Transcript M-theory, Topological Strings and the Black Hole Farey Tail

M-theory, Topological Strings
and the Black Hole Farey Tail
What the Topological String can (not) compute!
Erik Verlinde
University of Amsterdam
Based on work with Dijkgraaf and Vafa
and on work in progress with
Jan de Boer, Miranda Cheng,
Robbert Dijkgraaf & Jan Manschot
Outline
•
•
•
•
•
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The Black Hole Farey Tail. (Dijkgraaf, Moore,Maldacena, EV (2000) )
Topological Strings, GV-invariants and 5D BPS-counting.
4d/5d connection, DT-invariants. (Dijkgraaf, Vafa, EV)
4d Black Holes and Top. Strings: OSV conjecture.
M5-branes, MSW and AdS/CFT.
(Gaiotto, Strominger, Yin)
A New Black Hole Farey Tail.
(work in progress)
• Related work
(Denef, Moore )
The Rademacher Formula
Z  , y    c(n  m )e
2
Suppose
c )  ym
2 i ( n  24
n ,m
is a modular form of weight w
y 
 a  b
w
Z
,

(
c


d
)
e

 c  d c  d 
Then we have
Z  , y  
 (c  d )
cy 2
2 i
c  d
cy 2
2 i
w
c  d
e
 \ 
Z  , y 
 a  b y 
Z 
,

c


d
c


d



where
Z

 , y   
c
n  m2  24
c(n  m )e
2
c )  ym
2 i ( n  24
SL(2,Z) orbit of AdS Black Holes =AdS3 /
Different euclidean black holes distinguished by noncontractible cycle:
Euclidean action
i
S=
8GN
B  cA+dB
 a  b a  b 



 c  d c  d 
tE
Maldacena, Strominger
AdS3/CFT2
3
c
2GN
Thermal AdS3 =AdS3 /
2
2
dr
2
2
ds 2  (r 2  2 )dt E2  2

r
d

r  2
Periodic identification
tE
t E  i  t E  i  n
Thermal circle is non-contractible
Euclidean action
i
S=
  
8GN

The Euclidean BTZ Black Hole
=AdS3 /
ds 2  N 2 (r )dtE2  2 N 2 (r )dr 2  r 2 (d  N dtE )2
tE
2
2
2
2
(
r


)(
r


2
2
1 )
N (r ) 
r2
 1 2
N (r )  2
r
Euclidean time circle is contractible
cigar
Euclidean action
i
S=
8GN
1 1
  
  
Dijkgraaf, Maldacena,
Moore, EV
The Black Hole Farey Tail for N=4
Exact semi-classical expansion in terms of saddle-point contributions
Z  , y  

e
 cy 2
c a  b 
2 i 


c


d
24
c


d


(c  d )3
 \ 

c ( n  m 2 )e
y
 a  b

2 i 
n
m
 c  d c  d 
c
n  m2  24
including corrections due to ‘light’ (virtual) BPS particles
Proper length of
particle world line
tE
tE
Bound
n  m2 
c

24 16GN
tE
=> no BH-formation
Martinec
Topological Strings: A-model
GW invariants
Genus 0 free energy:

F0 (t )  d abct t t   N 0,na   k e
a
3  kt a na
a b c
na 
k 0
Higher genus:
   N
Fg t
a
na 

g ,na 
k

2 g 2
g , na
g 0

 N k
na

F t , 
a
1

2
e
k 0
GV invariants
Resummation of free energy:

2 g 3  kt a na
2 g 3  kt na
e
a
k 0
na ,

1  kt a na k 
  c  na ,   e
na ,
k 0 k
d abct t t   c  na ,
a b c
Gopakumar, Vafa
 log 1  e
 t a na 

Gopakumar, Vafa
M-theory interpretation:
1
Euclidean time as
11th
M-theory on CY x S x R
dimension
5D spin couples to graviphoton
T J 
SO(4)  SU (2) L  SU (2) R
Schwinger calculation of ‘D2-D0’ boundstate
  g sT 

 1 a
ds
1  kt a na k 
 
exp s  (t na  m)  T    e


s
m
 gs
 k 0 k
Free energy
F
 c  n ,  log 1  e
 t a na 
a
na ,
c  na ,
4
  # 5D BPS states with M2-charge

na and spin
Top. String describes gas of 5D BPS particles
spinning M2-branes
1


Ztop t ,   e
a
F ( t , )
a

e
a b b
t
t t d abc
2
 1  e
 t a na 
na ,

c na ,

Questions:
• Does this formula count all 5d BPS-states?
• Does it agree with the Bekenstein-Hawking formula for 5d black holes?
c(na , ) ~ exp  n3  14
2
?
• Does it have an interpretation in terms of 4D BPS black holes?
• What is the interpretation of the exponential pre-factor?
The 4d/5d connection and DT invariants
1
M-theory on CY x S x TN
The Taub-NUT geometry
ds 2
1
2
2

V
(
d




dx
)

Vdx
R2
    V ,
V  1
4
3
Interpolates from R to R x S
Gaiotto, Strominger, Yin
Dijkgraaf, Vafa, EV
1
|x|
1
SU (2) L  SU (2) R
and breaks
 U (1) L  SU (2) R
5D spin becomes KK-momentum
Gas of spinning M2’s => D2, D0 branes bound to D6 => DT-invariants
D
4D
qa , q0
(qa , q0 )e
 t a qa  q0

1
 1  e
na ,
 t a na 

c  na ,

4D Black Holes and Topological Strings
D 6, D 4, D 2, D 0
IIA on CY:
Entropy as Legendre transform
F
qI  I  p ,   ,

F
S  p, q   F  p ,    
p,  
I 

I
OSV partition function
p   
0
i
2
0
1

a
t
p a  2i  a 

 q

p
,
q
e
    Ztop t ,  
q
I  0, a
Cardoso, de Wit, Mohaupt
Ooguri, Strominger, Vafa
Connection with topological string
F  p,   2Re Ftop (t,  )
p0 , pa ,qa ,q0
2
S ( p, q)  log ( p, q)
GV/DT versus OSV partition function
GV/DT partition function
1

Ztop  , t
a

e
2
 1  e
t a t b t b d abc
 t na  
a
na ,
1

c  na ,

e

2
t a t b t b d abc
4D
 ( p , p , q , q )e
a
a
0
 t a qa   q0
(qa , q0 )e
qa , q0
pa  ita
t 
p0  i
a
OSV partition function
0
D
 t a qa   q0
 
 Ztop t , 
2

qa , q0
i
p0  i
• What is the explanation of the (absolute valued)-squared?
• What is the origin of the transformation
ta,  t a,
?
• Does the gas of 5D particles have an interpretation for 4D black holes?
Maldacena, Strominger, Witten
M-theory on CY (x S 1 )
4d Black Holes from (4,0) CFT
M5-brane wraps a 4-cycle
P  p a Pa in CY=> 5d black string
6d (2,0) theory => (4,0) 2d CFT
Contains chiral bosons
Lorentzian Narain lattice
c  6dabc pa pb pc  c2,a p a
H   a  a => metric
1,b2 ( M ) 1
Near-horizon geometry becomes
d ab  d abc p c
=> M2-branes charges
CYt a  p a  AdS3  S 2
p0  0
The OSV partition function equals the elliptic genus

tr (-1) e
F
2 i ( L0  y a qa )
   (0, p , q , q )e
a
a
qa , q0
0
 y a qa  q0
GV from MSW
The elliptic genus

Z pa  , y
a

Gaiotto, Strominger, Yin

 tr (-1) e
F

c ) ya q
2 i  ( L0  24
a
has a low temperature description in terms of
a gas of chiral primaries: wrapped
(anti-)M2-branes at ‘north’ and ‘south’ pole.

Z pa  , y
a

e
 1  e
na ,
2 i
c
(  )
24
 y a na  (  p a na )

c  na ,

2


OSV from MSW
The elliptic genus

Z pa  , y
a

Gaiotto, Strominger, Yin

 tr (-1) e
F

c ) ya q
2 i  ( L0  24
a


y 2  y a y b d abc p c
is a modular form of weight 0
 1 y 
 ,
e
   
a
Z pa
2 i
y2

Z pa  , y a 
High temperature expansion
Z pa  , y a   e
2 i  c
2
 y 
  24

a

(0,
p
, qa , q0 )e

qa , q0

ya

1
qa  q0

OSV from MSW
Gaiotto, Strominger, Yin
2 i  c

2
a b c
a b c
a
1

y

d
p
p
p

d
p
y
y

c
p
abc
abc


6 2, a
  24
 2



Re  d abc ( p a  iy a )( p a  iy b )( p a  iy c )  16 c2,a p a 
2
Corrections due to presence of (virtual) BPS-particles
1
t

 na 
1  e 


na , 
a
 p a na





c  na ,

2

Black Hole Farey Tail for N=2
tE
tE
Expected form of
exact semi-classical expansion
Z pa  , y  
 
e
a
 \  q  q 2  c
0
24
D( p , qa , q0 )
2
 cy 2
c a  b 
2 i 


 c  d 24 c  d 
(c  d )
e
w
tE
 a  b
y a qa 
2 i 
q 
 c  d 0 c  d 


Connection with topological string occurs in large-c limit, expect
lim Z pa  , y  
c

 \ 
 a  b p  iy 
Ztop 
,

 c  d c  d 
a
1
where

Z top  , y
a

e
2
de Boer, Cheng,
Dijkgraaf, Manschot,
EV. work in progress.
y a y b y b d abc
 1  e
na ,
2
 y a na 

c  na ,

Conclusions
Topological String Theory computes
• Leading semi-classical action of the saddle-points.
• Corrections due to particles below the BH-treshold for GN => 0
Open problems:
•Derivation of “no BH-formation”-bound on states:
seems to restrict genus of embedded M2-brane
2 g  2  d ab qa qb  pa q a ,
d ab  d abc p c
•Proof of the Rademacher expansion in this case.
•Other saddle points (black rings, multi-centered..)
•How to incorporate D6 branes…..,
de Boer, Cheng,
Dijkgraaf, Manschot, EV