I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on symplectic manifolds.

Equation 3.27 on page 47 reads as \begin{equation} H_V=\mathcal{L}_V-[\iota_V,\nabla]=\iota_V\theta=V^{\mu}\theta_{\mu}, \end{equation} where $H_V$ is the moment map evaluated on an element of the Lie algebra of $G$, with associated vector field $V$, and $\theta$ is the connection 1-form on the line bundle where $\nabla=d+\theta$. I am not completely sure how the above equality follows, and it seems to me that there is a sign error. Here is what I have: \begin{equation} \begin{aligned} H_V&=\{d,\iota_V\}-[\iota_V,\nabla]\\ &=\{d,\iota_V\}-[\iota_V,d+\theta]\\ &=d\iota_V+\nabla\iota_V-\iota_V\theta. \end{aligned} \end{equation}

I am not sure how the first two terms vanish, and why the last term has a minus sign. Any help would be much appreciated.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.