Robbert Dijkgraaf (UvA)

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Topological String Theory and Black
HolesEurostrings 2006, Cambridge, UK
- review - w/ C. Vafa, E.Verlinde, hep-th/0602087
- work in progress
Robbert DijkgraafUniversity of Amsterdam
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Topological Strings

• Toy model (cf topology versus geometry)
• Exact BPS sector of superstrings
• Mathematical experiments to testphysical
  intuition

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Geometry of Calabi-Yau manifolds
X
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Exact Effective Actions
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A-Model (IIA) Exact worldsheet instantons
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Gromov-Witten Invariants
Exact instanton sum
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B-Model (IIB string)
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Mirror Symmetry
A-model B-Model
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Fiberwise T-Duality Strominger, Yau, Zaslov
Dual Torus Fibrations
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D-branes Black holes
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D-Branes
homological mirror symmetry)
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Charge Lattice (B-model)
symplectic vector space
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Period Map Quantization
symplectic vector space
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Special Geometry
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Top String Partition Function Wave Function
Transforms as a wave function under Sp(2n,Z)
change of canonical basis (A,B)
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A-Model
symplectic vector space
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Charged objects D-branes
charged particles
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4d Black Holes
large charges
Attractor CY
Black Hole
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Attractor Mechanism
near-horizon moduli tI
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Quantization of Moduli Space
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Black Hole Partition Function
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Exact Black Hole EntropyG. Lopes Cardoso, T.
Mohaupt, B. de Wit Ooguri, Strominger, Vafa
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Mixed Ensemble
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Mixed Ensemble
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M-theory
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Gopakumar-Vafa
At strong coupling can integrate
out (light) electric charges D0-D2 to obtain
the effective action
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5d Black Holes in M-theory
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BPS degeneracies
space
self-dual flux rotation
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GV Partition function
Gas of 5d charged spinning black holes
m
N

³

Y
Q
(
)

Q



t
(
)

n
n
m
Z
1
1
2
t

e


n
n
1
2

Q
m

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61 dim SUSY Gauge Theory
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Donaldson-Thomas Invariants
Single D6 U(1) gauge theory singularities
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Lift to M-theory
Gaiotto, Strominger,Yin
D6 ? Taub-NUT geometry
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Bound states with D0-D2
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4 dim limit
Bound state of D6-D2
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5 dim limit
Free gas of M2-branes
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Topological String Triality
peturbative IIA stringsGromov-Witten
9-11 flip
strong-weak
Taub-NUT
M2-branesGopakumar-Vafa
D2-branesDonaldson-Thomas
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Simplest Calabi-Yau
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constant maps
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Stat-Mech 3d Partitions
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Melting Crystals
Reshetikhin,Okounkov,Vafa, Nekrasov,...
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UniversalWave Function
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Wave Function of String Theory
Compactify on a 9-space
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Topology Change
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Baby Universes
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Hawking-Hartle Wave Function
Sum over bounding geometries
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Entropic Principle
Natural probability density on moduli space of
string compactifications
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AdS/CFT duality
string theory on the near horizon geometry of
the black hole
supersymmetric gauge theory on the
brane superconformal quantum mechanics
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Hawking-Hartle Wave Function Ooguri,E.Verlinde,V
afa
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cf. open/closed worldsheet duality
string
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Index theorem
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time
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Supersymmetry breaking
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Space of AllCalabi-Yaus
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Topology of Calabi-Yau spaces
Core
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Non-Kahler CY are unique
Moduli space of complex structures
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Miles Reids FantasyThere is only one CY
space
All CY connected through conifold transitions S3
? S2
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SYZ fibrations by Slag T3
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Two Vertices

-
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3d Topological Gauge Theory
Wilson loops graphs
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Universal Moduli Space of CYs?
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Gell-Mann on renormalizationJust because its
infinite, it doesnt mean its zero
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Topological Strings

• Compute BPS black hole degeneracies(gauge-gravity
  dualities)
• Interesting probability distribution onthe
  moduli space of vacua
• Universal Calabi-Yau???
• Many more surprises...

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Happy 60th, Michael!