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Theo Johnson-Freyd

Publications and source records attributed to Theo Johnson-Freyd.

At least 19 recordsLinked to original sources
arXiv2026

Schopieray's Galois-modular extension conjecture

Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.

math.QA↗
arXiv2026

On the structure of Witt groups and minimal extension conjecture

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$. Then we show the Witt group $\mathcal{W}(\mathcal{E})$ is naturally a direct sum of Witt group $\mathcal{W}:=\mathcal{W}(\text{Vec})$ and the group $\text{H}^4(G,\mathbb{K}^\times)$. Consequently, for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$, there is a positive integer $n$ (e.g. $n=|G|$) such that $\mathcal{C}^{\boxtimes_\mathcal{E}^n}$ admits a minimal extension.

math.CT↗
arXiv2025

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

math.QA↗
arXiv2025

How to build a Hopf algebra

We construct a functor that inputs a retract in an $(\infty,3)$-category satisfying some adjunctibility conditions and outputs a Hopf algebra in a braided monoidal $(\infty,1)$-category. Provided the braided monoidal category is presentable, any Hopf algebra can be obtained in this way. Our functor specializes to - and provides a higher-categorical explanation for - the Tannakian reconstruction of a Hopf algebra from a monoidal category with duals and a fiber functor. Towards this end, we review and develop the lax (aka Gray) tensor product $\otimes$ of $(\infty,\infty)$-categories, and we analyze the "lax smash product" of pointed $(\infty,\infty)$-categories. We compute the lax-$\wedge$-square of the "walking adjunction" and show that its $3$-localization corepresents retracts with some adjunctibilty conditions, whereas the $3$-localization of the lax-$\wedge$-square of the "walking monad" corepresents bialgebras. In these terms, our functor is restriction along the lax-$\wedge$-square of the inclusion $\{\text{walking monad}\} \to \{\text{walking adjunction}\}$. After $3$-localization, we show that this restriction inverts a certain shear map, proving the existence of an antipode. We discuss generalizations of this construction to Hopf monads, analyze additional adjunctibility conditions and their interplay with integrals and cointegrals, and finally explain how variants of classical Tannakian reconstruction fit into our scheme.

math.CT↗
arXiv2024

The Classification of Fusion 2-Categories

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

math.CT↗
arXiv2024

On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing

We are aimed at giving a differential geometric, and accordingly physical, explanation of the 576-periodicity of TMF. In this paper, we settle the problem of giving the lower bound 576. We formulate the problem as follows: we assume a spectrum $\mathrm{SQFT}$ with some conditions, suggest from physical considerations about the classifying spectrum for two-dimensional $\mathcal{N}=(0,1)$-supersymmetric quantum field theories, and show that the periodicity of $\mathrm{SQFT}$ is no less than 576. The main tool for the proof is the analogue of the Anderson duality pairing introduced by the second-named author and Tachikawa. We do not rely on the Segal-Stolz-Teichner conjecture, so in particular we do not use any comparison map with TMF.

math.AT↗
arXiv2024

Dagger $n$-categories

Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.

math.CT↗
arXiv2023

Ground-state degeneracy of twisted sectors of Conway Moonshine SCFT

We calculate the ground state degeneracies of all twisted sectors in the "Conway Moonshine'' holomorphic SCFT $V^{f\natural}$. We find that almost all sectors have ground states of only a single parity: specifically, 66 twisted sectors have nontrivial ground states of a single parity, 39 twisted sectors have spontaneous supersymmetry breaking, and only 6 twisted sectors have ground states of both parities. Although "nontrivial ground states, all of the same parity'' is the expected behaviour for a generic SQM model without symmetry protection, it is surprising in the presence of a large symmetry group, as is the case in $V^{f\natural}$. This surprise hints that there are as-yet-undiscovered features of Conway Moonshine.

math-ph↗
arXiv2022

Topological Orders in (4+1)-Dimensions

We investigate the Morita equivalences of (4+1)-dimensional topological orders. We show that any (4+1)-dimensional super (fermionic) topological order admits a gapped boundary condition -- in other words, all (4+1)-dimensional super topological orders are Morita trivial. As a result, there are no inherently gapless super (3+1)-dimensional theories. On the other hand, we show that there are infinitely many algebraically Morita-inequivalent bosonic (4+1)-dimensional topological orders.

hep-th↗
arXiv2022

On the classification of topological orders

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion $n$-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such $n$-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion $n$-categories (i.e. multifusion $n$-categories with indecomposable unit) with centreless braided fusion $(n{-}1)$-categories. We then discuss the classification in low spacetime dimension, proving in particular that all $(1{+}1)$- and $(3{+}1)$-dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and proposals by X.G. Wen et al.

math.CT↗
arXiv2021

Minimal nondegenerate extensions

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in $\mathrm{H}^5(K(\mathbb{Z}_2, 2); k^\times)$ and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of Z(Mod-B) in terms of braided B-module categories, we develop a computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the M\"uger centre of B to the ground field.

math.QA↗
arXiv2021

Topological Mathieu Moonshine

We explore the Atiyah-Hirzebruch spectral sequence for the $tmf^\bullet[\frac12]$-cohomology of the classifying space $BM_{24}$ of the largest Mathieu group $M_{24}$, twisted by a class $ω\in H^4(BM_{24};Z[\frac12]) \cong Z_3$. Our exploration includes detailed computations of the $Z_3$-cohomology of $M_{24}$ and of the first few differentials in the AHSS. We are specifically interested in the value of $tmf^\bullet_ω(BM_{24})[\frac12]$ in cohomological degree $-27$. Our main computational result is that $tmf^{-27}_ω(BM_{24})[\frac12] = 0$ when $ω\neq 0$. For comparison, the restriction map $tmf^{-3}_ω(BM_{24})[\frac12]\to tmf^{-3}(pt)[\frac12] \cong Z_3$ is surjective for one of the two nonzero values of $ω$. Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjectural relationship between $TMF$ and supersymmetric quantum field theory, there is a canonically-defined $Co_1$-twisted-equivariant lifting $[\bar{V}^{f\natural}]$ of the class $\{24Δ\} \in TMF^{-24}(pt)$, where $Co_1$ denotes Conway's largest sporadic group. We conjecture that the product $[\bar{V}^{f\natural}] ν$, where $ν\in TMF^{-3}(pt)$ is the image of the generator of $tmf^{-3}(pt) \cong Z_{24}$, does not vanish $Co_1$-equivariantly, but that its restriction to $M_{24}$-twisted-equivariant $TMF$ does vanish. This conjecture answers some of the questions in Mathieu Moonshine: it implies the existence of a minimally supersymmetric quantum field theory with $M_{24}$ symmetry, whose twisted-and-twined partition functions have the same mock modularity as in Mathieu Moonshine. Our AHSS calculation establishes this conjecture "perturbatively" at odd primes. An appendix included mostly for entertainment purposes discusses "$\ell$-complexes" and their relation to $\mathrm{SU}(2)$ Verlinde rings. The case $\ell=3$ is used in our AHSS calculations.

math.AT↗
arXiv2020

(3+1)D topological orders with only a $\mathbb{Z}_2$-charged particle

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure $\mathbb{Z}_2$ gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin-$\mathbb{Z}_2$" gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion $2$-category $\mathcal{Z}_{(1)}(Σ\mathbf{SVec})$ of string and particle operators in pure spin-$\mathbb{Z}_2$ gauge theory.

math.QA↗
arXiv2020

Supersymmetry and the Suzuki chain

We classify $N{=}1$ SVOAs with no free fermions and with bosonic subalgebra a simply connected WZW algebra which is not of type $\mathrm{E}$. The latter restriction makes the classification tractable; the former restriction implies that the $N{=}1$ automorphism groups of the resulting SVOAs are finite. We discover two infinite families and nine exceptional examples. The exceptions are all related to the Leech lattice: their automorphism groups are the larger groups in the Suzuki chain ($\mathrm{Co}_1$, $\mathrm{Suz}{:}2$, $\mathrm{G}_2(4){:}2$, $\mathrm{J}_2{:}2$, $\mathrm{U}_3(3){:}2$) and certain large centralizers therein ($2^{10}{:}\mathrm{M}_{12}{:}2$, $\mathrm{M}_{12}{:}2$, $\mathrm{U}_4(3){:}D_8$, $\mathrm{M}_{21}{:}2^2$). Along the way, we elucidate fermionic versions of a number of VOA operations, including simple current extensions, orbifolds, and 't Hooft anomalies.

math.QA↗
arXiv2020

Galois action on VOA gauge anomalies

Assuming regularity of the fixed subalgebra, any action of a finite group $G$ on a holomorphic VOA $V$ determines a gauge anomaly $α\in \mathrm{H}^3(G; \boldsymbolμ)$, where $\boldsymbolμ \subset \mathbb{C}^\times$ is the group of roots of unity. We show that under Galois conjugation $V \mapsto {^γV}$, the gauge anomaly transforms as $α\mapsto γ^2(α)$. This provides an a priori upper bound of $24$ on the order of anomalies of actions preserving a $\mathbb{Q}$-structure, for example the Monster group $\mathbb{M}$ acting on its Moonshine VOA $V^\natural$. We speculate that each field $\mathbb{K}$ should have a "vertex Brauer group" isomorphic to $\mathrm{H}^3(\mathrm{Gal}(\bar{\mathbb{K}}/\mathbb{K}); \boldsymbolμ^{\otimes 2})$. In order to motivate our constructions and speculations, we warm up with a discussion of the ordinary Brauer group, emphasizing the analogy between VOA gauging and quantum Hamiltonian reduction.

math.QA↗
arXiv2019

How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism

The Batalin-Vilkovisky formalism in quantum field theory was originally invented to address the difficult problem of finding diagrammatic descriptions of oscillating integrals with degenerate critical points. But since then, BV algebras have become interesting objects of study in their own right, and mathematicians sometimes have good understanding of the homological aspects of the story without any access to the diagrammatics. In this note we reverse the usual direction of argument: we begin by asking for an explicit calculation of the homology of a BV algebra, and from it derive Wick's Theorem and the other Feynman rules for finite-dimensional integrals.

math-ph↗
arXiv2019

Heisenberg-picture quantum field theory

What we should mean by "Heisenberg-picture quantum field theory"? Atiyah--Segal-type axioms do a good job of capturing the "Schrödinger picture": these axioms define a "$d$-dimensional quantum field theory" to be a symmetric monoidal functor from an $(\infty,d)$-category of "spacetimes" to an $(\infty,d)$-category which at the second-from-top level consists of vector spaces, so at the top level consists of numbers. This paper argues that the appropriate parallel notion "Heisenberg picture" should also be defined in terms of symmetric monoidal functors from the category of spacetimes, but the target should be an $(\infty,d)$-category that in top dimension consists of pointed vector spaces instead of numbers; the second-from-top level can be taken to consist of associative algebras or of pointed categories. The paper ends by outlining two sources of such Heisenberg-picture field theories: factorization algebras and skein theory.

math-ph↗