D-brane Categories for Orientifolds -- The Landau-Ginzburg Case

TL;DR

This work develops a categorical framework for D-branes in orientifolds realized by Landau-Ginzburg models and their orbifolds, with parity treated as an anti-involution acting on matrix factorizations. It defines invariant orientifold categories MF_{P}^{\varepsilon}(W) and MF_{P_{\chi}}^{\pm c}(W) by selecting branes admitting a parity isomorphism, and it analyzes morphisms, gauge algebras, and Knörrer periodicity within this setting. A key result is the construction of topological crosscap states and a parity-twisted index in LG orientifolds, generalized to orbifolds via residue formulas and twisted sectors. The paper also discusses R-charge grading compatibility, and situates the orientifold data in a broader mathematical context via Hermitian/Real K-theory, suggesting connections to large-volume limits and Real-K-theoretic classifications of D-brane charges.

Abstract

We construct and classify categories of D-branes in orientifolds based on Landau-Ginzburg models and their orbifolds. Consistency of the worldsheet parity action on the matrix factorizations plays the key role. This provides all the requisite data for an orientifold construction after embedding in string theory. One of our main results is a computation of topological field theory correlators on unoriented worldsheets, generalizing the formulas of Vafa and Kapustin-Li for oriented worldsheets, as well as the extension of these results to orbifolds. We also find a doubling of Knoerrer periodicity in the orientifold context.

Figures (6)

• Figure 1: Orientation reversal of the open string worldsheet
• Figure 2: A disk correlator and its parity image. The existence of this symmetry does not (unfortunately) impose any restrictions on the parities of $\phi_1$, $\phi_2$, $\phi_3$. For example, the correlator could be non-zero for $P_1=-,P_2=+,P_3=+$.
• Figure 3: A topological field theory correlator on a general unoriented Riemann surface with boundary can be computed by replacing crosscaps (represented by a crossed circle) by the crosscap operator $C$. The open circle represents a boundary, with boundary condition labeled by $B$.
• Figure 4: The crosscap correlator we have computed is equivalent to a sphere correlator with insertion of the crosscap operator $C$.
• Figure 5: The Klein bottle correlator can be expressed in terms of the square of the crosscap operator on the sphere.