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]

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

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;

1. The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., ${\displaystyle c_2(M)=c_2(F)}$
2. A holomorphicn-form ${\displaystyle \mathrm{\Omega }}$ must exists, i.e., ${\displaystyle h^{n,0}=1}$ and ${\displaystyle c_1=0}$.

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

Once the solutions for the Strominger's equations are obtained, the warp factor ${\displaystyle \mathrm{\Delta }}$, dilaton ${\displaystyle \varphi }$ and the background flux H, are determined by

1. ${\displaystyle \mathrm{\Delta }(y)=\varphi (y)+\text{constant}}$,
2. ${\displaystyle \varphi (y)=\frac{1}{8}\text{ln}||\mathrm{\Omega }||+\text{constant}}$,
3. ${\displaystyle H=\frac{i}{2}(\overline{\mathrm{\partial }}-\mathrm{\partial })\omega .}$

References[edit]

1. ^Strominger, Superstrings with Torsion, Nuclear Physics B274 (1986) 253-284
2. ^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

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