1. M-theory, Topological Strings and the Black Hole Farey Tail Based on work with Dijkgraaf and Vafa and on work in progress with Jan de Boer, Miranda Cheng, Robbert Dijkgraaf & Jan Manschot Erik Verlinde University of Amsterdam What the Topological String can (not) compute!

2. Outline 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 )

3. The Rademacher Formula Suppose is a modular form of weight w Then we have where

4. SL(2,Z) orbit of AdS Black Holes Different euclidean black holes distinguished by non- contractible cycle: Euclidean action Maldacena, Strominger AdS 3 /CFT 2

5. Thermal AdS 3 Periodic identification Thermal circle is non-contractible Euclidean action

6. The Euclidean BTZ Black Hole cigar Euclidean time circle is contractible Euclidean action

7. The Black Hole Farey Tail for N=4 Dijkgraaf, Maldacena, Moore, EV including corrections due to ‘light’ (virtual) BPS particles Martinec Bound => no BH-formation Exact semi-classical expansion in terms of saddle-point contributions Proper length of particle world line

8. Topological Strings: A-model Higher genus: Genus 0 free energy: GW invariants GV invariants Gopakumar, Vafa Resummation of free energy:

9. Schwinger calculation of ‘D2-D0’ boundstate Euclidean time as 11 th dimension 5D spin couples to graviphoton M-theory interpretation: Free energy M-theory on CY x S x R 1 4 Gopakumar, Vafa

10. Top. String describes gas of 5D BPS particles 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

11. The Taub-NUT geometry Gaiotto, Strominger, Yin Dijkgraaf, Vafa, EV The 4d/5d connection and DT invariants Interpolates from R to R x S and breaks 5D spin becomes KK-momentum M-theory on CY x S x TN 1 4 3 1 Gas of spinning M2’s => D2, D0 branes bound to D6 => DT-invariants

12. 4D Black Holes and Topological Strings Cardoso, de Wit, Mohaupt Ooguri, Strominger, Vafa Entropy as Legendre transform OSV partition function IIA on CY: Connection with topological string

13. GV/DT versus OSV partition function GV/DT partition function 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?

14. 4d Black Holes from (4,0) CFT 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 M-theory on CY (x S ) 1 Maldacena, Strominger, Witten The OSV partition function equals the elliptic genus

15. GV from MSW 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.

16. OSV from MSW Gaiotto, Strominger, Yin The elliptic genus is a modular form of weight 0 High temperature expansion

17. OSV from MSW Gaiotto, Strominger, Yin Corrections due to presence of (virtual) BPS-particles

18. Black Hole Farey Tail for N=2 Expected form of exact semi-classical expansion Connection with topological string occurs in large-c limit, expect where de Boer, Cheng, Dijkgraaf, Manschot, EV. work in progress.

19. Conclusions Topological String Theory computes Leading semi-classical action of the saddle-points. Corrections due to particles below the BH-treshold for G N => 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