The actions for all classical (and consequently quantum) BF theories on n-manifolds is proven to be given by anti-commutators of hermitian, nilpotent, scalar fermionic charges with Grassmann-odd functionals. In order to show this, the space of fields in the theory must be enlarged to include “mass terms” for new, non-dynamical, Grassmann-odd fields. The implications of this result on observable...

For any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. This extends the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Our main tool is an e...