Effective Actions near Singularities

TL;DR

The paper addresses singularities in the four-dimensional $N=2$ effective action arising from $SU(2)$ gauge enhancement on the line $T=U$ in heterotic compactifications on $K3\times T^2$. By integrating in the light $W^{\pm}$ vector multiplets, the authors construct a non-singular, $SU(2)$-invariant effective action whose prepotential ${\cal F}_{\rm in}$ incorporates universal threshold corrections, notably a term $\delta{\cal F}\sim \frac{1}{2\pi^2} T_-^2\log T_-$. They demonstrate that the resulting theory respects a residual $SL(2,\mathbb{Z})$ symmetry on $T=U$ and reproduces the correct higher-gauge-enhancement singularities, with the large-radius limit matching the known five-dimensional action of M"obius-type gauged supergravity. The approach, combining symmetry, threshold analysis, and an integrating-in procedure, provides a coherent framework for non-singular effective actions near singular loci, with potential extensions to conifold points and higher-rank enhancements.

Abstract

We study the heterotic string compactified on K3 x T^2 near the line T=U, where the effective action becomes singular due to an SU(2) gauge symmetry enhancement. By `integrating in' the light W^\pm vector multiplets we derive a quantum corrected effective action which is manifestly SU(2) invariant and non-singular. This effective action is found to be consistent with a residual SL(2,Z) quantum symmetry on the line T=U. In an appropriate decompactification limit, we recover the known SU(2) invariant action in five dimensions.