This project has already resulted in a number of interesting results. Significant progress has been established in all 4 Aims.
Towards Aim 1, Coulomb gas type correlation functions were constructed for the full parameter range \kappa in (0,8) and they were proven to give SLE(\kappa) partition functions, and the associated global multiple SLE measures were constructed. At specific values, logarithmic behavior was revealed. The representation theoretical investigations are ongoing.
In the context of Aim 2, the functions were proven to be related to Ising, FK-Ising, uniform spanning tree, loop-erased random walk, and percolation models, and conjecturally to random-cluster and O(n)-loop models. In the special cases where the central charge equals c=1 or c=-2, correlation functions were constructed in terms of special functions: fused Specht polynomials and determinants involving excursion kernels. In particular, the operator content of the c=-2 boundary CFT relevant to uniform spanning trees and loop-erased random walks was developed fully in the first row and the first column of the Kac table. In a similar vein, a full c=1 theory was developed in the context of the Gaussian Free Field (GFF). An interesting, slightly surprising direction of research was revealed: one can also construct correlation functions pertaining to suitable multi-component GFFs related to extended symmetry in CFT (W-algebras) and multiple dimers.
Aim 3 has led in particular to developments of the real determinant line bundle associated to Riemann surfaces with parametrized boundary components and the Virasoro action therein, in intimate relation with loop Loewner energy (universal Liouville action). Here, we made an intriguing discovery: the natural action of the group of diffeomorphisms of the circle, which after a suitable central extension to a semigroup of annuli gives rise to the Virasoro action, is trivial for just diffeomorphisms but non-trivial for the full semigroup of annuli. (This answered in a surprising manner a question discussed earlier by A.Henriques and D.Thurston.) Furthermore, this framework seems appropriate for more conceptual definitions of Loewner energy, as was also demonstrated recently. On a related vein, large deviation principles (LDP) have been developed for variants of random SLE curves, in a variety of strong topologies. This also gives rise to new variants of the Loewner energy as the rate function of the LDPs.
Questions in Aim 4 have been ongoing.
