[Paper Review] Magnetic Flux in Toroidal Type I Compactification

Ralph Blumenhagen, Lars Goerlich+2 more|arXiv (Cornell University)|Oct 23, 2000

Black Holes and Theoretical Physics3 references11 citations

TL;DR

This paper investigates type I string compactification on a torus with background magnetic flux on D9-branes, showing that such flux induces noncommutative open string geometry and enables tadpole cancellation while generating chiral fermions and tunable gauge groups. The key result is a mechanism to engineer realistic gauge groups like U(3)×U(2)×U(1)² with four chiral generations and anomaly cancellation, though challenges remain in achieving large volume compactifications and chiral spectra.

ABSTRACT

We discuss the compactification of type I strings on a torus with additional background gauge flux on the D9-branes. The solutions to the cancellation of the RR tadpoles display various phenomenologically attractive features: supersymmetry breaking, chiral fermions and the opportunity to reduce the rank of the gauge group as desired. We also point out the equivalence of the concept of various different background fields and noncommutative deformations of the geometry on the individual D9-branes by constructing the relevant boundary states to describe such objects.

Motivation & Objective

• To understand the effects of background magnetic flux on D9-branes in type I string theory compactified on a torus.
• To establish the equivalence between magnetic flux backgrounds and noncommutative deformations of open string coordinates on D-branes.
• To derive and solve RR tadpole cancellation conditions in the presence of fluxed D9-branes, ensuring one-loop consistency.
• To construct phenomenologically viable models with chiral fermions, reduced gauge group rank, and potential for anomaly cancellation.
• To identify fundamental obstructions to achieving large-volume compactifications compatible with chiral spectra.

Proposed method

• Utilizes boundary state formalism to describe D-branes with constant magnetic flux, encoding the flux via boundary conditions on worldsheet fields.
• Derives the noncommutative open string coordinate algebra using the deformation parameters Gμij and θμij, showing [X₁,X₂] ∝ θμ¹².
• Applies T-duality to map fluxed D9-branes to D-branes at angles, providing intuitive insight into the geometry and spectrum.
• Computes one-loop Klein bottle amplitudes to derive tadpole cancellation conditions for D9-brane and D5-brane charges in various dimensions.
• Constructs a semi-realistic model with four chiral generations and U(3)×U(2)×U(1)² gauge group, analyzing anomaly structure and quantum numbers.
• Uses intersection number counting in the T-dual picture to determine fermion multiplicities and chirality.

Experimental results

Research questions

• RQ1 How does magnetic flux on D9-branes in type I compactification affect the noncommutative structure of open string coordinates?
• RQ2 What are the conditions for RR tadpole cancellation when D9-branes carry magnetic flux, and how do they constrain the allowed gauge groups and spectra?
• RQ3 Can chiral fermion spectra be consistently generated in this framework, and what determines their multiplicities?
• RQ4 To what extent can the gauge group rank be reduced or engineered through flux configurations?
• RQ5 Why is it impossible to achieve a large-volume compactification while preserving a chiral spectrum in this setup?

Key findings

• The presence of magnetic flux on D9-branes induces a noncommutative deformation of the open string coordinate algebra, with the commutator [X₁,X₂] = iθμ¹² proportional to the flux strength.
• RR tadpole cancellation in six dimensions requires ∑μ Nμnμ⁽¹⁾nμ⁽²⁾ = 16 for D9-brane charge and ∑μ Nμmμ⁽¹⁾mμ⁽²⁾ = 0 for D5-brane charge.
• The gauge group on a stack of N fluxed D9-branes is U(N), not SO(N) or Sp(N), due to the absence of Ω-projection.
• Chiral fermion spectra arise when two D9-branes have unequal flux on all compactified directions, with multiplicities determined by intersection numbers.
• Supersymmetry is always broken in this setup, even if subtly, and the one-loop amplitude is nonvanishing due to a surviving NSNS tadpole.
• A semi-realistic model with four chiral generations and U(3)×U(2)×U(1)² gauge group is constructed, where one U(1) factor can serve as hypercharge and anomaly cancellation is possible via a Green-Schwarz mechanism.