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Reconstruction of $g=1$ permutation equivariant quantum $K$-invariants

Dun Tang

TL;DR

The paper proves a genus-1 reconstruction theorem for permutation-equivariant quantum K-theory on general Kähler targets, extending prior work for point targets to arbitrary targets. It uses the permutation-equivariant Ancestor-Descendant correspondence, an $\mathcal{S}$-operator framework, and a residue calculus to separate base-curve and map contributions and to express the genus-1 potential as a combination of a no-descendant part and explicit permuted-cycle contributions. The core result provides a residue-based formula for $\mathcal{F}_1(\mathbf{t})$ in terms of genus-0 data and restricted genus-1 data, including explicit contributions from cycle types $M\in\{2,3,4,6\}$ and an algorithm to compute the descendant inputs $\tau$. The methods unify and generalize Tang1 and Tang2, enabling practical computation of $g=1$ perm-equivariant quantum K-invariants for broad target spaces and highlighting a Dijkgraaf-Witten–style reconstruction phenomenon in this setting.

Abstract

In this paper, we establish an analog of Dijkgraaf-Witten's theorem for $g=1$ invariants in permutation-equivariant quantum K-theory. This result generalizes the findings of \cite{Tang1} and \cite{Tang2}.

Reconstruction of $g=1$ permutation equivariant quantum $K$-invariants

TL;DR

The paper proves a genus-1 reconstruction theorem for permutation-equivariant quantum K-theory on general Kähler targets, extending prior work for point targets to arbitrary targets. It uses the permutation-equivariant Ancestor-Descendant correspondence, an -operator framework, and a residue calculus to separate base-curve and map contributions and to express the genus-1 potential as a combination of a no-descendant part and explicit permuted-cycle contributions. The core result provides a residue-based formula for in terms of genus-0 data and restricted genus-1 data, including explicit contributions from cycle types and an algorithm to compute the descendant inputs . The methods unify and generalize Tang1 and Tang2, enabling practical computation of perm-equivariant quantum K-invariants for broad target spaces and highlighting a Dijkgraaf-Witten–style reconstruction phenomenon in this setting.

Abstract

In this paper, we establish an analog of Dijkgraaf-Witten's theorem for invariants in permutation-equivariant quantum K-theory. This result generalizes the findings of \cite{Tang1} and \cite{Tang2}.

Paper Structure

This paper contains 19 sections, 8 theorems, 64 equations, 2 tables.

Table of Contents

  1. Definition of invariants
  2. Ancestor-Descendant correspondence in $g=1$
  3. Statement of the result
  4. Collection of known results
  5. Basic theory
  6. The permutation equivariant splitting axiom
  7. A formula for super-traces
  8. A vanishing theorem and non-vanishing cases
  9. Dilaton equations for permutable inputs
  10. A proposition on contribution from chains of rational components
  11. An algorithm to compute $\tau$
  12. Computations
  13. Separating base curve and map contributions
  14. Non-permutative part: Case 1
  15. Case $2a$
  16. ...and 4 more sections

Key Result

Proposition 1

(Ancestor-Descendant correspondence in $g=1$)

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 1
  • Proposition 1.1
  • proof
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 3 more