Superstrings with torsion

Compactifications of heterotic theory on non-Kähler complex manifolds, I

“…We have also taken into account, that the gamma matrices in the heterotic theory differ from the corresponding gamma matrices in the type-IIB theory by some powers of the warp factor. As discussed in [1] and [6], even in the presence of the dilaton in Ω we get 4) and the holomorphic three-form is a non-trivial function only of the complex structure τ ij . For a specific choice of complex structure, τ ij = iδ ij , we can use (2.35) to show, that Ω satisfies∂Ω = 0, i.e it is a∂ closed (3, 0) form with everywhere non-vanishing Ω ∧Ω > 0.…”

“…discussed in [1,2] and [3] some time ago. Yet, not much is known about such models in the literature, neither from the mathematics point of view nor from the physics point of view.…”

“…The constraint on the H-flux will be 6) where p i are the Pontryagin classes for the spin and gauge bundles respectively. Since H and J ab (the fundamental form) are related by the torsional constraints [2,1,3], the fundamental form is very constrained because of this and non-Kählerity. These issues have been already discussed in [6], where it was shown, how the fundamental form satisfies the torsional constraints ignoring the gauge fluxes.…”

“…As discussed in detail in [2,1], and [6], the covariantly constant complex structure (2.18) implies the following important relation between the H field and the metric (in terms of complex coordinates) 21) which is called the torsional constraint. In [6] we showed explicitly, how the background (2.7) satisfies the equations (2.21) (and its complex conjugates).…”

“…As explained in [1], as a consequence of the anomaly cancellation condition the fundamental form of a manifold with torsion has to satisfy…”