On Quantum Field Theories

We derive unitarity restrictions on the scaling dimensions of primary quantum operators in a superconformal quantum field theory, in d =3, 4, 5, 6. 1. Introduction Whereas classical massless field theories are often conformally invariant, the same is rarely true of their quantum counterparts. To define a quantum field theory, you need to specify both a Lagrangian and a cutoff. The cutoff introduces a hidden dimensional scale in the definition of the theory. A scale transformation relates a quantum theory with a given cutoff to another quantum theory with the scaled cutoff. Therefore a QFT with nontrivial cutoff dependence (i.e., nonzero beta function) is never scale invariant, and so never conformally invariant. Since quantum field theories generically have nonzero beta function, a study of the implications of exact conformal invariance on a quantum field theory might seem to be of limited interest. That is, however, not the case. A QFT is indeed not scale invariant at arbitrary cut off, but as you lower the cut off in your theory, a truly gapless theory flows, in the Wilsonian sense, to a fixed point of the renormalization group-i.e., a point with zero beta function. Long wavelength physics in such a theory is thus governed by an effective quantum theory that is exactly scale invariant. In many examples this effective quantum field theory is a free massless theory. In some cases of interest however, the low energy effective theory is an interacting scale invariant, and generically conformally invariant [1] quantum field theory. Low energy effective theories completely specify the vacuum structure of a theory, and the nature of its massless excitations, and hence are of interest. Supersymmetric field theories have proved to be particularly amenable to exact analysis in the infrared. Interacting gapless theories with unbroken supersymmetry typically exhibit both conformal and supersymmetric invariance in the infrared. The full symmetry of the (low energy effective) theory must then be generated by a superalgebra that contains in it, as sub super algebras, both the conformal algebra and the supersymmetry algebra. Such algebras are very constrained-in fact none exist in d ≥6, and their possible forms are known in d ≤ 6. These algebras are called superconformal algebras, and the symmetry they generate is called superconformal symmetry. Three examples of interacting superconformal theories of current interest are the intrinsic theories living on the world volumes of coincident M 2 , M 5 , D 3 …