Strominger's equations

In heterotic string theory, the Strominger's equations are the set of equations that are necessary and sufficient conditions for spacetime supersymmetry. It is derived by requiring the 4-dimensional spacetime to be maximally symmetric, and adding a warp factor on the internal 6-dimensional manifold.^[1]

Consider a metric $\omega$ on the real 6-dimensional internal manifold Y and a Hermitian metric h on a vector bundle V. The equations are:

The 4-dimensional spacetime is Minkowski, i.e., $g=\eta$ .
The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish $N=0$ .
The Hermitian form $\omega$ on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
1. $\partial {\bar {\partial }}\omega =i{\text{Tr}}F(h)\wedge F(h)-i{\text{Tr}}R^{{-}}(\omega )\wedge R^{{-}}(\omega ),$
2. $d^{{\dagger }}\omega =i(\partial -{\bar {\partial }}){\text{ln}}||\Omega ||,$
  where $R^{{-}}$ is the Hull-curvature two-form of $\omega$ , F is the curvature of h, and $\Omega$ is the holomorphic n-form; F is also known in the physics literature as the Yang-Mills field strength. Li and Yau showed that the second condition is equivalent to $\omega$ being conformally balanced, i.e., $d(||\Omega ||_{\omega }\omega ^{2})=0$ .^[2]
The Yang-Mills field strength must satisfy,
1. $\omega ^{{a{\bar {b}}}}F_{{a{\bar {b}}}}=0,$
2. $F_{{ab}}=F_{{{\bar {a}}{\bar {b}}}}=0.$

These equations imply the usual field equations, and thus are the only equations to be solved.

However, there are topological obstructions in obtaining the solutions to the equations;

The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., $c_{2}(M)=c_{2}(F)$
A holomorphic n-form $\Omega$ must exists, i.e., $h^{{n,0}}=1$ and $c_{1}=0$ .

In case V is the tangent bundle $T_{Y}$ and $\omega$ is Kähler, we can obtain a solution of these equations by taking the Calabi-Yau metric on $Y$ and $T_{Y}$ .

Once the solutions for the Strominger's equations are obtained, the warp factor $\Delta$ , dilaton $\phi$ and the background flux H, are determined by

$\Delta (y)=\phi (y)+{\text{constant}}$ ,
$\phi (y)={\frac {1}{8}}{\text{ln}}||\Omega ||+{\text{constant}}$ ,
$H={\frac {i}{2}}({\bar {\partial }}-\partial )\omega .$

References

↑ Strominger, Superstrings with Torsion, Nuclear Physics B274 (1986) 253-284
↑ Li and Yau, The Existence of Supersymmetric String Theory with Torsion, J. Differential Geom. Volume 70, Number 1 (2005), 143-181

Cardoso, Curio, Dall'Agata, Lust, Manousselis, and Zoupanos, Non-Kähler String Backgrounds and their Five Torsion Classes, hep-th/0211118

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