# Theo Johnson-Freyd

Dalhousie University

Areas of research:

### Overview

2016-present: Perimeter Institute (Postdoc) 2013-2016: Northwestern University (NSF Postdoc / Boas Assistant Professor) 2007-2013: UC Berkeley (PhD student) 2003-2007: Stanford (Undergraduate) prehistory-2003: Eugene, OR

### Research Interests

I study applications of higher category theory and homotopy theory to the quantum field theory and to condensed matter, and I study applications of condensed matter and quantum field theory to homotopy theory and higher category theory. Specific interests include anomalies, elliptic cohomology (aka topological modular forms), supersymmetric quantum field theory, sporadic groups, and "moonshine" phenomena.

### Positions Held

- 2016 - 2022 Perimeter Institute Senior Postdoctoral Fellow
- 2013 - 2016 Department of Mathematics, Northwestern University Boas Assistant Professor and NSF postdoctoral fellow

### Recent Publications

- Heisenberg-picture quantum field theory. Representation Theory, Mathematical Physics, and Integrable Systems. Progress in Mathematics volume in honour of Kolya Reshetikhin, 2021. arXiv: 1508.05908.
- Galois action on VOA gauge anomalies. Representation Theory, Mathematical Physics, and Integrable Systems. Progress in Mathematics volume in honour of Kolya Reshetikhin, 2021. arXiv: 1811.06495
- Supersymmetry and the Suzuki chain. Tunisian Journal of Mathematics, Vol 3, No 2, pp 309-359, 2021. DOI:10.2140/tunis.2021.3.309. arXiv: 1908.11012
- Fusion 2-categories with no line operators are grouplike. With Matthew Yu. Bulletin of the Australian Mathematical Society, 2021. arXiv: 2010.07950
- A note on some minimally supersymmetric models in two dimensions. With Davide Gaiotto and Edward Witten. In Integrability, Quantization, and Geometry II. Quantum Theories and Algebraic Geometry: Dedicated to the Memory of Boris Dubrovin 1950-2019, volume 103.2 of Proceedings of Symposia in Pure Mathematics, pp 203-222, Amer. Math. Soc., Providence, RI, 2021. arXiv: 1902.10249
- Holomorphic SCFTs with small index. With Davide Gaiotto. Canadian Journal of Mathematics, 2021:1-29. DOI:10.4153/S0008414X2100002X. arXiv: 1811.00589
- H4(Co0;Z)=Z/24. With David Treumann. International Mathematics Research Notices, 2020, no. 21, 7873-7907. DOI: 10.1093/imrn/rny219. arXiv: 1707.07587.
- Third homology of some sporadic finite groups. With David Treumann. Symmetry, Integrability and Geometry: Methods and Applications. 15 (2019), 059. arXiv: 1810.00463.
- Symmetry protected topological phases and generalized cohomology. With Davide Gaiotto. Journal of High Energy Physics. May 2019. DOI: 10.1007/JHEP05(2019)007. arXiv: 1712.07950.
- The Moonshine Anomaly. Communications in Mathematical Physics. February 2019, Volume 365, Issue 3, pp 943-970. DOI: 10.1007/s00220-019-03300-2. arXiv: 1707.08388.
- Spin, statistics, orientations, unitarity. Algebraic & Geometric Topology 17 (2017) 917-956 DOI: 10.2140/agt.2017.17.917. arXiv: 1507.06297
- (Op)lax natural transformations, twisted field theories, and the "even higher" Morita categories. With Claudia Scheimbauer. Advances in Mathematics, 307 (2017) 147-223. DOI: 10.1016/j.aim.2016.11.014. arXiv: 1502.06526.
- The quaternions and Bott periodicity are quantum Hamiltonian reductions. Symmetry, Integrability and Geometry: Methods and Applications, 12 (2016), 116, 6 pages. DOI: 10.3842/SIGMA.2016.116. arXiv: 1603.06603.
- Tree- versus graph-level quasilocal Poincaré duality on S1. Journal of homotopy and related structures, June 2016, Volume 11, Issue 2, pp 333-374. arXiv: 1412.4664
- Homological perturbation theory for nonperturbative integrals. Letters in Mathematical Physics, November 2015, Volume 105, Issue 11, pp 1605-1632. arXiv: 1206.5319
- Reflexivity and dualizability in categorified linear algebra. With Martin Brandenburg and Alexandru Chirvasitu. Theory and Applications of Categories, Vol. 30, No. 23, 2015, pp. 808-835. arXiv: 1409.5934
- Poisson AKSZ theories and their quantizations. In Proceedings of the conference String-Math 2013, volume 88 of Proceedings of Symposia in Pure Mathematics, pages 291--306, Providence, RI, 2014. Amer. Math. Soc. arXiv: 1307.5812
- The fundamental pro-groupoid of an affine 2-scheme. With Alex Chirvasitu. Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469-522. arXiv: 1105.3104
- The formal path integral and quantum mechanics. Journal of Mathematical Physics, 51, pp 122--103 (2010). arXiv: 1004.4305
- Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics. Letters in Mathematical Physics, November 2010, Volume 94, Issue 2, pp 123-149. arXiv: 1003.1156
- Minimal nondegenerate extensions. With David Reutter. arXiv: 2105.15167
- Topological Orders in (4+1)-Dimensions. With Matthew Yu. arXiv: 2104.04534
- (3+1)D topological orders with only a Z2-charged particle. arXiv: 2011.11165
- Topological Mathieu Moonshine. arXiv: 2006.02922
- On the classification of topological orders. arXiv: 2003.06663
- Condensations in higher categories. With Davide Gaiotto. arXiv: 1905.09566.
- Mock modularity and a secondary elliptic genus. With Davide Gaiotto. arXiv: 1904.05788.
- Exact triangles, Koszul duality, and coisotopic boundary conditions. arXiv: 1608.08598
- How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism. With Owen Gwilliam. Topology and Quantum Theory in Interaction, proceedings from the NSF-CBMS Regional Conference in the Mathematical Sciences that took place in Bozeman, Montana in 2017. arXiv: 1202.1554
- Berkeley Lectures on Lie Groups and Quantum Groups. R. Borcherds, M. Haiman, T. Johnson-Freyd, N. Reshetikhin, and V. Serganova. World Scientific. Accepted.

### Seminars

- Semisimple higher categories. Western Hemisphere Colloquium on Geometry and Physics.
- Operators and (higher) categories in quantum field theory. Seminar on Arithmetic Geometry and Quantum Field Theory, Korea Institute for Advanced Study.
- Higher S-matrices. TQFT Club, Instituto Superior Técnico.
- Minimal nondegenerate extensions. Fusion Friday, AIM.
- Minimal nondegenerate extensions and an anomaly indicator. Quantum Matter in Mathematics and Physics, CMSA, Harvard.
- Classification of topological orders. Quantum Mathematics: Quantum Matter and Quantum Information, 75+1 CMS Summer Meeting.
- Higher S-matrices. Higher Structures & Field Theory Seminar, UniVie/Erlangen/Würzburg/TUM.
- The classification of topological orders. Mathematics Department Colloquium, OSU.
- Fusion n-category Q&A. Fusion categories and tensor networks, AIM.
- Orbifolds. Moonshine Learning Seminar, IAS.
- Higher Galois closures. Seminar on Arithmetic Geometry and Quantum Field Theory, University of Warwick.
- Strongly-fusion 2-categories are grouplike. Representation Theory Seminar, UMass Amherst.
- Condensations and components. University Quantum Symmetries Lectures, NCSU.
- A topological umbral moonshine conjecture. Modularity in Quantum Systems, Kavli Institute for Theoretical Physics.
- Holomorphic SCFTs of small index. Topology, Algebraic Geometry, and Dynamics Seminar, George Mason University.
- Pseudounitary slightly degenerate braided fusion categories admit minimal modular extensions. Wales Mathematical Physics and Physical Mathematics, Cardiff University.
- Some examples in fusion 2-categories and 3+1D topological order. Global Categorical Symmetries Seminar.
- 3+1d topological orders with (only) an emergent fermion. Heidelberg-Munich-Vienna Seminar on Mathematical Physics.
- 3+1d topological orders with (only) an emergent fermion. Representation Theory and Mathematical Physics, UC Berkeley.
- Separable and central simple (higher) algebras. ATCAT Seminar, Dalhousie University.
- Algebraic definition of topological order. Topological Orders and Higher Structures, ESI.
- Mock modularity and a secondary invariant. String Math 2020, Stellenbosch University.
- SPT phases and generalized cohomology. Algebraic structures in quantum computation IV, UBC.
- Phases of SQFTs. Mathematics Colloquium, Dalhousie.
- Gapped condensation in higher categories. Tensor categories and topological quantum field theories, MSRI.
- A deformation invariant of 2D SQFTs. New High Energy Theory Center Seminar, Rutgers.
- A deformation invariant of 2D SQFTs. Geometry, Topology, and Physics, UC Santa Barbara.
- A deformation invariant of 2D SQFTs. Diff. Geom, Math. Phys., PDE Seminar, UBC.
- Spaces of quantum systems. Department Colloquium, UBC.
- Condensation in higher categories. Higher Structures in Geometry and Physics, Fields Institute.
- TMF and SQFT. High Energy Theory Seminar, IAS.
- Bott periodicity from quantum Hamiltonian reduction. Colloquium, Dalhousie University.
- Condensation and idempotent completion. ATCAT Seminar, Dalhousie University.
- Bott periodicity from quantum Hamiltonian reduction. Geometry Seminar, McGill.
- Phases of 2d SQFTs. Generalized Symmetries, Anomalies and Observables, Aspen Center for Physics.
- Exceptional Mathematics: from Egyptian fractions to heterotic strings. Colloquium, Canada/USA Mathcamp.
- Secondary invariants and mock modularity. Topology Seminar, Oxford.
- The Galois action on VOA anomalies. Higher Symmetries Conference 2019, Aspen Center for Physics.
- Galois actions on VOA gauge anomalies. Conference on Number Theory, Geometry, Moonshine & Strings III. Simons Foundation, NYC.
- Bott periodicity from quantum Hamiltonian reduction. Analysis & PDE Seminar, Stanford.
- Galois actions on VOA gauge anomalies. Algebra and Number Theory Seminar, UCSC.
- Bott periodicity from quantum Hamiltonian reduction. Math Department Colloquium, UCSC.
- Holomorphic SCFTs of small index. Quantum Algebra and Quantum Topology, OSU.
- Holomorphic SCFTs of small index. Mathematical Physics Seminar, UIUC.
- Galois action on gauge anomalies. Fusion Categories and Subfactors, BIRS, Banff, Canada.
- T-duality for finite groups. Representation Theory, Mathematical Physics and Integrable Systems, CIRM, Luminy, France.
- The fourth cohomology of some sporadic groups. Geometry, Symmetry and Physics, Yale.
- Moonshine anomalies. Algebra seminar, University at Buffalo
- Infinitely-categorified commutative algebra. Recent developments in noncommutative algebra and related areas, University of Washington
- Moonshine anomalies. QMAP seminar, UC Davis.
- Higher algebraic closures and phases of matter. Northeastern University
- Moonshine anomalies. UT Austin.
- Higher categories, generalized cohomology, and condensed matter. Representation Theory and Mathematical Physics, UC Berkeley
- 576 Fermions. Algebra Seminar, Emory
- Bott periodicity from Hamiltonian reduction. Number Theory and Algebraic Geometry. Boston College.
- Exceptional structures, fermions, anomalies, and Hamiltonian reduction. Research Seminar in Mathematics, Northeastern
- The Moonshine Anomaly. Higher Structures Lisbon, Instituto Superior Tecnico.
- The Moonshine Anomaly. Maximals Seminar, University of Edinburgh.
- Orbifolds of conformal field theories and cohomology of sporadic groups. Representation Theory, Geometry, and Combinatorics Seminar, UC Berkeley.
- Advanced integration by parts: the BV formalism. Geometric Structures Laboratory, Fields Institute.
- Fermionic hamiltonian reduction and periodicity. Geometry and Physics Seminar, Boston University.
- Ideals in derived algebra and boundary conditions in AKSZ-type field theories. Representation Theory, Geometry, and Combinatorics Seminar, UC Berkeley.
- 576 fermions, the Conway group, and tmf. Institute for Theoretical Physics, Stanford.
- Bott periodicity via quantum Hamiltonian reduction. Representation Theory, Geometry, and Combinatorics Seminar, UC Berkeley.
- PIRSA:20060018, On the classification of topological phases, 2020-06-01, Colloquium
- PIRSA:20050049, Welcome and Opening Remarks, 2020-05-25, Elliptic Cohomology and Physics
- PIRSA:20030111, Gapped condensation in higher categories, 2020-03-17, Mathematical Physics
- PIRSA:20010023, A deformation invariant of 1+1D SQFTs, 2020-01-14, Quantum Fields and Strings
- PIRSA:18120023, Poisson and coisotropic, 2018-12-12, Koszul Duality Seminar
- PIRSA:18110098, "Degenerate" AKSZ sigma models , 2018-11-21, Mathematical Physics
- PIRSA:18080042, Welcome and Opening Remarks, 2018-08-13, Higher Algebra and Mathematical Physics
- PIRSA:18010092, Higher algebraic closures and phases of matter, 2018-01-15, Mathematical Physics
- PIRSA:17050013, Welcome and Opening Remarks, 2017-05-08, Quantum Field Theory on Manifolds with Boundary and the BV Formalism
- PIRSA:16090051, Moonshine, topological modular forms, and 576 fermions., 2016-09-22, Mathematical Physics
- PIRSA:15100110, Spin--Statistics and Categorified Galois Groups, 2015-10-23, Condensed Matter Physics and Topological Field Theory