Recent progress in
2d Causal Dynamical Triangulations
Yuki Sato(KEK  Wits Univ., SA)
September 29th @ KEK Workshop

J. Ambjørn, L. Glaser, Y. S. and Y. Watabiki, Physics Letters B 722, 2013
J. Ambjørn, L. Glaser, A. Görlich and Y.S., Phys. Lett. B 712, 2012

Quantum Gravity
Question: How did the Universe at the very beginning look like?

Its energy was about 1019 GeV (short scale).
Its dynamics was governed by strong gravity.

Physics at short scale  quantum mechanics
Gravitational physics at large scale  general relativityTo answer the question,
one needs to combine two theories in a consistent manner…
 Quantum gravity
However, naïve construction doesn’t work
because quantum fluctuations cannot be handled.
Non-renormalizabe

Causal Dynamical Triangulations (CDT) A non-perturbative way to quantise Einstein gravity

lattice

Small scale structure
regularized by CDT

UVIR

Outcome:
1. 4d de Sitter Universe pops up(J. Ambjørn, Jurkiewicz and R. Loll, 2004)

2. 2nd-order phase transition(J. Ambjørn, A. Gorlich, S. Jordan, Jurkiewicz and R. Loll, 2010)

Causal Dynamical Triangulations (CDT) A non-perturbative way to quantise Einstein gravity

latticeQuestions:(1) What’s the field theory at IR?
(2) Matter coupling?

Small scale structure
regularized by CDT

UVIR

Outcome:
1. 4d de Sitter Universe pops up(J. Ambjørn, Jurkiewicz and R. Loll, 2004)

2. 2nd-order phase transition(J. Ambjørn, A. Gorlich, S. Jordan, Jurkiewicz and R. Loll, 2010)

2d model is a nice playground because…
 (1) CDT can be solved analytically only in 2 dimensions (so far).
 (2) 2d CDT may be a ``good’’ toy model for 4d CDT.

Recent progress in 2d CDT:

1. 2d CDT and 2d Horava-Lifshitz gravity are in the same universality class.
J. Ambjørn, L. Glaser, Y. S. and Y. Watabiki, 2013.

2. A generic method for coupling matters to 2d CDT has been proposed.
J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.

CDTHorava-Lifshitz

2d CDT J. Ambjørn, R. Loll, 1998
Lorentzian triangle lattice:
spacetimelattice spacing  fixed
space

space

triangulations (T)  dynamical(= how to divide geometry by triangles)

time
Metric path-integral  sum over triangulations (T):

2d CDT J. Ambjørn, R. Loll, 1998
Discrete action for CDT:
Area ()=#()=discretise
Lorentzian  Euclidean
rotation to Euclidian

After the rotation, CDT amplitude becomes: dimensionless cosmological constant2d CDT J. Ambjørn, R. Loll, 1998

Continuum limit&Quantum Hamiltonian2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013

isotropic limit

2D projectable H0rava-Lifshitz gravity (HL):
projectable lapse

where&“Hamiltonian constr.”

momentum constr.

2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013
Solve momentum constraint  System reduces to be of 1 dimension

Gauge fixing spatial Diff

Quantise the 1d system based on the following action:(from Hamiltonian constr.,)

2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013
Quantum amplitude (after a rotation to Euclidean signature):

whereintegrate

expandcompleteness:

2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013
Quantum amplitude (after a rotation to Euclidean signature):

whereintegratecompareexpandcompleteness:

2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013
Quantum Hamiltonian for HL

What is a ?
Completeness:

a=1

a=0

2d HL J. Ambjørn, L. Glaser, YS, Y. Watabiki, 2013
Quantum Hamiltonian for HL

What is a ?
Completeness:

a=1
 CDT Hamiltonian for an unmarked loop

a=0
 CDT Hamiltonian for a marked loop

Matter-coupled CDT

2d quantum gravity
Trivial classical dynamics:: Euler number

Gauss-Bonnet’s theorem, 1848Non-trivial quantum dynamics:

World sheet of string= 2d geometry
 Liouville gravity

2d quantum gravity
Dynamical Triangulation (DT)

Coupling to matters (Matrix Model)

Path-integral in DT = Sum over all triangulated geometries
space

space
space

Causal Dynamical Triangulation (CDT)

Coupling to matters (?)

Path-integral in CDT = Sum over all triangulated geometries w/ causality
space
time
space

2d quantum gravity
Define the matrix integral (Φ: N X N Hermitian matrix):
where
Feynman’s rule
Propagator:

3-point vertices:Loop:2d quantum gravity
Liouville
Vacuum diagram:dual

Triangulated surface:#(face)#(edge)#(vertex)Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Matter-coupled CDT can be obtained at least in a continuum limit:@ special critical point
continuum limit
plain
J. Ambjørn, et. al,
2007.

Discretized geometry by DT

CDT in continuum limit@ new multi-critical point

mattercoupledcontinuum limit

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Define the matrix integral (Φ, M: N X N Hermitian matrices):: string couplingwhereSample
diagramMatter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Define the matrix integral (Φ, M: N X N Hermitian matrices):: string couplingwhereDual picture

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Define the matrix integral (Φ, M: N X N Hermitian matrices):: string couplingwhereSample
geometry

Hard dimers ona dynamical latticeMatter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
After the Gaussian integration over M, we findwhere

In the large-N limit, the so-called loop equation is obtained:

whereOne-cut solution:

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
One-cut solution:

Since w(z) = 1/z + O(1) in |z| >> a-b,

where
5 equations

5 unknowns,
can be determined by 3 couplings,

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Approach to the multi-critical point!

5 unknowns
3 coupling constants+5 equations+Gravity limit

Matter limit

CDT limit
a new 3rd-ordermulti-critical point!

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Horava-Lifshitz w/c=-22/5[new, 2012]

Liouville w/ c=-22/5[M. Staudacher, 1990]

M2=0
gs=0

M1=0
Liouville w/ c=0
Horava-Lifshitz

Matter-coupled CDT[J. Ambjorn et al, 2007]

Plain CDT

Where to go
2d CDT has been understood quite well:[1] A corresponding field theory is the projectable Horava-Lifshitz gravity.
 2d black hole in CDT

K. Hirochi, T. Katsuragawa, H. Suenobu and Y.S.

 non-projectable HL = Generalised CDT?
[2] Matters can be coupled using matrix model techniques.
 CDT coupled to Ising model

What is an implication from 2d toy model to 4d world?
 4d CDT = 4d (projectable) Horava-Lifshitz?
It seems not… (CDT w/0 foliation, S. Jordan and R. Loll, 2013.)

CDT (global causality)

CDT w/o foliation (local causality)

Where to go
At least in 3 dimensions,
CDT and CDT w/o foliation are in the same universality class.
S. Jordan and R. Loll, arXiv: 1307.5469.

Status of 2d CDT:
GOOD: Exactly solvable (w/ or w/o matters)
BAD: May not be a good toy model for 4d CDT
Future directions:[1] Construct an analytic method for 2d CDT w/o foliation.

T. Tanaka and Y.S.

 Generalisation of Matrix Model[2] Construct an analytic method for higher-dimensional CDT.
N. Sasakura and Y.S.

 Tensor Models!!

Summary (2)Horava-Lifshitz(plain CDT)

-2

Horava-Lifshitz
coupled to non-unitary CFT(CDT coupled to dimers)

1/2

3/2

Magnetization:

Hausdorff dimension:

m=2 (plain), m=3 (dimer)Outlook for 4D CDT – a wild guess
J. Ambjørn, et. al,
Phys. Lett. B 690.

1st-order line

2nd-order line
λ= 1 ?
λ= 1 ?

4d CDT:

4d HL:

CDT without foliation

S. Jordan and R. Loll, 2013.

Four kinds of possible triangles with space-like and/or time-like edges:

CDTDT (Euclidean)

CDT w/o foliation (Lorentzian)Global causality vs. Local causality:
CDT (global causality)

Space-like edge:
CDT w/o foliation(local causality)
Time-like edge:

Light ray:

CDT without foliation

S. Jordan and R. Loll, 2013.

At least in 3 dimensions,
CDT and CDT w/o foliation are in the same universality class.
S. Jordan and R. Loll, arXiv: 1307.5469.

2d model may not be a good toy model for 4d CDT.
Same universality
2d CDT

2d Horava-Lifshitz

2d CDTw/o foliation

NOT Horava-Lifshitz
NOT LiouvilleShort Summary
2d CDT turns out to be the 2d projectable Horava-Lifshitz quantum gravity:
whereCDTDT

continuum
Liouville gravity
KnownProjectable
Horava-Lifshitz gravity
Our work

Causal Dynamical Triangulations vs. Horava-Lifshitz gravity

vs.

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.

M2=0
gs=0Solve&around small gs

Then we findMatter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S., 2012.
Continuum wave func. of the Universe:

Continuum limit

CDT background
Classical limit
Matter-coupled CDT

Plain CDT

FAQ
Q1. Is the diffeomorphism broken in a lattice approach?

No. Because a lattice gravity is quantum gravity without coordinates.
Q2. Is CDT a background-independent formulation?
It seems No. Because there is a global time direction.
Q3. So, then what’s the status of CDT?

Probably, an effective theory arising from integrating out baby universes.

Gauge fix

Integrate out0. INTRODUCTION
What is the IR (field-theoretical) description of CDT?
CDT looks like Horava-Lifshitz gravity (HL)…
CDT: Lattice quantum gravity w/ causality (foliation)common
HL: Quantum gravity w/ anisotropic scaling (foliation)

In our work, we have determined that
2D CDT is 2D projectable HL quantum gravity!!

space-time  space [time]
timespacefoliation (葉層)Horava-Lifshitz gravity (HL):

anisotropy

Causal Dynamical Triangulations (CDT):causality

0. INTRODUCTION
What is the IR description of CDT?

CDT looks like Horava-Lifshitz gravity (HL)…

CDT: Lattice quantum gravity w/ causality (foliation)common
HL: Quantum gravity w/ anisotropic scaling (foliation)

In our work, we have determined that
2D CDT is 2D projectable HL quantum gravity!!

2. 2D HL
ADM decomposition:time

3D space

4D Einstein (covariant) gravity:

2. 2D HL
4D Einstein (covariant) gravity:

time
space

[symmetry]changeableADM[Newton’s constant]

4D HL (anisotropic) gravity:[symmetry][Newton’s constant]higher spatial curvature

fixed
renormalizable2. 2D HL
4D HL:

2D HL:

No higher derivative terms

1. 2D CDT
Triangle lattice (UV cutoff):
space

lattice spacing (ε)  fixed

triangulations (T)  dynamicaltime(= how to divide geometry by triangles)
space
Metric path-integral  sum over triangulations (T):: cosmological constant#()=# (T) =2. 2D HL
Quantum Hamiltonian for HL CDT Hamiltonian for a marked loop

 CDT Hamiltonian for an unmarked loop

3. SUMMARY
2D CDT turns out to be the 2D projectable Horava-Lifshitz quantum gravity:
wherecontinuum
Liouville gravity
Known

Generalised CDT

CDT

DT

continuum
Projectable
Horava-Lifshitz gravity
Our work?

Non-projectable
Horava-Lifshitz gravity?
Wild guess3. SUMMARY & CONJECTURE
2D CDT turns out to be the 2D projectable Horava-Lifshitz quantum gravity:
where

Generalised CDT

CDT

continuum
Projectable
Horava-Lifshitz gravity
Our work?
Non-projectable
Horava-Lifshitz gravity?
Wild guess3. SUMMARY
2D CDT turns out to be the 2D projectable Horava-Lifshitz quantum gravity:
whereCDTDTFuture works:(1) Non-projectable HL= Generalised CDT?
(2) BH in 2D CDT?

continuum
Liouville gravity

Projectable
Horava-Lifshitz gravity

Infinite outgrowth ofbaby universes

No baby universe(3) What about 4D?

Matter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S.

M2=0
gs=0Solve&around small gs

Then we findMatter-coupled CDT J. Ambjørn, L. Glaser, A. Görlich and Y.S.
Continuum wave func. of the Universe:

whereCDT background
Classical limit

Pure caseWilsonian renormalization
If and only if a cutoff theory is fundamental, it should possess(1) Infinite UV cutoff limit characterized via a fixed point in RG flow:(A) Gaussian fixed point (GF) canonical scaling dim.

(B) Non-Gaussian fixed point (NGF) non-canonical scaling dim.

(2) Predictability

Finite number of couplings

Infinite number of couplings Predictable!!

 Non predictable

Summary (1)2d CDT turns out to be the 2d projectable Horava-Lifshitz quantum gravity:
whereDT

Integrate outbaby universes*

CDT?

continuum
Liouville gravity
Known

Generalised CDT

Projectable
Horava-Lifshitz gravity
Our work(*) J. Ambjørn, J. Correia, C. Kristjansen and R. Loll, 1999

Non-projectable
Horava-Lifshitz gravity?
Wild guess2d quantum gravity
Continuum limit:&# (trangle) = k (= F) = infinity

under the fixed volume V.
This can be realizedby fine-tuning the coupling constants to critical values.
Physical volume:

critical value