Compactification of 6d N 10quivers 4d SCFTs and their holographic dual Massive IIA backgrounds Paul Merrikin1 Carlos Nunez2and Ricardo Stuardo3

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Compactiﬁcation of 6d N= (1,0) quivers, 4d SCFTs and their

holographic dual Massive IIA backgrounds

Paul Merrikin 1, Carlos Nunez2and Ricardo Stuardo3

Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom

Abstract

In this paper we study an inﬁnite family of Massive Type IIA backgrounds that holographically

describe the twisted compactiﬁcation of N= (1,0) six-dimensional SCFTs to four dimensions. The

analysis of the branes involved motivates an heuristic proposal for a four dimensional linear quiver

QFT, that deconstructs the theory in six dimensions. For the case in which the system reaches

a strongly coupled ﬁxed point, we calculate some observables that we compare with holographic

results. Two quantities measuring the number of degrees of freedom for the ﬂow across dimensions

are studied.

1paulmerrikin@hotmail.co.uk, p.r.g.merrikin.2043506@swansea.ac.uk

2c.nunez@swansea.ac.uk

3ricardostuardotroncoso@gmail.com

arXiv:2210.02458v3 [hep-th] 24 Sep 2023

Contents

1 Introduction 2

2 Supergravity backgrounds 3

2.1 Pageﬂuxesandcharges ....................................... 5

2.2 The cases of H2and S2compactiﬁcations ............................. 9

3 Study of the dual ﬁeld theory 10

3.1 Free Energy and Holographic Central Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Flowcentralcharge.......................................... 14

3.3 A phenomenological proposal for the QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Conclusions 20

A Appendix 1 21

A.1 7D SU (2) Topologically Massive Gauged Supergravity . . . . . . . . . . . . . . . . . . . . . . 21

A.2 From SUSY Variations to BPS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A.3 Upliftto10DMassiveTypeIIA................................... 28

B Numerical solution of the BPS system 32

B.1 InfraredFixedPoint ......................................... 32

B.2 LinearPerturbations......................................... 33

C Analysis of the 4d QFT 33

C.1 Centralcharges............................................ 35

1 Introduction

Maldacena’s AdS/CFT conjecture [1] motivates the study of both gravity and ﬁeld theory topics.

In particular, the study of supersymmetric and conformal ﬁeld theories in diverse dimensions. In

the past few years we witnessed the deﬁnition of new, characteristically non-Lagrangian CFTs, by

the existence of a trustable background of Type II or M-theory, containing an AdS-factor.

In fact, this procedure has been applied to the possible space-time dimensions for which super

conformal ﬁeld theories exist (d+ 1 = 1, ...., 6). With eight Poincare supercharges, there exists a

classiﬁcation and an algorithmic way of associating a particular SCFTd+1 with a Type II background

containing an AdSd+2 factor. At present there seems to be exceptions to this statement for the

cases of supergravity solutions containing AdS3and AdS2spaces. See [2]-[14], for references working

details of the cases (d+ 1) = 1,2,3,4,5,6. A comprehensive summary of various aspects of SCFTs

in diverse dimensions can be found in [15].

A reasonable extension is the study of RG-ﬂows away from these SCFTsd+1. These ﬂows can

be between two conformal points or between a CFT and a gapped theory. Less conventional are

the ﬂows across dimensions, between a SCFTD+1 and a SCFTd+1 (there is also with the possibility

of ending in gapped systems). There are numerous case-studies of this in the bibliography, see

for example [16], for early examples working with twisted compactiﬁcations from the holographic

point of view. The topic progressed considerably after the paper [17]. This was followed by many

works studying compactiﬁcations (twisted or with ﬂuxes) from a purely QFT point of view. In

the particular case of compactiﬁcations of 6d to 4d systems (preserving minimal SUSY in both

dimensions), we ﬁnd the works [18]-[19]. For a very nice summary of these developments from a

ﬁeld theoretical perspective, see [20].

In this paper, we present an interesting example of ﬂow across dimensions involving a twisted

compactiﬁcation. In particular, we start from an inﬁnite family of six-dimensional N= (1,0)

SCFTs and compactify it on a two manifold of constant curvature. The end-point of the ﬂow is

an inﬁnite family of strongly coupled four dimensional N= 1 SCFTs (and possibly gapped QFTs,

that we leave for future studies). The holographic study of the family of 4d SCFTs occupies an

important part of this work, calculating observables that characterise it.

In more detail, the contents of the paper are distributed as follows.

In Section 2, we construct a new inﬁnite family of Massive Type IIA backgrounds that represent

the ﬂow between a family of six-dimensional N= (1,0) SCFTs and four dimensional N= 1

SCFTs. These ﬂows are new backgrounds, not present in the bibliography. The case of gapped four

dimensional systems leads to singular backgrounds, hence we leave it to future study. The charges

of the brane system are discussed, with emphasis on the eﬀects of the twisted-compactiﬁcation.

In Section 3, we present calculations of the holographic central charge in these supergravity

backgrounds (the free energy of the dual CFT). These are calculations at the AdS5ﬁxed point

and along the ﬂow. We also present a monotonic quantity interpolating between the conformal

points at low and high energies. After this, based on the branes charges discussed in Section 2, we

give a phenomenological proposal for a suitable quiver capturing the low energy dynamics. These

4d quiver QFTs are proposed to reach a conformal point at low energies, their strongly coupled

dynamics being described by the inﬁnite family of Massive Type IIA backgrounds with an AdS5

factor (discussed in Section 2). We emphasise on the heuristic character of this proposal. Indeed,

whilst the beta functions and R-symmetry anomalies of the proposed QFT are cancelled and the

scaling of the free energy with the quiver parameters (rank of gauge groups and number of nodes)

matches the holographic result, the precise coeﬃcient of the free energy does not exactly match the

one computed in the holographic dual. Hence the proposed quiver is only a ﬁrst step towards the

correct ﬁeld theory dual to our inﬁnite family of geometries. We discuss possible improvements in

the conclusions and Appendices.

In Section 4, we summarise, present conclusions and propose some ideas for further research.

Three very intensive appendices complement the presentation. The reader wishing to work on

these topics should beneﬁt from reading them in detail.

2 Supergravity backgrounds

We start this section by describing an inﬁnite family of supergravity solutions, the analysis of which,

is the main subject of the rest of this paper. This is a family of Massive Type IIA backgrounds,

preserving four supersymmetries (N=1 in four dimensional notation). The construction of these

backgrounds is described in great detail in Appendix A. From a quantum ﬁeld theoretical perspec-

tive, these backgrounds are dual to twisted compactiﬁcations of six dimensional N= (1,0) SCFTs

at the origin of their tensor branch. We discuss this in more detail in Section 3.

Let us present the family of backgrounds in Massive Type IIA. These are written in terms of

coordinates, parameters and functions,

Coordinates: (t, x1, x2, x3, r, θ1, ϕ1, z, θ2, ϕ2).Parameters: (Ψ0, k).(2.1)

Functions: α(z), f(r), h(r), X(r) = e2

5Φ(r), ω(r, z) = α′(z)2−2α(z)α′′(z)X(r)5

α′(z)2−2α(z)α′′(z).

The equations constraining these functions are written below. In terms of these coordinates and

functions, the string-frame spacetime metric reads

ds2

st = 2π√2s−α(z)

α′′(z)X(r)−1

2e−4Φ(r)

5e2f(r)dx2

3,1+dr2+e2h(r)dθ2

1+1

ksin2(√kθ1)dϕ2

1

+X(r)5/2"π√2s−α′′(z)

α(z)dz2+√2π

ω(r, z)p−α3(z)α′′(z)

2α(z)α′′(z)−α′2 dθ2

2+ sin2(θ2)dϕ2−1

kcos(√kθ1)dϕ12!#.

(2.2)

The Neveu-Schwarz (B2,Ψ) and Ramond (F0, F2, F4) background ﬁelds are,

B2=π

ω(r, z)

α(z)α′(z)

α′(z)2−2α(z)α′′(z)sin(θ2)dθ2−πcos(θ2)dz∧dϕ2−1

kcos(√kθ1)dϕ1,

e4Ψ(r,z)=X5(r)

ω2(r, z)−α(z)

α′′(z)3e2Ψ0

α′(z)2−2α(z)α′′(z)2

F0= 21

4e−Ψ0

√πα′′′(z),(2.3)

F2= 21

4√πe−Ψ0α′′(z)cos(θ2) Vol(Σk)−Vol(S2

c)+F0

ω(r, z)

α(z)α′(z)

α′(z)2−2α(z)α′′(z)Vol(S2

c),

F4= 21

4π3

2e−Ψ0

ω(r, z)!α(z)α′(z)α′′(z)

α′(z)2−2α(z)α′′(z)cos(θ2) Vol(Σk)∧Vol(S2)

+21

4π3

2e−Ψ0α′′(z) sin2(θ2)dz ∧dϕ2∧Vol(Σk).

We have deﬁned the volume elements,

Vol(S2) = sin(θ2)dθ2∧dϕ2,Vol(S2

c) = sin(θ2)dθ2∧dϕ2−1

kcos(√kθ1)dϕ1,

Vol(Σk) =

sin √kθ1

√kdθ1∧dϕ1.(2.4)

The functions f(r), h(r),Φ(r) must satisfy ﬁrst order (BPS) ordinary diﬀerential equations. De-

noting the derivative respect to the coordinate rwith a dot, they read,

f=±m

2e−2Φ,˙

h=±1

21

ke−2h+m e−2Φ,

Φ = ±−1 + 1

4ke−2h+m e−2Φ.(2.5)

We choose the positive sign from now on. The derivation of eqs.(2.5) and the origin of the parameter

mare explained in Appendix A. The remaining BPS equation for α(z) is already written in eq.(2.3).

In fact, the mass-parameter of massive Type IIA F0, that should be constant by pieces for an

interpretation in terms of localised D8 branes dictates that α′′′(z) must be piece-wise constant.

It is in the many possible choices for a piece-wise constant F0that the family of backgrounds is

originated. More on this in Section 2.1 below.

The conﬁgurations in eqs.(2.2)-(2.5), are new solutions to the equations of motion of Massive

IIA. In string frame these read,

4R+∇2Ψ−(∇Ψ)2−1

8H2

3= 0,

dFp+H3∧ ∗Fp−2= 0, d(e−2Ψ ∗H3)−(F0∗F2+F2∧ ∗F4+F4∧F4)=0,

RMN + 2∇M∇NΨ−1

2(H2

3)MN −1

4e2Ψ X

(F2

p)MN = 0.(2.6)

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Compactificationof6dN=(1,0)quivers,4dSCFTsandtheirholographicdualMassiveIIAbackgroundsPaulMerrikin1,CarlosNunez2andRicardoStuardo3DepartmentofPhysics,SwanseaUniversity,SwanseaSA28PP,UnitedKingdomAbstractInthispaperwestudyaninfinitefamilyofMassiveTypeIIAbackgroundsthatholographicallydescribethetwiste...

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Compactification of 6d N 10quivers 4d SCFTs and their holographic dual Massive IIA backgrounds Paul Merrikin1 Carlos Nunez2and Ricardo Stuardo3

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