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 ω 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 = η .
2. The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish N = 0 .
3. The Hermitian form ω on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
   1. ∂ ∂ ¯ ω = i Tr F ( h ) ∧ F ( h ) − i Tr R − ( ω ) ∧ R − ( ω ) ,
   2. d † ω = i ( ∂ − ∂ ¯ ) ln | | Ω | | ,
      where R − is the Hull-curvature two-form of ω , F is the curvature of h, and Ω 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 ω being conformally balanced, i.e., d ( | | Ω | | ω ω 2 ) = 0 .
4. The Yang-Mills field strength must satisfy,
   1. ω a b ¯ F a b ¯ = 0 ,
   2. F a b = F a ¯ 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 Ω must exists, i.e., h n , 0 = 1 and c 1 = 0 .

In case V is the tangent bundle T Y and ω 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 Δ , dilaton ϕ and the background flux H, are determined by

1. Δ ( y ) = ϕ ( y ) + constant ,
2. ϕ ( y ) = 1 8 ln | | Ω | | + constant ,
3. H = i 2 ( ∂ ¯ − ∂ ) ω .