Characterization of symplectic forms induced by some tangent G-structures of higher order

Let (M, ω) be a symplectic manifold induced by an integrable G-structure P on M . In this paper, we characterize the symplectic manifolds induced by the tangent lifts of higher order r ≥ 1 of G-structure P, from M to TrM .

Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

Integrable Systems on Singular Symplectic Manifolds: From Local to Global

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a $b$-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and $b$-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set $Z$ of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on $Z$.

A Family of Semitoric Systems with Four Focus–Focus Singularities and Two Double Pinched Tori

We construct a one-parameter family $$F_t=(J, H_t)_{0 \le t \le 1}$$ F t = ( J , H t ) 0 ≤ t ≤ 1 of integrable systems on a compact 4-dimensional symplectic manifold $$(M, \omega )$$ ( M , ω ) that changes smoothly from a toric system $$F_0$$ F 0 with eight elliptic–elliptic singular points via toric type systems to a semitoric system $$F_t$$ F t for $$ t^-< t < t^+$$ t - < t < t + . These semitoric systems $$F_t$$ F t have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at $$t= \frac{1}{2}$$ t = 1 2 , the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

Subcritical polarisations of symplectic manifolds have degree one

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni–Wood.

Spin Matrix theory string backgrounds and Penrose limits of AdS/CFT

Spin Matrix theory (SMT) limits provide a way to capture the dynamics of the AdS/CFT correspondence near BPS bounds. On the string theory side, these limits result in non-relativistic sigma models that can be interpreted as novel non-relativistic strings. This SMT string theory couples to non-relativistic U(1)-Galilean background geometries. In this paper, we explore the relation between pp-wave backgrounds obtained from Penrose limits of AdS5 × S5, and a new type of U(1)-Galilean backgrounds that we call flat-fluxed (FF) backgrounds. These FF backgrounds are the simplest possible SMT string backgrounds and correspond to free magnons from the spin chain perspective. We provide a catalogue of the U(1)-Galilean backgrounds one obtains from SMT limits of string theory on AdS5 × S5 and subsequently study large charge limits of these geometries from which the FF backgrounds emerge. We show that these limits are analogous to Penrose limits of AdS5 × S5 and demonstrate that the large charge/Penrose limits commute with the SMT limits. Finally, we point out that U(1)-Galilean backgrounds prescribe a symplectic manifold for the transverse SMT string embedding fields. This is illustrated with a Hamiltonian derivation for the SMT limit of a particle.

Floer-Novikov cohomology and symplectic fixed points, revisited

This note is mostly an exposition of a few versions of Floer-Novikov cohomology with a few new observations. For example, we state a lower bound for the number of symplectic fixed points of a non-degenerate symplectomorphism, which is symplectomorphic isotopic to the identity, on a compact symplectic manifold, more precisely than previous statements in [14,10].