# The State of Theory Comes Full Sphere

In Victor’s last talk, and the last talk of ST4, we turned to the computation of partition functions for gauge theories on $S^2$.

The 2d chiral and vector multiplets are simply dimensional reductions of their counterparts in 4d, and they correspond respectively to matter and gauge degrees of freedom. We used the constraining power of extended supersymmetry in two dimensions to write down actions for the super-Yang-Mills and matter sectors, and noted that they are $Q$-exact. This allows us to use the localization arguments we reviewed in earlier lectures.

We are interested in solutions to the fixed-point equations (which are easy to write down as the action is written down as a sum of squares!) and when the appropriate reality conditions are imposed, we find that the scalars $\sigma_i$ are required to be Cartan-valued, and further that the gauge field $F_{12}$ and auxiliary scalar $D$ are proportional to them. Dirac quantization then dictates that this gauge field flux is quantized.

Now that we have BPS solutions, we would like to compute 1-loop determinants of fluctuations about these solutions. Victor explained that there were many ways to do this, and the simplest of them is to decompose wavefunctions into spin spherical (or Wu-Yang) harmonics. In doing this, one encounters a generalized notion of spin, which is the usual spin minus a contribution coming from the quantum of flux piercing the sphere on which these gauge theories live. In the following discussions, it became clear that similar physics of flux attachment occurs when studying the quantum Hall effect.

Once the fields are decomposed into these “spin” spherical harmonics, it becomes straightforward to write down their determinant. However, when studying the 1-loop determinant of a matter multiplet for example, supersymmetry will ensure cancellations between factors contributing to the bosonic and fermionic determinants. There are, however, terms that do not cancel, and these come from chiral zero modes of the Dirac operator. Such cancellations are at the heart of any supersymmetric theory: all positive energy modes come in pairs — this follows from the SUSY algebra — but the zero modes are under no such algebraic restriction.

We played the same game with the vector multiplet and in the end, wrote down the most general partition function for a gauge theory on $S^2$ with matter. Victor concluded with some remarks on how the poles of the integrands that define the partition function encode information about non-perturbative sectors of the theory.

With this, all the main talks were concluded, and the audience broke into small groups that shared coffee, and boasts of who was more exhausted at the end of the workshop.

We ended a long, gruelling and satisfying two weeks with a four-hour discussion about future directions in the various sub-fields that are currently of interest in the community. We first listed out a bunch of fields and then went one by one and asked people to talk about what the interesting directions that would be worth exploring in the next five years would be. We feel safe saying that this session was extremely rewarding to all who came, helping the experts in these fields formulate what they thought was interesting, telling others what was worth expecting, and telling people who wanted to do something new what could be interesting. While we won’t try to summarise this entire discussion (for reasons of sanity), we did take a photo of the blackboard on which we listed out all the important points so that you, lovely reader, can look at it:

And, with that, we officially called to an end an amazing workshop that was our deepest honour to organise and attend.