# Localization for $K$-Contact Manifolds

## TL;DR

This paper extends localization formulas to $K$-contact manifolds, connecting equivariant basic integrals to contact quotients, and generalizes known symplectic results via contact geometry.

## Contribution

It proves an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology for $K$-contact manifolds, extending classical symplectic localization results.

## Key findings

- Established a localization formula for equivariant basic cohomology on $K$-contact manifolds.
- Derived analogues of Witten's nonabelian localization and Jeffrey-Kirwan residue formulas.
- Reduced to classical formulas in the case of Boothby-Wang fibrations over symplectic manifolds.

## Abstract

We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of $K$-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact manifold $M$ to basic integrals on the contact quotient $M_0 := \mu^{-1}(0)/G$, where $\mu$ denotes the contact moment map for the action of a torus $G$. In the special case that $M \to N$ is an equivariant Boothby-Wang fibration, our formulae reduce to the usual ones for the symplectic manifold $N$.