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String Loop Corrections to Kahler Potentials in Orientifolds

Marcus Berg, Michael Haack, Boris Kors

TL;DR

This work computes explicit one-loop string corrections to Kahler potentials in type IIB orientifolds with N=1 or N=2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes across sphere, disk, and one-loop world-sheets. It introduces Kahler-structure adapted vertex operators that mix closed and open moduli, derives the classical Kahler potential and metric, and then extracts one-loop corrections expressed in terms of Eisenstein-type sums E_2 with open-string moduli dependence. For the N=2 model, the authors show that the one-loop correction can be captured by a corrected Kahler potential that admits a prepotential F^{(1)} consistent with special geometry, fixing normalizations and confirming compatibility with supersymmetry. Extending to N=1 cases on T^6/(Z2×Z2) and hints for T^6/Z6′, the paper provides structured one-loop Kahler corrections involving momentum- and winding-type Eisenstein sums and analyzes how Wilson lines and D-brane moduli influence moduli stabilization and potential inflationary dynamics; duality to D3/D7 configurations is discussed, highlighting broader phenomenological relevance.

Abstract

We determine one-loop string corrections to Kahler potentials in type IIB orientifold compactifications with either N=1 or N=2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes.

String Loop Corrections to Kahler Potentials in Orientifolds

TL;DR

This work computes explicit one-loop string corrections to Kahler potentials in type IIB orientifolds with N=1 or N=2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes across sphere, disk, and one-loop world-sheets. It introduces Kahler-structure adapted vertex operators that mix closed and open moduli, derives the classical Kahler potential and metric, and then extracts one-loop corrections expressed in terms of Eisenstein-type sums E_2 with open-string moduli dependence. For the N=2 model, the authors show that the one-loop correction can be captured by a corrected Kahler potential that admits a prepotential F^{(1)} consistent with special geometry, fixing normalizations and confirming compatibility with supersymmetry. Extending to N=1 cases on T^6/(Z2×Z2) and hints for T^6/Z6′, the paper provides structured one-loop Kahler corrections involving momentum- and winding-type Eisenstein sums and analyzes how Wilson lines and D-brane moduli influence moduli stabilization and potential inflationary dynamics; duality to D3/D7 configurations is discussed, highlighting broader phenomenological relevance.

Abstract

We determine one-loop string corrections to Kahler potentials in type IIB orientifold compactifications with either N=1 or N=2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes.

Paper Structure

This paper contains 26 sections, 243 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The ${\cal N}=2$ orientifold $\mathbb T^4/\mathbb{Z}_2\times \mathbb T^2$
  3. The classical Lagrangian
  4. Classical Kähler metric
  5. Vertex operators for moduli fields
  6. Tree level diagrams
  7. One-loop diagrams
  8. One-loop Kähler potential and prepotential
  9. The ${\cal N}=1$ orientifold $\mathbb T^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$
  10. The classical Lagrangian
  11. One-loop amplitudes
  12. Wilson lines in $\mathbb T^6/(\mathbb Z_2\times \mathbb Z_2)$
  13. One-loop Kähler potential
  14. The ${\cal N} =1$ orientifold $\mathbb T^6/\mathbb{Z}_6'$
  15. Conclusions
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Computing a disk 3-point function using Kähler structure adapted vertex operators (denoted by $\bigcirc$ K ). The adapted vertex operators contain both open and closed vertex operators.
  • Figure 2: $\langle V_{S_2'}V_{S_2'}\rangle_{\sigma}$ for $\sigma={\cal A},{\cal M},{\cal K}$, with Kähler adapted vertex operators.
  • Figure 3: Annulus, Möbius strip and Klein bottle obtained from covering tori by the involutions (\ref{['app:is']}). Thick lines are boundaries (fixed lines under the involution). Each surface has one 1-cycle $C$. On the Klein bottle, the equality $CC'=D$ gives rise to the constraint (\ref{['constraint']}).
  • Figure 4: The 4-point function $\langle V_{S_2'}V_{A_j}V_{A_i}V_{\bar{A}_i}\rangle$. Unlike in fig \ref{['fig:disk']}, the $A_i$ vertex operators are drawn inserted on the boundary of the cylinder to show which D-brane they belong to. Since the vertex operators in (\ref{['vops']}) contain both open and closed string vertex operators, it is immaterial whether we draw them with dashed or wiggly lines.