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On quantum corrected Kähler potentials in F-theory

Iñaki García-Etxebarria, Hirotaka Hayashi, Raffaele Savelli, Gary Shiu

TL;DR

The paper addresses the problem of quantum corrections to the vector-multiplet Kähler potential in F-theory, delivering an exact result in string coupling $g_s$ and a perturbatively exact series in $oldsymbol{α'}$ for a specific $oldsymbol{ ext{N}=2}$ setup. By exploiting a web of dualities (heterotic, Type I, Type I′, M-theory/F-theory) in the $ ext{K3} imes ext{K3}'$ model, the authors disentangle $oldsymbol{α'}$-towers and show that each even $oldsymbol{α'}$-order carries an $oldsymbol{SL(2,oldsymbol{Z})}$-invariant set of $g_s$ corrections. They derive explicit forms for the quantum Kähler potential, including the function $oldsymbol{h(S,U)}$ expressed via polylogarithms and Eisenstein series, and discuss non-perturbative $oldsymbol{α'}$ corrections and Wilson-line effects. The results illuminate how quantum corrections act in F-theory, offering a framework to study moduli stabilization in $oldsymbol{ ext{N}=1}$ contexts and guiding future work on the hypermultiplet sector and more general elliptic fibrations.

Abstract

We work out the exact in string coupling and perturbatively exact in α' result for the vector multiplet moduli Kähler potential in a specific N=2 compactification of F-theory. The well-known correction cubic in α' is absent, but there is a rich structure of corrections at all even orders in α'. Moreover, each of these orders independently displays an SL(2,Z) invariant set of corrections in the string coupling. This generalizes earlier findings to the case of a non-trivial elliptic fibration. Our results pave the way for the analysis of quantum corrections in the more complicated N=1 context, and may have interesting implications for the study of moduli stabilization in string theory.

On quantum corrected Kähler potentials in F-theory

TL;DR

The paper addresses the problem of quantum corrections to the vector-multiplet Kähler potential in F-theory, delivering an exact result in string coupling and a perturbatively exact series in for a specific setup. By exploiting a web of dualities (heterotic, Type I, Type I′, M-theory/F-theory) in the model, the authors disentangle -towers and show that each even -order carries an -invariant set of corrections. They derive explicit forms for the quantum Kähler potential, including the function expressed via polylogarithms and Eisenstein series, and discuss non-perturbative corrections and Wilson-line effects. The results illuminate how quantum corrections act in F-theory, offering a framework to study moduli stabilization in contexts and guiding future work on the hypermultiplet sector and more general elliptic fibrations.

Abstract

We work out the exact in string coupling and perturbatively exact in α' result for the vector multiplet moduli Kähler potential in a specific N=2 compactification of F-theory. The well-known correction cubic in α' is absent, but there is a rich structure of corrections at all even orders in α'. Moreover, each of these orders independently displays an SL(2,Z) invariant set of corrections in the string coupling. This generalizes earlier findings to the case of a non-trivial elliptic fibration. Our results pave the way for the analysis of quantum corrections in the more complicated N=1 context, and may have interesting implications for the study of moduli stabilization in string theory.

Paper Structure

This paper contains 24 sections, 144 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting up the problem
  3. Review of the model
  4. Generalities
  5. Duality dictionary
  6. 10d duality
  7. 4d duality
  8. Threshold corrections and $SL(2,\mathbb{Z})$ invariance
  9. Ignoring Wilson lines
  10. Inclusion of Wilson lines
  11. Non-perturbative $\alpha'$ corrections
  12. F-theory picture
  13. Preliminaries
  14. F-theory lifts
  15. Classical theory
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Quantum corrections to the prepotential of F-theory on $K3\times K3$. The horizontal axis labels the degree in $\alpha'$ of the correction, and the vertical one the degree in $g_s$. The circles represent the non-vanishing perturbative terms in the prepotential. The solid band on top represents the set of non-perturbative corrections in $g_s$; Notice that there are no such corrections at tree level in $\alpha'$, as discussed in the text.
  • Figure 2: Atlas of dualities used to compute directly in M-theory the quantum corrections to the vector multiplet moduli space of F-theory on K3$\times$K3. In particular, the c-map is used to get a trivial F-theory fibration (top right corner) from a non trivial one (top left corner). The information of the non-trivial fibration is all encoded in the geometry of $X_3$.