We study the constraints of spacetime supersymmetry for perturbative threeand two-dimensional Minkowski vacua of the critical heterotic string. Assuming a standard RNS construction of the spacetime supersymmetry generators and a compact unitary internal superconformal worldsheet theory, we describe the worldsheet structures associated to various spacetime supersymmetries. In three dimensions we show that there are no CFT surprises: each allowed spacetime supersymmetry is realized by a supergravity compactification. As a recent orbifold construction shows, in two dimensions there are more exotic possibilities, and we discuss how these fit into our analysis. 1 There is a classification [2][3][4] of the SCAs that only have currents with spins in {0, 1/2, 1, 3/2, 2} and a unique spin 2 current. All such SCAs have N ≤ 4, and those with N ≤ 3 are unique up to isomorphism; for N = 4 our notation A 4 c denotes the so-called "small" N=4 algebra. 2 The investigations of the worldsheet structure of Calabi-Yau compactifications and related theories [5,6] showed that A 2 9 SCA with integral R-charges was a sufficient condition for spacetime supersymmetry, and the work of [1] went on to show this to be necessary as well. 4 z 2 12 , J αβ 1 J γ δ 2 ∼ 1 z 2 12 δ αγ δ βδ − δ αδ δ βγFrom the last line it follows that the currents J α are chiral primary with respect to A 2 12 , while the J †α are their anti-chiral primary conjugates. Moreover, for any α the triplet {J α , J †α , J R } forms an su(2) 2 Kac-Moody algebra. As a result, we have the additional non-trivial OPEs