Editing BF model

{{Short description|Topological field}}
The '''BF model''' or '''BF theory''' is a topological [[field (physics)|field]], which when [[quantization (physics)|quantized]], becomes a [[topological quantum field theory]]. BF stands for background field '''B''' and '''F''', as can be seen below, are also the variables appearing in the [[Lagrangian (field theory)|Lagrangian]] of the theory, which is helpful as a mnemonic device.
 
We have a 4-dimensional [[differentiable manifold]] M, a [[gauge group]] G, which has as "dynamical" fields a [[2-form]] '''B''' taking values in the [[Adjoint representation of a Lie group|adjoint representation]] of G, and a [[connection form]] '''A''' for G.

The [[action (physics)|action]] is given by

:<math>S=\int_M K[\mathbf{B}\wedge \mathbf{F}]</math>

where K is an invariant [[nondegenerate]] [[bilinear form]] over <math>\mathfrak{g}</math> (if G is [[semisimple Lie algebra|semisimple]], the [[Killing form]] will do) and '''F''' is the [[curvature form]]

:<math>\mathbf{F}\equiv d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}</math>

This action is [[diffeomorphism|diffeomorphically]] invariant and [[gauge invariance|gauge invariant]]. Its [[Euler–Lagrange equation]]s are

:<math>\mathbf{F}=0</math> (no curvature)

and

:<math>d_\mathbf{A}\mathbf{B}=0</math> (the [[covariant exterior derivative]] of '''B''' is zero).

In fact, it is always possible to gauge away any local degrees of freedom, which is why it is called a topological field theory.

However, if M is topologically nontrivial, '''A''' and '''B''' can have nontrivial solutions globally.

In fact, BF theory can be used to formulate discrete gauge theory. One can add additional twist terms allowed by group cohomology theory such as [[Robbert Dijkgraaf|Dijkgraaf]]–[[Edward Witten|Witten]] topological gauge theory.<ref name="Dijkgraaf-Witten">{{cite journal | last1=Dijkgraaf | first1=Robbert| last2=Witten | first2=Edward| title= Topological Gauge Theories and Group Cohomology | doi=10.1007/BF02096988 |volume=129|journal=Commun. Math. Phys.|pages=393–429|date=1990| issue=2| bibcode=1990CMaPh.129..393D| s2cid=2163226| url=http://projecteuclid.org/euclid.cmp/1104180750}}</ref> There are many kinds of modified BF theories as [[topological quantum field theory|topological field theories]], which give rise to [[Linking number|link invariants]] in 3 dimensions, 4 dimensions, and other general dimensions.<ref name="1612.09298">{{cite journal | arxiv=1612.09298  | last1=Putrov | first1=Pavel| last2=Wang | first2=Juven| last3=Yau | first3=Shing-Tung|title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions | doi=10.1016/j.aop.2017.06.019 |volume=384C|journal=Annals of Physics|pages=254–287|bibcode=2017AnPhy.384..254P|date=September 2017| s2cid=119578849 }}</ref>

== See also ==
* [[Background field method]]
* [[Barrett–Crane model]]
* [[Dual graviton]]
* [[Plebanski action]]
* [[Spin foam]]
* [[Tetradic Palatini action]]

==References==
{{Reflist}}

==External links==
* http://math.ucr.edu/home/baez/qg-fall2000/qg2.2.html

{{Quantum field theories}}

[[Category:Quantum field theory]]


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