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Blowup relations and $q$-Painlevé VI

Artem Stoyan

TL;DR

The work develops blowup relations for partition functions of $5d$ $\mathcal{N}=1$ SU$(2)$ quiver theories with four flavors and uses them to derive bilinear relations for $q$-Painlevé VI tau functions. By identifying Weyl-group actions on blowup data and interpreting relation parameters as Lie-algebra weights, the authors connect gauge-theory symmetries to isomonodromic tau structures. Through a $q_2\to1$ degeneration, they express $q$-Painlevé VI tau functions in terms of a generating function $S$ and obtain a complete set of eight bilinear tau-identities, supplemented by Higgsed two-term relations to supply the remaining four. The results illuminate a path from $5d$ gauge-theory partition functions to Painlevé isomonodromy data and suggest extensions to quantum (and perhaps affine) Painlevé systems and their wavefunctions.

Abstract

We propose and study blowup relations obeyed by the partition functions of $5d$ $\mathcal{N}=1$ (quiver) SYM theories with $SU(2)$ gauge group and four flavours. By analyzing the Weyl group action on the sets of blowup relations, we identify the integer parameters of a blowup relation with the weights of a corresponding Lie algebra. We also explain how this action of the Weyl group follows from the Weyl group symmetry of the partition function. Finally, we use these relations to derive bilinear relations on the $q$-Painlevé VI tau functions.

Blowup relations and $q$-Painlevé VI

TL;DR

The work develops blowup relations for partition functions of SU quiver theories with four flavors and uses them to derive bilinear relations for -Painlevé VI tau functions. By identifying Weyl-group actions on blowup data and interpreting relation parameters as Lie-algebra weights, the authors connect gauge-theory symmetries to isomonodromic tau structures. Through a degeneration, they express -Painlevé VI tau functions in terms of a generating function and obtain a complete set of eight bilinear tau-identities, supplemented by Higgsed two-term relations to supply the remaining four. The results illuminate a path from gauge-theory partition functions to Painlevé isomonodromy data and suggest extensions to quantum (and perhaps affine) Painlevé systems and their wavefunctions.

Abstract

We propose and study blowup relations obeyed by the partition functions of (quiver) SYM theories with gauge group and four flavours. By analyzing the Weyl group action on the sets of blowup relations, we identify the integer parameters of a blowup relation with the weights of a corresponding Lie algebra. We also explain how this action of the Weyl group follows from the Weyl group symmetry of the partition function. Finally, we use these relations to derive bilinear relations on the -Painlevé VI tau functions.

Paper Structure

This paper contains 15 sections, 1 theorem, 124 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Partition functions and their symmetries
  4. A-1-partition-function
  5. A-2-partition-function
  6. A-3-partition-function
  7. Blowup relations
  8. A1
  9. A2
  10. Two-term-A2
  11. Higgsed-A2
  12. A3
  13. Tau functions and bilinear relations
  14. Conclusion
  15. Coefficients

Key Result

Proposition 1

The extended Weyl group $\widetilde{W}(D_5)$ acts on the set of blowup relations. The generators act on the integer parameters $(j_1, j_2, n, j_3, j_4, d)$ by $\mathfrak{s}_i \mapsto s_i$ where $f$-factors in the l.h.s. of (blowup4) transform according to where $\mathcal{A}_4$ is defined in (A_4) and $S_i$ act by (A1:S1-A1:S5).

Theorems & Definitions (6)

  • Conjecture 1
  • Proposition 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5