Correction to: The Dependence on the Monodromy Data of the Isomonodromic Tau Function

1 Correction to: Commun. Math. Phys. https://doi.org/10.1007/s00220-009-0961-7

2 Introduction

With this note I wish to correct two mistakes of loc. cit.; one of them (and its consequences) is of trivial nature and it is quickly disposed of. The second is significant, but not fatal, and requires rather extensive additional material, which is the reason of the relatively large size of this erratum.

I also wish to thank Oleg Lisovyy for pointing out certain inconsistencies that convinced me that indeed there was a mistake in need of fixing. The mistake was observed during the preparation of [8] and [9] because it was not consistent with their explicit computations: this prompted one of the authors to contact me.

The correction only affects certain types of setup; if the contours supporting the jumps of the Riemann–Hilbert Problem do not intersect, then the result is correct as it stands. In the subsequent works of my collaborators and myself only this type of problems were actually considered and therefore the following papers are essentially unaffected (although any reporting of the original formula is not correct in the stated generality): [2,3,4,5,6].

3 Minimal Setup

The original setup requires to consider a Riemann–Hilbert Problem (RHP) with the following data (here reformulated with greater detail than in the original paper)

The Riemann–Hilbert data

• 1. a finite collection of smooth oriented arcs $\gamma _\nu ,\ \nu =1\dots K$, possibly meeting at a finite number of points but always in non-tangential way. We denote collectively these arcs by the symbol $\displaystyle \Sigma \gamma = \bigcup \gamma _\nu$.

• 2. a collection of $r\times r$ matrices $M_\nu (z)$, each of which analytic at each interior point of its corresponding arc $\gamma _\nu$. We will denote collectively by M(z) the matrix defined on $\Sigma \gamma$ that coincides with $M_\nu (z)$ on $\gamma _\nu$,

  $$\begin{aligned} M: \Sigma \gamma\rightarrow & {} SL_r({{\mathbb {C}}})\nonumber \\ z\mapsto & {} \sum _{\nu } M_\nu (z) \chi _{\gamma _\nu }(z) \end{aligned}$$

  (2.1)

  where, for a set S, $\chi _S$ denotes its indicator function.

• 3. At each point c where several arcs meet, denoting by $\gamma _1,\dots , \gamma _\ell$ the arcs entering a suitably small disk at c, we require that the arcs approach c along distinct, well-defined directions and we impose that the jump matrices along its corresponding arc either

  • admit local analytic extension within said disk. In this case, if we denote by $\gamma _1,\dots , \gamma _n$ the contours incident at c, oriented outwards, and labelled counterclockwise, and $M_\ell (z) = M(z)\bigg |_{z\in \gamma _\ell }$, we require that the aforementioned analytic extensions satisfy

    $$\begin{aligned} M_1(z)\cdot M_2(z) \cdots M_n(z) \equiv \mathbf{1}, \end{aligned}$$

    (2.2)

    and this equality holds (locally) identically also with respect to the deformation parameters. Such an intersection point will be referred to as “essential" later on, for lack of better word.

  • tend to the identity matrix as ${\mathcal {O}}((z-c)^\infty )$ (faster than any power) in an open sector containing the direction of approach (this applies also to any jump matrix on contours extending to infinity, where $(z-c)$ is replaced by 1/z) and admit analytic continuation on the universal cover of the punctured disk around c. Such an intersection point will be referred to as “inessential".

Problem 2.1

(RHP). Find a holomorphic matrix $\Gamma :{{\mathbb {C}}}{\setminus } \Sigma \gamma \rightarrow GL_n({{\mathbb {C}}})$ such that

• $\Gamma _+(z) = \Gamma _-(z) M(z)$ $z\in \Sigma \gamma$;

• $\Gamma (z),\ \Gamma ^{-1}(z)$ are uniformly bounded in ${{\mathbb {C}}}$;

• $\Gamma (z_0)=\mathbf{1}$

Assuming that the solution exists for given initial data, [1] considered the deformations of the jump matrices (respecting the conditions listed above).

Remark 2.1

The conditions on the jump matrices laid out above ensure that the solution $\Gamma (z)$ admits analytic continuation in a neighbourhood of the intersection point c, or at least in the open sector around the direction of approach mentioned above. In this latter case the decay condition guarantees that the solution admits an asymptotic expansion near c in the same sector, and that the expansion coefficients do not depend on the sector. The conditions are modelled upon the case of RHPs associated to rational ODEs in the complex plane.

3.1 Corrections

The overall minus sign in (2.7) of Def. 2.1 [1] should be removed. While this is a definition, the purpose was to extend the Jimbo-Miwa-Ueno definition, and the correct sign should have been the opposite one. For convenience, here is the corrected definition. Of course the sign should be changed also in the subsequent formulæ.

Definition 2.1

(Def 2.1 in [1]) Let $\partial$ denote the derivative w.r.t. one of the parameters s and assume that the Riemann–Hilbert Problem 2.1 admits a solution in an open subset of the deformation-parameter space.¹ Then we define Malgrange’s form $\omega _{_M}$

$$\begin{aligned} \omega _{_M}(\partial )= & {} \omega _{_{M}}(\partial ; [\Gamma ]) := \int _{\Sigma \gamma } \mathrm {Tr}\bigg (\Gamma ^{-1}_-(z) \Gamma _-'(z) \Xi _\partial (z)\bigg ) \frac{{\mathrm{d }}z}{2i{\pi }}\nonumber \\ \Xi _\partial (z):= & {} \partial M(z) M^{-1}(z)\ . \end{aligned}$$

(2.3)

New notation In order to deal more expeditiously with the correction we shall also use the matrix-valued forms (Maurer-Cartan like) $\Xi (z):= \delta M(z) M^{-1}(z)$, where $\delta$ shall denote henceforth the exterior derivative with respect to the deformation parameters $\mathbf {t}$ (not to confuse it with $\,\mathrm dz$ of the spectral variable). We shall also retain the notation $\Xi _\partial$ for the contraction of said form with a vector field $\partial$.

Proposition 2.1 in [1] offers an incomplete formula for the exterior derivative of $\omega _{_{M}}$ and we correct it now. The additional term in the following Theorem is present only when there are points of $\Sigma \gamma$ with several incident arcs; we call this the “set of vertices” of $\Sigma \gamma$ and denote it by ${\mathfrak {V}}$. If $\Sigma \gamma$ consists in the union of smooth disjoint arcs, or all the jump matrices tend to the identity at all the vertices, then the original statement stands correct.

Theorem 2.1

(Replaces Prop. 2.1 of [1]). Denote by ${\mathfrak {V}}\ni v$ the vertices of the graph $\Sigma \gamma$; let ${\mathcal {E}}_v = \bigcup _{j=1}^{n_v} \gamma _j^{(v)}$ be the set of arcs incident to v, oriented outwards and enumerated counterclockwise. Then exterior derivative of $\omega _{_{M}}$ is

$$\begin{aligned} \delta \omega _{_{M}} = - \frac{{1}}{2} \int _{\Sigma \gamma } \frac{{\mathrm{d }}z}{2i{\pi }}\mathrm {Tr}\bigg ( \frac{\,\mathrm d}{\,\mathrm dz} \Xi (z) \wedge \Xi (w) \bigg )\bigg |_{w=z} + \eta _{_{{\mathfrak {V}}}} \end{aligned}$$

(2.4)

with

$$\begin{aligned} \eta _{_{{\mathfrak {V}}}}:= & {} \frac{\mathbf -1}{4i\pi }\sum _{v\in {\mathfrak {V}}} \sum _{\ell =2}^{n_v} \sum _{m=1}^{\ell -1} \mathrm {Tr}\bigg ( M_{[1:m-1]}^{(v)} \Xi ^{(v)}_{m}M^{(v)}_{[m:\ell -1]}\wedge \Xi ^{(v)}_{\ell } M^{(v)}_{[\ell :n_v]} \bigg ) \end{aligned}$$

(2.5)

$$\begin{aligned} \Xi ^{(v)}_\ell= & {} \lim _{z\rightarrow v} \delta M_{\ell }^{(v)} (M_{\ell }^{(v)})^{-1}\bigg |_{z\in \gamma _{\ell }^{(v)}}\qquad M_{\ell }^{(v)} = \lim _{z\rightarrow v\atop z\in \gamma _{\ell }^{(v)}} M^{\epsilon _\ell } (z). \end{aligned}$$

(2.6)

where the power $\epsilon _\ell = 1$ if the contour $\gamma _\ell ^{(v)}$ is oriented away from v and $\epsilon _\ell = -1$ if oriented towards. Here the subscript $_{[m:\ell -1]}$ is a shorthand to signify the product of the corresponding matrices over the range of indices $m,m+1,\dots , \ell -1$.

The complete proof is reported in Sect. 3. The "modified Malgrange form" $\Omega$ (Def. 2.2 in [1]) is (with the corrected sign)

Definition 2.2

The modified Malgrange differential is defined as $\Omega := \omega _{_M} +\vartheta$ with

$$\begin{aligned} \vartheta (\partial ):= \frac{{1}}{2} \int _{\Sigma \gamma } \mathrm {Tr}\left( M'(z)M^{-1}(z) \partial M(z) M^{-1}(z)\right) \frac{{\mathrm{d }}z}{2i{\pi }} \end{aligned}$$

(2.7)

Equivalently (see (2.32) in [1])

$$\begin{aligned}&\Omega (\partial ;[\Gamma ]) \nonumber \\&\quad =\frac{{1}}{2} \int _{\Sigma \gamma } \mathrm {Tr}\left( \Gamma _-^{-1} (z)\Gamma _-' (z)\partial M(z) M^{-1}(z) +\Gamma _+^{-1}(z) \Gamma _+'(z) M^{-1}(z) \partial M (z) \right) \frac{{\mathrm{d }}z}{2i{\pi }}\nonumber \\ \end{aligned}$$

(2.8)

Consequent to the correction of Prop. 2.1, the ancillary result below (Prop. 2.2 in [1]) is also similarly modified

Proposition 2.1

The curvature of the modified Malgrange form is

$$\begin{aligned} \delta \Omega =- \frac{{1}}{2} \int _{\Sigma \gamma } \mathrm {Tr}\bigg (M'(z) M^{-1}(z) \Xi (z) \wedge \Xi (z)\bigg ) \frac{{\mathrm{d }}z}{2i{\pi }}+ \eta _{_{{\mathfrak {V}}}}\end{aligned}$$

(2.9)

3.2 Rational differential equations (amended)

In the setup of Sec. 2.2, the term $\eta _{_{{\mathfrak {V}}}}$ is closed and admits a potential $\theta _{_{{\mathfrak {V}}}}$; consequently in Sec. 2.2 of [1] the title should read: "Submanifolds of ${\mathcal {G}}$ where $\delta \Omega - \eta _{_{{\mathfrak {V}}}}=0$". Sections 2.3, 2.4 are unaffected.

The main application of the original paper was to Riemann–Hilbert problems related to the setting of [10], i.e. the generalized monodromy data associated to a (generic) rational connection on ${{\mathbb {C}}}{\mathbb {P}}^1$.

The statement that $\Omega$ is a closed one-form is incorrect in the stated generality and needs to be corrected.

To explain the necessary modifications we keep the same setup of Sections 3,4,5 (and Fig. 5, 6 of [1]).

Theorem 2.2

(Replaces Thm. 5.1 in [1]) There exists a locally defined one form $\theta _{_{{\mathfrak {V}}}}= \theta _{_{{\mathfrak {V}}}}(\mathbf {L}, \mathbf {S}, \mathbf {C})$ on the manifold of generalized monodromy data (independent of the Birkhoff invariants and the position of the poles) such that the Jimbo–Miwa–Ueno tau function satisfies

$$\begin{aligned} \delta \ln \tau (\mathbf {T}, \mathbf {a}, \mathbf {L}, \mathbf {S}, \mathbf {C}) = \omega _{_M} - \theta _{_{{\mathfrak {V}}}}, \end{aligned}$$

(2.10)

where $\delta \theta _{_{{\mathfrak {V}}}}= \eta _{_{{\mathfrak {V}}}}$ in (2.52.6). This function is defined up to nonzero multiplicative constant and it vanishes precisely when the Riemann–Hilbert problem is not solvable, namely, on the Malgrange Theta-divisor.

Remark 2.2

Note that the theorem is now stated directly in terms of $\omega _{_{M}}$ rather than the "modified" form $\Omega$ used in the original paper. The two forms differ by an explicit one form so there is little simplification in choosing one over the other, since neither is closed by itself in the relevant case.

Remark 2.3

Since $\theta _{_{{\mathfrak {V}}}}$ is only locally defined on the monodromy manifold, the formula (2.10) allows to identify $\tau$ as a section of a line bundle on said manifold. The transition functions are given by $\delta \ln g = \widetilde{\theta }_{_{{\mathfrak {V}}}}- \theta _{_{{\mathfrak {V}}}}$ on the overlap of two open charts. This observation, which stems from the correction term in Thm. 2.1 seems to be of interest in applications that are arising from recent works [8] and deserves further study.

Rather than chasing a complete generality we illustrate the statement in several significant cases.

Fig. 1

The arrangement of disks for the scalar case

3.3 Example 0: Scalar Fuchsian case

Consider the scalar RHP (see Fig. 1)

$$\begin{aligned} \Gamma (z)= & {} \left\{ \begin{array}{ll} \prod _{j=1}^n \left( z-a_j\right) ^{\theta _j} &{} {{\mathbb {C}}}{\setminus } \bigcup {\mathbb {D}}_j\\ \prod _{j=1, j\ne k}^n \left( z-a_j\right) ^{\theta _j} &{} z\in {\mathbb {D}}_k\\ &{}\sum \theta _j =0. \end{array} \right. \end{aligned}$$

(2.11)

$$\begin{aligned} \Gamma _-^{-1} \Gamma _+= & {} \left\{ \begin{array}{ll} (z-a_k)^{-\theta _k} &{} z\in \gamma _k = \partial {\mathbb {D}}_k\\ \mathrm{e}^{-2i\pi \theta _k} &{} z\in \gamma _k^0 \end{array} \right. \end{aligned}$$

(2.12)

where $\gamma ^0_k$ is a contour $[\beta _k, z_0]$, with $\beta _k$ chosen and fixed on $\gamma _k$ and $z_0$ a fixed basepoint outside, chosen in such a way that no two points $a_j$ are not on the same ray from $z_0$. The Malgrange one-form is

$$\begin{aligned} \omega _{_M}= & {} \int \Gamma _{-}^{-1} \Gamma _-' \delta M M^{-1} \frac{{\mathrm{d }}z}{2i{\pi }}\nonumber \\= & {} \sum _{j=1}^n \int \frac{\theta _j}{z-a_j} \sum _{k}\left( \left( \frac{ \theta _k \,\mathrm da_k }{z-a_k} - \ln (z-a_k)\,\mathrm d\theta _k \right) \chi _{_{\partial {\mathbb {D}}_k}} - {2i\pi }\,\mathrm d\theta _k \chi _{_{\gamma ^0_k}}\right) \nonumber \\ \frac{{\mathrm{d }}z}{2i{\pi }}= & {} \hbox {(contour deformation)}\nonumber \\= & {} \sum _{j=1}^{n}\left[ \sum _{k\ne j}\left( \frac{ \theta _j \theta _k \,\mathrm da_k}{a_k- a_j} -\theta _j \,\mathrm d\theta _k \int _{a_k}^{z_0} \frac{\,\mathrm dz }{z-a_j} \right) - \frac{1}{4i\pi } \theta _j\,\mathrm d\theta _j {\ln ^2} (z-a_j)\bigg |_{\beta _j}^{\beta _j + \gamma _j}\right. \nonumber \\&\left. - \theta _j \,\mathrm d\theta _j \int _{\beta _j}^{z_0} \frac{\,\mathrm dz}{(z-a_j)}\right] \nonumber \\= & {} \sum _{j=1}^{n}\Bigg [ \sum _{k\ne j}\left( \frac{ \theta _j \theta _k \,\mathrm da_k}{a_k- a_j} -\theta _j \,\mathrm d\theta _k \big (\ln (z_0-a_j) - \ln (a_k-a_j)\big ) \right) \nonumber \\&+\, -\frac{1}{4i\pi } \theta _j\,\mathrm d\theta _j (4i\pi \ln (\beta _j - a_j) + (2i\pi )^2) - \theta _j \,\mathrm d\theta _j (\ln (z_0-a_j) - \ln (\beta _j - a_j))\Bigg ]\nonumber \\ \end{aligned}$$

(2.13)

Here all logarithms are principal; the term involving $z_0$ drop out because $\sum \,\mathrm d\theta _k =0$, as well as the dependence on $\beta _j$. We are left with

$$\begin{aligned} \omega _{_{M}} = \sum _{j=1}^{n}\Bigg [ \sum _{k\ne j}\left( \frac{ \theta _j \theta _k \,\mathrm da_k}{a_k- a_j} +\theta _j \,\mathrm d\theta _k \ln (a_k-a_j)\right) - i\pi \theta _j\,\mathrm d\theta _j\Bigg ] \end{aligned}$$

(2.14)

Here the logarithms are all principal. The exterior derivative of the above expression is

$$\begin{aligned} \delta \omega _{_{M}}= & {} \eta _{_{{\mathfrak {V}}}}= \sum _j\sum _{k\ne j} \,\mathrm d\theta _j\wedge \,\mathrm d\theta _k(\ln (a_k-a_j)\nonumber \\&-\, \ln (a_j-a_k)) ={ i\pi \sum _j\sum _{k< j} \,\mathrm d\theta _j\wedge \,\mathrm d\theta _k = i\pi \,\mathrm d\left( \sum _{j}\sum _{k<j} \theta _j \,\mathrm d\theta _k\right) }\nonumber \\ \end{aligned}$$

(2.15)

In this case the Tau function is explicit

$$\begin{aligned} \delta \ln \tau = \omega _{_{M}} +i\pi \sum _{k<\ell } \theta _k \,\mathrm d\theta _\ell \ ,\ \ \ \tau (\mathbf {a}, \mathbf {\theta }) = \prod _{\ell =1}^n\prod _{k<\ell } (a_k-a_\ell )^{\theta _k\theta _\ell } \prod _{k=1}^n \mathrm{e}^{-\frac{i\pi }{2} \theta _k^2} \end{aligned}$$

(2.16)

To be noted, there is an ambiguity in the above writing because of the determinations of the logarithm; the ambiguity is what defines the line bundle of which $\tau$ is a section.

3.4 Fuchsian singularities with nontrivial monodromy (e.g. Painlevé VI)

Fig. 2

The arrangement of jumps for a generic Fuchsian system

Suppose that the RHP corresponds to the solution of a generic Fuchsian ODE with simple poles at $a_1,\dots , a_K$ of the form

$$\begin{aligned} \Psi '(z) = \sum _{j=1}^K\frac{A_j}{z-a_j} \Psi (z)\ ,\ \ \ A_j = O_j L_j O_j^{-1}, \ \ \ L_j = \hbox { diagonal }. \end{aligned}$$

(2.17)

We set $\Lambda _j = \mathrm{e}^{2i\pi L_j}$ (diagonal) and the monodromy matrices are ${\mathcal {M}}_j := C_j^{-1} \Lambda _j C_j$. The enumeration is counterclockwise from the basepoint $z_0$ as indicated in Fig. 2. We have the condition

$$\begin{aligned} {\mathcal {M}}_1 \cdots {\mathcal {M}}_K = \mathbf{1} \end{aligned}$$

(2.18)

Then a direct computation using Thm. 2.1 yields

$$\begin{aligned} \delta \omega _{_{M}}= & {} \frac{-1}{4i\pi } \Bigg ( \sum _{\ell =2}^K \sum _{1\le \ell < k} \mathrm {Tr}\bigg ( {\mathcal {M}}_{{[1:\ell -1]}} \delta {\mathcal {M}}_{\ell } {\mathcal {M}}_{{[\ell +1: k-1]}}\wedge \delta {\mathcal {M}}_{{k}}{\mathcal {M}}_{{[ k+1:K]}} \bigg ) \nonumber \\&+\, \sum _{\ell =1}^K \mathrm {Tr}\bigg ( \Lambda _\ell \delta C_{\ell }C_{\ell } ^{-1}\wedge \Lambda _\ell ^{-1} \delta C_\ell C_\ell ^{-1}+2 \Lambda _\ell ^{-1} \delta \Lambda _\ell \wedge \delta C_\ell C_\ell ^{-1} \bigg ) \Bigg ) \end{aligned}$$

(2.19)

As announced in Thm. 2.2, $\delta \omega _{_{M}}$ is independent of the poles’ positions. It is a closed two-form on the monodromy variety (2.18) itself. It is not immediate to verify directly from the formula that the two-form in (2.19) is actually closed, but it is a consequence of the Theorem 2.1.²

On the other hand, since it is a closed two-form on the monodromy variety (2.18), it follows that there is a locally defined one-form on the monodromy variety, which we shall denote $\theta _{_{{\mathfrak {V}}}}$, such that $\delta \theta _{_{{\mathfrak {V}}}}= \delta \omega _{_M}$.

We remark for the reader that the left hand side of (2.19) is the result of a partial cancellation of terms between the two terms in (2.4).

Example 2.1

The simplest example of Painlevé VI requires to describe explicitly the one-form $\theta _{_{{\mathfrak {V}}}}$.

We shall assume that the monodromies ${\mathcal {M}}_{1,..4}$ are non-resonant (i.e. the eigenvalues of $L_j$ do not differ by integers). This, however, proves to be too complicated to handle explicitly in complete generality, so we consider, by the way of example, the following particular submanifold of (2.18);

$$\begin{aligned} C_1 = \left[ \begin{array}{cc} 1&{} s_1\\ 0&{}1 \end{array} \right] ,\ \ C_2 = \left[ \begin{array}{cc} 1&{} 0\\ s_2&{}1 \end{array} \right] ,\ \ C_3 = \left[ \begin{array}{cc} 1&{} s_3\\ 0&{}1 \end{array} \right] ,\ \ C_4 = \left[ \begin{array}{cc} 1&{} 0\\ s_4&{}1 \end{array} \right] \end{aligned}$$

(2.20)

We set $\Lambda _j = \mathrm{diag}(\lambda _j, \lambda _j^{-1})$; then we can solve the condition (2.18) for $\lambda _4, s_2, s_4$ on a suitable open subset of the above submanifold of the monodromy variety

$$\begin{aligned}&\bigg \{ \lambda _{{4}}=-{\frac{s_{{3}}\lambda _{{1}}\lambda _{{2}} \left( \lambda _{{3}}^2-1 \right) }{ \lambda _{{3}}s_{{1}} \left( \lambda _{{1}}^2-1 \right) }},\ \ s_{{2}}={\frac{s_{{3}}{\lambda _{{1}}}^{2}{\lambda _{{ 2}}}^{2}{\lambda _{{3}}}^{2}-s_{{3}}{\lambda _{{1}}}^{2}{\lambda _{{2}}}^ {2}+{\lambda _{{1}}}^{2}s_{{1}}-s_{{1}}}{s_{{3}}s_{{1}} \left( \lambda _ {{3}}^2-1 \right) \left( \lambda _{{2}}^2- 1 \right) \left( \lambda _{{1}}^2-1 \right) }}, \nonumber \\&s_{{4}}=-{\frac{ \left( s _{{3}}{\lambda _{{1}}}^{2}{\lambda _{{2}}}^{2}{\lambda _{{3}}}^{2}-s_{{3} }{\lambda _{{1}}}^{2}{\lambda _{{2}}}^{2}+{\lambda _{{1}}}^{2}s_{{1}}-s_{ {1}} \right) {\lambda _{{3}}}^{2}}{ \left( {\lambda _{{1}}}^{2}\lambda _{ {3}}s_{{1}}-\lambda _{{3}}s_{{1}}+s_{{3}}\lambda _{{1}}\lambda _{{2}}-s_{ {3}}\lambda _{{1}}\lambda _{{2}}{\lambda _{{3}}}^{2} \right) \left( { \lambda _{{1}}}^{2}\lambda _{{3}}s_{{1}}-\lambda _{{3}}s_{{1}}+s_{{3}} \lambda _{{1}}\lambda _{{2}}{\lambda _{{3}}}^{2}-s_{{3}}\lambda _{{1}} \lambda _{{2}} \right) }} \bigg \}\nonumber \\ \end{aligned}$$

(2.21)

so that the monodromy variety is now locally coordinatized $\lambda _1,\lambda _2,\lambda _3, s_1,s_3$. Then the two form $\delta \omega _{_{M}}$ is

$$\begin{aligned} \delta \omega _{_{M}}= & {} {\frac{( {\lambda _{{1}}}^{2}+1 )d\lambda _1\wedge d\lambda _2 }{2i\pi \lambda _{{2}} \left( {\lambda _{{1}}}^{2}-1 \right) \lambda _{{1}}}}\nonumber \\&+\, \frac{({\lambda _{{1}}}^{2}{\lambda _{{3}}}^{2}+{\lambda _{{3}}}^{2}+{\lambda _{{1}}}^{2}+1) d\lambda _1\wedge d\lambda _3 }{ 2i\pi \left( {\lambda _{{3}}}^{2}-1 \right) \lambda _{{3}}\lambda _{{1}} \left( {\lambda _{{1}}}^{2}-1 \right) } - {\frac{ d\lambda _1\wedge ds_1}{2i\pi \lambda _{{1}}s_{{1}}}} + \frac{({\lambda _{{1}}}^{2}+1) d\lambda _1\wedge ds_3}{2i\pi \lambda _{{1}}s_{{3}} \left( {\lambda _{{1}}}^{2}-1 \right) }\, +\nonumber \\&-\,{\frac{({\lambda _{{3}}}^{2}+1) d\lambda _2\wedge d\lambda _3 }{2i\pi \lambda _{{2}}\lambda _{{3}} \left( {\lambda _{{3}}}^{2}-1 \right) }} - {\frac{ d\lambda _2\wedge ds_1}{2i\pi s_{{1}}\lambda _{{2}}}}\nonumber \\&-\,{\frac{d\lambda _2\wedge ds_3}{2i\pi s_{{3}}\lambda _{{2}}}} - {\frac{({\lambda _{{3}}}^{2}+1)d\lambda _3\wedge ds_1}{2i\pi s_{{1}} \left( {\lambda _{{3}}}^{2}-1 \right) \lambda _{{3}}}} - {\frac{ d\lambda _3\wedge ds_3}{2i\pi s_{{3}}\lambda _{{3}}}} +{\frac{ ds_1\wedge ds_3}{2i\pi s_{{1}}s_{{3}}}} \end{aligned}$$

(2.22)

and then a direct computation using the DeRham homotopy operator (after checking that the form above is indeed closed), we obtain

$$\begin{aligned} \theta _{_{{\mathfrak {V}}}}= & {} {\frac{ \left( { \lambda _{{3}}}^{2}\ln \left( s_{{1}} s_3 \right) +\ln \left( \frac{ s_{{1}}}{s_3} \right) \right) \!\! { \,\mathrm d\lambda }_{{3}}}{2i\pi \lambda _{{3}} \left( { \lambda _{{3}}}^{2}-1 \right) }} \\&-\, {\frac{ \ln \left( s_{{3}} \right) \! {\,\mathrm ds}_{{1}}}{2i\pi s_{{1}} }} \\&+\, {\frac{ \left( \ln \left( \frac{s_{{3}} s_1 (\lambda _3^2-1)}{\lambda _3}\right) \right) \!\!\,\mathrm d\lambda _{{2}}}{2i\pi \lambda _{{2}}}} - {\frac{ \left( {\lambda _{{1}}}^{2}\ln \left( \frac{ s_3(\lambda _{{3}}^2-1 )\lambda _2}{s_1 \lambda _3}\right) +\ln \left( \frac{s_3 s_{{1}} \lambda _2(\lambda _3^2-1)}{\lambda _3}\right) \right) \!\! { \,\mathrm d\lambda }_{{1 }}}{2i\pi \left( {\lambda _{{1}}}^{2}-1 \right) \lambda _{{1}}}} \end{aligned}$$

3.5 Higher Poincaré rank singularities: the case of Painlevé II

If the system has also poles of higher order (under the same original genericity assumption that the leading coefficient matrix of the singular part of the connection is semi-simple), then the corresponding RHP has additional contours of jumps to account for the Stokes’ phenomenon. In view of the correction in Thm. 2.1 we slightly modify their definition within the “toral circle” (Fig. 5 in [1]) as in Fig. 3. We consider the case of only one singularity of higher Poincaré rank for clarity. Supposing that $S_1,\dots S_{2r}$ are the Stokes’ matrices and L the diagonal matrix of the exponents of formal monodromy at a the singularity, they must satisfy

$$\begin{aligned} S_1\cdots S_{2r} { \mathrm e}^{2i\pi L}=\mathbf{1} \end{aligned}$$

(2.23)

Fig. 3

The contours of the RHP within a “toral circle” (in the terminology of [1]). In the figure, $M_\nu = (z-a)^L \mathrm{e}^{T(z)} S_\nu \mathrm{e}^{-T(z)} (z-a)^{-L}$, the cut of the function $(z-a)^L$ is along the blue contour, T(z) is of the form $T(z) = \sum _{j=1}^{r+1} T_j (z-a)^{-j} + T_0$ and $T_0$ is a constant diagonal matrix chosen so that $(\rho -a)^L \mathrm{e}^{T(\rho )}= \mathbf{1}$, where $\rho$ is the point on the boundary of the toral circle where the various Stokes’ contours meet

and there is a contribution to $\eta _{_{{\mathfrak {V}}}}$ in the form (we denote $S_{2r+1}= \mathrm{e}^{2i\pi L}$)

$$\begin{aligned} \eta _{_{{\mathfrak {V}}}}= \frac{-1}{4i\pi } \sum _{\ell =1}^{2r+1} \sum _{1\le k<\ell } \mathrm {Tr}\bigg (S_{[1:k-1]} \delta S_{k} S_{[k+1;\ell -1]} \wedge \delta S_{\ell } S_{[\ell +1:2r+1]}\bigg ) \end{aligned}$$

(2.24)

which is a closed two-form on the manifold (2.23). This type of contributions comes one for each higher Poincaré rank singularity. As an illustration, the example of Painlevé II is instructive. We follow the general formulation of ([11], App.C); in this case we have six rays along $\varpi _\ell = \mathrm{e}^{ (\ell -1) i\pi /3 }{{\mathbb {R}}}_+$ and $\varpi _7 = \mathrm{e}^{-i\pi / 6} {{\mathbb {R}}}_+$ with jumps

$$\begin{aligned} M_{1,3,5} = \left[ \begin{array}{cc} 1 &{}\quad s_{1,3,5}\\ 0&{}\quad 1 \end{array}\right] \ ,\ \ \ M_{2,4,6} = \left[ \begin{array}{cc} 1 &{}\quad 0\\ s_{2,4,6} &{}\quad 1 \end{array}\right] \ ,\ \ M_7 = \left[ \begin{array}{cc} \lambda &{}\quad 0\\ 0 &{}\quad \lambda ^{-1} \end{array}\right] \end{aligned}$$

(2.25)

subject to the condition (2.2). We can solve for $s_2,s_4,s_6$ in in the subset $\{s_1s_3s_5 \lambda \ne 0\}$ to give

$$\begin{aligned} \left\{ s_{{2}}=-{\frac{s_{{5}}+s_{{3}}\lambda +\lambda \,s_{{1}}}{s_{ {3}}\lambda \,s_{{1}}}},s_{{4}}=-{\frac{\lambda \,s_{{1}}+s_{{5}}+s_{{3 }}}{s_{{3}}s_{{5}}}},s_{{6}}=-{\frac{s_{{5}}+s_{{3}}\lambda +s_{{1}}{ \lambda }^{2}}{s_{{1}}{\lambda }^{2}s_{{5}}}} \right\} \end{aligned}$$

(2.26)

Then $\eta _{_{{\mathfrak {V}}}}$ becomes

$$\begin{aligned} \eta _{_{{\mathfrak {V}}}}= & {} {\frac{\,\mathrm ds_1 \wedge \,\mathrm d\lambda }{2i\pi s_{{1}}\lambda }} - {\frac{ \,\mathrm ds_3\wedge \,\mathrm d\lambda }{2i\pi \lambda \,s_{{3}}}} \nonumber \\&+\, {\frac{\,\mathrm ds_5\wedge \,\mathrm d\lambda }{2i\pi \lambda \,s_{{5}}}} - {\frac{ \,\mathrm ds_3\wedge \,\mathrm ds_1}{2i\pi s_{{3}}s_{{1}}}} + {\frac{ \,\mathrm ds_5\wedge \,\mathrm ds_1 }{2i\pi s_{{1}}s_{{5}}}} - {\frac{ \,\mathrm ds_5\wedge \,\mathrm ds_3}{2i\pi s_{{3}}s_{{5}}}} \end{aligned}$$

(2.27)

Note that $\eta _{_{{\mathfrak {V}}}}$ defines a symplectic form on the Stokes’ manifold, whereby all coordinates $\lambda ,s_1,s_3,s_5$ are log-canonical and the form $\theta _{_{{\mathfrak {V}}}}$ is

$$\begin{aligned} \theta _{_{{\mathfrak {V}}}}= & {} \ln \left( {\frac{s_{{5}}s_{{1}}}{s_{{3}}}} \right) \frac{ \,\mathrm d\lambda }{4i\pi \lambda } - \ln \left( {\frac{s_{{3}}\lambda }{s_{{5}}}} \right) \nonumber \\&\frac{ \,\mathrm ds_1}{4i\pi s_1} + \ln \left( {\frac{s_{{1}}\lambda }{s_{{5}}}} \right) \frac{ \,\mathrm ds_3}{4i\pi s_3} - \ln \left( {\frac{s_{{1}}\lambda }{s_{{3}}}} \right) \frac{ \,\mathrm ds_5}{4i\pi s_5} \end{aligned}$$

(2.28)

The special case of PII as reported in [1] consists in

$$\begin{aligned}&\left\{ \lambda =1,s_{{1}}=s_{{1}},s_{{2}}=-{\frac{s_{{3}}+s_{{1}}}{s _{{1}}s_{{3}}+1}},s_{{3}} \right. \nonumber \\&\quad \left. =s_{{3}},s_{{4}}=s_{{1}},s_{{5}}=-{\frac{s_{ {3}}+s_{{1}}}{s_{{1}}s_{{3}}+1}},s_{{6}}=s_{{3}} \right\} \end{aligned}$$

(2.29)

$$\begin{aligned}&\quad \eta _{_{{\mathfrak {V}}}}= \frac{ \,\mathrm ds_1\wedge \,\mathrm ds_3}{i\pi (s_1s_3+1)}\ ,\qquad \theta _{_{{\mathfrak {V}}}}= \frac{\ln (s_1s_3+1)}{2i\pi } \left( \frac{\,\mathrm ds_3}{s_3} - \frac{\,\mathrm ds_1}{s_1} \right) \end{aligned}$$

(2.30)

4 Proof of Thm. 2.1

Lemma 3.1

Let $\gamma$ be an oriented smooth arc without self-intersection and let $\varphi : \gamma \times \gamma \rightarrow {{\mathbb {C}}}$ be a function which is locally analytic in each variable and such that $\varphi (z,w)= - \varphi (w,z)$. Then

$$\begin{aligned} \int _\gamma \frac{{\mathrm{d }}z}{2i{\pi }}\int _\gamma \frac{\mathrm{d }w}{2i\pi }\frac{\varphi (w,z)}{(w-z_-)^2} = -\frac{{1}}{2} \int _\gamma \frac{{\mathrm{d }}z}{2i{\pi }}\partial _w \varphi (w,z)\bigg |_{w=z} . \end{aligned}$$

(3.1)

Proof

It is shown in Section 7 of [7] that if A(w, z) is Hölder (jointly) for $z,w\in \gamma$, then

(3.2)

where denotes the Cauchy’s principal value integral. We will use Sokhotski-Plemelj’s formula

(3.3)

Since $\varphi (w,z)$ is (locally) jointly analytic in z, w, we can use Gakhov’s result (3.2) with $A(w,z) := \frac{\varphi (w,z)}{(w-z)}$, which is now also a jointly analytic function of its variables in a neighbourhood of $\gamma$. Note that $A(z,z) = \partial _w \varphi (w,z)|_{w=z}$ is well defined and $A(z,w)=A(w,z)$. Now

(3.4)

We now show that the principal value integral is zero; to this end we use $A(z,w)=A(w,z)$, which holds for our case. Then

(3.5)

Thus the contribution of the principal value integral in (3.4) is zero. $\square$

Proof of Thm. 2.1

We shall denote by ${\Sigma \gamma } _\epsilon = \Sigma \gamma {\setminus } \bigcup _{v\in {\mathfrak {V}}} {\mathbb {D}}_\epsilon ^{(v)} =: \Sigma \gamma {\setminus } {\mathbb {D}}_\epsilon$ the support of the jumps minus small $\epsilon$-disks around the point of self-intersection of ${\Sigma \gamma }$; we use the notation

$$\begin{aligned} \varphi (w,z):= & {} \mathrm {Tr}\bigg (\Gamma _-(w) \Xi (w) \Gamma _-^{-1}(w)\wedge \Gamma _-(z) \Xi (z) \Gamma _-^{-1}(z) \bigg )\, \nonumber \\&\Rightarrow \ \ \ \varphi (z,w) = -\varphi (w,z). \end{aligned}$$

(3.6)

Lemma 2.1 in [1] yields the formula (in the new notation)

$$\begin{aligned} \delta \left( \Gamma _-^{-1}(z) \Gamma _-'(z)\right) = \int _{\Sigma \gamma } \frac{\mathrm{d }w}{2i\pi }\frac{\Gamma _-^{-1}(z)\Gamma _-(w) \Xi (w) \Gamma _-^{-1}(w) \Gamma _-(z)}{(z_--w)^2} \end{aligned}$$

(3.7)

whence we compute the exterior derivative as follows

$$\begin{aligned} \delta \omega _{_{M}}= & {} \delta \int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\mathrm {Tr}\left( \Gamma _-^{-1}(z) \Gamma _-'(z) \Xi (z)\right) = \end{aligned}$$

(3.8)

$$\begin{aligned}= & {} \int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{\Sigma \gamma } \frac{\mathrm{d }w}{2i\pi }\mathrm {Tr}\left( \frac{\Gamma _-^{-1}(z)\Gamma _-(w) \Xi (w) \Gamma _-^{-1}(w)\wedge \Gamma _-(z) \Xi (z)}{(z_--w)^2}\right) \nonumber \\&+\, \int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\mathrm {Tr}\left( \Gamma _-^{-1}(z) \Gamma _-'(z) \delta \Xi (z)\right) \end{aligned}$$

(3.9)

Note that $\Xi = \delta M M^{-1}$ satisfies $\delta \Xi =\Xi \wedge \Xi$ (i.e. $\delta \Xi \lfloor _{\partial ,\widetilde{\partial }} = \partial \Xi _{\widetilde{\partial }} -\widetilde{\partial }\Xi _{\partial } = [\Xi _\partial , \Xi _{\widetilde{\partial }}]$) and hence (using the cyclicity of the trace)

$$\begin{aligned} \delta \omega _{_{M}}= & {} \int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{\Sigma \gamma } \frac{\mathrm{d }w}{2i\pi }\frac{ \varphi (w,z)}{(z_--w)^2}\nonumber \\&+\, \int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\mathrm {Tr}\left( \Gamma _-^{-1}(z) \Gamma _-'(z) \Xi (z) \wedge \Xi (z) \right) \end{aligned}$$

(3.10)

The issue is the computation of the iterated integral; as an iterated integral it is convergent but its value depends on the order of integration. To see why it is convergent, we need to make sure that the inner integral does not have too severe singularities; these may occur at the intersection points of the arcs because the derivative of the Cauchy transform may have poles. For an “inessential” intersection point (where the jump matrices tend exponentially to the identity) we can excise a small disk and easily evaluate the contribution to be infinitesimal as the radius tends to zero (this procedure will be considered in more detail later on). So let us consider the issue of local convergence near an “essential" intersection point $v\in {\mathfrak {V}}$. On the incident arcs $\gamma _1,\dots , \gamma _n$ (in counterclockwise order) define

$$\begin{aligned} \Gamma _j(z) := \Gamma _-(z)\bigg |_{z\in \gamma _j}\ ,\ \ \varphi _{k j }(w,z) = \varphi (w,z) \bigg |_{z\in \gamma _j\atop w\in \gamma _k}\ \ \hbox { etc.} \end{aligned}$$

(3.11)

Without loss of generality we can assume that all arcs $\gamma _j$ are oriented away from v. If a ray is incident, then this requires us to locally re-define $\Gamma _j \mapsto \Gamma _j M_j$, $M_j \mapsto M_j^{-1}$ and reverse the orientation in the integral, thanks to the obvious formula

$$\begin{aligned} \Gamma _k\partial M_k M_k^{-1} \Gamma ^{-1}_k =- (\Gamma _kM_k)\partial (M_k^{-1}) M_k(\Gamma _kM_k)^{-1} \end{aligned}$$

(3.12)

(note that $\Gamma _- M = \Gamma _+$ becomes the − boundary value in the reversed orientation).

Under Assumption (2.2), we can locally express the analytic extensions of each $\Gamma _j(z)$ in terms of the analytic extension of $\Gamma _1(z)$ to a full neighbourhood of $v\in {\mathfrak {V}}$;

$$\begin{aligned} \Gamma _k (z)= \Gamma _\ell (z) M_{[\ell :k-1]}(z) = \Gamma _1(z)M_{[1:k-1]}(z)\ ,\ \ \ell <k. \end{aligned}$$

(3.13)

where $M_{[a:b]}(z) = M_a (z)M_{a+1} (z)\cdots M_b(z)$. Taking the differential $\delta$ of the local condition (2.2) we find (evaluation being understood at $z=v$)

$$\begin{aligned}&\sum _{j=1}^n M_{[1:j-1]} \Xi _{j} M_{[j:n]} =\sum _{j=1}^n \Gamma _1^{-1} \Gamma _j \Xi _{j}\Gamma _j^{-1} \Gamma _1 \equiv 0 \ \ \Rightarrow \ \ \sum _{j=1}^n \Gamma _j \Xi _{j}\Gamma _j^{-1} \equiv 0 \nonumber \\ \end{aligned}$$

(3.14)

$$\begin{aligned}&\quad \Rightarrow \ \ \forall \ell = 1,\dots , n\ \ \ \ \sum _{k=1}^n \varphi _{k\ell }(v,w) = \sum _{ k=1}^n \varphi _{\ell k} (z,v) \equiv 0 \end{aligned}$$

(3.15)

Now, consider the part of the inner integral along $\gamma _\ell$ for $z\not \in {\Sigma \gamma }$ near v; since $\varphi _{\ell ,m}(z,w)$ extends to a locally analytic function in the neighborhood of v, the properties of the Cauchy transform immediately imply the following local identity of analytic functions

$$\begin{aligned} J_{\ell m}(z) = \int _{\gamma _\ell } \frac{\varphi _{\ell m}(w,z)}{(w-z)^2} \frac{\mathrm{d }w}{2i\pi }= \frac{\varphi _{ \ell m}(v,v)}{z-v} + \ln _{\ell } (z-v) F_{\ell m}(z) + G_{\ell m}(z). \end{aligned}$$

(3.16)

Here $F_{\ell m}, G_{ \ell m}$ are locally analytic functions and $\ln _\ell (z-v)$ stands for the logarithm with the branch-cut extending along $\gamma _\ell$. Now we see that the total integral over $\Sigma \gamma$ involves summing over the incident arcs at v and then (3.15) implies that the pole in (3.16) cancels out in the summation so that the integral is locally convergent in the ordinary sense (irrespectively of the boundary values of the logarithms). Having established the convergence of the integral, we now choose $\epsilon$ sufficiently small so that the various disks ${\mathbb {D}}_\epsilon ^{(v)}$ are disjoint. We then have

$$\begin{aligned}&\int _{{\Sigma \gamma } } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{\Sigma \gamma } \frac{\mathrm{d }w}{2i\pi }\frac{ \varphi (w,z)}{(z_--w)^2} = \underbrace{\int _{{\Sigma \gamma }_\epsilon } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{{\Sigma \gamma }_\epsilon } \frac{\mathrm{d }w}{2i\pi }\frac{ \varphi (w, z)}{(z_--w)^2} }_{A_\epsilon } \nonumber \\&\qquad +\, \underbrace{\left( \int _{{\Sigma \gamma }_\epsilon } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{{\Sigma \gamma }\cap {\mathbb {D}}_\epsilon } \frac{\mathrm{d }w}{2i\pi }+ \int _{{\Sigma \gamma }\cap {\mathbb {D}}_\epsilon } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{{\Sigma \gamma }_\epsilon } \frac{\mathrm{d }w}{2i\pi }\right) \frac{ \varphi (w, z)}{(z -w)^2} }_{B_\epsilon }\nonumber \\&\qquad +\, \underbrace{ \int _{{\Sigma \gamma }\cap {\mathbb {D}}_\epsilon } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{{\Sigma \gamma }\cap {\mathbb {D}}_\epsilon } \frac{\mathrm{d }w}{2i\pi }\frac{ \varphi (w,z)}{(z_--w)^2} }_{C_\epsilon } \end{aligned}$$

(3.17)

The expression $A_\epsilon$ consists only of integrations over non-intersecting arcs and thus we can apply Lemma 3.1

$$\begin{aligned} A_\epsilon = -\frac{{1}}{2}\int _{{\Sigma \gamma }_\epsilon } \frac{{\mathrm{d }}z}{2i{\pi }}\partial _w \varphi (w,z) \bigg |_{w=z} \end{aligned}$$

(3.18)

This term clearly admits a limit as $\epsilon =0$ equal to $A_0$; a short computation gives

$$\begin{aligned}&\partial _w\varphi (w,z)\bigg |_{w=z} \mathop {=}^{(3.6)} 2 \mathrm {Tr}\bigg ( \Gamma _-^{-1}(z) \Gamma _-'(z)\Xi (z) \wedge \Xi (z) \bigg ) \nonumber \\&\quad +\, \mathrm {Tr}\bigg (\Xi '(z) \wedge \Xi (z)\bigg ) \end{aligned}$$

(3.19)

and hence

$$\begin{aligned} A_0= & {} -\int _{\Sigma \gamma } \frac{{\mathrm{d }}z}{2i{\pi }}\nonumber \\&\left\{ \mathrm {Tr}\bigg ( \Gamma _-^{-1}(z) \Gamma _-'(z)\Xi (z) \wedge \Xi (z) \bigg ) + \frac{{1}}{2}\mathrm {Tr}\bigg (\Xi '(z) \wedge \Xi (z)\bigg ) \right\} \end{aligned}$$

(3.20)

The remaining issue is the evaluation of $B_\epsilon , C_\epsilon$: as for $B_\epsilon$ we now show that it is identically zero. The inner integration in w and the outer integration in z have common points at ${\Sigma \gamma } \cap \partial {\mathbb {D}}_\epsilon$; let c be one of these points.

The integrand in $B_\epsilon$ is actually $L^1$ integrable over $\Sigma \gamma _\epsilon \times (\Sigma \gamma \cap {\mathbb {D}}_\epsilon )$ (and the reversed) because near the points common to those two sets (on the boundary of ${\mathbb {D}}_\epsilon$) the behaviour of the integrand is

$$\begin{aligned} \frac{\varphi (w,z)}{(z-w)^2} = \frac{C}{(z-w)} + {\mathcal {O}}(1) \end{aligned}$$

(3.21)

and hence the local nature of the integral is the same as the convergent integral $\int _0^\epsilon \int _{-\epsilon }^0 \frac{\,\mathrm dx \,\mathrm dy}{x+y}$. Thus the interchange of order of integral is allowed by Fubini’s theorem and we conclude that $B_\epsilon \equiv 0$ (using $\varphi (z,w) = -\varphi (w,z)$) for all $\epsilon$ (sufficiently small).

It remains to analyze the term $C_\epsilon$; it is clear (due to the skew-symmetry of $\varphi$) that the only contributions to the double integral may come from (z, w) in a neighborhood of the same vertex $v\in {\mathfrak {V}}$. Moreover, a simple estimate shows that if v is an “inessential" vertex, then the contribution tends to zero as $\epsilon \rightarrow 0$. For this reason we focus below only on the “essential” vertices.

Consider one of them and denote the incident arcs in $\Sigma \gamma \cap \{|z-v|<\epsilon \} = \bigcup _{\ell =1}^{n_v} \gamma _\ell ^{(v)}$ by $\gamma _\ell , \ \ell =1,\dots , n$; denote by $\sigma _\ell$ the distal endpoints of the arcs (at distance $\epsilon$ from v). We denote with $\varphi _{\ell m}^{(v)} =\displaystyle \lim _{z, w \rightarrow v\atop z\in \gamma _\ell , w\in \gamma _m }\varphi _{\ell m}(z,w)$ for brevity (we also omit the superscript $^{(v)}$ since we consider one vertex at a time). Then

$$\begin{aligned}&\int _{\Sigma \gamma \cap {\mathbb {D}}_\epsilon ^{(v)}} \frac{{\mathrm{d }}z}{2i{\pi }}\int _{\Sigma \gamma \cap {\mathbb {D}}_\epsilon ^{(v)}} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi (w,z)}{(z_--w)^2} \nonumber \\&= \sum _{\ell =1}^{n} \sum _{m=1}^n \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{ m \ell }(w,z)}{(z_--w)^2} \end{aligned}$$

(3.22)

The term with $\ell =m$ yields a convergent integral that is handled by Lemma 3.1 and tends to zero as $\epsilon \searrow 0$ since the length of $\gamma _\ell$ is ${\mathcal {O}}(\epsilon )$ (and the integrand is bounded). Let us now consider the remaining terms;

$$\begin{aligned} {\mathbf {J}}:= & {} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{ m \ell }(w,z)}{(w-z)^2} = \nonumber \\= & {} \underbrace{\sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{ m \ell }(w,z) - \varphi _{m \ell }}{(w-z)^2}}_{(\star )}\nonumber \\&+\, \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{m \ell }}{(w-z)^2} \end{aligned}$$

(3.23)

Each double integral in the sum marked $(\star )$ is a regularly convergent integral because of the regularization constant that we have added and subtracted; the other integral, on the contrary, is a singular integral and it depends on the order of integration. On the integral $(\star )$ we can swap order of integration and relabel $z\leftrightarrow w, \ell \leftrightarrow m$, and then use the skew symmetry $\varphi _{\ell m}(z,w) = -\varphi _{m \ell } (w,z)$, like so

$$\begin{aligned} {\mathbf {J}}= & {} \sum _m \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\sum _{\ell \ne m} \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\frac{\varphi _{m\ell }(w,z) -\varphi _{m\ell }}{(w-z)^2}\nonumber \\&+\, \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{m \ell }}{(w-z)^2} \mathop {=}^{z\leftrightarrow w \atop \ell \leftrightarrow m} \end{aligned}$$

(3.24)

$$\begin{aligned}= & {} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m \ne \ell } \int _{\gamma _m } \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{\ell m }(z,w) -\varphi _{\ell m }}{(z-w)^2} \nonumber \\&+\, \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{m \ell }}{(w-z)^2} . \end{aligned}$$

(3.25)

In the step above we only have re-labeled the dummy variables of integration. Now we can use the skew-symmetry $\varphi _{\ell m }(z,w) =-\varphi _{m\ell }(w,z)$ and $\varphi _{\ell m}=-\varphi _{m\ell }$ in the first integral. The term containing $\varphi _{\ell m}(z,w)$ yields back $-{\mathbf {J}}$ while the term containing $\varphi _{m\ell }$ adds to the last integral in (3.25). Thus we continue the chain of equalities:

$$\begin{aligned}= & {} -{\mathbf {J}} +2 \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\varphi _{m\ell }}{(w-z)^2} = -{\mathbf {J}} \nonumber \\&+\, 2 \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \frac{\varphi _{ m \ell }}{2i\pi (z-{\sigma }_m)} \end{aligned}$$

(3.26)

Fig. 4

Arcs near a vertex

Fig. 5

Vertex contribution to the exterior derivative

Here ${\sigma }_m = \gamma _m \cap {\mathbb {D}}_\epsilon$ (Fig. 4). Solving for ${\mathbf {J}}$ we obtain finally:

$$\begin{aligned} {\mathbf {J}} = \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \frac{\varphi _{ m \ell }}{2i\pi (z-{\sigma }_m)} =\sum _\ell \sum _{m\ne \ell }\frac{\varphi _{m \ell }}{(2i\pi )^2} \ln _\ell \left( \frac{{\sigma }_\ell - {\sigma }_m}{v-{\sigma }_m}\right) \end{aligned}$$

(3.27)

To compute the last expression we proceed as follows; first we observe that it is independent of v and ${\sigma }_\ell$’s. Indeed, differentiating we get

$$\begin{aligned} \partial _v {\mathbf {J}}= & {} \sum _\ell \sum _{m\ne \ell }\frac{\varphi _{m \ell }}{(2i\pi )^2} \frac{1}{v-{\sigma }_m} \mathop {=}^{(3.15) + (\varphi _{\ell \ell }=0)} 0 \nonumber \\ \partial _{{\sigma }_j} {\mathbf {J}}= & {} \sum _{m\ne j} \frac{\varphi _{ m j }}{(2i\pi )^2} \frac{1}{{\sigma }_j - {\sigma }_m} + \sum _{\ell \ne j} \frac{\varphi _{j \ell }}{(2i\pi )^2} \left( \frac{1}{{\sigma }_j - {\sigma }_\ell } - \frac{1}{v-{\sigma }_j}\right) \mathop {=}^{ }(3.15)\nonumber \\= & {} \sum _{m\ne j} \frac{\varphi _{ m j}}{(2i\pi )^2} \frac{1}{{\sigma }_j - {\sigma }_m} \nonumber \\&+\, \sum _{m \ne j} \frac{\varphi _{j m }}{(2i\pi )^2} \left( \frac{1}{{\sigma }_j - {\sigma }_m}\right) \mathop {=}^{\varphi _{m \ell } = -\varphi _{\ell m}} 0 \end{aligned}$$

(3.28)

Thus we can compute ${\mathbf {J}}$ by arranging ${\sigma }_\ell$’s as we wish; to do so, we re-write it back as a double integral

$$\begin{aligned} {\mathbf {J}}= & {} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{{\varphi _{m \ell }}}{(z-w)^2} = \nonumber \\= & {} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{{\varphi _{m \ell }}\left( 1 - \frac{v-c}{2(w-c)} - \frac{v-c}{2(z-c)}\right) }{(z-w)^2}\nonumber \\&+\, \frac{{1}}{2} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } {\varphi _{m \ell }}\int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{ \frac{v-c}{w-c} + \frac{v-c}{z-c}}{(z-w)^2} \end{aligned}$$

(3.29)

The point c is an arbitrary point not on any of the segments. The first integral is a regular convergent integral (the integrand is in $L^1$ near $z=w=v$) and exchanging the order of integration and then renaming the variables yields the same expression with a minus sign: hence we conclude that it is zero. We are thus left with the second term, which we now know is also independent of $\sigma _\ell$’s. To compute it more conveniently, we send all ${\sigma }_\ell$’s to infinity along distinct directions: the value of the expression, as we know, is independent of ${\sigma }$’s, so that now the arcs $\gamma _\ell$ are simply pairwise distinct rays issuing from v. In the limit we obtain the following

$$\begin{aligned} {\mathbf {J}}= & {} \underbrace{ \frac{{1}}{2} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } {\varphi _{m \ell }}\int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{ \frac{v-c}{z-c}}{(z-w)^2}}_{(\dagger )} \nonumber \\&+\,\underbrace{\frac{{1}}{2} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } {\varphi _{m \ell }}\int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{ \frac{v-c}{w-c}}{(z-w)^2}}_{(\heartsuit )} \end{aligned}$$

(3.30)

The term $(\dagger )$ is zero: indeed it is

$$\begin{aligned} (\dagger ) = \frac{{1}}{2} \sum _\ell \int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\sum _{m\ne \ell } {\varphi _{m \ell }} \frac{ \frac{v-c}{z-c}}{2i\pi (z-v)} \mathop {=}^{(3.15)}0. \end{aligned}$$

(3.31)

In the remaining term $(\heartsuit )$ , we write the inner integral in partial fractions

$$\begin{aligned} \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\frac{v-c}{w-c}}{(w-z)^2}= & {} \frac{v-c}{(z-c)^2} \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\left( \frac{1}{w-c} - \frac{1}{w-z} \right) \nonumber \\&+\, \frac{c-v}{c-z} \underbrace{\int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{1}{(w-z)^2}}_{=-\frac{1}{z-v} }\ \ z\in \gamma _\ell \end{aligned}$$

(3.32)

The first integral depends on m; to make the dependence manifest, we choose c on the right of $\gamma _1$ but to the left of $\gamma _n$ (the final result is independent of this choice) and rotate the contour of integration in (3.32) counterclockwise to a direction between c and $\gamma _n$ (denoted $\gamma _0$, see Fig. 5):

$$\begin{aligned}&\frac{v-c}{(z-c)^2} \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\left( \frac{1}{w-c} - \frac{1}{w-z} \right) \nonumber \\&\quad =\, \frac{v-c}{(z-c)^2} \int _{\gamma _0} \frac{\mathrm{d }w}{2i\pi }\left( \frac{1}{w-c} - \frac{1}{w-z} \right) -\frac{v-c}{(z-c)^2} \left\{ \begin{array}{cc} 1 &{} m<\ell \\ 0 &{} m>\ell \end{array} \right. \end{aligned}$$

(3.33)

The last term is due to the residue that we picked up while rotating $\gamma _m$ counterclockwise, and the fact that $z\in \gamma _{\ell }$. Thus, summarizing

$$\begin{aligned} \int _{\gamma _m} \frac{\mathrm{d }w}{2i\pi }\frac{\frac{v-c}{w-c}}{(w-z)^2} = \frac{v-c}{(z-c)^2} \ln _0\left( \frac{ v-z}{v-c} \right) + \frac{c-v}{(z-c)(z-v)} -\frac{v-c}{(z-c)^2} \left\{ \begin{array}{cc} 1 &{} m<\ell \\ 0 &{} m>\ell \end{array} \right. \nonumber \\ \end{aligned}$$

(3.34)

The first two terms are independent of m and hence they give a zero contribution to the term $(\heartsuit )$ in (3.30) because of the condition (3.15). We are thus left with

$$\begin{aligned} {\mathbf {J}} =-\frac{{1}}{2} \sum _\ell \sum _{m< \ell } {\varphi _{m \ell }}\int _{\gamma _\ell } \frac{{\mathrm{d }}z}{2i{\pi }}\frac{v-c}{(z-c)^2} = -\frac{1}{4i\pi } \sum _\ell \sum _{m< \ell } {\varphi _{m \ell }} \end{aligned}$$

(3.35)

Finally, one reads off the definition of $\varphi _{m\ell }$,

$$\begin{aligned} \varphi _{m\ell } = \lim _{z,w\rightarrow v\atop z\in \gamma _m, w\in \gamma _\ell }\mathrm {Tr}\left( \Gamma _m\Xi _m \Gamma _m^{-1}\wedge \Gamma _\ell \Xi _\ell \Gamma _\ell ^{-1} \right) \end{aligned}$$

(3.36)

and using (3.13) we obtain the terms in the sum appearing in (2.52.6). $\square$