Relationship between the index-based Bott periodicity model and classical models

Determine the precise relationship between the homotopy equivalence Ω(U/O) ≃ Z × BO constructed via the index map that models Ω(U/O) as a space of Cauchy–Riemann operators on bundle pairs over the disk with totally real boundary conditions and the classical models of real Bott periodicity; in particular, ascertain whether this index-based equivalence agrees with standard constructions and identify explicit comparison maps between the models.

Background

In the paper’s discussion of Lagrangian Floer theory, the authors construct a model of real Bott periodicity by identifying Ω(U/O) with the space of Cauchy–Riemann operators on bundle pairs (E,F) over the disk with totally real boundary conditions. The resulting index map induces a homotopy equivalence Ω(U/O) ≃ Z × BO.

While this gives a concrete and geometrically motivated model of Bott periodicity that is well suited to the orientation theory appearing in Floer-theoretic flow categories, the authors note that the connection of this index-based model with other standard constructions of real Bott periodicity (e.g., via the J-homomorphism or clutching constructions) has not been fully clarified. Resolving this would situate their model within the established landscape and enable direct comparison of induced structures and invariants across approaches.