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

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Erik Verlinde

University of Amsterdam

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

What the Topological String can (not) compute!

Based on work with Dijkgraaf and Vafa and on work in progress with

Jan de Boer, Miranda Cheng,

Robbert Dijkgraaf & Jan Manschot

- 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 )

Then we have

where

Suppose

is a modular form of weight w

Different euclidean black holes distinguished by non-contractible cycle:

Euclidean action

Maldacena, Strominger

AdS3/CFT2

Periodic identification

Thermal circle is non-contractible

Euclidean action

Euclidean time circle is contractible

cigar

Euclidean action

Proper length of particle world line

Martinec

Bound => no BH-formation

Dijkgraaf, Maldacena,

Moore, EV

Exact semi-classical expansion in terms of saddle-point contributions

including corrections due to ‘light’ (virtual) BPS particles

GW invariants

GV invariants

Gopakumar, Vafa

Resummation of free energy:

Genus 0 free energy:

Topological Strings: A-model

Higher genus:

1 4

M-theory on CY x S x R

Gopakumar, Vafa

M-theory interpretation:

Euclidean time as 11th dimension

5D spin couples to graviphoton

Schwinger calculation of ‘D2-D0’ boundstate

Free energy

Questions:

- Does this formula count all 5d BPS-states?
- Does it agree with the Bekenstein-Hawking formula for 5d black holes?
- ?
- Does it have an interpretation in terms of 4D BPS black holes?
- What is the interpretation of the exponential pre-factor?

spinning M2-branes

Top. String describes gas of 5D BPS particles

1

M-theory on CY x S x TN

The Taub-NUT geometry

The 4d/5d connection and DT invariants

Gaiotto, Strominger, Yin

Dijkgraaf, Vafa, EV

4 3 1

Interpolates from R to R x Sand breaks

5D spin becomes KK-momentum

Gas of spinning M2’s => D2, D0 branes bound to D6 => DT-invariants

OSV partition function

IIA on CY:

Entropy as Legendre transform

Cardoso, de Wit, Mohaupt

Ooguri, Strominger, Vafa

Connection with topological string

OSV partition function

- What is the explanation of the (absolute valued)-squared?
- What is the origin of the transformation ?
- Does the gas of 5D particles have an interpretation for 4D black holes?

GV/DT partition function

Maldacena, Strominger, Witten

The OSV partition function equals the elliptic genus

M-theory on CY (x S )

1

M5-brane wraps a 4-cycle in CY=> 5d black string

6d (2,0) theory => (4,0) 2d CFT

Contains chiral bosons => metric

Lorentzian Narain lattice => M2-branes charges

Near-horizon geometry becomes

Gaiotto, Strominger, Yin

The elliptic genus

has a low temperature description in terms of

a gas of chiral primaries: wrapped

(anti-)M2-branes at ‘north’ and ‘south’ pole.

High temperature expansion

Gaiotto, Strominger, Yin

The elliptic genus

is a modular form of weight 0

Corrections due to presence of (virtual) BPS-particles

Gaiotto, Strominger, Yin

Expected form of

exact semi-classical expansion

Connection with topological string occurs in large-c limit, expect

de Boer, Cheng,

Dijkgraaf, Manschot,

EV. work in progress.

where

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
- 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