Gradings of Lie algebras, magical spin geometries and matrix factorizations

<p>We describe a remarkable rank <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="14">
  <mml:semantics>
    <mml:mn>14</mml:mn>
    <mml:annotation encoding="application/x-tex">14</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> matrix factorization of the octic <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal p normal i normal n Subscript 14">
  <mml:semantics>
    <mml:msub>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="normal">S</mml:mi>
        <mml:mi mathvariant="normal">p</mml:mi>
        <mml:mi mathvariant="normal">i</mml:mi>
        <mml:mi mathvariant="normal">n</mml:mi>
      </mml:mrow>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mn>14</mml:mn>
      </mml:mrow>
    </mml:msub>
    <mml:annotation encoding="application/x-tex">\mathrm {Spin}_{14}</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z">
  <mml:semantics>
    <mml:mrow class="MJX-TeXAtom-ORD">
      <mml:mi mathvariant="double-struck">Z</mml:mi>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-grading of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German e 8">
  <mml:semantics>
    <mml:msub>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="fraktur">e</mml:mi>
      </mml:mrow>
      <mml:mn>8</mml:mn>
    </mml:msub>
    <mml:annotation encoding="application/x-tex">\mathfrak {e}_8</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>. Intriguingly, the whole story can in fact be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal p normal i normal n Subscript 14">
  <mml:semantics>
    <mml:msub>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="normal">S</mml:mi>
        <mml:mi mathvariant="normal">p</mml:mi>
        <mml:mi mathvariant="normal">i</mml:mi>
        <mml:mi mathvariant="normal">n</mml:mi>
      </mml:mrow>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mn>14</mml:mn>
      </mml:mrow>
    </mml:msub>
    <mml:annotation encoding="application/x-tex">\mathrm {Spin}_{14}</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.</p>