arXiv subjects

Agnes Beaudry

Publications and source records attributed to Agnes Beaudry.

At least 19 recordsLinked to original sources

arXiv2025

A Classifying Space for Phases of Matrix Product States

We construct a topological space $\mathcal{B}$ consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions and show that it has the weak homotopy type $K(\mathbb{Z}, 2) \times K(\mathbb{Z}, 3)$. The implication is that the phase of a family of such states parametrized by a space $X$ is completely determined by two invariants: a class in $H^2(X; \mathbb{Z})$ corresponding to the Chern number per unit cell and a class in $H^3(X; \mathbb{Z})$, the so-called Kapustin-Spodyneiko (KS) number. The space $\mathcal{B}$ is defined as the quotient of a contractible space $\mathcal{E}$ of MPS tensors by an equivalence relation describing gauge transformations of the tensors. We prove that the projection map $p:\mathcal{E} \rightarrow \mathcal{B}$ is a quasifibration, and this allows us to determine the weak homotopy type of $\mathcal{B}$. As an example, we review the Chern number pump-a family of MPS parametrized by $S^3$-and prove that it generates $\pi_3(\mathcal{B})$.

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arXiv2023

Homotopical Foundations of Parametrized Quantum Spin Systems

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite dimensional Hilbert space to the pure state space of the quasi-local algebra of the quantum spin system with this Hilbert space at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to $\mathscr{E}_\infty$-spaces for an operad we call the "multiplicative" linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step towards understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

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arXiv2022

The Exotic $K(2)$-Local Picard Group at the Prime $2$

We calculate the group $\kappa_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group.

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arXiv2022

Cohomology of the Morava stabilizer group through the duality resolution at $n=p=2$

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.

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arXiv2021

Quotient rings of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

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arXiv2020

Transchromatic extensions in motivic and Real bordism

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $\pi_{(\ast,\ast)}MGL$-module structure of the motivic Morava $K$-theories, including non-trivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $\pi_{\star}MU_\mathbb R$-module structure on the Real Morava $K$-theories.

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arXiv2020

Models of Lubin-Tate spectra via Real bordism theory

We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.

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arXiv2019

The telescope conjecture at height 2 and the tmf resolution

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end we compute the structure of the tmf-resolution for a specifc type 2 complex Z. We find that, analogous to the height 1 case, the E1-page of the tmf-resolution possesses a decomposition into a v2-periodic summand, and an Eilenberg-MacLane summand which consists of bounded v2-torsion. However, unlike the height 1 case, the E2-page of the tmf-resolution exhibits unbounded v2-torsion. We compare this to the work of Mahowald-Ravenel-Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v2-torsion: either the unbounded v2-torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2-parabolas and the telescope conjecture is false. We also study how to use the tmf-resolution to effectively give low dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)-local Adams-Novikov spectral sequence for Z collapses.

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arXiv2019

Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra

We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order 32 generated by $ E\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\wedge S^{7+\sigma}$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum $E$.

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arXiv2018

A Guide for Computing Stable Homotopy Groups

This paper contains an overview of background from stable homotopy theory used by Freed--Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.

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arXiv2018

Computations of Orbits for the Lubin-Tate Ring

We take a direct approach to computing the orbits for the action of the automorphism group $\mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{\mathbb{G}_2} \cong \mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $p\geq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.

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arXiv2017

The $\alpha$-Family in the $K(2)$-Local Sphere at the Prime $2$

In this note, we compute the image of the $\alpha$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.

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arXiv2017

Chromatic splitting for the $K(2)$-local sphere at $p=2$

We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1L_{K(2)}S^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H^\ast_c(\mathbb{G}_2,E_0)$ where $\mathbb{G}_2$ is the Morava stabilizer group and $E_0 = \mathbb{W}[[u_1]]$ is the ring of functions on the height $2$ Lubin-Tate space. We show that the inclusion of the constants $\mathbb{W} \to E_0$ induces an isomorphism on group cohomology, a radical simplification.

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arXiv2017

Gross-Hopkins Duals of Higher Real K-theory Spectra

We determine the Gross-Hopkins duals of certain higher real $K$-theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy fixed point spectra $E_n^{hG}$ are Gross-Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups $G$ which contain $p$-torsion. This generalizes previous results for $n=2$ and $p=3$.

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arXiv2017

On the E2-term of the bo-Adams spectral sequence

The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

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arXiv2016

Motivic Homotopical Galois Extensions

We establish a formal framework for Rognes's homotopical Galois theory and adapt it to the context of motivic spaces and spectra. We discuss examples of Galois extensions between Eilenberg-MacLane motivic spectra and between the Hermitian and algebraic K-theory spectra.

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arXiv2015

The Algebraic Duality Resolution at $p=2$

The goal of this paper is to develop some of the machinery necessary for doing $K(2)$-local computations in the stable homotopy category using duality resolutions at the prime $p=2$. The Morava stabilizer group $\mathbb{S}_2$ admits a norm whose kernel we denote by $\mathbb{S}_2^1$. The algebraic duality resolution is a finite resolution of the trivial $\mathbb{Z}_2[[\mathbb{S}_2^1]]$-module $\mathbb{Z}_2$ by modules induced from representations of finite subgroups of $\mathbb{S}_2^1$. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial $\mathbb{Z}_3[[\mathbb{G}_2^1]]$-module $\mathbb{Z}_3$ at the prime $p=3$. The construction was never published and it is the main result in this paper. In the process, we give a detailed description of the structure of Morava stabilizer group $\mathbb{S}_2$ at the prime $2$. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

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arXiv2015

The Chromatic Splitting Conjecture at n=p=2

We show that the strongest form of Hopkins' chromatic splitting conjecture, as stated by Hovey, cannot hold at chromatic level n=2 at the prime p=2. More precisely, for V(0) the mod 2 Moore spectrum, we prove that the kth homotopy group of L_1L_{K(2)}V(0) is not zero when k is congruent to -3 modulo 8. We explain how this contradicts the decomposition of L_1L_{K(2)}S predicted by the chromatic splitting conjecture.

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