Supersymmetric Field Theories and Orbifold Cohomology

<p>Using the Stolz-Teichner framework of supersymmetric Euclidean field theories (EFTs), we provide geometric interpretations of some aspects of the algebraic topology of orbifolds.</p><p>We begin with a classification of 0|1-dimensional twists for EFTs over an orbifold <em>X</em>, and show that the collection of concordance classes of twisted EFTs over the inertia <em>ΛX</em> is in natural bijection with the delocalized twisted cohomology of <em>X</em> (which is isomorphic to its complexified <em>K</em>-theory). Then, turning to 1|1-dimensional considerations, we construct a (partial) twist functor over <em>X</em> taking as input a class in <em>H</em>³(<em>X</em>; ℤ).</p><p>Next, we define a dimensional reduction procedure relating the 0|1-dimensional Euclidean bordism category over <em>ΛX</em> and its 1|1-dimensional counterpart over <em>X</em>, and explore some applications. As a basic example, we show that dimensional reduction of untwisted EFTs over a global quotient orbifold <em>X//G</em> 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 <em>K</em>-theory.</p>