Symplectic representation

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.

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:

1. The 4-dimensional spacetime is Minkowski, i.e., $g=\eta$.
2. The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish $N=0$.
3. 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., $d(||\Omega ||_{\omega }\omega ^{2})=0$.
4. 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;

1. The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., $c_{2}(M)=c_{2}(F)$
2. 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

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

References

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