Intersection numbers in quasi-Hamiltonian reduced spaces
Lisa Jeffrey, Joon-Hyeok Song
备注：
29 pages; this article presents the main results of the 2004 University of Toronto Ph.D. thesis of J.-H. Song supervised by L.C. Jeffrey. In this version, a close analogue of Alekseev-Meinrenken-Woodward's localization theorem for quasi-Hamiltonian spaces is proved. The summation in Theorem 6.1 is a summation over dominant weights, correcting an error in the previous version, with the result that Theorem 5.5 is analogous to Theorem 6.1 but not equivalent to it. We are posting the final version of the article (as it is published)
In this paper we prove a residue formula for intersection pairings of reduced spaces of certain quasi-Hamiltonian G-spaces, by constructing the corresponding Hamiltonian G-space. Our argument closely follows the methods of a 1998 paper of the first author and F. Kirwan on intersection numbers in moduli spaces (for G=SU(n)).
For the more general class of compact Lie groups treated by Alekseev, Meinrenken and Woodward, we rely on results of Szenes and Brion-Vergne concerning diagonal bases. Our result is a close analogue of the result of Alekseev-Meinrenken-Woodward.
We study the problem of determining which diffeomorphism classes of Kähler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.
In this paper we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (non-transversal) case.
Two degenerate intersections of Clifford tori in \(S^2\times S^2\) and \(\CP^2\)
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University（数学与复杂系统教育部重点实验室，北京师范大学数学科学学院）
In this paper, we prove that if a continuous Hamiltonian flow fixes the points in an open subset $U$ of a symplectic manifold $(M,ω)$, then its associated Hamiltonian is constant at each moment on $U$. As a corollary, we prove that the Hamiltonian of compactly supported continuous Hamiltonian flows is unique both on a compact $M$ with smooth boundary $\del M$ and on a non-compact manifold bounded at infinity. An essential tool for the proof of the locality is the Lagrangian intersection theorem for the conormals of open subsets proven by Kasturirangan and the author, combined with Viterbo's sc…
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We consider the Hamiltonian flow on complex complete intersection surfaces with isolated singularities, equipped with the Jacobian Poisson structure. More generally we consider complete intersections of arbitrary dimension equipped with Hamiltonian flow with respect to the natural top polyvector field, which one should view as a degenerate Calabi-Yau structure. Our main result computes the coinvariants of functions under the Hamiltonian flow. In the surface case this is the zeroth Poisson homology, and our result generalizes those of Greuel, Alev and Lambre, and the authors in the quasihomogen…
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We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the $C^0$-metric and the Hofer metric. This answers a variant of a question of Entov--Polterovich--Py which is one of the open problems listed in the monograph of McDuff--Salamon. Throughout the proof, we make extensive use of the idea of working with different coefficient fields in quantum cohomology rings. As a by-product of the arguments in the paper, we answer a question of Polterovich--…
In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians $C^0$-converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that $C^0$-convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force …
In this note we observe that Arnold conjecture for the Hamiltonian maps still holds on weighted projective spaces $\CP^n({\bf q})$, and that Arnold conjecture for the Lagrange intersections for $(\CP^n({\bf q}), \RP^n({\bf q}))$ is also true if each weight $q_i\in {\bf q}=\{q_1,..., q_{n+1}\}$ is odd.
We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian G-spaces.
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We classify all momentum polytopes of rank one for multiplicity free quasi-Hamiltonian $K$-manifolds for simple and simply connected Lie groups $K$ by using the methods developed in a recent paper by F. Knop. This leads to lots of new concrete examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.
We interpret the chiral WZNW model with general monodromy as an infinite dimensional quasi-Hamiltonian dynamical system. This interpretation permits to explain the totality of complicated cross-terms in the symplectic structures of various WZNW defects solely in terms of the single concept of the quasi-Hamiltonian fusion. Translated from the WZNW language into that of the moduli space of flat connections on Riemann surfaces, our result gives a compact and transparent characterisation of the symplectic structure of the moduli space of flat connections on a surface with k handles, n boundaries a…
