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Gauge Threshold Corrections for Local Orientifolds

Joseph P. Conlon, Eran Palti

TL;DR

This work computes gauge threshold corrections for fractional branes at local orientifold singularities and compares the results with the Kaplunovsky–Louis formula for locally supersymmetric N=1 theories. Using a CFT construction for Z_4, Z_6, and Z'_6 orientifolds and a background-field one-loop method, it reveals a two-phase running: N=2 sectors induce running from the bulk winding scale to the string scale, while N=1 plus N=2 sectors drive running from the string scale to the IR. A key finding is that a 1-loop redefinition of twisted moduli Re(M_k) = m_k − α_k ln R^2 is required to restore agreement with KL, reflecting non-universal couplings of twisted moduli to different gauge nodes. The paper provides explicit Z_4 results and checks in related models, illustrating when gauge unification at an enhanced scale M_X arises and when it does not, and highlighting the role of global geometry and U(1) masses in local models. These insights clarify threshold corrections in local string constructions and have implications for local GUT scenarios and their global completions.

Abstract

We study gauge threshold corrections for systems of fractional branes at local orientifold singularities and compare with the general Kaplunovsky-Louis expression for locally supersymmetric N=1 gauge theories. We focus on branes at orientifolds of the C^3/Z_4, C^3/Z_6 and C^3/Z_6' singularities. We provide a CFT construction of these theories and compute the threshold corrections. Gauge coupling running undergoes two phases: one phase running from the bulk winding scale to the string scale, and a second phase running from the string scale to the infrared. The first phase is associated to the contribution of N=2 sectors to the IR beta functions and the second phase to the contribution of both N=1 and N=2 sectors. In contrast, naive application of the Kaplunovsky-Louis formula gives single running from the bulk winding mode scale. The discrepancy is resolved through 1-loop non-universality of the holomorphic gauge couplings at the singularity, induced by a 1-loop redefinition of the twisted blow-up moduli which couple differently to different gauge nodes. We also study the physics of anomalous and non-anomalous U(1)s and give a CFT description of how masses for non-anomalous U(1)s depend on the global properties of cycles.

Gauge Threshold Corrections for Local Orientifolds

TL;DR

This work computes gauge threshold corrections for fractional branes at local orientifold singularities and compares the results with the Kaplunovsky–Louis formula for locally supersymmetric N=1 theories. Using a CFT construction for Z_4, Z_6, and Z'_6 orientifolds and a background-field one-loop method, it reveals a two-phase running: N=2 sectors induce running from the bulk winding scale to the string scale, while N=1 plus N=2 sectors drive running from the string scale to the IR. A key finding is that a 1-loop redefinition of twisted moduli Re(M_k) = m_k − α_k ln R^2 is required to restore agreement with KL, reflecting non-universal couplings of twisted moduli to different gauge nodes. The paper provides explicit Z_4 results and checks in related models, illustrating when gauge unification at an enhanced scale M_X arises and when it does not, and highlighting the role of global geometry and U(1) masses in local models. These insights clarify threshold corrections in local string constructions and have implications for local GUT scenarios and their global completions.

Abstract

We study gauge threshold corrections for systems of fractional branes at local orientifold singularities and compare with the general Kaplunovsky-Louis expression for locally supersymmetric N=1 gauge theories. We focus on branes at orientifolds of the C^3/Z_4, C^3/Z_6 and C^3/Z_6' singularities. We provide a CFT construction of these theories and compute the threshold corrections. Gauge coupling running undergoes two phases: one phase running from the bulk winding scale to the string scale, and a second phase running from the string scale to the infrared. The first phase is associated to the contribution of N=2 sectors to the IR beta functions and the second phase to the contribution of both N=1 and N=2 sectors. In contrast, naive application of the Kaplunovsky-Louis formula gives single running from the bulk winding mode scale. The discrepancy is resolved through 1-loop non-universality of the holomorphic gauge couplings at the singularity, induced by a 1-loop redefinition of the twisted blow-up moduli which couple differently to different gauge nodes. We also study the physics of anomalous and non-anomalous U(1)s and give a CFT description of how masses for non-anomalous U(1)s depend on the global properties of cycles.

Paper Structure

This paper contains 22 sections, 146 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction and Summary of Results
  2. Orientifold constructions
  3. Orientifolds of orbifold singularities
  4. Tadpole amplitudes
  5. The canonical example: $\mathbb{Z}_4$
  6. Threshold corrections: the string calculation
  7. Magnetised amplitudes
  8. The $\mathbb{Z}_4$ case
  9. Threshold corrections: matching the field theory
  10. Extracting closed string couplings
  11. The $\mathbb{Z}_4$ case
  12. More examples
  13. The $\mathbb{C}^3/\mathbb{Z}_6$ orientifold
  14. The $\mathbb{C}^3/\mathbb{Z}'_6$ orientifold
  15. D3-D7 Orbifolds
  16. ...and 7 more sections

Figures (2)

  • Figure 1: From a supergravity perspective this represents an $\left(\hbox{Re}M_k\right) F_{\mu \nu} F^{\mu \nu}$ vertex sourcing an $M_k$ field which propagates and is then absorbed by the vacuum tadpole. By factoring out the vacuum tadpole we can infer the coefficient of the $\left(\hbox{Re}M_k\right) F_{\mu \nu} F^{\mu \nu}$ coupling.
  • Figure 2: The $T^6/\mathbb{Z}_4$ orbifold. Dark circles correspond to $\theta$ fixed points and hollow squares correspond to $\theta^2$ fixed points.