Strominger's equations explained

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

\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 =η .

1. The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish

N =0 .\omega on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,\partial\bar{\partial}\omega = i Tr F(h)\wedge F(h) - i Tr R ^{-} (\omega)\wedge R ^{-} (\omega), d ^{\dagger} \omega = i(\partial - \bar{\partial}) ln ||\Omega ||,
where R ^{-} is the Hull-curvature two-form of \omega , F is the curvature of h, and \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(||\Omega || _{\omega} \omega ^{2)=0} .^[2]

1. The Yang–Mills field strength must satisfy,

\omega ^{a\bar{b} } F_=0,F _{ab} = F _{\bar{a} \bar{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)}

1. 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 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

\Delta , dilaton \phi and the background flux H, are determined by\Delta(y) = \phi(y) + constant ,\phi(y) = (1)/(8) ln ||\Omega|| + constant ,H = (i)/(2) (\bar{\partial} - \partial)\omega.

References

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

Notes and References

1. Superstrings with torsion . 10.1016/0550-3213(86)90286-5 . 1986 . Strominger . Andrew . Nuclear Physics B . 274 . 2 . 253–284 . 1986NuPhB.274..253S .
2. Li and Yau, The Existence of Supersymmetric String Theory with Torsion, J. Differential Geom. Volume 70, Number 1 (2005), 143-181