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The supersymmetry of near-horizon geometries in heterotic supergravity is considered. A necessary and sufficient condition for a solution to preserve more than the minimal N=2 supersymmetry is obtained. A supersymmetric near-horizon solution is constructed which is a U(1) fibration of AdS3 over a particular Aloff-Wallach space. It is proven that this solution preserves the conditions required for N=2 supersymmetry, but does not satisfy the necessary condition required for further supersymmetry enhancement. Hence, there exist supersymmetric near-horizon heterotic solutions preserving exactly N=2 supersymmetry.
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We consider supersymmetric near-horizon geometries in heterotic supergravity up to two loop order in sigma model perturbation theory. We identify the conditions for the horizons to admit enhancement of supersymmetry. We show that solutions which undergo supersymmetry enhancement exhibit an sl(2,R) symmetry, and we describe the geometry of their horizon sections. We also prove a modified Lichnerowicz type theorem, incorporating $alpha$ corrections, which relates Killing spinors to zero modes of near-horizon Dirac operators. Furthermore, we demonstrate that there are no AdS2 solutions in heterotic supergravity up to second order in $alpha$ for which the fields are smooth and the internal space is smooth and compact without boundary. We investigate a class of nearly supersymmetric horizons, for which the gravitino Killing spinor equation is satisfied on the spatial cross sections but not the dilatino one, and present a description of their geometry.
We classify the geometries of the most general warped, flux AdS backgrounds of heterotic supergravity up to two loop order in sigma model perturbation theory. We show under some mild assumptions that there are no $AdS_n$ backgrounds with $n ot=3$. Moreover the warp factor of AdS$_3$ backgrounds is constant, the geometry is a product $AdS_3times M^7$ and such solutions preserve, 2, 4, 6 and 8 supersymmetries. The geometry of $M^7$ has been specified in all cases. For 2 supersymmetries, it has been found that $M^7$ admits a suitably restricted $G_2$ structure. For 4 supersymmetries, $M^7$ has an $SU(3)$ structure and can be described locally as a circle fibration over a 6-dimensional KT manifold. For 6 and 8 supersymmetries, $M^7$ has an $SU(2)$ structure and can be described locally as a $S^3$ fibration over a 4-dimensional manifold which either has an anti-self dual Weyl tensor or a hyper-Kahler structure, respectively. We also demonstrate a new Lichnerowicz type theorem in the presence of $alpha$ corrections.
We prove that Killing horizons in massive IIA supergravity preserve an even number of supersymmetries, and that their symmetry algebra contains an $mathfrak{sl}(2, R)$ subalgebra, confirming the conjecture of [5]. We also prove a new class of Lichnerowicz type theorems for connections of the spin bundle whose holonomy is contained in a general linear group.
Perturbative heterotic string theory develops a single complex tachyonic mode beyond the Hagedorn temperature. We calculate the quartic effective potential for this tachyonic mode at the critical temperature. Equivalently, we determine the quartic effective potential for strong supersymmetric breaking via anti-perdiodic boundary conditions for fermions on a small circle. We give many details of the heterotic tachyon scattering amplitudes, including a unitarity check to fix all normalization constants. We discuss difficulties in obtaining an effective action valid at all radii. We argue that in certain variables, the quartic term in the potential is radius independent. Speculations on the properties of a new strongly curved phase that could occur after tachyon condensation are offered.
The strongly coupled heterotic M-theory vacuum for both the observable and hidden sectors of the $B-L$ MSSM theory is reviewed, including a discussion of the bundle constraints that both the observable sector $SU(4)$ vector bundle and the a hidden sector bundle induced from a line bundle must satisfy. Gaugino condensation is then introduced within this context, and the hidden sector bundles that exhibit gaugino condensation are presented. The condensation scale is computed, singling out one line bundle whose associated condensation scale is low enough to be compatible with the energy scales available at the LHC. The corresponding region of Kahler moduli space where all bundle constraints are satisfied is presented. The generic form of the moduli dependent $F$-terms due to a gaugino superpotential - which spontaneously break $N=1$ supersymmetry in this sector - is presented and then given explicitly for the unique line bundle associated with the low condensation scale. The moduli dependent coefficients for each of the gaugino and scalar field soft supersymmetry breaking terms are computed leading to a low-energy effective Lagrangian for the observable sector matter fields. We then show that at a large number of points in Kahler moduli space that satisfy all bundle constraints, these coefficients are initial conditions for the renormalization group equations which, at low energy, lead to completely realistic physics satisfying all phenomenological constraints. Finally, we show that a substantial number of these initial points also satisfy a final constraint arising from the quadratic Higgs-Higgs conjugate soft supersymmetry breaking term.
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