What is the Topologically Twisted Index? I know that one can take a supersymmetric theory defined on $\mathbb{R}^n$  and topologically twist it by redefining the rotation group of the theory into a mixture of the (spacetime) rotation group and the R-symmetry group. However, what I'm a bit confused about is: what is a topologically twisted index? What does it physically mean? I can't seem to find a definition anywhere. Most papers seem to assume that the reader already knows the definition.
 A: It would be great that you sharpen your question by asking for a specific case.
But the general intuition is as follows: In a non-twisted theory, an index compute the dimension (possibly the virtual dimension) of the space of solutions of some differential equations that preserve some amount of supersymmetry.
Twisted theories dosen't have propagating degrees of freedom and the path integral (when a lagrangian description is avaliable) of those theories typically localizes to contributions coming from instantons. A twisted index does exactly what a "non-twisted" idex does because they typically come from twisting operators from non-twisted theories. They are indices for twisted theories.
But is important to specify a particular case because twisted indices can be defined for non-twisted theories by composing them with some projector that localizes the space of solutions of some equations to a subset of them that are non-dynamical and preserve some supersymmetry or transform well under some symmetry.
The most general definition that can be stated is that a topologically twisted index for a non-twisted theory is the composition of an index with a possible twist projection operator for the theory (when a twist is avaliable in the theory). What is interesting of this generality is that it makes manifest the fact that a non-twisted theory actually receives non-perturbative contributions from all its possible twists.
In the case of a twisted theory, a topologically twisted theory is just any index.
