Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory

We develop a general theory of pushforward operations for principal $G$-bundles equipped with a certain type of orientation.

In the case $G=BU(1)$ and orientations in twisted K-theory we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation.

As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible $BU(1)$-action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.