We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi-and ind-coherent sheaves on X and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.

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Given a monoidal ∞-category C equipped with a monoidal recollement, we give a simple criterion for an object in C to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified ∞-category in terms of stratumwise dualizability and a projection formula for the links.Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism ∞-category that corepresents dualizable objects in symmetric monoidal recollements.We also give a number of examples and applications of our criterion drawn from algebra and homotopy theory, including equivariant and cyclotomic spectra and a multiplicative form of the Thom isomorphism.