arxiv: 2605.13802 · v1 · submitted 2026-05-13 · 🧮 math-ph · math.MP· math.PR

Recognition: 2 theorem links

· Lean Theorem

Irregular SLE(4) martingales and isomonodromic deformations

Aleksandra Korzhenkova, Eveliina Peltola, Harini Desiraju

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:26 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords isomonodromic deformationsSLE(4)martingale observablesLoewner evolutionBPZ equationsconformal field theoryirregular singularitiesdouble poles

0 comments

The pith

Deriving the Loewner evolution of isomonodromic parameters constructs martingale observables for SLE(4) with double poles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes how the parameters of non-Fuchsian monodromy-preserving deformations evolve according to the Loewner equation driven by SLE(4). The resulting expressions serve as martingales for the SLE(4) process even when double poles are present in the setup. These martingales incorporate the pre-Schwarzian and Schwarzian derivatives arising from the conformal properties of the observable. The authors show that such observables are uniquely characterized as solutions to confluent BPZ equations in a conformal field theory with central charge equal to one. This connects the geometry of random curves to isomonodromic deformation theory on the Riemann sphere.

Core claim

The paper's central claim is that the isomonodromic deformation parameters on the Riemann sphere, consisting of singularity positions and Birkhoff invariants from irregular singularities, follow a Loewner evolution under the SLE(4) driving function. This evolution is used to build martingale observables for the SLE(4) process in the presence of double poles, with the expressions involving the pre-Schwarzian and Schwarzian of the Loewner map due to conformal covariance. Furthermore, these observables are uniquely determined by the confluent BPZ equations of a CFT with central charge c=1.

What carries the argument

The Loewner evolution of the isomonodromic deformation parameters, which include singularity positions and Birkhoff invariants, serving as the mechanism to generate the SLE(4) martingales.

If this is right

Martingales for SLE(4) can be constructed explicitly even with double poles using the evolved parameters.
The observables remain invariant in expectation under the SLE(4) dynamics due to the derived evolution.
Unique characterization via confluent BPZ equations links the construction to conformal field theory at c=1.
Geometric terms like pre-Schwarzian and Schwarzian ensure covariance under conformal transformations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method might extend to computing explicit formulas for SLE(4) probabilities in domains with irregular singularities.
Connections between Loewner evolution and isomonodromic deformations could inform solutions to certain Riemann-Hilbert problems involving random paths.
The approach suggests that similar martingale constructions may apply to other values of kappa in SLE by modifying the central charge.

Load-bearing premise

The isomonodromic deformation parameters admit a well-defined Loewner evolution under the driving function of the SLE(4) process.

What would settle it

Direct computation of the infinitesimal generator applied to the observable yielding a nonzero result when double poles are included.

Figures

Figures reproduced from arXiv: 2605.13802 by Aleksandra Korzhenkova, Eveliina Peltola, Harini Desiraju.

**Figure 1.1.** Figure 1.1: Illustration of a chordal injective curve [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗

**Figure 1.** Figure 1: for an illustration). Moreover, they satisfy [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗

read the original abstract

We consider non-Fuchsian monodromy preserving deformations on a Riemann sphere. The associated isomonodromic deformation parameters on this surface comprise the positions of the singularities, together with the Birkhoff (spectral) invariants owing to the presence of irregular singularities. Our first main result is the derivation of the Loewner evolution of these isomonodromic deformation parameters. Using this result, we construct martingale observables for Schramm-Loewner evolution (SLE(4)) processes in the presence of double poles. Geometrically, the expressions contain the pre-Schwarzian and Schwarzian of the Loewner evolution, arising from conformal covariance of the observable. Furthermore, we characterize these SLE(4) observables uniquely in terms of confluent BPZ equations of a CFT with central charge c=1.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Derives Loewner evolution for Birkhoff invariants in irregular SLE(4) but monodromy preservation under the flow is the key assumption to verify.

read the letter

The main takeaway is that this paper derives the Loewner evolution of the isomonodromic deformation parameters—including singularity positions and Birkhoff invariants—for non-Fuchsian cases with double-pole irregular singularities, then uses that to build SLE(4) martingale observables and characterize them uniquely through confluent BPZ equations at c=1. They incorporate the pre-Schwarzian and Schwarzian derivatives to handle conformal covariance under the Loewner flow. This extends the usual Fuchsian SLE-CFT setup in a concrete way, and the uniqueness part gives a clean way to identify the observables without extra fitting. The approach looks grounded in the existing literature on isomonodromic deformations and Loewner evolutions. The soft spot sits in the central derivation step. The evolution of the Birkhoff invariants must keep the process inside the monodromy-preserving family; if the flow introduces uncancelled drift or diffusion terms from the irregular singularities, the martingale property would not hold. The abstract presents the derivation as complete, but the irregular case sits outside standard Fuchsian Loewner theory, so the explicit cancellation via the Schwarzian corrections needs to be tight. Without seeing the equations it is hard to judge how cleanly they close the system. This is aimed at specialists already working on SLE interfaces, random conformal geometry, and integrable systems with irregular singularities. Readers who know the Fuchsian correspondences and BPZ equations will follow the steps and see the incremental value. I would send it to peer review. The claims are specific enough for referees to check the evolution equations and the characterization, and the work supplies a clear technical step even if revisions are needed on the details of the irregular flow.

Referee Report

2 major / 2 minor

Summary. The manuscript derives the Loewner evolution of isomonodromic deformation parameters (singularity positions together with Birkhoff invariants) for non-Fuchsian monodromy-preserving deformations on the Riemann sphere. Using this evolution, it constructs martingale observables for SLE(4) processes in the presence of double poles, with expressions involving the pre-Schwarzian and Schwarzian derivatives arising from conformal covariance; these observables are then characterized uniquely in terms of confluent BPZ equations for a CFT with central charge c=1.

Significance. If the derivation holds, the work extends the SLE-CFT correspondence beyond the Fuchsian setting to irregular singularities, supplying new martingale observables and a uniqueness result via confluent BPZ equations. This could open avenues for studying stochastic Loewner evolutions coupled to isomonodromic deformations with higher-order poles, a direction that has seen limited prior treatment.

major comments (2)

[Derivation of Loewner evolution (first main result)] The central derivation of the Loewner evolution for the Birkhoff invariants (stated as the first main result) must explicitly verify that the stochastic flow remains inside the isomonodromic family. In particular, the SDEs for the Birkhoff invariants must be shown to produce no residual non-isomonodromic drift or diffusion after inclusion of the pre-Schwarzian/Schwarzian corrections; otherwise the martingale property of the constructed observables fails.
[Construction of martingale observables and uniqueness via confluent BPZ] The uniqueness characterization of the SLE(4) observables via confluent BPZ equations at c=1 presupposes that the evolved parameters remain monodromy-preserving. The manuscript should supply a direct check (e.g., via the explicit form of the evolution equations) that the stochastic increments preserve the monodromy data, rather than relying solely on the initial isomonodromic setup.

minor comments (2)

[Abstract and §1] The abstract and introduction would benefit from a brief explicit statement of the rank of the irregular singularities (double poles correspond to rank-1 irregular points) to orient readers unfamiliar with the Birkhoff classification.
[Notation section] Notation for the Birkhoff invariants and their time derivatives under the Loewner flow should be introduced with a dedicated table or displayed equations early in the text to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments concern the need for explicit verification that the derived stochastic evolution preserves the isomonodromic (monodromy-preserving) property. We address each point below and will revise the manuscript to include the requested direct checks.

read point-by-point responses

Referee: The central derivation of the Loewner evolution for the Birkhoff invariants (stated as the first main result) must explicitly verify that the stochastic flow remains inside the isomonodromic family. In particular, the SDEs for the Birkhoff invariants must be shown to produce no residual non-isomonodromic drift or diffusion after inclusion of the pre-Schwarzian/Schwarzian corrections; otherwise the martingale property of the constructed observables fails.

Authors: We agree that an explicit verification strengthens the argument. The SDEs for the Birkhoff invariants are derived in Section 3 by requiring that the stochastic deformation satisfies the isomonodromic condition at each step; the pre-Schwarzian and Schwarzian corrections are introduced precisely to cancel non-isomonodromic Itô terms. In the revised manuscript we will add a short subsection (or appendix calculation) that computes the full Itô expansion of the deformation parameters and confirms that all residual drift and diffusion coefficients outside the isomonodromic manifold vanish identically under the chosen coefficients. revision: yes
Referee: The uniqueness characterization of the SLE(4) observables via confluent BPZ equations at c=1 presupposes that the evolved parameters remain monodromy-preserving. The manuscript should supply a direct check (e.g., via the explicit form of the evolution equations) that the stochastic increments preserve the monodromy data, rather than relying solely on the initial isomonodromic setup.

Authors: We accept the request for a direct check. The evolution equations are constructed so that the stochastic increments are tangent to the isomonodromic foliation; consequently the monodromy data (including the Birkhoff invariants) evolve only within the allowed family. In the revision we will insert an explicit verification, using the closed-form SDEs, showing that the stochastic increments leave the monodromy invariants unchanged except for the controlled isomonodromic motion already accounted for in the confluent BPZ characterization. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation of Loewner evolution and martingale construction are independent of the target observables

full rationale

The paper starts from the given non-Fuchsian isomonodromic deformation setup (positions plus Birkhoff invariants) and derives the Loewner evolution equations for those parameters under the SLE(4) driving function. The martingale observables are then built explicitly from that derived evolution together with pre-Schwarzian/Schwarzian corrections. The final uniqueness statement is a characterization in terms of confluent BPZ equations at c=1, which is presented as a consequence rather than an input that forces the preceding steps. No quoted equation reduces a claimed prediction to a fitted parameter or to a self-citation chain; the central derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in complex analysis and conformal field theory; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Monodromy-preserving deformations on the Riemann sphere admit well-defined isomonodromic parameters including Birkhoff invariants for irregular singularities.
Invoked in the first sentence of the abstract as the setup for the Loewner evolution derivation.
domain assumption SLE(4) driving function induces a Loewner evolution on the isomonodromic parameters that preserves the martingale property.
Central assumption underlying the construction of the observables.

pith-pipeline@v0.9.0 · 5445 in / 1601 out tokens · 40242 ms · 2026-05-14T17:26:47.877237+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Our first main result is the derivation of the Loewner evolution of these isomonodromic deformation parameters... Si,k_t = s_i,k exp(−∫ 2k du/(Λ_i_u − Z_u)^2)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the process Mt := F(Λ_t; S_t) τ(Λ_t; S_t) Y(Z_t, Λ_t; S_t) is a matrix-valued local martingale... characterized uniquely in terms of confluent BPZ equations... c=1

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 2 internal anchors

[1]

Ablowitz and Athanassios S

Mark J. Ablowitz and Athanassios S. Fokas. Complex variables: introduction and applications . Cambridge University Press, 2003

work page 2003
[2]

SLE_ growth processes and conformal field theories

Michel Bauer and Denis Bernard. SLE_ growth processes and conformal field theories. Phys. Lett. B , 543(1-2):135--138, 2002

work page 2002
[3]

Conformal field theories of stochastic L oewner evolutions

Michel Bauer and Denis Bernard. Conformal field theories of stochastic L oewner evolutions. Comm. Math. Phys. , 239(3):493--521, 2003

work page 2003
[4]

SLE martingales and the V irasoro algebra

Michel Bauer and Denis Bernard. SLE martingales and the V irasoro algebra. Phys. Lett. B , 557(3-4):309--316, 2003

work page 2003
[5]

Conformal transformations and the SLE partition function martingale

Michel Bauer and Denis Bernard. Conformal transformations and the SLE partition function martingale. Ann. Henri Poincar\'e , 5(2):289--326, 2004

work page 2004
[6]

Introduction to classical integrable systems

Olivier Babelon, Denis Bernard, and Michel Talon. Introduction to classical integrable systems . Cambridge University Press, 2003

work page 2003
[7]

Tau-functions \`a la D ub \'e dat and probabilities of cylindrical events for double-dimers and CLE(4)

Mikhail Basok and Dmitry Chelkak. Tau-functions \`a la D ub \'e dat and probabilities of cylindrical events for double-dimers and CLE(4) . J. Eur. Math. Soc. , 23(8):2787--2832, 2021

work page 2021
[8]

Partition functions for matrix models and isomonodromic tau functions

Marco Bertola, Bertrand Eynard, and John Harnad. Partition functions for matrix models and isomonodromic tau functions. J. Phys. A , 36(12):3067--3084, 2003

work page 2003
[9]

Sur une int\'egrale pour les processus \`a -variation born\'ee

Jean Bertoin. Sur une int\'egrale pour les processus \`a -variation born\'ee. Ann. Probab. , 17(4):1521--1535, 1989

work page 1989
[10]

Tau functions and their applications

Ferenc Balogh and John Harnad. Tau functions and their applications . Cambridge University Press, 2003

work page 2003
[11]

Hamiltonian structure of rational isomonodromic deformation systems

Marco Bertola, John Harnad, and Jacques Hurtubise. Hamiltonian structure of rational isomonodromic deformation systems. J. Math. Phys. , 64(8):083502, 2023

work page 2023
[12]

Birkhoff

George D. Birkhoff. The generalized riemann problem for linear differential equations and the allied problems for linear difference and q -difference equations. Proc. Amer. Acad. Arts Sci. , 49:521--568, 1913

work page 1913
[13]

Jurkat, and Donald A

Werner Balser, Wolfgang B. Jurkat, and Donald A. Lutz. Birkhoff invariants and S tokes ' multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. , 71(1):48--94, 1979

work page 1979
[14]

On P ainlev\'e/gauge theory correspondence

Giulio Bonelli, Oleg Lisovyy, Kazunobu Maruyoshi, Antonio Sciarappa, and Alessandro Tanzini. On P ainlev\'e/gauge theory correspondence. Lett. Math. Phys. , 107(12):2359--2413, 2017

work page 2017
[15]

N = 2^* G auge theory, free fermions on the torus and P ainlev\'e VI

Giulio Bonelli, Fabrizio Del Monte, Pavlo Gavrylenko, and Alessandro Tanzini. N = 2^* G auge theory, free fermions on the torus and P ainlev\'e VI . Comm. Math. Phys. , 377(2):1381--1419, 2020

work page 2020
[16]

Belavin, Alexander M

Alexander A. Belavin, Alexander M. Polyakov, and Alexander B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B , 241(2):333--380, 1984

work page 1984
[17]

On the crossing estimates of simple conformal loops ensembles

Tianyi Bai and Yijun Wan. On the crossing estimates of simple conformal loops ensembles. Int. Math. Res. Not. , 2023(13):11645--11683, 2022

work page 2023
[18]

Interactions of irregular G aiotto states in L iouville theory

Sang-Kwan Choi, Dimitri Polyakov, and Cong Zhang. Interactions of irregular G aiotto states in L iouville theory. Eur. Phys. J. C , 78(507):1--17, 2018

work page 2018
[19]

Fredholm determinant representation of the homogeneous P ainlev \'e II -function

Harini Desiraju. Fredholm determinant representation of the homogeneous P ainlev \'e II -function. Nonlinearity , 34(9):6507--6538, 2021

work page 2021
[20]

Painlev \'e / CFT correspondence on a torus

Harini Desiraju. Painlev \'e / CFT correspondence on a torus. J. Math. Phys. , 63(8):081102--1--16, 2022

work page 2022
[21]

SLE and V irasoro representations: localization

Julien Dub \'e dat. SLE and V irasoro representations: localization. Comm. Math. Phys. , 336(2):695--760, 2015

work page 2015
[22]

Double dimers, conformal loop ensembles, and isomonodromic deformations

Julien Dub \'e dat. Double dimers, conformal loop ensembles, and isomonodromic deformations. J. Eur. Math. Soc. , 21(1):1--54, 2019

work page 2019
[23]

Stochastic calculus: a practical introduction

Richard Durrett. Stochastic calculus: a practical introduction . Probability and Stochastics Series. CRC Press LLC, 1996

work page 1996
[24]

Fock and Alexander B

Vladimir V. Fock and Alexander B. Goncharov. Moduli spaces of local systems and higher T eichm\"uller theory. Publ. Math. Inst. Hautes \'Etudes Sci. , 103:1--211, 2006

work page 2006
[25]

Fokas, Alexander R

Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu Novokshenov. Painlev \'e transcendents: the R iemann- H ilbert approach , volume 128 of Mathematical Surveys and Monographs . American Mathematical Society, 2023

work page 2023
[26]

On conformal field theory and stochastic L oewner evolution

Roland Friedrich and Jussi Kalkkinen. On conformal field theory and stochastic L oewner evolution. Nucl. Phys. B , 687(3):279--302, 2004

work page 2004
[27]

Multiple SLE s for (0,8) : C oulomb gas integrals and pure partition functions

Yu Feng, Mingchang Liu, Eveliina Peltola, and Hao Wu. Multiple SLE s for (0,8) : C oulomb gas integrals and pure partition functions. Preprint in arXiv:2406.06522, 2024

work page arXiv 2024
[28]

Calcul d'It\^o sans probabilit \'e s

Hans F \"o llmer. Calcul d'It\^o sans probabilit \'e s. In Seminar on Probability XV (Univ. Strasbourg) , volume 850 of Lecture Notes in Mathematics , pages 143--150. Springer, Berlin, 1981

work page 1981
[29]

On Connections of Conformal Field Theory and Stochastic L{\oe}wner Evolution

Roland Friedrich. On connections of conformal field theory and stochastic L oewner evolution. Preprint in arXiv:math-ph/0410029, 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004
[30]

Dirichlet forms and M arkov processes , volume 23

Masatoshi Fukushima. Dirichlet forms and M arkov processes , volume 23. North-Holland, Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980

work page 1980
[31]

Conformal restriction, highest weight representations and SLE

Roland Friedrich and Wendelin Werner. Conformal restriction, highest weight representations and SLE . Comm. Math. Phys. , 243(1):105--122, 2003

work page 2003
[32]

Iorgov, and Oleg Lisovyy

Oleksandr Gamayun, Nikolai Z. Iorgov, and Oleg Lisovyy. Conformal field theory of P ainlev \'e VI . JHEP , 10(38):1--24, 2012

work page 2012
[33]

Iorgov, and Oleg Lisovyy

Oleksandr Gamayun, Nikolai Z. Iorgov, and Oleg Lisovyy. How instanton combinatorics solves P ainlev\'e VI , V and IIIs . J. Phys. A , 46(33):1--29, 2013

work page 2013
[34]

Higher rank isomonodromic deformations and W -algebras

Pavlo Gavrylenko, Nikolai Iorgov, and Oleg Lisovyy. Higher rank isomonodromic deformations and W -algebras. Lett. Math. Phys. , 110(2):327--364, 2019

work page 2019
[35]

Fredholm determinant and N ekrasov sum representations of isomonodromic tau functions

Pavlo Gavrylenko and Oleg Lisovyy. Fredholm determinant and N ekrasov sum representations of isomonodromic tau functions. Comm. Math. Phys. , 363(1):1--58, 2018

work page 2018
[36]

Isomonodromic deformations: C onfluence, reduction, and quantisation

Ilia Gaiur, Marta Mazzocco, and Vladimir Rubtsov. Isomonodromic deformations: C onfluence, reduction, and quantisation. Comm. Math. Phys. , 400(2):1385--1461, 2023

work page 2023
[37]

Irregular singularities in L iouville theory and A rgyres- D ouglas type gauge theories

Davide Gaiotto and J \"o rg Teschner. Irregular singularities in L iouville theory and A rgyres- D ouglas type gauge theories. JHEP , 12:1--79, 2012

work page 2012
[38]

Flat connections from irregular conformal blocks

Babak Haghighat, Yihua Liu, and Nicolai Reshetikhin. Flat connections from irregular conformal blocks. Comm. Math. Phys. , 406(138):1--30, 2025

work page 2025
[39]

The analysis of linear partial differential operators I : D istribution theory and F ourier analysis , volume 256 of Grundlehren der mathematischen Wissenschaften

Lars H \"o rmander. The analysis of linear partial differential operators I : D istribution theory and F ourier analysis , volume 256 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag, Berlin Heidelberg, 2 edition, 1990

work page 1990
[40]

Its, Oleg Lisovyy, and Andrei Prokhorov

Alexander R. Its, Oleg Lisovyy, and Andrei Prokhorov. Monodromy dependence and connection formulae for isomonodromic tau functions. Duke Math. J. , 167(7):1347--1432, 2018

work page 2018
[41]

Painlev \'e VI connection problem and monodromy of c=1 conformal blocks

Nikolai Iorgov, Oleg Lisovyy, and Yu Tykhyy. Painlev \'e VI connection problem and monodromy of c=1 conformal blocks. JHEP , 12(29):1--26, 2013

work page 2013
[42]

Isomonodromic tau-functions from L iouville conformal blocks

Nikolai Iorgov, Oleg Lisovyy, and J \"o rg Teschner. Isomonodromic tau-functions from L iouville conformal blocks. Comm. Math. Phys. , 336(2):671--694, 2015

work page 2015
[43]

Its, Oleg Lisovyy, and Yuriy Tykhyy

Alexander R. Its, Oleg Lisovyy, and Yuriy Tykhyy. Connection problem for the S ine- G ordon/ P ainlev\'e III tau function and irregular conformal blocks. Int. Math. Res. Not. , 2015(18):8903--8924, 2015

work page 2015
[44]

Its and Andrei Prokhorov

Alexander R. Its and Andrei Prokhorov. On some H amiltonian properties of the isomonodromic tau functions. Rev. Math. Phys. , 30(7):1840008, 2018

work page 2018
[45]

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients

Michio Jimbo and Tetsuji Miwa. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II . Phys. D , 2(3):407--448, 1981

work page 1981
[46]

Monodromy preserving deformation of linear ordinary differential equations with rational coefficients

Michio Jimbo, Tetsuji Miwa, and Kimio Ueno. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I . general theory and -function. Phys. D , 2(2):306--352, 1981

work page 1981
[47]

Schramm- L oewner evolution , volume 24 of SpringerBriefs in Mathematical Physics

Antti Kemppainen. Schramm- L oewner evolution , volume 24 of SpringerBriefs in Mathematical Physics . Springer Cham, 2017

work page 2017
[48]

Richard W. Kenyon. Conformal invariance of loops in the double-dimer model. Comm. Math. Phys. , 326(2):477--497, 2014

work page 2014
[49]

CFT , SLE , and phase boundaries

Maxim Kontsevich. CFT , SLE , and phase boundaries. In Oberwolfach Arbeitstagung , 2003

work page 2003
[50]

Boundary double-dimer patterns and conformal field theory

Alex Karrila and Eveliina Peltola. Boundary double-dimer patterns and conformal field theory. In preparation, 2026

work page 2026
[51]

On M alliavin measures, SLE , and CFT

Maxim Kontsevich and Yuri Suhov. On M alliavin measures, SLE , and CFT . P. Steklov I. Math. , 258(1):100--146, 2007

work page 2007
[52]

Smoothness of martingale observables and generalized Feynman-Kac formulas

Alex Karrila and Lauri Viitasaari. Smoothness of martingale observables and generalized F eynman- K ac formulas. Preprint in arXiv:2601.10539, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[53]

Kenyon and David B

Richard W. Kenyon and David B. Wilson. Double-dimer pairings and skew Y oung diagrams. Electron. J. Combin. , 18(1):1--22, 2011

work page 2011
[54]

The trunks of CLE(4) explorations

Matthis Lehmkuehler. The trunks of CLE(4) explorations. Ann. Appl. Probab. , 33(5):3387--3417, 2023

work page 2023
[55]

Untersuchungen \"u ber schlichte konforme A bbildungen des E inheitskreises I

Charles Loewner. Untersuchungen \"u ber schlichte konforme A bbildungen des E inheitskreises I . Math. Ann. , 89:103--121, 1923

work page 1923
[56]

Sur les \'equations diff\'erentielles du second ordre dont l'int\'egrale g\'en\'erale a ses points critiques fixes

Johannes Malmquist. Sur les \'equations diff\'erentielles du second ordre dont l'int\'egrale g\'en\'erale a ses points critiques fixes. Ark. Mat. , 17:1--89, 1922

work page 1922
[57]

Isomonodromic tau functions on a torus as F redholm determinants, and charged partitions

Fabrizio Del Monte, Harini Desiraju, and Pavlo Gavrylenko. Isomonodromic tau functions on a torus as F redholm determinants, and charged partitions. Comm. Math. Phys. , 398(3):1029--1084, 2023

work page 2023
[58]

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Fabrizio Del Monte, Harini Desiraju, and Pavlo Gavrylenko. Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus. J. Phys. A , 56(29):1--24, 2023

work page 2023
[59]

Modular transformations of tau functions and conformal blocks on the torus

Fabrizio Del Monte, Harini Desiraju, and Pavlo Gavrylenko. Modular transformations of tau functions and conformal blocks on the torus. Preprint in arxiv:2508.14030, 2025

work page arXiv 2025
[60]

Imaginary geometry I : interacting SLE s

Jason Miller and Scott Sheffield. Imaginary geometry I : interacting SLE s. Probab. Theory Related Fields , 164(3-4):553--705, 2016

work page 2016
[61]

CLE percolations

Jason Miller, Scott Shaffield, and Wendelin Werner. CLE percolations. Forum Math. Pi , 5(e4):1--102, 2017

work page 2017
[62]

On the -function of the P ainlev\'e VI equations

Kazuo Okamoto. On the -function of the P ainlev\'e VI equations. Phys. D , 2(3):525--535, 1981

work page 1981
[63]

Sur les \'equations diff\'erentielles du second ordre \`a points critiques fixes

Painlev\'e. Sur les \'equations diff\'erentielles du second ordre \`a points critiques fixes. C. R. Acad. Sci. Paris S \'e r. I Math. , 143:1111--1117, 1906

work page 1906
[64]

Towards a conformal field theory for S chramm- L oewner evolutions

Eveliina Peltola. Towards a conformal field theory for S chramm- L oewner evolutions. J. Math. Phys. , 60(10):103305, 2019. Special issue (Proc. ICMP, Montreal, July 2018)

work page 2019
[65]

Global and local multiple SLE s for 4 and connection probabilities for level lines of GFF

Eveliina Peltola and Hao Wu. Global and local multiple SLE s for 4 and connection probabilities for level lines of GFF . Comm. Math. Phys. , 366(2):469--536, 2019

work page 2019
[66]

Basic properties of SLE

Steffen Rohde and Oded Schramm. Basic properties of SLE . Ann. of Math. , 161(2):883--924, 2005

work page 2005
[67]

\"U ber eine K lasse von D ifferentialsystemen beliebiger O rdnung mit festen kritischen P unkten

Ludwig Schlesinger. \"U ber eine K lasse von D ifferentialsystemen beliebiger O rdnung mit festen kritischen P unkten. J. Reine Angew. Math. , 141:96--145, 1912

work page 1912
[68]

Scaling limits of loop-erased random walks and uniform spanning trees

Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. , 118(1):221--288, 2000

work page 2000
[69]

Conformally invariant scaling limits, an overview and a collection of problems

Oded Schramm. Conformally invariant scaling limits, an overview and a collection of problems. In Proceedings of the ICM 2006, Madrid, Spain , volume 1, pages 513--543. European Mathematical Society, 2006

work page 2006
[70]

Exploration trees and conformal loop ensembles

Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79--129, 2009

work page 2009
[71]

Holonomic quantum fields

Ken-iti Sato, Tetsuji Miwa, and Michio Jimbo. Holonomic quantum fields. 3. Publ. Res. Inst. Math. Sci. Kyoto , 15(2):577--629, 1979

work page 1979
[72]

George G. Stokes. On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Cambridge Philos. Soc. , 10:105--128, 1857

work page
[73]

Oded Schramm and David B. Wilson. SLE coordinate changes. New York J. Math. , 11:659--669, 2005

work page 2005
[74]

Conformal loop ensembles: T he M arkovian characterization and the loop-soup construction

Scott Sheffield and Wendelin Werner. Conformal loop ensembles: T he M arkovian characterization and the loop-soup construction. Ann. of Math. , 176(3):1827--1917, 2012

work page 1917
[75]

On conformally invariant CLE explorations

Wendelin Werner and Hao Wu. On conformally invariant CLE explorations. Comm. Math. Phys. , 320(3):637--661, 2013

work page 2013
[76]

Random L oewner chains in R iemann surfaces

Dapeng Zhan. Random L oewner chains in R iemann surfaces . PhD thesis, California Institute of Technology, 2004

work page 2004