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arXiv · 1403.3307

When is a symplectic quotient an orbifold?

Hans-Christian Herbig·Gerald W. Schwarz·Christopher Seaton
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Abstract

Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary $\operatorname{SU}_2$-modules yield symplectic quotients that are $\mathbb{Z}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.

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Hans-Christian Herbig, Gerald W. Schwarz, Christopher Seaton. 2014-03-31. When is a symplectic quotient an orbifold?. https://doi.org/10.1016/j.aim.2015.04.016

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