String Threshold corrections in models with spontaneously broken supersymmetry

TL;DR

The work analyzes one-loop gauge and gravitational threshold corrections in four-dimensional heterotic vacua with $N=2$ supersymmetry, realized via spontaneous breaking of $N=4$ on SU(2) holonomy manifolds locally like $K3\times T^2$. Thresholds are computed from helicity-generating partition functions and their dependence on the elliptic genus, leading to a universal decomposition in terms of moduli-dependent functions $\Delta^w(T,U), H^w(T,U), V^w(T,U), Y^w(T,U)$ weighted by low-energy data $b_i, k_i$ and beta-function discontinuities; in particular, the shift vector $w$ classifies universality classes. In models with freely-acting $Z_2$ shifts ($\lambda=0,1$), the masses of the two gravitinos depend on $(T,U)$, and decompactification can restore $N=4$ supersymmetry, with distinct behaviors for $\lambda=0$ versus $\lambda=1$; the analysis extends to orbifolds where extra massless states arise along rational lines, modifying both gauge and gravitational thresholds. The results reveal that universality established for standard $K3\times T^2$ compactifications is modified but retained in spirit, with threshold functions governed by modular invariance and the elliptic genus, yet increasingly model-dependent in the gravitational sector. The framework yields explicit expressions and asymptotics, providing a tool to study non-perturbative links via dualities and potential phenomenological implications for orbifold constructions.

Abstract

We analyse a class of four-dimensional heterotic ground states with N=2 space-time supersymmetry. From the ten-dimensional perspective, such models can be viewed as compactifications on a six-dimensional manifold with SU(2) holonomy, which is locally but not globally K3 x T^2. The maximal N=4 supersymmetry is spontaneously broken to N=2. The masses of the two massive gravitinos depend on the (T,U) moduli of T^2. We evaluate the one-loop threshold corrections of gauge and R^2 couplings and we show that they fall in several universality classes, in contrast to what happens in usual K3 x T^2 compactifications, where the N=4 supersymmetry is explicitly broken to N=2, and where a single universality class appears. These universality properties follow from the structure of the elliptic genus. The behaviour of the threshold corrections as functions of the moduli is analysed in detail: it is singular across several rational lines of the T^2 moduli because of the appearance of extra massless states, and suffers only from logarithmic singularities at large radii. These features differ substantially from the ordinary K3 x T^2 compactifications, thereby reflecting the existence of spontaneously-broken N=4 supersymmetry. Although our results are valid in the general framework defined above, we also point out several properties, specific to orbifold constructions, which might be of phenomenological relevance.

Figures (3)

• Figure 1: Decompactification scheme of generic models with $\lambda=0$ shifted $(2,2)$ lattice.
• Figure 2: Decompactification scheme of models with $\lambda=1$ shifted $(2,2)$ lattice.
• Figure 3: Decompactification scheme for models with $\lambda=0$ shifted $(2,2)$ lattice of class (ii).