GRAVITY AS AN EMERGENT FORCE Erik Verlinde University of Amsterdam
The Topological String Partition Function as a Wave Function (of the Universe) Erik Verlinde...
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Transcript of The Topological String Partition Function as a Wave Function (of the Universe) Erik Verlinde...
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