Can everyone remember the Simpsons episode Treehouse of Horror IV where Homer walks through the wall behind the bookcase into another dimension?
Lisa: Well, where's my Dad?
Frink: Well, it should be obvious to even the most dim-witted individual who holds an advanced degree in hyperbolic topology, n'gee, that Homer Simpson has stumbled into...the lights go off the third dimension.
Lisa: turning the lights back on Sorry.
Frink: drawing on a blackboard Here is an ordinary square --
Wiggum: Whoa, whoa -- slow down, egghead!
Frink: -- but suppose we exte-end the square beyond the two dimensions of our universe along the hypothetical Z axis, there.
Frink: This forms a three-dimensional object known as a "cube", or a "Frinkahedron" in honor of its discoverer, n'hey, n'hey.
Homer: disembodied Help me! Are you helping me, or are you going on and on?
Frink: Oh, right. And, of course, within, we find the doomed individual.
(http://www.snpp.com/episodes/3F04.html ... see site for copyright and disclaimers.)
Many ordinary quantum field theories involve mathematical equations that deal with quantum structures in one-dimensional metric space. Metric space is your one-dimensional "flat" or "curved" surface.
A Topological Quantum Field Theory (also known as 3-dimensional TQFT) is a quantum field theory you can define on a 3-dimensional spacetime manifold, that is, a space without your usual one-dimensional metric structure. In the 3-dimensional space the previously identified quantum structure is "smeared", ie: from a one-dimensional loop to a 3-dimensional knot.
TQFT is where theories such as quantum gravity come in to play.
The basic theoretical tools of ordinary quantum field theory (creation and annihilation operators, canonical commutation relations, gaussian measures, propagators) cannot be used in quantum field over a manifold. They require a background metric to work.
A different set of mathematical equations therefore have to be used to work without the background metric. In the infinite space of 3 dimensions, integers turn into infinities, and from apparently simple geometry complex categorical and algebraic structures appear.
This is a horrendously difficult section of quantum physics to deal with, even for those physicists experienced in Topological Field Theory. There have been cases such as that of Igor and Grichka Bogdanov whose work on TQFT was so obtuse and difficult to understand it was later stated to be false by peers...but only after being published in several significant physics journals.