Localization and traces in open-closed topological Landau-Ginzburg models

We reconsider the issue of localization in open-closed B-twistedLandau-Ginzburg models with arbitrary Calabi-Yau target. Through carefulanalsysis of zero-mode reduction, we show that the closed model allows for aone-parameter family of localization pictures, which generalize the standardresidue representation. The parameter $\lambda$ which indexes these picturesmeasures the area of worldsheets with $S^2$ topology, with the residuerepresentation obtained in the limit of small area. In the boundary sector, wefind a double family of such pictures, depending on parameters $\lambda$ and$\mu$ which measure the area and boundary length of worldsheets with disktopology. We show that setting $\mu=0$ and varying $\lambda$ interpolatesbetween the localization picture of the B-model with a noncompact target spaceand a certain residue representation proposed recently. This gives a completederivation of the boundary residue formula, starting from the explicitconstruction of the boundary coupling. We also show that the variouslocalization pictures are related by a semigroup of homotopy equivalences.Comment: 36 page