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

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., [math]\displaystyle{ c_2(M)=c_2(F) }[/math]
2. A holomorphic n-form [math]\displaystyle{ \Omega }[/math] must exists, i.e., [math]\displaystyle{ h^{n,0}=1 }[/math] and [math]\displaystyle{ c_1=0 }[/math].

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

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

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

References

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