arXiv subjects

Lisa Jeffrey

Publications and source records attributed to Lisa Jeffrey.

21 records · Page 2Linked to original sources

arXiv2004

Group-valued Implosion and Parabolic Structures

The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian $K$-manifolds, where $K$ is a simply connected compact Lie group. The imploded cross-section of the double $K\times K$ turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian $K$-manifolds, such as the ``spinning $2n$-sphere'' for $K=\SU(n)$. Secondly we construct a universal (``master'') moduli space of parabolic bundles with structure group $K$ over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of $K$ and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for $K=\SU(2)$), the master moduli space turns out to be a complex algebraic variety.

math.SG↗

arXiv2003

Localization theorems by symplectic cuts

Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M_0 the symplectic reduction at zero. Denote by κ_0 the Kirwan map H^*_T(M)-> H^*(M_0). For an equivariant cohomology class η\in H^*_T(M) we present new localization formulas which express \int_{M_0} κ_0(η) as sums of certain integrals over the connected components of the fixed point set M^T. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T) we give a new proof of the Jeffrey-Kirwan localization formula.

math.SG↗

arXiv2001

Symplectic implosion

Let $K$ be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian $K$-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of $K$. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of $K$, which turns out to be identical to an affine variety studied by Gelfand, Vinberg, Popov, and others. Finally we show that ``quantization commutes with implosion''.

math.SG↗