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Suppose we couple $N$ colors of fermions to an $SU(N)$ gauge field $A$, but instead of a Yang-Mills action, there is a BF theory that restricts the gauge field to be flat $dA+A\wedge A\equiv F=0$ (by using a Lagrange multiplier field called $B$, hence the name). In some sense the BF theory is like an 'asymptotically free' limit of Yang-Mills where the gauge coupling is taken to zero.

If we are considering the fermions to live in a topologically trivial space like infinite volume Minkowski space, what is the difference of this BF coupled theory from a theory of $N$ free fermions?

It seems there must be some difference in the sense that if we have a finite temperature we can have non-trivial flat gauge fields twisting around the thermal circle, and this should lead to different thermal correlation functions. Is coupling to the BF theory just equivalent to just filtering out the non-gauge invariant states from the spectrum of the free theory?

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    $\begingroup$ What is a "BF theory"? $\endgroup$
    – flippiefanus
    Commented Apr 24 at 3:49
  • $\begingroup$ "Background Field" $\endgroup$
    – Gabriel Ybarra Marcaida
    Commented Apr 24 at 7:38
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    $\begingroup$ I don't think BF really was originally an abbreviation for "background field." See for instance the review Topological field theory, Phys Rept. 209 (1991) 129, where BF is never defined as an abbreviation. B and F are just the names of the fields in the action. F is the field strength tensor for the gauge field, and B is like a Lagrange multiplier constraining the field strength to zero. $\endgroup$
    – octonion
    Commented Apr 24 at 17:03
  • $\begingroup$ I know BF as ‘best friend’, ‘best focus’ and so forth. It is preferable not to use abbreviations before you explain them, unless they are well known and universally accepted. $\endgroup$
    – my2cts
    Commented Apr 24 at 17:13
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    $\begingroup$ @my2cts, I put a little bit of explanation now, but my understanding is that BF is simply the name of the model and not an abbreviation for anything (like "M theory"). Both the Wikipedia and nLab articles on it use "BF theory" or "BF model" in the title. $\endgroup$
    – octonion
    Commented Apr 24 at 17:22

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