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It is my understanding that a connection form on a principal G-bundle over a manifold X is defined to be a Lie algebra-valued 1-form $\alpha$ which reproduces the Lie algebra generators of the fundamental vector fields at every point. That is, if at $p$ a vector field $v$ takes the value $\frac{d}{dt}(exp(tg)p) |_{t=0}$, then $\alpha(v)|_p = g$.
I have noticed, however, that the fundamental vector fields defined for $\mathbb{C}^*$ bundles in Sniatycki's book on geometric quantization are not defined as $\frac{d}{dt}(exp(tg)p) |_{t=0}$ but rather as $\frac{d}{dt}(exp(2\pi itg)p) |_{t=0}$, where $g$ is any complex number.
1. Is my initial definition of a connection form correct?
2. Assuming the answer is yes, what will the relationship be between connection forms defined in the two different conventions?
3. Apart from the fact that Sniatycki's convention makes the real line generate the fundamental vector fields corresponding to the flow of the $U(1)$ action (if I am not in error here), is there some additional benefit to this convention? It appears to make $2\pi i$'s show up everywhere.

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I think that you are missing an equivariance condition on the connection 1-form. Without it, the horizontal distribution defined as the kernel of $\alpha$ is not necessarily $G$-invariant. –  José Figueroa-O'Farrill Sep 20 '11 at 18:10
If you start by defining the connection as a $G$-invariant horizontal distribution, then the connection 1-form is defined in the way you have done with the additional condition that it should annihilate the horizontal distribution. So either way, I think that there is something missing in your definition. –  José Figueroa-O'Farrill Sep 20 '11 at 18:11
Agreed, I just forgot to mention it. –  Blake Sep 21 '11 at 12:52

1 Answer 1

up vote 4 down vote accepted

First off, Prof. Figueroa-O'Farrill is correct in noting that you omitted the $G$-equivariance condition for the connection, which reduces to $G$-invariance in the case where $G$ is abelian (in particular, when $G=\mathbb{C}^*$).

  1. The choice of normalization conventions really just comes down to how you identify the Lie algebra of $\mathbb{C}^* $ with $\mathbb{C}$. Śniatycki's convention amounts to identifying the $\mathrm{U}(1)$-generator (i.e. the vector field $2\pi\frac{\partial}{\partial \theta}$ in the Lie algebra of $\mathbb{C}^* $) with $1\in\mathbb{C}$, whereas the seemingly more logical choice would be to identify it with $2\pi i\in\mathbb{C}$. Śniatycki does this because it's also Kostant's convention (who got it from Weil I believe).

  2. The relationship between your $\alpha$ and Śniatycki's $\alpha$ should be just $$ \alpha_{\mathrm{Blake}} = 2\pi i~\alpha_{\mathrm{Śniatycki}} $$

  3. The condition for the existence of a line bundle $L$ over $X$ with connection $\alpha$ and $\alpha$-compatible pairing $\langle\cdot,\cdot\rangle$ can formulated in terms of the curvature $\omega$ of $\alpha$, defined by $d\alpha = \pi^*\omega$, where $\pi$ is the bundle projection. The big advantage of defining the connection $\alpha$ à la Kostant/Śniatycki is that the condition becomes that $\omega$ is integral, i.e. gives an integer when integrated over closed two-cycles in the base manifold $X$. For this reason, the $2\pi i$ is a common normalization in the theory of Chern classes.

From personal experience, normalization conventions can drive you mad, especially when comparing results from different authors. For example, Guillemin and Sternberg seem to favor the convention that $\frac{\partial}{\partial \theta}\in\textrm{Lie algebra of }\mathbb{C}^* \leftrightarrow 1\in\mathbb{C}$. So best of luck :)

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Amen to the last two sentences. The advice I got from my advisor (Phillip Griffiths) was to make my own choices for my own notation and conventions and then learn to translate anybody else's to mine. Actually, usually you just look at something just enough to get the idea of what to do and then you write out the proof and calculations using your own notation and conventions. –  Deane Yang Sep 21 '11 at 0:47

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