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.[1]
Consider a metric on the real 6-dimensional internal manifold Y and a Hermitian metric h on a vector bundle V. The equations are:
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- The 4-dimensional spacetime is Minkowski, i.e., .
- The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish .
- The Hermitian form on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
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., .[2]
- The Yang-Mills field strength must satisfy,
These equations imply the usual field equations, and thus are the only equations to be solved.
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However, there are topological obstructions in obtaining the solutions to the equations;
- The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e.,
- A holomorphicn-form must exists, i.e., and .
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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
- ,
- ,
References[edit]
- ^Strominger, Superstrings with Torsion, Nuclear Physics B274 (1986) 253-284
- ^Li and Yau, The Existence of Supersymmetric String Theory with Torsion, J. Differential Geom. Volume 70, Number 1 (2005), 143-181
- Cardoso, Curio, Dall'Agata, Lust, Manousselis, and Zoupanos, Non-Kähler String Backgrounds and their Five Torsion Classes, hep-th/0211118
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Strominger%27s_equations&oldid=890165766'
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