Periodic Reporting for period 1 - ISCoURaGe (Interplay of structures in conformal and universal random geometry)
Período documentado: 2023-01-01 hasta 2025-06-30
The overall goal of this project is to provide novel conceptual understanding of persistent challenges in mathematical physics. The emphasis is especially in finding connections at the crossroads of different areas of mathematics and theoretical physics. The first two aims of the project concern statistical mechanics and mathematical formulations of (logarithmic) conformal field theory (CFT), on the one hand algebraically and on the other hand probabilistically. The last two aims focus on connections and interplay of structures arising in statistical mechanics, such as Schramm-Loewner evolutions (SLE), with algebro-geometric formulations of CFT.
Specifically, Aim 1 of the project focuses on CFT correlation functions aiming in particular to reveal non-semisimple and logarithmic behavior, poorly understood even in the physics literature. Aim 2 combines CFT with probability theory by investigating non-local quantities in statistical mechanics models and relating them rigorously to specific CFT correlation functions and to SLE. The overall challenge here is to build rigorous foundations for 2D (log-)CFT, building on algebraic formulations (like potentially non-semisimple representation theory) and probabilistic constructions from critical lattice models of statistical mechanics and their scaling limits.
Aim 3 develops generalizations of so-called Loewner energy of curves in order to reveal their rich interplay with SLE and CFT. The main objective is to shed light on the hidden geometric interpretation of Loewner energy from the point of view of formulations of CFT in terms of Riemann surfaces, and eventually also to find its role within geometric quantization. To elaborate the latter goal, Aim 4 combines these ideas with related structures in the theory of isomonodromic deformations and semiclassical CFT.
Specifically, Aim 1 of the project focuses on CFT correlation functions aiming in particular to reveal non-semisimple and logarithmic behavior, poorly understood even in the physics literature. Aim 2 combines CFT with probability theory by investigating non-local quantities in statistical mechanics models and relating them rigorously to specific CFT correlation functions and to SLE. The overall challenge here is to build rigorous foundations for 2D (log-)CFT, building on algebraic formulations (like potentially non-semisimple representation theory) and probabilistic constructions from critical lattice models of statistical mechanics and their scaling limits.
Aim 3 develops generalizations of so-called Loewner energy of curves in order to reveal their rich interplay with SLE and CFT. The main objective is to shed light on the hidden geometric interpretation of Loewner energy from the point of view of formulations of CFT in terms of Riemann surfaces, and eventually also to find its role within geometric quantization. To elaborate the latter goal, Aim 4 combines these ideas with related structures in the theory of isomonodromic deformations and semiclassical CFT.
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.
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.
Overall, this project has already revealed intricate interplay of various universal structures in conformal geometry and random geometry. New methods for proving scaling limit results, verifying CFT properties, and proving LDPs have been developed. New conceptual understanding and natural constructions pertaining to Loewner energy have been established. Together with the findings on how these objects relate to models emerging from mathematical physics, the project has resulted in inspiration of new research directions in an unusually wide range of areas.
All of the results of the project are beyond the state-of-the-art. There has already been a significant impact in the relevant research area, and the project has progressed the careers of 6 postdocs at Aalto University (of which 3 directly hired to the project), and 5 PhD students at Aalto University and University of Bonn (of which 2 directly hired to the project).
Results emerging from this project have been presented at various venues: conferences, summer schools, colloquia, seminars, and posters, as well as in public presentations targeted to audiences outside of academia.
All of the results of the project are beyond the state-of-the-art. There has already been a significant impact in the relevant research area, and the project has progressed the careers of 6 postdocs at Aalto University (of which 3 directly hired to the project), and 5 PhD students at Aalto University and University of Bonn (of which 2 directly hired to the project).
Results emerging from this project have been presented at various venues: conferences, summer schools, colloquia, seminars, and posters, as well as in public presentations targeted to audiences outside of academia.