<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> duality in higher-derivative corrections of heterotic supergravity

This study investigates the consistency of heterotic supergravity, where the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>B</a:mi></a:math>-field gauge transformation becomes anomalous due to the Green-Schwarz mechanism, under <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>T</c:mi></c:math>-duality transformations. We establish that preserving <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>T</e:mi></e:math> duality necessitates an infinite tower of higher-derivative couplings proportional to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>e</g:mi><g:mrow><g:mo>−</g:mo><g:mn>2</g:mn><g:mi mathvariant="normal">Φ</g:mi></g:mrow></g:msup></g:math>. For spacetimes featuring a Killing self-dual circle, these couplings remain immune to quantum corrections. Leveraging <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>S</j:mi></j:math> duality in heterotic or type I string theory, we then identify the corresponding couplings in type I theory. Notably, our analysis reveals that standard <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>S</l:mi></l:math>-duality transformations do not involve higher-derivative corrections. Building on this insight, we derive the explicit form of the effective action for type I theory at the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>α</n:mi><n:mo>′</n:mo></n:msup></n:math> order, adhering to a scheme that omits derivatives of the dilaton.