Table Of ContentWarped AdS , dS and flows from N = (0,2) SCFTs
3 3
Eoin O´ Colg´ain1,2
1C.N.Yang Institute for Theoretical Physics, SUNY Stony Brook, NY 11794-3840, USA
2Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
We present the general form of all timelike supersymmetric solutions to 3D U(1)3 gauged su-
pergravity, a known consistent truncation of string theory. We uncover a rich vacuum structure,
including an infinite class of new timelike-warped AdS (Go¨del) and timelike-warped dS critical
3 3
points. We outline the construction of supersymmetric flows, driven by irrelevant scalar operators
in the SCFT, which interpolate between critical points. For flows from AdS to Go¨del, the natural
3
candidate for the central charge decreases along the flow. Flows to timelike-warped dS exhibit
3
topology change.
5
1 INTRODUCTION ticclosed-timelike-curves(CTCs),signalingabreakdown
0 in unitarity in the dual theory. Along with the G¨odel
2 universe [26], which is not ruled out by supersymmetry
ItisaremarkablefactthatthegeometricalBekenstein-
y Hawking (BH) entropy of black holes with AdS near- [27, 28], a version of Hawking’s Chronology Protection
3
a Conjecture [29] is expected for timelike-warped dS . See
horizonscanbederivedfromthecentralchargeofatwo- 3
M [30–33] for related works in the AdS/CFT context.
dimensional(2D)CFT[1,2]. Thisresult, akeyprecusor
to AdS/CFT [3], rests on the Brown-Henneaux analysis We construct numerical supersymmetric flows from
5
AdS to timelike-warped critical points and identify the
1 of the asymptotic symmetries of AdS3 [4] and the Cardy 3
flowsasdeformationsofthe2DSCFTbyirrelevantscalar
formula [5], which permits one to determine the asymp-
] toticdensityofstatesinaCFTinthesemi-classicallimit. operators. We show that the inverse of the real superpo-
h
tential monotonically decreases along flows to timelike-
t It is well-known that Kerr black holes, candidates for
- warped AdS vacua and calculate an expression for the
p astrophysical black holes, e.g. Cygnus X-1 [6], exhibit 3
candidate central charge in terms of twist parameters.
e warped AdS near-horizons [7]. In recent years, the
h matchingofB3HentropythroughtheCardyformulalead For flows to timelike-warped dS3, the curvature of the
[ Riemann surface changes sign and the topology changes.
to a bold conjecture that there is a (warped) CFT dual
Since the 2D SCFTs correspond to twisted compactifi-
3 to Kerr black holes [8] (see [9] for a review). A greater
v cations of N = 4 super-Yang-Mills, our 3D flows can be
understanding of the putative dual QFT, if it is even a
5 uplifted to 5D, where they may be interpreted as defor-
CFT [10–14], requires a theory with a UV completion,
5 mations of N = 4 super-Yang-Mills. This short letter
3 such as string theory.
highlights the existence of novel warped critical points;
4 In this work we take a step in this direction by iden-
further examples of supersymmetric flows, the 5D uplift
0 tifying warped AdS vacua of N = 2 U(1)3 gauged su-
. 3 andthegeneralisationtonullspacetimescanbefoundin
1 pergravity [15], a consistent truncation of string theory
[34].
0 [16, 17], and offering evidence that they can be con-
5
nected to understood AdS vacua by supersymmetric
1 3
: flows. This places holography on a firmer footing, since 3D U(1)3 GAUGED SUPERGRAVITY
v at one end of the flow, the supersymmetric AdS vacua
3
i
X are dual to 2D N =(0,2) SCFTs [18–21], whose central
We consider 3D N = 2 gauged supergravity [15],
charge and R symmetry can be determined exactly us-
r which uplifts on a constant curvature Riemann surface
a ing c-extremization [20, 21] and agree with holographic
of genus g, Σ , to well-known 5D U(1)3 gauged super-
calculations (see [22] for subleading terms). g
gravity,whereitcanbefurtherembeddedconsistentlyin
Following a review in the next section, we make the
higher dimensions [16, 17]. Examples of consistent trun-
following novel contributions. Firstly, we present the
cations of string theory with warped AdS vacua have
3
general form - dictated by the bps conditions - of all
appeared previously in [24] (see also [25]).
supersymmetric timelike solutions to 3D U(1)3 gauged
The action for the theory may be written as
supergravity, including an infinite class of new half-bps
critical points, going under the moniker timelike-warped 3
AdS (G¨odel) and timelike-warped dS in the literature. L = R∗ 1− 1(cid:88)(cid:2)dW ∧∗ dW +e2WiGi∧∗ Gi(cid:3)
3 3 3 3 2 i 3 i 3
Indeed, the latter is a known solution to Topologically i=1
Massive Gravity with a positive cosmological constant (cid:32) 3 (cid:33)
(cid:88)
[23] and here we provide potentially the first example in + 8 T2− (∂WiT)2 ∗31
both a supersymmetric and string theory context. Be- i=1
ing timelike-warped, the geometries exhibit characteris- − a B2∧dB3−a B3∧dB1−a B1∧dB2, (1)
1 2 3
2
with the field content comprising three scalars, W , and The field strengths Gi are completely determined and
i
three gauge fields, Gi = dBi, which may be rewritten break supersymmetry by one-half when non-zero,
in the canonical form of 3D gauged supergravity [35]. T
Gi =e−Wi(−4∂ T ∗ P +P ∧dW ), (7)
denotes the superpotential Wi 3 0 0 i
meaning Gi =0 at AdS vacua. This appears to contra-
3 (cid:18) (cid:19) 3
T =(cid:88) 1e−Wi − aieWi+K , (2) dict the existence of well known holographic flows from
2 4 AdS to AdS [18], but in our conventions these fall into
i=1 5 3
the null class of spacetimes, P ·P =0.
a b
(cid:80)
K =− W is the K¨ahler potential of the scalar man- Given these conditions, it is a straightforward exercise
i i
ifold, and ai, i = 1,2,3 denote constants that are con- to introduce coordinates P0 ≡ ∂τ, Pz = eD−21K(dx1 +
strained by the curvature κ of Σg idx2), so that the spacetimes take the form:
a +a +a =−κ. (3) ds2 = −(dτ +ρ)2+e2D−K(dx2+dx2),
1 2 3 3 1 2
(cid:20)
WenotethereisthefreedomtochangethesignofT and Gi = e−Wi −4∂WiTe2D−Kdx1∧dx2
the potential does not change [54].
(cid:21)
ThroughAdS/CFT[3],AdS3vacuaoftheabovesuper- +(dτ +ρ)∧dWi , (8)
gravitycorrespondto2DSCFTsarisingthroughtwisted
compactifications of 4D N = 4 super Yang-Mills with whereρisaone-formconnectionontheRiemannsurface
gaugegroupU(N)onΣg [36,37]. Topreservesupersym- parametrised by (x1,x2), with dρ=4Te2D−Kdx1∧dx2.
metry one“twists” thetheory by turningon gaugefields Here D(x ,x ) is modulo a convenient factor of the
1 2
coupled to the SO(6) R symmetry of the 4D theory. For K¨ahler potential, a warp factor parametrising the vec-
twists involving the SO(2)3 Cartan subgroup of the R tor P , and in turn the Riemann surface. Inserting
z
symmetry, the twist parameters, ai, must satisfy (3), a the expressions for Gi into the flux equations of motion
necessary condition for N = 2 supergravity. Supersym- (EOMs), one can derive the following equation for the
metry is enhanced to N = (2,2) and N = (4,4), when scalars:
one or two of the a vanish, respectively.
i (cid:20)4T (cid:88) (cid:89) (cid:21)
Supersymmetric AdS3 vacua of the action (1) corre- ∇2eWi =2e2D eK − aj(eWj +eWi)+ aj . (9)
spond to the critical points, ∂ T =0 [25],
Wi j(cid:54)=i j(cid:54)=i
(cid:81) a From(6),onederivesadifferentialconditionforthewarp
eWi =− j(cid:54)=i j. (4) factor,
κ+2a
i
3
Thesevacuahavefeaturedinaseriesofworks[18–21,38]. ∇2D =4(cid:88)e−Wi(∂ T +T)e2D−K. (10)
From extrema of T, one can see there is no good AdS Wi
3 i=1
vacuum dual to N = (4,4) SCFTs and that N = (2,2)
We note for a given constant value of W , this equation
vacua only exist when g>1. i
reducestotheLiouvilleequationontheRiemannsurface,
∇2D =−Ke2D, with Gaussian curvature K.
ALL TIMELIKE SOLUTIONS Using these four equations, it is possible to show that
the scalar and Einstein EOMs are satisfied. It can be in-
Given a supergravity theory, it is feasible to invoke dependently checked that the EOMs are consistent with
Killing spinor techniques to find all supersymmetric so- the integrability conditions [34], as expected.
lutions, e. g. [27, 39, 40] in 5D. Here we present the
NEW CRITICAL POINTS
timelike solutions to the theory (1). Full details of the
classification exercise appear elsewhere [34].
Inthissection,wegetorientedbyrecoveringtheAdS
The class of supersymmetric geometries is charac- 3
vacua (4). For simplicity, we introduce a radial direction
terised by a real timelike Killing vector P0, LP0Wi = r = (cid:112)x2+x2 and a U(1) isometry parametrised by ϕ.
L Gi = 0 and an additional complex vector, P = 1 2
P0 z A general solution to (10) exists where
P +iP . Suitably normalised, we have P ·P = η ,
1 2 a b ab
ηab =(−1,1,1)anda=0,1,2. TheexistenceofaKilling 2(cid:112)|K|
spinor is equivalent to the differential conditions [34]: eD = , (11)
|K|+Kr2
dP = 4T ∗ P , (5) resulting in a spacetime metric of the form:
0 3 0
3
e−12Kd[e12KPz] = (cid:88)(cid:0)e−Wi ∗3+iBi(cid:1)∧Pz. (6) ds2 = −(cid:96)2(cid:18)dτ − sgn(K)r2 dϕ(cid:19)2
3 [1+sgn(K)r2]
i=1
3
1.0 R = 2(4T2 +KeK), changes sign as one crosses this lo-
0.8 cus,anobservationthatjustifiesthebilling“deSitter”in
the internal region, but not de Sitter in the conventional
0.6
sense, since the geometry is supersymmetric. Uplifting
0.4 thewarpedcriticalpointsto10Dor11D[16,17]onecan
showthatCTCsappearforlargevaluesofr[34]. Finally,
0.2
againsuppressinga ,weremarkthatthereisanexternal
1
(cid:45)0.4 (cid:45)0.2 0.2 0.4 0.6 0.8 1.0 locus,illustratedinFIG.2,wherecriticalpointscoalesce
(cid:45)0.2 and only the supersymmetric AdS3 vacuum exists
(cid:45)0.4 −1+2a −a2±(cid:112)a −2a2+a4
a = 3 3 3 3 3. (15)
2 2(a −1)
3
FIG. 1: The range of parameters in the (a ,a ) plane where
2 3
thescalarsWiremainrealforAdS3vacua(cream)andwarped At (14) the Gaussian curvature may be written:
AdS vacua (green). The dotted red line separates external
3
(K<0) from internal regions (K>0). K=2(a a +a a +a a )−a2−a2−a2 = 2a1a2a3. (16)
1 2 2 3 3 1 1 2 3 (cid:96)
e−K (cid:20)4(dr2+r2dϕ2)(cid:21) For K < 0, the critical points are easy to identify and
+ . (12) correspond to supersymmetric G¨odel spacetimes [26], a
|K| (1+sgn(K)r2)2
healthy collection of which can be found in 3D [41–44].
For AdS3 vacua, K = −4Te−K(cid:80)ie−Wi|∂WiT=0, which To see this, we can recast the solution in the form [45]
uponredefinition,r =tanhρandashiftϕ→ϕ−τ,leads
to the usual form of global AdS3 (radius (cid:96)= T2|∂WiT=0), ds23 = −(cid:18)dτ + m4Ω2 sinh2(cid:16)m2ρ(cid:17)dϕ(cid:19)2
ds2 =(cid:96)2[−cosh2ρdτ2+dρ2+sinh2ρdϕ2]. (13) sinh2(mρ)
3 +dρ2+ dϕ2, (17)
m2
Wenowpresentakeyobservationofthisletter,namely
that (9) has a second critical point, i. e. solutions with where in our notation, one has ρ = 2 tanh−1(r), Ω =
m
∂ W = 0, supported by fluxes. In addition to (4), the (cid:96)|K|eK| , m2 = |K|eK| . Written in the above form
a i 4 crit crit
RHS of (9) vanishes when (17), the homogeneity and causal structure of the G¨odel
solution holds in the range 0 ≤ m2 < 4Ω2 [46], with
(cid:81)
eWi =(cid:88)a + κ + j(cid:54)=iaj. (14) the original G¨odel solution at m2 = 2Ω2 and AdS3 at
j 2 κ m2 = 4Ω2. We plot 4Ω2 − m2 in FIG. 2, noting the
j(cid:54)=i
yellow (zero valued) AdS locus and steadily increasing
3
Thisexhauststhepossibilityforadditionalcriticalpoints contours outwards towards the boundaries.
beyond the supersymmetric AdS vacuum. For W ∈R,
3 i 2.0
a requirement for real solutions, necessarily κ < 0, so
without loss of generality we set κ = −1. Furthermore, 1.8
the range in parameter space where good vacua is con-
1.6
strained,asdepictedinFIG.1. FromFIG.1,suppressing
a throughthesupersymmetryconditiona =1−a −a ,
1 1 2 3 1.4
we recognise that within the range of parameters where
good AdS vacua exist (cream), there are regions where 1.2
3
additional new critical points exist (green). Points in
1.0
parameter space where supersymmetry is enhanced to
N = (2,2) (K = 0) e. g. (a ,a ) = (1,0),(0,1),(1,1) 0.8
2 3 2 2 2 2
on the dashed red locus are excluded, meaning new crit-
0.6
ical points only exist for N =(0,2) supersymmetry.
As one crosses the dashed red locus in FIG. 1, the (cid:45)4 (cid:45)3 (cid:45)2 (cid:45)1 0
topology of the Riemann surface parmetrised by (x1,x2) FIG. 2: A contour plot in the (a2,a3) plane of stretching,
changes from H2 externally to S2 internally. We note 4Ω2−m2,inasampleK<0region. Yellowcurvecorresponds
that (cid:96)2eK|K| ≥ 0 for critical points, so that the time- to the AdS3 locus.
4
like fibration in the metric (12) is stretched, and thus
warped. This inequality is saturated only for the su- For N = (0,2) SCFTs, the central charge is propor-
persymmetric AdS vacua, where no stretching occurs. tional to T−1 [15], making it the natural candidate for a
3
Moreover, the Ricci scalar of the overall 3D spacetime, holographic c-function [47, 48]. Indeed, T−1 > T−1 ,
AdS3 Go¨del
4
so for flows from AdS to G¨odel, this observation sug- tion that the periods of the first Chern class be integer
3
gestsananalogueofZamolodchikov’sc-theorem[49]. Re- valued, or
cent work [14] on the asymptotic symmetries of warped
1 (cid:90)
AtwdoS3c,opiniecsludofintgheG¨oVdierla,sodreomosynmstmraettersy,thraetsuolntiengcainnfitnhde 2π ΣgdAi =2ai(g−1)∈Z. (19)
expected central charge c = 3(cid:96) for a 2D CFT. Unfortu-
2G For g > 1 (κ = −1), where new critical points exist,
nately, the analysis in [14] and earlier [50] only holds for
this constraint poses little obstacle since we can ensure
warpedAdS vacuawherethecosmologicalconstantdoes
3
that the regions in FIG. 1 are populated by increasing
not change. In the language of 3D gauged supergravity,
the genus.
this means we are confined to warped AdS vacua that
3
co-exist with unwarped AdS partners at constant value
3
of the scalars in the potential.
SUPERSYMMETRIC FLOWS
Here,oursettingismoregeneral,sincethevacuaexist
at different values of the scalar fields, and these results
In this section we focus solely on parameters in the
donotapply. Itisanopenproblemtorepeattheanalysis
internal region of FIG. 1, where the timelike-warped de
ofRef. [14]toseehowthecentralchargedependsonthe
Sitter vacua exist, and construct a sample numerical so-
scalar potential. Given the limitations of the literature,
lution to show flows from N = (0,2) fixed-points exist.
it is fitting to speculate that the inverse of the superpo-
In this region topology changes from H2 to S2, and T
tential, asintheAdS case, istherelevantquantitythat
3
changes sign making its c-function interpretation prob-
encodes the central charge of the dual QFT. On this as-
lematic. Wenotethatlinearising(9)aboutitsAdS val-
sumption [55], we can determine c at G¨odel fixed-points 3
ues, thereisaninstabilitytoflowsinthedirectionofthe
in terms of twist parameters of N =4 super-Yang-Mills:
timelike-warped dS point. In contrast, flows to G¨odel
3
3 3 are perturbatively stable.
(cid:89) 1 (cid:88)
c=3|g−1|N2 (2a2− a2). (18)
a i k
i
i=1 k=1 1.0
ItwillbeinterestingtorepeattheanalysisofRef. [14]to
0.8
determine the central charge for gauged supergravities.
When K > 0, little is known about these solutions,
0.6
other than they exist as solutions to Topologically Mas-
sive Gravity [23] and suffer from CTCs. Since they are
0.4
topologicallyR×H2,itispossiblethattheycanbeana-
lyticallycontinuedalongthelinesof[51]togivespacelike-
0.2
warpedAdS withtopologyS1×AdS ,ontheprovisowe
3 2
changethesignofT. Wenowshowthatthisisnotpossi-
0.2 0.4 0.6 0.8 1.0
ble. To see this, we send ds2(S2)=−ds2(AdS ) through
2
redefining r →iρ, ϕ→iτ˜ and τ →iϕ˜. This would leave FIG. 3: Supersymmetric flow for (a ,a ,a ) = ( 3 , 3 ,2),
1 2 3 10 10 5
us with a signature problem, however this is overcome interpolating between AdS3 values W1 = W2 = 130 (Red),
by Wi → W˜i +iπ, an analytic continuation that allows W3 = 290 (Blue) at r = 0 and warped AdS3 values, W1 =
(with ai → −ai) flip the sign of T. Note, this leaves W2 = 225, W3 = 1100, at r =1. The Green curve corresponds
(9), (10) and (16) unchanged. From the uplifted 5D per- to D.
spective, this analytic continuation sends κ → −κ, thus
Given a sample point in this region (a ,a ,a ) =
changing the genus g of the Riemann surface used in the 1 2 3
( 3 , 3 , 4 ), we can use a shooting method, i.e. vary-
5D to 3D reduction. Unfortunately, the price one pays 10 10 10
forthisoperationisthatGi becomescomplex, sotheso- ing the initial conditions in the vicinity of r = 0, so
thattheinterpolatingsolutionarrivesatthesecondcrit-
lution is not real. One can potentially overcome this by
ical point at r = 1. We have checked that the output
sending T → iT, but then one sacrifices the consistent
of mathematica in FIG. 3 leads to an error of order
truncation, essentially by complexifying the theory.
1 × 10−7 when reinserted in the EOMs over the same
Before outlining the construction of numerical solu-
range. Stiffness is encountered beyond r = 1, but this
tions in the next section, we end with a final remark
that we have only discussed classical supergravity vacua is due to T blowing up as eWi → 0. We have lin-
earised the scalar EOMs about the AdS vacuum to
and the a should be quantised. To see this, we recall √ 3
i
that the embedding in string theory is through a U(1)3 extract the masses, m2φ±(cid:96)2 = 12(4 ± 3 3), for scalars
fibration of S5. For each U(1) isometry ∂ϕi, the cor- φ± = ±√13W1 + 12(1∓ √13)W3, and they are consistent
responding gauge field, Ai, must be a connection on a with one relevant and one irrelevant operator. Tracing
bona fide U(1)-fibration. This is equivalent to the condi- the fluctuation to the boundary of AdS at r =1, it can
3
5
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i 3
withequation(3)unchanged,andanalyticallycontinuing
W =W˜ +iπ, with W˜ ∈R. Equation (3) is unaffected.
i i i
