# The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere

**ABSTRACT** We present the gravity dual to a class of three-dimensional N=2

supersymmetric gauge theories on a biaxially squashed three-sphere, with a

non-trivial background gauge field. This is described by a 1/2 BPS Euclidean

solution of four-dimensional N=2 gauged supergravity, consisting of a

Taub-NUT-AdS metric with a non-trivial instanton for the graviphoton field. The

holographic free energy of this solution agrees precisely with the large N

limit of the free energy obtained from the localized partition function of a

class of Chern-Simons quiver gauge theories. We also discuss a different

supersymmetric solution, whose boundary is a biaxially squashed Lens space

S^3/Z_2 with a topologically non-trivial background gauge field. This metric is

of Eguchi-Hanson-AdS type, although it is not Einstein, and has a single unit

of gauge field flux through the S^2 cycle.

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**ABSTRACT:**We construct three-dimensional = 2 supersymmetric field theories on conic spaces. Built upon the fact that the partition function depends solely on the Reeb vector of the Killing vector, we propose that holographic dual of these theories are four-dimensional, supersymmetric charged topological black holes. With the supersymmetry localization technique, we study conserved supercharges, free energy, and supersymmetric Rényi entropy. At planar large N limit, we demonstrate perfect agreement between the superconformal field theories and the supersymmetric charged topological black holes.Journal of High Energy Physics 02/2014; · 5.62 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We find the gravity dual of \( \mathcal{N} \) = 2* super-Yang-Mills theory on S 4 and use holography to calculate the universal contribution to the corresponding S 4 free energy at large N and large ’t Hooft coupling. Our result matches the expression previously computed using supersymmetric localization in the field theory. This match represents a non-trivial precision test of holography in a non-conformal, Euclidean signature setting.Journal of High Energy Physics 07/2014; 2014(7). · 6.22 Impact Factor - SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We study gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) x U(1) symmetry. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localization. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories with a large N limit, defined on a general class of background three-manifold geometries.04/2014;

Page 1

arXiv:1111.6930v1 [hep-th] 29 Nov 2011

November 30, 2011

The nuts and bolts of supersymmetric gauge

theories on biaxially squashed three-spheres

Dario Martelli1and James Sparks2

1Department of Mathematics, King’s College, London,

The Strand, London WC2R 2LS, United Kingdom

2Mathematical Institute, University of Oxford,

24-29 St Giles’, Oxford OX1 3LB, United Kingdom

Abstract

We present the gravity dual to a class of three-dimensional N = 2 supersym-

metric gauge theories on a biaxially squashed three-sphere, with a non-trivial

background gauge field. This is described by a 1/2 BPS Euclidean solution of

four-dimensional N = 2 gauged supergravity, consisting of a Taub-NUT-AdS

metric with a non-trivial instanton for the graviphoton field. The holographic

free energy of this solution agrees precisely with the large N limit of the free

energy obtained from the localized partition function of a class of Chern-Simons

quiver gauge theories. We also discuss a different supersymmetric solution, whose

boundary is a biaxially squashed Lens space S3/Z2 with a topologically non-

trivial background gauge field. This metric is of Eguchi-Hanson-AdS type, al-

though it is not Einstein, and has a single unit of gauge field flux through the

S2cycle.

Page 2

1Introduction

Supersymmetric gauge theories on compact curved backgrounds are interesting for var-

ious reasons. For example, supersymmetry may be combined with localization tech-

niques, allowing one to perform a variety of exact computations in strongly coupled field

theories. The authors of [1] presented a construction of N = 2 supersymmetric gauge

theories in three dimensions in the background of a U(1) × U(1)-invariant squashed

three-sphere and R-symmetry gauge field. The gravity dual of this construction was

recently given in [2]. It consists of a 1/4 BPS Euclidean solution of four-dimensional

N = 2 gauged supergravity, which in turn may be uplifted to a supersymmetric solu-

tion of eleven-dimensional supergravity. In particular, the bulk metric in [2] is simply

AdS4, and the graviphoton field is an instanton with (anti)-self-dual field strength. The

asymptotic metric and gauge field then reduce to the background considered in [1].

The purpose of this letter is to present the gravity dual to a different field theory

construction, obtained recently in [3]. In this reference the authors have constructed

three-dimensional N = 2 supersymmetric gauge theories in the background of the

SU(2)×U(1)-invariant squashed three-sphere (which we refer to as biaxially squashed)

and a non-trivial background U(1) gauge field, and have computed the corresponding

partition functions using localization. Differently from a superficially similar construc-

tion discussed briefly in [1], this partition function depends non-trivially on the squash-

ing parameter. As we will see, the gravity dual to this set-up will have some distinct

features with respect to the solution in [2]. In particular, the metric is not simply AdS4,

although it will again be an Einstein metric, and there is a self-dual graviphoton.

As an illustration of the generality of our constructions, in section 4 we discuss

yet another supersymmetric solution, consisting of a non-Einstein metric and a non-

instantonic graviphoton field. This has the topology of M4= T∗S2, with a single unit

of gauge field flux through the S2. We conjecture that this solution is dual to a class

of supersymmetric gauge theories on S3/Z2, with a background gauge field that is a

connection on the non-trivial torsion line bundle over S3/Z2.

1

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2Supersymmetric gauge theories on the biaxially

squashed S3

In the construction of [3] the metric on the three-sphere is, up to an irrelevant overall

factor, given by

ds2

3

= σ2

1+ σ2

2+1

v2σ2

3, (2.1)

where σi are the standard SU(2) left-invariant one-forms on S3, defined as iσiτi =

−2g−1dg, where τidenote the Pauli matrices and g ∈ SU(2). The background U(1)

gauge field reads

√v2− 1

A(3)

=

2v2

σ3,(2.2)

and the spinors in the supersymmetry transformations obey the equation (setting the

radius r = 2 in [3])

∇(3)

αχ −

i

4vγαχ − A(3)

βγαβχ = 0 ,(2.3)

where ∇(3)

(2.1), and γαgenerate Cliff(3,0). There are two linearly independent solutions to (2.3),

transforming as a doublet under SU(2), whose explicit form is given in [3]. This will

α , α = 1,2,3, is the spinor covariant derivative constructed from the metric

be important for identifying the gravity dual.

In [3] the authors constructed Chern-Simons, Yang-Mills, and matter Lagrangians for

the N = 2 vector multiplets V = (Aα,σ,λ,D) and chiral multiplets Φ = (φ,ψ,F), in

the background of the metric (2.1) and R-symmetry gauge field (2.2). These are invari-

ant under a set of supersymmetry transformations, provided the spinorial parameters

obey the equation (2.3). The supersymmetric completion of the Chern-Simons La-

grangian contains new terms, in addition to those appearing in flat space, proportional

to σ2and σA(3)∧ dA (cf. eq. (32) of [3]). The Yang-Mills and matter Lagrangians

are total supersymmetry variations (cf. eq. (31) of [3]) and therefore can be used to

compute the partition function using localization. In particular, the partition function

localizes on supersymmetric configurations obeying

Aα = D = 0 ,σ = u = constant ,(2.4)

with the matter fields all being zero. Notice that although D = 0, the Chern-Simons

Lagrangian is non-zero because of the new term proportional to σ2, and therefore it

2

Page 4

contributes classically to the localized partition function, as in previous constructions.

The Yang-Mills and matter terms contribute one-loop determinants from the Gaussian

integration about the the classical solutions (2.4). The final partition function may be

expressed again in terms of double sine functions sb(z), and for a U(N) gauge theory

at Chern-Simons level k ∈ Z reads

Z=

?

?

Cartan

du exp

?iπk

v2Tru2

?

?

Rootsαsb

?

α(u)−i

v

?

?

?

Chirals,repRa

?

ρ∈Rasb

ρ(u)−i(1−∆a)

v

? , (2.5)

where b = (1 + i√v2− 1)/v. The exponential term is the classical contribution from

the Chern-Simons Lagrangian, evaluated on (2.4); the numerator is the one-loop vector

multiplet determinant and involves a product over the roots α of the gauge group G;

while the denominator is the one-loop matter determinant and involves a product over

chiral fields of R-charge ∆ain representations Ra, with ρ running over weights in the

weight-space decomposition of Ra. Following [4], one can easily extract the large N

behaviour of this partition function for a class of non-chiral N = 2 quiver Chern-

Simons-matter theories. The calculation was done in [3], and the result is that the

leading contribution to the free energy (defined as F = −logZ) is given by

Fv

=

1

v2Fv=1,(2.6)

and thus depends very simply on the squashing parameter v. In the next section we

will present the supergravity dual to this construction, in particular showing that the

holographic free energy precisely agrees with the field theory result (2.6).

3The gravity dual

As anticipated in [2], we will show that the gravity dual to the set-up described in the

previous section is a supersymmetric solution of d = 4, N = 2 gauged supergravity. In

Lorentzian signature, the bosonic part of the action is given by

SLorentzian

=

1

16πG4

?

d4x?−detgµν

?R + 6g2− (FL)2?

.(3.1)

Here R denotes the Ricci scalar of the four-dimensional metric gµν, and the cosmo-

logical constant is given by Λ = −3g2. The graviphoton is an Abelian gauge field

ALwith field strength FL= dAL; here the superscript L emphasizes that this is a

3

Page 5

Lorentzian signature object. A solution to the equations of motion derived from (3.1)

is supersymmetric if there is a non-trivial spinor ǫ satisfying the Killing spinor equation

?∇µ+1

2gΓµ− igAL

µ+i

4FL

νρΓνρΓµ

?ǫ = 0 . (3.2)

Here Γµ, µ = 0,1,2,3, generate the Clifford algebra Cliff(1,3), so {Γµ,Γν} = 2gµν.

Since the background of [3] preserves half of the maximal supersymmetry in three

dimensions, we should seek a 1/2 BPS Euclidean solution of d = 4, N = 2 gauged

supergravity, whose metric has as conformal boundary the biaxially squashed metric

on S3(2.1), and whose background U(1) gauge field restricted to this asymptotic

boundary reduces to (2.2). This very strongly suggests that the appropriate solution is

a Euclideanized version of the 1/2 BPS Reissner-Nordstr¨ om-Taub-NUT-AdS solution

discussed in [5].

We will first present this Euclidean solution, and then discuss the Wick rotation that

leads to it. The metric reads

r2− s2

Ω(r)

ds2

4

=

dr2+ (r2− s2)(σ2

1+ σ2

2) +4s2Ω(r)

r2− s2σ2

3,(3.3)

where

Ω(r) = (s − r)2?1 + g2(r − s)(r + 3s)?

,(3.4)

and s is the NUT parameter.1The SU(2) left-invariant one-forms σimay be written

in terms of angular variables as

σ1+ iσ2 = e−iψ(dθ + isinθdϕ) ,σ3 = dψ + cosθdϕ .(3.5)

The graviphoton field is

A = sr − s

r + s

?

1 − 4g2s2σ3. (3.6)

In the orthonormal frame

e1 =

√r2− s2σ1,

?

e2 =

√r2− s2σ2,

?

Ω(r)

e3 = 2s

Ω(r)

r2− s2σ3,e4 =

r2− s2

dr ,(3.7)

the curvature may be written as

F= dA = −s?1 − 4g2s2

(r + s)2

(e12+ e34) .(3.8)

1This is denoted N in [5].

4

Page 6

Thus the gauge field is an instanton, as in the solution discussed in [2]. In particular,

with our choice of orientation the curvature is self-dual, and the on-shell gauge field

action is finite. Since the stress-energy tensor of an instanton vanishes, the metric (3.3)

is accordingly an Einstein metric. However, differently from the solution in [2], one can

check that this metric is not locally AdS4. It is in fact a Euclidean version of the well-

known Taub-NUT-AdS metric, with a special value of the mass parameter. This metric

is locally asymptotically AdS4, and therefore it can be interpreted holographically [6].

Notice that the NUT parameter |s| ≤ 1/(2g), and for |s| = 1/(2g) the gauge field

instanton vanishes and the metric reduces to Euclidean AdS4.

For large r the metric becomes

ds2

4

≈

dr2

g2r2+ r2?σ2

1+ σ2

2+ 4g2s2σ2

3

?

, (3.9)

while to leading order the gauge field reduces to

A ≈ A(3)≡ s

?

1 − 4g2s2σ3. (3.10)

We see that the conformal boundary may be identified precisely with the metric (2.1),

and the background gauge field with (2.2), by setting s =

1

2gv. Recall here that in

order to uplift to eleven-dimensional supergravity one should also set g = 1 [2]. Notice

that when |v| = 1 the boundary metric reduces to the round metric on S3, and the

background gauge field vanishes.Correspondingly, in the bulk the instanton field

vanishes, and the metric becomes AdS4.

Wick rotation and regularity

Let us discuss briefly how this solution was obtained. The reader not interested in these

details may safely jump to the discussion of the Killing spinors and the holographic

free energy.

As we are interested in a 1/2 BPS solution, we may begin by appropriately Wick

rotating the solution (2.1), (2.4) of [5]. We take their parameter ℵ = +1, so as to

obtain a biaxially squashed S3as constant r surface. The Wick rotation may then be

taken to be t → iτ, N → is, Q → iQ, together with a change in sign of the metric.

This leads to the following metric and gauge field

ds2

4

=

r2− s2

Ω(r)

sP − Qr

dr2+ (r2− s2)(dθ2+ sin2θdϕ2) +

r2− s2dτ +P(r2+ s2) − 2sQr

Ω(r)

r2− s2(dτ + 2scosθdϕ)2,

cosθdϕ ,AL

=

r2− s2

(3.11)

5

Page 7

where

Ω(r) = g2(r2− s2)2+ (1 − 4g2s2)(r2+ s2) − 2Mr + (P2− Q2) .(3.12)

This depends on the parameters s,g,M,P,Q. Notice we have kept a Lorentzian su-

perscript on ALin (3.11) – the reason for this will become clear momentarily.

For the 1/2 BPS solution of interest, the Euclideanized BPS equations of [5] imply

that

M2

= (1 − 4g2s2)?s2(1 − 4g2s2) + P2− Q2?

s2P(1 − 4g2s2) = sMQ − P(P2− Q2) ,

,

(3.13)

and the corresponding 1/2 BPS solution then depends on only two parameters. We

take these to be s and Q, with

P = is

?

1 − 4g2s2,M = −iQ

?

1 − 4g2s2,(3.14)

then solving (3.13). The factors of i in (3.14) may look problematic, but there are (at

least) two different ways of obtaining real solutions. We require s and M to be real

in order that the metric in (3.11) is real. If |s| ≤ 1/(2g) then P and Q will be purely

imaginary, and we may write P = ip, Q = −iq to obtain the real gauge field

sp + qr

r2− s2dτ +p(r2+ s2) + 2sqr

On the other hand, if |s| ≥ 1/(2g) then P and Q will be real; we shall examine this

class of solutions in section 4.

A ≡ −iAL=

r2− s2

cosθdϕ . (3.15)

Redefining τ = 2sψ, in terms of standard Euler angles (θ,ϕ,ψ) notice that the metric

(3.11) takes the form presented in (3.3), albeit with a more general form of the function

Ω(r), given by (3.12) and (3.14). That (3.3) has only one free parameter s, and not

the two we have above, follows from imposing regularity of the Euclidean metric. At

any fixed r > s that is not a root of Ω(r), we obtain a smooth biaxially squashed S3

metric. In order to obtain a complete metric, the space must “close off” at the largest

root r0of Ω(r), so that Ω(r0) = 0. More precisely, if r0> s this should be a single

root, while if r0= s the metric will be regular only if r0= s is a double root of Ω(r).

We shall return to the former case in section 4, here focussing on the case r0= s. The

condition Ω(r0= s) = 0 immediately fixes

q = −s

?

6

1 − 4g2s2, (3.16)

Page 8

so that now (see also [7])

p = s

?

1 − 4g2s2= −q ,M = s(1 − 4g2s2) . (3.17)

It is then in fact automatic that r = s is a double root of Ω.

In conclusion, we end up with the metric (3.3), with Ω(r) given in (3.4), and gauge

field (3.6). The gauge field is manifestly non-singular and one can check that the metric

indeed smoothly closes off at r = s, giving the topology M4= R4.

Killing spinors

In this subsection we briefly discuss the supersymmetry of the Euclidean solution (3.3),

(3.6), in particular reproducing the three-dimensional spinor equation (2.3) asymptot-

ically.

In Lorentzian signature the Killing spinor equation is (3.2). However, in Wick ro-

tating we have introduced a factor of i into the gauge field in (3.15), so that AL= iA.

Thus the appropriate Killing spinor equation to solve in this case is

?∇µ+1

2gΓµ+ gAµ−1

4FνρΓνρΓµ

?ǫ = 0 .(3.18)

This possibility of Wick rotating the gauge field (or not) was also discussed in [8]. In

particular, the authors of [8] pointed out that any Euclidean solution with a real gauge

field that solves (3.18) will automatically be 1/2 BPS. The reason is simple: if ǫ solves

(3.18), then so does its conjugate ǫc. We shall see this explicitly below.

We introduce the following representation for the generators of Cliff(4,0)

ˆΓ4 =

?

0iI2

0

−iI2

?

,

ˆΓα =

?

0

τα

τα

0

?

, (3.19)

where α ∈ 1,2,3, ταare the Pauli matrices, and hats denote tangent space quantities.

Decomposing the Dirac spinor ǫ into positive and negative chirality parts as

ǫ =

?

ǫ+

ǫ−

?

, (3.20)

where ǫ±are two-component spinors, it is then straightforward, but tedious, to verify

that in the orthonormal frame (3.7)

ǫ+ =

?

λ(r)χ+

λ∗(r)χ−

?

,ǫ− = i

?r − s

r + s

?

λ∗(r)χ+

λ(r)χ−

?

,(3.21)

7

Page 9

is the general solution to the µ = r component of (3.18), where χ±are independent of

r and we have defined

λ(r) ≡

?

g(r + s) − i

?

1 − 4g2s2?1/2

. (3.22)

If we now define the charge conjugate spinor ǫc≡ Bǫ∗, where B is the charge conjuga-

tion matrix defined in [2], then it is straightforward to see that taking the conjugate

ǫ → ǫcsimply maps χ+→ −χ∗

Let us analyze the large r asymptotics of the Killing spinor equation (3.18), and its

−, χ−→ χ∗

+.

solutions (3.21). We begin by expanding

ǫ+ =

√gr1/2

?

I2+

?s

?s

2I2−

i

2g

?

?

1 − 4g2s2τ3

?

?

r−1+ O(r−2)

?

χ ,

ǫ− = i√gr1/2

?

I2−

2I2−

i

2g

1 − 4g2s2τ3

r−1+ O(r−2)

?

χ ,(3.23)

where we have defined the r-independent two-component spinor

χ ≡

?

χ+

χ−

?

. (3.24)

We then write the asymptotic expansion of the metric as

ds2

4

=

dr2

g2r2

?1 + O(r−2)?+r2

≡ g2?σ2

g2

?

?ds2

.

3+ O(r−2)?

, (3.25)

ds2

3

1+ σ2

2+ 4g2s2σ2

3

(3.26)

It is then straightforward to extract the coefficient of r1/2in the Killing spinor equation

(3.18). One finds that the positive and negative chirality projections lead to the same

equation for χ, namely

∇(3)

αχ + gA(3)

αχ −is

2γαχ −1

2g

?

1 − 4g2s2γατ3χ = 0 ,(3.27)

where ∇(3)denotes the spin connection for the three-metric (3.26), and A(3)is defined

in (3.10). Using the explicit form for A(3)in (3.10), the identity γαγβ = γαβ+ g(3)

and recalling that s = 1/(2gv), g = 1, we precisely obtain the spinor equation (2.3).

Finally, one can verify that the d = 4 spinors (3.21), with χ satisfying (3.27), do indeed

αβ,

solve (3.18).

8

Page 10

The holographic free energy

The holographic free energy of the Taub-NUT-AdS solution was discussed in [7], but

of course in this latter reference there was no instanton field, which is crucial for

supersymmetry. The calculation proceeds essentially as in section 2.5 of [2], except for

the following caveat. The integrability condition for the Killing spinor equation (3.18)

gives the equations of motion following from the action

SEuclidean = −

1

16πG4

?

d4x?detgµν

?R + 6g2+ F2?

, (3.28)

which has opposite (relative) sign for the gauge field term compared with (3.1) (see

also [8]). This is clear from the fact that our equation (3.18) was obtained from the

Lorentzian form of the equation by sending A → iA. It is therefore natural to expect

that in the computation of the holographic free energy we have to evaluate the action

SEuclideanon shell.

Setting g = 1 and cutting off the space at r = R, the bulk gravity contribution is

given by

Igrav

bulk

=

3

8πG4

?

d4x?detgµν =

4πsR3

G4

−12πs3R

G4

+8πs4

G4

. (3.29)

Denoting by R[γ] the scalar curvature of the boundary metric, and by K the trace of

its second fundamental form, the combined gravitational boundary terms

Igrav

ct

+ Igrav

bdry

=

1

8πG4

?

d3x?detγαβ

?

2 +1

2R[γ] − K

?

(3.30)

have the following asymptotic expansion

Igrav

ct

+ Igrav

bdry

= −4πsR3

G4

+12πs3R

G4

+4πs2(1 − 4s2)

G4

+ O(1/R) ,(3.31)

where in particular notice there is a non-zero finite contribution. The instanton action

is

IF

bulk

= −

1

16πG4

?

d4x?detgµνFµνFµν= −2πs2(1 − 4s2)

G4

.(3.32)

Therefore the total on-shell action SEuclidean, obtained after removing the cut-off (R →

∞), is given by

I= Igrav

bulk+ Igrav

ct

+ Igrav

bdry+ IF

bulk=

2πs2

G4

. (3.33)

9

Page 11

Since the round sphere result2is s = 1/2, we thus see that

Is =

2πs2

G4

= (2s)2Is=1/2,(3.34)

which since v = 1/(2s) precisely agrees with the field theory result (2.6).

4A supersymmetric Eguchi-Hanson-AdS solution

In the previous section we assumed that 1 − 4g2s2> 0, which led to an “imaginary”

gauge field in the Euclidean regime. However, we may also consider the case 4g2s2−1 >

0, or equivalently |s| ≥ 1/(2g). In this case the Euclidean supersymmetry equation

takes the same form as the Lorentzian equation (3.2), namely

?∇µ+1

2gΓµ− igAµ+i

4FνρΓνρΓµ

?ǫ = 0 . (4.1)

In this section we show that there is a one-parameter family of regular solutions in this

class, of topology M4= T∗S2, for which there are Killing spinors solving (4.1).

When |s| ≥ 1/(2g) we may rewrite (3.14) as

P= −s

which are now real. Again setting τ = 2sψ, the metric takes the form given in (3.3)

?

4g2s2− 1 ,M = Q

?

4g2s2− 1 ,(4.2)

where now

Ω(r) = g2(r2− s2)2−

?

r

?

4g2s2− 1 + Q

?2

. (4.3)

It will be useful to note that the four roots of Ω(r) in (4.3) are

?

?

r4

r3

?

?

=

1

2g

??

?

4g2s2− 1 ±

?

8g2s2+ 4gQ − 1

?

,

r2

r1

=

1

2g

−

?

4g2s2− 1 ±

?

8g2s2− 4gQ − 1

?

. (4.4)

The gauge field is given by (after a suitable gauge transformation)

s

r2− s2

As r → ∞ this tends to

A ≈ A(3)≡ −s

which is (up to analytic continuation) what we had in the previous example (3.10).

A = −

?

2Qr + (r2+ s2)

?

4g2s2− 1

?

σ3. (4.5)

?

4g2s2− 1σ3, (4.6)

2To recover the result for S2×S1boundary, one should first change coordinates back to the form in

(3.11), and then set s = 0 there. In these coordinates, with τ ∈ [0,2π] the gravitational contribution

to the free energy is half that of the round sphere.

10

Page 12

Killing spinors

Taking the same Clifford algebra and spinor conventions as the previous section, and

again using the orthonormal frame (3.7), one can verify that the integrability condition

for the Killing spinor equation (4.1) leads to the algebraic relation

ǫ− = i

?r − s

r + s

?

(r−r3)(r−r4)

(r−r1)(r−r2)

0

0

?

(r−r1)(r−r2)

(r−r3)(r−r4)

ǫ+.(4.7)

Here recall that ǫ±are two-component spinors, and ri, i = 1,2,3,4, are the four roots

of Ω in (4.4). Substituting into the µ = r component of (4.1) then leads to decoupled

first order ODEs, which may be solved to give

ǫ+ =

?

?

(r−r1)(r−r2)

(r−s)

χ+

(r−r3)(r−r4)

(r−s)

χ−

,ǫ− = i

?

?

(r−r3)(r−r4)

(r+s)

χ+

(r−r1)(r−r2)

(r+s)

χ−

,(4.8)

where χ±are independent of r. The large r expansion of these is given by

ǫ+ = r1/2

?

I2+

?s

?s

2I2+1

2I2+1

2g

?

?

4g2s2− 1τ3

?

?

r−1+ O(r−2)

?

χ , (4.9)

ǫ− = ir1/2

?

I2−

2g

4g2s2− 1τ3

r−1+ O(r−2)

?

χ ,(4.10)

where the two-component spinor χ is again given by (3.24). Notice this is the same as

(3.23), up to analytic continuation. Again using the metric expansion and three-metric

in (3.26), we may extract the coefficient of r1/2in (4.1). A very similar computation to

that in the previous section then leads to the three-dimensional Killing spinor equation

∇(3)

αχ −is

2γαχ + igA(3)

βγβ

αχ = 0 .(4.11)

Setting g = 1 and again identifying the squashing parameter v = 1/(2s), notice this

is identical to our original equation (2.3), but where we have replaced A(3)→ −iA(3).

Of course, given the relative difference in Wick rotations of the gauge field in two the

cases, this was precisely to be expected. In fact, comparing the A(3)(4.6) in this section

with its counterpart (3.10) in the previous section, we see that equation (4.11) is in

fact identical to (2.3), due to the factor of i difference in (4.6), (3.10).

The solution to (4.11) is therefore given by an appropriate analytic continuation of

the solution presented in [3], and reads

χ = eητ3/2g−1χ0,(4.12)

11

Page 13

where g ∈ SU(2), χ0is a constant two-component spinor, and

v =

1

coshη,

(4.13)

where v = 1/(2s). In terms of Euler angles (ψ,θ,ϕ), recall that

g =

?

cosθ

−sinθ

2ei(ψ+ϕ)/2

2ei(ψ−ϕ)/2

sinθ

cosθ

2e−i(ψ−ϕ)/2

2e−i(ψ+ϕ)/2

?

. (4.14)

This will be important below when we consider global properties of the solution.

Regularity of the metric

We must again consider regularity of the metric (3.3). A complete metric will neces-

sarily close off at the largest root r0of Ω(r), which must satisfy r0≥ s. From (4.4) we

see that either r0= r+or r0= r−, where it is convenient to define

r+ ≡ r4,r− ≡ r2.(4.15)

A priori the coordinate ψ must have period 2π/n, for some positive integer n, so that

the surfaces of constant r are Lens spaces S3/Zn. Assuming that r0> s is strict, then

the metric (3.3) will have the topology of a complex line bundle M4= O(−n) → S2

over S2, where r − r0 is the radial direction away from the zero section. However,

the form of the Killing spinors in the previous subsection implies we must take either

n = 1 or n = 2 for a supersymmetric solution: only when ψ has period 4π or 2π is

(4.14), and hence (4.12), single-valued (or single-valued up to sign, respectively). We

may thus rule out the existence of n > 2 supersymmetric solutions.

Regularity of the metric near to the S2zero section at r = r0requires

????

r2

sΩ′(r0)

0− s2

????

=

2

n.

(4.16)

This conditon ensures that near to r = r0the metric (3.3) takes the form

ds2

4

≈ dρ2+ ρ2

?

d

?nψ

2

?

+n

2cosθdϕ

?2

+ (r2

0− s2)(dθ2+ sin2θdϕ2) , (4.17)

near to ρ = 0. Here note that nψ/2 has period 2π. Imposing (4.16) at r0= r±gives

Q = Q±(s) ≡ ∓128g4s4− 16g2s2− n2

64g3s2

. (4.18)

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Page 14

In turn, substituting Q = Q±(s) into (4.15) one then finds

r±(Q±(s)) =

1

8g

?n

gs± 4

?

4g2s2− 1

?

. (4.19)

Recall that in order to have a smooth metric, we require r0 > s. Imposing this for

r0= r±(Q±(s)) gives

r±(Q±(s)) − s =

1

2gf±

n(2gs) ,(4.20)

where the function

f±

n(x) ≡

n

2x− x ±√x2− 1(4.21)

is required to be positive for a smooth metric with s = x/(2g). Notice here that

s ≥ 1/(2g) implies x ≥ 1. It is straightforward to show that f−

decreasing on x ∈ [1,∞). The analysis then splits into the cases {n = 1}, {n = 2},

which have a qualitatively different behaviour:

n(x) is monotonic

n = 1

It is easy to see that f±

be made regular in this case. Specifically, f±

decreasing, this rules out taking r0= r−(Q−(s)) given by (4.19); on the other hand

f+

r0= r+(Q+(s)) in (4.19).

1(x) < 0 on x ∈ [1,∞), and thus the metric (3.3) cannot

1(1) = −1/2: since f−

1(x) is monotonic

1(x) monotonically increases to zero from below as x → ∞, and we thus also rule out

n = 2

It is easy to see that f−

which means we must set

2(x) < 0 for x ∈ (1,∞), while f+

2(x) > 0 on the same domain,

Q ≡ Q+(s) = −(4g2s2− 1)(1 + 8g2s2)

16g3s2

,(4.22)

and

r0(s) =

1

4g

?1

gs+ 2

?

4g2s2− 1

?

, (4.23)

may then be shown to be the largest root of Ω(r), for all s ≥ 1/(2g). In particular,

this involves showing that r0(s)−r−(Q+(s)) > 0 for all s ≥ 1/(2g), which follows since

r0(s) − r−(Q+(s)) =

1

2gh(2gs) , (4.24)

13

Page 15

where we have defined

h(x) ≡

1

x+ 2√x2− 1 −

?

4x2− 2 −1

x2. (4.25)

It is a simple exercise to prove that h(x) > 0 on x ∈ (1,∞).

After this slightly involved analysis, for n = 2 we end up with a smooth complete

metric on M4= T∗S2, given by (3.3), (4.3) with Q = Q+(s) given by (4.22), for all

s > 1/(2g). The S2zero section is at r = r0(s) given by (4.23). The metric is thus of

Eguchi-Hanson-AdS type, although we stress that it is not Einstein for any s > 1/(2g).

The large r behaviour is again given by (3.9), so that the conformal boundary is a

squashed S3/Z2. The s = 1/2g limit gives a round S3/Z2 at infinity with the bulk

metric being the singular AdS4/Z2, albeit with a non-trivial torsion gauge field, as we

shall see momentarily.

It follows that another interesting difference to the Taub-NUT-AdS solution of the

previous section is that the gauge field (4.5) no longer has (anti)-self-dual field strength

F = dA; moreover, the latter has a non-trivial flux. Indeed, although the gauge

potential in (4.5) is singular on the S2at r = r0, one can easily see that the field

strength F = dA is a globally defined smooth two-form on our manifold. One computes

the period of this through the S2at r0(s) to be

g

2πr0(s)2− s2

= 1 ,

?

S2F= −

2gs

?

−2Q+(s)r0(s) − (r0(s)2+ s2)

?

4g2s2− 1

?

(4.26)

the last line simply being a remarkable identity satisfied by the largest root r0(s).

Setting g = 1, we thus see that we have precisely one unit of flux through the S2! It

follows that the gauge field A is a connection on the non-trivial line bundle O(1) →

T∗S2. The corresponding first Chern class c1 = [F/2π] ∈ H2(T∗S2;Z)∼= Z is the

generator of this group. Moreover, the map H2(T∗S2;Z) → H2(S3/Z2;Z)∼= Z2that

restricts the gauge field to the conformal boundary is reduction modulo 2. Hence at

infinity the background gauge field is more precisely given by the global one-form (4.6)

plus the flat non-trivial Wilson line that represents the element 1 ∈ H2(S3/Z2;Z)∼=

H1(S3/Z2;Z)∼= Z2. One would be able to see this explicitly by writing the gauge

field A as a one-form that is locally well-defined in coordinate patches, and undergoes

appropriate gauge transformations between these coordinate patches. It follows that

the gauge field at infinity is more precisely a connection on the non-trivial torsion line

bundle over S3/Z2.

14

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- Available from James Sparks · May 28, 2014
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