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Relative quantum field theory

Daniel S. Freed, Constantin Teleman

TL;DR

This paper formalizes the concept of relative quantum field theory, where a theory F in n dimensions is defined relative to an extended (n+1)-dimensional theory α, often capturing anomalies or boundary data. It develops concrete realizations through relative σ-models and relative gauge theories, illustrating how partition functions and state spaces are constrained by the bulk α, and how absolute theories can be recovered by summing over substructures. The framework is then applied to Theory X, a 6D (0,2) SCFT, outlining a finite, self-dual abelian structure that governs its dimensional reductions and dualities via a 7D topological theory α_g. The work provides a cohesive picture of how relative theories encode higher-dimensional data, domain walls, and disorder operators, with a geometric and categorical underpinning for fields and their locality. It also sets the stage for further development of relative theories in more general nonabelian and self-dual contexts.

Abstract

We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions.

Relative quantum field theory

TL;DR

This paper formalizes the concept of relative quantum field theory, where a theory F in n dimensions is defined relative to an extended (n+1)-dimensional theory α, often capturing anomalies or boundary data. It develops concrete realizations through relative σ-models and relative gauge theories, illustrating how partition functions and state spaces are constrained by the bulk α, and how absolute theories can be recovered by summing over substructures. The framework is then applied to Theory X, a 6D (0,2) SCFT, outlining a finite, self-dual abelian structure that governs its dimensional reductions and dualities via a 7D topological theory α_g. The work provides a cohesive picture of how relative theories encode higher-dimensional data, domain walls, and disorder operators, with a geometric and categorical underpinning for fields and their locality. It also sets the stage for further development of relative theories in more general nonabelian and self-dual contexts.

Abstract

We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions.

Paper Structure

This paper contains 6 sections, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Relative quantum theories
  3. Warmup: relative $\sigma$-models
  4. Relative gauge theories
  5. Expectations for Theory $\mathscr{X}$
  6. What is a classical field?

Figures (4)

  • Figure 1: A geometric bordism $X\colon Y_0\to Y_1$
  • Figure 2: A domain wall
  • Figure 3: The vector bundle $F(Y)\to B\pi (Y)$
  • Figure 4: A groupoid equivalent to $B\pi (Y)$

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 3.4
  • Example 4.2
  • Definition 4.6
  • Remark 4.7
  • Claim 5.7
  • Remark 5.10: Reduction to four dimensions
  • Remark 5.11
  • Remark 5.12
  • Definition A.1
  • ...and 1 more