Comments on heterotic flux compactifications

In heterotic flux compactification with supersymmetry, three differentconnections with torsion appear naturally, all in the form $\omega+a H$.Supersymmetry condition carries $a=-1$, the Dirac operator has $a=-1/3$, andhigher order term in the effective action involves $a=1$. With a view towardthe gauge sector, we explore the geometry with such torsions. After reviewingthe supersymmetry constraints and finding a relation between the scalarcurvature and the flux, we derive the squared form of the zero mode equationsfor gauge fermions. With $\d H=0$, the operator has a positive potential term,and the mass of the unbroken gauge sector appears formally positive definite.However, this apparent contradiction is avoided by a no-go theorem that thecompactification with $H\neq 0$ and $\d H=0$ is necessarily singular, and theformal positivity is invalid. With $\d H\neq 0$, smooth compactificationbecomes possible. We show that, at least near smooth supersymmetric solution,the size of $H^2$ should be comparable to that of $\d H$ and the consistenttruncation of action has to keep $\alpha'R^2$ term. A warp factor equation ofmotion is rewritten with $\alpha' R^2$ contribution included precisely, andsome limits are considered.Comment: 31 pages, a numerical factor correcte