Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noninvertible symmetries in the B model TFT

A. Caldararu, T. Pantev, E. Sharpe, B. Sung, X. Yu

TL;DR

This work develops a concrete, derived-category–based framework for noninvertible symmetries in B-models on Calabi–Yau manifolds, describing line operators as kernels in $D^b(X\times X)$ and detailing their fusion, adjoints, and junctions.It demonstrates that this framework extends beyond finite fusion categories, with explicit analyses on elliptic curves (Tambara–Yamagami structures, $\mathbb{Z}_n$ lines, and the absence of Fibonacci/Haagerup structures), and discusses geometric realizations on K3s and general Calabi–Yau manifolds, including noninvertible B-field transformations and quantum symmetries in orbifolds.The paper also clarifies how decomposition and condensation defects fit into this language and explores interface/folding pictures such as T-duality, highlighting the broader landscape of noninvertible symmetries in string compactifications.Overall, the results provide a concrete toolkit for computing defect fusion, bulk actions, and correlation data in Calabi–Yau sigma models, with implications for noninvertible gauged symmetries in quantum gravity contexts.

Abstract

In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a general target space Calabi-Yau X, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on X \times X, and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.

Noninvertible symmetries in the B model TFT

TL;DR

This work develops a concrete, derived-category–based framework for noninvertible symmetries in B-models on Calabi–Yau manifolds, describing line operators as kernels in $D^b(X\times X)$ and detailing their fusion, adjoints, and junctions.It demonstrates that this framework extends beyond finite fusion categories, with explicit analyses on elliptic curves (Tambara–Yamagami structures, $\mathbb{Z}_n$ lines, and the absence of Fibonacci/Haagerup structures), and discusses geometric realizations on K3s and general Calabi–Yau manifolds, including noninvertible B-field transformations and quantum symmetries in orbifolds.The paper also clarifies how decomposition and condensation defects fit into this language and explores interface/folding pictures such as T-duality, highlighting the broader landscape of noninvertible symmetries in string compactifications.Overall, the results provide a concrete toolkit for computing defect fusion, bulk actions, and correlation data in Calabi–Yau sigma models, with implications for noninvertible gauged symmetries in quantum gravity contexts.

Abstract

In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a general target space Calabi-Yau X, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on X \times X, and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.

Paper Structure

This paper contains 52 sections, 3 theorems, 231 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Topological defects in the B model
  3. Review of defects and fusion in the B model
  4. Integral transforms as defects
  5. Defects as generalized symmetries in B models
  6. Orientation reversal as adjoints
  7. Junction and defect operators
  8. Basic identities
  9. Junction operators and cyclic identities
  10. Defect local operators
  11. Grading
  12. Examples
  13. Comparison to axioms for defects and junction operators
  14. Actions of defects on bulk states
  15. Nonfiniteness and semisimplicity
  16. ...and 37 more sections

Key Result

Lemma 4.1

We have the following

Figures (1)

  • Figure 1: Topological interfaces of B model TFTs can be folded into boundaries of the produced theory, which can naturally be captured by derived categories. The interface is promoted to a defect in the special case $X=Y$.

Theorems & Definitions (3)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3