Narrow Results

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to $C^{\infty }$ potentials.

We give a construction of stationary discs and the indicatrix for manifolds of higher codimension which is a partial analog of L. Lempert's theory of stationary discs for strictly convex hypersurfaces. This leads to new invariants of the CR structure in higher codimension linked with the contact structure of the conormal bundle.

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kähler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group $H$ of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant $\eta$-invariants which play a fundamental role in the equivariant analog of Getzler’s spectral flow formula. As a consequence, we establish a relation between equivariant $\eta$-invariants and equivariant Maslov triple indices in the splitting of manifolds.

Within extensions of the new quantization rule approach in arbitrary dimensions, the Maslov indices and energy spectra of some exactly solvable potentials are presented. We find that the Maslov index for the harmonic oscillator in three dimensions agrees well with those obtained by other methods.

The present article delves into some symplectic features arising in basic knot theory. An interpretation of the writhing number of a knot (with reference to a plane projection thereof) is provided in terms of a phase function analogous to those encountered in geometrical optics, its variation upon switching a crossing being akin to the passage through a caustic, yielding a knot theoretical analogue of Maslov's theory, via classical fluidodynamical helicity. The Maslov cycle is given by knots having exactly one double point, among those having a fixed plane shadow and lying on a semi-cone issued therefrom, which turn out to build up a Lagrangian submanifold of Brylinski's symplectic manifold of (mildly) singular knots. A Morse family (generating function) for this submanifold is determined and can be taken to be the Abelian Chern–Simons action plus a source term (knot insertion) appearing in the Jones–Witten theory. The relevance of the Bohr–Sommerfeld conditions arising in geometric quantization are investigated and a relationship with the Gauss linking number integral formula is also established, together with a novel derivation of the so-called Feynman–Onsager quantization condition. Furthermore, an additional Chern–Simons interpretation of the writhe of a braid is discussed and interpreted symplectically, also making contact with the Goldin–Menikoff–Sharp approach to vortices and anyons. Finally, a geometrical setting for the ground state wave functions arising in the theory of the Fractional Quantum Hall Effect is established.

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π₁(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S³. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π₁(S³ \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

We investigate a Floer type cohomology on cosymplectic manifolds M. To do this, we study a symplectic type action functional on the universal covering space of the loop space of contractible loops in M and the moduli space of gradient flow lines of the functional. The cochain complex induced by the critical points of the functional produces Floer type cohomology of M which is naturally isomorphic to a quantum type cohomology of M. We have an Arnold type theorem for Hamiltonian cosymplectomorphisms on compact semipositive cosymplectic manifolds. As an example, we consider the product of a Calabi–Yau 3-fold and the unit circle.

We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in ℝ²ⁿ is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behavior of Oh's spectral sequence with respect to this product. As further applications, we prove existence of holomorphic disks with boundaries on Lagrangians as well as new results on Lagrangian intersections.