Abstract We consider a six-dimensional gauge theory compactified on  with magnetic flux. The configurations of models are classified by winding numbers at the fixed points. Requiring the existence of generation numbers and Yukawa coupling, we see that allowed and forbidden configurations are described by geometry of winding numbers.

Japan

Hiroki Imai

Nobuhito Maru

Toward Realistic Models in /ℤ Flux Compactification

We analyze the number of independent chiral zero modes and the winding numbers at the fixed points on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> (<math><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math>) orbifolds with magnetic flux. In the case of <math><mi>N</mi><mo>=</mo><mn>2</mn></math>, we derive the index formula <math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mi>M</mi><mo>/</mo><mn>2</mn><mo>+</mo><mo>(</mo><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mn>4</mn><mo>=</mo><mi>M</mi><mo>/</mo><mn>2</mn><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> by using the trace formula, where <math><msub><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msub></math> are the numbers of the ± chiral zero modes and <math><msub><mrow><mi>V</mi></mrow><mrow><mo>±</mo></mrow></msub></math> are the sums of the winding numbers at the fixed points on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. We also obtain the formula <math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mi>M</mi><mo>/</mo><mi>N</mi><mo>+</mo><mo>(</mo><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mo>(</mo><mn>2</mn><mi>N</mi><mo>)</mo><mo>=</mo><mi>M</mi><mo>/</mo><mi>N</mi><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>/</mo><mi>N</mi><mo>+</mo><mn>1</mn></math> for <math><mi>N</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math> under an assumption.