Supersymmetric Field Theories and Orbifold Cohomology

Using the Stolz-Teichner framework of supersymmetric Euclidean field theories (EFTs), we provide geometric interpretations of some aspects of the algebraic topology of orbifolds.

We begin with a classification of 0|1-dimensional twists for EFTs over an orbifold X, and show that the collection of concordance classes of twisted EFTs over the inertia ΛX is in natural bijection with the delocalized twisted cohomology of X (which is isomorphic to its complexified K-theory). Then, turning to 1|1-dimensional considerations, we construct a (partial) twist functor over X taking as input a class in H³(X; ℤ).

Next, we define a dimensional reduction procedure relating the 0|1-dimensional Euclidean bordism category over ΛX and its 1|1-dimensional counterpart over X, and explore some applications. As a basic example, we show that dimensional reduction of untwisted EFTs over a global quotient orbifold X//G recovers the equivariant Chern character. Finally, we describe the dimensional reduction of the 1|1-twist built earlier, showing that it has the expected relation to twisted K-theory.