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Moore-Tachikawa Varieties: Beyond Duality

Veronica Pasquarella

TL;DR

The paper generalizes Moore-Tachikawa varieties to targets that are hyperkähler quotients, necessitating a generalization of cobordism operators when reparametrisation invariance is absent and linking this to the Drinfeld center of composite class S theories. It analyzes how duality underpins the standard Moore-Segal framework and develops a composite identity structure to describe theories connected by non-invertible defects, thereby extending S-duality concepts. By connecting Higgs branches to hyperkähler quotients and interpreting the modifications through tensor products over the original identity, the work lays groundwork for a broader mathematical description of composite class S theories and their Coulomb/Higgs sectors, with anticipated applications to magnetic quivers and 3D N=4 mirror symmetry. Overall, the approach clarifies how non-invertible defects alter the algebraic and geometric structure of class S theories and proposes a path toward a rigorous absolute theory in non-dual settings.

Abstract

We propose a generalisation of the Moore-Tachikawa varieties for the case in which the target category of the 2D TFT is a hyperk$\ddot{\text{a}}$hler quotient. The setup requires generalising the bordism operators of Moore and Segal to the case involving lack of reparametrisation-invariance on the Riemann surface, ultimately enabling to relate this to the issue of defining a Drinfeld center for composite class ${\cal S}$ theories.

Moore-Tachikawa Varieties: Beyond Duality

TL;DR

The paper generalizes Moore-Tachikawa varieties to targets that are hyperkähler quotients, necessitating a generalization of cobordism operators when reparametrisation invariance is absent and linking this to the Drinfeld center of composite class S theories. It analyzes how duality underpins the standard Moore-Segal framework and develops a composite identity structure to describe theories connected by non-invertible defects, thereby extending S-duality concepts. By connecting Higgs branches to hyperkähler quotients and interpreting the modifications through tensor products over the original identity, the work lays groundwork for a broader mathematical description of composite class S theories and their Coulomb/Higgs sectors, with anticipated applications to magnetic quivers and 3D N=4 mirror symmetry. Overall, the approach clarifies how non-invertible defects alter the algebraic and geometric structure of class S theories and proposes a path toward a rigorous absolute theory in non-dual settings.

Abstract

We propose a generalisation of the Moore-Tachikawa varieties for the case in which the target category of the 2D TFT is a hyperkhler quotient. The setup requires generalising the bordism operators of Moore and Segal to the case involving lack of reparametrisation-invariance on the Riemann surface, ultimately enabling to relate this to the issue of defining a Drinfeld center for composite class theories.
Paper Structure (13 sections, 58 equations, 3 figures)

This paper contains 13 sections, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Cochain level theories
  3. Cochain complexes
  4. The Moore-Segal setup
  5. Cobordism operators
  6. Moore-Tachikawa varieties
  7. Categorical structure assuming duality
  8. Duality from an algebraic perspective
  9. Algebraic Varieties
  10. Moore-Tachikawa varieties beyond duality
  11. Categorical structure without duality
  12. Interesting shortcomings
  13. Conclusions and Outlook

Figures (3)

  • Figure 1: Partial reproduction of a diagram displayed in Moore. The first part of our treatment focuses on the functorial field theory description of class ${\cal S}$ theories and their Higgs branches in terms of 2D TFT cobordism constructions.
  • Figure 2: Adaptation of a correspondence first proposed in Pasquarella:2023deo playing a key role towards generalising Moore:2011ee to the hyperk$\ddot{\text{a}}$hler target category case. As explained in section \ref{['sec:new1234']}, this also requires the generalisation of cobordism operators, Moore:2006dw, due to the lack of reparametrisation-invariance of the Riemann surface on which the compactification of the 6D ${\cal N}=(2,0)$ SCFT is performed.
  • Figure 3: Basic bordisms assuming duality of, both, the source and target categories leading to the definition of the identity element, $V_{_{G_{_{\mathbb{C}}}}}$, and the maximal dimensional Higgs branch, $W_{_{G_{_{\mathbb{C}}}}}$.