TITLE: 1-dimensional pure gauge theory
QUESTION [6 upvotes]: 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.

REPLY [6 votes]: Just to add something about Gauss' law to the excellent previous answer: The Hamiltonian of a pure gauge theory, as initially derived from the Lagrangean, typically has the structure $H=E^2 +B^2 - A_0 \nabla \cdot E$, where $E$ and $B$ are the electric and magnetic fields, and $A_0 $ is the temporal component of the gauge field. In addition, the Lagrangean contains no time derivative of $A_0 $, and therefore there is no conjugate momentum for $A_0 $ - it is not a dynamical degree of freedom. Instead, it acts as a Lagrange multiplier enforcing $\nabla \cdot E =0$. That is Gauss' law, a constraint on the remaining dynamical fields.
There are distinct ways of dealing with this. One avenue is to use the gauge freedom to choose $A_0 =0$. Then the non-dynamical degree of freedom is gone, but one has lost the knowledge about the Gauss law constraint. In $A_0 =0$ gauge, one must impose the constraint in addition to the dynamical equations generated by the Hamiltonian.
Another avenue is to choose a different gauge, but then the formalism still contains $A_0 $. The "equation of motion" for $A_0 $ is, again, not dynamical, but a constraint on $A_0 $, which one can then solve to eliminate $A_0 $ (this, e.g., generates the Coulomb interaction). In this manner of speaking, Gauss' law doesn't play quite such a central role - one thinks of $A_0 $ being constrained, as opposed to the other fields being constrained.
Either way, one eliminates two components of the gauge field: One by choice of gauge, one by constraint. Only the remaining components are dynamical degrees of freedom. Thus, light comes with two polarizations, even though one starts out with a 4-component field $A_{\mu } $.