Orbifolds from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mrow><mml:mi mathvariant="normal">Sp</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and their modular symmetries

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond SL(2,Z). In the simplest case with one Wilson line Z, Kähler modulus T and complex structure modulus U, we are led to the Siegel modular group Sp(4,Z). It includes SL(2,Z)T×SL(2,Z)U as well as Z2 mirror symmetry, which interchanges T and U. Possible applications to flavor physics of the Standard Model require the study of orbifolds of Sp(4,Z) to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of Sp(4,Z). Some cases correspond to symmetric orbifolds that extend previously discussed cases of SL(2,Z). Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of Sp(4,Z) reveal exciting new aspects of modular symmetries with promising applications to flavor model building.