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Theo Johnson-Freyd

Publications and source records attributed to Theo Johnson-Freyd.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2019

Mock modularity and a secondary elliptic genus

The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of some of these invariants. The theory of mock modular forms makes a surprise appearance, shedding light on the integrality properties of some well-known examples.

hep-th↗
arXiv2018

Holomorphic SCFTs with small index

We observe that every self-dual ternary code determines a holomorphic N=1 superconformal field theory. This provides ternary constructions of some well-known holomorphic N=1 SCFTs, including Duncan's "supermoonshine" model and the fermionic "beauty and the beast" model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by 24; we are able to prove this conjecture except in the case when the code has length 12 mod 24. Lastly, we discuss a conjecture of Stolz and Teichner relating N=1 SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic N=1 SCFT of central charge 12k and index 24/gcd(k,24). We give ternary code constructions of SCFTs realizing this suggestion for k \leq 5.

hep-th↗
arXiv2018

The Moonshine Anomaly

The anomaly for the Monster group $\mathbb{M}$ acting on its natural (aka moonshine) representation $V^\natural$ is a particular cohomology class $ω^\natural \in \mathrm{H}^3(\mathbb{M},\mathrm{U}(1))$ that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that $ω^\natural$ has order exactly $24$ and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate $ω^\natural$ to the anomaly for the Leech lattice CFT.

math.QA↗
arXiv2018

Symmetry Protected Topological phases and Generalized Cohomology

We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.

hep-th↗
arXiv2017

Exact triangles, Koszul duality, and coisotopic boundary conditions

We develop a theory of "arrowed" (operads and) dioperads, which are to exact triangles as dioperads are to vector spaces. A central example to this paper is the arrowed operad controlling "derived ideals" for any operad. The Koszul duality theory of arrowed dioperads interacts well with rotation of exact triangles, and in particular with "exact Stars of David," which are pairs of exact triangles drawn on top of each other in an interesting way. Using this framework, we give a cochain-level lift of the "relative Poincaré duality" enjoyed by oriented manifolds with boundary; moreover, our cochain-level lift satisfies a natural locality-type condition, and is uniquely determined by this property. We discuss the meaning of the words "relative orientation" and "coisotropic." We extend the AKSZ construction to bulk-boundary settings with Poisson bulk fields and coisotropic boundary conditions.

math.QA↗
arXiv2016

The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions

We show that the Morita equivalences $\mathrm{Cliff}(4) \simeq {\mathbb H}$, $\mathrm{Cliff}(7) \simeq \mathrm{Cliff}(-1)$, and $\mathrm{Cliff}(8) \simeq {\mathbb R}$ arise from quantizing the Hamiltonian reductions ${\mathbb R}^{0|4} // \mathrm{Spin}(3)$, ${\mathbb R}^{0|7} // G_2$, and ${\mathbb R}^{0|8} // \mathrm{Spin}(7)$, respectively.

math.SG↗
arXiv2016

Spin, statistics, orientations, unitarity

A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation. A field theory satisfies spin-statistics if it is both spin and super, and $360^\circ$-rotation of the spin structure agrees with the operation of flipping the signs of all fermions. We set up a framework in which these two notions are precisely analogous. In this framework, field theories are defined over $\mathrm{Vect}_{\mathbb R}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $\mathrm{Spec}(\mathbb R)$. Bundles of tangential structures may be etale-locally equivalent without being equivalent, and Hermitian field theories are nothing but the field theories controlled by the unique nontrivial bundle of tangential structures that is etale-locally equivalent to Orientations. This bundle owes its existence to the fact that $π_1^{et}(\mathrm{Spec}(\mathbb R)) = π_1{BO}$. We interpret Deligne's "existence of super fiber functors" theorem as implying that in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings, $π_2^{et}(\mathrm{Spec}(\mathbb R)) = π_2{BO}$. There are eight bundles etale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of that distinguished tangential structure, one arrives at field theories that are both Hermitian and satisfy spin-statistics. Finally, we formulate a notion of "reflection-positivity" and prove that if an etale-locally-oriented field theory is reflection-positive then it is necessarily Hermitian, and if an etale-locally-spin field theory is reflection-positive then it necessarily both satisfies spin-statistics and is Hermitian. The latter result is a topological version of the famous Spin-Statistics Theorem.

math-ph↗
arXiv2016

(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong $(\infty, n)$-functors. We construct a double $(\infty,n)$-category built out of the target $(\infty, n)$-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty, n)$-category $\mathcal{C}$ to an $(\infty, n+d)$-category using the higher morphisms in $\mathcal{C}$.

math.CT↗
arXiv2015

Homological perturbation theory for nonperturbative integrals

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler--Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.

math-ph↗
arXiv2015

Tree- versus graph-level quasilocal Poincare duality on S^1

Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reasonable to ask for homotopically-constant families of operations that become local "in the limit." To make the question concrete, we take the manifold to be the one-dimensional circle. The answer to whether a quasilocal homotopy-Frobenius algebra structure exists turns out to depend on the choice of context in which to do homotopy algebra. There are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations: one can work at "tree level," corresponding roughly to the world of operadic homotopy algebras and their homotopy modules; or one can work at "graph level," corresponding to the world of PROPs. For the tree-level version of our question, the answer is the unsurprising "Yes, such a structure exists" --- indeed, it is unique up to a contractible space of choices. But for the graph-level version, the answer is the surprising "No, such a structure does not exist." Most of the paper consists of computing explicitly this nonexistence, which is controlled by the numerical value of a certain obstruction, and we compute this value explicitly via a sequence of integrals.

math.AT↗
arXiv2014

Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure

The embedding Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaningful algebraic structure on homology that does not lift in a geometrically meaningful way to the chain level.

math.AT↗
arXiv2014

Reflexivity and dualizability in categorified linear algebra

The "linear dual" of a cocomplete linear category $\mathcal C$ is the category of all cocontinuous linear functors $\mathcal C \to \mathrm{Vect}$. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category $\mathrm{Qcoh}(X)$ of quasi-coherent sheaves on a stack $X$ is not dualizable if $X$ is the classifying stack of a semisimple algebraic group in positive characteristic or if $X$ is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if $X$ is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset.

math.CT↗
arXiv2014

Poisson AKSZ theories and their quantizations

We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.

math-ph↗