Abstract
We present new anomalies in two-dimensional mathcal{N}=left(2,2right) 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}=left(2,2right) spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond mathcal{N}=left(2,2right) . These anomalies explain why the conformal manifolds of the K3 and T4 sigma models are not Kähler and do not factorize into chiral and twisted chiral moduli spaces and why there are no mathcal{N}=left(2,2right) 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.
Highlights
We present new anomalies in two-dimensional N = (2, 2) superconformal theories
We provide a complementary perspective on the anomalies by studying the Riemann curvature of the conformal manifold in N = (2, 2) SCFT’s, extending the previous work [8, 19, 32]
Our aim is to determine to what extent these shortening conditions can be maintained as we explore the conformal manifold of an N = (2, 2) SCFT
Summary
The OPE (2.18) may lead to some contact terms (2.3). To understand these we need to study the superspace derivatives (the derivatives with respect to the second argument follow from these). In order to define these derivatives we need to specify the behavior of Green’s function in superspace for the left-movers, and different choices yield different answers This ambiguity is a manifestation of the fact that the pole 1/z1− ̄2− in N = (2, 2) superspace is too singular to yield an unambiguous distribution.. In order to establish our shortening anomaly we must show that it is not possible to tune the contact term r such that the shortening conditions for a chiral and a twisted chiral multiplet can be maintained simultaneously. Since a + b = 1, it is impossible to solve both equations This implies that we cannot simultaneously preserve the chiral and twisted chiral shortening conditions along the conformal manifold. We emphasize that the violations in (3.12) depend on the fermionic components of the multiplet to which the couplings have been promoted
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