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Consistency of M-Theory on nonorientable manifolds

Daniel S. Freed, Michael J. Hopkins

TL;DR

The paper shows that the parity/time-reversal anomaly cancels for M-theory in the low-energy limit on nonorientable, pin^+-structured 11-manifolds equipped with a w1-twisted lift of w4. By formulating Wick-rotated M-theory as a pair of invertible 12D field theories (one from the Rarita-Schwinger field and one from the C-field) and analyzing their product on generators of the 12D bordism group π_{12} M m_c, the authors prove trivialization of the total anomaly. This entails a hybrid computational approach combining eta-invariants, cubic refinements of the C-field, Adams spectral sequence computations, and explicit geometric bordism calculations on carefully chosen manifolds, establishing anomaly cancellation in the pin^+ setting. The work also develops techniques of broader interest, including computational eta-invariants on pin manifolds, a KO-theoretic view of cubic forms, and a concrete framework for translating invertible 12D topological field theories into topological data via Thom spectra. While the product α_RS × α_C is shown to be trivial, the paper notes an intrinsic ambiguity in the quantum integrand, leaving a canonical trivialization as an open problem and pointing to a conjectural 2-torsion obstruction in π_{11} M m_c. These results supply the mathematical groundwork for a consistent, time-reversal invariant M-theory in the low-energy regime on nonorientable manifolds and introduce tools applicable to related anomalies in high-energy and condensed-matter contexts.

Abstract

We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine generators for the 12-dimensional bordism group of pin manifolds with a w_1-twisted integer lift of w_4; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific eta-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of: computational techniques for eta-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques, and anomalies for spinor fields and Rarita-Schwinger fields.

Consistency of M-Theory on nonorientable manifolds

TL;DR

The paper shows that the parity/time-reversal anomaly cancels for M-theory in the low-energy limit on nonorientable, pin^+-structured 11-manifolds equipped with a w1-twisted lift of w4. By formulating Wick-rotated M-theory as a pair of invertible 12D field theories (one from the Rarita-Schwinger field and one from the C-field) and analyzing their product on generators of the 12D bordism group π_{12} M m_c, the authors prove trivialization of the total anomaly. This entails a hybrid computational approach combining eta-invariants, cubic refinements of the C-field, Adams spectral sequence computations, and explicit geometric bordism calculations on carefully chosen manifolds, establishing anomaly cancellation in the pin^+ setting. The work also develops techniques of broader interest, including computational eta-invariants on pin manifolds, a KO-theoretic view of cubic forms, and a concrete framework for translating invertible 12D topological field theories into topological data via Thom spectra. While the product α_RS × α_C is shown to be trivial, the paper notes an intrinsic ambiguity in the quantum integrand, leaving a canonical trivialization as an open problem and pointing to a conjectural 2-torsion obstruction in π_{11} M m_c. These results supply the mathematical groundwork for a consistent, time-reversal invariant M-theory in the low-energy regime on nonorientable manifolds and introduce tools applicable to related anomalies in high-energy and condensed-matter contexts.

Abstract

We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine generators for the 12-dimensional bordism group of pin manifolds with a w_1-twisted integer lift of w_4; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific eta-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of: computational techniques for eta-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques, and anomalies for spinor fields and Rarita-Schwinger fields.

Paper Structure

This paper contains 47 sections, 34 theorems, 290 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Time-reversal, anomalies, and bordism
  3. The Rarita-Schwinger operator and $\eta$-invariants
  4. Brief recollection of free fermionic path integrals
  5. The Rarita-Schwinger anomaly
  6. Properties of the $\eta$-invariant
  7. Cubic forms and the $C$-field
  8. Motivation: $\mathop{\mathrm{spin^{c}}}\nolimits$ manifolds
  9. Algebraic theory of cubic forms
  10. The cubic form on spin 12-manifolds
  11. The cubic form on $\mathop{\mathrm{pin}}\nolimits^{+}\!$ 12-manifolds
  12. The $C$-field and its anomaly; cancellation on spin manifolds
  13. Some spin and pin manifolds
  14. $K3$ surface
  15. Quaternionic projective plane
  16. ...and 32 more sections

Key Result

Theorem 2.2

The total anomaly theory $\alpha ^{\newline} _{RS}\otimes \alpha ^{\newline} _{C}$ is trivializable.

Figures (2)

  • Figure 1: The Adams spectral sequence for $\pi_{\ast}{M}\mathfrak m_{c}$
  • Figure 2: The map from $\mathop{\mathrm{Spin}}\nolimits$ bordism to ${M}\mathfrak m_{c}$-bordism in dimension $12$

Theorems & Definitions (87)

  • Definition 2.1
  • Theorem 2.2
  • Conjecture 2.6
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.14
  • proof : Proof
  • Corollary 3.15
  • Proposition 3.18
  • proof
  • ...and 77 more