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Kaehler metrics and gauge kinetic functions for intersecting D6-branes on toroidal orbifolds - The complete perturbative story

Gabriele Honecker

TL;DR

This work provides a comprehensive, perturbatively exact mapping from open-string gauge data to 4D ${\rm \mathcal{N}=1}$ supergravity by deriving the holomorphic gauge kinetic functions, open-string Kaehler metrics, and closed-string Kaehler potentials for intersecting D6-branes on toroidal orientifolds and orbifolds. By systematically computing and reexpressing open-string one-loop gauge thresholds across bulk, fractional, and rigid D6-branes, the authors identify three distinct one-loop correction types—moduli-dependent at vanishing angles, complex-structure dependent for fractional/rigid branes, and constant contributions from O6-plane intersections—and demonstrate their universal applicability to phenomenologically relevant models. The paper provides explicit results for several models, including complete gauge-kinetic and Kaehler data for T^6/\Z_2\times\Z_2 with rigid branes and for the Standard-Model-like fractional D6-brane stacks on T^6/\Z_6', illustrating how moduli, Wilson lines, displacements, and discrete torsion shape the effective field theory. Through detailed comparisons with T-dual constructions (e.g., magnetised D9/D5 setups), the work validates the perturbative framework and offers practical tools for constructing and analyzing globally consistent string vacua with realistic gauge sectors. These results enable precise computations of gauge couplings, Yukawa textures, and potential kinetic-mixing effects crucial for connecting string theory to low-energy phenomenology.

Abstract

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kaehler metrics and closed string Kaehler potential on intersecting D6-branes by matching open string one-loop computations of gauge thresholds with field theoretical gauge couplings in N=1 supergravity. We consider all cases of bulk, fractional and rigid D6-branes on T6/OR and the orbifolds T6/(Z(N)xOR) and T6/(Z(2)xZ(2M)xOR) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kaehler potential. Our analysis includes all supersymmetric configurations of vanishing and non-vanishing angles among D6-branes and O6-planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kaehler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one-loop corrections: a Kaehler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6-branes, and finally a constant term from intersections with O6-planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D-brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kaehler metrics for some T6/Z(2)xZ(2) models with rigid D-branes of arXiv:0902.1694 [hep-th] by Angelantonj et al. As a second class of examples, the Kaehler metrics and gauge kinetic functions for the fractional QCD and leptonic D6-brane stacks of the Standard Model on T6/Z(6') from arXiv:0806.3039 [hep-th] by Gmeiner and Honecker are given.

Kaehler metrics and gauge kinetic functions for intersecting D6-branes on toroidal orbifolds - The complete perturbative story

TL;DR

This work provides a comprehensive, perturbatively exact mapping from open-string gauge data to 4D supergravity by deriving the holomorphic gauge kinetic functions, open-string Kaehler metrics, and closed-string Kaehler potentials for intersecting D6-branes on toroidal orientifolds and orbifolds. By systematically computing and reexpressing open-string one-loop gauge thresholds across bulk, fractional, and rigid D6-branes, the authors identify three distinct one-loop correction types—moduli-dependent at vanishing angles, complex-structure dependent for fractional/rigid branes, and constant contributions from O6-plane intersections—and demonstrate their universal applicability to phenomenologically relevant models. The paper provides explicit results for several models, including complete gauge-kinetic and Kaehler data for T^6/\Z_2\times\Z_2 with rigid branes and for the Standard-Model-like fractional D6-brane stacks on T^6/\Z_6', illustrating how moduli, Wilson lines, displacements, and discrete torsion shape the effective field theory. Through detailed comparisons with T-dual constructions (e.g., magnetised D9/D5 setups), the work validates the perturbative framework and offers practical tools for constructing and analyzing globally consistent string vacua with realistic gauge sectors. These results enable precise computations of gauge couplings, Yukawa textures, and potential kinetic-mixing effects crucial for connecting string theory to low-energy phenomenology.

Abstract

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kaehler metrics and closed string Kaehler potential on intersecting D6-branes by matching open string one-loop computations of gauge thresholds with field theoretical gauge couplings in N=1 supergravity. We consider all cases of bulk, fractional and rigid D6-branes on T6/OR and the orbifolds T6/(Z(N)xOR) and T6/(Z(2)xZ(2M)xOR) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kaehler potential. Our analysis includes all supersymmetric configurations of vanishing and non-vanishing angles among D6-branes and O6-planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kaehler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one-loop corrections: a Kaehler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6-branes, and finally a constant term from intersections with O6-planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D-brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kaehler metrics for some T6/Z(2)xZ(2) models with rigid D-branes of arXiv:0902.1694 [hep-th] by Angelantonj et al. As a second class of examples, the Kaehler metrics and gauge kinetic functions for the fractional QCD and leptonic D6-brane stacks of the Standard Model on T6/Z(6') from arXiv:0806.3039 [hep-th] by Gmeiner and Honecker are given.

Paper Structure

This paper contains 36 sections, 172 equations, 2 figures, 35 tables.

Table of Contents

  1. Introduction
  2. The gauge thresholds revisited
  3. Geometry, three-cycles and RR tadpole cancellation
  4. Orbifolds, compactification lattices and one-cycles
  5. Three-cycles and RR tadpole cancellation
  6. $SU(N)$, $SO(2N)$ and $Sp(2N)$ gauge thresholds on fractional and rigid D6-branes
  7. Bulk D6-branes on $T^6/\Omega\mathcal{R}$ and $T^6/(\mathbb{Z}_3 \times \Omega\mathcal{R})$
  8. Fractional D6-branes on $T^6/(\mathbb{Z}_2 \times \mathbb{Z}_{2M} \times \Omega\mathcal{R})$ without discrete torsion
  9. Fractional D6-branes on $T^6/(\mathbb{Z}_{2N}\times \Omega\mathcal{R})$
  10. Example I: the adjoints of $SU(N_a)$
  11. Example 2: (anti)symmetric matter of $SO(2M_x)$ and $Sp(2M_x)$ gauge groups
  12. Comments on anomaly-free $U(1)$ gauge groups
  13. Kähler metrics, Kähler potential and holomorphic gauge kinetic function at one loop
  14. Tree level gauge kinetic function
  15. One-loop results for $T^6$ and $T^6/\mathbb{Z}_2$ and $T^6/\mathbb{Z}_2 \times \mathbb{Z}_2$ with $\eta=\pm 1$
  16. ...and 21 more sections

Figures (2)

  • Figure 1: The $\mathbb{Z}_2$ invariant 'untilted' a-type (left) and 'tilted' b-type (right) tori, which are parameterised by $b=0,\frac{1}{2}$, respectively. The $\mathbb{Z}_2$ fixed points are depicted in blue. The points 1,4 are invariant under ${\cal R}$, whereas the other two points are on the tilted torus exchanged under ${\cal R}$, $2 \stackrel{\cal R}{\leftrightarrow} 2+2b$ and $3 \stackrel{\cal R}{\leftrightarrow} 3-2b$. The untilted torus with $R_1=R_2=r$ corresponds to the A-type $\mathbb{Z}_4$ invariant lattice in figure \ref{['Fig:Z4-Z6lattice']}. The tilted torus for $R_2/R_1=2,2\sqrt{3},2/\sqrt{3}$ corresponds to the B-type $\mathbb{Z}_4$ lattice and the A- and B-type $\mathbb{Z}_3$ (and $\mathbb{Z}_6$) invariant lattices in figure \ref{['Fig:Z4-Z6lattice']} with radii $r=\sqrt{2} R_2, R_2/\sqrt{3},R_2$, respectively.
  • Figure 2: The $\mathbb{Z}_4$ (left) and $\mathbb{Z}_3$ (right) invariant lattices. For the A orientation (with green coordinate axes), $\pi_{2i-1}$ spans the $\Re(z)$ axis, and on the B lattice (axes in yellow), the $\Re(z)$ axis extends along $\pi_{2i-1} + \pi_{2i}$. The $\mathbb{Z}_4$ invariant points 1,2 (left, in red) are ${\cal R}$ invariant, the additional $\mathbb{Z}_2$ fixed points $3 \stackrel{\mathbb{Z}_4}{\leftrightarrow} 4$ (left, in blue) are ${\cal R}$ invariant on the A-lattice, but are permuted under ${\cal R}$ on the B-lattice. The $\mathbb{Z}_3$ invariant points $2\stackrel{\mathbb{Z}_2}{\leftrightarrow} 3$ (right, in blue) are exchanged under ${\cal R}$ on the A orientation and are invariant on the B-lattice. The $\mathbb{Z}_2$ fixed points $6 \stackrel{\mathbb{Z}_3}{\rightarrow} 5 \stackrel{\mathbb{Z}_3}{\rightarrow} 4$ (right, in red) contain one point fixed under ${\cal R}$, the other two are exchanged under ${\cal R}$. The origin 1 is fixed under the full ${\cal R}$ and $\mathbb{Z}_6$ symmetry.