Virtual vector bundles and graded Thom spectra
  We introduce a convenient framework for constructing and analyzing orthogonal Thom spectra arising from virtual vector bundles. This framework enables us to set up a theory of orientations and graded Thom isomorphisms with good multiplicative properties. The theory is applied to the analysis of logarithmic structures on commutative ring spectra. 
Steffen Sagave and Christian Schlichtkrull
math.AT