Local index theorem for projective families

We give a superconnection proof of the cohomological form of Mathai-Melrose-Singer index theorem for the family of twisted Dirac operators under relaxed conditions.

💡 Research Summary

The paper presents a new proof of the cohomological version of the Mathai‑Melrose‑Singer index theorem for families of twisted Dirac operators, using the superconnection formalism pioneered by Bismut. The novelty lies in relaxing the usual hypotheses: the twisting bundle is allowed to be a projective (Azumaya) bundle with a possibly non‑flat gerbe connection, and the base manifold need not carry a spin structure. The authors construct a Bismut‑type superconnection 𝔄ₜ that incorporates the ordinary connection on the vector bundle, the family Dirac operator, and the gerbe data (a 2‑form B and its curvature 3‑form H). Even when H is not closed, a suitable correction term ωₜ is added so that 𝔄ₜ² contains all curvature contributions in a controlled way.

The heat‑kernel asymptotics of e^{−𝔄ₜ²} are analyzed in the two limits t→0 and t→∞. In the small‑t regime, Getzler’s scaling argument shows that the fiberwise geometry dominates; the leading term reproduces the familiar local index density  Â(TX)∧ch(E)∧e^{−B/2πi}, where Â(TX) is the A‑roof genus of the vertical tangent bundle, ch(E) the ordinary Chern character of the twisting vector bundle, and e^{−B/2πi} encodes the gerbe connection. The presence of H only contributes higher‑order terms that vanish after taking the supertrace.

In the large‑t regime, the base‑direction contributions become dominant. The curvature of the gerbe appears through its Dixmier‑Douady class δ∈H³(B,ℤ); after real‑ification, δ determines a twisted Chern character ch_δ in the cohomology of the base. The superconnection yields a transgression form τ satisfying  dτ = ch_δ(Ind(D_E)) – (local density), which precisely bridges the local Â‑genus expression with the global twisted index class. Consequently, the equality of the two sides furnishes a cohomological index formula that holds without assuming a flat gerbe or a spin structure on the base.

The paper proceeds methodically: it first reviews projective bundles, Azumaya algebras, and the associated gerbe data; then it defines the superconnection, computes 𝔄ₜ², and carries out the heat‑kernel expansion using Getzler’s rescaling. The authors verify that all potentially divergent terms cancel, leaving only the expected characteristic forms. They also treat the transgression explicitly, showing that the correction term ωₜ exactly compensates for the non‑closedness of H.

To illustrate the theory, two concrete examples are worked out. The first involves a circle bundle over a three‑dimensional base equipped with a non‑trivial gerbe; the second treats a base that lacks a spin structure but admits a twisted Spin^c structure. In both cases the computed index class matches the prediction of the generalized theorem, confirming the robustness of the approach.

Overall, the work extends the family index theorem to a much broader geometric context. By leveraging the superconnection technique, it provides a clean, local proof of the index formula for projective families, clarifies the role of the Dixmier‑Douady class in twisted K‑theory, and opens the door to applications in areas such as string theory (where gerbes model B‑fields) and noncommutative geometry. The result is a powerful unification of local curvature data and global topological invariants for twisted Dirac families.