Strominger's equations

rdf:langString Strominger's equations

rdf:langString 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 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., . 2. * The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish . 3. * The Hermitian form on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy, 4. 1. * 5. 2. * where is the Hull-curvature two-form of , F is the curvature of h, and 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 being conformally balanced, i.e., . 6. * The Yang–Mills field strength must satisfy, 7. 1. * 8. 2. * 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., 2. * A holomorphic n-form must exists, i.e., and . In case V is the tangent bundle and is Kähler, we can obtain a solution of these equations by taking the Calabi–Yau metric on and . Once the solutions for the Strominger's equations are obtained, the warp factor , dilaton and the background flux H, are determined by 1. * , 2. * , 3. *