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. 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