Reflection Positivity and Invertible Phases

Spring 2022 (online)

This is a learning seminar about Freed-Hopkins's work on reflection positivity and invertible phases. The current goal is to go through Freed's CBSM book, a draft of which can be found here.

The first half of the seminar will be quite math-heavy as we develop the necessary formalism. The second half will be focused on physical applications, namely recovering a ten fold way and the classification of topological insulators and superconductors (anomalies of free fermions). If there is more time, we may also go through some newer work on anomalies based on the results of Freed-Hopkins.

This is organized by Cameron Krulewski and Leon Liu.

The usual time for the seminar is Mondays 3:00 PM - 4:00 EST, and the zoom link is here. Dan's special talk will be Friday Feb. 11th at 2:30 - 3:30 EST. The talk may or may not be recorded at the discretion of the speaker.

• • Jan 312022
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  Organization meeting

  Leon and Cameron

  I will give a 15 minutes quick overview over the content of the paper and the notes. Then we will decide on logistics and sign up for talks.
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• • Feb 112022

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  Overview and motivation. Video

  Dan Freed

  Prof. Freed is going to give a proper overview of/ motivation for their work. Note that it is at a special time Friday Feb 11th at 2:30-3:30 EST.

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• • Feb 142022
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  Bordism and Topological Field Theories. Video

  Charlie Reid

  We will first introduce bordism, and bordism invariants. Then we will categorify the story by defining the bordism category and Topological Field Theories. We will finish by talking about the Pontryagin-Thom theorem which converts the problem of finding bordism invariants into a problem in homotopy theory. This is a warm up for the categorified story where classification of invertible TQFTs can be achieved through homotopy theory.

  References:

  • Freed. Lectures in field theory and topology Lecture 1.
  • Freed, Hopkins. Reflection positivity and invertible phases
  • Debray. Bordism and invertible field theory lecture notes
  • Atiyah. Topological quantum field theory
  • Dijkgraaf, Witten. Topological gauge theories and group cohomology
  • Freed, Quinn. Chern-Simons theory wth finite gauge groups.
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• • Feb 212022
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  Quantum Mechanics. Video

  Justin Kulp

  I will provide a lightning fast review of quantum mechanics before describing gapped Hamiltonians and gapped "quantum systems." After this, I will move on to the (gapped) quantum phases and "space of theories" which are central to this seminar, and discuss their relationship to TFT. Time permitting, I will outline some features of Kitaev's Toric Code, following his original work, and then show how the pieces are formalized following Freed's notes.

  References:

  • Freed. Lectures in field theory and topology Lecture 2.
  • Freed, Hopkins. Reflection positivity and invertible phases section 3.1.
  • Kitaev. Fault-tolerant quantum computation by anyons for Toric code.
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• • Feb 282022
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  Wick-Rotated Quantum Field Theory. Video

  Luuk Stehouwer

  A prominent goal of Freed and Hopkins is to study relativistic QFT through the lens of TQFT. In this talk, we look for physically reasonable assumptions on the symmetries of a QFT on Minkowski space. We then study the resulting `symmetry type' after Wick-rotation to Euclidean signature. This provides the data we need to define the corresponding differential geometric structures on curved Riemannian spaces, which we will later use as an input to define TQFTs in the way we learned during talk 1. The main result in this section is a Coleman-Mandula type theorem: every symmetry type `almost' splits into a product of the internal symmetry group and the Lorentz group. We will pay extra attention to time-reversing symmetries, as they play a special role in relativistic QFT; they act anti-unitarily on Hilbert space and interplay with the positivity of energy. Special care has to be taken when Wick-rotating these structures, as they will be essential to define reflection-positivity in future talks. I will provide a construction of the resulting symmetry type and its extension to anti-unitary symmetries given the `spatially/nonrelativistically internal symmetry groups' that are often encountered in condensed matter physics.

  References:

  • Freed. Lectures in field theory and topology Lecture 3.
  • Freed, Hopkins. Reflection positivity and invertible phases section 2, 3, Appendix A.
  • Segal. Felix Klein lectures 2011 video.
  • Freed, Moore. Twisted equivariant matter.
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• • Mar 72022
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  Classification Theorems. Video

  Leon Liu

  As a warm up to our main theorem, we classify 1 dimensional field theories and 2 dimensional area-dependent field theories using Morse/Cerf theory.

  References:

  • Freed. Lectures in field theory and topology Lecture 4.
  • Freed, Hopkins. Reflection positivity and invertible phases
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• • Mar 142022
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  Extended Locality. Video

  William Stewart

  In this talk we will explore one of the two pillars of QFT: the notion of locality. We will demonstrate how this leads to the concept of an extended topological field theory. Along the way we will explore heuristically the concept of higher categories, which is the mathematical tool used to encode full locality. We will finish by exploring some of the structure of extended TQFTs and a powerful classification result: the cobordism hypothesis.​

  References:

  • Freed. Lectures in field theory and topology Lecture 5.
  • Freed, Hopkins. Reflection positivity and invertible phases section 5.1.
  • Freed. The cobordism hypothesis
  • Lurie. On the classification of topological field theories.
  • MIT Juvitop seminar fall 2020.
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• • Mar 212022
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  No talk

  Spring break
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• • Mar 282022
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  Invertibility and Stable Homotopy Theory. Video

  Julius Frank

  This talk is all about invertibility. We’ll see what it means for a topological field theory to be invertible and why we should care. On the technical side, we’ll be able to switch from higher categories to spectra, which are more amenable to computations. We’ll have a rough introduction to spectra, and then discuss the spectra which are involved with TFTs: For the domain, the bordism categories will be replaced by certain Thom spectra. For the target, we’ll suddenly have a canonical choice—this was not the case for non-invertible theories.

  References:

  • Freed. Lectures in field theory and topology Lecture 6.
  • Freed, Hopkins. Reflection positivity and invertible phases section 5.2, 7.1.
  • Galatius, Madsen, Tillman, Weis. The homotopy type of the cobordism category
  • Beaudry-Campell. A guide for computing stable homotopy groups
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• • Apr 062022
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  Wick-Rotated Unitarity

  Natalia Pacheco-Tallaj

  We quickly review unitarity in QFT and then discuss reflection positivity, the manifestation of unitarity after Wick rotation, in the non-extended case.​

  References:

  • Freed. Lectures in field theory and topology Lecture 7.
  • Freed, Hopkins. Reflection positivity and invertible phases section 3 and Appendix A.
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• • Apr 112022
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  Extended Positivity and Stable Homotopy Theory. Video

  Juan Moreno

  In this talk we will implement the notion of reflection positivity (Lecture 7) in the context of invertible extended TFTs (Lecture 6). This will require us to introduce some tools from Borel equivariant stable homotopy theory. The result is a determination of the homotopy type of the space of reflection positive invertible extended TFTs. ​

  References:

  • Freed. Lectures in field theory and topology Lecture 8.
  • Freed, Hopkins. Reflection positivity and invertible phases 6, 7, 8.
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• • Apr 182022
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  Non-Topological Invertible Field Theories

  Arun Debray. Video

  Though Freed-Hopkins showed that reflection-positive invertible TFTs are classified by the torsion subgroup of [MTH, \Sigma^{n+1} IZ], they went further and conjectured that the entire group classifies reflection-positive invertible field theories that aren't necessarily topological. In this talk, we will discuss this conjecture and its relationship with differential cohomology. We will go over some examples including classical Chern-Simons theory.​​

  References:

  • Freed. Lectures in field theory and topology Lecture 9.
  • Freed, Hopkins. Reflection positivity and invertible phases
  • Freed. Short-ranged entanglement and invertible field theories
  • Kitaev. On the classification of short-ranged entangled states video.
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• • Apr 252022
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  Computations for Electron Systems. Video

  Cameron Krulewski

  References:

  • Freed. Lectures in field theory and topology Lecture 10.
  • Freed, Hopkins. Reflection positivity and invertible phases section 9.3.
  • Kitaev. Periodic table for topological insulators and superconductors.
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• • May 22022
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  Anomalies in Field Theory

  Sanath Devalapurkar/Leon Liu. Video

  References:

  • Freed. Lectures in field theory and topology Lecture 11.
  • Freed, Hopkins. Reflection positivity and invertible phases
  • Freed. Anomalies and invertible field theories
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