Riemann-Roch and index formulae in twisted K-theory

In this paper, we establish the Riemann-Roch theorem in twisted K-theory extending our earlier results. As an application, we prove a twisted index formula and show that D-brane charges in Type I and Type II string theory are classified by twisted KO-theory and twisted K-theory respectively in the presence of B-fields as proposed by Witten.

💡 Research Summary

The paper “Riemann‑Roch and index formulae in twisted K‑theory” provides a comprehensive extension of classical Riemann‑Roch and Atiyah‑Singer index theorems to the setting of twisted K‑theory, and uses these results to substantiate Witten’s proposal that D‑brane charges in the presence of a B‑field are classified by twisted KO‑theory for Type I strings and by twisted complex K‑theory for Type II strings.

The authors begin by recalling that a background B‑field on a space X is mathematically encoded by a degree‑three integral cohomology class α∈H³(X,ℤ). This class determines a “twist” of the ordinary K‑theory spectrum, giving rise to a twisted K‑theory group K^α(X). In contrast to the untwisted case, elements of K^α(X) are represented by α‑twisted vector bundles (or, equivalently, projective modules over a continuous‑trace C∗‑algebra with Dixmier‑Douady invariant α). The paper first establishes the functorial machinery needed for such groups: push‑forward maps f_* for proper K‑oriented maps f:X→Y, pull‑back maps f^* for arbitrary continuous maps, and a well‑behaved product structure. Crucially, the authors prove a projection formula and a base‑change compatibility that hold in the twisted context, thereby ensuring that the usual formalism of K‑theory extends without loss of coherence.

A central technical achievement is the construction of a twisted Chern character ch^α:K^α(X)→H^{even}(X;ℚ). By adapting the Chern‑Weil construction to connections on projective bundles with curvature modified by the B‑field, the authors obtain a rational characteristic class that respects the twist: the image of a twisted K‑class under ch^α lands in the ordinary de Rham cohomology, but the class encodes the α‑dependence through additional differential form terms. This twisted Chern character remains an isomorphism after tensoring with ℚ, just as in the untwisted case, and it is the key bridge between K‑theoretic and cohomological statements.

With these tools in place, the paper proves a twisted Grothendieck‑Riemann‑Roch (GRR) theorem. Let f:X→Y be a proper map between smooth, compact, complex manifolds equipped with a K‑orientation and a compatible twist α on X and β on Y satisfying f^*β=α+δ for some class δ∈H³(X,ℤ). The authors show that for any x∈K^α(X) one has

  ch^β(f_(x)) = f_(ch^α(x)·Td^δ(T_f)),

where Td^δ(T_f) is a twisted Todd class incorporating the correction term δ. The proof proceeds by reducing to the case of an embedding, applying the Thom isomorphism in twisted K‑theory, and then using the naturality of the twisted Chern character together with the classical GRR for the underlying manifolds. This formula demonstrates that the push‑forward in twisted K‑theory is compatible with integration of characteristic forms, exactly as in the untwisted situation, but with the twist manifesting as an extra multiplicative factor.

From the twisted GRR the authors derive a twisted Atiyah‑Singer index theorem. For a compact spin^c manifold X equipped with a twist α and a twisted complex vector bundle E∈K^α(X), consider the twisted Dirac operator D_E acting on sections of the spinor bundle tensored with E. The index of D_E, now an integer valued in ℤ, is given by

  Ind^α(D_E) = ∫_X Â(X) ∧ ch^α(E).

Here Â(X) is the usual A‑roof genus, and the integral is taken in ordinary cohomology after applying the twisted Chern character. The formula reduces to the classical index theorem when α=0, confirming that the twist does not alter the analytic nature of the Dirac operator but only modifies the topological term through ch^α.

The final part of the paper connects these mathematical results to string theory. Witten’s conjecture states that D‑brane charges in the presence of a B‑field are classified by twisted K‑theory: for Type II strings by K^α, and for Type I strings by twisted KO‑theory KO^α. The authors construct the real analogue of the twisted Chern character, mapping KO^α(X) to appropriate real cohomology groups, and they verify that the charge quantization condition matches the index formula derived earlier. By examining the world‑volume anomaly cancellation conditions and the Freed‑Witten anomaly, they show that the twisted KO‑theory classification naturally incorporates the orientifold projection and the real structure required for Type I strings. In the Type II case, the complex twisted K‑theory classification follows directly from the complex index theorem.

Overall, the paper achieves three major goals: (1) it establishes a rigorous functorial framework for twisted K‑theory, including push‑forward, pull‑back, and product structures; (2) it proves a twisted Grothendieck‑Riemann‑Roch theorem and a corresponding twisted Atiyah‑Singer index formula; (3) it applies these results to confirm Witten’s physical proposal concerning D‑brane charge classification in the presence of B‑fields. The work not only fills a gap in the mathematical literature on twisted K‑theory but also provides a solid foundation for further investigations into higher twists, non‑commutative backgrounds, and the role of twisted K‑theory in M‑theory and related quantum gravity frameworks.