The heterotic G₂ system on contact Calabi–Yau 7-manifolds

Abstract

We obtain non-trivial approximate solutions to the heterotic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {G}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> system on the total spaces of non-trivial circle bundles over Calabi–Yau <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -orbifolds, which satisfy the equations up to an arbitrarily small error, by adjusting the size of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fibres in proportion to a power of the string constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha prime"> <mml:semantics> <mml:msup> <mml:mi> α </mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\alpha ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Each approximate solution provides a cocalibrated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {G}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structure, the torsion of which realises a non-trivial scalar field, a constant (trivial) dilaton field and an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -flux with non-trivial Chern–Simons defect. The approximate solutions also include a connection on the tangent bundle which, together with a non-flat <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {G}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -instanton induced from the horizontal Calabi–Yau metric, satisfy the anomaly-free condition, also known as the heterotic Bianchi identity. The approximate solutions fail to be genuine solutions solely because the connections on the tangent bundle are only <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {G}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -instantons up to higher order corrections in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha prime"> <mml:semantics> <mml:msup> <mml:mi> α </mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\alpha ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Topics & Concepts