Groupes et algébres de lie

Introduction to lie algebras and representation theory

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.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0002/CNTP-2007-0001-0002-a004.pdf

Invariants of algebraic curves and topological expansion

Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these integrals from the standard descendent potential (for any target X). We use virtual localization and classical degeneracy calculations to find trigonometric closed form solutions for special Hodge integrals over the moduli space of pointed curves. These formulas are applied to two computations in Gromov-Witten theory for Calabi-Yau 3-folds. The genus g, degree d multiple cover contribution of a rational curve is found to be simply proportional to the Euler characteristic of M_g. The genus g, degree 0 Gromov-Witten invariant is calculated (in agreement with recent string theoretic calculations of Gopakumar-Vafa and Marino-Moore). Finally, with Zagier's help, our Hodge integral formulas imply a general genus prediction of the punctual Virasoro constraints applied to the projective line.

Hodge integrals and Gromov-Witten theory

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov - Witten classes and their descendents

Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

An explicit reciprocal transformation between a two-component generalization of the Camassa–Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established. This transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented

A Two-component Generalization of the Camassa-Holm Equation and its Solutions

An explicit reciprocal transformation between a 2-component generalization of the Camassa-Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented.

A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions

We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov–Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.

/pdf/bihamiltonian-hierarchies-in-2d-topological-field-theory-at-2usjqb7lfb.pdf

Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs vt +[ φ(v)]x = 0. Under certain genericity assumptions it is proved that any bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so-called quasiMiura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi-Hamiltonian structure of hydrodynamic type (the quasi-triviality theorem). We also describe, following [35], the invariants ofsuchbi-Hamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives. c

/pdf/on-hamiltonian-perturbations-of-hyperbolic-systems-of-3ivh6jzic8.pdf

Youjin Zhang

Papers

Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

A Two-component Generalization of the Camassa-Holm Equation and its Solutions

A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions

Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws I: Quasi-Triviality of Bi-Hamiltonian Perturbations