Orbifold boundary states from Cardy's condition

TL;DR

This work extends Cardy’s boundary-state construction to D-branes at orbifold fixed points, explicitly deriving boundary states for fractional branes in $\,\mathbb{C}^2/\Gamma$ and clarifying the origin of the accompanying coefficients through modular properties of chiral blocks. It develops the orbifold CFT machinery, including twisted sectors, chiral blocks, and modular transformations, and then uses a generalized Fourier-like transformation to map Ishibashi data to consistent boundary states, uncovering a clear link to wrapped branes on vanishing cycles via the McKay correspondence. The paper further generalizes to discrete torsion, showing how projective representations of the orbifold group encode discrete torsion phases in boundary states and verifying their consistency with the closed-string channel. By connecting fractional branes to geometric cycles, twisted RR couplings, and ALE/Gepner points, the work provides a coherent framework for interpreting D-branes in singular geometries and emphasizes the consistency conditions required by open-closed duality in orbifold backgrounds.

Abstract

Boundary states for D-branes at orbifold fixed points are constructed in close analogy with Cardy's derivation of consistent boundary states in RCFT. Comments are made on the interpretation of the various coefficients in the explicit expressions, and the relation between fractional branes and wrapped branes is investigated for $\mathbb{C}^2/Γ$ orbifolds. The boundary states are generalised to theories with discrete torsion and a new check is performed on the relation between discrete torsion phases and projective representations.