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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]

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Consider a metric ω{displaystyle 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{displaystyle g=eta }.
  2. The internal manifold Y must be complex, i.e., the Nijenhuis tensor must vanish N=0{displaystyle N=0}.
  3. The Hermitian formω{displaystyle omega } on the complex threefold Y, and the Hermitian metric h on a vector bundle V must satisfy,
    1. ¯ω=iTrF(h)F(h)iTrR(ω)R(ω),{displaystyle partial {bar {partial }}omega =i{text{Tr}}F(h)wedge F(h)-i{text{Tr}}R^{-}(omega )wedge R^{-}(omega ),}
    2. dω=i(¯)lnΩ,{displaystyle d^{dagger }omega =i(partial -{bar {partial }}){text{ln}} Omega ,}
      where R{displaystyle R^{-}} is the Hull-curvature two-form of ω{displaystyle omega }, F is the curvature of h, and Ω{displaystyle 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 ω{displaystyle omega } being conformally balanced, i.e., d(Ωωω2)=0{displaystyle d( Omega _{omega }omega ^{2})=0}.[2]
  4. The Yang-Mills field strength must satisfy,
    1. ωab¯Fab¯=0,{displaystyle omega ^{a{bar {b}}}F_{a{bar {b}}}=0,}
    2. Fab=Fa¯b¯=0.{displaystyle F_{ab}=F_{{bar {a}}{bar {b}}}=0.}

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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., c2(M)=c2(F){displaystyle c_{2}(M)=c_{2}(F)}
  2. A holomorphicn-form Ω{displaystyle Omega } must exists, i.e., hn,0=1{displaystyle h^{n,0}=1} and c1=0{displaystyle c_{1}=0}.

In case V is the tangent bundle TY{displaystyle T_{Y}} and ω{displaystyle omega } is Kähler, we can obtain a solution of these equations by taking the Calabi-Yau metric on Y{displaystyle Y} and TY{displaystyle T_{Y}}.

Once the solutions for the Strominger's equations are obtained, the warp factor Δ{displaystyle Delta }, dilaton ϕ{displaystyle phi } and the background flux H, are determined by

  1. Δ(y)=ϕ(y)+constant{displaystyle Delta (y)=phi (y)+{text{constant}}},
  2. ϕ(y)=18lnΩ+constant{displaystyle phi (y)={frac {1}{8}}{text{ln}} Omega +{text{constant}}},
  3. H=i2(¯)ω.{displaystyle H={frac {i}{2}}({bar {partial }}-partial )omega .}

References[edit]

  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

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

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