Related papers: The twisting index in semitoric systems
About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vu Ngoc by means of five invariants. A standard example of such a system is the coupled spin-oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$.…
About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2…
The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they…
Semitoric systems are a special class of four-dimensional completely integrable systems where one of the first integrals generates an $\mathbb{S}^1$-action. They were classified by Pelayo & Vu Ngoc in terms of five symplectic invariants…
Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vu Ngoc about a decade ago in terms of five symplectic invariants. If a semitoric…
Since simple semitoric systems were classified about fifteen years ago, and semitoric systems five years ago, we want to move a step forward to almost-toric systems. We give a classification of compact almost-toric systems in dimension four…
Dissipative mechanical systems on the torus with a friction that is proportional to the velocity are modeled by conformally symplectic maps on the annulus, which are maps that transport the symplectic form into a multiple of itself (with a…
This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…
Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and…
Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with…
In the first part of this note, we review and compare various instances of the notion of twisted coefficient system, a.k.a. polynomial functor, appearing in the literature. This notion hinges on how one defines the degree of a functor from…
We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors…
We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural…
In this note we present a formula for the equivariant index of the cohomological complex obtained from localization of $\mathcal{N}=2$ SYM on simply-connected compact four-manifolds with a $T^2$-action. Knowledge of said index is essential…
Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the…
This paper is a continuation of our 2005 paper on complex topology and its implication on invertibility (or non-invertibility). In this paper, we will try to classify the complexity of inversion into 3 different classes. We will use…
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…
A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…
We introduce the topologically twisted index for four-dimensional $\mathcal N=1$ gauge theories quantized on ${\rm AdS}_2 \times S^1$. We compute the index by applying supersymmetric localization to partition functions of vector and chiral…
The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require…