1. Holography and Topological Strings
Cumrun Vafa
Oct. 31, 2017
20 Years Later: The Many Faces of AdS/CFT
Princeton University

2. The AdS/CFT is perhaps the most beautiful duality discovered in string theory.
It is the most powerful realization of the idea of holography in a physical system, leading to a solution of strong coupling gauge fields at large N in terms of a gravitational theory.
It has also shed light on some aspects of quantum gravity and in particular its unitarity.

3. In the spirit of taking stock of many faces of holography, in this talk I review holographic aspects of topological strings.
-What is Topological Strings
-Large N Duality for Topological Strings
-A Derivation of Large N Duality
-Extensions to Non-Compact Calabi-Yau
-Applications to Superstrings
-Topological Quantum Gravitational Foam and the Missing Corner

4. What Is Topological Strings?
A Baby version of superstring theory:
Topological A-model [W]. There is also a mirror version known as topological B-model.

5. Number of hololmorphic curves of genus g in class d

6. Including D-branes:
Lagrangian submanifolds
Leads to U(N) Chern-Simons Theory on M [W]

7. Large N Duality: [GV]
Chern-Simons on S^3 —> topological strings on resolved conifold

8. Idea:
Torically:

9. Wilson loop observables can be accommodated by including probe branes [OV]
Large N duality,
checked to all orders.

10. Proposal for a worldsheet derivation of large N duality
[OV]
Start from closed string side via a GLSM description
[W]
2d, U(1) gauge theory with 4 fields (1,1,-1,-1)
Higgs branch leads to resolved conifold. FI parameter gives the area. At zero area, Coulomb/Higgs coexist.
Coulomb branch—>Fields are all massive, c=0->holes.

11. Large N duality extended to include all toric CY [AKMV]

12. Topological Vertex:

13. Feynman-like diagrams

14. There is also a mirror version (B-model version)
B-branes wrapped 2-cycles —> deformed 3-cycle, no branes
Matrix model at large N —> Topological B-gravity
(Gaussian matrix model) (Semi-circle law)

15. Applications to Superstrings [CIV,DV]
D-branes wrapped around cycles of CY 3-folds, filling 4d—>
N=1 supersymmetric theories in 4d
Topological string computes Glueball superpotential W(S)
Large N duality —> Confinement for N=1 SUSY
Planar diagrams —> exact superpotential computations

16. A Topological Quantum Gravitational Foam
So far:
1-Perturbative string definition of topological gravity
2-An exact holographic definition via a large N duality
Missing: A direct exact definition of quantum gravity
How about a `Planckian’ definition of topological quantum gravity, taking into account fluctuations at the `Planck scale’?

17. In fact there is [ROV]:
Consider as an example toric picture of C^3:

18. In fact there is [ROV]:
Consider as an example toric picture of C^3:

19. In fact there is [ROV]:
Consider as an example toric picture of C^3:

20. The quantum gravitational foam can be viewed as a (topologically twisted) non-commutative U(1) gauge theory. Certain gauge configurations in this U(1) theory (ch2,ch3) correspond to quantum gravity foam.
The gauge theory completes it to a theory [INOV].
Amounts to
where some coefficients are forced to be zero.
This can be generalized for compact CY as well giving an formulation of topological string in terms of Donaldson-Thomas invariants [OMNP].

21. In the context of Superstrings this suggests that perhaps holography should also be viewed as an equivalence of two independent definitions of the gravity theory:
1) Large N Gauge theory
2) Planckian gravitational foam.
The (2) is currently a missing corner for Superstring dualities.