Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces
This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and the self-dual Yang-Mills equations, and describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.
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algebraic curve algebraic geometry coefﬁcients cohomology groups commutative compact support conserved quantities constant coordinate curvature deﬁned Deﬁnition denote described det(w differential equation differential operator dimension dimensional dimH1(M eigenfunction eigenvalues equa example fact ﬁbre ﬁnd ﬁnite ﬁrst ﬁxed ﬂat ﬂow gauge geodesic given gives Grassmannian Grres Hamiltonian harmonic maps holo holomorphic function holomorphic line bundle holomorphic map holomorphic sections holomorphic vector bundle holonomy identiﬁed inﬁnite integrable systems isomorphism KdV equation kernel Lax form Lax pair Lax pair equations Lie algebra long exact sequence loop group matrix-valued meromorphic function morphic multiplication non-linear open sets parameter poles proof Proposition quotient rapidly decreasing rational functions real axis Riemann sphere rigid body satisﬁes SDYM equations Segal self-dual Serre duality sheaf short exact sequence solitons spectral curve spectrum subspace symplectic manifold tangent theorem tion topology torus transition functions trivial bundle upper half-plane values vanishes vector ﬁeld vector space zero