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I'm reading about the Yang-Mills heat flow, and I'm curious how adding a Chern-Simons term alters its solutions. This is probably elementary or folklore, but I don't know well enough to say.

Background: The path integral measure for Yang-Mills theory should live on some distributional completion $\mathcal{F} = \overline{\mathcal{A}/\mathcal{G}}$ of quotient of the space of smooth connections by the group of gauge transformations. Constructing the path integral via lattice regularizations makes it hard to tell what this space is; the distributions are too obscured to get a grip on. Charalambous & Gross suggested that one can find a good model for $\mathcal{F}$ by looking at gauge-equivalence classes of $t \gt 0$ solutions to the YM gradient flow. What's nice about this approach is that gradient flow behaves like heat flow, smearing out short distance structure. If one regards the $t > 0$ solutions as regularized distributions, then flowing to larger $t$ is a lot like integrating out degrees of freedom.

This is very nicely explained in some recent papers by Cao & Chatterjee and Chandra, Chevyrev, Hairer, and Shen, which is what got me interested in the subject. There's also a lot of related literature in the hep-lat world.

Reading constructive gauge theory literature, I see a convention that strikes me as a little weird: Most work on 3d YM theory ignores the CS-terms, even though the CS term dominates the YM term at large distances.

So I'm wondering: Suppose we're on a 3-manifold, maybe compact, maybe of the form $\Sigma \times I$. Suppose we're only looking at connections on flat bundles. If we add a Chern-Simons term to the 3d YM flow, deforming the flow equation to $$ \partial_t A_t = -d_{A_t}^* F_{A_t} - g^2 c \star F_{A_t}, $$ do we significantly alter the character of the allowed solutions?

I'm specifically interested in the case where the solutions are distributional, with prescribed roughly-Gaussian irregularity at $t=0$. What I'm really wondering is: does the Yang-Mills measure with a Chern-Simons term live on the same space as the Yang-Mills measure without a Chern-Simons term? Or at least on a space that maps to it in some reasonable fashion?

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  • $\begingroup$ Can you clarify your questions? What do you mean by a "correspondence between the finite $t$ solutions? The proof of existence and uniqueness for solutions with sufficiently regular ($H^{1/2}$ in this dimension) initial conditions for this modified flow should follow easily from a standard implicit function theorem argument for the a heat flow (as you say the added term is lower order) once gauge fixing is accounted for (via say deTurck's trick or other methods). Although flat connections are fixed points for the flow it's not clear to me that they are the only critical points. $\endgroup$
    – Tom Mrowka
    Commented Jan 25, 2022 at 1:48
  • $\begingroup$ @TomMrowka Thanks for the feedback. I've made edits that hopefully prune this question to an answerable one. I've deleted (and hopefully will re-ask in a new question) some aspects of the original question that relate to renormalization flow. $\endgroup$
    – user1504
    Commented Jan 25, 2022 at 5:41

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