The physical moduli of heterotic G2 string compactifications

A bstract In previous works, an operator was developed for heterotic compactifications on ℝ 2 , 1 × G 2 and AdS 3 × G 2 , which preserves N = 1 d = 3 supersymmetry and whose kernel is related to the moduli of the compactification. The operator is described in terms of non-physical spurious degrees of freedom, specifically, deformations of a connection on the tangent bundle. In this paper, we eliminate these spurious degrees of freedom by linking deformations of the spin connection to the moduli of the G 2 manifold Y . This results in an operator $$ \overset{\check{} }{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mi>ˇ</mml:mi> </mml:mover> </mml:math> that captures the physical moduli space of the G 2 heterotic string theory. When Y = X × S 1 , with X an SU(3) manifold, we show $$ \overset{\check{} }{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mi>ˇ</mml:mi> </mml:mover> </mml:math> produces results that align with existing literature. This allows us to propose a G 2 moduli space metric. We check that this metric reduces to the SU(3) moduli metric constructed in the literature. We then define an adjoint operator $$ {\overset{\check{} }{D}}^{\dagger } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mi>ˇ</mml:mi> </mml:mover> <mml:mo>†</mml:mo> </mml:msup> </mml:math> . We show the G 2 moduli correspond to the intersection of the kernels of $$ \overset{\check{} }{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mi>ˇ</mml:mi> </mml:mover> </mml:math> and $$ {\overset{\check{} }{D}}^{\dagger } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mi>ˇ</mml:mi> </mml:mover> <mml:mo>†</mml:mo> </mml:msup> </mml:math> . These kernels reduce to the SU(3) F-terms and D-terms respectively on X × S 1 . This gives two non-trivial consistency checks of our proposed moduli space metric. Working perturbatively in α ‵, we also demonstrate that the heterotic G 2 moduli problem can be characterised in terms of a double extension of ordinary bundles, just like in the SU(3) case.