Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models

Abstract

We review the compactification to five‐dimensional heterotic M‐theory on a Schoen Calabi–Yau threefold and the specific SU(4) vector bundle leading to the “heterotic standard model” in the observable sector. Within strongly coupled heterotic M‐theory, a formalism for consistent hidden‐sector bundles associated with a single line bundle is presented, and a specific line bundle is introduced as a concrete example. Anomaly cancellation and the associated bulk space five‐branes are discussed in this context, as is slope‐stability of both the observable and hidden sector bundles. The further compactification to a four‐dimensional effective theory on a linearized BPS double domain wall is then presented to order κ114/3. Specifically, the generic constraints required for anomaly cancellation and the restrictions imposed by positive squared gauge couplings to order κ114/3 are presented in detail. Three additional constraints are imposed, one guaranteeing that the S1/double-struckZ2 orbifold length is sufficiently larger than the average Calabi–Yau radius, and two enforcing that the hidden sector be compatible with both the unification mass scale and unified gauge coupling of the SO(10) group in the observable sector. Finally, the expression for the Fayet–Iliopoulos term associated with an anomalous U(1) symmetry is presented and its role in N=1 supersymmetry in the low‐energy effective theory is discussed. It is shown that N=1 supersymmetry can be preserved by cancelling the tree‐level and genus‐one contributions against each another. As a check on our results, we calculate several quantities to order κ116/3 and show that they remain physically acceptable, and even improve, when computed to higher order.

Moduli and Hidden Matter in Heterotic M-Theory with an Anomalous $U(1)$ Hidden Sector

“…Let us continue to work under the assumption that πβ S /3 = 1. Furthermore, based on our results in [26], the magnitude of the FI-term is expected to be of the order of the compactification scale squared. Therefore, for simplicity, we will take…”

“…See, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In [24][25][26], it was shown that for a heterotic vacuum to be phenomenogically viable, the hidden sector must be consistent with a series of constraints: 1) allowing for five-branes in the S 1 /Z 2 orbifold interval, the entire theory must be anomaly-free [6,27]; 2) the unified gauge coupling parameters must be positive in both the observable and hidden sectors; 3) the regions of Kähler moduli space where both the observable and hidden sector bundles are slope-stable must overlap and 4) prior to a specified supersymmetry breaking mechanism being introduced, N = 1 SUSY must be preserved at the compactification scale. Early attempts [24] to build such a hidden sector were valid only in the weakly coupled heterotic string regime in which, however, one cannot obtain reasonable values for the observable sector unification scale and unified gauge coupling [28][29][30].…”

“…This problem was rectified in [26], where we proposed that the hidden sector gauge bundle be a rank-two Whitney sum L ⊕ L −1 constructed from a line bundle L. The associated U (1) structure group is then embedded into the E 8 gauge group via the mapping U (1) → SU (2) ⊂ E 8 . Importantly, within a substantial region of Kähler moduli space, the genus-one corrected slope of the bundle L vanishes, yielding a poly-stable L ⊕ L −1 bundle.…”

“…In [36], we analyzed a possible SUSY-breaking mechanism for the vacua found in [26], namely gaugino condensation in the hidden sector. The condensate induces non-zero F-terms in the 4D effective theory which break SUSY globally [37][38][39][40][41][42][43][44][45][46][47].…”

“…We think that our method, as well as our conclusions, become more clear in this reduced set-up. Furthermore, once this analysis is well understood, it is straightforward to extend it to more complicated heterotic vacua constructions, such as those studied in [26].…”