# What are marginal fields in CFT?

In this article they call weight $(h,\bar{h})=(1,1)$ fields marginal.

Why are these fields called marginal? Why are they to be distinguished.

This terminology comes from renormalization group flow, where one has relevant, marginal, and irrelevant operators.

In CFT, operators with conformal weight $(1, 1)$ are known as marginal operators. More generally, operators of conformal weight $(h, \bar{h})$ are said to be relevant if $h + \bar{h} < 2$ and irrelevant if $h + \bar{h} > 2$. A (necessarily marginal) operator that preserves conformality is called truly marginal, or exactly marginal, etc, cf. Ref. 1.

References:

1. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; Section 8.6.

They are called marginal because they correspond to "slight deformations" of a CFT, which do not break conformal invariance. Given a marginal field $\phi$, one can add to the action a term

$$\delta S = \Delta \int_\Sigma \phi$$

which is just the operator integrated over the worldsheet, modulated by a deformation parameter $\Delta$. Since the (2D) integration measure transforms exactly oppositely to the marginal field, this term is conformally invariant, and hence produces a new theory that is still conformally invariant.

Therefore, it is possible to study new CFTs by producing them by disturbing those already known - the idea is to "probe" the CFT landscape starting from "nice" theories, e.g. rational CFTs.