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stub for moduli space of connections, started to collect some references
Hey Urs,
Was just looking at this. Are there references differentiating between the moduli space of connections and that of flat connections? The page sort of mixes the two, and has a link to itself.
-Jon
Yes, the entry is pretty stubby.
The world of references on this topic has to pronounced peaks at discusssion of non-flat but harmonic connections in 4d (for YM) and of flat connections in 2d (for CS and WZW). For the latter there is an entry Hitchin connection.
But this is a huge subject and the entries are stubs. What is it you are looking into, specifically?
(If what you mean to say is: “hey, this entry sucks, I have energy to work on it, may I?” then the answer is: Yes!)
Well, in particular I was thinking about the fact that flat connections very generally should be identifiable as descent data, and as such, they should be in bijection with the totalization of a certain cosimplicial space (or cosimplicial ring, I guess, depending on whether or not you put $(\infty,1)$ in front of things). I thought about writing something about this, but as usual am not sure that I’m thinking about things in the right way.
Or rather, I should say, in certain cases we can think of descent data as flat connections on the generalized Amitsur complex of the sweedler coring of an extension. I’m not sure about whether or not a given flat connection can be thought of descent data for some cover.
I’m also really interested in knowing whether or not there is some operator on the moduli space of connections whose kernel is the moduli space of flat connections, or something along these lines.
Hi Jon,
just one boring comment to start with: of course everything that has a local description (hence a moduli stack) may be thought of in terms of descent data.
The universal characterization of flat local structure, also called local systems or the like, is that the structure is not just local, but also locally constant, hence that the corresponding moduli stack is a “constant $\infty$-stack” (the stackification of a constant presheaf, hence in the inverse image of the terminal geometric morphism).
Concretely if one is talking about principal infinity-connections for some geometric $\infty$-group $G$, then the universal moduli stack of flat connections is $\flat \mathbf{B}G$, where $\flat$ (the “flat modality”) projects out (or rather co-projects) the constant part.
Now the construction of moduli stacks of connections on a fixed space $X$ is more subtle than that: there is the mapping stack $[X,\flat \mathbf{B}G]$, but that is typically not quite what one wants to mean: over a given $U$ this classifies maps $U \to [X,\flat \mathbf{B}G]$ and hence maps $U \times X \to \flat\mathbf{B}G$ and hence flat $\infty$-connection on $U \times X$, while one would really want just a $U$-parameterized collection of flat connections on $X$, hence one would want that the connection data along $U$ is trivial.
In Higher geometric prequantum theory (schreiber) (arXiv:1304.0236) it is discussed how the actual moduli stack of (flat or not) connections may be obtained from that naive mapping stack by some canonical process (called “differential concretification” there) using the sharp modality.
I have added to the derivation of the formula for the moduli space of flat connections on a torus here a pointer to the MO comment by Jacob Lurie here which cautions about the interpretation of this formula in equivariant elliptic cohomology (and I have also added the pointer to that discussion).
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