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Extended V-systems and almost-duality for extended affine Weyl orbit spaces

Richard Stedman, Ian A. B. Strachan

TL;DR

The paper develops extended ∨-systems by augmenting a given ∨-system with a perpendicular direction via small orbits, and then enforces the ∨-conditions to obtain consistent higher-dimensional configurations. It shows that Legendre transformations along the extended direction map rational extended ∨-systems to trig ∨-systems and links these constructions to almost-dual Frobenius manifolds associated with extended affine Weyl orbit spaces. For classical types A and B/C/D, the authors establish that the almost-dual prepotentials of the orbit spaces correspond (up to Legendre transforms) to the extended ∨-systems, with explicit data for the extension and small-orbit structures. The work provides a unified framework connecting rational, trigonometric, and almost-dual perspectives and reveals symmetry enhancements tied to extended affine Weyl groups, with Hurwitz-space interpretations and potential extensions to generalized root systems and beyond.

Abstract

Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as $\bigvee$-conditions. The simplest examples of such configuration are the root systems of finite Coxeter groups. In this paper conditions are derived which ensure that an extended configuration - a configuration in a space one-dimension higher -satisfy these $\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit, as defined by Serganova. Symmetries of such resulting solutions to the WDVV-equations are studied; in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions and, for certain values of the free data, one obtains a transformation from extended $\bigvee$-systems to the trigonometric almost dual solutions corresponding to the classical extended affine Weyl groups.

Extended V-systems and almost-duality for extended affine Weyl orbit spaces

TL;DR

The paper develops extended ∨-systems by augmenting a given ∨-system with a perpendicular direction via small orbits, and then enforces the ∨-conditions to obtain consistent higher-dimensional configurations. It shows that Legendre transformations along the extended direction map rational extended ∨-systems to trig ∨-systems and links these constructions to almost-dual Frobenius manifolds associated with extended affine Weyl orbit spaces. For classical types A and B/C/D, the authors establish that the almost-dual prepotentials of the orbit spaces correspond (up to Legendre transforms) to the extended ∨-systems, with explicit data for the extension and small-orbit structures. The work provides a unified framework connecting rational, trigonometric, and almost-dual perspectives and reveals symmetry enhancements tied to extended affine Weyl groups, with Hurwitz-space interpretations and potential extensions to generalized root systems and beyond.

Abstract

Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as -conditions. The simplest examples of such configuration are the root systems of finite Coxeter groups. In this paper conditions are derived which ensure that an extended configuration - a configuration in a space one-dimension higher -satisfy these -conditions. Such a construction utilizes the notion of a small-orbit, as defined by Serganova. Symmetries of such resulting solutions to the WDVV-equations are studied; in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions and, for certain values of the free data, one obtains a transformation from extended -systems to the trigonometric almost dual solutions corresponding to the classical extended affine Weyl groups.

Paper Structure

This paper contains 16 sections, 9 theorems, 107 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Outline
  4. $\bigvee$-systems and small orbits
  5. Extended $\bigvee$-systems
  6. Extended configurations
  7. Imposition of the $\bigvee$-conditions
  8. Generalized Root systems and their extensions
  9. The series $A(m,n)$
  10. The series $B(m,n)$
  11. Symmetries of the extended configurations
  12. Legendre transformations and Trigonometric $\bigvee$-systems
  13. Extended $\bigvee$-systems and almost duality for extended affine Weyl orbit spaces
  14. Extended Affine Weyl orbit spaces of type $A$
  15. Extended Affine Weyl orbit spaces of type $B\,,C\,,D$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $W$ be a finite irreducible classical Coxeter group of rank $l$ and let $\widetilde{W}$ be the extended affine Weyl group of $W$ with arbitrary marked node. Then up to a Legendre transformation, the almost dual prepotentials of the classical extended affine Weyl group orbit spaces $\mathbb{C}^{l

Figures (7)

  • Figure 1: The geometry of the $\bigvee$-configuration
  • Figure 2: The geometry of the $\bigvee$-configuration, rotated with superimposed light cone
  • Figure 3: The roots of $A_2$ and the small orbit superimposed on the roots
  • Figure 4: The (partial) construction of an extended $\bigvee$-system
  • Figure 5: An extended $\bigvee$-system
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Example 2
  • Definition 3
  • Example 4
  • Definition 5
  • Theorem 6
  • Example 7
  • Definition 8
  • Lemma 9
  • proof
  • ...and 18 more