The closed-path Berry phase defined above can be expressed as γ γ n = ∫ ∫ C d R ⋅ ⋅ A n ( R ) where A n ( R ) = i ⟨ ⟨ n ( R ) | ∇ ∇ R | n ( R ) ⟩ ⟩ is a vector-valued function known as the Berry connection (or Berry potential). The Berry connection is gauge-dependent, transforming as A ~ ~ n ( R ) = A n ( R ) + ∇ ∇ R β β ( R ) . Hence the local Berry connection A n ( R ) can never be physically observable. However, its integral along a closed path, the Berry phase γ γ n , is gauge-invariant up to an integer multiple of 2 π π . Thus, e i γ γ n is absolutely gauge-invariant, and may be related to physical observables.