Gerbes and time-reversal-invariant topological insulators

Definition. A bundle gerbe G with unitary connection (below, gerbe for short) over manifold M is a triple (Y,B,L) s.t. Y is a manifold equipped with a surjective submersion π : Y →M B is a 2-form on Y L is a line bundle (always with hermitian structure and unitary connection) over Y [2] ≡ Y ×M Y p1 −→ −→ p2 Y with the curvature p2B − p1B L comes with a groupoid multiplication L(y1,y2) × L(y2,y3)−→L(y1,y3) respecting the structure of L. • Necessarily, dB = π∗H where H is a closed 3-form on M called the curvature of the gerbe G. • Gerbes over M form a 2-category with 1-morphisms η : G1 −→ G2 and 2-morphisms μ : η1 −→ η2 between a pair of 1-morphisms η : G1 −→ G2 [6]. • As line bundles, gerbes may be tensored dualized or pulled back. • The abelian group (under ⊗) G(M) of 1-isomorphism classes of gerbes over M is isomorphic to the (real) Deligne hyper-cohomology group H(M) [8] and to the group Ĥ(M) of Cheeger-Simons differential characters [9].