superconformal

Superconformal refers to theories or algebras that combine supersymmetry with conformal symmetry. Conformal symmetry extends Poincaré symmetry by including dilations and special conformal transformations, leaving the theory invariant under angle-preserving transformations. Supersymmetry adds fermionic generators relating bosons and fermions. A superconformal theory thus has a graded symmetry algebra that includes the conformal algebra so(d,2), supersymmetry generators Q and their conformal partners S, and an R-symmetry algebra.

Depending on spacetime dimension d and the amount of supersymmetry, the resulting superconformal algebras vary; common four-dimensional examples use su(2,2|N). In two dimensions the superconformal algebra is infinite-dimensional in the local case, giving rich structures such as N=(p,q) superconformal algebras. Theories with this symmetry are highly constrained: correlation functions of superconformal primaries are fixed up to a few coefficients, and the operator spectrum is organized into superconformal multiplets.

Notable examples include four-dimensional N=4 super Yang-Mills theory, a maximally supersymmetric and conformal field theory, and certain three-dimensional theories like ABJM at specific levels, which are superconformal. In two dimensions, superconformal field theories with various N have played central roles in string theory and statistical mechanics.

In modern physics, superconformal symmetry underpins the AdS/CFT correspondence, where a superconformal field theory on the boundary corresponds to a supergravity or string theory in Anti-de Sitter space. The study of these theories—through representation theory, operator algebra, and bootstrap methods—advances understanding of quantum field theory, critical phenomena, and quantum gravity.