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

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. Outline.

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

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  1. 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

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

  4. SL(2,Z) orbit of AdS Black Holes Different euclidean black holes distinguished by non-contractible cycle: Euclidean action Maldacena, Strominger AdS3/CFT2

  5. Thermal AdS3 Periodic identification Thermal circle is non-contractible Euclidean action

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

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

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

  9. 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

  10. 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

  11. 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

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

  13. 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 versus OSV partition function GV/DT partition function

  14. Maldacena, Strominger, Witten The OSV partition function equals the elliptic genus 4d Black Holes from (4,0) CFT 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

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

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

  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 de Boer, Cheng, Dijkgraaf, Manschot, EV. work in progress. where

  19. 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 • 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

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