Reflection positivity and invertible topological phases

Abstract: We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.

• The paper introduces an extended notion of reflection positivity that systematically classifies invertible TQFTs using stable homotopy theory.
• It employs Madsen-Tillmann spectra and the Adams spectral sequence to rigorously analyze fermionic phases and associated bordism groups.
• The findings offer a solid mathematical framework with significant implications for understanding symmetry and phase transitions in quantum systems.

Reflection Positivity and Invertible Topological Phases: A Theoretical Overview

The paper by Freed and Hopkins explores the intricate relationship between reflection positivity, invertible topological quantum field theories (TQFTs), and stable homotopy theory to provide a classification framework for invertible topological phases of matter. The discussion is rooted in a formal yet comprehensive treatment of these intersections, targeting experienced researchers in fields such as mathematical physics and topology. The focus is on offering a systematic characterization of the constraints and implications arising from symmetry and deformation classes in quantum systems, particularly in relation to condensed matter applications like topological insulators and superconductors.

Reflection Positivity in TQFTs

The concept of reflection positivity is central to ensuring unitarity in Wick-rotated field theories. Freed and Hopkins propose an extended version of reflection positivity for invertible topological quantum field theories (iTQFTs). This involves establishing a correspondence between the deformation classes of these theories and homotopy classes of maps in stable homotopy theory. Reflection positivity here is dealt with through a unique group extension involving symmetry groups, leading to an involution on the bordism categories used to describe TQFTs.

Homotopy Theory and Invertible Field Theories

The authors explore Madsen-Tillmann spectra and utilize them to model the domain of invertible topological theories. This approach moves from abstract multicategories to topological spaces, enabling the classification of these theories through spectra. The core theorem derived in this domain is an equivalence:

$$\left\{\text{deformation classes of reflection positive invertible iTQFTs}\right\} \cong [MTH, n+1],$$

where $MTH$ denotes a Thom spectrum associated with symmetry and bordism groups.

Analysis of Fermionic Phases

Freed and Hopkins focus particularly on understanding fermionic phases through symmetry groups akin to those relevant for topological insulators and superconductors. The real and complex symmetries are systematically classified, linking these systems to the so-called "ten-fold way" classifications common in condensed matter physics. They examine both free and interacting fermionic systems, predicting phase transitions and anomalies by mapping the boundary theories of free fermions to low-energy field theories.

Computational Framework

The mathematical machinery employed involves robust computational tools like the Adams spectral sequence, allowing a granular examination of bordism groups at play. This is crucial for transforming theoretical ideals into tangible predictions or classifications of topological phases that align with empirical investigations.

Theoretical Implications and Future Prospects

This research offers a formal basis for understanding the symmetry and topological structure underlying quantum systems. It concludes with a conjecture regarding the broader applicability of these models to non-topological field theories, potentially impacting how symmetry and locality are approached in quantum field theory. Future research might extend these findings to explore even more complex systems, potentially refining our understanding of phase stability and transitions in quantum materials.

In summary, Freed and Hopkins leverage the interplay between advanced homotopy theory and the robust framework of reflection positivity to explore the underpinning mathematical structures of invertible topological phases. This paper is exacting in its treatment and lays foundational ground for further investigations into the topology and symmetry of quantum systems.