Hamiltonian diffeomorphisms of toric manifolds and flag manifolds

In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.

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Nonlinear flag manifolds as coadjoint orbits

Annals of Global Analysis and Geometry ◽

10.1007/s10455-020-09725-6 ◽

2020 ◽

Vol 58 (4) ◽

pp. 385-413

Author(s):

Stefan Haller ◽

Cornelia Vizman

Keyword(s):

Finite Sequence ◽

Flag Manifolds ◽

Coadjoint Orbits ◽

Hamiltonian Diffeomorphisms

Abstract A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

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Invariant generalized complex geometry on maximal flag manifolds and their moduli

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2021.104108 ◽

2021 ◽

pp. 104108

Author(s):

Elizabeth Gasparim ◽

Fabricio Valencia ◽

Carlos Varea

Keyword(s):

Complex Geometry ◽

Flag Manifolds ◽

Generalized Complex

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Gauge theories on compact toric manifolds

Letters in Mathematical Physics ◽

10.1007/s11005-021-01419-9 ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Giulio Bonelli ◽

Francesco Fucito ◽

Jose Francisco Morales ◽

Massimiliano Ronzani ◽

Ekaterina Sysoeva ◽

...

Keyword(s):

Partition Function ◽

Gauge Theories ◽

Blow Up ◽

Gauge Coupling ◽

Piecewise Constant Function ◽

Partition Functions ◽

Piecewise Constant ◽

Toric Manifolds ◽

Holomorphic Anomaly ◽

The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

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