In this article, I will introduce the concept of isomonodromic deformations and explain the role they play in the study of random curves.
1.Isomonodromic deformations
Isomonodromic deformations study the deformations of complex functions such that their monodromy is preserved. To understand this, let me first motivate the concept of monodromy for a given complex function.
Consider a function $f(z)=z^{a}$, where $z$ is a complex variable and $a$ is a complex number. If I take a loop around $z=0$, i.e. make the transformation $z\to e^{2\pi i} z $. The function $f(z)$ under this transformation $f(e^{2\pi i}z) = z^a e^{2\pi i a} = f(z) e^{2\pi i a}$. Notice that the value of $e^{2\pi i a}\neq 1$ if $a\neq \mathbb{Z}$. For such non-trivial values of $a$, $f(z)$ is not single-valued, but instead acquires a phase $e^{2\pi i a}$. This is called the monodromy of the function.
An important component of my research surrounds a special class of matrix valued differential equations on Riemann surfaces with punctures, called isomonodromic deformation equations [FIKN86, JMU81]. These study the deformations of some matrix valued function, as opposed to a scalar like the example above, such that its monodromy stays constant as the punctures move around. These equations have the following structure
$$\frac{d Y}{dz} = \left(\sum_{i=1}^n \frac{A_i}{z-\lambda_i} \right)Y, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)$$
where $Y \equiv Y(z,\lambda_1,..., \lambda_n)$ and $A_i\equiv A_i(\lambda_1,..., \lambda_n)$ are traceless $2\times 2$ matrices of determinant 1, with $z\in \mathbb{C}\cup \left\lbrace \infty\right\rbrace\setminus \left\lbrace \lambda_1,..., \lambda_n \right\rbrace$ meaning that $z$ is a coordinate on the Riemann sphere $\mathbb{C}\cup \left\lbrace \infty\right\rbrace$ with the points $\lambda_1,...,\lambda_n$ removed, and $\lambda_1,...,\lambda_n$ denote the positions of the punctures.
Let $\pm \theta_i$ be the eigenvalues of $A_i$, and $\Theta_i= \text{diag}(\theta_i, - \theta_i)$. We can then show that, traversing the solution $Y$ in (1) around a singularity $z=\lambda_i$ gives the transformation $Y\to Y M_i$, where $M_i := C_i^{-1} e^{2\pi i \Theta_i}C_i$. The eigenvalues of these monodromy matrices are $e^{\pm 2\pi i \theta_i}$ and the matrices $C_i$ diagonalize $M_i$.
The condition that the eigenvalues $\theta_i$ remain constant under deforming $a_i$, i.e. $\frac{d \theta_i}{d \lambda_i}=0$ gives rise to the following equation for the function $Y$ in (1):
\begin{align}
\frac{d Y}{d \lambda_i} = -\frac{A_i}{z-\lambda_i} Y. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ (2)
\end{align}
The equations (1) and (2) together are called the system of isomonodromic deformation equations (IDEs). Note that $\lambda_i$'s are the deformation parameters.
An important quantity in the study of IDEs is the so-called tau-function [HB21, JMU81], which depends only on the deformation parameters $\lambda_i$, $i=1,...,n$, and is defined as
\begin{align}
\left( \tfrac{\partial}{\partial \lambda_i} \log \tau(\lambda_1,...,\lambda_n) \right) d\lambda_i = \frac{1}{2} \, \underset{z=\lambda_i}{\text{res}}\, \text{tr} \left(\frac{A_i^2}{(z-\lambda_i)^2} \right) dz. \,\,\,\,\,\,\,\,\,\,\,\,\,\, (3)
\end{align}
While the above definition may seem arbitrary at first, the tau-function is in fact a fundamental quantity for IDEs. From a geometric point of view tau-functions describe line bundles on the moduli space of the associated Riemann surfaces. Analytically, they form a distinguished class of special functions that describe a range of objects in mathematics and physics from correlation functions in conformal field theories, solutions of certain dynamical systems, to moments of random matrices and certain invariants in enumerative geometry.
The focus of my upcoming paper with Eveliina Peltola and Aleksandra Korzhenkova [DKP26] is a result by Dubédat in 2014 [Dub14] which describes the observables called Martingales of certain random curves that go by the name Schramm-Lowner evolutions (SLEs) in terms of the solution $Y$ in (1) and its tau-function (3). Let me briefly describe this connection.
2. Schramm-Lowner evolutions
SLEs are random curves in two-dimensional space with specific properties which I will describe shortly. Historically, the study of SLEs has been closely tied to critical interfaces of lattice model. The breakthrough result of Schramm in 2000 [Sch00, Sch06] showed that all such random curves which describe the scaling limit of 2D lattice models at criticality are SLE curves. The upshot of this identification is that SLE then provides a description of the lattice model’s correlation functions in terms of observables, known as martingales. These martingales, roughly speaking, encode the invariants of the random curve evolution. Since their introduction, SLEs have also established deep connections with various areas of mathematics including geometry [Law10, Wang23] and Conformal Field Theory (CFT) [BB03, Car05, Pel19]. Our object of interest will be SLE martingales. We now define SLE and describe these martingales.
Consider a simply connected domain ${D}$, a random curve $\gamma$ between the points $a, b \in \partial D$ on the boundary of the domain $D$, and a measure $\mu(\gamma; {D}, a,b)$. SLE is a random curve that satisfy the following two properties:
1. Conformal invariance: Choose two simply connected domains ${D}$ and ${D}'$, and the boundary points $a, b\in \partial D$, and $a', b' \in \partial D'$. By Riemann mapping theorem, there exists a map $\Phi: D\to D'$, such that $\Phi(a) =a'$, and $\Phi(b) = b'$. Then the measure
\begin{align}
(\Phi \backslash\mu)(\gamma; D, a, b) = \mu \left(\Phi(\gamma); D', a', b' \right).
\end{align}
2. Domain Markov property: Let $\gamma = \gamma_1 \cup \gamma_2$. Then the conditional measure
$$ \mu(\gamma_2 \mid \gamma_1; D, a, b) = \mu(\gamma_2; D \setminus \gamma_1, a, b)$$
The above statements effectively state that the evolution of the random curve depends only on the initial point of the curve.
Let us now set the domain to be the upper half-plane $D= \mathbb{H}$, and understand the evolution in time $t$ of the curve $\gamma$. Consider a time dependent map $g_t(x)$, $x\in \mathbb{H}$ which maps the curve $[0, \gamma(t)]$ to the real line with the normalisation $g_t(x) \sim x + \frac{2t}{x}+\mathcal{O}(x^2)$ as $x\to \infty$. Then, we find that $\gamma(t)$ should be the Brownian motion with the 'driving function' $Z_t = \sqrt{\kappa} B_t$, where $\kappa$, loosely speaking, parametrises the speed of the growth. Furthermore, $g_t(x)$ solves the differential equation, called the Lowner equation [Loe23]
\begin{align}\label{eq:Lowner}
d g_t(x) = \frac{2}{g_t(x)-Z_t} dt. \,\,\,\,\,\,\,\,\,\,\,\,\, (4)
\end{align}
With the above, let us recall Ito formula for a function of the driving function and $g_t(x)$
\begin{align}
d f(Z_t, g_t) = \sqrt{\kappa} \partial_{Z_t} f dZ_t + \partial_{g_t} dg_t+ \frac{\kappa}{2} \partial_{Z_t}^2 f dt, \label{Id:Ito} \,\,\,\,\,\,\,\,\,\,\,\,\, (5)
\end{align}
where the $dt$ dependent term coming from $\partial_{g_t} dg_t+ \frac{\kappa}{2} \partial_{Z_t}^2 f dt$ in the above expression is called the drift term. The last term comes from the quadratic variation $<Z_t, Z_t>=\kappa<B_t, B_t> = \kappa t$, and $<g_t, g_t>=0$. Finally, heuristically, a martingale $M_t$ is a stochastic process whose expected future value, given all available past information, is equal to its current value. Specifically, $\frac{d M_t}{M_t}$ has no drift term.
3. From SLE martingales to the tau-function
In our article, we focus on the following observation from the paper of Dubedat. Let us consider a stochastic equivalent of the system of IDEs (1), (2), i.e. $(z, \lambda_i)\to (Z_t, \Lambda_t^{(i)})$, where $\Lambda_t^{(i)} = g_t(\lambda_i)$ and the evolution of these stochastic deformation parameters is given by the Lowner equation (4).
The martingale, for $\kappa=4$ then has the form
\begin{align}
M_{t}(Z_t, \Lambda_t^{i}) = \prod_{i=1}^{n} |g_t'(\lambda_i)|^{\text{tr} A_i^2/2} \tau(\Lambda_t^i) Y(Z_t, \Lambda_i).\,\,\,\,\,\,\,\,\,\,\, (6)
\end{align}
A quick check is to see that there is no drift $dt$ term in the expression $\frac{d M_t}{M_t}$. This follows from substituting the stochastic equivalents of (1) and (2). In the paper [Dub14], the author uses the above construction to then study correlation functions for Conformal loop ensembles and double dimers.
The result above is inspiring for the following reasons. Firstly, the tau-function naturally comes out of the stochastic derivative of the solution $Y$, particularly through the quadratic variation term in (5). Secondly, the prefactor $\prod_{i=1}^{n} |g_t'(\lambda_i)|^{\text{tr} A_i^2/2}$ in (6) coincides with the conformal transformation of the tau-function known in the literature [GIL12].
4. Irregular martingales
The results of the previous section are limited to $SL_2(\mathbb{C})$ valued IDEs on a punctured Riemann sphere. However, these make up only the simplest case of IDEs. One may consider IDEs with higher order singularities, i.e. instead of (1), we would have
\begin{align}
\frac{dY}{dz} = \left(\sum_{i=1}^n \sum_{k=1}^{r_i} \frac{A_{i, k}}{(z-\lambda_i)^k} \right).\label{eq:linsys-irr} \,\,\,\,\,\,\,\,\,\,\,\,\, (7)
\end{align}
As opposed to the case of simple poles in (1), the above equation depends on higher order singularities, with the highest order of pole for a singularity $\lambda_i$ denoted by $r_i$. The deformation parameters of the solution to the equation (6) then come not only from the positions of the poles $\lambda_i$, but also the eigenvalues of the matrices $A_{i,k}$ for $k\geq 2$. These additional parameters are called Birkhoff invariants [BJL79], which we denote by $s_{i,k}$.
As one may anticipate, the geometry associated to (7) is exceptionally rich, while requiring novel tools for its analysis. In our upcoming paper [DKP26], we derive the stochastic evolution of these Birkhoff invariants.
Theorem 1.
Let $(g_t)_{t \geq 0}$ be the Loewner flow with driving function $(Z_t)_{t \geq 0}$. Consider the Loewner evolution $S_t^{i,k} := \hat{g}_t(s_{i,k})$ of the Birkhoff invariant $s_{i,k}$ associated to the differential equation (7) corresponding to the pole $\lambda_i$ of order $k$. Then, we have
\begin{align}
S_t^{i,k} = s_{i,k} \exp \left( -\int_0^t \frac{2 k du}{(\Lambda_u^i - Z_u)^2} \right), \,\,\,\,\,\,\,\,\,\,\,\,\, (8)
\end{align}
or equivalently, $\partial_t S_t^{i,k} = -\frac{2 k S_t^{i,k}}{(\Lambda_t^i-Z_t)^2} $, where $\Lambda_u^i = g_u(\lambda_i)$ represents the time-evolution of the singularities $\bf\lambda=(\lambda_1,\ldots,\lambda_n)$.
With the above evolution, our second result is that we construct the following irregular martingale when $r_i=2$ for $i=1,...,n$.
Theorem 2.
Let $(g_t)_{t \geq 0}$ be the Loewner flow corresponding to a chordal SLE$_4$ in $(\mathbb{H}, 0,\infty)$. Then, the following process is a matrix-valued local martingale
\begin{align}
M_t :=\left(\prod_{i=1}^{n} (g_t'(\lambda_i))^{\alpha_i^2} \exp \left(\frac{s_i^2}{6} \mathscr{S}(g_t)(\lambda_i) + s_i \alpha_i \mathscr{A}(g_t)(\lambda_i) \right)\right)
\tau({\bf\Lambda_t}; {\bf S_t}) Y(Z_t, {\bf\Lambda_t}; {\bf S_t}) , \,\,\,\,\,\,\,\,\,\,\,\,\, (9)
\end{align}
where $Z_t = 2 B_t$ is the Loewner driving function, $\bf\Lambda_t = (\Lambda_t^1, \ldots,\Lambda_t^{n})$ are the time-evolutions $\Lambda_t^i = g_t(\lambda_i)$ of the punctures and ${\bf S_t} = (S_t^1, \ldots,S_t^{n})$ the time-evolutions $S_t^i = g_t(s_i)$ of the associated Birkhoff invariants, and where $\mathscr{A} (g) := g''/g'$ is the pre-Schwarzian derivative and $\mathscr{S}(g) :=(g''/g')' - \frac12 (g''/g')^2$ is the Schwarzian derivative of a function $g$.
A few remarks on the above results follow.
1. Note that the analytic structure of the stochastic evolution of the Birkhoff invariants (8) is independent of the rank $r_i$ or the order of the singularity. This indicates that this expression is fundamental to the geometry of the equation (7), which is the moduli space of Riemann sphere with higher order punctures.
2. The pre-factor in the martingale (9): $\bigg(\prod_{i=1}^{n} (g_t'(\lambda_i))^{\alpha_i^2} \exp \bigg(\frac{s_i^2}{6} \, \mathscr{S}(g_t)(\lambda_i) + s_i \alpha_i \, \mathscr{A}(g_t)(\lambda_i) \bigg)\bigg) $ precisely dictates the conformal transformation of the tau-function associated to (7). This expression hints at a deeper link between the martinagales and hyperbolic geometry through Schwarzians and higher Schwarzians.
3. Certain correlation functions of CFTs solve a PDE called the Belavin-Polyakov-Zamolodchikov (BPZ) equation. We can show that the martingale is related to a solution of a BPZ equation, thereby establishing a connection with CFTs. Moreover, we prove previously conjectured behaviour of the correlation functions under conformal transformations, which is precisely given by the pre-factor in (9).
4. With the above results, we hope to study new structures in lattice models with the presence of higher order poles.
Dr Harini Desiraju is a Research Fellow here in Oxford Mathematics.
References
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[DKP26] Desiraju, H., Korzhenkova, A. and Peltola, E, 2026. Irregular SLE$_4$ martingales and isomonodromic deformations. In preparation.
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