Symplectic representation

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

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 $\mathrm{\Omega }$ must exists, i.e., $h^{n,0}=1$ and $c_1=0$.

In case V is the tangent bundle $T_Y$ and $\omega$ 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 $\mathrm{\Delta }$, dilaton $\varphi$ and the background flux H, are determined by

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

References

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• Cardoso, Curio, Dall'Agata, Lust, Manousselis, and Zoupanos, Non-Kähler String Backgrounds and their Five Torsion Classes, hep-th/0211118