[Paper Review] The Full Orbifold $K$-theory of Abelian Symplectic Quotients

Rebecca Goldin, Megumi Harada+2 more|ArXiv.org|Dec 29, 2008

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper develops an explicit algorithm for computing the full orbifold K-theory of abelian symplectic quotients using inertial K-theory and integral K-theoretic Kirwan surjectivity. It proves that the full orbifold K-theory of weighted projective spaces arising as symplectic quotients is torsion-free, providing a complete description over the integers via explicit presentation of the ring structure using generators and relations from fixed-point data and sector contributions.

ABSTRACT

In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold $K$-theory of an orbifold ${\mathfrak X}$, analogous to the Chen-Ruan orbifold cohomology of ${\mathfrak X}$ in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when ${\mathfrak X}$ arises as an abelian symplectic quotient. Our methods are integral $K$-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the $K$-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold $K$-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of weighted projective spaces, we show that the associated invariant is torsion-free.

Motivation & Objective

• To develop a systematic method for computing the full orbifold K-theory of orbifolds arising as abelian symplectic quotients.
• To generalize inertial K-theory to non-locally free actions, extending prior work on stringy K-theory.
• To establish an integral K-theoretic analogue of Kirwan surjectivity for symplectic quotients.
• To provide explicit presentations of the full orbifold K-theory ring for weighted projective spaces.
• To prove that the full orbifold K-theory of such weighted projective spaces is torsion-free over ℤ.

Proposed method

• Introduce inertial K-theory $NK^{ullet}_{T}(M)$ as a generalization of stringy K-theory to non-locally free actions on stably complex manifolds.
• Use the $T$-fixed point sets and their normal bundles to reformulate the stringy product via a $ atural$-product, analogous to orbifold cohomology.
• Apply integral K-theoretic Kirwan surjectivity theorems to relate the inertial K-theory of the total space to the orbifold K-theory of the quotient.
• Construct the full orbifold K-theory ring as a quotient of a polynomial ring over $\mathbb{Z}[u,u^{-1}]$ by ideals $\mathcal{I}$ (product relations) and $\mathcal{J}$ (kernel of Kirwan maps).
• Leverage localization techniques in equivariant K-theory to compute the K-theory of fixed-point submanifolds and their contributions to the orbifold ring.
• Use explicit computation on $\mathbb{P}^2_{(1,2,4)}$ to verify the general framework and confirm torsion-freeness.

Experimental results

Research questions

• RQ1 How can the full orbifold K-theory of abelian symplectic quotients be computed explicitly over the integers?
• RQ2 What is the structure of the full orbifold K-theory ring for weighted projective spaces obtained as symplectic quotients?
• RQ3 Does the full orbifold K-theory of such weighted projective spaces contain torsion elements?
• RQ4 How does inertial K-theory generalize stringy K-theory in the non-locally free case?
• RQ5 Can integral K-theoretic Kirwan surjectivity be used to compute orbifold K-theory for non-global quotient stacks?

Key findings

• The full orbifold K-theory of a weighted projective space $\mathbb{P}^n_b$ is isomorphic to $\mathbb{Z}[u,u^{-1},\alpha_0,\ldots,\alpha_\ell]/(\mathcal{I} + \mathcal{J})$, where $\mathcal{I}$ encodes product relations and $\mathcal{J}$ encodes the kernels of Kirwan maps.
• The ring structure is fully determined by the fixed-point data of the circle action and the sector contributions from roots of unity.
• For the specific example $\mathbb{P}^2_{(1,2,4)}$, the full orbifold K-theory is explicitly computed as $\mathbb{Z}[u,u^{-1},\alpha_0,\alpha_1,\alpha_2,\alpha_3]/(\mathcal{I} + \langle\alpha_0 - 1\rangle + \mathcal{J})$, with $\mathcal{I}$ and $\mathcal{J}$ derived from multiplication tables and kernel computations.
• The full orbifold K-theory of weighted projective spaces is torsion-free over $\mathbb{Z}$, as shown by analyzing the structure of the summands in the decomposition.
• The inertial K-theory ring $NK^\diamond_T(M)$ is a new invariant when the $T$-action is not locally free, and it surjects onto the full orbifold K-theory of the quotient.
• The computation confirms that the full orbifold K-theory is isomorphic to the $G$-invariant part of the stringy K-theory in the global quotient case, but is more general for non-global quotient stacks.