Many results in 4d N=2 theories come from the combination of two principles: renormalization group and supersymmetry. Computing a protected observable in the UV and the IR gives different expressions for the same quantity, and this can be interesting. Let us see an example, following a talk Clay Cordova gave at String-Maths 2017 (video available on the website of the conference).

The physics of a generic RG flow goes as follows:

• UV is asymptotically free or CFT. Characterized by spectrum of local operators and OPE.
• IR is gapped of free. Characterized by spectrum of one-particle states and S-matrix.
What is the relation between these concepts? We will see an explicit formula relating them in a supersymmetric context.

Look at the Schur index $\mathcal{I}(q)$, that can be associated to any N=2 theory.

• UV side : the index is a sum over local operators $$\mathcal{I}(q) = \sum (-1)^F q^{R+J_1 + J_2}$$ where $R$ is the R-charge and the $J$s are the two Lorentz spins. If UV theory is conformal, then it is the character of a $2d$ chiral algebra, as I explained in a previous post.
• IR : involves the rank $r$ of the Coulomb branch and the generating function of BPS particles. Let's explain briefly what this is. The construction of BPS particles depends on a central charge $Z$ which can be varied in a finite dimensional complex moduli space. The BPS spectrum is locally constant but jumps across walls of real codimension one. What are the wall-crossing invariants? An invariant is $\mathcal{O}(q)$ (constructed in Kontsevitch-Soibelman and understood in physics by Gaiotto, Moore and Neitzke), valued in a quantum torus algebra. A simpler invariant is obtained by taking its trace $\mathrm{Tr} \, \mathcal{O} (q)$.

The conjecture of Cordova and Shao is then that for any $\mathcal{N}=2$ theory, $$\mathcal{I}(q) = (q)^{2r}_{\infty} \mathrm{Tr} \, \mathcal{O} (q) \, .$$ The left-hand side is UV, the right-hand side is IR. In general, they look very different from each other (see for instance the BPS generating function for the $A_2$ Argyres-Douglas theory, and the $(2,5)$ Virasoro minimal model vacuum character), and it is a miracle that the formula works!

Note that what has been done here for particles can be generalized to lines, or surfaces! This leads to the Cordova-Gaiotto-Shao conjecture.