Orientifolds and Twisted KR Theory

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I will report on some work in progress with Dan Freed and Greg Moore. In an orientifold background, D-brane charge takes values in a certain twisted version of KR Theory. Moreover, there is a nontrivial background charge (\'tadpole\'). Up \'til now, this background charge has only been calculated rationally -- i.e., ignoring torsion. We derive a formula for it, over the integers. Only after \'inverting 2\', does the charge localize to the fixed point sets of the orientifold action, and we can give a compact formula for it. This reproduces the previously known rational results, but contains new information.