Double space T dualization and coordinate dependent RR field

In this article, we examine <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>T</a:mi></a:mrow></a:math> dualization in the double space formalism of type II superstring theory in pure spinor formulation. All background fields are constant except the Ramond-Ramond field, which depends infinitesimally on bosonic coordinates <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>x</c:mi><c:mi>μ</c:mi></c:msup></c:math>. In double space, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>T</e:mi></e:mrow></e:math> dual transformations are represented as permutations of the starting <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>x</g:mi><g:mi>μ</g:mi></g:msup></g:math> and dual coordinates <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>y</i:mi><i:mi>μ</i:mi></i:msub></i:math>. Combining these two sets of coordinates into the double coordinate <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>Z</k:mi><k:mi>M</k:mi></k:msup><k:mo>=</k:mo><k:mo stretchy="false">(</k:mo><k:msup><k:mi>x</k:mi><k:mi>μ</k:mi></k:msup><k:mo>,</k:mo><k:msub><k:mi>y</k:mi><k:mi>μ</k:mi></k:msub><k:mo stretchy="false">)</k:mo></k:math>, while demanding that the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi>T</o:mi></o:mrow></o:math> dual double coordinate has the same <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:mi>T</q:mi></q:mrow></q:math> dual transformation law as the initial ones, we obtain how background fields transform under <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mrow><s:mi>T</s:mi></s:mrow></s:math> duality. Comparing these results with ones obtained using the Buscher <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>T</u:mi></u:mrow></u:math> dualization procedure, we conclude that these two approaches are equivalent for the considered choice of the background fields. Published by the American Physical Society 2024