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Relative field theories via relative dualizability

Claudia Scheimbauer, Thomas Stempfhuber

TL;DR

The work develops a unified higher-categorical framework for relative or twisted topological field theories by analyzing adjunctibility and dualizability in $(\infty,N)$-categories. It shows that in even dimensions, $n$-times left adjunctibility and $n$-times right adjunctibility coincide via an interchange lemma, while odd dimensions retain a distinction; mixed adjunctibility collapses to two parity classes (even^n and odd^n). It introduces dexterity functions and dexterity-trees to classify all higher adjunctibility notions, derives precise reduction results, and applies these ideas to Morita categories and to the classification of relative TFTs, arguing that oplax natural transformations are the natural data for element-in-$T$ field theories. The paper also provides Morita-category examples illustrating dualizability and adjointability conditions, and counts the vast landscape of generalized adjunctibility notions through binary-tree structures, revealing deep combinatorial structure with connections to well-known sequences. These results illuminate the correct categorical framework for relative/defect TFTs and clarify how dualizability interacts with higher adjunctions in a rich categorical setting.

Abstract

We investigate relative versions of dualizability designed for relative versions of topological field theories (TFTs), also called twisted TFTs, or quiche TFTs in the context of symmetries. In even dimensions we show an equivalence between lax and oplax fully extended framed relative topological field theories valued in an $(\infty , N)$-category in terms of adjunctibility. Motivated by this, we systematically investigate higher adjunctibility conditions and their implications for relative TFTs. Summarizing we arrive at the conclusion that oplax relative TFTs is the notion of choice. Finally, for fun we explore a tree version of adjunctibility and compute the number of equivalence classes thereof.

Relative field theories via relative dualizability

TL;DR

The work develops a unified higher-categorical framework for relative or twisted topological field theories by analyzing adjunctibility and dualizability in -categories. It shows that in even dimensions, -times left adjunctibility and -times right adjunctibility coincide via an interchange lemma, while odd dimensions retain a distinction; mixed adjunctibility collapses to two parity classes (even^n and odd^n). It introduces dexterity functions and dexterity-trees to classify all higher adjunctibility notions, derives precise reduction results, and applies these ideas to Morita categories and to the classification of relative TFTs, arguing that oplax natural transformations are the natural data for element-in- field theories. The paper also provides Morita-category examples illustrating dualizability and adjointability conditions, and counts the vast landscape of generalized adjunctibility notions through binary-tree structures, revealing deep combinatorial structure with connections to well-known sequences. These results illuminate the correct categorical framework for relative/defect TFTs and clarify how dualizability interacts with higher adjunctions in a rich categorical setting.

Abstract

We investigate relative versions of dualizability designed for relative versions of topological field theories (TFTs), also called twisted TFTs, or quiche TFTs in the context of symmetries. In even dimensions we show an equivalence between lax and oplax fully extended framed relative topological field theories valued in an -category in terms of adjunctibility. Motivated by this, we systematically investigate higher adjunctibility conditions and their implications for relative TFTs. Summarizing we arrive at the conclusion that oplax relative TFTs is the notion of choice. Finally, for fun we explore a tree version of adjunctibility and compute the number of equivalence classes thereof.
Paper Structure (17 sections, 43 theorems, 128 equations, 2 algorithms)

This paper contains 17 sections, 43 theorems, 128 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Higher duals and adjoints
  3. Adjoints in bicategories
  4. Adjoints in higher categories
  5. Duals
  6. Variants of adjunctibility
  7. Higher left and right adjunctibility
  8. Mixed adjunctibility and reducing conditions
  9. Adjoints in opposite categories
  10. Adjunctibility in Morita categories
  11. Applications to field theories
  12. Comparing lax and oplax natural transformations and variants
  13. Dualizable relative topological field theories
  14. More general notions of higher adjunctibility and binary trees
  15. Some basic lemmas about adjunctions
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $p(a^n, b^n)$ is even, then $f$ is $a^n$-adjunctible if and only if $f$ is $b^n$-adjunctible.

Theorems & Definitions (121)

  • Theorem 1.1: \ref{['Prop:AnReducesToTwoClasses']}
  • Theorem 1.2: \ref{['Prop:AnReducesToTwoClasses']} and \ref{['Theo:fromMixedToNadjunctibility']}
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • ...and 111 more