4.4 Time-like and null solutions in the bulk
4.4.2 Null case
In the null case, f = 0 and the bulk metric can be written as [74]
ds2 = −2 ˆH−1du dy + 12F du + ˆH2γmndxmdxn, (4.91)
where ˆH, γmn and F depend only on u and xm, m = 1, 2, 3, but not on y. Here
V = ˆH−1du (4.92)
and by comparison with (4.76) we see that ˆH−1 = r2H−1 + . . . , and z = H−1du, in agreement with the results of section 4.2.3.
Explicit asymptotically locally AdS solutions in the null case are also discussed in [74]. These are the magnetic string solutions of [111, 112]. The boundary is R1,1 × M
2, with
metric (after some obvious rescaling)
ds2 = 2 dudy + ds2(M2) , (4.93)
and the gauge field is
F = −k
2vol(M2) . (4.94)
Here, M2 is S2 if k > 0 (with radius k−1/2), T2 if k = 0, or the hyperbolic space H2 if
k < 0 (with radius (−k)−1/2). The bulk space-time has a regular horizon when k < 0, while it has a naked singularity when k > 0. Setting H = 1, F = 0, we find that the formulae in our section4.2.3 are consistent with v = 0 and F = da.
Notice that these bulk solutions can be easily Wick-rotated to Euclidean signature, giving boundary metrics onR2× M2, orT2× M2. In the case M2 =H2, the Wick-rotated
bulk solution is non-singular, and interpolates between Euclidean AdS5 asymptotically
and H3×H2 in the interior. Indeed in [62] these four-dimensional geometries were shown
Chapter 5
The geometry of N = 2 in three
Euclidean dimensions
A big part of the excitement that recent years have seen about exact results in super- symmetric field theories has been due to the ability of computing partition functions of various theories on three dimensional curved spaces, see the discussing in the introduc- tion & motivations chapter. One in the context of holography particularly interesting result are the gravity duals [113–116] to field theories on various squashed three-spheres [117, 118] and Lens spaces . These are amongst the first examples of AdS/CFT dualities with a conformally non-flat boundary.
In this chapter we want to address the same question as in the last two chapters, namely on which general curved spaces we can define a supersymmetric field theory with some residue fermionic symmetry. Our interest will be in Euclidean space-times and three dimensional N = 2 theories. The analysis will parallel closely the one of chapters 3 and
4and we will hence be brief with the introduction.
Again we will start with the superconformal case. We have seen in section2.2that also in three dimensions, holography tells us that supersymmetry is equivalent with the exis- tence of a charged conformal Killing spinor (2.21). Alternatively, following the strategy of [43] we can couple the theory directly on the boundary to three dimensional confor- mal supergravity [119]. In consistency with the holographic approach, the condition for unbroken supersymmetry is again the existence of a conformal Killing spinor. We find that such a spinor exists on every manifold with dreibein ei, that fulfils the condition
do = W ∧ o where o = e1+ ie2 and W some one-form. The condition we find is very simi-
to what is known in the mathematic literature as transversely holomorphic foliation with transversely hermitian metric.
We can also get the three dimensional analogue of new minimal supergravity by di- mensional reduction of (3.31) and study the condition for unbroken supersymmetry with- out necessary conformal invariance.1 As it turns out the geometrical condition that one finds is exactly the same as for the conformal case. We also determine the values of the background fields in terms of the geometrical data. As an application for our formulae we discuss various examples, such as round and squashed spheres, which had been so popular in the context of supersymmetric partition functions.
The rest of the chapter is as follows. In section5.1 we classify manifolds with charged conformal Killing spinors in 3d. In section 5.2 we discuss the reduction of new mini- mal supergravity to three dimensions and solve again the supersymmetry equations. To illustrate our findings we provide various examples.
This chapter is based on the paper [1], a detailed analysis with some overlap has also appeared in [58].
5.1
Geometry of conformal Killing spinors
In this section, we will deal with equation (3.8) in d = 3. The arguments are very similar to those in d = 4, and we will be brief.
Given a spinor χ, we can complete it to a basis with its complex conjugate:
χ , χC ≡ Cχ∗ , (5.1)
where C−1σmC = −σmT. Any nowhere-vanishing χ defines an identity structure. If it is
charged this is true up to a phase. We can indeed construct the bispinors χ ⊗ χ†= 1 2e B (e3− ivol3) , χ ⊗ χ = − i 2e B o (o ≡ e1+ ie2) , (5.2)
where ea are a vielbein for the metric on M3. We defined χ = χTC−1. Notice that in
odd dimensions the map between bispinors and forms is not bijective; a bispinor can be identified both with an even or an odd differential form. In writing (5.2) we opted for odd forms. In terms of this vielbein, one can also show
σmχ = em3 χ − iomχC , m = 1, 2, 3 . (5.3)
We can now define “intrinsic torsions” by expanding ∇mχ in the basis (5.1):
∇mχ ≡ pmχ + qmχC . (5.4)
Alternatively, we can simply use the “anholonomy coefficients” ca
bc defined by dea =
ca
bceb∧ ec. It is more convenient to work with e3 and o = e1 + ie2, and to organise the
ca bc as
de3 ≡ Re(w1e3∧ o) + iw2o ∧ ¯o ,
do ≡ w3e3∧ ¯o + w4o ∧ ¯o + w5e3∧ o .
(5.5)
Here, w2is real, while all the other wi are complex, which gives a total of nine (which is the
correct number for the ca
bc). Together with dB, these are in one-to-one correspondence
with the p and q in (5.4):
dB = 2Rep , w1 = −2i¯q · e3 , w2 = Re(q · ¯o)
w3 = iq · o , w4 = −iImp · o , w5 = iq · ¯o + 2iImp · e3 .
(5.6)
We are now ready to impose (3.8). Using (5.4) and (5.3), we get
2pA· e3 = iq · ¯o , pA· o = −2iq · e3 , pA· ¯o = 0 = q · o . (5.7)
The first three simply determine A. The last can be written as w3 = 0, which means that
the sole geometrical constraint is that
do = w ∧ o (5.8)
for some w, in analogy to (3.26).