Rozansky-Witten theory, Localised then Tilted
Author: Jian Qiu
Preprint number UUITP 47/20
The paper is motivated by a very curious preprint Gukov:2020lqm, where the equivariant index formula for the dimension of the Hilbert space of the Rozansky-Witten theory is interpreted as some Verlinde formula. In this interpretation, the fixed points of target HyperKähler geometry correspond to certain 'states'. In the first part of the current paper, we apply the localisation technique to the Rozansky-Witten theory via first reformulating the latter as some supersymmetric sigma-model. We obtain the exact formula for the partition function on S1×Sigmag and the lens spaces. The second part extends the formalism to incorporate equivariance on the target geometry. We then apply the tilting theory to the derived category of coherent sheave on the non-compact HyperKähler variety, which allows us to pick a 'basis' for the Wilson loops in the theory. We can then compute the fusion products in this basis and we show that the objects that have diagonal fusion rules are intimately related to the fixed points of the geometry. Using these objects as basis to compute the dimension of the Hilbert space leads back to the equivariant index theorem, thus answering the question that motivated the paper.