Related papers: Integrable systems and holomorphic curves

It has been proved by S.L.Ziglin, for a large class of 2-degree-of-freedom (d.o.f) Hamiltonian systems, that transverse intersections of the invariant manifolds of saddle fixed points imply infinite branching of solutions in the complex…

This manuscript describes in detail the symplectic sum formulas in Gromov-Witten theory and related topological and analytic issues. In particular, we analyze and compare two analytic approaches to these formulas. The Ionel-Parker formula…

We develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we construct Gibbons-Tsarev systems associated to moduli space of algebraic…

We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures emerging in the Gromov-Witten theory of…

Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi} \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of…

We prove a non-squeezing result for infinite-dimensional Hamiltonian flows using non-standard model theory. For this we prove the existence of a corresponding family of pseudoholomorphic spheres and characterize the maximal time in terms of…

We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian…

We establish a correspondence between an infinite set of special solutions of the (classical) modified sinh-Gordon equation and a set of stationary states in the finite-volume Hilbert space of the integrable 2D QFT invented by V.A. Fateev.…

In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes",…

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a…

We consider the moduli space of flat connections on the Riemann surface with marked points. The new efficient parametrization is suggested and used to construct an integrable model on the moduli space. A family of commuting Hamiltonians is…

We show that the fundamental group of every enumeratively rationally connected closed symplectic manifold is finite. In other words, if a closed symplectic manifold has a non-zero Gromov-Witten invariant with two point insertions, then it…

We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these…

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in our…

It is known that the Schr\"odinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schr\"odinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this…

Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…

We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of…

With the help of the Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field, we find an integrable subsystem of SU(2) Yang-Mills theory coupled to the dilaton. Here integrability means the existence of infinitely many…

Let M be a connected, symplectic 4-manifold. A semitoric integrable system on M essentially consists of a pair of independent, real-valued, smooth functions J and H on the manifold M, for which J generates a Hamiltonian circle action under…

We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\Omega$-spectrum $\mathit{IP}_*$…