exact path integral for 3d quantum gravityentangle2016/yiizuka.pdf · 3d pure gravity 3d...

Exact Path Integral for 3D Quantum Gravity

Nori Iizuka @ Osaka U.

u Phys. Rev. Lett. 115, 161304 (2015):arXiv:1504.05991 with A. Tanaka, and S. Terashima. Quantum Matter, Spacetime and Information,

YKIS 2016, Kyoto

Can we calculate Quantum Gravity partition function?

2

Yes or No.

Zgravity =

ZDgµ⌫e

�Sgravity


3

Yes or No.

Zgravity =

ZDgµ⌫e

�Sgravity

“Directly in the Bulk”

In this talk, we try to calculate pure quantum gravity partition

function directly in 3D (2+1) bulk,

4

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤)

Zgravity =

ZDgµ⌫e

�Sgravity

Why 3D pure Gravity?

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤)


•  Simplest! •  No gravitons •  Still black holes (Banados-Teitelboim-Zanelli

or BTZ BH in short) exist, so interesting enough!

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤)


•  No string theory compactification (i.e., stringy derivation) to obtain pure 3D gravity

•  This makes it difficult to construct concrete holographic setting for pure gravity

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤)


•  BTZ BH sol’ns in pure gravity (8GN = 1 unit )

•  In order to have horizons; we need

ds2 = �f(r)dt2 + f�1(r)dr2 + r2(J(r)dt+ d�)2

f(r) = �M +r2

`2+

J2

4r2J(r) = � J

2r2

M � 0

M � J

`

⇣rH = ±`

pM

r1±

q1� J2

`2M2

p2

⌘

(Banados-Teitelboim-Zanelli ’92)


•  Consider J=0 for simplicity.

•  BH exsits only •  If we set , then it becomes AdS3 (no

horizon, no sing., homogeneous, isotropic) •  AdS3 vacuum is separated from the continuous

black hole spectrum by a mass gap.

ds2 = �(�M +r2

`2)dt2 +

dr2

(�M + r2

`2 )+ r2d2�

(Banados-Teitelboim-Zanelli ’92)

M � 0

M = �1

H

H < 0

H > 0

H

H < 0

H > 0

Gap

H = � 1

8GN


12

Zgravity =

ZDgµ⌫e

�Sgravity

“Directly in the Bulk”

Our strategy

•  3D pure gravity (in the bulk)

•  3D Chern-Simons theory (in the bulk)

•  3D super-Chern-Simons theory (in the bulk)

Step 1

Step 2

(NI – Tanaka - Terashima ’15)

Our strategy

•  3D super-Chern-Simons theory (in the bulk) •  Exact calculation is possible by localization •  The result for c = 24 is;

•  This agrees with the extremal CFT partition function (Frenkel, Lepowsky, Meurman), predicted by Witten for 3D pure gravity

Zgravity = J(q)J(q)


(with mild assumptions)

Plan of my talk:

•  Introduction •  Quick 3D pure gravity overview •  Witten’s proposal •  Our strategy in detail •  Localization and Exact results •  Discussions of “boundary fermions”

3D pure gravity overview

•  Our theory = 3D pure gravity

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤) + SGH + Sc

gravity counter-term

Gibbons-Hawking boundary term

3D pure gravity overview

•  Our theory = 3D pure gravity

•  ( = AdS scale)

Sgravity =1

16⇡GN

Zd

3x

pg (R� 2⇤) + SGH + Sc

⇤ = �1/`2 `

SGH =1

8⇡GN

Zd

2x

phK

gravity counter-term

Gibbons-Hawking boundary term

Sc = � 1

8⇡GN

Zd

2x

ph

boundary metric

extrinsic curvature

3D pure gravity 3D Chern-Simons

•  defining gauge fields as; SL(2, C)✓!

aµ � i

`

e

aµ

◆i

2�adx

µ = A

✓!

aµ +

i

`

e

aµ

◆i

2�adx

µ = A ,

Pauli matrix

Step 1

(Achucarro-Townsend ’88, Witten ’88)


•  defining gauge fields as;

•  One can show, by defining

where

SL(2, C)✓!

aµ � i

`

e

aµ

◆i

2�adx

µ = A

✓!

aµ +

i

`

e

aµ

◆i

2�adx

µ = A ,

Pauli matrix

SCS [A] =

Z

MTr

⇣AdA+

2

3A3

⌘

Step 1


Sgravity =ik

4⇡SCS [A]� ik

4⇡SCS [A] ⌘ Sgauge

k ⌘ `/4GN


•  Since action is decomposed into holomorphic part and anti-holomorphic part, regarding and are independent variables;

•  Then partition function “naturally” factorizes:


Step 1

Sgravity =ik

4⇡SCS [A]� ik


AA

Z = Zhol

⇥ Zanti�hol


•  A few more comments: •  Asym AdS3 allows deformation of the metric;

which is represented by the Virasoro algebra; •  Its central charge is obtained as

•  Using this, Cardy formula gives BTZ BH entropy in the large limit

(Brown - Henneaux ’86)

Step 1

c =3`

2GN= 6k

c (Strominger ’97)

Witten’s proposal (Witten ’06)

H

H < 0

H > 0

Gap

H = � 1

8GN

H

H < 0

H > 0

Virasoro descendant of the vacuum i.e., Verma module of the vacuum

|0 >

L�2|0 >

L�3|0 >··

H = � 1

8GN

Witten’s proposal

•  Since

•  If the dual CFT exists, and assuming holomorphic factorization,

•  Then with

(Witten ’06)

Zgravity

= Zhol

(q)⇥ Zanti�hol

(q)

Hvacuum = � 1

8GN= � c

12`= �1

`

⇣ c

24+

c

24

⌘

(Hhol

= L0 �c

24)

q ⌘ e2⇡i⌧

Witten’s proposal

•  Its holomorphic partition function

should have the following forms: Zhol

= Tr�qL0� c

24�

(Witten ’06)

“Verma module of the vacuum”

“Verma module from “BTZ BHs = primary states”

Zhol

(q) = q�c24

1Y

n=2

1

1� qn+ O(q)

H

Virasoro descendant of the vacuum i.e., Verma module of the vacuum

|0 >

L�2|0 >

L�3|0 >··

Hvac = � 1

8GN= �1

`

⇣ c

24+

c

24

⌘

L0 �c

24< 0

L0 �c

24= 0

L0 �c

24> 0

Witten’s proposal

•  CS quantization:

holomorphic partition fn contains poles;

(Witten ’06)


“Verma module from “BTZ BHs as primary states”

Zhol

(q) = q�c24

1Y

n=2

1

1� qn+ O(q)

All poles are uniquely fixed by the central charge

“Extremal CFT”

c

24=

k

4=

`

16GN2 Z

Witten’s proposal

•  For c = 24 case, we have

•  With the modular invariance, we have J(q),

(Witten ’06)

Zhol

= Tr�qL0� c

24�=

1

q+ 0 +O(q)

q = e2⇡i⌧ ⌧ ! a⌧ + b

c⌧ + d

= q�1 + 196884q + 21493760q2 + . . .

Zhol

= J(q) = j(q)� 744

Witten’s proposal

•  Similarly, for c = 48,

(Witten ’06)

Zhol

= J(q)2 � 393767

= q�2 + 1 + 42987520q + 40491909396q2 + · · ·

Witten’s proposal

•  For c=24 case, J(q) function as a partition function is constructed by Frenkel, Lepowsky, and Meurman

(Witten ’06)

( ’84, ‘86)

We would like to calculate the bulk quantum pure gravity partition function and “re-derive” these

results (under mild assumptions)

Our strategy




Step 1

Step 2



•  In step 1, we have; CS theory; SL(2, C)

✓!

aµ +

i

`

e

aµ

◆i

2�adx

µ = A ,

SCS [A] =

Z

MTr

⇣AdA+

2

3A3

⌘

Step 1


Sgravity =ik

4⇡SCS [A]� ik


k ⌘ `/4GN

We supersymmetrize CS theory

•  By introducing auxiliary fields to construct 3D vector multiplet,

•  we supersymmetrize the Chern-Simons action

•  Note that additional fermions and bosons are only auxiliary fields

Step 2

N = 2

V = (A,�, D, �,�)

SSCS [V ] = SCS [A] +

Zd

3x

pg Tr

⇣� ��+ 2D�

⌘.


We supersymmetrize CS theory

•  Therefore we expect

to holds.

Step 2

ZDAe�SCS [A] ⇡

ZDV e�SSCS [V ]


Our strategy




Step 1

Step 2


In step 1, we have discussed action equivalence,

i.e., classical equivalence. But we have not discussed

the path-integral measure yet.

Let’s discuss it, since we are doing quantum theory rather than

classical theory

We need to define the measure for quantum gravity in terms of the

Chern-Simons theory, such that gravity path-integral can be

calculated in terms of the Chern-Simons theory path-integral.

The Chern-Simons theory lives on three-dimensional Euclidian space

which we call ‘M’. Since the Chern-Simons theory is topological, it depends on only the

topology of M and the boundary metric of M

Since we solve the quantum gravity in asym AdS b.c.,

and asym Euclidean AdS is parametrized by the torus, with its complex moduli τ,

M should have a boundary torus with moduli τ.

Note that in gravity, radial rescaling changes the size of torus,

so the size of torus cannot be a parameter for the boundary.

Note that different space-time topology in gravity side should be mapped into

different topology for M in the Chern-Simons side. And in the gauge theory,

one regards the Chern-Simons theory on different topology M

as a different theory.

So for the correspondence to work, we decompose the metric path-integral

into each “sector” distinguished by the topology of M,

and then we need to sum over different topology for M,

where all of them should have a boundary torus with τ.

Furthermore, in the CS theory, boundary conditions related by the modular transformation for τ, are

regarded as giving different theory.

Therefore we need to sum over different b.c. for the CS theory, where all of the b.c. with complex structure τ

are related by the modular transformation.

All of these gives sequence of transformation as

where,

Zgravity

=

ZDg

µ⌫

e�Sgravity !XZ

Dgsectorµ⌫

e�Sgravity

= Zhol

⇥ Zanti�hol

Zhol

=XZ

DV e�ik4⇡SSCS [V ]

!✓XZ

DAe�ik4⇡SCS [A]

◆✓XZDAe+

ik4⇡SCS [A]

◆

!✓XZ


◆✓XZDV e+

ik4⇡SSCS [V ]

◆

Furthermore, we need to evaluate

Zhol

=

X

all topology having the

boundary torus w./ ⌧ and

its modular transformation

Z

M


How can we calculate this quantity?

Localization

•  One can show that

is actually t - independent since SSYM is Q-exact

Z[t] =

Z

MDV e�

ik4⇡SSCS [V ]+tSSY M

SSYM =

Z

MTr

⇣14Fµ⌫F

µ⌫ +1

2Dµ�D

µ� +1

2(D +

�

l)2

+i

4��µDµ�+

i

4��µDµ�+

i

2�[�,�]� 1

4l��

⌘

Localization

•  Setting , only SSYM = 0 configuration is expected to contribute to the path-integral for

•  Then if only contributes, by writing this in terms of the metric, we use that this eq is nothing but the Einstein equation!

Z[t] =

Z

MDV e�

ik4⇡SSCS [V ]+tSSY M

t ! 1V

Fµ⌫ = 0

(Witten ’88)

So we have,

Zhol

=

X

all topology having the



Z

M


Only solutions of Einstein eqs contributes, and it is known that all of the solutions of the Einstein eq in 3D

have solid torus topology

So we have,

Only solutions of Einstein eqs contributes, and it is known that all of the solutions of the Einstein eq in 3D

have solid torus topology

Zhol

=

X

solid torus only having the



Z

M


(Dijkgraaf-Maldacena-Moore-Verlinde ’00, Maloney-Witten ’07, Manschot-Moore ’07)

Solid torus

：Non-Contractable circle

：Contractable circle

：radial direction (emergent direction)

: Boundary torus

++ · · ·

CS theory living on this is equivalent to AdS space

CS theories living on these are equivalent to various BHs

Z=：Euclidean time ：Spatial circle

related by modular tr.


Using localization technique

•  Asym AdS = Dirichlet b.c. •  Metric maps => gauge boson A •  SUSY Dirichlet b.c. at the boundary torus is

given by;

A' ! a' , AtE ! atE ,

� ! 0 , � ! e�i('�tE)�✓�

(Sugishita - Terashima ’13)

Using localization technique

•  The classical contribution & the one-loop determinant gives exact (non-perturbative) answer:

Zclassical = eik⇡Tr(a'atE)

Zone�loop

=Y

m2Z

⇣m� ↵(a

'

)⌘= ei⇡↵(a') � e�i⇡↵(a'),

Zeta-function regularization

Z(c,d) =

Z

b.c.DV e�

ik4⇡SSCS [V ]+tSSY M = Z

classical

⇥ Zone�loop

(Sugishita - Terashima ’13)

Exact results

•  For the non-rotating BTZ black hole, b.c. from the metric gives:

•  With this, Z is

a' =1

2i��3 =

1

2⌧�3 , atE =

1

2�3

Z(c=1,d=0) = e14 (k+2) 2⇡i

⌧ � e14 (k�2) 2⇡i

⌧


Exact results

•  Non-rotating BTZ is related to the thermal AdS with by the modular transf. with c=1, d=0,

•  For all generic case are then given by

we need to sum over (c, d)

⌧ ! a⌧ + b

c⌧ + d

Z(c,d) = e�2⇡i 14 (k+2) a⌧+b

c⌧+d � e�2⇡i 14 (k�2) a⌧+b

c⌧+d


++ · · ·

CS theory living on this is equivalent to AdS space

CS theories living on these are equivalent to various BHs

Z=：Euclidean time ：Spatial circle

related by modular tr.

Exact results

•  sum over (c, d): there is a good way to perform such summation with appropriate regularization, called the “Rademacher sum”

•  Taking such regularization we obtain:


Exact results

•  The resultant holomorphic partition function:

•  where

= R(�keff/4)(q)�R(�keff/4+1)(q) ,

Zhol

[q] ⌘ Z(0,1)(⌧) +X

c>0,(c,d)=1

⇣Zc,d

(⌧)� Zc,d

(1)⌘

keff ⌘ k + 2q ⌘ e2⇡i⌧

Quantum shift c = 6keff


Exact results

•  And is given by:

R(m)(q) ⌘ e2⇡im⌧ +X

c>0,(c,d)=1

⇣e2⇡im

a⌧+bc⌧+d � e2⇡im

ac

⌘

= qm + (const.) +

1X

n=1

c(m,n)qn

c(m,n) ⌘X

c>0,(c,d)=1,dmod c

e2⇡i(mac +n

dc )

1X

⌫=0

⇣2⇡c

⌘2⌫+2

⌫!(⌫ + 1)!(�m)⌫+1n⌫


R(m)(q)

Exact results

•  For c = 24 ( ), we obtain

•  Except for the const. term which we can choose to be any value we want by regularization, we obtain J(q) function!

c = 6keff

Zhol

(q) = R(m=�1)(q)�R(m=0)(q)

= J(q) + const.


Summary:

•  Exact calculation is possible by localization technique, and the results for c = 24 is;

•  This agrees with the extremal CFT partition function of Frenkel, Lepowsky, and Meurman, predicted by Witten for 3D pure gravity!

•  Furthermore, localization gives “justification” of semi-classical approximation

Zgravity = J(q)J(q)


Boundary fermions discussions

Boundary fermion

•  Dirichlet b.c. (for AdS3) and SUSY yields b.c. for fermion as

•  This b.c. is incompatible with the gaugino “mass term” in

��bdry

= e�i('�tE)�3��bdry

,

SSCS [V ] = SCS [A] +

Zd

3x

pg Tr

⇣� ��+ 2D�

⌘.

(NI – Honda - Tanaka - Terashima ’15)

Boundary fermion

•  This means that SCS’s mass term for gaugino vanishes at the boundary and therefore there are “boundary localized fermions”.

•  We “guessed’’ its contribution to partition function as;

•  And we define

ZB-fermion

(q) =1Y

n=1

(1� qn)

Zhol

(q)

ZB-fermion

(q)⌘ Z large k

gravity

(q)


“bulk pure gravity” partition function

•  We can show that;

Z large kgravity(q)

= q�keff4

1Y

n=2

1

1� qn+

1X

�=1

c(keff)� q�

1Y

n=1

1

1� qn




“bulk pure gravity” partition function

•  With

•  This is always positive intergers: satisfying Cardy formula:

c(keff)� = c

✓�ke↵

4,�

◆� c

✓�ke↵

4+ 1,�

◆

c(4)�=1 = 196884, c(4)�=2 = 21493760,

c(8)�=1 = 42790636, c(8)�=2 = 40470415636,

c(12)�=1 = 2549912390, c(12)�=2 = 12715577990892,


lim

k!1log c

(keff )� = 2⇡

pkeff� = 2⇡

rcQ�

6

Summary 1:

•  Exact calculation is possible by localization technique, and the results for c = 24 is;

•  This agrees with the extremal CFT partition function of Frenkel, Lepowsky, and Meurman, predicted by Witten for 3D pure gravity.

•  Furthermore, localization gives “justification” of semi-classical approximation

Zgravity = J(q)J(q)


Summary 2:

•  The resultant partition function for the holomorphic part is


Zhol

[q] = R(�keff/4)(q)�R(�keff/4+1)(q) ,

c(m,n) ⌘X

c>0,(c,d)=1,dmod c

e2⇡i(mac +n

dc )

1X

⌫=0

⇣2⇡c

⌘2⌫+2

⌫!(⌫ + 1)!(�m)⌫+1n⌫

R(m)(q) = qm + (const.) +

1X

n=1

c(m,n)qn

Summary 3:

•  Taking into account “boundary fermions” (NI – Honda - Tanaka - Terashima ’15)

Z large kgravity(q)

= q�keff4

1Y

n=2

1

1� qn+

1X

�=1

c(keff)� q�

1Y

n=1

1

1� qn



(⌘ Zhol

(q)

ZB-fermion

(q)) Satisfying Cardy formula for BH

Summary 4:

•  All of these results can be generalized to higher spin gravity theory w/ SL(N,C):

•  There, instead of Virasoro algebra, we have a good expression for the partition function in terms of characters for the vacuum and pri- maries in 2D unitary CFT with WN symmetry.

•  Again exhibiting Cardy formula in the large central charge limit.


exact path integral for 3d quantum gravityentangle2016/yiizuka.pdf · 3d pure gravity 3d...

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