Abstract
Heterotic string compactifications on a $K3$ surface $\mathfrak{S}$ depend on a choice of hyperk\"ahler metric, anti-self-dual gauge connection and Kalb-Ramond flux, parametrized by hypermultiplet scalars. The metric on hypermultiplet moduli space is in principle computable within the $(0,2)$ superconformal field theory on the heterotic string worldsheet, although little is known about it in practice. Using duality with type II strings compactified on a Calabi-Yau threefold, we predict the form of the quaternion-K\"ahler metric on hypermultiplet moduli space when $\mathfrak{S}$ is elliptically fibered, in the limit of a large fiber and even larger base. The result is in general agreement with expectations from Kaluza-Klein reduction, in particular the metric has a two-stage fibration structure, where the $B$-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli. A more precise match must await a detailed analysis of $R^2$-corrected ten-dimensional supergravity.
Highlights
Supersymmetry requires additional higher-derivative couplings in the D = 10 supergravity Lagrangian [5], which greatly complicate the Kaluza-Klein reduction
The result is in general agreement with expectations from KaluzaKlein reduction, in particular the metric has a two-stage fibration structure, where the B-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli
The main goal of this paper is to investigate the structure of this two-stage fibration and to understand how it can be compatible with the quaternion-Kahler (QK) property of the total hypermultiplet moduli space MH, which is a necessary requirement for supersymmetry [18]
Summary
We discuss qualitative aspects of the hypermultiplet moduli space in compactifications of the heterotic string on a K3 surface S. The same hypermultiplet moduli space appears in compactifications on K3 × T 2 down to 4 dimensions, as the additional scalar fields coming from the metric and gauge bundle on T 2 all lie in vector multiplets. We restrict our analysis to elliptically fibered K3 surfaces and the gauge group G = E8 ×E8, so that heterotic/type II duality applies, but most of the considerations below hold more generally. As mentioned in the introduction, vacua with unbroken N = 2 supersymmetry are characterized by a hyperkahler metric g on S, a bundle F on S with second Chern class c2(F ) = χ(S) = 24 (as follows from the Bianchi identity (1.2)) equipped with an anti-self dual connection A such that F = dA + A ∧ A, and a two-form B on S satisfying the equation of motion (1.2). In the following we discuss each of these contributions separately
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