  In this paper we introduce two new ways to split ramification of Brauer
classes on surfaces using stacks. Each splitting method gives rise to a new
moduli space of twisted stacky vector bundles. By studying the structure of
these spaces we prove new results on the standard period-index conjecture. The
first yields new bounds on the period-index relation for classes on curves over
higher local fields, while the second can be used to relate the Hasse principle
for forms of moduli spaces of stable vector bundles on pointed curves over
global fields to the period-index problem for Brauer groups of arithmetic
surfaces. We include an appendix by Daniel Krashen showing that the local
period-index bounds are sharp.
