Periodic Reporting for period 1 - RaConTeich (Connecting Random Conformal Geometry and Teichmüller theory)
Berichtszeitraum: 2024-01-01 bis 2026-06-30
Random conformal geometry deals with the analysis of conformally invariant systems using probabilistic methods. Random geometric objects, such as the random fractal SLE curves, arise from statistical mechanics models and are of central interest in modern probability theory and mathematical physics.
Teichmüller theory studies complex structures on a surface. Teichmüller spaces carry a rich geometric structure and have been an important research topic since the mid-twentieth century. These two fields are traditionally far apart. However, recent developments, by groups of researchers including the PI suggest that the connection between the two areas is stronger than currently known and we believe exploring the connection is of major scientific interest.
This project aims to break new ground in establishing links between fundamental concepts in random conformal geometry and Teichmüller theory by combining techniques from probability, complex analysis, geometric analysis, Kähler geometry, spectral theory, and representation theory. We focus on three objectives:
1) Advance the understanding of the deep reason behind the link between SLE and the Kahler geometry of Weil-Petersson universal Teichmüller space.
2) Establish the link between Liouville actions and projective structures in a systematic way.
3) Identify the holography of Loewner energy and explore the uncharted territory of holographic principles of random conformal geometry.
The successful completion of this program will substantially reshape our understanding of both areas and has the potential to bring revolutionary development to them both.
Teichmüller theory studies complex structures on a surface. Teichmüller spaces carry a rich geometric structure and have been an important research topic since the mid-twentieth century. These two fields are traditionally far apart. However, recent developments, by groups of researchers including the PI suggest that the connection between the two areas is stronger than currently known and we believe exploring the connection is of major scientific interest.
This project aims to break new ground in establishing links between fundamental concepts in random conformal geometry and Teichmüller theory by combining techniques from probability, complex analysis, geometric analysis, Kähler geometry, spectral theory, and representation theory. We focus on three objectives:
1) Advance the understanding of the deep reason behind the link between SLE and the Kahler geometry of Weil-Petersson universal Teichmüller space.
2) Establish the link between Liouville actions and projective structures in a systematic way.
3) Identify the holography of Loewner energy and explore the uncharted territory of holographic principles of random conformal geometry.
The successful completion of this program will substantially reshape our understanding of both areas and has the potential to bring revolutionary development to them both.
During the first 18 months of the project, breakthroughs and/or initial results have been obtained for the three goals. We now discuss them one by one. The letter refers to the goal, and the number after the goal only serves to distinguish different results.
(A.1) Jointly with Marco Carfagnini, we show that the Loewner energy is the Onsager-Machlup functional of the SLE loop measure. The paper "Onsager-Machlup functional for SLE loop measures" is published in Communications in Mathematical Physics.
(A.2) Jointly with Jinwoo Sung, we derived the quasiconformal deformation formula for the Loewner driving function. From this, we derived a new proof of the variational formula of the Loewner energy. These results also allow us to connect the tangent spaces of the driving functions to the tangent spaces of the universal Teichmüller space, and this is the first step needed to understand the Kähler structure on driving functions. The work, "Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy," is published in Mathematische Annalen.
(A.3) Jointly with Maria Gordina and Wei Qian and based on the works (A.1) and (A.2) we used the SLE loop measures to construct representations of the Virasoro algebra. This result provides a precise link between the SLE loop and Malliavin's measures, which was raised 20 years ago and has been a topic of wonder in the random conformal geometry community. This is one of the major potential breakthroughs listed in Goal A. The work, "Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra," is published in Journal de Mathématiques Pures et Appliquées.
(B.1) Jointly with Mario Bonk, Janne Junnila, and Steffen Rohde, we studied the projective structure on the sphere arising from Jordan curve with geodesic property and the associated accessory parameters. We showed that these projective structures and their accessory parameters are expressed in terms of the Loewner energy (which coincides with the universal Liouville action). This task provides prototypical examples to study the link between Liouville actions and projective structures. It resulted in a preprint "Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters".
(B.2) Yan Luo (ERC graduate student) and Sid Maibach studied the generalization of Loewner energy to two loops and discussed the link to partition functions of statistical mechanics models. By studying the first non trivial generalization (2 loops), this work exhibits the effect of moduli space on the possible generalizations of Liouville action given by curves. Their work resulted in the preprint "Two-loop Loewner potential".
(C.1) Jointly with Martin Bridgeman, Kenneth Bromberg, and Franco Vargas-Pallete (ERC postdoc), we obtained the holographic expression of the Loewner energy as renormalized volume of a hyperbolic 3-manifold. This result is a major breakthrough as it is the first time that the Loewner energy, as arising from 2D random conformal geometry, to be related exactly to a quantity in a hyperbolic 3 space. The work, "Universal Liouville action as a renormalized volume and its gradient flow," is accepted in the Duke Mathematical Journal.
(C.2) Conformal welding is a procedure that encodes a Jordan curve on the sphere into a circle homeomorphism. I point out that conformal welding should be viewed as a correspondence between curves on the boundary of hyperbolic 3-space and Anti de Sitter 3-space. I studied two optimization problems related to the Loewner energy and their connection to pleated planes in these two 3-spaces. This work presents an alternative perspective on the holography of the Loewner energy. The paper "Two optimization problems for the Loewner energy" is published in the Journal of Mathematical Physics.
(C.3) Jointly with Franco Vargas-Pallete (ERC postdoc) and Catherine Wolfram, we considered the construction of renormalized area in hyperbolic 2-space, and showed an identity with the Schwarzian action for circle homeomorphisms. This correspondence uses a construction (Epstein's map) mostly studied in one dimension higher (correspondence between Riemann surfaces and hyperbolic 3-manifolds). However, the seemingly easier lower-dimensional analog, as we demonstrate, provides a neat approach to the connection between Schwarzian field theory and JT gravity at the action level, highlighting the robustness of Epstein's construction in the study of holography. This resulted in a preprint, "Epstein curves and holography of the Schwarzian action".
(A.1) Jointly with Marco Carfagnini, we show that the Loewner energy is the Onsager-Machlup functional of the SLE loop measure. The paper "Onsager-Machlup functional for SLE loop measures" is published in Communications in Mathematical Physics.
(A.2) Jointly with Jinwoo Sung, we derived the quasiconformal deformation formula for the Loewner driving function. From this, we derived a new proof of the variational formula of the Loewner energy. These results also allow us to connect the tangent spaces of the driving functions to the tangent spaces of the universal Teichmüller space, and this is the first step needed to understand the Kähler structure on driving functions. The work, "Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy," is published in Mathematische Annalen.
(A.3) Jointly with Maria Gordina and Wei Qian and based on the works (A.1) and (A.2) we used the SLE loop measures to construct representations of the Virasoro algebra. This result provides a precise link between the SLE loop and Malliavin's measures, which was raised 20 years ago and has been a topic of wonder in the random conformal geometry community. This is one of the major potential breakthroughs listed in Goal A. The work, "Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra," is published in Journal de Mathématiques Pures et Appliquées.
(B.1) Jointly with Mario Bonk, Janne Junnila, and Steffen Rohde, we studied the projective structure on the sphere arising from Jordan curve with geodesic property and the associated accessory parameters. We showed that these projective structures and their accessory parameters are expressed in terms of the Loewner energy (which coincides with the universal Liouville action). This task provides prototypical examples to study the link between Liouville actions and projective structures. It resulted in a preprint "Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters".
(B.2) Yan Luo (ERC graduate student) and Sid Maibach studied the generalization of Loewner energy to two loops and discussed the link to partition functions of statistical mechanics models. By studying the first non trivial generalization (2 loops), this work exhibits the effect of moduli space on the possible generalizations of Liouville action given by curves. Their work resulted in the preprint "Two-loop Loewner potential".
(C.1) Jointly with Martin Bridgeman, Kenneth Bromberg, and Franco Vargas-Pallete (ERC postdoc), we obtained the holographic expression of the Loewner energy as renormalized volume of a hyperbolic 3-manifold. This result is a major breakthrough as it is the first time that the Loewner energy, as arising from 2D random conformal geometry, to be related exactly to a quantity in a hyperbolic 3 space. The work, "Universal Liouville action as a renormalized volume and its gradient flow," is accepted in the Duke Mathematical Journal.
(C.2) Conformal welding is a procedure that encodes a Jordan curve on the sphere into a circle homeomorphism. I point out that conformal welding should be viewed as a correspondence between curves on the boundary of hyperbolic 3-space and Anti de Sitter 3-space. I studied two optimization problems related to the Loewner energy and their connection to pleated planes in these two 3-spaces. This work presents an alternative perspective on the holography of the Loewner energy. The paper "Two optimization problems for the Loewner energy" is published in the Journal of Mathematical Physics.
(C.3) Jointly with Franco Vargas-Pallete (ERC postdoc) and Catherine Wolfram, we considered the construction of renormalized area in hyperbolic 2-space, and showed an identity with the Schwarzian action for circle homeomorphisms. This correspondence uses a construction (Epstein's map) mostly studied in one dimension higher (correspondence between Riemann surfaces and hyperbolic 3-manifolds). However, the seemingly easier lower-dimensional analog, as we demonstrate, provides a neat approach to the connection between Schwarzian field theory and JT gravity at the action level, highlighting the robustness of Epstein's construction in the study of holography. This resulted in a preprint, "Epstein curves and holography of the Schwarzian action".
The first 18 months of the project allowed us to obtain preliminary results and major milestone achievements, pushing the front towards all three goals.
In particular, we understood the precise link between SLE and Malliavin's measure which provides a global description of SLE (related to goal A); we studied thoroughly the first few examples confirming the conjectural link (outlined in the proposal) between projective structures arising from curves, and various Liouville action (related to goal B); we obtained the precise holographic expression of the Loewner energy using Epstein's surfaces (related to goal C).
We also obtained an unexpected result expressing the Schwarzian action using Epstein's construction (we call it Epstein's curve) in one dimension (related to goal C), which shows the robustness of Epstein’s construction for the holography. Another unexpected result was obtained while studying the random geometry on higher genus surfaces. We use Brownian loop measure to study the length spectrum of hyperbolic surfaces and obtain surprising identities between the length spectra of surfaces, showing another unpredicted link between random geometry and Teichmuller theory.
In particular, we understood the precise link between SLE and Malliavin's measure which provides a global description of SLE (related to goal A); we studied thoroughly the first few examples confirming the conjectural link (outlined in the proposal) between projective structures arising from curves, and various Liouville action (related to goal B); we obtained the precise holographic expression of the Loewner energy using Epstein's surfaces (related to goal C).
We also obtained an unexpected result expressing the Schwarzian action using Epstein's construction (we call it Epstein's curve) in one dimension (related to goal C), which shows the robustness of Epstein’s construction for the holography. Another unexpected result was obtained while studying the random geometry on higher genus surfaces. We use Brownian loop measure to study the length spectrum of hyperbolic surfaces and obtain surprising identities between the length spectra of surfaces, showing another unpredicted link between random geometry and Teichmuller theory.