Invariants of algebraic curves and topological expansion

@article{Eynard2007InvariantsOA,
  title={Invariants of algebraic curves and topological expansion},
  author={Bertrand Eynard and Nicolas Orantin},
  journal={Communications in Number Theory and Physics},
  year={2007},
  volume={1},
  pages={347-452}
}
  • B. EynardN. Orantin
  • Published 14 February 2007
  • Mathematics
  • Communications in Number Theory and Physics
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed… 

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References

SHOWING 1-10 OF 84 REFERENCES

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural

Intersection theory, integrable hierarchies and topological field theory

The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory,

Topological Strings and Integrable Hierarchies

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using -algebra symmetries which encode the symmetries of holomorphic

Topological String Theory on Compact Calabi–Yau: Modularity and Boundary Conditions

The topological string partition function Z(λ,t,t) =exp(λ2 g-2 Fg(t, t)) is calculated on a compact Calabi–Yau M. The Fg(t, t) fulfil the holomorphic anomaly equations, which imply that ψ=Z

2D gravity and random matrices

Two-matrix model with semiclassical potentials and extended Whitham hierarchy

We consider the two-matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals (hard

Complex curve of the two-matrix model and its tau-function

We study the Hermitian and normal two-matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general

Exact solution of the O(n) model on a random lattice

...