Symplectic formulation of the type IIB scalar potential with U -dual fluxes

We present a symplectic formulation of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>1</a:mn></a:math> four-dimensional type IIB scalar potential arising from a flux superpotential which has four <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>S</c:mi></c:math>-dual pairs of fluxes demanded by the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>U</e:mi></e:math>-dual completion arguments. Our symplectic formulation presents a very compact and concise way of expressing the generic scalar potential in just a few terms via using a set of symplectic identities along with the so-called “axionic-flux” combinations. We demonstrate the utility of our symplectic master formula by considering an underlying four-dimensional type IIB supergravity model based on a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msup><g:mrow><g:mi mathvariant="double-struck">T</g:mi></g:mrow><g:mrow><g:mn>6</g:mn></g:mrow></g:msup><g:mo>/</g:mo><g:mo stretchy="false">(</g:mo><g:msub><g:mrow><g:mi mathvariant="double-struck">Z</g:mi></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msub><g:mo>×</g:mo><g:msub><g:mrow><g:mi mathvariant="double-struck">Z</g:mi></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msub><g:mo stretchy="false">)</g:mo></g:mrow></g:math> orientifold, in which the scalar potential induced by the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>U</n:mi></n:math>-dual flux superpotential results in a total of 76 276 terms involving 128 flux parameters. Given that our symplectic formulation does not need the information about the metric of the internal background, it is applicable to the models beyond the toroidal compactifications such as to those which use orientifolds of the Calabi-Yau threefolds. Published by the American Physical Society 2024