The Topological String Partition Function

as a Wave Function (of the Universe)

Erik Verlinde

Institute for Theoretical Physics

University of Amsterdam

Topological Strings = BPS Type II Stringswith 8 supercharges (N=2 in 4d)

• Introduction to Topological Strings

• A-model Partition Function and BPS counting in 5D

• B-model Partition Function as a Wave-function

• 4D Black Hole Entropy and the OSV Conjecture

• A Hartle-Hawking wave function for Flux compactifications: “The Entropic Principle”

= twisted N=2 SCFT

JTT 21 JTT 2

1

GQT ,

Nilpotent BRST-charge: 02 Q

BRST-exact stress energy:

Topological CFT

Physical operators 0, IOQ ,QOO II

Chiral ringK

KIJJI OCOO

Topological Strings on a Calabi-Yau

......)( jiijjiji XXXgS

Topological Sigma model

Operators become forms

Physical operators <=> closed forms on the Calabi-Yau

nn

mnmn

jjiijjii XOO ....)( 11

11),( ....

BRST charge = exterior derivative iiXQ ,

Chiral ring = “quantum” cohomology ring of CY.

A- and B-model

A-model: physical operators are (n,m)-forms with n=m

(1,1)-forms => Kahler deformations of CY “size”

JTT 21

JTT 21

B-model: physical operators are (n,m)-forms with n=3-m

(2,1)-forms => Complex structure deformations of CY

1

00

11

00

1

1,1

2,11,2

2,2

b

bb

b

Hodge diamond

),(. dim mnmn Hb Mirror symmetry:

A-model

B-model

“shape”

Free Energy

String amplitude: integrated correlation function

nn IIIggIII OOOSdXOOO ...exp......

2121

II

gI

g OtSdXtF exp...

Free energy:

= generating function

gIIIgIII FOOOnn

......2121

computes F-terms in space time effective action of the form

222)( RTtF gIg

fieldn graviphotoT

Partition Function

Full Free energy:

Partition function:

,exp, II tFt

0

22,g

Ig

gI tFtF

coupling string ltopologica

Coupling constants: parametrize background

A-model: complexified Kahler moduli B-model: complex structure moduli

It

A-model amplitudes

3-point function: intersection form + worldsheet instantons

KJCY IIJK OOOd

KJInt

nt

nnIJKKJI nnn

e

eNdOOO

II

II

I

I

1,00

Genus 0 free energy: obtained by integrating

00 KJIKJI OOOF

Higher genus:

I

II

In k

nktgngg

Ig ekNctF

0

32,

counts the number of holomorphic curves in homology class nI

Counting 5D BPS States

M-theory on CY => 5D SUGRA

3

)7(

SC

I

I

FQWrapped membranes = Charged BPS States

We like to know

JQJQD spin and charge with states BPS#),(5

Bekenstein-Hawking entropy of extremal spinning Black Holes predicts

235 ),(log JQJQD

KJIIJK tttdQ 2/3

KJIJKI ttdQ 3

Schwinger calculation of single D2-D0 boundstate in graviphoton field

Gopakumar, Vafa

0

1)(

1exp

k

knkt

mI

I

s

II

ek

Tmntg

ss

ds

JT

Take Euclidean time circle as 11th dimension in M-theory.

Spin couples to graviphoton

Counting 5D BPS States

suggests rewritting of free energy

, 00

32,

0

22 1,~

I

II

I

II

In k

knktI

n k

nktgngg

g

g ek

ncekNcF

II nte1log

Gopakumar, Vafa

spin and charge with states BPS single #, II nnc

Total free energy can be rewritten in terms of integer invariants

as

Counting 5D BPS States

,

2 1log,,I

II

n

ntIIJK

KJII encdttttF

,

,

1

1,

2

I

I

II

IJKKJI

nc

n

nt

dttt

Itop

e

et

For the partition function this gives the product formula

III

I

II Qt

JQD

nc

n

nt eJQe,

5

,

,

),(1

Counting 5D BPS States

Conjecture

The l.h.s. describes a “free” gas of “single” BPS states.

,),(2

1

,5

Itop

dtttQt

JQD teeJQ

IJKKJI

II

If true the 5D black hole partition function equals

B-model amplitudes

3-point function: obtained by differentiation

Genus 0 free energy: from periods of holomorphic 3-form

IJKKJIKJI CFOOO 00

XFB

0

XA

)( ItXX

MBA

,#

Higher genus: from holomorphic anomaly

1 gKJhKhgJJK

IgI FFFCF

B-model partition sum as a wave function

Holomorphic anomaly in terms of partition function

,, ItKJ

JKI

ItI tCt

Background independent wave functions

,lim Itttop tX

XtX I ,

expresses background dependence, exactly like

a wavefunction obtained by quantizing the 3rd cohomology

31 Ht II

WittenDijkgraaf, Vonk, EV

B-model partition sum as a wave function

The 3rd cohomology

)3,0()2,1()1,2()0,3(3 HHHHH

The decomposition

leads to background dependent wave functions ,It t

11 II

II tt

CY

2121 ),( has a natural symplectic form

EV

Background independent decomposition

leads to real wavefunctions

pq

q

4D Black Hole Entropy from Topological Strings

)(Im2, 2 itop qFqF

,,, qF

qFpqS

,qF

p

Cardoso, de Wit, MohauptOoguri, Strominger, Vafa

Entropy as Legendre transform

pF

qX

Re

Re pqFXFXpqS ,,

Semiclassical entropy

XiFX toptop exp

Mixed partition function factorizes as

2

2, itop

p

p qepq

),(log),( pqpqS

Exact Counting of 4D Black Hole States?

21

21

4 ,

qeqdqp topip

topD

OSV-conjecture: # BPS states is Wigner function

Is this exact? Can one use product formula to obtain integral numbers? No!

Recent connection with 5D black holes using Taub-NUT

Shih, Strominger, Xi

,exp, qFeqpp

p

,),(2

1

5I

topQt

dttt

D teedtdJQ IIIJK

KJI

For these our conjectured formula is

Cheng,Dijkgraaf, Manschot, EVwork in progress

,

,5 1),(

I

I

II

II nc

n

ntQtD eedtdJQ

Flux Compactifications

qFA

)3(

pFB

)3(

Fluxes through cycles

)(0 XFB

XA

Type IIB string on CY

qpXX ,

pqF 3

)()( 0, XFpXqXW qp

Superpotential for moduli fields Moduli stabilization

0)(, XWD qp

CY

FW 3

BPS Black holes as Flux Vacua

Entropy

FXFXS

qFSA 2

)5(

pFSB 2

)5(

Electric and magnetic charges

FqXpFWSCY 2

)5(

Graviphoton charge

Attractor Mechanism

0WDI

pF

qX

Re

Re

Attractor Equations

Type IIB string on CY

qpXX ,

Near Horizon Geometry as Cosmological Model

Euclidean metric

22)(22))((222S

UU ddededs

with gauge choice

FXFXe U )(2

0

WDgd

dX

Attractor flow equation

)(X

Black Hole Entropy

FXFXpqS ),()(2

2

4

)(),( Ue

SApqS

Ferrara, Gibbons, Kallosh

qp,qp,

Hartle-Hawking wave function

                     

22222 : ddedsAdS

0, qpWDWHWDWWDW

qpWDW

HQ

Q

2

, 0

qe topip

qp,

qe topip

qp*

,

The wave functions

qpSd qpqpqpqp ,exp ,,,, obey

)( topiFtop e

Ooguri, Vafa, EV

• Flux vacua as wave functions on moduli space

• Relative probability determined by entropy

Flux Wave Functions

. ..

.. .

...

.. .

qpX ,

Xqp,

• Moduli fixed by fluxes : discrete points.

The Entropic Principle

Flux Vacua

),(exp,, qpSqpqp

Entropic Principle

• Nature is (most likely) described by state of maximal entropy

• Constructive way to select vacua (in contrast with “Anthropic Principle”)

The Entropic Principle: A Hartle-Hawking Wave Function for String Compactification*

Erik Verlinde   

Institute for Theoretical Physics

University of Amsterdam

* based on work with H. Ooguri and C. Vafa

Physics 2005 ConferenceWarwick, April 12, 2005

A-model partition sum: a product formula

Resummation of free energy

,

,

1

1,

2

I

I

II

IJKKJI

nc

n

nt

dttt

Itop

e

et

Gopakumar, Vafa

, 0

2 1,,

I

II

n k

knktIIJK

KJII ek

ncdttttF

I

II

In k

nktgngg

g

gIJK

KJII ekNcdttttF0

32,

0

222,

In terms of integral invariants

gives the product formula

                     

qp,

0, qpWDWH

2222222 2 :

SddeddsSAdS

qp,WDWWDW

qpWDW

HQ

Q

2

, 0

• Flux vacua and moduli stabilization • Cosmological model: type IIB on • Attractor flow and the Wheeler-de Witt equation• `Exact´ Hartle-Hawking wave function and topological strings

22 AdSSCY

Outline

Wheeler-De Witt equation

0, Iqp

II

XXd

dX

Quantizing the BPS flow equation XXqp ,,

JIJ

I

Xg

d

dX

gives the BPS WDW equation

qpqp XXXXXXqp Ce ,,

,0,,

qpI

qpI

JIJ XX

Xg

+c.c

Probality density

qpqpqp

qp

SXXXXXXXdXd ee ,,,2

,

2 ||

peaked near Attractror value

Natural Normalization => Entropy

Wave functions

qp,qp,

qpqpqpqp dqp ,,,, , obey

Exact Hartle-Hawking wave function

)( topiFtop e

pe topqi

qp,

pe topqi

qp*

,

• Evidence has been given for the identification of the topological string partition function with the `exact’ euclidean Hartle-Hawking wave function in mini superspace for Type IIB theory on a CY x S2.

• Our description leads for each flux vacuum to a probability density on the moduli space. Relative probalities between different flux vacua is determined by an `entropic’ instead of `anthropic’ principle.

• The continuation to Minkowski signature is presumably possible if one allows supersymmetry to be broken, but needs further investigation.

• The implications for more general 4d flux compactifications are worth studying.

Conclusion

• Flux vacua as discrete points in the moduli space

• Each point has a priori equal probability

Discrete Flux Vacua

• Flux vacua as wave functions on the moduli space

• Relative probability determined by entropy

Flux Wave Functions

. ..

.. .

...

...

qpX ,

Xqp,

• Moduli determined by fluxes qp,X

Flux vacua

qpqp S ,

2

, exp 2, ||

,qpXX

qp CeX

A Hartle-Hawking Wavefunction for Flux Vacua

Outline

• Flux vacua and BPS black holes• Moduli stabilization and attractor mechanism• Cosmological model: type IIB on • Attractor flow and the Wheeler-de Witt equation• Exact Entropy and topological strings• Attractor equations as canonical transformation• `Exact´ Hartle-Hawking wave function

22 AdSSCY

                     

qp,

r

0, qpWDWH

22222 : drdedsAdS r

qp,WDWWDW

qpWDW

HQ

Q

2

, 0

Flux Vacua

I

B

FI

I

A

XI

)(XFF II 0

Type IIB string on CY

I

A

pFI )3(

I

B

qFI

)3(

Fluxes through 3-cycles Complex structure moduli

Kahler potential

I

II

IK FXFXe

III

I FpXqFW )3(

Superpotential for moduli fields

Scalar potential

WWWDWDgeV JIIJK 3

WKWD III

Kg JIIJ

Moduli stabilization

0WDI

Moduli Stabilization

BPS condition

0 WKWD III

II

II

qCF

pCX

Re

Re

Attractor Equations

I

II

IK FXFXe

JJI

I FpXqW

021 KWWD III

WeC K 1

0FJIIJ

gives

IJJ

IIJJ

I pqXFC

II

II

qF

pX

Re

ReIJJI K Im2

Kahler metric on Moduli Space

XXXXK JIIJ Im2

Gauge choice

11 C

I

II

I FXFXK

Cosmological model

Euclidean metric

Type IIB string on CYxS2xS1

22)(22))((222S

rUrrU ddrededs

Gauge choice

UeK 2

0)(Im XW

BPS flow equations

WKedr

dU U 21

1 0 WDdr

dXg I

J

IJ

0Im 21 WK

dr

dXII

J

IJ

Combined BPS flow equation

r

)(rX I

Wheeler-De Witt equation

0Im 21 WK

dr

dXII

J

IJ

Quantizing the BPS flow equationr XXqp ,,

I

J

IJ Xdr

dX

Im

Normalization => Entropy

qp

qp

SXWXWXXKXdXd ee ,2

,

)(2)(2),( ||

gives the BPS WDW equation

)()(),(,

21 XWXWXXK

qp e 0,21

qpIIIWK

X +c.c

2, ||

,qpXX

qp Ce

Peaked near Attractor value

Reduced BPS phase space

0, qpIC

WKX

C IIII

21BPS condition = Constraint

Dirac bracket

LK

JL

LKK

IJIDirac

JI XCCC

CXXXXX,

* ,,

1,,,

WKX

C IIII

21

)(, )( XWqp eX

                     

)()( 21),(

21 XXeXdXd XXK

Holomorphic wave functions with inner product)(X

IJJI K Im2

Non-commutative moduli

IJ

DiracJI XX 1

21 Im,

Attractor equations as canonical transformation

represent canonical transformation

IJJI XX 121 Im, J

IJ

I i ,

IIII FX Re , Re

Attractor equations

1)(0,0 X )(0,0

0)( iFe Topological string partition function

Quantization of 3rd cohomology

II

IX )(Re I 2121 ),( Q

• Topological Strings have “real” physical applications in 4D (and 5D) type II (and M-theory) on a Calabi-Yau space, in particular in describing the entropy of BPS black holes.

• A proof that the 5D BPS states counted by the topological string is sufficient to explain the 5D black hole entropy is still missing.

• An interesting connection between 4D and 5D black holes suggest

•Our description leads for each flux vacuum to a probability density on the moduli space. Relative probalities between different flux vacua is determined by an `entropic’ instead of `anthropic’ principle.

• The continuation to Minkowski signature is presumably possible if one allows supersymmetry to be broken, but needs further investigation.

• The implications for more general 4d flux compactifications are worth studying.

Summary and Conclusion

Partition Function

Partition function:

0

22exp,g

Ig

gI tFtZ

Partition Function

String amplitude: integrated correlation function

nn IIIggIII OOOSdXOOO ...exp......

2121

=> generating function of string amplitudes

I

Ig

Ig OtSdXtF exp...

Free energy:

gIIIgIII FOOOnn

......2121

Coupling constants: A-model: Kahler moduli B-model: Complex structure moduli

0

22exp,g

Ig

gI tFtZ

coupling string

Partition Function

4D Black Hole Entropy from Topological Strings

)(Im2, 2 itop qFqF

,,, qF

qFpqS

,qF

p

Cardoso, de Wit, MohauptOoguri, Strominger, Vafa

Entropy as Legendre transform

pF

qX

Re

Re pqFXFXpqS ,,

Semiclassical entropy

pqSqeqd topip

top,exp 2

121

topiFtop e

# BPS states as Wigner function