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Gauge theories with time-dependent
couplings and cosmological
singularities
K. Narayan
Chennai Mathematical Institute (CMI), Chennai
[ hep-th/0602107, hep-th/0610053,
Sumit Das, Jeremy Michelson, KN, Sandip Trivedi;
arXiv:0711.2994, Adel Awad, Das, KN, Trivedi;
arXiv:0807.1517, Awad, Das, Suresh Nampuri, KN, Trivedi.]
• basic setup: AdS/CFT with cosmological singularities
• gauge theories with time-dep coupling sources
• Spacelike cosmological singularities, BKL etc
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.1/25
Related references:
Craps, Hertog, Turok, arXiv:0712.4180 [hep-th] ,
Chu, Ho, hep-th/0602054, arXiv:0710.2640 [hep-th],
Hertog, Horowitz, hep-th/0503071 : cosmological generalizations of
AdS/CFT framework.
B. Craps, S. Sethi, E. Verlinde, hep-th/0506180, and followup work by
various people: Matrix theory duals of cosmological singularities.
...
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.2/25
Cosmology, time dependence, . . .
Tempting to think very early Universe has deep repercussions on
various aspects of physics.
• Big Bang singularities, time, in string theory models?
Understand spacelike, null singularities — events in time.
General Relativity breaks down at singularities: curvatures, tidal forces
divergent. Want “stringy” description, eventually towards smooth
quantum (stringy) completion of classical spacetime geometry.
Previous examples: “stringy phases” in e.g. 2-dim worldsheet (linear
sigma model) descriptions (including time-dep versions, e.g. tachyon
dynamics in (meta/)unstable vacua), dual gauge/Matrix theories, . . .
In what follows, we’ll use the AdS/CFT framework.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.3/25
AdS/CFT and deformations
Nice stringy playground: AdS/CFT. Bulk string theory on AdS5 × S 5
with dilaton (scalar) Φ = const, and metric
ds2 = z12 (ηµν dxµ dxν + dz 2 ) + ds2S 5 ,
(Poincare coords) with 5-form field strength, dual to boundary d = 4
N =4 (large N ) SU (N ) Super Yang-Mills theory.
Want: time-dependent deformations of AdS/CFT.
Bulk subject to time-dependent sources classically evolves in time (thro
Einstein eqns), eventually giving rise to a cosmological singularity, and
breaks down. Avoid any bulk investigation near singularity.
Boundary: Gauge theory dual is a sensible Hamiltonian quantum
system in principle, subject to time-dependent sources. Response ?
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.4/25
AdS cosmologies
Start with AdS5 × S 5 and turn on non-normalizable deformations for
the metric and dilaton (also nontrivial 5-form):
ds2 =
1
µ dxν
(g̃
dx
µν
2
z
+ dz 2 ) + ds2S 5 ,
Φ = Φ(xµ ) .
This is a solution in string theory if
R̃µν =
1
2 ∂µ Φ∂ν Φ
,
√1
−g̃
√
∂µ ( −g̃ g̃ µν ∂ν Φ) = 0 ,
i.e. if it is a solution to a 4-dim Einstein-dilaton system.
Time dep: Φ = Φ(t) or Φ = Φ(x+ ) .
More later on cosmological solutions.
General family of solutions: (Z(xm ) harmonic function)
ds2 = Z −1/2 g̃µν dxµ dxν + Z 1/2 gmn dxm dxn ,
Φ = Φ(xµ ),
gmn (xm ) is Ricci flat, and g̃µν = g̃µν (xµ ). [µ = 0123, m = 4 . . . 9.]
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.5/25
AdS cosmologies cont’d
In many cases, possible to find new coordinates such that boundary
metric ds24 = limz→0 z 2 ds25 is flat, at least as an expansion about the
boundary (z = 0) if not exactly.
These are Penrose-Brown-Henneaux (PBH) transformations: subset of
bulk diffeomorphisms leaving metric invariant (in Fefferman-Graham
form), acting as Weyl transformation on boundary.
1
f (x+ ) η dxµ dxν +
(e
µν
z2
w2 f ′
−f
/2
−
−
ze
, y =x − 4 ,
E.g. null cosmologies ds2 =
dz 2 ) , Φ(x+ ).
The coord. transf. w =
gives
ds2
=
1
+ dy −
[−2dx
2
w
+
dx2i
+
1 2
′ )2 (dx+ )2 ]
w
(Φ
4
+
dw2
w2
,
using R++ = 21 (f ′ )2 − f ′′ = 12 (Φ′ )2 , the constraint on these solutions.
Now boundary at w = 0 manifestly flat 4D Minkowski spacetime.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.6/25
Gauge theories with time-dep
couplings
Thus dual gauge theory lives on flat space. So sharp sub-question:
Gauge theory with time-dependent coupling gY2 M = eΦ . Response?
We would like to study sources that are trivial in the far past (bulk is
AdS5 × S 5 ) and smoothly turn on: this means the gauge theory begins
in vacuum state and is subject to Hamiltonian time evolution through
this external time-dependent source. Basic expectation: time-dep
source excites vacuum to higher energy state.
Want to consider sources that approach eΦ → 0 at some finite point in
time: e.g. gY2 M = eΦ → (−t)p , p > 0 [t < 0].
We’d specially like to understand gauge theory response near t = 0.
This point in time corresponds to a singularity in the bulk:
Rtt = 12 Φ̇2 ∼ t12 . Curvatures, tidal forces diverge near t = 0.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.7/25
Gauge theories, time-dep couplings
Gauge theory kinetic terms e−Φ F 2 not canonical.
In usual perturbation theory, we absorb the coupling gY2 M into the
definition of the gauge field Aµ so that gY M appears only in the
interaction terms. Let’s do something similar here.
R
First, consider a simpler toy scalar theory
L = −e−Φ 21 (∂ X̃)2 + X̃ 4 . Redefining X̃ = eΦ/2 X gives
L = −(∂X)2 − m2 (Φ)X 2 − eΦ X 4 ,
dropping a boundary term. The mass term is
m2 (Φ) = 14 ∂µ Φ∂ µ Φ − 21 ∂µ ∂ µ Φ .
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.8/25
Null time-dependence
Null cosmologies: Φ = Φ(x+ ) . No nonzero contraction so the mass
term vanishes i.e. m2 (Φ) = 0.
Similar story for gauge theory using lightcone gauge for convenience.
Suppressing many details, but briefly, cubic/quartic interaction terms:
multiplied by powers of gY M = eΦ/2 , unimportant near eΦ → 0.
Thus we obtain weakly coupled Yang-Mills theory at the location in
null time (x+ = 0) corresponding to the bulk singularity [e.g.
eΦ = gs (−x+ )p ].
This suggests that lightcone Hamiltonian time evolution of the gauge
theory is sensible.
In other words, null cosmological singularities seem to be an artifact of
using bad (classical bulk sugra) variables.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.9/25
Null time-dependence
Is energy pumped in by null-time-dependent source ? No.
Since x− -translations are symmetries, there is in fact no particle
production. This suggests that continuing past singularity at x+ = 0 is
OK, and late-time state is vacuum.
Thus late-time bulk is AdS5 × S 5 (dual to vacuum state of N =4 gauge
theory, with Φ → const for large x+ ).
Bulk: since eΦ → 0 near singularity, no large gs effects. Preliminary
calculations suggest stringy (α′ ) effects (beyond GR) are becoming
important.
In some still simpler toy models with no RR-flux, dilaton, the
singularity is purely gravitational so possibly more tractable.
I am studying these to understand worldsheet effects near null
singularities.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.10/25
Time-dependent couplings
The story is very different with time-dependent couplings.
1
2 + X̃ 4 . The redefinition
(∂
X̃)
2
−(∂X)2 − m2 (Φ)X 2 − eΦ X 4 ,
Toy scalar theory L = −e−Φ
X̃ = eΦ/2 X gives
L=
dropping a boundary term, the mass term being
m2 (Φ) = − 41 (Φ̇)2 + 21 Φ̈ .
.
For gY2 M = eΦ → (−t)p , p > 0 [t < 0] , we have m2 = − p(p+2)
4 t2
Tachyonic mass term, divergent as t → 0.
* X variables canonical: analysing them shows that the mass term
1
, so that extra information as X → ∞ is required:
forces X ∼ tp/2
X description not good.
* X̃ variables finite near t = 0: interaction terms e−Φ X̃ 4 |t∼0 large
(unlike the null case).
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.11/25
Time-dep quantum mechanics
In more detail: first ignore interactions, quantize quadratic theory.
For a single momentum-k mode, this is time-dep quantum mechanics:
R
Sk = dt ( 21 Ẋ 2 − ω 2 (t)X 2 ) , ω 2 (t) = k 2 + m2 (t) −→t→−∞ ω02 .
√
Generic classical solutions: X = −t [AJν (−t) + BNν (−t)] ,
p+1
2
. Diverge as t → 0 : i.e. generic trajectory driven to large X.
p πω0 √
1
¨ + ω2 f = 0 ,
−tH
as
the
solution
of
f
Take f (t) =
(−ω
t)
0
ν
2
ν=
with f → e−iω0 t , t → −∞ . Expand X =
√ 1 [af (t)
2ω0
+ a† f ∗ (t)] .
Using the Schrodinger equation: the ground state wave-function is
ψ(t, x) = √ A∗
f (t)
˙∗
e
2
i( ff ∗ ) x2
.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.12/25
Time-dep quantum mechanics
X:
wave-fn
ψ(t, x) = √ A∗
Wave-fn phase ∼
f (t)
1
t
˙∗
e
2
i( ff ∗ ) x2
,
, “wildly” oscillating near t = 0.
t → 0− : f ∼ (−t)−p .
Probability density: |ψ(t, x)|2 =
|A|2
|f |
e
ω0 x2
−
|f |2
.
Gaussian, width |f |2 → ∞ as t → 0.
Wave packet infinitely spread out as t → 0.
X variables spread out infinitely: need extra information at X ∼ ∞.
X description not good.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.13/25
Time-dep quantum mechanics
Original X̃ = eΦ/2 X variables better defined: finite near t = 0.
√
Φ/2
X̃ = e
−t[AJν (−t) + BNν (−t)] ∼t→0 tp/2 t1/2 t−ν/2 .
Wave-fn, probability:
A
e
f ∗ (t)eΦ/2
ψ(t, x̃) = √
f˙∗
x̃2
i( f ∗ + Φ̇
)
2 2eΦ
t → 0− : f ∼ (−t)−p ,
X̃: wave-fn phase ∼
,
|ψ(t, x̃)|2 =
|A|2
|f |eΦ/2
e
ω0 x̃2
− 2 Φ
|f | e
.
|f |2 eΦ ∼ const .
1
(−t)p−1
,
prob. width const .
p > 1: wave-fn ill-defined near t ∼ 0. “Wildly” oscillating phase.
p < 1: X̃ wave fn phase regular near t ∼ 0 , |ψ(t, x̃)|2 finite.
Quadratic approximation shows interactions are important near t = 0.
Perturbation theory insufficient.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.14/25
Time-dep field theory wave-fn
More general Schrodinger picture analysis in full field theory is
possible near t = 0.
R 3 −Φ 1
L = d x e ( 2 (∂t X̃)2 − 21 (∂i X̃)2 − X̃ 4 ) .
Lagrangian
R 3
1 δ2
−Φ
Φ
Field theory Hamiltonian: H = e V [X̃] + e
d x(− 2 δX̃ 2 ) ,
R 3 1
where V [X̃] = d x ( 2 (∂i X̃)2 + X̃ 4 ) [replacing Π(x) → 1i δδX̃ ] .
Schrodinger eqn:
i∂t ψ[X̃(x), t] = Hψ[X̃(x), t] .
Near t = 0, the potential term e−Φ V dominates in the Hamiltonian ⇒
i∂t ψ = e−Φ(t) V [X̃(x)]ψ . This gives the wave-fn (generic state)
ψ[X̃(x), t] = e
Phase as before ∼
(t → 0).
−i( dt e−Φ(t) )V [X̃(x)]
R
(−t)1−p
1−p V
ψ0 [X̃(x)] .
[X̃(x)] . If p > 1, “wildly” oscillating
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.15/25
Energy divergence
Analyzing KE terms shows they are indeed subleading near t = 0.
This means energy pumped in by time-dep source diverges as
R
1
−Φ
hHi ≃ e hV i = (−t)p DX̃ V [X̃] |ψ0 [X̃(x)]|2 ,
since no time-dep in hV i. Oscillating phase cancels in |ψ0 |2 .
This holds for generic states. For special states with hV i = 0 , energy
may be finite (subleading KE terms do not diverge unless p > 2).
Even for these special states, hH 2 i will diverge (if hV i = 0,
generically hV 2 i does not vanish).
Thus fluctuations non-negligible about states with hV i = 0 .
Note: this is not perturbation theory. Interactions important.
Diverging energy since coupling strictly vanishes near t = 0.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.16/25
The gauge theory
Scalars, fermions: no dilaton coupling in KE terms. Fermion Yukawa
and scalar quartic terms come with powers of gY M = eΦ/2 , vanish
near t = 0.
R −Φ
Gauge fields: KE terms have dilaton coupling e TrF 2 .
Since eΦ = (−t)p near t = 0, the gauge field terms determine the
behaviour of the system near t ∼ 0. Focus on this.
Consider non-interacting theory first.
Convenient (Coulomb) gauge A0 = 0 , ∂j Aj = 0 (longitudinal part of
gauge field time-indep from Gauss law: ∂0 (∂j Aj ) = 0 ).
Residual action for two physical transverse components Ai becomes
R −Φ
e (∂Ai )2 , (i.e. two copies of the scalar theory earlier).
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.17/25
The gauge theory
Cubic/quartic interactions: no time derivatives.
Contribute only to potential energy terms (from magnetic field), not to
R 3
1
i
KE terms (from electric field). PE V [A (x)] = 4 d x TrFij2 .
R
Lg = 41 d3 x e−Φ Tr (∂t Ai )2 − Fij2 .
Schrodinger quantization: E i →
i
ψ[A (x), t] = e
1 δ
i δAi
R
−i( dt e−Φ )V [Ai (x)]
. Then wave-fn near t = 0:
ψ0 [Ai (x)] .
Wave-fn phase as before: “wildly” oscillating as t → 0 (for p > 1).
R
−Φ
Energy diverges hHi ≃ e
DAi V [Ai (x)] |ψ0 [Ai ]|2 (if hV i =
6 0) .
Thus if eΦ → 0 strictly, gauge theory response singular.
For cutoff eΦ , large energy production due to time-dep source.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.18/25
AdS cosmologies with spacelike
singularities
Recall:
1
µ dxν + dz 2 ) + ds2 , Φ = Φ(xµ ) .
(g̃
dx
µν
2
S5
z
√
1
1
R̃µν = 2 ∂µ Φ∂ν Φ , √−g̃ ∂µ ( −g̃ g̃ µν ∂ν Φ) = 0
ds2 =
Solution if:
.
Solutions with spacelike
Big-Bang (Crunch) singularities:
h
i
P
*
ds2 = z12 dz 2 − dt2 + 3i=1 t2pi (dxi )2 ,
√
P 2
P
p
)
2(1−
Φ
i i ,
[Kasner cosmologies]
e = |t|
i pi = 1 .
i
h
2
dr
2
2
2
2
* ds2 = z12 dz 2 + | sinh(2t)|(−dt2 + 1+r
2 + r (dθ + sin θdφ )) ,
√
eΦ = gs | tanh t|
3
.
[k = −1 (hyperbolic) FRW boundary]
Dilaton bounded, approaching constant at early/late times: asymptotic
spacetime is AdS5 × S 5 (using a coord transformation).
The k = 0 (flat) FRW is the same as symmetric Kasner (pi = 31 ).
(There is also a k = +1 (spherical) FRW solution.)
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.19/25
AdS BKL-cosmologies
In fact, larger family of cosmological solutions where spatial metric is
one of the homogenous spaces in the Bianchi classification:
i
h
ds2 = z12 dz 2 − dt2 + ηab (t)(eaα dxα )(ebβ dxβ ) , eΦ = eΦ(t) .
eaα dxα are a triad of 1-forms defining symmetry directions. Spatially
a vanish, and R0 = 1 (∂ Φ)2 .
homogenous dilaton means spatial R(a)
0
2 0
i
h
Bianchi-IX: ds2 = z12 dz 2 − dt2 + ηi2 (t)eiα eiβ dxα dxβ , eΦ = |t|α .
Approximate Kasner-like solution ηi (t) ≃ tpi with
P 2
P
α2
i pi = 1 − 2 .
i pi = 1 ,
If all pi > 0, cosmology “stable”. Else, spatial curvatures force BKL
bounces between distinct Kasner regimes. With each bounce, α
increases — dilaton-driven attractor-like behaviour.
Attractor basin: generic Kasner-like solution with all pi > 0.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.20/25
More on AdS BKL-cosmologies
Bianchi IX: symmetry algebra of Xa = eαa ∂α is SU (2).
a
Spatial Ricci, decomposing along triad R(a)
= Rαa eαa :
1 =
R(1)
∂t (η2 η3 ∂t η1 )
η1 η2 η3
−
1
2
[(η
2
2
2(η1 η2 η3 )
− η32 )2 − η14 ] = 0 ,
....
Say p1 < 0: then η14 ∼ t−4|p1 | non-negligible at some time. This forces
metric to transit from one Kasner regime to another. As long as some
pi < 0, these bounces continue as:
(n+1)
pi
(n)
=
−p−
(n)
1+2p−
,
(n+1)
pj
(n)
=
(n)
p+ +2p−
(n)
1+2p−
,
α(n+1) =
αn
(n)
1+2p−
,
for the bounce from the (n)-th to the (n + 1)-th Kasner regime.
−2p−
If p− < 0 , then αn+1 > αn . Also αn+1 − αn = αn ( 1+2p
),
−
i.e., α increases slowly for small α: attractor-like behaviour. Finite
number of bounces. If all pi > 0, no bounce: cosmology “stable”.
For no dilaton (α = 0), BKL bounces purely oscillatory.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.21/25
More on AdS BKL-cosmologies
Parametrization: p1 = x, p2,3 =
√
2
1−x
2
±
√
1−α2 +2x−3x2
2
.
4
2
.
Solution
existence
forces
α
≤
Lower bound: p1 ≥ 1− 4−3α
3
3 .
Under bounces, α increases, window of allowed pi shrinks. Lower
bound hits p1 ≥ 0 ⇒ α2 ≥ 1. Bounces stop, cosmology “stabilizes”.
Attractor-like behaviour: e.g.: {p01 = x0 = 0.3, α0 = 0.001}, flows
(initially slowly) to {pi > 0} after 15 oscillations (α15 = 1.0896).
9 33
5 7 19
3 5 9
E.g.: (− 15 , 35
, 35 ) → (− 21
, 21 , 21 ) → (− 11
, 11 , 11 ) →
(− 15 , 35 , 53 ) → ( 31 , 13 , 31 ) . [multiple flows with same endpoint]
Chaotic behaviour: 7% change to smallest exponent − 15 gives
13 9 65
2 13 39
3 2 23
1 3 17
(− 70
, 35 , 70 ) → (− 11
, 44 , 44 ) → (− 28
, 7 , 28 ) → ( 11
, 22 , 22 ) ,
drastically different endpoint.
Note also that dilatonic (α 6= 0) [attractor-like] and non-dilatonic
(α = 0) [oscillatory] flows drastically different.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.22/25
Universal behaviour near
singularities
Consider symmetric Kasner-like AdS BKL-cosmologies. Near
singularity, spatial curvatures unimportant. Leading singular behaviour
is essentially dilaton-driven, symmetric Kasner spacetime. Holographic
N2
stress tensor has similar leading behaviour (Tµν ∼ t4 ).
Consider families of such AdS cosmologies which are of the form of
the symmetric Kasner-like solution i.e. pi = 13 : (ds23 spatial metric)
√
2
1
2
2
2
Φ
ds = z 2 dz + |2t|(−dt + ds3 ) ,
e = |t| 3 .
Ignoring subleading curvature effects, spatial metric approximately flat
i.e. ds23 ∼ f lat. Then boundary metric is conformally flat, to leading
order. [we’ve used a different time coordinate here.]
Can use PBH transformations to recast boundary metric to be flat
spacetime.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.23/25
The gauge theory
Now
gY2 M
=
eΦ
√
= (−t)
3
√
(t < 0). That is, p = 3 > 1 .
From earlier: wave-fn phase “wildly” oscillating, ill-defined.
Energy production divergent if coupling vanishes strictly near t = 0.
* In gauge theory, deform gauge coupling so that gY2 M = eΦ is small
but nonzero near t = 0. Now finite but large phase oscillation, finite
but large energy production.
Eventual gauge theory endpoint ? Depends on details of energy
production at coupling O(1).
On long timescales, expect gauge theory thermalizes: then reasonable
to imagine that late-time bulk is AdS-Schwarzschild black hole.
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.24/25
Conclusions, open questions
* If gY2 M (t) → 0 strictly, then gauge theory response singular: energy
diverges. Deform gY2 M to be small but nonzero near t = 0. Now finite
but large phase oscillation and energy production.
finite now, so bulk also nonsingular.
Φ̇ ∼ gġYY M
M
Sugra may still not be valid of course.
* Gauge theory on S 3 : We’re investigating this and other issues
currently (in part with Archisman Ghosh, Jae Oh).
* Explore AdS BKL-cosmologies/duals further
...
Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.25/25