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Quantum invariants arising from $U_h\mathfrak{sl}(2|1)$ are q-holonomic

Jennifer Brown, Nathan Geer

TL;DR

The paper addresses establishing q-holonomicity for quantum invariants arising from typical representations of $U_h(\mathfrak{sl}(2|1))$. It develops a framework where the generators, R-matrix, dualities, and modified dimensions act by q-holonomic maps within a ribbon category of topologically free modules, and uses closure properties to propagate q-holonomicity to the entire invariant. The main result shows that the knot invariant $F'(L)$, defined via a modified dimension and a cutting construction, is q-holonomic as a function of representation parameters, enabling a recursion- and quasi-periodicity-based understanding of these supergroup invariants. This provides evidence for a field-theoretic interpretation of the invariants as partition functions and suggests avenues toward a generalized volume conjecture, as well as potential skein-theoretic realizations via parabolic defects. The work also outlines conjectures for broader Lie superalgebras and sketches the relationship between skein theory and q-Weyl actions in this context.

Abstract

We show that the quantum invariants arising from typical representations of the quantum group $U_h\mathfrak{sl}(2|1)$ are q-holonomic. In particular, this implies the existence of an underlying field theory for which this family of invariants are partition functions.

Quantum invariants arising from $U_h\mathfrak{sl}(2|1)$ are q-holonomic

TL;DR

The paper addresses establishing q-holonomicity for quantum invariants arising from typical representations of . It develops a framework where the generators, R-matrix, dualities, and modified dimensions act by q-holonomic maps within a ribbon category of topologically free modules, and uses closure properties to propagate q-holonomicity to the entire invariant. The main result shows that the knot invariant , defined via a modified dimension and a cutting construction, is q-holonomic as a function of representation parameters, enabling a recursion- and quasi-periodicity-based understanding of these supergroup invariants. This provides evidence for a field-theoretic interpretation of the invariants as partition functions and suggests avenues toward a generalized volume conjecture, as well as potential skein-theoretic realizations via parabolic defects. The work also outlines conjectures for broader Lie superalgebras and sketches the relationship between skein theory and q-Weyl actions in this context.

Abstract

We show that the quantum invariants arising from typical representations of the quantum group are q-holonomic. In particular, this implies the existence of an underlying field theory for which this family of invariants are partition functions.
Paper Structure (23 sections, 9 theorems, 58 equations)

This paper contains 23 sections, 9 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Beyond $\mathfrak{sl}(2|1)$
  4. A quantum torus from skein theory
  5. Acknowledgements
  6. The Lie superalgebra $\mathfrak{sl}(2|1)$
  7. Representations of $\mathfrak{sl}(2|1)$ and $U_h(\mathfrak{sl}(2|1))$
  8. Typical $U_h(\mathfrak{sl}(2|1))$-modules
  9. $h_1$ and $h_2$ actions
  10. $E_1$ and $E_2$ actions
  11. $F_1$ and $F_2$ actions
  12. The ribbon structure on ${\mathop{\mathrm{Rep}}\nolimits_h^{tf}\mathfrak{sl}(2|1)}$
  13. Quantum Invariants from $\mathfrak{sl}(2|1)$
  14. q-Holonomic systems
  15. Non-discrete variables
  16. ...and 8 more sections

Key Result

Proposition 2.6

The above pivotal structure and braiding give a ribbon structure on ${\mathop{\mathrm{Rep}}\nolimits_h^{tf}\mathfrak{sl}(2|1)}$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 16 more