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Towards the One-loop Kahler Metric of Calabi-Yau Orientifolds

Marcus Berg, Michael Haack, Jin U. Kang, Stefan Sjors

TL;DR

The work addresses one-loop corrections to the Kähler metric of closed-string moduli in Calabi–Yau orientifolds with D-branes, focusing on the intricate N=1 sectors that couple to all massive string states. By computing the two-point function of closed-string moduli on all one-loop worldsheet topologies and employing a covering-torus lifting, the authors separate four- and eight-fermion contributions, perform spin-structure sums, and manage the integrals via analytic continuation and generalized Mellin transforms. They demonstrate exact UV divergence cancellation under tadpole-consistency conditions and extract a finite N=1 sector contribution to the τ-modulus kinetic term, expressed in terms of Clausen functions and special theta-function combinations. The results provide a concrete, moduli-dependent one-loop correction in the string frame, with an Einstein-frame interpretation that exhibits expected suppression by the string coupling, and lay groundwork for completing the full one-loop effective action including Einstein-Hilbert corrections and N=2-sector contributions.

Abstract

We evaluate string one-loop contributions to the Kahler metric of closed string moduli in toroidal minimally supersymmetric (Calabi-Yau) orientifolds with D-branes. We focus on the poorly understood N=1 sectors that receive contributions from all massive string states.

Towards the One-loop Kahler Metric of Calabi-Yau Orientifolds

TL;DR

The work addresses one-loop corrections to the Kähler metric of closed-string moduli in Calabi–Yau orientifolds with D-branes, focusing on the intricate N=1 sectors that couple to all massive string states. By computing the two-point function of closed-string moduli on all one-loop worldsheet topologies and employing a covering-torus lifting, the authors separate four- and eight-fermion contributions, perform spin-structure sums, and manage the integrals via analytic continuation and generalized Mellin transforms. They demonstrate exact UV divergence cancellation under tadpole-consistency conditions and extract a finite N=1 sector contribution to the τ-modulus kinetic term, expressed in terms of Clausen functions and special theta-function combinations. The results provide a concrete, moduli-dependent one-loop correction in the string frame, with an Einstein-frame interpretation that exhibits expected suppression by the string coupling, and lay groundwork for completing the full one-loop effective action including Einstein-Hilbert corrections and N=2-sector contributions.

Abstract

We evaluate string one-loop contributions to the Kahler metric of closed string moduli in toroidal minimally supersymmetric (Calabi-Yau) orientifolds with D-branes. We focus on the poorly understood N=1 sectors that receive contributions from all massive string states.

Paper Structure

This paper contains 64 sections, 342 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. String effective action
  3. Two-point function of closed strings
  4. Four fermions
  5. Evaluation by analytic continuation
  6. Eight-fermion term: General arguments
  7. First piece of eight-fermion term
  8. Spin structure sum
  9. Second piece of eight-fermion term
  10. Pole $\nu_1=\nu_2$ from twisted correlators
  11. Pole $\nu_1=I_\sigma(\nu_2)$ from untwisted correlators
  12. Adding up eight-fermion: $\sigma={\mathcal{A}},{\mathcal{M}},{\mathcal{K}}$
  13. Eight-fermion term: torus
  14. Adding up orbifold $k$ sectors
  15. Computing the $t$ integral
  16. ...and 49 more sections

Figures (7)

  • Figure 1: The one-loop two-point function on all four worldsheets of Euler character zero.
  • Figure 2: The standard way to make the worldsheet surfaces with crosscaps and boundaries, and their Green's functions, from that of the worldsheet torus is by identifying under a certain involution.
  • Figure 3: A lifting that gives amplitudes for open and unoriented strings from closed strings.
  • Figure 4: The integrand is not invariant under a half-integer shift, but the integral over the fundamental domain is. In this figure, each quarter-piece of integration domain is mapped by an integer lattice shift.
  • Figure 5: One possible factorization of the one-loop two-point function on the annulus.
  • ...and 2 more figures