Invariants of algebraic curves and topological expansion

By B. Eynard, N. Orantin. 2007

247 citations recorded by the source corpus.

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 in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1cc9d7704fa6c0a109af0bf9443de4a70e02cc0e

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Citations in the corpus

Invariants of spectral curves and intersection theory of moduli spaces of complex curves (2011, methodology): “Graphical representation In [10, 13], it was observed that the definition of W (g) n can be written diagramatically:”
Liouville quantum gravity and KPZ (2008, background): “More recently, a purely combinatorial approach to discretized quantum gravity has been successful [12, 15–19, 21–28, 33, 47, 75, 106–108, 113–115, 117, 128], as well as the so-called topological expansion involving higher-genus random surfaces [35, 36, 38, 69 , 70, 72].”
Open string amplitudes and large order behavior in topological string theory (2006, methodology): “Of course, if our formalism holds in general, one should be able to prove direclty in Gromov– Witten theory that the generating functionals of open (or relative) Gromov–Witten invariants satisfy the recursion relations typical of loop equations found in [29, 32 ].”
A-polynomial, B-model, and quantization (2011, methodology): “ndices J denote p~J = {pi}i∈J. Then, for N= {1,...,n} and the corresponding set of points p~N = {p 1,...,pn} 4For reasons that will become clear later, we choose a sign opposite to the conventions of [12]. – 8 – p 1 p n p 1 p n ... = ... + J,m Σ p p g g−1 . . . p . . . . . . . . . m g−m p p J N/J Fi”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, methodology): “The known methods for quantizing H(x, y) involve matrix models [24, 25], and express Ĥ(x̂, ŷ; q) not as a finite polynomial in its three arguments (cf.”
Remodeling the B-Model (2007, methodology): “As a first guess, one could try to apply directly Eynard and Orantin to the mirror curve and see what the objects correspond to in topological string theory.”