Shortening Anomalies in Supersymmetric Theories

TL;DR

This paper uncovers new anomalies in two-dimensional ${ m extbf{N}}=(2,2)$ superconformal theories that affect the structure of conformal manifolds and challenge previous assumptions about supersymmetric marginal couplings.

Contribution

It identifies anomalies that obstruct shortening conditions in ${ m extbf{N}}=(2,2)$ theories, revealing limitations of standard spurion analysis and explaining the non-K"ahler nature of certain conformal manifolds.

Findings

Anomalies obstruct chiral multiplet shortening at coincident points.

Conformal manifolds of K3 and T^4 sigma models are not K"ahler.

No ${ m extbf{N}}=(2,2)$ gauged linear sigma models cover these manifolds.

Abstract

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.