The $${\textbf{H}}$$-flux on flag manifolds generated by infinitesimal $${\textbf{T}}$$-duality

Abstract We define a new correspondence for pairs $$({\mathbb {F}},H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> formed by a flag manifold $${\mathbb {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> together with an H -flux on $${\mathbb {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . Given its role within our correspondence, infinitesimal T -duality may be viewed as a source of H -flux, in the sense that it contributes towards taking fluxless pairs $$({\mathbb {F}},0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to pairs $$({\mathbb {F}}^\vee , H^\vee )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mo>∨</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∨</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> carrying nonzero flux $$H^\vee \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∨</mml:mo> </mml:msup> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We also illustrate how our correspondence exchanges complex structures with symplectic ones up to B -transformations.