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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.
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1 Introduction
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.
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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
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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).
3 The 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
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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].
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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)
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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)
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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)
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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).
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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)
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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).
4 A 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.
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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)
12
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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