Why is it interesting?

It is an old idea in particle physics that, in some sense, at sufficiently high energies the masses of the elementary particles should become unimportant. In recent years this somewhat vague hope has acquired a more definite form in the theory of scale transformations, or dilatations.

Aspects of Symmetry: Selected Erice Lectures by Sidney Coleman

In particle physics, one longstanding hope has been that at high energies, particle masses can be neglected, so that the physics would become scale invariant. It turns out that in a local field theory, it is true, more or less in general, that scale invariance typically leads to conformal invariance. (This is because the violation of scale invariance and conformal invariance are both determined by the trace $T^\mu_\mu$ of the energy momentum tensor.)

page 621 Einstein Gravity in a Nutshell - A. Zee

studying the space of CFTs basically amounts to studying the space of all well-defined QFTs

Lectures on AdS/CFT from the Bottom Up by Jared Kaplan

See also: https://www.quantamagazine.org/using-the-bootstrap-physicists-uncover-geometry-of-theory-space-20170223 and Critical Exponents

Layman

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Researcher

Common Question 1

Common Question 2

Examples

Example1

Example2:

History