Generation structures and Yukawa couplings in magnetized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>g</mml:mi> </mml

We study fermion zero-mode wave functions with various chiralities in the magnetized <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>T</a:mi> <a:mrow> <a:mn>2</a:mn> <a:mi>g</a:mi> </a:mrow> </a:msup> </a:math> , ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mrow> <c:mi>g</c:mi> <c:mo>=</c:mo> <c:mn>2</c:mn> </c:mrow> <c:mo>,</c:mo> <c:mn>3</c:mn> </c:mrow> </c:math> ), torus. First, we consider the wave functions satisfying the Dirac equation and the boundary conditions on the magnetized torus. Second, we introduce the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>S</e:mi> <e:mi>O</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>3</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> (or parity) transformations and derive the wave functions under the modular transformation. Additionally, we calculate the Yukawa couplings taking chirality into account. Lastly, we briefly review how to construct the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>T</i:mi> <i:mn>4</i:mn> </i:msup> <i:mo>/</i:mo> <i:msub> <i:mi mathvariant="double-struck">Z</i:mi> <i:mi>N</i:mi> </i:msub> </i:math> , ( <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mrow> <l:mi>N</l:mi> <l:mo>=</l:mo> <l:mn>2</l:mn> <l:mo>,</l:mo> <l:mn>3</l:mn> <l:mo>,</l:mo> <l:mn>4</l:mn> <l:mo>,</l:mo> <l:mn>6</l:mn> </l:mrow> </l:math> ), and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mi>T</n:mi> <n:mn>6</n:mn> </n:msup> <n:mo>/</n:mo> <n:msub> <n:mi mathvariant="double-struck">Z</n:mi> <n:mn>12</n:mn> </n:msub> </n:math> twisted orbifold. Also, we explicitly analyze the number of the wave functions in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi mathvariant="double-struck">Z</q:mi> <q:mi>N</q:mi> </q:msub> </q:math> sectors.