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Holography and Topological Strings
Cumrun Vafa Oct. 31, 2017 20 Years Later: The Many Faces of AdS/CFT Princeton University
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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.
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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
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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.
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Number of hololmorphic curves of genus g in class d
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Including D-branes: Lagrangian submanifolds Leads to U(N) Chern-Simons Theory on M [W]
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Large N Duality: [GV] Chern-Simons on S^3 —> topological strings on resolved conifold
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Idea: Torically:
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Wilson loop observables can be accommodated by including probe branes [OV]
Large N duality, checked to all orders.
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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.
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Large N duality extended to include all toric CY [AKMV]
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Topological Vertex:
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Feynman-like diagrams
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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)
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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
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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’?
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In fact there is [ROV]: Consider as an example toric picture of C^3:
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In fact there is [ROV]: Consider as an example toric picture of C^3:
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In fact there is [ROV]: Consider as an example toric picture of C^3:
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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].
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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.
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