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Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

P. Gavrylenko, O. Lisovyy

TL;DR

The authors derive a constructive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems on the sphere by decomposing the punctured sphere into pairs of pants and wiring the tau function through 3-point Riemann–Hilbert problems. The main formula $\tau_{ ext{JMU}}(a)=\Upsilon(a)\det(\mathbb{1}-K)$ expresses the tau function in terms of elementary building blocks, enabling a detailed Fourier-based determinant expansion whose principal minors are labeled by Maya diagrams and charged partitions, i.e., Nekrasov-type sums. In the rank-two case, the 3-point problems collapse to Gauss hypergeometric forms, yielding explicit Garnier tau representations and, for $n=4$, a Painlevé VI determinant with a hypergeometric kernel. The work also establishes connections with conformal blocks, $W_N$ algebras, and gauge theory partition functions, offering a versatile framework for both exact representations and asymptotic analyses of isomonodromic tau functions. Extensions to irregular singularities, higher genus, and q-difference (q-isomonodromy) are outlined as promising directions.

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlevé VI equation.

Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

TL;DR

The authors derive a constructive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems on the sphere by decomposing the punctured sphere into pairs of pants and wiring the tau function through 3-point Riemann–Hilbert problems. The main formula expresses the tau function in terms of elementary building blocks, enabling a detailed Fourier-based determinant expansion whose principal minors are labeled by Maya diagrams and charged partitions, i.e., Nekrasov-type sums. In the rank-two case, the 3-point problems collapse to Gauss hypergeometric forms, yielding explicit Garnier tau representations and, for , a Painlevé VI determinant with a hypergeometric kernel. The work also establishes connections with conformal blocks, algebras, and gauge theory partition functions, offering a versatile framework for both exact representations and asymptotic analyses of isomonodromic tau functions. Extensions to irregular singularities, higher genus, and q-difference (q-isomonodromy) are outlined as promising directions.

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with regular singular points on the Riemann sphere and generic monodromy in . The corresponding operator acts in the direct sum of copies of . Its kernel has a block integrable form and is expressed in terms of fundamental solutions of elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant -point system via a decomposition of the punctured sphere into pairs of pants. For these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to gives a series representation of the general solution to Painlevé VI equation.

Paper Structure

This paper contains 18 sections, 24 theorems, 203 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Motivation and some results
  3. Notation
  4. Outline of the paper
  5. Perspectives
  6. Tau functions as Fredholm determinants
  7. Riemann-Hilbert setup
  8. Auxiliary $3$-point RHPs
  9. Plemelj operators
  10. Tau function
  11. Example: $4$-point tau function
  12. Fourier basis and combinatorics
  13. Structure of matrix elements
  14. Combinatorics of determinant expansion
  15. Rank two case
  16. ...and 3 more sections

Key Result

Lemma 2.2

We have $\left(\mathcal{P}^{[k]}\right)^2=\mathcal{P}^{[k]}$ and $\operatorname{ker}\mathcal{P}^{[k]}=\mathcal{H}^{[k]}_{\mathrm{in},+}\oplus\mathcal{H}^{[k]}_{\mathrm{out},-}$. Moreover, $\mathcal{P}^{[k]}$ can be explicitly written as where the operators $\mathsf a^{[k]}$, $\mathsf b^{[k]}$, $\mathsf c^{[k]}$, $\mathsf d^{[k]}$ are defined by

Figures (12)

  • Figure 1: Young diagram associated to the partition $\lambda=\left\{6,5,4,2\right\}$.
  • Figure 6: Contour $\Gamma$ for $n=5$
  • Figure 7: Generators of $\pi_1\left( \mathbb{P}^1\backslash a\right)$
  • Figure 8: Labeling of trinions, annuli and boundary curves
  • Figure 9: Contour $\Gamma^{[k]}$ (left) and $\hat{\Gamma}$ for $n=5$ (right)
  • ...and 7 more figures

Theorems & Definitions (33)

  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Corollary 2.10
  • Theorem 2.11
  • Theorem 2.12
  • ...and 23 more