Interpretation of the OSFT 2-form connection as a Berry connection

Determine whether the 2-form connection B_ij(λ) defined in bosonic open string field theory by B_ij(λ) = ∫ Ψ * (∂Ψ/∂λ^i) * (∂Ψ/∂λ^j) − (i ↔ j), where Ψ is a classical open string field and * denotes the open string star product with integration ∫, can be regarded as a Berry connection associated to an adiabatic process in open string field theory.

Background

The paper defines a gauge-invariant 2-form connection B_ij on a parameter space X of classical solutions Ψ(λ) in bosonic open string field theory, using the open string star product and integration. This construction yields higher holonomies and a 3-form curvature H that are invariant under the infinite-dimensional gauge algebra of OSFT.

The authors draw an analogy to Berry phases in quantum mechanics and to recent developments in higher Berry phases in condensed matter and quantum field theory, noting structural similarities but emphasizing key differences (e.g., Ψ is a classical string field, the construction is cubic, and a 2-form is natural due to ghost number considerations). Against this backdrop, they explicitly state uncertainty about whether B_ij can be interpreted as an actual Berry connection for an adiabatic process in OSFT.