The finite regime is β β 2 < 4 π π , where no counterterms are needed to render the theory well-posed. The super-renormalizable regime is 4 π π < β β 2 < 8 π π , where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold n n + 1 8 π π passed. For β β 2 > 8 π π , the theory becomes ill-defined (Coleman 1975). The boundary values are β β 2 = 4 π π and β β 2 = 8 π π , which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an affine sl 2 subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).