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.

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,
4. \partial\bar{\partial}\omega=i\text{Tr}F(h)\wedge F(h)-i\text{Tr}R^{-}(\omega)\wedge R^{-}(\omega),
5. 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.
6. The Yang–Mills field strength must satisfy,
7. \omega^{a\bar{b}} F_{a\bar{b}}=0,
8. 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.