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Lisa C. Jeffrey

Publications and source records attributed to Lisa C. Jeffrey.

At least 19 recordsLinked to original sources
arXiv2025

Poisson structure on character varieties. II

Let G be a complex reductive group and D a finite subset of a compact Riemann surface X. It was shown in [BJ] that the moduli space of G-characters of the complement of D in X has a natural Poisson structure. We show that the moduli space of logarithmic G-connections on X singular over D has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic G-connections to the moduli space of G-characters is Poisson structure preserving.

math.SG↗
arXiv2022

Poisson maps between character varieties: gluing and capping

Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.

math.AG↗
arXiv2021

Flat connections and the commutator map for SU(2)

We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).

math.SG↗
arXiv2018

The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.

math.AT↗
arXiv2017

The SU(2)-character variety of the closed surface of genus 2

We study the symplectic geometry of the SU(2)-representation variety of the compact oriented surface of genus 2. We use the Goldman flows to identify subsets of the moduli space with corresponding subsets of $\mathbb P^3(\mathbb C)$. We also define and study two antisymplectic involutions on the moduli space and their fixed point sets.

math.SG↗
arXiv2017

Hyperfunctions, the Duistermaat-Heckman theorem, and Loop Groups

In this article we investigate the Duistermaat-Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite dimensional manifolds, this more general theory seems to be necessary in order to accomodate the existence of the infinite order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. We will quickly review of the theory of hyperfunctions and their Fourier transforms. We will then apply this theory to construct a hyperfunction analogue of the Duistermaat-Heckman distribution. Our main goal will be to study the Duistermaat-Heckman hyperfunction of $ΩSU(2)$, but in getting to this goal we will also characterize the singular locus of the moment map for the Hamiltonian action of $T\times S^1$ on $ΩG$. The main goal of this paper is to present a Duistermaat-Heckman hyperfunction arising from a Hamiltonian action on an infinite dimensional manifold.

math.SG↗
arXiv2013

The module structure of the equivariant K-theory of the based loop group of SU(2)

Let $G=SU(2)$ and let $ΩG$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^*(ΩG)$ and in particular show that it is a direct product of copies of $K^*_G(\pt) \cong R(G)$. (We intend to describe in detail the $R(G)$-algebra (i.e. product) structure of $K^*_G(ΩG)$ in a forthcoming companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant $K$-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience.

math.AT↗
arXiv2012

Symplectic Quantum Mechanics and Chern-Simons Gauge Theory I

In this article we describe the relation between the Chern-Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional.

math-ph↗
arXiv2012

The product structure of the equivariant K-theory of the based loop group of SU(2)

Let G=SU(2) and let ΩG denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H^*(ΩG) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k goes to infinity of the symmetric subalgebra in the exterior algebra Λ(x_1, ...,x_k) in the variables x_1, ..., x_k. We compute the R(G)-algebra structure of the G-equivariant K-theory of ΩG in a way which naturally generalizes the classical computation of the ordinary cohomology ring of ΩG as a divided polynomial algebra Γ[x]. Specifically, we prove that K^*_G(ΩG) is an inverse limit of the symmetric (S_{2r}-invariant) subalgebra of K^*_G((P^1)^{2r}), where the symmetric group S_{2r} acts in the natural way on the factors of the 2r-fold product (P^1)^{2r} and G acts diagonally via the standard action on each complex projective line P^1.

math.KT↗
arXiv2009

Real loci of based loop groups

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $σ$. This induces an involutive automorphism $τ$ of the based loop space $Ω(G)$. There exists a maximal torus $T\subset G$ such that the canonical action of $T\times S^1$ on $Ω(G)$ is compatible with $τ$ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of $Ω(G)$ and $Ω(G)^τ$ (fixed point set of $τ$) under the $T\times S^1$ moment map on $Ω(G)$ are equal. The space $Ω(G)^τ$ is homotopy equivalent to the loop space $Ω(G/K)$ of the Riemannian symmetric space $G/K$. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in $\mathbb{Z}_2$ of $Ω(G)$ and $Ω(G/K)$. Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe.

math.DG↗
arXiv2008

Goldman flows on the Jacobian

We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian.

math.DG↗
arXiv2006

Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of singularities of M(n,d). The case when n=2 is discussed in detail. Finally Witten's integral is considered for this singular case.

math.AG↗
arXiv2005

Cohomology pairings on singular quotients in geometric invariant theory

In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant theoretic quotients for which semistability is not necessarily the same as stability (although we make some weaker assumptions on the action). We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.

math.AG↗
arXiv2005

The Kirwan map, equivariant Kirwan maps, and their kernels

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map $κ_K$. We start with the case when K is a torus T: we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S in T, and show how to recover from this the kernel of $κ_T$, as described by Tolman and Weitsman. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [Je], [Je-Ma]). For a compact nonabelian Lie group K we will use the ``non-abelian localization formula'' of [Je-Ki1] and [Je-Ki2] to establish relationships -- some of them obtained by Tolman and Weitsman in [To-We] -- between $\ker(κ_K)$ and $\ker(κ_T)$, where T is a maximal torus in K. An Appendix generalizes Theorem 1.8 to the case of singular values of $κ_T$.

math.SG↗
arXiv2003

Distinguishing the Chambers of the Moment Polytope

Let M be a compact manifold with a Hamiltonian T action and moment map Phi. The restriction map in equivariant cohomology from M to a level set Phi^{-1}(p) is a surjection, and we denote the kernel by I_p. When T has isolated fixed points, we show that I_p distinguishes the chambers of the moment polytope for M. In particular, counting the number of distinct ideals I_p as p varies over different chambers is equivalent to counting the number of chambers.

math.SG↗