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The Euler characteristic correction to the Kaehler potential - revisited

Federico Bonetti, Matthias Weissenbacher

TL;DR

The authors provide a direct, explicit derivation of the leading α'^3 Euler-characteristic correction to the 4d Kaehler potential in type IIB orientifold compactifications by performing a complete KK reduction on an α'^3-corrected, SU(3)-structure internal geometry. They construct the explicit corrected background metric in terms of the non-harmonic part of the third Chern form, showing the internal space becomes almost Calabi-Yau with a Weyl factor multiplying the 10d metric. Their analysis yields the BBHL-type correction to the 4d Kaehler potential in the N=1 theory and confirms the corresponding Euler-number–driven correction to the N=2 Kaehler moduli prepotential, with leading corrections to Kaehler coordinates vanishing. The results establish a transparent topological origin for the correction and set the stage for including localized sources and full α'^3 couplings in future work.

Abstract

We confirm the leading $α'^3$ correction to the 4d, $\mathcal N = 1$ Kähler potential of type IIB orientifold compactifications, proportional to the Euler characteristic of the Calabi-Yau threefold (BBHL correction). We present the explicit solution for the $α'^3$-modified internal background metric in terms of the non-harmonic part of the third Chern form of the leading order Calabi-Yau manifold. The corrected internal manifold is almost Calabi-Yau and admits an $SU(3)$ structure with non-vanishing torsion. We also find that the full ten-dimensional Einstein frame background metric is multiplied by a non-trivial Weyl factor. Performing a Kaluza-Klein reduction on the modified background we derive the $α'^3$-corrected kinetic terms for the dilaton and the Kähler deformations of the internal Calabi-Yau threefold for arbitrary $h^{1,1}$. We analyze these kinetic terms in the 4d, $\mathcal N = 2$ un-orientifolded theory, confirming the expected correction to the Kähler moduli space prepotential, as well as in the 4d, $\mathcal N = 1$ orientifolded theory, thus determining the corrections to the Kähler potential and Kähler coordinates.

The Euler characteristic correction to the Kaehler potential - revisited

TL;DR

The authors provide a direct, explicit derivation of the leading α'^3 Euler-characteristic correction to the 4d Kaehler potential in type IIB orientifold compactifications by performing a complete KK reduction on an α'^3-corrected, SU(3)-structure internal geometry. They construct the explicit corrected background metric in terms of the non-harmonic part of the third Chern form, showing the internal space becomes almost Calabi-Yau with a Weyl factor multiplying the 10d metric. Their analysis yields the BBHL-type correction to the 4d Kaehler potential in the N=1 theory and confirms the corresponding Euler-number–driven correction to the N=2 Kaehler moduli prepotential, with leading corrections to Kaehler coordinates vanishing. The results establish a transparent topological origin for the correction and set the stage for including localized sources and full α'^3 couplings in future work.

Abstract

We confirm the leading correction to the 4d, Kähler potential of type IIB orientifold compactifications, proportional to the Euler characteristic of the Calabi-Yau threefold (BBHL correction). We present the explicit solution for the -modified internal background metric in terms of the non-harmonic part of the third Chern form of the leading order Calabi-Yau manifold. The corrected internal manifold is almost Calabi-Yau and admits an structure with non-vanishing torsion. We also find that the full ten-dimensional Einstein frame background metric is multiplied by a non-trivial Weyl factor. Performing a Kaluza-Klein reduction on the modified background we derive the -corrected kinetic terms for the dilaton and the Kähler deformations of the internal Calabi-Yau threefold for arbitrary . We analyze these kinetic terms in the 4d, un-orientifolded theory, confirming the expected correction to the Kähler moduli space prepotential, as well as in the 4d, orientifolded theory, thus determining the corrections to the Kähler potential and Kähler coordinates.

Paper Structure

This paper contains 19 sections, 105 equations.

Table of Contents

  1. Introduction
  2. Four-dimensional $\alpha'^3$ Lagrangian
  3. Type IIB higher-derivative action
  4. Supersymmetric background
  5. Corrections to the background
  6. $SU(3)$ structure on the internal manifold
  7. Fluctuations associated to Kähler moduli
  8. Reduction results
  9. Results and uplift
  10. Weyl rescaling
  11. Correction to the $\mathcal{N} = 2$ prepotential
  12. Translation into $\mathcal{N} = 2$ field variables
  13. Implications for $\mathcal{N} =2$ prepotential
  14. $\mathcal{N} = 1$ Kähler coordinates and Kähler potential
  15. Correction to the Kähler potential and coordinates
  16. ...and 4 more sections