Modular symmetry in magnetized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>g</mml:mi></mml:mrow></mml:msup></mml:math> torus and orbifold models

Abstract

We study the modular symmetry in 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> torus and orbifold models. The <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>T</c:mi><c:mrow><c:mn>2</c:mn><c:mi>g</c:mi></c:mrow></c:msup></c:math> torus has the modular symmetry <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi mathvariant="normal">Γ</e:mi><e:mi>g</e:mi></e:msub><e:mo>=</e:mo><e:mi>S</e:mi><e:mi>p</e:mi><e:mo stretchy="false">(</e:mo><e:mn>2</e:mn><e:mi>g</e:mi><e:mo>,</e:mo><e:mi mathvariant="double-struck">Z</e:mi><e:mo stretchy="false">)</e:mo></e:math>. The magnetic flux background breaks the modular symmetry to a certain normalizer <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>N</k:mi><k:mi>g</k:mi></k:msub><k:mo stretchy="false">(</k:mo><k:mi>H</k:mi><k:mo stretchy="false">)</k:mo></k:math>. We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msup><o:mi>T</o:mi><o:mrow><o:mn>2</o:mn><o:mi>g</o:mi></o:mrow></o:msup></o:math> and certain orbifolds. It is found that wave functions on magnetized <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>T</q:mi><q:mrow><q:mn>2</q:mn><q:mi>g</q:mi></q:mrow></q:msup></q:math> as well as its orbifolds behave as the Siegel modular forms of weight <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mn>1</s:mn><s:mo>/</s:mo><s:mn>2</s:mn></s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mover accent="true"><u:mi>N</u:mi><u:mo stretchy="false">˜</u:mo></u:mover><u:mi>g</u:mi></u:msub><u:mo stretchy="false">(</u:mo><u:mi>H</u:mi><u:mo>,</u:mo><u:mi>h</u:mi><u:mo stretchy="false">)</u:mo></u:math>, which is the metaplectic congruence subgroup of the double covering group of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>N</ab:mi><ab:mi>g</ab:mi></ab:msub><ab:mo stretchy="false">(</ab:mo><ab:mi>H</ab:mi><ab:mo stretchy="false">)</ab:mo></ab:math>, <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mover accent="true"><eb:mi>N</eb:mi><eb:mo stretchy="false">˜</eb:mo></eb:mover><eb:mi>g</eb:mi></eb:msub><eb:mo stretchy="false">(</eb:mo><eb:mi>H</eb:mi><eb:mo stretchy="false">)</eb:mo></eb:math>. Then, wave functions transform nontrivially under the quotient group, <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mover accent="true"><kb:mi>N</kb:mi><kb:mo stretchy="false">˜</kb:mo></kb:mover><kb:mrow><kb:mi>g</kb:mi><kb:mo>,</kb:mo><kb:mi>h</kb:mi></kb:mrow></kb:msub><kb:mo>=</kb:mo><kb:msub><kb:mover accent="true"><kb:mi>N</kb:mi><kb:mo stretchy="false">˜</kb:mo></kb:mover><kb:mi>g</kb:mi></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mi>H</kb:mi><kb:mo stretchy="false">)</kb:mo><kb:mo>/</kb:mo><kb:msub><kb:mover accent="true"><kb:mi>N</kb:mi><kb:mo stretchy="false">˜</kb:mo></kb:mover><kb:mi>g</kb:mi></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mi>H</kb:mi><kb:mo>,</kb:mo><kb:mi>h</kb:mi><kb:mo stretchy="false">)</kb:mo></kb:math>, where the level <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>h</wb:mi></wb:math> is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional chiral fields also transform nontrivially under <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mover accent="true"><yb:mi>N</yb:mi><yb:mo stretchy="false">˜</yb:mo></yb:mover><yb:mrow><yb:mi>g</yb:mi><yb:mo>,</yb:mo><yb:mi>h</yb:mi></yb:mrow></yb:msub></yb:math> modular flavor transformation with modular weight <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mo>−</cc:mo><cc:mn>1</cc:mn><cc:mo>/</cc:mo><cc:mn>2</cc:mn></cc:math>. We also study concrete modular flavor symmetries of wave functions on magnetized <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msup><ec:mi>T</ec:mi><ec:mrow><ec:mn>2</ec:mn><ec:mi>g</ec:mi></ec:mrow></ec:msup></ec:math> orbifolds. Published by the American Physical Society 2024

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