No manifest <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi></mml:math> duality at order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow><mml:mi>α</mml:mi></mml

When reduced from 10 to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>10</a:mn><a:mo>−</a:mo><a:mi>d</a:mi></a:math> dimensions tree-level string theory exhibits an <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>O</c:mi><c:mo stretchy="false">(</c:mo><c:mi>d</c:mi><c:mo>,</c:mo><c:mi>d</c:mi><c:mo stretchy="false">)</c:mo></c:math> symmetry. This symmetry, which is closely related to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>T</g:mi></g:math> duality, appears only after certain field redefinitions. We find a simple form for a subset of these redefinitions at order <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mrow><i:mi>α</i:mi></i:mrow><i:mrow><i:mo>′</i:mo><i:mn>3</i:mn></i:mrow></i:msup></i:math> and show that they cannot be lifted to ten dimensions. This is inconsistent with “manifestly <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>T</k:mi></k:math> duality invariant” approaches such as generalized geometry (in the uncompactified setting). Such formulations therefore seem not to be the correct language to describe string theory.