The Pfaffian-Grassmannian derived equivalence

Abstract

We argue that there exists a derived equivalence between Calabi–Yau threefolds obtained by taking dual hyperplane sections (of the appropriate codimension) of the Grassmannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G left-parenthesis 2 comma 7 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {G}(2,7)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Pfaffian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P bold f left-parenthesis 7 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">P</mml:mi> <mml:mi mathvariant="bold">f</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>7</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Pf}(7)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The existence of such an equivalence has been conjectured by physicists for almost ten years, as the two families of Calabi–Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between non-birational Calabi–Yau threefolds.