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Anomalies and Invertible Field Theories

Daniel S. Freed

TL;DR

The paper develops a modern geometric framework in which anomalies are themselves invertible quantum field theories, local on spacetime and described as spectrum-valued maps. It illustrates these ideas in supersymmetric quantum mechanics on a target $X$, showing the lagrangian anomaly from the Pfaffian line and the hamiltonian anomaly from a family of complex Clifford algebras are controlled by transgressions of $w_2(X)$ in suitable KO-theoretic contexts. A central contribution is a spectrum-level construction of the anomaly as an invertible 2D extended QFT $\alpha$, with a concrete topological model via $\Sigma^{2} MTO_2 \wedge X_+ \to \Sigma^{2} H\mathbb{Z}/2$ and spin-structure refinements that trivialize it; the framework extends to general targets with a KO-theory–based apparatus and explicit 2D cobordism data encoded by the Arf invariant. These results have implications for resolving worldsheet orientifold anomalies by providing a rigorous, topological approach to anomaly cancellation and multiplicative structure in invertible field theories.

Abstract

We give a modern geometric viewpoint on anomalies in quantum field theory and illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This is background for the resolution of worldsheet anomalies in orientifold superstring theory.

Anomalies and Invertible Field Theories

TL;DR

The paper develops a modern geometric framework in which anomalies are themselves invertible quantum field theories, local on spacetime and described as spectrum-valued maps. It illustrates these ideas in supersymmetric quantum mechanics on a target , showing the lagrangian anomaly from the Pfaffian line and the hamiltonian anomaly from a family of complex Clifford algebras are controlled by transgressions of in suitable KO-theoretic contexts. A central contribution is a spectrum-level construction of the anomaly as an invertible 2D extended QFT , with a concrete topological model via and spin-structure refinements that trivialize it; the framework extends to general targets with a KO-theory–based apparatus and explicit 2D cobordism data encoded by the Arf invariant. These results have implications for resolving worldsheet orientifold anomalies by providing a rigorous, topological approach to anomaly cancellation and multiplicative structure in invertible field theories.

Abstract

We give a modern geometric viewpoint on anomalies in quantum field theory and illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This is background for the resolution of worldsheet anomalies in orientifold superstring theory.

Paper Structure

This paper contains 14 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Anomalies
  3. Fields and field theories: formal view
  4. Anomalies: traditional view
  5. Anomalies: modern view
  6. Supersymmetric quantum mechanics
  7. Lagrangian anomaly
  8. Hamiltonian anomaly
  9. Trivializing the lagrangian and hamiltonian anomalies
  10. The anomaly as an invertible field theory
  11. Central simple algebras and topology
  12. Some $ko$-modules
  13. Some maps between $ko$-modules
  14. Supersymmetric QM with a general target

Key Result

Theorem 3.4

Given Assumption thm:2, the topological equivalence class in $H^1(\mathscr{F}'(M);{\mathbb Z}/2{\mathbb Z})$ of the lagrangian anomaly $\mathop{\mathrm{Pfaff}}\nolimits D\to\mathscr{F}'(M)$ is the transgression of $w_2(X)\in H^2(X;{\mathbb Z}/2{\mathbb Z})$.

Theorems & Definitions (20)

  • Example 2.4
  • Theorem 3.4
  • proof
  • Remark 3.11
  • Remark 3.12
  • Theorem 3.19
  • Corollary 3.21
  • proof : Proof of Corollary \ref{['thm:8']}
  • proof : Proof of Theorem \ref{['thm:7']}
  • Remark 3.24
  • ...and 10 more