Physics: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 [math]\displaystyle{ \omega }[/math] 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., [math]\displaystyle{ g=\eta }[/math].
- The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish [math]\displaystyle{ N=0 }[/math].
- The Hermitian form [math]\displaystyle{ \omega }[/math] on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
- [math]\displaystyle{ \partial\bar{\partial}\omega=i\text{Tr}F(h)\wedge F(h)-i\text{Tr}R^{-}(\omega)\wedge R^{-}(\omega), }[/math]
- [math]\displaystyle{ d^{\dagger}\omega=i(\partial-\bar{\partial})\text{ln}||\Omega ||, }[/math]
where [math]\displaystyle{ R^{-} }[/math] is the Hull-curvature two-form of [math]\displaystyle{ \omega }[/math], F is the curvature of h, and [math]\displaystyle{ \Omega }[/math] 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 [math]\displaystyle{ \omega }[/math] being conformally balanced, i.e., [math]\displaystyle{ d(||\Omega ||_\omega \omega^2)=0 }[/math].[2]
- The Yang–Mills field strength must satisfy,
- [math]\displaystyle{ \omega^{a\bar{b}} F_{a\bar{b}}=0, }[/math]
- [math]\displaystyle{ F_{ab}=F_{\bar{a}\bar{b}}=0. }[/math]
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., [math]\displaystyle{ c_2(M)=c_2(F) }[/math]
- A holomorphic n-form [math]\displaystyle{ \Omega }[/math] must exists, i.e., [math]\displaystyle{ h^{n,0}=1 }[/math] and [math]\displaystyle{ c_1=0 }[/math].
In case V is the tangent bundle [math]\displaystyle{ T_Y }[/math] and [math]\displaystyle{ \omega }[/math] is Kähler, we can obtain a solution of these equations by taking the Calabi–Yau metric on [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ T_Y }[/math].
Once the solutions for the Strominger's equations are obtained, the warp factor [math]\displaystyle{ \Delta }[/math], dilaton [math]\displaystyle{ \phi }[/math] and the background flux H, are determined by
- [math]\displaystyle{ \Delta(y)=\phi(y)+\text{constant} }[/math],
- [math]\displaystyle{ \phi(y)=\frac{1}{8} \text{ln}||\Omega||+\text{constant} }[/math],
- [math]\displaystyle{ H=\frac{i}{2}(\bar{\partial}-\partial)\omega. }[/math]
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
![]() | Original source: https://en.wikipedia.org/wiki/Strominger's equations.
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