Shortening Anomalies in Supersymmetric Theories

We present new anomalies in two-dimensional ${\mathcal N} =(2, 2)$ superconformal theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background superfields in short representations. Therefore, standard results that follow from ${\mathcal N} =(2, 2)$ spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond ${\mathcal N} =(2, 2)$. These anomalies explain why the conformal manifolds of the K3 and $T^4$ sigma models are not K\"ahler and do not factorize into chiral and twisted chiral moduli spaces and why there are no ${\mathcal N} =(2, 2)$ gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds.