and not the full black string solution.

Solutions of this type were classified in Couzens:2017nnr , whilst AdS3
solutions preserving other amounts of supersymmetry have been studied in
Martelli:2003ki ; Tsimpis:2005kj ; Kim:2005ez ; Kim:2007hv ; Figueras:2007cn ;
Donos:2008hd ; Colgain:2010wb ; DHoker:2008lup ; Estes:2012vm ; Bachas:2013vza
; Jeong:2014iva ; Lozano:2015bra ; Kelekci:2016uqv ; Couzens:2017way ;
Eberhardt:2017uup ; Dibitetto:2018iar ; Dibitetto:2018ftj ; Macpherson:2018mif
; Legramandi:2019xqd ; Lozano:2019emq ; Lozano:2019jza ; Lozano:2019zvg ;
Lozano:2019ywa ; Couzens:2019mkh ; Couzens:2019iog ; Passias:2019rga ;
Filippas:2019ihy ; Speziali:2019uzn ; Lozano:2020bxo ; Farakos:2020phe ;
Couzens:2020aat ; Rigatos:2020igd ; Faedo:2020nol ; Dibitetto:2020bsh ;
Filippas:2020qku ; Passias:2020ubv ; Faedo:2020lyw ; Eloy:2020uix ;
Legramandi:2020txf ; Zacarias:2021pfz ; Emelin:2021gzx ; Couzens:2019wls . The
conditions for preserving supersymmetry and satisfying the equations of
motion, which we review in section 2, are very similar to the conditions for
other AdS solutions which are identified as wrapped brane solutions. In
particular the form of the solutions are similar to the AdS5 solutions arising
from M5-branes on a Riemann surface Bah:2013qya  and AdS4 solutions in massive
type IIA obtained from the D4-D8 bound state on a Riemann surface,
Passias:2018zlm ; Bah:2018lyv . We will reduce the conditions of
Couzens:2017nnr  to a Monge–Amperé like equation for a potential. We will
present two distinct classes of solutions. The first has a constant curvature
metric on the Riemann surface and we find that the solution is a deformation
of the Maldacena–Nu$\tilde{\text{n}}$ez solution Maldacena:2000mw . The
benefit of our formulation is that the inclusion of punctures has a natural
interpretation in this description: they are interpreted as sources to the
Monge–Amperé equation. The second class of solutions we study has a non-
constant curvature metric on the Riemann surface wrapped by the D3-branes.
Topologically the space is a disk with a $\mathbb{Z}_{k}$ orbifold singularity
at the centre and smeared D3-branes on the boundary.

The paper is organised as follows. In section 2 we will review the
construction of AdS3 solutions in type IIB preserving $\mathcal{N}=(2,2)$
supersymmetry before reducing the conditions under the assumption of the
existence of a flavour symmetry. We next consider solutions where the branes
wrap a constant curvature Riemann surface in section 3. We begin by reviewing
the field theory in section 3.1 before constructing a simple holographic dual
in section 3.2 and analysing the regularity and central charge of the solution
in section 3.3. We generalise this solution further in section 3.4 before
computing additional observables of the theory and comparing to the field
theory results. In section 4 we consider the holographic duals of 4d
$\mathcal{N}=4$ SYM on a topological disk equipped with a non-constant
curvature Riemann surface, analysing the solutions in detail. Some technical
material is relegated to two appendices.

##  2 $\mathcal{N}=(2,2)$ AdS3 solutions

We now turn our attention to constructing the supergravity dual of the 2d
$\mathcal{N}=(2,2)$ SCFTs discussed in the introduction. We will reduce
$\mathcal{N}=4$ SYM on a Riemann surface, breaking the SU$(4)_{\rm R}$
R-symmetry to its U$(1)^{3}$ Cartan in order to perform a topological twist.
As such, the resultant two-dimensional field theory should admit three U$(1)$
symmetries: two will furnish the U$(1)_{L}\times$U$(1)_{R}$ R-symmetry of the
$\mathcal{N}=(2,2)$ superconformal algebra whilst the third will be a flavour
symmetry of the theory. Given these considerations, the metric of the gravity
solution should take the form

$\text{AdS}_{3}\times\left(\Sigma_{2}\times\text{S}^{1}_{R}\times\text{S}^{1}_{L}\times\text{S}^{1}\times
I_{1}\times I_{2}\right)\,,$ (2.1)

where $I_{1}$, $I_{2}$ are two line intervals. Solutions of this form,
supported by five-form flux were classified in Couzens:2017nnr .555For
convenience we make a few redefinitions of the results in Couzens:2017nnr , in
particular we redefine the warp factor via
$\mathrm{e}^{-4\Delta}y=\sin^{2}\zeta$. The metric is given by

$\frac{1}{L^{2}}\mathrm{d}s^{2}=\frac{\sqrt{y}}{\sin\zeta}\left[\mathrm{d}s^{2}(\text{AdS}_{3})+\mathrm{d}s^{2}(X_{7})\right]\,,$
(2.2)

with AdS3 having unit radius and $L$ a constant. Supersymmetry fixes the
internal metric to take the form

$\mathrm{d}s^{2}(X_{7})=\cos^{2}\zeta(\mathrm{d}\psi_{1}+\sigma)^{2}+\sin^{2}\zeta\mathrm{d}\psi_{2}^{2}+\frac{\sin^{2}\zeta}{4y^{2}\cos^{2}\zeta}\mathrm{d}y^{2}+\frac{\sin^{2}\zeta}{y}g^{(4)}(y,x)_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}\,.$
(2.3)

The coordinate $y$ defines an almost-product structure, implying that the one-
form $\sigma$ has “legs” only along the four-dimensional base, though it may
still depend on $y$ non-trivially. The metric $g^{(4)}$ on the base admits an
SU$(2)$-structure $(J,\,\Omega)$ satisfying666One can reduce the number of
these conditions further by reinterpreting the conditions in terms of
derivatives of the determinant of the metric, see Couzens:2017nnr . This
reformulation is reminiscent of the ones appearing for AdS5 solutions in
M-theory Gauntlett:2004zh  and AdS4 solutions in massive type IIA
Passias:2018zlm . However, for our purposes the most convenient formulation of
the conditions are the ones we present here.

$\displaystyle\partial_{\psi_{1}}J$ $\displaystyle=\partial_{\psi_{2}}J=0\,,$
(2.4) $\displaystyle\partial_{y}J$
$\displaystyle=\frac{1}{2}\mathrm{d}_{4}\sigma\,,$ (2.5)
$\displaystyle\mathrm{d}_{4}J$ $\displaystyle=0\,,$ (2.6)

and

$\displaystyle\partial_{\psi_{1}}\Omega$ $\displaystyle=-\mathrm{i}\Omega\,,$
(2.7) $\displaystyle\partial_{\psi_{2}}\Omega$ $\displaystyle=0\,,$ (2.8)
$\displaystyle\partial_{y}\Omega$
$\displaystyle=\frac{1}{2}\tan^{2}\zeta\partial_{y}\left(\log\left(\frac{\sin^{2}\zeta}{y}\right)\right)\Omega\,,$
(2.9) $\displaystyle\mathrm{d}_{4}\Omega$
$\displaystyle=-\left[\mathrm{i}\sigma+\mathrm{d}_{4}(\log\cos\zeta)\right]\wedge\Omega\,.$
(2.10)

The metric is supported by the five-form flux

$\frac{1}{L^{4}}F_{5}=(1+\star)\mathrm{d}\mathrm{vol}(\mathrm{AdS}_{3})\wedge
F^{(2)}\,,$ (2.11)

with

$F^{(2)}=y\mathrm{d}\sigma-2J_{4}-\mathrm{d}\left(\frac{y}{\sin^{2}\zeta}(\mathrm{d}\psi_{1}+\sigma)\right)\,.$
(2.12)

Supersymmetry implies that all equations of motion are satisfied provided all
the torsion conditions given above hold. Contrast this with the
$\mathcal{N}=(0,2)$ case Kim:2005ez , where one must also impose the Maxwell
equation for $F^{(2)}$.

Consistent with these geometries preserving $\mathcal{N}=(2,2)$ supersymmetry,
which has R-symmetry group U(1)${}_{L}\times$U(1)R, the metric admits two
independent Killing vectors: $\partial_{\psi_{1}}$, $\partial_{\psi_{2}}$. The
generator dual to the left-moving R-symmetry is
$\partial_{\psi_{1}}-\partial_{\psi_{2}}$ whilst the right moving is dual to
$\partial_{\psi_{1}}+\partial_{\psi_{2}}$. Recall that the solution we are
looking for should have three U$(1)$ isometries. The general form of the
internal metric accounts for two of the three required, and therefore we
require that the metric $g^{(4)}$ should admit a single Killing vector.

###  2.1 Embedding into the $(0,2)$ classification

Before we proceed, it is useful to study the embedding of the
$\mathcal{N}=(2,2)$ conditions into the classification of $\mathcal{N}=(0,2)$
solutions Kim:2005ez . We first want to isolate the right-moving R-symmetry
vector field, to do this we make the change of coordinates

$\psi_{1}=\frac{1}{2}(\phi_{R}+\phi_{L})\,,\quad\psi_{2}=\frac{1}{2}(\phi_{R}-\phi_{L})\,.$
(2.13)

A small rearrangement of terms after the coordinate transformation, and
identifying the warp factor to be

$y\mathrm{e}^{-4\Delta}=\sin^{2}\zeta\,,$ (2.14)

puts the solution into the form

$\mathrm{d}s^{2}(X_{7})=\frac{1}{4}\left[\mathrm{d}\phi_{R}+(1-2y\mathrm{e}^{-4\Delta})\mathrm{d}\phi_{L}+2(1-y\mathrm{e}^{-4\Delta})\sigma\right]^{2}+\mathrm{e}^{-4\Delta}\mathrm{d}s^{2}(Y_{6})\,,$
(2.15)

where

$\mathrm{d}s^{2}(Y_{6})=\frac{1}{4y(1-y\mathrm{e}^{-4\Delta})}\mathrm{d}y^{2}+y(1-y\mathrm{e}^{-4\Delta})(\mathrm{d}\phi_{L}+\sigma)^{2}+g(y,x)_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}\,.$
(2.16)

This is the same form for the metric in the $\mathcal{N}=(0,2)$ classification
Kim:2005ez .777We use the conventions in Couzens:2017nnr  rather than
Kim:2005ez , the difference is a minor change of orientation. The one-form
dual to the R-symmetry vector is

$\eta=\frac{1}{2}\left[\mathrm{d}\phi_{R}+(1-2y\mathrm{e}^{-4\Delta})\mathrm{d}\phi_{L}+2(1-y\mathrm{e}^{-4\Delta})\sigma\right]\,,$
(2.17)

and the Kähler base $Y_{6}$ is defined above. The SU$(3)$-structure forms, in
terms of the SU$(2)$-structure ones are given by

$J_{6}=\frac{1}{2}\mathrm{d}y\wedge(\mathrm{d}\phi_{L}+\sigma)+J\,,\quad\Omega_{6}=\frac{1}{2\sqrt{y(1-y\mathrm{e}^{-4\Delta})}}\left[\mathrm{d}y+2\mathrm{i}y(1-y\mathrm{e}^{-4\Delta})(\mathrm{d}\phi_{L}+\sigma)\right]\wedge\Omega\,,$
(2.18)

and it is simple to see that they satisfy the $\mathcal{N}=(0,2)$ conditions,

$\displaystyle\mathrm{d}J_{6}=0\,,\quad\mathrm{d}\Omega_{6}=-2\mathrm{i}\eta\wedge\Omega_{6}\,,$
(2.19)

provided the $\mathcal{N}=(2,2)$ torsion conditions (2.4)-(2.10) hold.
Moreover, since $\mathcal{N}=(2,2)$ supersymmetry imposes the Maxwell equation
for $F^{(2)}$ it follows that the $\mathcal{N}=(0,2)$ master equation is also
satisfied.

###  2.2 Canonical form of solutions with $\mathbb{T}^{3}$ fibration

In appendix A we have reduced the torsion conditions (2.4)-(2.10) under the
assumption that $g^{(4)}$ contains an additional flavour U(1) direction. The
solution is determined by a potential $D$ satisfying a Mongé–Ampere like
equation (2.20), which is a generalisation of a Toda equation. Specifically
this takes the form

$\square
D=16y^{2}\Big{(}\partial_{y}^{2}D\partial_{\Theta}D-(\partial_{y}\partial_{\Theta}D)^{2}\Big{)}\mathrm{e}^{\partial_{y}D}\,,$
(2.20)

with $\square\equiv\partial_{X_{1}}^{2}+\partial_{X_{2}}^{2}$. Once a solution
to (2.20) has been provided the metric is given by

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}$
$\displaystyle=\sqrt{\frac{yg}{h}}\left[\mathrm{d}s^{2}(\text{AdS}_{3})+\frac{h}{g}\mathrm{d}\psi_{2}^{2}+\frac{h\mathrm{e}^{2A}}{yg}(\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2})+\frac{h}{yg}\mathrm{d}s^{2}(\mathcal{M}_{4})\right]\,,$
$\displaystyle\mathrm{d}s^{2}(\mathcal{M}_{4})$
$\displaystyle=\frac{1}{4}g_{ij}\mathrm{d}u^{i}\mathrm{d}u^{j}+h^{ij}\eta_{i}\eta_{j}\,,\quad\mathrm{e}^{2A}=4y^{2}\mathrm{e}^{\partial_{y}D}g\,,$
$\displaystyle g_{ij}$ $\displaystyle\equiv-\partial_{i}\partial_{j}D\,,\quad
h_{ij}\equiv-\partial_{i}\partial_{j}\left(D+y(\log y-1)\right)\,,$
$\displaystyle\eta_{1}$
$\displaystyle\equiv\mathrm{d}\psi_{1}+\star_{2}\mathrm{d}_{2}(\partial_{y}D)\,,\quad\eta_{2}\equiv\frac{1}{2}\left(\mathrm{d}\phi+\star_{2}\mathrm{d}_{2}(\partial_{\Theta}D)\right)\,,$
(2.21)

with

$u^{i}=\\{y,\Theta\\}\,,\quad g\equiv\det(g_{ij})\,,\quad
h\equiv\det(h_{ij})\,.$ (2.22)

The form of the metric suggests that one should look for solutions with
D3-branes wrapped on the $(X_{1},\,X_{2})$ directions; we will do just that in
section 3. However, we should stress that these directions can lie in the co-
dimensions of the wrapped D3-branes as we shall find in section 4. Our focus
in this work are cases where $(X_{1},\,X_{2})$ span a Riemann surface of
constant curvature. Going away from the constant curvature metric and
including punctures on the Riemann surface is beyond the scope of this work,
however the generalisation of our ansatz in this direction is obvious, though
technically challenging.

##  3 D3-branes wrapping a constant curvature Riemann surface

In this section we will consider the compactification of 4d $\mathcal{N}=4$
SYM on a constant curvature Riemann surface $\Sigma_{\mathfrak{g}}$, of genus
$\mathfrak{g}$, preserving $\mathcal{N}=(2,2)$ supersymmetry. We begin in
section 3.1 by reviewing the construction of the 2d SCFT arising from the
compactification. In particular, we will compute the central charge and the
R-charges of baryonic operators with which we may compare to a dual gravity
computation. Our gravity analysis begins with the reduction of the general
$\mathcal{N}=(2,2)$ supersymmetry conditions of (2.2) for a constant curvature
Riemann surface. We find that there is no solution for $\mathfrak{g}=0$,
whilst the solution for $\mathfrak{g}=1$ has an enhancement of supersymmetry
to $\mathcal{N}=(4,4)$ and is the double T-dual of the D1-D5 system. For
$\mathfrak{g}>1$ we find a family of solutions which generalise the
Maldacena–Nu$\tilde{\text{n}}$ez solution Maldacena:2000mw , breaking the
SU$(2)$ symmetry of their solution to U$(1)$. We present the regularity
analysis of these solutions before computing their central charge and
R-charges, which match with the field theory analysis in section 3.1.

###  3.1 Twisted D3-branes

Let us consider compactifications of 4d $\mathcal{N}=4$ SYM on a Riemann
surface. As is well known when the Riemann surface has non-trivial curvature
it is necessary to perform a topological twist of the theory in order to
preserve supersymmetry.888Recently, compactifications which twist with the
isometry of the compactification space have been investigated, see for example
Bah:2019rgq .  Let us begin by defining our conventions for 4d $\mathcal{N}=4$
SYM. For the moment we will use $\mathcal{N}=4$ representations, however as we
will explain later, in order to compare to the R-charges computed in gravity
it is useful to rewrite the theory in terms of $\mathcal{N}=1$ multiplets.

As is well known 4d $\mathcal{N}=4$ SYM contains a single $\mathcal{N}=4$
gauge multiplet, which in $\mathcal{N}=1$ language consists of a single
$\mathcal{N}=1$ gauge multiplet and three chiral superfields $\Phi_{i}$. The
bosonic symmetry group of the theory is

$\text{SO}(1,3)\times\text{SU}(4)_{\rm R}\,,$ (3.1)

with SU$(4)_{\rm R}$ the R-symmetry of the theory. The 16 supersymmetry
transformation parameters (Killing spinors) transform under the bosonic
symmetry group as

$\epsilon:({\bf 2},{\bf 1},{\bf 4})\,,\quad\tilde{\epsilon}:({\bf 1},{\bf
2},\bar{{\bf 4}})\,,$ (3.2)

whilst the gauginos transform in

$\Psi:({\bf 2},{\bf 1},{\bf 4})\,,\quad\tilde{\Psi}:({\bf 1},{\bf 2},\bar{{\bf
4}})\,.$ (3.3)

We now want to place the theory on a Riemann surface of constant curvature
$\Sigma_{\mathfrak{g}}$, such that the Ricci scalar is
$R_{\Sigma_{\mathfrak{g}}}=2\kappa$ with $\kappa=\\{-1,0,1\\}$, for
$\Sigma_{\mathfrak{g}>1}=\text{H}^{2}$,
$\Sigma_{\mathfrak{g}=1}=\mathbb{T}^{2}$, and
$\Sigma_{\mathfrak{g}=0}=\text{S}^{2}$ respectively. For H2 this is meant
locally, globally it is a compact quotient H${}^{2}/\Gamma$, where $\Gamma$ is
an element of the Fuschian subgroup of the PSL(2, $\mathbb{R}$) isometry group
of H2. The theory then flows to a two-dimensional SCFT in the IR. The presence
of the Riemann surface breaks the four-dimensional Lorentz symmetry to

$\text{SO}(1,3)\rightarrow\text{SO}(1,1)\times\text{U}(1)_{\Sigma_{\mathfrak{g}}}\,.$
(3.4)

Generically this does not preserve any supersymmetry unless the Riemann
surface is the two-torus $\mathbb{T}^{2}$. Since we are interested in
preserving $(2,2)$ supersymmetry in the resultant 2d SCFT we must perform a
topological twist. We will twist with the Cartan of the SU$(4)_{\rm R}$
R-symmetry, namely U$(1)^{3}$. Under the Cartan the representations of
SU$(4)_{\rm R}$ that we need, decompose as (see for example Lawrie:2016axq ):

$\displaystyle\text{SU}(4)_{\rm R}$
$\displaystyle\rightarrow\text{U}(1)_{1}\times\text{U}(1)_{2}\times\text{U}(1)_{3}$
$\displaystyle{\bf 4}$ $\displaystyle\rightarrow{\bf
1}_{\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2}}\oplus{\bf
1}_{\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2}}\oplus{\bf
1}_{-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2}}\oplus{\bf
1}_{-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2}}$ (3.5) $\displaystyle{\bf 6}$
$\displaystyle\rightarrow{\bf 1}_{1,0,0}\oplus{\bf 1}_{0,1,0}\oplus{\bf
1}_{0,0,1}\,.$

We now want to study the possible twists of the theory which preserve
$\mathcal{N}=(2,2)$ supersymmetry. Such twists have been classified previously
in Maldacena:2000mw ; Benini:2013cda ; Amariti:2017cyd ; Bobev:2019ore ,
however for completeness and to extract out the R-charges, we will perform the
analysis here as well. We turn on a background gauge field $A$ along the
generator $T$

$T=a_{1}T_{1}+a_{2}T_{2}+a_{3}T_{3}\,,$ (3.6)

with $T_{i}$ the generator of the $i$’th U$(1)$ in the U$(1)^{3}$ Cartan.
Supersymmetry of the theory is preserved subject to a Killing spinor equation
being satisfied. By turning on a background gauge field, the Killing spinor
equation on the Riemann surface is given by

$\left(\partial_{\mu}-\frac{\mathrm{i}}{2}s\omega_{\mu}-\mathrm{i}A_{\mu}\right)\epsilon=0\,.$
(3.7)

The gauge field $A$ has field-strength $F=-\kappa T\mathrm{d}\omega$, for
$\kappa\neq 0$, with $\omega$ the spin connection on the Riemann surface,
whilst for $\kappa=0$ we have
$F=-T\frac{2\pi}{\mathrm{vol}(\mathbb{T}^{2})}\mathrm{d}\mathrm{vol}(\mathbb{T}^{2})$.
The constant $s$ is $\pm 1$ for positive/negative chirality spinors
respectively. In order for supersymmetry to be preserved we must tune the
parameters $a_{i}$ such that the Killing spinor equation admits constant
spinor solutions. This requires the contribution from the spin connection to
be cancelled by the background gauge field.

Using the decomposition in (3.5) and denoting the spinors as
$\epsilon_{\pm}^{m}$, $m\in\\{1,2,3,4\\}$ we see that constant spinors satisfy
the Killing spinor equation provided

$\displaystyle\mp\frac{\kappa}{2}=a_{1}+a_{2}+a_{3}\,,\quad\mp\frac{\kappa}{2}=a_{1}-a_{2}-a_{3}\,,\quad\mp\frac{\kappa}{2}=-a_{1}+a_{2}-a_{3}\,,\quad\mp\frac{\kappa}{2}=-a_{1}-a_{2}+a_{3}\,,$
(3.8)

for each of the spinors $\epsilon_{\pm}^{m}$ respectively.

$\mathcal{N}=(2,2)$ supersymmetry is preserved if the $a_{i}$ are chosen such
that two of the conditions above are satisfied, one for a positive chirality
spinor and one for a negative chirality spinor. Without loss of generality, we
can choose these spinors to be $\epsilon_{+}^{1}$ and $\epsilon_{-}^{2}$,
which implies that the $a_{i}$’s satisfy

$a_{1}=0\,,\quad a_{2}+a_{3}=-\kappa\,.$ (3.9)

Recall that a 2d $\mathcal{N}=(2,2)$ SCFT admits a U$(1)_{L}\times$U$(1)_{R}$
R-symmetry. We may identify the right-moving R-symmetry by requiring that the
positive chirality spinor has R-charge 1 and the negative chirality spinor has
vanishing R-charge under it. Conversely under the left-moving R-symmetry the
positive chirality spinor has vanishing R-charge whilst the negative chirality
spinor has R-charge 1. The most general right-moving U$(1)$ R-symmetry with
these properties is given by999These choices of R-symmetry generalise those
taken in Maldacena:2000mw .

${\rm R}_{R}=T_{1}+\cos^{2}\alpha_{R}T_{2}+\sin^{2}\alpha_{R}T_{3}\,,$ (3.10)

whilst for the left-moving R-symmetry we have

${\rm R}_{L}=T_{1}-\cos^{2}\alpha_{L}T_{2}-\sin^{2}\alpha_{L}T_{3}\,.$ (3.11)

Note in addition that

$T_{\rm F}=T_{2}-T_{3}\,,$ (3.12)

parametrises a flavour symmetry of the theory. For a theory which is conformal
at the IR fixed point it is necessary that the ‘t Hooft anomaly101010The
indices $I,J$ run over all U$(1)$’s of the theory. Here these are a flavour
symmetry, and a left- and right-moving R-symmetry. The $Q$’s are the charges
of the Weyl fermion under the generator of the symmetry and $\gamma^{3}$ is
the chirality matrix in 2d.

$k_{IJ}=\sum_{\text{Weyl fermions}}\gamma_{3}Q_{I}Q_{J}\,,$ (3.13)

of a left- and right-moving current vanishes Benini:2013cda . In addition at
the fixed point the ‘t Hooft anomaly for a flavour and R-symmetry must vanish,
therefore altogether we must impose

$k_{R{\rm F}}=k_{L{\rm F}}=k_{LR}=0\,.$ (3.14)

Note that the first two are the conditions imposed by $c$-extremization,
whilst the latter is an additional constraint which imposes
$\mathcal{N}=(2,2)$ supersymmetry. As an aside, if one naively performs
$c$-extremization preserving $\mathcal{N}=(0,2)$ supersymmetry and then tunes
the parameters to preserve $\mathcal{N}=(2,2)$ supersymmetry as in (3.9) one
finds the non-sensical result that the central charge vanishes. By imposing
the additional constraint that $k_{LR}=0$ we alleviate this problem.

In order to evaluate the ‘t Hooft anomalies we need to know the multiplicities
of the 2d fermions. This can be computed by using the Riemann–Roch index
theorem. The difference between the number of zero modes on
$\Sigma_{\mathfrak{g}}$ is given by Maldacena:2000mw

$n^{+}-n^{-}=\frac{1}{2\pi}\int_{\Sigma_{\mathfrak{g}}}F=2T(\mathfrak{g}-1)\,.$
(3.15)

The constraints from (3.14) are satisfied provided

$a_{2}=a_{3}=-\frac{\kappa}{2}\,,$ (3.16)

which gives the following ‘t Hooft anomalies for the left- and right-moving
R-symmetries

$k_{RR}=k_{LL}=N^{2}(\mathfrak{g}-1)\,.$ (3.17)

Consequently the central charges are given by

$c_{L}=c_{R}=3N^{2}(\mathfrak{g}-1)\,,$ (3.18)

where we have used $c_{R}=3k_{RR}$ and $c_{L}=3k_{LL}$. Note that the central
charge is only well-defined for $\mathfrak{g}>1$, the compactification on the
two-sphere with this twist does not give rise to a well defined IR SCFT. Note
in particular that the parameters of the R-symmetry drop out of the central
charge. To see the contribution of these parameters we should compute the
R-charges of the reduced scalar fields. Recall that the scalars of 4d
$\mathcal{N}=4$ SYM transform in the $\bf 6$ of the SU$(4)_{\rm R}$
R-symmetry. From (3.5) we have that the $i$’th $\mathcal{N}=1$ chiral
multiplet may be arranged to have charge $1$ under the $i$’th U$(1)$ and
uncharged under the others. It then follows from (3.10) that the (right-
moving) R-charges are

${\rm R}[\Phi_{1}]=1\,,\quad{\rm R}[\Phi_{2}]=\cos^{2}\alpha_{R}\,,\quad{\rm
R}[\Phi_{3}]=\sin^{2}\alpha_{R}\,.$ (3.19)

We will see later that these are indeed reproduced by the gravity solution of
section 3.4.5.

###  3.2 A simple ansatz for constant curvature Riemann surfaces

In this section we consider a simple ansatz for D3-branes wrapping Riemann
surfaces of constant curvature. As we shall see only the cases of
$\mathbb{T}^{2}$ and H2 give rise to sensible holographic duals, the former
reproduces the D3-D3 near-horizon while the latter generalises the ${\cal
N}=(2,2)$ solution of Maldacena:2000mw . Our starting point is the system
presented in section 2.2.

In order to make the comparison with the field theory more manifest it is
useful to first make the change of coordinates

$y=m^{+}t^{+}+m^{-}t^{-}\,,\quad\Theta=t^{+}-t^{-}\,,$ (3.20)

where $m^{\pm}$ are constrained to satisfy

$m^{+}+m^{-}=1\,.$ (3.21)

This change of coordinates puts the defining equation into the form

$\square
D=16(m^{+}t^{+}+m^{-}t^{-})^{2}\Big{(}\partial_{t^{+}}^{2}D\partial_{t^{-}}^{2}D-(\partial_{t^{+}}\partial_{t^{-}}D)^{2}\Big{)}\mathrm{e}^{(\partial_{t^{+}}+\partial_{t^{-}})D}\,.$
(3.22)

We will make the assumption that the Riemann surface is of constant curvature,
as such we assume that the warp factor $\mathrm{e}^{2A}$ appearing in front of
the Riemann surface takes the factorised form

$\mathrm{e}^{2A}=f(t^{+},t^{-})\mathrm{e}^{2A_{0}(X_{1},X_{2})}\,,$ (3.23)

where the potential $A_{0}$ satisfies111111In order to generalise to include
punctures on the Riemann surface one should add source terms to the right-hand
side of equation (3.24).

$\square A_{0}=-\kappa\mathrm{e}^{2A_{0}}\,.$ (3.24)

Given the definition of $e^{2A}$, after substituting this ansatz into (2.20),
the potential $D$ is required to take the form

$D=\mu_{g}^{2}\tilde{A}_{0}+I(t^{+},t^{-})\,,$ (3.25)

where

$\mu_{g}^{2}\equiv 2(a^{+}t^{+}+a^{-}t^{-})+c\,,$ (3.26)

with $c$ an integration constant and $a^{\pm}$ constants satisfying

$a^{+}+a^{-}=\kappa\,.$ (3.27)

In addition we have defined the scalar $\tilde{A}_{0}$ which satisfies

$\square\tilde{A}_{0}=-\mathrm{e}^{2A_{0}}\,,$ (3.28)

and has solution

$\tilde{A}_{0}=\begin{cases}-\frac{1}{4}(X_{1}^{2}+X_{2}^{2})&\text{for
}\kappa=0\,,\\\\[5.69054pt] \kappa A_{0}&\text{for }\kappa\neq
0\,.\end{cases}$ (3.29)

With these definitions equation (3.22) reduces to

$-\mu_{g}^{2}=16(m^{+}t^{+}+m^{-}t^{-})^{2}\mathrm{e}^{(\partial_{t^{+}}+\partial_{t^{-}})I(t^{+},t^{-})}\Big{(}\partial_{t^{+}}^{2}I\partial_{t^{-}}^{2}I-(\partial_{t^{+}}\partial_{t^{-}}I)^{2}\Big{)}\,,$
(3.30)

which we must solve for $I$. Notice that this equation is left invariant when
$I$ is shifted by any linear function of $(t^{+}-t^{-})$, so we can without
loss of generality set such terms to zero. We will take a similar ansatz to
that used in Bah:2018lyv , namely

$\displaystyle I=$
$\displaystyle-\mu_{0}^{2}(\log\mu_{0}^{2}-1)-\frac{1}{c^{+}}\mu_{+}^{2}(\log\mu_{+}^{2}-1)-\frac{1}{c^{-}}\mu_{-}^{2}(\log\mu_{-}^{2}-1)+2\nu(t^{+}+t^{-})\,,$
(3.31)

where we define

$\mu^{2}_{\pm}=b^{\pm}-c^{\pm}t^{\pm}\,,\quad\mu_{0}^{2}=m^{+}t^{+}+m^{-}t^{-}=y\,.$
(3.32)

In general the constants are fixed as

$a^{\pm}=\frac{1}{2}\kappa\,,\quad m^{\pm}=\frac{1\pm\epsilon}{2}\,,\quad
c^{\pm}\neq 0\,,$ (3.33)

with additional constraints that depend on the curvature of the Riemann
surface, namely

$\displaystyle\kappa=0:$ $\displaystyle~{}~{}~{}~{}\epsilon=\pm
1\,,~{}~{}~{}~{}c=-16b^{\pm}c^{\mp}e^{4\nu}\,,$ (3.34)
$\displaystyle\kappa\neq 0:$
$\displaystyle~{}~{}~{}~{}\epsilon=\frac{\kappa(b^{-}c^{+}-b^{+}c^{-})\pm\sqrt{c^{+}c^{-}(c^{2}c^{+}c^{-}-4b^{-}b^{+})}}{\kappa(b^{-}c^{+}+b^{+}c^{-})+c\,c^{-}c^{+}}\,,~{}~{}~{}~{}\mathrm{e}^{4\nu}=-\frac{\kappa}{4c^{+}c^{-}(1-\epsilon^{2})}\,.$

Notice that we have

$c^{+}c^{-}\mu_{0}^{2}+m^{+}\mu^{2}_{+}+m^{-}\mu^{2}_{-}=c^{+}b^{-}m^{-}+c^{-}b^{+}m^{+}\,,$
(3.35)

so that when $m^{\pm}\neq 0$ and likewise the right-hand side of this
expression, $(\mu_{0},\,\mu_{+},\,\mu_{-})$ define an embedding of a surface
into a three-dimensional ambient space. The metric of the constant curvature
solutions take a simple form in terms of these coordinates, we have in general
that

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}$
$\displaystyle=\sqrt{\Lambda}\left[\mathrm{d}s^{2}(\text{AdS}_{3})+4e^{4\nu}c^{+}c^{-}\mathrm{e}^{2A_{0}}(\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2})\right]+\frac{1}{\sqrt{\Lambda}}\mathrm{d}s^{2}(\mathcal{M}_{5})\,,$
(3.36) $\displaystyle\mathrm{d}s^{2}(\mathcal{M}_{5})$
$\displaystyle=\mathrm{d}\mu_{0}^{2}+\frac{1}{c^{+}}\mathrm{d}\mu_{+}^{2}+\frac{1}{c^{-}}\mathrm{d}\mu_{-}^{2}+\frac{1}{c^{+}}\mu_{+}^{2}\eta_{+}^{2}+\frac{1}{c^{-}}\mu_{-}^{2}\eta_{-}^{2}+\mu_{0}^{2}\mathrm{d}\psi_{2}^{2}\,,$

where we define the warp factor

$\Lambda\equiv\mu_{0}^{2}+\frac{(m^{+})^{2}}{c^{+}}\mu^{2}_{+}+\frac{(m^{-})^{2}}{c^{-}}\mu_{-}^{2}\,,$
(3.37)

and the fibered terms

$\eta_{\pm}\equiv\frac{1}{2}\left[(1\pm\epsilon)\mathrm{d}\psi_{1}\pm\mathrm{d}\phi+V\right]\,,\quad
V=\kappa\left(\partial_{X_{2}}\tilde{A}_{0}\,\mathrm{d}X_{1}-\partial_{X_{1}}\tilde{A}_{0}\,\mathrm{d}X_{2}\right)\,.$
(3.38)

Clearly, positivity of $\mathcal{M}_{5}$ demands $c^{\pm}>0$ and this manifold
has the appearance of a U(1)3 preserving deformed five-sphere. However, as we
will see shortly, the actual topology of this space depends on that of the
Riemann surface. We shall study the distinct solutions in more detail in the
the next section.

###  3.3 Analysis of solutions

We saw in the previous section that the solution has the appearance of a
U$(1)^{3}$ preserving deformed five-sphere, fibered over a Riemann surface,
subject to the embedding condition (3.35). In this section we will study the
regularity of the solutions for all choices of Riemann surface.

####  3.3.1 $\mathbb{T}^{2}$ case

We begin by considering the case $\kappa=0$, taking the solution

$\epsilon=1\,,\quad c=-16e^{4\nu}b^{+}c^{-}\,,$ (3.39)

of (3.34) without loss of generality. The condition (3.35) then informs us
that we should fix

$c^{+}\mu_{0}^{2}=b^{+}\sin^{2}\alpha\,,\quad\mu_{+}^{2}=b^{+}\cos^{2}\alpha\,,$
(3.40)

and one finds that $\eta_{\pm}$ are merely two independent linear combinations
of $(\psi_{1},\,\phi)$, that we shall identify as $\eta_{\pm}=d\tau_{\pm}$ so
that

$\mathrm{d}s^{2}(\mathcal{M}_{5})=\frac{b^{+}}{c^{+}}\left(\mathrm{d}\alpha^{2}+\cos^{2}\alpha\mathrm{d}\tau_{+}^{2}+\sin^{2}\alpha\mathrm{d}\psi_{2}^{2}\right)+\frac{1}{c^{-}}\left(\mathrm{d}\mu_{-}^{2}+\mu_{-}^{2}\mathrm{d}\tau^{2}_{-}\right)\,,\quad\Lambda=\frac{b^{+}}{c^{+}}\,,$
(3.41)

in other words the warp factor is constant and the internal five-manifold is
S${}^{3}\times\mathbb{R}^{2}$. Locally there is of course no difference
between this and S${}^{3}\times\mathbb{T}^{2}$, making the entire space
locally AdS${}_{3}\times{\rm S}^{3}\times\mathbb{T}^{4}$ – clearly then we
have reproduced the D3-D3 near-horizon121212I.e. the D1-D5 near-horizon twice
T-dualised on different U(1)’s within $\mathbb{T}^{4}$., one can check that
the flux is consistent with this. There are no further solutions with
$\kappa=0$.

####  3.3.2 S2 case

In this section we shall study the case $\kappa>0$, i.e. the two-sphere case.
When $\kappa\neq 0$, positivity of the Riemann surface factor in (3.36)
demands that

$-\frac{\kappa}{1-\epsilon^{2}}>0\,,$ (3.42)

so when $\kappa>0$ we must have $\epsilon^{2}>1$, the symmetry of the solution
means we can take this to be $\epsilon>1$ and so $m^{-}<0$ without loss of
generality. Turning our attention to (3.35) we see that when the right-hand
side is positive/negative it becomes an embedding equation for dS2/AdS2, both
of which are non-compact and pseudo-Riemannian – however
$(\mu_{\pm},\,\mu_{0})$ do not appear in the definition of ${\cal M}_{5}$ with
the correct signs to give rise to sub-manifolds of these topologies and in
fact ${\cal M}_{5}$ remains positive in both these cases. One can parametrise

$\displaystyle
c^{+}b^{-}m^{-}+c^{-}b^{+}m^{+}=+1\,:~{}~{}~{}~{}c^{+}c^{-}\mu_{0}^{2}=\cos^{2}\beta\cosh^{2}r\,,~{}~{}~{}~{}m^{+}\mu^{2}_{+}=\sin^{2}\beta\cosh^{2}r\,,$
$\displaystyle
c^{+}b^{-}m^{-}+c^{-}b^{+}m^{+}=-1\,:~{}~{}~{}~{}c^{+}c^{-}\mu_{0}^{2}=\cos^{2}\beta\sinh^{2}r\,,~{}~{}~{}~{}m^{+}\mu^{2}_{+}=\sinh^{2}\beta\cosh^{2}r\,,$

without loss of generality, however in either case $0\leq r<\infty$, and the
upper limit is at infinite proper distance in the ten-dimensional space, so
the solutions following from these tuning are unbounded. There is of course
one final option, fix

$c^{+}b^{-}m^{-}+c^{-}b^{+}m^{+}=0\,.$ (3.43)

Here one can parameterise

$c^{+}c^{-}\mu_{0}^{2}=r^{2}\cos^{2}\beta,\quad
m^{+}\mu^{2}_{+}=r^{2}\sin^{2}\beta\,,$ (3.44)

but once again $0\leq r<\infty$, with the upper bound at infinite proper
distance. Hence while solutions exist with S2, they do not represent good
holographic duals to CFTs.

####  3.3.3 H2 case

In this section we shall study the case $\kappa<0$, where the Riemann surface
is a compact quotient of H2. Positivity of the metric in this case requires
that $\epsilon^{2}<1$, meaning that $m^{\pm}>0$ and so (3.35) embeds a two-
sphere in $\mathbb{R}^{3}$. We thus fix

$c^{+}b^{-}m^{-}+c^{-}b^{+}m^{+}=2c^{+}c^{-}\,,~{}~{}~{}~{}~{}m^{+}\mu_{+}^{2}=2c^{+}\cos^{2}\alpha\cos^{2}\beta\,,~{}~{}~{}~{}m^{-}\mu_{-}^{2}=2c^{-}\cos^{2}\alpha\sin^{2}\beta\,,$

without loss of generality which makes the warp factor become

$\Lambda=1+\sin^{2}\alpha+\epsilon\cos^{2}\alpha\cos(2\beta)\,,$ (3.45)

which has no zeros as $-1<\epsilon<1$. The internal five-manifold on the other
hand can be written as

$\displaystyle\mathrm{d}s^{2}({\cal M}_{5})$
$\displaystyle=2\frac{\Lambda}{{\cal
Q}}\mathrm{d}\alpha^{2}+2\sin^{2}\alpha\mathrm{d}\psi_{2}^{2}+4\cos^{2}\alpha\bigg{(}\frac{{\cal
Q}}{1-\epsilon^{2}}D\beta^{2}+\frac{1}{1+\epsilon}\cos^{2}\beta
D\tau^{2}_{+}+\frac{1}{1-\epsilon}\sin^{2}\beta D\tau^{2}_{-}\bigg{)}\,,$
$\displaystyle{\cal Q}$ $\displaystyle\equiv
1+\epsilon\cos(2\beta)\,,~{}~{}~{}~{}D\tau_{\pm}\equiv\mathrm{d}\tau_{\pm}+\frac{V}{2}\,,~{}~{}~{}~{}D\beta\equiv\mathrm{d}\beta-\epsilon\frac{\tan\alpha\sin(2\beta)}{\cal
Q}\mathrm{d}\alpha\,,$ (3.46)

which is topologically a five-sphere with
$0\leq\alpha<\frac{\pi}{2},~{}0\leq\beta<\frac{\pi}{2},~{}0\leq\psi_{2}<2\pi$
and the U(1) directions $\tau_{\pm}$ are defined as

$2\mathrm{d}\tau_{\pm}\equiv(1\pm\epsilon)\mathrm{d}\psi_{1}\pm\mathrm{d}\phi\,,$
(3.47)

and have period $2\pi$. Notice that when $\epsilon=0$ we have
$D\beta=\mathrm{d}\beta$ and the metric becomes that of the AdS3 solution of
Maldacena:2000mw . The general solution is a parametric deformation breaking
the U(1)$\times$U(1)$\times$SU(2) isometry to U(1)3, whilst preserving ${\cal
N}=(2,2)$ supersymmetry. One can show that the two form flux is given by

$\displaystyle F^{(2)}$
$\displaystyle=-2\sin(2\alpha)\mathrm{d}\alpha\wedge\bigg{(}(1-\epsilon)\frac{\cos^{2}\beta}{{\cal
Q}}\mathrm{d}\tau_{+}+(1+\epsilon)\frac{\sin^{2}\beta}{{\cal
Q}}\mathrm{d}\tau_{-}+\left(\frac{1}{{\cal Q}}-\frac{1}{2}\right)V\bigg{)}$
(3.48)
$\displaystyle-2\cos^{2}\alpha\sin(2\beta)D\beta\wedge\bigg{(}\frac{1-\epsilon}{1+\epsilon}\mathrm{d}\tau_{+}-\frac{1+\epsilon}{1-\epsilon}\mathrm{d}\tau_{-}-\frac{2\epsilon}{1-\epsilon^{2}}V\bigg{)}$
$\displaystyle+e^{2A_{0}}\bigg{(}\frac{2\Lambda}{(1-\epsilon^{2})}+\cos^{2}\alpha\bigg{)}\mathrm{d}X_{1}\wedge\mathrm{d}X_{2}\,.$

####  3.3.4 Flux quantisation

In order to compare to CFT quantities we must first quantise the flux. To this
end it is helpful to know the internal part of the five-form flux explicitly,
we find

$\displaystyle\frac{1}{L^{4}}F^{\text{int.}}_{5}$
$\displaystyle=\mathrm{d}\psi_{2}\wedge\bigg{[}8\frac{\sin\alpha\cos^{3}\alpha}{1-\epsilon^{2}}\frac{\sin(2\beta)}{\Lambda^{2}}\bigg{(}2\Lambda+(1-\epsilon^{2})\cos^{2}\alpha\bigg{)}\mathrm{d}\alpha\wedge
D\beta\wedge D\tau_{-}\wedge D\tau_{+}$
$\displaystyle+2\sin(2\alpha)e^{2A_{0}}\mathrm{d}X_{1}\wedge\mathrm{d}X_{2}\wedge\mathrm{d}\alpha\wedge\bigg{(}\frac{\cos^{2}\beta}{(1-\epsilon){\cal
Q}}\mathrm{d}\tau_{+}+\frac{\sin^{2}\beta}{(1+\epsilon){\cal
Q}}\mathrm{d}\tau_{-}\bigg{)}$
$\displaystyle+\frac{\sin^{2}(2\alpha)\sin(2\beta)}{(1-\epsilon^{2})\Lambda}e^{2A_{0}}\mathrm{d}X_{1}\wedge\mathrm{d}X_{2}\wedge
D\beta\wedge(\mathrm{d}\tau_{+}-\mathrm{d}\tau_{-})\bigg{]}\,.$ (3.49)

The relevant part of this is the first line. We should impose

$\frac{1}{(2\pi)^{4}}\int_{{\cal M}_{5}}F^{\text{int.}}_{5}=N\,,$ (3.50)

in units where $g_{s}=\alpha^{\prime}=1$. The integral is non-trivial but can
be performed exactly, ultimately we find that we must tune

$\frac{2L^{4}}{\pi(1-\epsilon^{2})}=N\,.$ (3.51)

####  3.3.5 Central charge

In this section we will study the central charge with which we may compare to
the field theory results presented in section 3.1. Since the R-charges may be
obtained from a specification of the more general solution we present later in
section 3.4.5, we will suppress the calculation of them here. The
Brown–Henneaux formula Brown:1986tm

$c_{\text{sugra}}=\frac{3}{2G_{N}^{(3)}}$ (3.52)

computes the sum of the left- and right- moving central charges of the field
theory

$c_{\text{sugra}}=\frac{c_{L}+c_{R}}{2}\,.$ (3.53)

Since the theories we are considering preserve $\mathcal{N}=(2,2)$
supersymmetry we have $c_{L}=c_{R}$, and therefore the gravitational anomaly
should vanish. The dimensionally reduced Newton’s constant appearing in the
Brown–Henneaux formula is computed by reducing the full 10d solution on the
compact internal manifold down to 3d. For a type II AdS3 solution with metric
of the form

$\mathrm{d}s^{2}=e^{2A_{3}}\mathrm{d}s^{2}(\text{AdS}_{3})+\mathrm{d}s^{2}(\mathcal{M}_{7})\,,$
(3.54)

the central charge is given by as131313This expression is in Einstein frame,
though we have set the dilaton to zero, so this is identical to the string
frame for us.

$c_{\text{\text{sugra}}}=\frac{3L}{2^{4}\pi^{6}}\int_{\mathcal{M}_{7}}e^{A_{3}}\mathrm{d}\text{vol}(\mathcal{M}_{7})\,.$
(3.55)

For the H2 solution we have

$e^{A_{3}}=L\Lambda^{\frac{1}{4}}\,,\quad\mathrm{d}\text{vol}(\mathcal{M}_{7})=\frac{16L^{7}}{1-\epsilon^{2}}\Lambda^{-\frac{1}{4}}e^{2A_{0}}\mathrm{d}X_{1}\wedge\mathrm{d}X_{2}\wedge\mathrm{d}\text{vol}(\text{S}^{5})\,,$
(3.56)

where by $\mathrm{d}\text{vol}(\text{S}^{5})$, we mean the volume form on the
unit norm round five-sphere, integrating to $\pi^{3}$. We thus find

$c_{\text{sugra}}=3N^{2}(\mathfrak{g}-1)\,,$ (3.57)

where we have used that the volume of the Riemann surface is given by

$\mathrm{vol}(\Sigma_{\mathfrak{g}})=\int_{\Sigma_{\mathfrak{g}}}\mathrm{e}^{2A_{0}}\mathrm{d}X_{1}\wedge\mathrm{d}X_{2}=\int_{\Sigma_{\mathfrak{g}}}\rho=4\pi(\mathfrak{g}-1)\,,$
(3.58)

where $\rho$ is the Ricci form, which follows since the chosen metric on the
Riemann surface is Einstein.

###  3.4 A more general solution

In this section we will generalise the previous solution by adding two
additional free parameters. We generalise the previous solution by taking as
ansatz for $I$

$\displaystyle
I=-\mu_{0}^{2}(\log\mu_{0}^{2}-1)-\frac{1}{2R^{+}}\mu_{+}^{2}(\log\mu_{+}^{2}-1)-\frac{1}{2R^{-}}\mu_{-}^{2}(\log\mu_{-}^{2}-1)+2\nu(t^{+}+t^{-})\,,$
(3.59)

where

$\mu_{\pm}^{2}=b^{\pm}-R^{\pm}\left[(1+r^{\pm})t^{+}+(1-r^{\pm})t^{-}\right]\,.$
(3.60)

Note that the previous ansatz for $I$ is recovered for $r^{+}=-r^{-}=1$ and
$c^{\pm}=2R^{\pm}$. As before the solution is split into two distinct cases
depending on the genus. The universal sector fixes the constants

$\displaystyle a^{+}=\frac{\kappa}{4}(2+r^{+}+r^{-})\,,\quad
m^{\pm}=\frac{1\pm\epsilon}{2}\,,\quad R^{\pm}\neq 0$ (3.61)

whilst the curvature dependent part satisfies

$\displaystyle\kappa$ $\displaystyle=0:\quad\epsilon=r^{\mp}\,,\quad
c=-8b^{\mp}R^{\pm}(r^{\pm}-\epsilon)^{2}\mathrm{e}^{4\nu}\,,$ (3.62)
$\displaystyle\kappa$ $\displaystyle\neq
0:\quad\mathrm{e}^{4\nu}=\frac{\kappa}{16R^{+}R^{-}(\epsilon-r^{-})(\epsilon-r^{+})}\,,$
$\displaystyle\quad\quad\epsilon=\frac{cR^{+}R^{-}(r^{+}+r^{-})+\kappa(b^{+}R^{-}r^{-}+b^{-}R^{+}r^{+})\pm\sqrt{R^{+}R^{-}(r^{+}-r^{-})^{2}(c^{2}R^{+}R^{-}-b^{-}b^{+})}}{\kappa(b^{+}R^{-}+b^{-}R^{+})+2cR^{+}R^{-}}\,.$

Note that these reduce to (3.34) for $r^{+}=-r^{-}=1$ and $c^{\pm}=2R^{\pm}$
as they should. Moreover, we have a similar constraint on the functions
$\mu_{0}$, $\mu_{\pm}$ as in (3.35) given by

$2R^{+}R^{-}(r^{+}-r^{-})\mu_{0}^{2}+R^{-}(\epsilon-r^{-})\mu_{+}^{2}+R^{-}(r^{+}-\epsilon)\mu_{-}^{2}=b^{-}R^{+}(r^{+}-\epsilon)+b^{+}R^{-}(\epsilon-r^{-})\,.$
(3.63)

As before, this defines an embedding equation of a surface into a three-
dimensional ambient space. Note that the inclusion of the parameters $r^{\pm}$
shift the coefficients of the functions $\mu_{\pm}$ so that it is no longer
simply $m^{\pm}$ as in (3.35). We may now assemble the full metric of the
solution. We have

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}$
$\displaystyle=\sqrt{\Lambda}\Big{[}\mathrm{d}s^{2}(\text{AdS}_{3})+4R^{+}R^{-}(r^{+}-r^{-})^{2}\mathrm{e}^{4\nu}\mathrm{e}^{2A_{0}}(\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2})\Big{]}$
(3.64)
$\displaystyle+\frac{1}{\sqrt{\Lambda}}\Big{[}\mathrm{d}\mu_{0}^{2}+\frac{1}{2R^{+}}\mathrm{d}\mu_{+}^{2}+\frac{1}{2R^{-}}\mathrm{d}\mu_{-}^{2}+\Lambda\eta^{2}+\frac{\mu_{-}^{2}\mu_{+}^{2}}{2(R^{-}(r^{-}-\epsilon)^{2}\mu_{+}^{2}+R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2})}D\phi^{2}$
$\displaystyle+\frac{\mu_{0}^{2}[R^{-}(r^{-}-\epsilon)^{2}\mu_{+}^{2}+R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}]}{2R^{+}R^{-}(r^{+}-r^{-})^{2}\mu_{0}^{2}+R^{-}(r^{-}-\epsilon)^{2}\mu_{+}^{2}+R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}}D\phi_{2}^{2}\Big{]}\,,$

where

$\displaystyle D\phi$
$\displaystyle\equiv\mathrm{d}\phi-\left(\epsilon-\frac{r^{+}+r^{-}}{2}\right)\kappa
V\,,$ $\displaystyle D\phi_{2}$ $\displaystyle\equiv\mathrm{d}\phi_{2}+\kappa
V-\frac{R^{-}(r^{-}-\epsilon)\mu_{+}^{2}+R^{+}(r^{+}-\epsilon)\mu_{-}^{2}}{R^{-}(r^{-}-\epsilon)^{2}\mu_{+}^{2}+R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}}D\phi\,,$
(3.65) $\displaystyle\eta$
$\displaystyle\equiv\frac{1}{2}\left(\mathrm{d}\phi_{1}+\frac{R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}+R^{-}(r^{-}-\epsilon)^{2}\mu_{+}^{2}}{R^{+}R^{-}(r^{+}-r^{-})^{2}\Lambda}D\phi_{2}\right)\,,$

and

$\Lambda\equiv\mu_{0}^{2}+\frac{(r^{-}-\epsilon)^{2}}{2R^{+}(r^{+}-r^{-})^{2}}\mu_{+}^{2}+\frac{(r^{+}-\epsilon)^{2}}{2R^{-}(r^{+}-r^{-})^{2}}\mu_{-}^{2}\,.$
(3.66)

From the presentation above we see that the metric looks locally like a
U$(1)^{3}$ preserving deformed five-sphere. In addition, from the
$\mathcal{N}=(0,2)$ perspective of section 2.1, one can identify the one-form
$\eta$ as the one-form dual to the R-symmetry vector. This puts the metric on
the five-sphere in the form of a U$(1)$-fibration over $\mathbb{CP}^{2}$
equipped with a non-Einstein metric. The two-form flux is given by

$\displaystyle F^{(2)}=2J_{6}-\mathrm{d}(\Lambda\eta)\,,$ (3.67)

with

$\displaystyle J_{6}$
$\displaystyle=4\mathrm{e}^{4\nu}R^{+}R^{-}(r^{+}-r^{-})^{2}\Lambda\mathrm{d}V-\frac{1}{4}\left[(1-\epsilon)\mathrm{d}t^{-}+(1+\epsilon)\mathrm{d}t^{+}\right]\wedge
D\phi_{2}$
$\displaystyle+\frac{(1+r^{+})R^{+}(r^{+}-\epsilon)\mu_{-}^{2}-(1+r^{-})R^{-}(\epsilon-r^{-})\mu_{+}^{2}}{4(R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}+R^{-}(\epsilon-r^{-})^{2}\mu^{2}_{+})}\mathrm{d}t^{+}\wedge
D\phi$
$\displaystyle+\frac{(1-r^{+})R^{+}(r^{+}-\epsilon)\mu_{-}^{2}+(r^{-}-1)R^{-}(\epsilon-r^{-})\mu_{+}^{2}}{4(R^{+}(r^{+}-\epsilon)^{2}\mu_{-}^{2}+R^{-}(\epsilon-r^{-})^{2}\mu^{2}_{+})}\mathrm{d}t^{-}\wedge
D\phi\,.$ (3.68)

####  3.4.1 $\mathbb{T}^{2}$ case

As in section 3.3.1 the $\mathbb{T}^{2}$ example reduces to the
AdS${}_{3}\times{\rm S}^{3}\times\mathbb{T}^{4}$ solution, as such we shall
suppress further analysis of this case.

####  3.4.2 S2 case

Let us turn to the S2 case. It is clear that we require $R^{\pm}>0$ and
therefore positivity of the warp factor of the Riemann surface,
$\mathrm{e}^{4\nu}$ implies that

$(\epsilon-r^{-})(\epsilon-r^{+})>0\,.$ (3.69)

Studying the embedding equation (3.63) we see that it defines an embedding
into either three-dimensional anti-de Sitter, Minkowski or de Sitter space for
choices of the parameters. In each case the solutions will be non-compact and
therefore they do not give rise to good holographic duals as in section 3.3.2.

####  3.4.3 H2 case

Instead, let us consider the $\mathfrak{g}>1$ case. Positivity of the warp
factor requires

$(r^{+}-\epsilon)(r^{-}-\epsilon)<0\,.$ (3.70)

Due to the symmetry of the solution we may assume without loss of generality
that $r^{+}>r^{-}$; if this is not the case then we may relabel as
$+\leftrightarrow-$. We see that we must therefore require

$r^{+}>\epsilon\,,\quad r^{-}<\epsilon\,,$ (3.71)

and consequently the embedding equation is that of a two-sphere in
$\mathbb{R}^{3}$. We may then parametrise

$\displaystyle\mu_{0}^{2}$
$\displaystyle=\frac{b^{-}R^{+}(r^{+}-\epsilon)+b^{+}R^{-}(\epsilon-r^{-})}{2R^{+}R^{-}(r^{+}-r^{-})}\sin^{2}v\,,\quad\mu_{-}^{2}=\frac{b^{-}R^{+}(r^{+}-\epsilon)+R^{-}b^{+}(\epsilon-r^{-})}{R^{+}(r^{+}-\epsilon)}\cos^{2}v\cos^{2}u\,,$
$\displaystyle\mu_{+}^{2}$
$\displaystyle=\frac{b^{-}R^{+}(r^{+}-\epsilon)+R^{-}b^{+}(\epsilon-r^{-})}{R^{-}(\epsilon-r^{-})}\cos^{2}v\sin^{2}u\,,$
(3.72)

giving rise to a compact internal manifold.

####  3.4.4 Toric data and three-cycles

We now want to analyse the regularity of these generalised solutions. We saw
earlier that regularity of the metric can be ensured by correctly identifying
the embedding coordinates. In this section we will take an alternative
approach using the toric data of the five-sphere at fixed point on the Riemann
surface. Note that the compact part of the metric in (3.64), without the
Riemann surface, has been written as a U$(1)^{3}$ fibration over a two-
dimensional space defined by the embedding condition (3.63). One can then use
the tools of toric geometry to analyse the regularity of the metric. Moreover,
this analysis allows us to identify the three-cycles, each corresponding to a
degeneration surface of the metric, on which one may wrap D3-branes giving
rise to di-baryonic operators of the dual SCFT. Of course, since on the five-
sphere $b_{3}=0$ defining such charges is subtle. From the toric viewpoint
these three-cycles wrapped by D3-branes are obtained when one goes to the edge
of the polytope over which the U$(1)^{3}$ fibration is defined and where one
of the U$(1)$’s shrinks. With this in hand we may compute the R-charges of
these operators and compare to the field theory. In the following we will
consider just the $\kappa\neq 0$ case since the $\kappa=0$ case reduces to the
previously studied AdS${}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ solution.

The form of the ansatz for the function $I$ is reminiscent of the canonical
potential for a symplectic toric manifold. Here, the $\mu_{\bullet}^{2}$’s
play the role of the functions defining the edges of the polytope. Note that
the metric has a singular like behaviour when any of $\mu_{0}$ or $\mu_{\pm}$
vanish, the goal is to obtain constraints such that the degeneration is
smooth.141414For a more detailed discussion as to why the following works see
Couzens:2019mkh  for example.  To this end we construct Killing vectors which
have zero norm on some locus of the solution. The degeneration locus is
associated to the edges of a 2d polytope over which the U$(1)^{3}$ is fibered.
Requiring that the surface gravity of these Killing vectors is normalised to 1
on the respective degeneration surface, and giving the dual coordinate period
$2\pi$ leads to a smooth degeneration and a regular metric. In the case at
hand there are three Killing vectors, which, after the prescribed
normalisation condition discussed above, are given by

$k_{0}=2\partial_{\phi_{1}}-\partial_{\phi_{2}}\,,\quad
k_{\pm}=(r^{\pm}-\epsilon)\partial_{\phi}+\partial_{\phi_{2}}\,.$ (3.73)

Defining a new $2\pi$-periodic coordinate for each of these three vectors via,
$k_{\bullet}=\partial_{\psi_{\bullet}}$, we find that the metric is regular.
In order to compute various integrals it is useful to perform a change of
coordinates,

$\displaystyle\phi_{1}=2\psi_{0}\,,\quad\phi_{2}=-\psi_{0}+\psi_{+}+\psi_{-}\,,\quad\phi=(r^{+}-\epsilon)\psi_{+}+(r^{-}-\epsilon)\psi_{-}\,.$
(3.74)

These new $2\pi$-periodic coordinates give a free action for the U$(1)^{3}$
torus action.

To extend this regularity analysis to the full manifold we now need to make
sure that the fibration over the Riemann surface is well defined. Viewing the
fibration as a gauging of the U(1)3 torus action via

$\mathrm{d}\psi_{\bullet}\rightarrow
D\psi_{\bullet}\equiv\mathrm{d}\psi_{\bullet}+A_{\bullet}\,,$ (3.75)

with the $A_{\bullet}$ gauge fields on the Riemann surface. The fibration is
well-defined provided that the field-strength of the gauge fields
$A_{\bullet}$ over the Riemann surface have integral period. Explicit
computation gives

$A_{0}=0\,,\quad A_{+}=A_{-}=\frac{\kappa}{2}\,V\,,$ (3.76)

and therefore the fibration is well-defined since

$\frac{1}{2\pi}\int_{\Sigma_{\mathfrak{g}}}\mathrm{d}V=2(\mathfrak{g}-1)\,.$
(3.77)

From the non-trivial gauge fields we can read off the topological twist of the
dual field theory. We see that the topological twist is performed by turning
on a background gauge field of the form

$A=\frac{\kappa}{2}(T_{+}+T_{-})V\,.$ (3.78)

Finally we may identify the (right-moving) R-symmetry of the solution from the
expression for the one-form $\eta$. We find

${\rm
R}_{R}=\partial_{\psi_{0}}+\frac{r^{-}-\epsilon}{r^{+}-r^{-}}\partial_{\psi_{+}}+\frac{\epsilon-r^{+}}{r^{+}-r^{-}}\partial_{\psi_{-}}\,.$
(3.79)

Note in particular that the R-symmetry does _not_ mix with the isometries of
the Riemann surface. This is a distinction between the solutions discussed in
this section and the ones in the following section 4.

####  3.4.5 Flux quantisation and observables

Having discussed the geometry of the solution and the various three-cycles let
us first consider the quantisation of the five-form flux. We must ensure that
this is an integer over all integral five-cycles in the geometry according to
the quantisation condition (3.50). Since the geometry is a five-sphere fibered
over a Riemann surface the only integral five-cycle is the five-cycle at fixed
point on the Riemann surface. The internal part of the five-form is

$\displaystyle\frac{1}{L^{4}}F_{5}^{\text{int.}}$
$\displaystyle=\frac{(r^{+}-r^{-})(b^{+}R^{-}(\epsilon-r^{-})+b^{-}R^{+}(r^{+}-\epsilon))^{2}}{2(R^{+}R^{-})^{2}(r^{+}-r^{-})(r^{+}-\epsilon)(\epsilon-r^{-})\Lambda^{2}}\cos
u\sin u\cos^{3}v\sin v\times$
$\displaystyle\Big{(}\Lambda+\cos^{2}v\frac{(r^{+}-\epsilon)(\epsilon-r^{-})\left[b^{+}R^{-}(\epsilon-r^{-})+b^{-}R^{+}(r^{+}-\epsilon)\right]}{2R^{+}R^{-}(r^{+}-r^{-})^{3}}\Big{)}\mathrm{d}u\wedge\mathrm{d}v\wedge\mathrm{d}\psi_{0}\wedge\mathrm{d}\psi_{+}\wedge\mathrm{d}\psi_{-}$
$\displaystyle+\dots\,,$ (3.80)

where the suppressed terms do not have support on the topological five-sphere.
Integrating the flux over the five-sphere we find the quantisation condition

$N=\frac{L^{4}}{8\pi}\frac{(r^{+}-r^{-})[b^{+}R^{-}(r^{-}-\epsilon)-b^{-}R^{+}(r^{+}-\epsilon)]}{R^{+}R^{-}(r^{+}-\epsilon)(\epsilon-r^{-})}\,.$
(3.81)

We can now compute the R-charges of baryonic operators in the SCFT. These
correspond to wrapping D3-branes on three-cycles on the five-sphere. The
three-cycles, let us denote them by $S_{\bullet}$, are the ones obtained by
going to the facet of the polytope, upon which a U$(1)$ shrinks.151515We
emphasise again that these three-cycles are not elements of
$H^{3}($S${}^{5},\mathbb{Z})$ since $b_{3}($S${}^{5})=0$. Following
Couzens:2017nnr  (see also Couzens:2018wnk ) the R-charge is given by

${\rm
R}[S_{\bullet}]=\frac{L^{4}}{(2\pi)^{3}}\int_{S_{\bullet}}\mathrm{e}^{4\Delta}\mathrm{d}\mathrm{vol}(S_{\bullet})=\frac{L^{4}}{(2\pi)^{3}}\int_{S_{\bullet}}\eta\wedge
J\,,$ (3.82)

where the last inequality has been written using $(0,2)$ language. Explicit
computation gives

$\displaystyle{\rm R}[S_{0}]$
$\displaystyle=-\frac{L^{4}}{8\pi}\frac{(r^{+}-r^{-})[b^{+}R^{-}(r^{-}-\epsilon)-b^{-}R^{+}(r^{+}-\epsilon)]}{R^{+}R^{-}(r^{+}-\epsilon)(r^{-}-\epsilon)}\,,$
(3.83) $\displaystyle{\rm R}[S_{+}]$
$\displaystyle=\frac{L^{4}}{8\pi}\frac{b^{+}R^{-}(r^{-}-\epsilon)-b^{-}R^{+}(r^{+}-\epsilon)}{R^{+}R^{-}(r^{+}-\epsilon)}\,,$
$\displaystyle{\rm R}[S_{-}]$
$\displaystyle=\frac{L^{4}}{8\pi}\frac{b^{-}R^{+}(r^{+}-\epsilon)-b^{+}R^{-}(r^{-}-\epsilon)}{R^{+}R^{-}(r^{-}-\epsilon)}\,,$

which after using the definition of $N$ become

${\rm R}[S_{0}]=N\,,\quad{\rm
R}[S_{+}]=\frac{N(\epsilon-r^{-})}{r^{+}-r^{-}}\,,\quad{\rm
R}[S_{-}]=\frac{N(r^{+}-\epsilon)}{r^{+}-r^{-}}\,.$ (3.84)

Note that the R-charges are all positive and moreover they sum to $2N$ as
expected Couzens:2018wnk ,

${\rm R}[S_{0}]+{\rm R}[S_{+}]+{\rm R}[S_{-}]=2N\,.$ (3.85)

In addition, from the form of the R-symmetry vector as given in (3.79), it is
clear that the dual operators in the field theory have charge 1 under the
associated degenerating Killing vector, and vanishing charge under the others
directions. We may compare this with (3.19) and find agreement upon
identifying

$\cos^{2}\alpha_{R}=\frac{\epsilon-r^{-}}{r^{+}-r^{-}}\,.$ (3.86)

Finally, let us compute the central charge. As before we use the
Brown–Henneaux formula as given in (3.55), and as expected from the field
theory analysis, the result is

$\displaystyle c=3(\mathfrak{g}-1)N^{2}\,.$ (3.87)

Note in particular that the newly introduced parameters drop out of the
central charge, despite not dropping out of the R-charges of the fields.

##  4 D3-branes wrapping a topological disk

In Boido:2021szx  a solution with D3-branes wrapping a spindle is obtained in
5d U(1)3 gauged supergravity then lifted to ten dimensions. The result locally
coincides with a solution originally derived in Gauntlett:2006ns , but the
interpretation in terms of a spindle and CFT analysis is new. The solution
supports multiple charges and generically preserves ${\cal N}=(0,2)$
supersymmetry, however, as we shall establish in this section, for a certain
tuning of these charges it experiences an enhancement to ${\cal N}=(2,2)$,
where the D3-branes now actually wrap a Riemann surface with the topology of a
disk and includes additional source D3-branes. This solution can be embedded
in the canonical form of section 2.2, interestingly the Riemann surface which
the branes wrap is not that of the $(X_{1},\,X_{2})$ directions. Instead, as
we establish in appendix B, $(X_{1},\,X_{2})$ span a two-sphere in the co-
dimensions of the wrapped D3-branes.

###  4.1 Summary of Boido:2021szx

The metric of the solution of Boido:2021szx  takes the form161616The flux can
be found in this reference.

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}$
$\displaystyle=\sqrt{W}H(x)^{\frac{1}{3}}\left[\mathrm{d}s^{2}(\text{AdS}_{3})+\mathrm{d}s^{2}(\Sigma_{2})\right]+\frac{1}{\sqrt{W}}\sum_{I=1}^{3}(X^{I})^{-1}\left[\mathrm{d}\mu^{2}_{I}+\mu^{2}_{I}(\mathrm{d}\phi_{I}+A_{I})^{2}\right]\,,$
$\displaystyle W$
$\displaystyle=\sum_{I=1}^{3}X^{I}\mu_{I}^{2}\,,~{}~{}~{}~{}A_{I}=\frac{x-x_{0}}{x+3K_{I}}\mathrm{d}\varphi\,,~{}~{}~{}~{}X^{I}=\frac{H(x)^{\frac{1}{3}}}{x+3K_{I}}\,,$
$\displaystyle\mathrm{d}s^{2}(\Sigma_{2})$
$\displaystyle=\frac{1}{4P(x)}\mathrm{d}x^{2}+\frac{P(x)}{H(x)}\mathrm{d}\varphi^{2}\,,$
(4.1)

where $\mu_{I}$ embed a unit radius two-sphere into $\mathbb{R}^{3}$ and the
functions of $x$ are

$H=(x+3K_{1})(x+3K_{2})(x+3K_{3})\,,\quad P=H-(x-x_{0})^{2}\,,$ (4.2)

where Boido:2021szx  constrain $K_{1}+K_{2}+K_{3}=0$, which kills the
$O(x^{2})$ term in $H$. The $x$ direction can then be bounded between two real
roots of the cubic polynomial $P$ for which $H\neq 0$, and the surface becomes
$\Sigma_{2}=\mathbb{WCP}^{1}_{[n_{+},n_{-}]}$, which is an orbifold known as a
spindle. Here $n_{\pm}\in\mathbb{N}$, are related to the period of $\varphi$.
When $K_{1}=K_{2}=K_{3}=0$ the solution of Ferrero:2020laf  is recovered. In
this section we shall consider a different tuning of these parameters, not
considered in Boido:2021szx .

###  4.2 An ${\cal N}=(2,2)$ tuning

We would like to extract an ${\cal N}=(2,2)$ solution from the local solution
of the previous section. The first thing to appreciate is that
$K_{1}+K_{2}+K_{3}=0$ is not a requirement for supersymmetry (at least in ten
dimensions), so let us instead tune the parameters as

$K_{2}=K_{1}=K\,,\quad K_{3}=-\frac{1}{3}x_{0}\,.$ (4.3)

The connection $A_{3}$ then becomes topologically trivial,
$\\{A_{1}=A_{2},\,X_{1}=X_{2}=X_{3}^{-2}\\}$ and so the U(1)3 isometry of the
five-dimensional internal space gets enhanced to SU(2)$\times$U(1) – similar
to the ${\cal N}=(2,2)$ solution of Maldacena:2000mw . This solution
experiences an enhancement of supersymmetry to ${\cal N}=(2,2)$, as we prove
in appendix B. The two functions defining the solution become

$H=(3K+x)^{2}(x-x_{0})\,,\quad P=(x-x_{0})Q\,,\quad Q=(3K+x)^{2}-x+x_{0}\,.$
(4.4)

For certain tunings of $(K,x_{0})$ the polynomial $Q$ contains two real roots
$x=x_{\pm}$ namely

$x_{\pm}=\frac{1}{2}\left(1-6K\pm\sqrt{1-12K-4x_{0}}\right)\,.$ (4.5)

$\Sigma_{2}$ exhibits spindle-like behaviour between these loci, however a
physical metric should be real and positive which requires
$\\{Q>0,~{}x>x_{0},~{}3K+x>0\\}$, and $Q$ is strictly negative for
$x_{-}<x<x_{+}$. It is however possible to achieve a physical metric and to
bound $x$ to the interval $[x_{0},x_{-}]$ provided that
$-\frac{1}{3}x_{0}<K<\frac{1}{12}(1-4x_{0})$. At $x=x_{0}$ $H$ and $P$ both
vanish, but $P/H$ remains finite, so similar to the solution of Bah:2021hei ,
the topology of $\Sigma_{2}$ is now that of a disk with
$\mathbb{R}^{2}/\mathbb{Z}_{k}$ orbifold singularity at $x=x_{-}$. Close to
$x=x_{-}$ only $P$ exhibits a zero, defining $u=(x_{-}-x)$ we find

$\mathrm{d}s^{2}(\Sigma_{2})=\frac{1}{x_{+}-x_{-}}\bigg{(}\mathrm{d}u^{2}+\frac{(x_{+}-x_{-})^{2}(x_{-}-x_{0})^{2}}{H(x_{-})}u^{2}\mathrm{d}\varphi^{2}\bigg{)}\,,$
(4.6)

at this loci, so that we have a $\mathbb{R}^{2}/\mathbb{Z}_{k}$ orbifold when
the period of $\varphi$ is
$\Delta\varphi=\frac{2\pi\sqrt{H(x_{-})}}{k(x_{+}-x_{-})(x_{-}-x_{0})}$ for
$k\in\mathbb{N}$ or a regular zero for $k=1$. The behaviour close to $x=x_{0}$
in the full space needs more care to disentangle: first one should note that
warp factor $W$ can now be written as

$H^{\frac{2}{3}}W=(3K+x)\bigg{(}(x-x_{0})+(3K+x_{0})\mu_{3}^{2}\bigg{)}\,,$
(4.7)

which provided $\mu_{3}\neq 0$, remains finite as $x$ approaches $x_{0}$.
Similarly around $x=x_{0}$ when $\mu_{3}\neq 0$ we find
$W(X^{1,2})^{-2}=\mu_{3}^{-2}$ and
$W(X^{3})^{-2}=(x-x_{0})^{2}(3K+x_{0})^{-2}\mu_{3}^{-2}$. We see then that the
sub-metric spanned by $(x,\,\phi_{3}+\varphi)$ vanishes as $\mathbb{R}^{2}$ in
polar coordinates. The behaviour as both $\mu_{3},(x-x_{0})\to 0$ is a bit
subtle, one can study it by changing coordinates as

$\mu_{3}^{2}=r\cos^{2}\left(\frac{\tilde{\theta}}{2}\right),~{}~{}~{}~{}\mu_{2}+\text{i}\mu_{1}=e^{\frac{\text{i}}{2}\beta}\sqrt{1-\mu_{3}^{2}}\,,~{}~{}~{}~{}x=x_{0}+(3K+x_{0})r\sin^{2}\left(\frac{\tilde{\theta}}{2}\right)\,,$
(4.8)

and then expanding about $r=0$. To leading order the metric then becomes

$\frac{1}{L^{2}}\mathrm{d}s^{2}=\sqrt{r}(3K+x_{0})\left[\mathrm{d}s^{2}(\text{AdS}_{3})+\mathrm{d}\varphi^{2}\right]+\frac{1}{4\sqrt{r}}\left[\mathrm{d}r^{2}+r^{2}\mathrm{d}s^{2}(\tilde{\text{S}}^{2})+4\mathrm{d}s^{2}(\text{S}^{3})\right]\,,$
(4.9)

where $(\tilde{\theta},\,\phi_{3}+\varphi)$ span the two-sphere and
$(\beta\,,\phi_{1},\,\phi_{2})$ span the three-sphere (in topological joint
coordinates). It is not hard to see that this behaviour is singular, it is in
fact that of a partially localised stack of D3-branes with worldvolume
$(\text{AdS}_{3},\,\varphi)$ that are smeared over S3. Thus we see that at
generic points of the deformed two-sphere spanned by $\mu_{I}$ the solution is
bounded between a regular zero at $x=x_{0}$ and a
$\mathbb{R}^{2}/\mathbb{Z}_{k}$ orbifold at $x=x_{-}$, with flavour D3-branes
at the loci $(x=x_{0},\,\mu_{3}=0)$.

To further study the solution let us fix

$\displaystyle\mu_{3}$
$\displaystyle=\sin\alpha\,,~{}~{}~{}~{}\mu_{2}+\text{i}\mu_{1}=\cos\alpha
e^{\frac{\text{i}}{2}\beta}\,,$ $\displaystyle\phi_{1}$
$\displaystyle=-\frac{1}{2}(\tau_{1}-\tau_{2})\,,~{}~{}~{}~{}\phi_{2}=\frac{1}{2}(\tau_{1}+\tau_{2})\,,~{}~{}~{}~{}\phi_{3}=\psi-\varphi\,,$
$\displaystyle\Lambda$ $\displaystyle=x-x_{0}+(3K+x_{0})\sin^{2}\alpha\,,$
(4.10)

we can now write the metric as

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}$
$\displaystyle=\sqrt{3K+x}\sqrt{\Lambda}\left[\mathrm{d}s^{2}(\text{AdS}_{3})+\frac{1}{4(x-x_{0})Q}\mathrm{d}x^{2}+\frac{Q}{(3K+x)^{2}}\mathrm{d}\varphi^{2}\right]+\frac{1}{\sqrt{\Lambda}}\mathrm{d}s^{2}({\cal
M}_{5})\,,$ $\displaystyle\mathrm{d}s^{2}({\cal M}_{5})$
$\displaystyle=\frac{\Lambda}{\sqrt{3K+x}}\mathrm{d}\alpha^{2}+\frac{x-x_{0}}{\sqrt{3K+x}}\sin^{2}\alpha\mathrm{d}\psi^{2}$
$\displaystyle+\frac{1}{4}\sqrt{3K+x}\cos^{2}\alpha\left[\mathrm{d}\beta^{2}+\sin^{2}\beta\mathrm{d}\tau_{2}^{2}+\left(\mathrm{d}\tau_{1}+\cos\beta\mathrm{d}\tau_{2}+2\frac{x-x_{0}}{3K+x}\mathrm{d}\varphi\right)^{2}\right]\,,$
(4.11)

where ${\cal M}_{5}$ is topologically a five-sphere when
$0\leq\alpha<\frac{\pi}{2},~{}0\leq\beta<\pi,~{}0\leq\psi<2\pi,~{}0\leq\tau_{1}<2\pi,~{}~{}0\leq\tau_{2}<4\pi$.

###  4.3 Flux quantisation and central charge

In order to compute some properties of the dual CFT we should first quantise
the flux on ${\cal M}_{5}$, the relevant part is

$\frac{1}{L^{4}}F_{5}=\frac{1}{4}\sin\alpha\sin\beta\mathrm{d}\alpha\wedge\mathrm{d}\psi\wedge\mathrm{d}\beta\wedge\mathrm{d}\tau_{1}\wedge\mathrm{d}\tau_{2}+\dots$
(4.12)

where $\dots$ either vanishes on ${\cal M}_{5}$ or does so when it is
integrated over it. We then have

$\frac{1}{(2\pi)^{4}}\int_{{\cal
M}_{5}}F_{5}=N~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}L^{4}=2\pi N\,.$ (4.13)

We should also quantise the connections of the fibrations such that we have a
well defined orbifold fibration, this demands that we impose that the field
strengths $F_{I}=\mathrm{d}A_{I}$ satisfy171717Recall $dA_{3}=0$.,

$M/k=\frac{1}{2\pi}\int_{\Sigma_{2}}F_{1}=\frac{1}{2\pi}\int_{\Sigma_{2}}F_{2}=\frac{\Delta\varphi}{4\pi}[1-(x_{+}-x_{-})]\,,~{}~{}~{}~{}\Delta\varphi=\frac{\pi[1-(x_{+}-x_{-})]}{k(x_{+}-x_{-})\sqrt{x_{-}-x_{0}}}\,,$
(4.14)

for $M\in\mathbb{N}$, which fixes

$3K+x_{0}=\frac{M(M\pm 1)}{(2M\pm 1)^{2}}\,.$ (4.15)

Given the constraints on $(K,\,x_{0})$ discussed below (4.5) the positive
branch is valid for $M\geq 1$ and the negative one for $M\geq 2$, so they are
equivalent – to be concrete we take the former. We can compute the holographic
central charge using (3.55), we find after substituting for $L$ and $3K+x_{0}$
this takes the form

$c_{\text{sugra}}=3N^{2}\frac{M^{2}}{4k(1+2M)}\,,$ (4.16)

where again we expect the central charges of the dual CFT to be
$c_{L}=c_{R}=c_{\text{sugra}}$. Clearly we have a deviation from the behaviour
derived in section 3.1, but this should be no surprise as the D3-branes are
now wrapping a topological disk which does not have constant curvature.

###  4.4 Discussion

In appendix B we show how the solution of this section is embedded into the
canonical form of section 2.2. To our knowledge this is the first time an
example with branes wrapping a surface of non-constant curvature has been
embedded into one of the various AdS classifications that are defined in terms
of a Mongé–Ampere like equation – it may be possible to construct similar
solutions for other wrapped brane scenarios. Interestingly, for this example,
the Riemann surface that the branes wrap and that of the classification are
not the same – indeed the $(X_{1},\,X_{2})$ directions of (2.21) actually span
a two-sphere. This may provide a hint towards finding solutions with branes
wrapping non-constant Riemann surfaces more broadly. It would be interesting
to generalise this example as section 3.3.3 generalises Maldacena:2000mw , but
we shall not attempt that here.

The dual field theory computation in this case is somewhat subtle. One cannot
simply specialise the field theory result in Boido:2021szx  to our current
case. Concretely, one finds that after $c$-extremization has been performed,
despite the trial central charge having a well defined extremal point, the
central charge is identically 0. Clearly this is inconsistent. This is in fact
the same problem one would encounter if one naively specialises the general
$(0,2)$ twist of 4d $\mathcal{N}=4$ SYM in Benini:2013cda  to a $(2,2)$ twist
in the constant curvature Riemann surface case. The problem arises because
$c$-extremization mixes the holomorphic (right-moving) and anti-holomorphic
(left-moving) R-symmetries. We leave recovering the supergravity result from a
field theory computation to the future.

## Acknowledgments

C. Couzens is supported by the Netherlands Organisation for Scientific
research (NWO) under the VICI grant 680-47-602. N. Macpherson is supported by
the Spanish government grant PGC2018-096894-B100. A. Passias is supported by
the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the
“First Call for H.F.R.I. Research Projects to support Faculty members and
Researchers and the procurement of high-cost research equipment grant” (MIS
1857, Project Number: 16519), and by the LabEx ENS-ICFP:
ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.

##  Appendix A Reducing the conditions

We have seen that the gravity solution is fixed by determining a four-
dimensional metric satisfying the torsion conditions (2.4)-(2.10). Our goal in
this section is to obtain the reduced conditions after substituting in an
ansatz for the four-dimensional base. We will take the most general 4d metric
admitting a single U$(1)$ isometry as used in Bah:2013qya  to study AdS5
solutions in M-theory. Explicitly, the metric is

$\mathrm{d}s^{2}_{4}=\mathrm{e}^{2A}(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2})+\mathrm{e}^{2B}\left[(\mathrm{d}\theta+\mathrm{e}^{C}V^{R})^{2}+\mathrm{e}^{2C}(\mathrm{d}\phi+V^{I})^{2}\right]\,,$
(A.1)

where $V^{I}$ and $V^{R}$ have legs only along the Riemann surface
parametrised by the coordinates $(x_{1},\,x_{2})$. The vector
$\partial_{\phi}$ is taken to be a Killing vector and thus the three scalars
$A,B,C$ are independent of $\phi$, though they may depend on the other three
coordinates and $y$. We will use conventions in which the Hodge dual on the
Riemann surface satisfies

$\star_{2}\mathrm{d}x_{1}=-\mathrm{d}x_{2}\,,\quad\star_{2}\mathrm{d}x_{2}=\mathrm{d}x_{1}\,,$
(A.2)

and the volume form is given by

$\mathrm{d}\mathrm{vol}_{2}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\,.$ (A.3)

In addition, let us define the one-forms

$\eta_{\theta}\equiv\mathrm{d}\theta+\mathrm{e}^{C}V^{R}\,,\quad\eta_{\phi}\equiv\mathrm{d}\phi+V^{I}\,,$
(A.4)

and the twisted exterior derivative on the Riemann surface

$\hat{\mathrm{d}}_{2}\equiv\mathrm{d}_{2}-\mathrm{e}^{C}V^{R}\partial_{\theta}\,.$
(A.5)

With the above metric ansatz the SU$(2)$-structure forms are

$J=\mathrm{e}^{2A}\mathrm{d}\mathrm{vol}_{2}+\mathrm{e}^{2B+C}\eta_{\theta}\wedge\eta_{\phi}\,,\quad\Omega=\mathrm{e}^{A+B}\mathrm{e}^{-\mathrm{i}\psi_{1}}(\mathrm{d}x_{1}+\mathrm{i}\mathrm{d}x_{2})\wedge(\eta_{\theta}+\mathrm{i}\mathrm{e}^{C}\eta_{\phi})\,.$
(A.6)

###  A.1 Reducing the conditions

Let us reduce the torsion conditions (2.4)-(2.10) on the ansatz above. We
begin with the conditions for the holomorphic volume form before moving on to
the Kähler form.

### Reducing (2.9): $\partial_{y}\Omega$ equation

From equation (2.9) we obtain the conditions

$\displaystyle\partial_{y}C$ $\displaystyle=0\,,$ (A.7)
$\displaystyle\star_{2}V^{R}+V^{I}$
$\displaystyle=V^{0}\,,\quad\partial_{y}V^{0}=0\,,$ (A.8) $\displaystyle
y\partial_{y}\log\Big{(}\mathrm{e}^{2(A+B)}\cos^{2}\zeta\Big{)}$
$\displaystyle=-\tan^{2}\zeta\,.$ (A.9)

Since the first condition implies that the scalar $C$ is independent of the
$y$ coordinate it follows that it may be removed by a change of coordinates.
We will therefore set $C=0$ in the following. Moreover we may solve the final
condition by introducing the scalar $\Lambda$:

$\cos^{2}\zeta=-\frac{1}{y\partial_{y}\Lambda}\,,\quad\mathrm{e}^{2(A+B)}=-y^{2}\partial_{y}\mathrm{e}^{\Lambda}\,.$
(A.10)

### Reducing (2.10): $\mathrm{d}_{4}\Omega$ equation

Next consider (2.10). Once the dust settles we obtain two conditions

$\displaystyle 0$ $\displaystyle=\partial_{\theta}V^{0}\,,$ (A.11)
$\displaystyle\sigma$
$\displaystyle=-\frac{1}{2}\star_{2}\hat{\mathrm{d}}_{2}\log\Big{(}\mathrm{e}^{2(A+B)}\cos^{2}\zeta\Big{)}+\frac{1}{2}\partial_{\theta}\log\Big{(}\mathrm{e}^{2(A+B)}\cos^{2}\zeta\Big{)}\eta_{\phi}\,,$
(A.12)

the latter of which implies that $\sigma$ is determined by the function
$\Lambda$ introduced in the previous section,

$\sigma=-q\mathrm{d}\phi-\frac{1}{2}\star_{2}\hat{\mathrm{d}}_{2}\Lambda+\frac{1}{2}\partial_{\theta}\Lambda\eta_{\phi}\,.$
(A.13)

This exhausts the non-trivial conditions arising from the holomorphic volume
form, and we turn our attention to the Kähler form.

### Reducing (2.5): $\partial_{y}J$ equation

The conditions resulting from imposing (2.5) on the above ansatz are

$\displaystyle 4\partial_{y}\mathrm{e}^{2B}$
$\displaystyle=\partial_{\theta}^{2}\Lambda\,,$ (A.14) $\displaystyle
4\mathrm{e}^{2B}\partial_{y}V^{R}$
$\displaystyle=\hat{\mathrm{d}}_{2}\partial_{\theta}\Lambda\,,$ (A.15)
$\displaystyle 4\partial_{y}\mathrm{e}^{2A}\mathrm{d}\mathrm{vol}_{2}$
$\displaystyle=\partial_{\theta}\Lambda\hat{\mathrm{d}}_{2}V^{I}-\hat{\mathrm{d}}_{2}\star_{2}\hat{\mathrm{d}}_{2}\Lambda\,,$
(A.16) $\displaystyle 4\mathrm{e}^{2B}\partial_{y}V^{I}$
$\displaystyle=\partial_{\theta}\Lambda\partial_{\theta}V^{I}-\partial_{\theta}\star_{2}\hat{\mathrm{d}}_{2}\Lambda\,.$
(A.17)

Note that the final condition, (A.17) is implied by (A.15) above and equation
(A.8).

### Reducing (2.6): $\mathrm{d}_{4}J$ equation

Finally imposing that the base is Kähler for fixed $y$ is equivalent to the
three conditions

$\displaystyle\partial_{\theta}\mathrm{e}^{2A}\mathrm{d}\mathrm{vol}_{2}$
$\displaystyle=\mathrm{e}^{2B}\hat{\mathrm{d}}_{2}V^{I}\,,$ (A.18)
$\displaystyle\hat{\mathrm{d}}_{2}\mathrm{e}^{2B}$
$\displaystyle=\mathrm{e}^{2B}\partial_{\theta}V^{R}\,,$ (A.19)
$\displaystyle\hat{\mathrm{d}}_{2}V^{R}$ $\displaystyle=0\,.$ (A.20)

###  A.2 Simplifying the conditions

Restricting to the independent equations we are left with six independent
conditions to solve, namely (A.14)-(A.16) and (A.18)-(A.20) subject to the two
constraints (A.8) and (A.10).

First note that equation (A.20) implies that the twisted differential operator
$\hat{\mathrm{d}}_{2}$ is nilpotent. Moreover, it follows that $V^{R}$ is
locally exact with respect to this twisted exterior derivative and we may
write it (locally) as

$V^{R}=\hat{\mathrm{d}}_{2}\Gamma\,,$ (A.21)

for some function $\Gamma$. To proceed it is useful to make the change of
coordinates Bah:2013qya

$X^{i}(x)=x^{i}\,,\quad-\Gamma(X,\Theta,y)=\theta\,.$ (A.22)

With this change of coordinates the derivatives are given by

$\displaystyle\partial_{\theta}$
$\displaystyle=-\frac{1}{\partial_{\Theta}\Gamma}\partial_{\Theta}\,,\quad\partial_{y}\rightarrow\partial_{y}-\frac{\partial_{Y}\Gamma}{\partial_{\Theta}\Gamma}\partial_{\Theta}\,,\quad\partial_{x}=\partial_{X}-\frac{\partial_{X}\Gamma}{\partial_{\Theta}\Gamma}\partial_{\Theta}\,,$
(A.23)

whilst the twisted differential becomes

$\hat{\mathrm{d}}_{2}=\mathrm{d}^{X}_{2}\equiv\mathrm{d}X^{i}\wedge\partial_{X^{i}}\,.$
(A.24)

With this change of coordinates (and dropping the superscript $X$ on
$\mathrm{d}^{X}_{2}$ from now on) the one-form $\eta_{\theta}$ takes the
simple form

$\eta_{\theta}=-\partial_{\Theta}\Gamma\mathrm{d}\Theta-\partial_{y}\Gamma\mathrm{d}Y\,.$
(A.25)

We can now solve (A.19) as

$\mathrm{e}^{2B}=-\frac{G}{\partial_{\Theta}\Gamma}\,,\quad\mathrm{d}_{2}G=0\,,$
(A.26)

with $G$ a scalar function. Inserting this into (A.10) fixes the warp factor
of the Riemann surface to be

$\mathrm{e}^{2A}=\frac{y^{2}\mathrm{e}^{\Lambda}g}{G}\,,$ (A.27)

where we have defined

$g\equiv\partial_{y}\Lambda\partial_{\Theta}\Gamma-\partial_{y}\Gamma\partial_{\Theta}\Lambda\,.$
(A.28)

Since $V^{0}$ is independent of both $\theta$ and $y$ it follows that for a
suitable $X$ dependent gauge choice for $\Gamma$ we may, without loss of
generality set $V^{0}=0$. It then follows that (A.18) is equivalent to

$G\Big{(}\partial_{X_{1}}^{2}+\partial_{X_{2}}^{2}\Big{)}\Gamma=y^{2}\partial_{\Theta}\bigg{(}\frac{\mathrm{e}^{\Lambda}g}{G}\bigg{)}\,.$
(A.29)

Next consider equations (A.14) and (A.15). They are equivalent to

$\displaystyle\partial_{y}G$ $\displaystyle=\partial_{\Theta}\widetilde{G}\,,$
(A.30) $\displaystyle\mathrm{d}_{2}\widetilde{G}$ $\displaystyle=0\,,$ (A.31)

where we have defined the function

$\widetilde{G}\equiv\frac{1}{4\partial_{\Theta}\Gamma}\Big{(}4G\partial_{y}\Gamma-\partial_{\Theta}\Lambda\Big{)}\,.$
(A.32)

The final equation to solve is (A.16) which implies

$(G\partial_{y}-\widetilde{G}\partial_{\Theta})\mathrm{e}^{2A}=\frac{G}{4}(\partial_{X_{1}}^{2}+\partial_{X_{2}}^{2})\Lambda\,.$
(A.33)

Having further reduced the conditions in terms of scalar potentials let us
study the structure of the metric. First we define

$h\equiv\partial_{y}\tilde{\Lambda}\partial_{\Theta}\Gamma-\partial_{\Theta}\tilde{\Lambda}\partial_{y}\Gamma\,,\quad\text{with}\quad\tilde{\Lambda}=\Lambda+\log
y\,,$ (A.34)

then the warp factor is given by

$\cos^{2}\zeta=-\frac{\partial_{\Theta}\Gamma}{yg}\,,\quad\text{and}\quad\sin^{2}\zeta=\frac{h}{g}\,.$
(A.35)

The ten-dimensional metric becomes

$\frac{1}{L^{2}}\mathrm{d}s^{2}=\sqrt{\frac{yg}{h}}\bigg{[}\mathrm{d}s^{2}(\text{AdS}_{3})+\frac{h}{g}\mathrm{d}\psi_{2}^{2}+\mathrm{e}^{-4\Delta}\mathrm{e}^{2A}(\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2})+\mathrm{e}^{-4\Delta}\mathrm{d}s^{2}(\mathcal{M}_{4})\bigg{]}\,,$
(A.36)

where

$\mathrm{d}s^{2}(\mathcal{M}_{4})=-\bigg{[}\frac{g}{4\partial_{\Theta}\Gamma}\mathrm{d}y^{2}+\frac{\partial_{\Theta}\Gamma}{h}\Big{(}\eta_{\Theta}+\frac{\partial_{\Theta}\Lambda}{\partial_{\Theta}\Gamma}\eta_{y}\Big{)}^{2}+\partial_{\Theta}\Gamma
G\Big{(}\mathrm{d}\Theta+\frac{\partial_{y}\Gamma}{\partial_{\Theta}\Gamma}\mathrm{d}y\Big{)}^{2}+\frac{4G}{\partial_{\Theta}\Gamma}\eta_{y}^{2}\bigg{]}$
(A.37)

and we have defined

$\eta_{\Theta}\equiv\mathrm{d}\psi_{1}-\frac{1}{2}\star_{2}\mathrm{d}_{2}\Lambda\,,\quad\eta_{y}\equiv-\frac{1}{2}(\mathrm{d}\phi-\star_{2}\mathrm{d}_{2}\Gamma)\,,$
(A.38)

where the reason for this labelling of the one-forms will become apparent
soon. If we make the change of coordinates specified by

$\mathrm{d}\bar{\theta}=G\mathrm{d}\Theta+\widetilde{G}\mathrm{d}y\,,$ (A.39)

under which the partial derivatives transform as

$\partial_{\Theta}=G\partial_{\bar{\theta}}\,,\quad\partial_{y}\rightarrow\partial_{y}+\widetilde{G}\partial_{\bar{\theta}}\,,$
(A.40)

the functions $G$ and $\widetilde{G}$ are eliminated from both the metric and
the remaining conditions to solve. We may therefore set $G$ to any non-zero
constant, set $\widetilde{G}=0$ and revert back to the previous set of
coordinates. We fix $G=\frac{1}{4}$ so that the condition from setting
$\widetilde{G}=0$ implies the integrability condition

$\partial_{y}\Gamma=\partial_{\Theta}\Lambda\,.$ (A.41)

We may solve this without loss of generality by introducing a potential $D$
such that

$\Gamma=\partial_{\Theta}D\,\quad\Lambda=\partial_{y}D\,.$ (A.42)

The two remaining conditions (A.29) and (A.33) then imply

$\square D-16gy^{2}\mathrm{e}^{\partial_{y}D}=f(X_{1},X_{2})\,.$ (A.43)

However, by suitably redefining the potential $D$ we can set the function
$f(X_{1},X_{2})$ to zero without loss of generality, so we do so. We must
therefore solve the Mongé–Ampere like equation

$\square
D=16y^{2}\Big{(}\partial_{y}^{2}D\partial_{\Theta}^{2}D-(\partial_{y}\partial_{\Theta}D)^{2}\Big{)}\mathrm{e}^{\partial_{y}D}\,.$
(A.44)

The final metric is

$\displaystyle\frac{1}{L^{2}}\mathrm{d}s^{2}=$
$\displaystyle\,\sqrt{\frac{yg}{h}}\bigg{[}\mathrm{d}s^{2}(\text{AdS}_{3})+\frac{h}{g}\mathrm{d}\psi_{2}^{2}+\frac{h}{yg}\mathrm{e}^{2A}(\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2})+\frac{h}{yg}\mathrm{d}s^{2}(\mathcal{M}_{4})\bigg{]}\,,$
(A.45) $\displaystyle\mathrm{d}s^{2}(\mathcal{M}_{4})=$
$\displaystyle\,\frac{1}{4}g_{ij}\mathrm{d}u^{i}\mathrm{d}u^{j}+h^{ij}\eta_{i}\eta_{j}\,,$
(A.46) $\displaystyle g_{ij}\equiv$
$\displaystyle\,-\partial_{i}\partial_{j}D\,,$ (A.47) $\displaystyle
h_{ij}\equiv$ $\displaystyle\,-\partial_{i}\partial_{j}\tilde{D}\,,$ (A.48)

where

$u^{i}=\\{y,\Theta\\}\,,\quad\eta_{1}=\eta_{\Theta}\,,\quad\eta_{2}=-\eta_{y}\,,$
(A.49)

and

$\tilde{D}\equiv D+y(\log y-1)\,,\quad e^{2A}=4y^{2}e^{\partial_{y}D}g.$
(A.50)

Note that the functions $g$ and $h$ are now the determinants of the matrices
$g_{ij}$ and $h_{ij}$ respectively.

The flux is given by

$F^{(2)}=-2J+y\mathrm{d}\Big{(}\eta_{\Theta}-\frac{\partial_{\Theta}\Lambda}{\partial_{\Theta}\Gamma}\eta_{y}\Big{)}-\mathrm{d}\bigg{(}\frac{yg}{h}\Big{(}\eta_{\Theta}-\frac{\partial_{\Theta}\Lambda}{\partial_{\Theta}\Gamma}\eta_{y}\Big{)}\bigg{)}\,,$
(A.51)

with $J$ as given in (A.6).

##  Appendix B Embedding the topological disk solution into section 2.2

In this appendix we prove that the solution of section 4 does indeed preserve
${\cal N}=(2,2)$ supersymmetry. We shall do so by embedding it into the
general form of section 2.2. Much of this can be reverse-engineered comparing
(4.11) to (2.3) and (2.21). This is sufficient to establish how the
coordinates are related, extract the values of
$(\partial^{2}_{\Theta}D,\,\partial_{y}D)$ and to identify the Riemann surface
of the local ansatz which turns out not to be $\Sigma_{2}$. In fact it is not
hard to show that the $(X_{1},\,X_{2})$ directions in (2.21) correspond to the
two-sphere spanned by $(\beta,\,\tau_{1})$ in (4.11), as such this is another
example in the constant curvature ansatz, our first with a two-sphere. We must
identify the coordinates of (2.21) as

$\displaystyle X_{1}$
$\displaystyle=\tan\left(\frac{\beta}{2}\right)\cos\tau_{1}\,,~{}~{}~{}~{}X_{2}=\tan\left(\frac{\beta}{2}\right)\sin\tau_{1}\,,$
$\displaystyle y$
$\displaystyle=(x-x_{0})\sin^{2}\alpha\,,~{}~{}~{}~{}\Theta=3K+2y-x_{0}-\Lambda\,,$
$\displaystyle\psi_{1}$
$\displaystyle=-\varphi-\tau_{1}-\tau_{2}\,,~{}~{}~{}~{}\psi_{2}=\psi\,,~{}~{}~{}~{}\phi=\tau_{1}+\tau_{2}\,,$
(B.1)

and decompose the the potential defining the system as

$D=I(y,\theta)+(2y-\theta)A_{0}\,,~{}~{}~{}~{}A_{0}=-\log\left(\frac{1}{2}\left(1+X_{1}^{2}+X_{2}^{2}\right)\right)\,,$
(B.2)

so that the partial differential equation of (2.20) reduces to

$\partial_{y}^{2}I\partial_{\Theta}^{2}I-(\partial_{y}\partial_{\Theta}I)^{2}+\frac{(2y-\theta)}{16y^{2}}e^{-\partial_{y}I}=0\,.$
(B.3)

It is possible to extract the following derivative of $I$ from the
metric181818We can also extract an expression for
$\partial_{y}\partial_{\Theta}I$ which is consistent with these.

$\displaystyle e^{\partial_{y}I}$
$\displaystyle=\frac{1}{32\tilde{K}}\Delta_{3}\,,$ (B.4) $\displaystyle
2\Delta_{2}\partial_{\Theta}^{2}I$
$\displaystyle=-\tilde{K}\Theta-(1-2\tilde{K})y+\frac{\Delta_{2}+\tilde{K}\sqrt{\Delta_{1}}}{2y-\Theta}-\frac{y(1-3\tilde{K}+y)}{\sqrt{\Delta_{1}}}-\frac{y(1-\tilde{K}-y)(\tilde{K}+y)}{(2y-\Theta)\sqrt{\Delta_{1}}}\,,$

where we introduce

$\displaystyle\tilde{K}$ $\displaystyle\equiv
3K+x_{0}\,,~{}~{}~{}~{}\Delta_{1}\equiv
4\tilde{K}y+(\tilde{K}+y-\Theta)^{2}\,,$ $\displaystyle\Delta_{2}$
$\displaystyle\equiv\tilde{K}^{2}+y(\Theta-1-y)+\tilde{K}[4y(1+y)-(1+4y)\Theta+\Theta^{2}]\,,$
$\displaystyle\Delta_{3}$ $\displaystyle\equiv
2\Delta_{2}-(\tilde{K}+y)(\sqrt{\Delta_{1}}-2-y+\Theta)-\tilde{K}(2+\tilde{K}+5y-2\Theta)\,.$
(B.5)

The definitions (B.4) are self consistent (i.e. they give rise to the same
$\partial_{\Theta}^{2}\partial_{y}I$) and are already enough to confirm that
(B.3) is satisfied, which is an important consistency check. Further, though
non trivial, the definition of $\partial_{y}I$ in (B.4) can be integrated to
give $I$ up to a function of $\Theta$ which can then be fixed191919(B.3) is
invariant under $I\to I+f(\Theta)$ when $f^{\prime\prime}(\Theta)=0$, so we
set terms linear in $\Theta$ to zero. by imposing consistency with the
expression for $\partial_{\Theta}^{2}I$. The final expression is rather
complicated but closed form, we find it convenient to decompose it as

$\displaystyle I$ $\displaystyle=I_{0}+I_{+}+I_{-}+I_{\Theta}\,,$
$\displaystyle I_{\Theta}$
$\displaystyle=\frac{1}{4}\left((\Theta-1)+\frac{1-2\tilde{K}-\Theta}{\sqrt{1-4\tilde{K}}}\right)\log(\Theta-2\tilde{K}),~{}~{}~{}~{}I_{\pm}=\frac{1}{4\sqrt{1-4\tilde{K}}}q_{\pm}\left[\log\left(l_{\mp}+p_{\pm}\sqrt{\Delta_{1}}\right)-\log
m_{\pm}\right]\,,$ $\displaystyle I_{0}$
$\displaystyle=-y\big{[}1+\log(32\tilde{K}y)\big{]}+y\log\Delta_{3}-\Theta\bigg{[}\frac{1}{2}\log\bigg{(}\left(\tilde{K}+y+\sqrt{\Delta_{1}}\right)\left(\tilde{K}-y+\Theta+\sqrt{\Delta_{1}}\right)\bigg{)}-\log(2y-\Theta)\bigg{]}$
$\displaystyle-\frac{1}{\sqrt{1-4\tilde{K}}}\left[2\tilde{K}-(1-\Theta)\left(1+\sqrt{1-4\tilde{K}}\right)\right]\log\Delta_{2}-\frac{1}{2}\log\left(3\tilde{K}+y-\Theta+\sqrt{\Delta_{1}}\right)$
$\displaystyle-\frac{1-2\tilde{K}-\Theta}{2\sqrt{1-4\tilde{K}}}\log\left((1-\Theta)\left(1-\sqrt{1-4\tilde{K}}\right)-2y\sqrt{1-4\tilde{K}}-2\tilde{K}\right)\,,$
(B.6)

where $q_{\pm}\equiv q_{1}\pm q_{2}$ and so on for $p_{\pm}$, $l_{\pm}$ and
$m_{\pm}$, with

$\displaystyle q_{1}$
$\displaystyle\equiv-1+2\tilde{K}+\Theta\,,~{}~{}~{}p_{1}\equiv
1+\Theta-6\tilde{K}\,,~{}~{}~{}q_{2}\equiv\sqrt{1-4\tilde{K}}(1-\Theta)\,,~{}~{}~{}p_{2}\equiv\sqrt{1-4\tilde{K}}q_{1}\,,$
$\displaystyle l_{1}$
$\displaystyle\equiv\sqrt{1-4\tilde{K}}\big{[}\Delta_{1}+y(1+y)+\Theta(1+\tilde{K}+y)+\tilde{K}(\tilde{K}-2)\big{]}\,,~{}~{}~{}l_{2}\equiv(3\tilde{K}+y-\Theta)q_{1}\,,$
$\displaystyle m_{1}$
$\displaystyle\equiv\sqrt{1-4\tilde{K}}(1+2y-\Theta)\,,~{}~{}~{}m_{2}\equiv
q_{1}\,.$ (B.7)

We have confirmed that (B.6) does indeed solve (B.3), so the solution
preserves ${\cal N}=(2,2)$ supersymmetry as claimed.

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