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 manifold.[1]
Consider a metric 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.,
.
- The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish
.
- The Hermitian form
on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
-
-
whereis the Hull-curvature two-form of
, F is the curvature of h, and
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
being conformally balanced, i.e.,
.[2]
-
- The Yang-Mills field strength must satisfy,
-
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.,
- A holomorphic n-form
must exists, i.e.,
and
.
In case V is the tangent bundle and
is Kähler, we can obtain a solution of these equations by taking the Calabi-Yau metric on
and
.
Once the solutions for the Strominger's equations are obtained, the warp factor , dilaton
and the background flux H, are determined by
-
,
-
,
-
References
- ↑ Strominger, Superstrings with Torsion, Nuclear Physics B274 (1986) 253-284
- ↑ 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