Localisation in Equivariant Cohomology

Catherine C. Notman; Muaadh A. Sanabani

Home/Archive/arXiv

arXiv · 2309.11405

Localisation in Equivariant Cohomology

Catherine C. Notman·Muaadh A. Sanabani

View original source ↗Read source document ↗Export metadata ↓

Abstract

Equivariant cohomology, a captivating fusion of symmetry and abstract mathematics, illuminates the profound role of group actions in shaping geometric structures. At its core lies the Atiyah-Bott Localization Theorem, a mathematical jewel unveiling the art of localization. This theorem simplifies intricate integrals on symplectic manifolds with Lie group actions, revealing the hidden elegance within complexity. Our paper embarks on a journey to explore the theoretical foundations and practical applications of equivariant cohomology, demonstrating its transformative power in diverse fields, from theoretical physics to geometry. As we delve into the symphonic interplay between geometry and symmetry, readers are invited to witness the beauty of mathematical patterns emerging from abstraction. This mathematical voyage unveils the harmonious marriage of symmetry, topology, and elegance in the captivating realm of equivariant cohomology.

Explore related subjects

math.SG math-ph math.AG math.AT math.DG math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Catherine C. Notman, Muaadh A. Sanabani. 2023-09-20. Localisation in Equivariant Cohomology. https://arxiv.org/abs/2309.11405

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

KEEP EXPLORING

Related papers

arXiv2026

Darboux's Theorem in $p$-adic symplectic geometry

We prove a non Archimedean Darboux's Theorem: any two symplectic forms on a $p$-adic analytic manifold are locally isomorphic. Understanding local problems such as the existence of flows or the normalization of singularities in the theory of integrable systems, is essential to understand the physics behind these systems. Our result tells us that the phase space defined by a $p$-adic manifold is locally standard, allowing us to concentrate on the equations defining the dynamics rather than on the space itself. Our proof uses a non Archimedean version of Moser's Path Method to push one symplectic form onto another one by a flow. A central technical contribution of the paper is the proof that the flow is given by a power series with non zero radius of convergence, which requires geometric analytic estimates and does not follow from algebraic considerations. The main difficulty of the paper is analytic: an ODE will not have a long-time solution even in a neighborhood of a fixed point but we are able to find a condition for a long-time solution to exist, which allows us to tweak the ODE in Moser's Path Method so that it satifies the condition. As a global application, we derive a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume, which generalizes a classical theorem of J-P. Serre.

math.SG↗

arXiv2026

A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

math.SG↗

arXiv2026

The Legendrian Hopf Link has exactly two Lagrangian fillings

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such Lagrangian fillings: we show these are all. Specifically, we use a type of neck-stretching procedure to construct a pseudoholomorphic conic fibration that makes a given arbitrary exact Lagrangian filling fiber over a real curve, under a global pseudoholomorphic Lefschetz fibration. This then allows for an explicit Hamiltonian isotopy to be constructed from any given Lagrangian filling to one of two known standard fillings.

math.SG↗