Real Inertia, Phase Obstructions, and de Rham Algebras for Orientifolds

Abstract

In this paper, we study de Rham realizations for finite orientifold quotients. On the orientation preserving inertia of an almost complex global quotient we construct an anti-linear involution of the Chen-Hu algebra whose fixed subalgebra is a real form of the Chen-Ruan algebra. For genuine orientation reversing fixed sectors we prove that no coordinate invariant single sector fractional age exists. When a Calabi-Yau volume line is preserved, the relative phases define a canonical class $[μ_{\mathbb R}]\in H^2(\widehat G;\mathbb Z_ε)$. We identify this class with the Bockstein obstruction to a coherent real phase lift and prove that its vanishing is equivalent, after rephasing the volume form, to a phase trivial orientifold action. We then formulate a general Euler-Gysin realization theorem using relative Fredholm bundles and determinant-line sewing. For phase trivial Calabi-Yau surfaces we verify the admissibility conditions directly and obtain a canonically defined associative graded cohomology algebra containing the Chen-Hu even sector and the odd fixed surfaces. For an Eisenstein abelian surface with an order six dihedral action we compute the centralizer invariant algebra explicitly; it is a $16$ dimensional Frobenius algebra whose odd sector multiplication is noncommutative in the shifted grading. We also explain the relation with earlier unoriented orbifold cohomology constructions.