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 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. #
  5. #
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.,.
  1. The Yang-Mills field strength must satisfy,
  2. #
  3. #
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. ,
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