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.



1-dimensional pure gauge theory
A gauge theory in mathematical terms (as I understand it) is a field theory whose fields include a gauge field: a $G$-bundle with connection. In this case, $G$ is a finite group, so there are no non-trivial connections. Thus a gauge field is simply a $G$-bundle.


1) Here, by pure gauge theory, I think the authors mean that this is a theory whose fields are precisely $G$-bundles and nothing more (in particular, no "matter" fields).


2,3) The path-integral is one approach to quantizing a field theory. It says, in particular, we should compute the partition function of the theory by a suitably weighted integral over a space of fields of the theory. The weight is prescribed by the action, a certain function on the space of fields.


Fields in this theory are $G$-bundles. The collection of $G$ bundles on a manifold $M$ (spacetime) can naturally be organized in to a groupoid, whose objects are $G$-bundles and morphisms are isomorphisms of $G$-bundles. For example, the groupoid of $G$-bundles on the circle $S^1$ may naturally be identified with the quotient groupoid $G//G$ (where $G$ acts on itself by conjugation). This identification assigns to a $G$-bundle on $S^1$ its holonomy (which is well-defined up to conjugation).


The (exponentiated) action functional is given by the abelian character $\lambda:G \to T$. This assigns to a $G$-bundle on $S^1$ with holonomy $g$, the function $\lambda(g)$. Thus the partition function on $S^1$ is given by an ''integral'' over the groupoid $G//G$ weighted by the function $\lambda$. This really just means the sum of $\lambda(g)$ as $g$ ranges over $G$, and one must divide by $|G|$ to get the right notion of volume (we are integrating over $G//G$ not $G$). You can check that one gets the stated results.


There is a similar idea happening for the $0$-manifold $pt\_+$. In general QFT, one expects to assign to a codimension 1 submanifold a vector space (the "Hilbert space" of the theory). Roughly speaking one might think of elements of this vector space as being functions on the space of fields - or better, sections of a line bundle (which again is associated to the action). 


In this case, the function $\lambda$ is actually the same thing as a line bundle on the space of fields on a point (which is the groupoid $pt//G$). The space of sections of this line bundle is either 1-dimensional if the bundle is trivial ($\lambda =1$) or $0$-dimensional if the bundle is non-trivial ($\lambda \neq 1$).


I wish I had something reasonable to add about the Gauss Law. Perhaps someone more knowledgeable could weigh in?


