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 manifoldStrominger, " [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVC-4718X2H-16M&_user=126524&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=126524&md5=c045d0aabfb064c58379a5efdf05e008 Superstrings with Torsion] ", Nuclear Physics B274 (1986) 253-284] .

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:

# The 4-dimensional spacetime is Minkowski, i.e., $g=eta$.
# The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish $N=0$.
# The Hermitian form $omega$ on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
##
##
where "R" is the Ricci form, "F" is the Hermitian curvature, also known in the physics literature as the Yang-Mills field strength, and $Omega$ is the holomorphic "n"-form. Li and Yau showed that the second condition is equivalent to $omega$ being conformally balanced, i.e., $d(||Omega\; ||\_omega\; omega^2)=0$Li and Yau, " [http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jdg/1143572017&page=record The Existence of Supersymmetric String Theory with Torsion] ", J. Differential Geom. Volume 70, Number 1 (2005), 143-181] .
# The Yang-Mills field strength must satisify,
##
##

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;

# The second Chern class of the manifold, and the second Chern class of the gauge field must be equal, i.e., $c\_2(M)=c\_2(F)$
# A holomorphic "n"-form $Omega$ must exists, i.e., $h^\{n,0\}=1$ and $c\_1=0$.

In case V is the tangent bundle $T\_Y$ and $omega$ is Kahler, 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 $Delta$, dilaton $phi$ and the background flux "H", are determined by
# $Delta(y)=phi(y)+\; ext\{constant\}$,
# $phi(y)=frac\{1\}\{8\}\; ext\{ln\}||omega||+\; ext\{constant\}$,
#

References

* Cardoso, Curio, Dall'Agata, Lust, Manousselis, and Zoupanos, " [http://www.arxiv.org/pdf/hep-th/0211118 Non-Kahler String Backgrounds and their Five Torsion Classes] ", hep-th/0211118