Chromatic structures in stable homotopy theory

In this survey, we review how the global structure of the stable homotopy category gives rise to the chromatic filtration. We then discuss computational tools used in the study of local chromatic homotopy theory, leading up to recent developments in the field. Along the way, we illustrate the key methods and results with explicit examples.Comment: To appear in the Handbook of Homotopy Theory. All comments welcom

The topological modular forms of RP2\mathbb{R}P^2 and RP2∧CP2\mathbb{R}P^2 \wedge \mathbb{C}P^2

In this paper, we study the elliptic spectral sequence computing $tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, $\eta$, and $\nu$. For $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$, we also compute the effect of the $v_1$-self maps of $\mathbb{R} P^2 \wedge \mathbb{C} P^2$ on $tmf$-homology

On the slice spectral sequence for quotients of norms of Real bordism

In this paper, we study equivariant quotients of the multiplicative norm $MU^{((C_{2^n}))}$ of the Real bordism spectrum by permutation summands, a concept defined here. These quotients are interesting because of their relationship to the so-called "higher real $K$-theories". We provide new tools for computing the equivariant homotopy groups of such quotients of $MU^{((C_{2^n}))}$ and quotients of the closely related spectrum $BP^{((C_{2^n}))}$. As a new example, we study spectra denoted by $BP^{((C_{2^n}))}\langle m,m\rangle$, which have non-trivial chromatic localizations only at heights equal to $rm$ where $0\leq r\leq 2^{n-1}$. These spectra are natural equivariant generalizations of integral Morava-$K$-theories. For $\sigma$ the real sign representation of $C_{2^{n-1}}$, we give a complete computation of the $a_{\sigma}$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$. We do this by establishing a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. We also give a complete computation of the $a_{\lambda}$-localized slice spectral sequence of $BP^{((C_{4}))}\langle 2,2\rangle$ for $\lambda$ a rotation of $\mathbb{R}^2$ by an angle of $\pi/2$. The non-localized slice spectral sequences can be recovered completely from these localizations.Comment: 55 pages, 15 figures. Comments welcome

Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction

We consider how the outputs of the Kadison transitivity theorem and Gelfand-Naimark-Segal construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation $(\mathcal{H}, \pi)$ of a $C^*$-algebra $\mathfrak{A}$ and $n \in \mathbb{N}$, there exists a continuous function $A:X \rightarrow \mathfrak{A}$ such that $\pi(A(\mathbf{x}, \mathbf{y}))x_i = y_i$ for all $i \in \{1, \ldots, n\}$, where $X$ is the set of pairs of $n$-tuples $(\mathbf{x}, \mathbf{y}) \in \mathcal{H}^n \times \mathcal{H}^n$ such that the components of $\mathbf{x}$ are linearly independent. Versions of this result where $A$ maps into the self-adjoint or unitary elements of $\mathfrak{A}$ are also presented. Regarding the Gelfand-Naimark-Segal construction, we prove that given a topological $C^*$-algebra fiber bundle $p:\mathfrak{A} \rightarrow Y$, one may construct a topological fiber bundle $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ whose fiber over $y \in Y$ is the space of pure states of $\mathfrak{A}_y$ (with the norm topology), as well as bundles $\mathscr{H} \rightarrow \mathscr{P}(\mathfrak{A})$ and $\mathscr{N} \rightarrow \mathscr{P}(\mathfrak{A})$ whose fibers $\mathscr{H}_\omega$ and $\mathscr{N}_\omega$ over $\omega \in \mathscr{P}(\mathfrak{A})$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $\omega$. When $p:\mathfrak{A} \rightarrow Y$ is a smooth fiber bundle, we show that $\mathscr{P}(\mathfrak{A}) \rightarrow Y$ and $\mathscr{H}\rightarrow \mathscr{P}(\mathfrak{A})$ are also smooth fiber bundles; this involves proving that the group of $*$-automorphisms of a $C^*$-algebra is a Banach-Lie group. In service of these results, we review the geometry of the topology and pure state space. A simple non-interacting quantum spin system is provided as an example

Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems

This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems, rather than considering a single Hamiltonian, one considers a family of Hamiltonians that depend continuously on some parameters. After discussing the notion of phases of parametrized systems, we formulate a bulk-boundary correspondence for an important bulk quantity, the Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and then in arbitrary dimension. This clarifies the physical interpretation of the higher Berry curvature, which in one spatial dimension is a flow of (ordinary) Berry curvature. In d dimensions, the higher Berry curvature is a flow of (d-1)-dimensional higher Berry curvature. Based on this, we discuss one-dimensional systems that pump Chern number to/from spatial boundaries, resulting in anomalous boundary modes featuring isolated Weyl points. In higher dimensions, there are pumps of the analogous quantized invariants obtained by integrating the higher Berry curvature. We also discuss the consequences for parametrized systems of Kitaev's proposal that invertible phases are classified by a generalized cohomology theory, and emphasize the role of the suspension isomorphism in generating new examples of parametrized systems from known invertible phases. Finally, we present a pair of general quantum pumping constructions, based on physical pictures introduced by Kitaev, which take as input a d-dimensional parametrized system, and produce new (d+1)-dimensional parametrized systems. These constructions are useful for generating examples, and we conjecture that one of the constructions realizes the suspension isomorphism in a generalized cohomology theory of invertible phases.Comment: 35 pages, 11 figures. v2: references adde

Charting the space of ground states with tensor networks

We employ matrix product states (MPS) and tensor networks to study topological properties of the space of ground states of gapped many-body systems. We focus on families of states in one spatial dimension, where each state can be represented as an injective MPS of finite bond dimension. Such states are short-range entangled ground states of gapped local Hamiltonians. To such parametrized families over $X$ we associate a gerbe, which generalizes the line bundle of ground states in zero-dimensional families (\emph{i.e.} in few-body quantum mechanics). The nontriviality of the gerbe is measured by a class in $H^3(X, \mathbb{Z})$, which is believed to classify one-dimensional parametrized systems. We show that when the gerbe is nontrivial, there is an obstruction to representing the family of ground states with an MPS tensor that is continuous everywhere on $X$. We illustrate our construction with two examples of nontrivial parametrized systems over $X=S^3$ and $X = \mathbb{R} P^2 \times S^1$. Finally, we sketch using tensor network methods how the construction extends to higher dimensional parametrized systems, with an example of a two-dimensional parametrized system that gives rise to a nontrivial 2-gerbe over $X = S^4$.Comment: 25 pages, 2 figure