Conformal Field Theory On Quantum Spectral Curves Conference in honor of 60th birthday of Professor

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Conformal Field TheoryOn Quantum Spectral
CurvesConference in honor of 60th birthday of
Professor Eguchi, Kyoto
Robbert Dijkgraaf University of Amsterdam
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Dear Tohru, Welcome to the crowd of wise old
men. Only at the respectable age of 60 the
capacity comes to digest and interpret all those
deep thoughts that we may have had earlier. My
sincere congratulations, Gerard t Hooft  
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30 Years of Geometry and Physics
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CFT on Riemann Surfaces
Geometry
Algebra
Vertex algebras Frobenius algebras
Riemann surfaces Moduli spaces
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4d Gauge Theories vs 2d CFT

• SL(2,Z) S-duality in N4 gauge theories ? modular
  invariance of a quantum CFT on 2-torus
  Montonen-Olive

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N 4 gauge theories on ALE spaces

• Resolved singularity

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N 4 gauge theories on ALE spaces
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Seiberg-Witten solution of N 2

• Period matrix of a Riemann surface (SW spectral
  curve)
• Masses of dyons are given by periods of
  meromorphic 1-form.

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Gravitational Terms

• Gravitational coupling

Quantum CFT chiral determinant
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N 1 Supersymmetry

• N 1 SUSY superpotential
• Only planar diagrams contribute D.-Vafa

Effective geometry
Large N matrix models
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Hermitian Matrix Model
Filling fractions
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Genus Expansion
g0, large N limit is captured by effective
geometry spectral curve H(x,y) 0 of matrix
model pair (C,w) (Riemann surface, meromorphic
one-form) Higher genus corrections? Quantum
deformation of CFT on Riemann surfaces
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Reduction to eigenvalues
Dysons Coulomb gas model Effective
force/resolvent
x
complex plane
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N?8 Effective Geometry
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N?8 Effective Geometry
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Loop Equations
Hyperelliptic spectral curve
filling fractions ti
x
Meromorphic one-form y(x)dx
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Phase Space
algebraic curve level set Hamilton-Jacobi
theory branch pts turning pts
Liouville form
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Eynard-Orantin Solution
Recursion relations for correlators, completely
geometric! Lowers genus, adds more
insertions. Reduces to Bergmann kernel
w
z
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Graphical interpretation
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Matrix models and chiral CFT
eigenvalue density collective
field eigenvalues are fermions one-loop
correction chiral determinant
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Quantization of spectral curve
Variations of complex structure Trivial
variations Cohomology class of one-form does
not change This fixes
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Chiral quantum boson
Deformation of complex structure by
stress-tensor Here self-coupling
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Interacting chiral boson theory
Action Total derivative sum over
singularities (branch points) Extra term
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Matrix models and chiral bosons

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Two-point function Bergmann kernel Close to
branch point LHôpitals rule
w
z
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Recursion relations of Eynard-Orantin Deriv
e the partition function
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Hidden affine SL(2,R) structure
First order formalism spin (0,1) (b,g)
system Action SL(2,R) currents (twisted
Wakimoto representation at critical level)
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Twistor-like construction Clearly suggest a
3d relation modified SL(2,R) Chern-Simons
theory, 3d quantum gravity?
Chiral boson ? Chiral Fermions
Free Fermions associated to a D-module
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Quantum Spectral Curves
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N 4 Gauge Theories on ALE Spaces
Taub-NUT space
ALE singularity
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M-Theory Duality
D.-Hollands-Sulkowski-Vafa
M Theory 11 dim
Type IIA 10 dim
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Intersecting Branes
Itzhaki-Kutasov-Seiberg
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Free Fermions

• 4-6 strings are 2-dim chiral fermions on
  intersection 2-torus
• Action
• Symmetry

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Level-Rank Duality

• Conformal embedding
• Central charge WZW model
• Level-rank duality

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D-branes and D-modules

• Intersecting branes

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Quantization

• What is role of string coupling ?
• Magnetic flux

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Non-Commutative Geometry

• Heisenberg algebra

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D-modules

• Module for open string algebra
• Non-commutative CFT, fermions are elements
  D-module

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6
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String Algebras

• Geometry, closed strings commutative algebra

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Classical Quantum Curves

• Classical curve

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Example double-scaled matrix models
Genus zero curve/KP hierarchy Douglas
Isomorphism (uniformization)
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Example c1 string

• Classical case
• Quantum case
• Toda hierarchy, two-matrix model

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Sato Grassmannian
state in fermion Fock space
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Circle of Ideas
CFT
Riemann surfaces Algebraic curves
D-modules
Integrable hierarchies
Free fermions
Hartelijk Gefeliciteerd! Happy Birthday!
Non-commutative geometry
Matrix Models
TFT
SUSY Gauge Theories