Brane space localization and genus-0 T-duality with $H$-flux

Abstract

We construct a geometric theory on the brane space $S^2M=C^\infty(S^2,M)$ associated to a gerbe with connection on $M$. Normalized logarithmic holonomy around the latitude circles defines an average holonomy line bundle on the brane space, whose connection and averaged flux give an equivariantly flat superconnection. We prove that the completed periodic exotic twisted cohomology defined by this superconnection localizes to the twisted cohomology of $M$. This {\bf brane space localization} happens in two stages: circle localization lands on the free path space of $M$, and contracting the paths, after a correction of the twisting form, brings us back to $M$. Parallel transport along the meridians lets us average gerbe module connections, after which the circular iterated-integral construction applies. The resulting {\bf twisted brane Bismut-Chern forms} restrict to the twisted Chern forms on $M$; their variation is described by {\bf twisted brane Bismut-Chern-Simons forms.} For a T-dual pair, transporting the classical Hori morphisms through localization yields {\bf graded genus-0 T-duality} on brane cohomology. The composite of each morphism with its dual is minus the Euler operator in the level variable, so these morphisms are isomorphisms at every nonzero level. At level one these morphisms are compatible with the brane character and twisted $K$-theory T-duality.