Topological Open Strings on Orbifolds

Abstract

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C 3 /Z 3 orbifold point of local P 2 , where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold GromovWitten invariants of C 3 /Z 3 , or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

Crepant resolutions and open strings

“…A natural question one can ask is whether similar relations exist in the context of open Gromov-Witten theory. Within the toric realm, physics arguments based on open mirror symmetry [8,9,16] have given strong indications that some version of the Bryan-Graber [18] statement of the crepant resolution conjecture should hold at the level of open invariants. This was proven explicitly for the crepant resolution of the Calabi-Yau orbifold [C 3 /Z 2 ] in [19].…”

Effective Superpotentials for Compact D5-Brane Calabi-Yau Geometries

“…This manifests in this example the existence of the variational sub-system (2.19) within the variation of mixed Hodge structure (2.18). Finally it remains to expand the element, ∂ ψ π 0,3 , for which we obtain 18) in terms of the polynomials…”

“…[16,17,18], in the vicinity of orbifold points the superpotential encodes rational orbifold disk invariants [19,20,21].…”

“…Then, similarly to the analysis performed for D-branes in local CalabiYau geometries [16,17,18], we extract (up to overall numerical normalizations) from the flat superpotential a tower of orbifold disk invariants for the mirror D-brane configuration in the compact mirror Calabi-Yau geometry. Finally, as a bonus and as a highly non-trivial consistency check we reproduce for our two examples the domain wall tension computed in refs.…”

“…As for local Calabi-Yau geometries in refs. [16,17,18], we determined (up to overall normalizations) from this superpotential a tower of disk orbifold invariants of the corresponding mirror geometry in the topological A-model. The resulting superpotential reproduces the correct critical locus in agreement with the leading order behavior of the superpotential computed by matrix-factorization techniques in ref.…”