I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie, and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard even in the first section ($n = 1$), which was "trivial but included for completeness".

In particular, I have several unfamiliar terms while it discusses "1-dimensional pure gauge theory". Each term will require some explanations, if not too long. Any pointers to places where I can learn more about the terms will be highly appreciated.

### 1. 1-dimensional "pure" gauge theory

I have an impression that gauge theory is just bundle theory in math term. But how about the adjective "pure"?

### 2. The standard quantization procedure

Given a compact Lie group $G$, an abelian character $\lambda: G \to U(1)$, and a $G$-bundle with connection over the circle ($g$ being its holonomy), they define a 1d-TQFT by assigning to the circle the number $\lambda(g)$. Then "by the standard quantization procedure", they assign to the positively oriented point a subspace of $\mathbb{C}$, depending on if the abelian character is trivial or not.

I know little about geometric quantization - all I have read is J. Baez's informal introduction (http://www.math.ucr.edu/home/baez/quantization.html). But I have know idea how these two relate, and the article even claimed that this procedure relates to "the Gauss law in physics", making it even more mysterious for me..

### 3. Path integral over the groupoid $G//G$

Now the value assigned to the circle is just the dimension of the vector space assigned to the positively oriented point, and as above this value is either 0 or 1 depending on whether the character is nontrivial or not. The authors claimed that this may be understood as the result of the **path integral over the groupoid $G//G$ of connections on $S^1$ with respect to Haar measure:

$$ \frac{1}{|G|} \int_G \lambda(g)dg = 0 \mbox{ or } 1$$

.. This has nothing to do with what I think a path integral is: to me, a path integral is an integral over all path/section space with a suitable weight.

I hope I express my questions clear. If there's any confusion, please let me know. Thank you.