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:

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., $$d(||\Omega ||_\omega \omega^2)=0$$.
 * 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) $$d^{\dagger}\omega=i(\partial-\bar{\partial})\text{ln}||\Omega ||,$$
 * 1) The Yang–Mills field strength must satisfy,
 * 2) $$\omega^{a\bar{b}} F_{a\bar{b}}=0,$$
 * 3) $$F_{ab}=F_{\bar{a}\bar{b}}=0.$$

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) $$H=\frac{i}{2}(\bar{\partial}-\partial)\omega.$$