Why is conformal field theory so important? I just started escaping the world of quantum mechanics and looking to study quantum field theory. I heard of AdS/CFT and also heard that CFT is of much importance. Now I do not get why having conformal invariance is so important in quantum field theory. Can anyone explain this part?
 A: Here's a quick answer. There are a few reasons why CFT's are very interesting to study. The first is that at fixed points of RG flows, or at second order phase transitions, a quantum field theory is scale invariant. Scale invariance is a weaker form of conformal invariance, and it turns out in all cases that we know of (or at least the ones I know of) scale invariance of a quantum field theory actually ends up implying the larger symmetry of conformal invariance. So, if you care about a field theory near a phase transition or at the fixed point of an RG flow, you should care about conformal field theories.
The second reason is that the requirement that a theory is conformally invariant is so restrictive that many things can be solved for that would otherwise be intractable. As an example, conformal invariance fixes 2- and 3-point functions entirely. In an ordinary quantum field theory, especially one at strong coupling, these would be hard or impossible to calculate at all. So even if you're not interested in CFT's per se, the QFT phenomenon you're interested in might be most tractable in a CFT.
And I suppose a third reason is string theory. In string theory, the worldsheet theory describing the string's excitations is a CFT, so if string theory is correct, then in some sense conformal invariance is really one of the most fundamental features of the elemental constituents of reality. And through string theory we have the most precise and best-understood gauge/gravity dualities (the AdS/CFT dualities) that also involve CFT's. So if you are interested in using these dualities to try to gain some insight into quantum gravity (for example, if you are interested in the black hole information problem), then the precise AdS/CFT dualities derived from string theory are your best bet.
