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 ${\displaystyle \omega }$ 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., ${\displaystyle g=\eta }$.
2. The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish ${\displaystyle N=0}$.
3. The Hermitian form ${\displaystyle \omega }$ on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
   1. ${\displaystyle \partial {\bar {\partial }}\omega =i{\text{Tr}}F(h)\wedge F(h)-i{\text{Tr}}R^{-}(\omega )\wedge R^{-}(\omega ),}$
   2. ${\displaystyle d^{\dagger }\omega =i(\partial -{\bar {\partial }}){\text{ln}}||\Omega ||,}$
      where ${\displaystyle R^{-}}$ is the Hull-curvature two-form of ${\displaystyle \omega }$, F is the curvature of h, and ${\displaystyle \Omega }$ 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 ${\displaystyle \omega }$ being conformally balanced, i.e., ${\displaystyle d(||\Omega ||_{\omega }\omega ^{2})=0}$.^[2]
4. The Yang–Mills field strength must satisfy,
   1. ${\displaystyle \omega ^{a{\bar {b}}}F_{a{\bar {b}}}=0,}$
   2. ${\displaystyle F_{ab}=F_{{\bar {a}}{\bar {b}}}=0.}$

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., ${\displaystyle c_{2}(M)=c_{2}(F)}$
2. A holomorphic n-form ${\displaystyle \Omega }$ must exists, i.e., ${\displaystyle h^{n,0}=1}$ and ${\displaystyle c_{1}=0}$.

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

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

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

References

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