## Everything You Always Wanted to Know About BPS States (but were afraid to ask)

**Ingmar Saberi (Caltech)**

There are very few ways to consistently truncate a quantum field theory. The most common example of such a truncation occurs in gauge theories, where one declares that the physical states and operators must be gauge-invariant. Another well-known kind of truncation is a twist of a supersymmetric theory, in which the physical states in the twisted theory are declared to be cohomology classes of a nilpotent supercharge acting in the untwisted theory. These two examples are not actually independent: the BRST construction reformulates the first as an instance of the second. Moreover, both exemplify the general principle that a theory may be consistently truncated by requiring invariance under a chosen (bosonic or fermionic) symmetry. For fermionic symmetries of supersymmetric field theories, the surviving states or operators are commonly called "BPS." In this talk, I'll review the various ways people usually characterize BPS states, and remark on both why these characterizations are related and subtle ways in which they differ. I'll then touch on recent work that centers around the question of how the BPS states in a theory change as its moduli are perturbed.