Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1910.04752

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions

Benjamin Küster·Pablo Ramacher
View original source ↗Read source document ↗Export metadata ↓

Abstract

We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden-Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Benjamin Küster, Pablo Ramacher. 2021-03-18. Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions. https://arxiv.org/abs/1910.04752

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Symplectic Yang-Mills Theory

On a symplectic manifold, any differential two-form has a natural decomposition into two components: a primitive part and a non-primitive one. Applying this decomposition to the curvature two-form of a principal bundle over a symplectic manifold, we obtain a natural splitting of the Yang-Mills (YM) functional into two functionals that intrinsically depend on the symplectic structure: the primitive Yang-Mills (PYM) functional and the trace Yang-Mills (TYM) functional. We work out the basic properties of the critical solutions of these two functionals. The PYM functional in particular exhibits many of the desirable properties of the YM functional, including the ellipticity of its Euler-Lagrange equations and an algebraic classification of its flat solutions on G-bundles. We also prove a monotonicity formula for the PYM functional as a first step towards characterizing its moduli space of solutions.

math.SG↗
arXiv2026

Existence of a positive hyperbolic orbit in three-dimensional Reeb flows

Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with $b_1=0$ such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for $b_1>0$, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero $J$-holomorphic curves counted by the $U$-map. Under the contrary assumption, the number of simple orbits in each action interval $[L,2L]$ is uniformly bounded with respect to $L$. We use this property to study ECH generators and genus zero $U$-curves.

math.SG↗
arXiv2026

New systems of log-canonical coordinates on $SL(2, \mathbb{C})$ character varieties of compact Riemann surfaces

We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.

math.SG↗