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Markus J. Pflaum

Publications and source records attributed to Markus J. Pflaum.

At least 19 recordsLinked to original sources
arXiv2026

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 $π_3(\mathcal{B})$.

math-ph↗
arXiv2025

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 $X = S^4$.

cond-mat.str-el↗
arXiv2024

Algebraic Machine Learning with an Application to Chemistry

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominately on the manifold hypothesis, which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities. In this paper we develop a machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gröbner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety.

math.AG↗
arXiv2024

On the Weak Contractibility of the Space of Pure States

We prove that the space $\mathscr{P}(\mathfrak{A})$ of pure states of a nonelementary, simple, separable, real rank zero $C^*$-algebra $\mathfrak{A}$ has trivial homotopy groups of all orders when $\mathscr{P}(\mathfrak{A})$ is equipped with the weak* topology. The convex-valued and finite-dimensional selection theorems of Michael are used to deform a family of pure states via the action of a homotopy of unitaries so that the entire family evaluates to one on a given projection $P \in \mathfrak{A}$. Then, the excision theorem of Akemann, Anderson, and Pedersen is used to iterate this deformation for a sequence of projections in $\mathfrak{A}$ excising a base point of the family of pure states, thereby contracting the family to the base point. Finally, we compare our weak contractibility result to the spaces of pure states of commutative $C^*$-algebras and rational rotation algebras, and compute the homotopy groups of the latter in terms of the homotopy groups of spheres.

math.OA↗
arXiv2022

Localization in Hochschild homology

Localization methods are ubiquitous in cyclic homology theory, but vary in detail and are used in different scenarios. In this paper we will elaborate on a common feature of localization methods in noncommutative geometry, namely sheafification of the algebra under consideration and reduction of the computation to the stalks of the sheaf. The novelty of our approach lies in the methods we use which mainly stem from real instead of complex algebraic geometry. We will then indicate how this method can be used to determine the Hochschild homology theory of more complicated algebras out of simpler ones.

math.KT↗
arXiv2022

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney B stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.

math.DG↗
arXiv2022

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 $(\mathcal{H}, π)$ of a $C^*$-algebra $\mathfrak{A}$ and $n \in \mathbb{N}$, there exists a continuous function $A:X \rightarrow \mathfrak{A}$ such that $π(A(\mathbf{x}, \mathbf{y}))x_i = y_i$ 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 $\mathscr{H}_ω$ and $\mathscr{N}_ω$ over $ω\in \mathscr{P}(\mathfrak{A})$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $ω$. 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.

math.OA↗
arXiv2022

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.

cond-mat.str-el↗
arXiv2020

Deformation quantization and homological reduction of a lattice gauge model

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on $G$. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $G = \operatorname{SU} (2)$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the - in general singular - symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

math-ph↗
arXiv2020

On the Hochschild homology of convolution algebras of proper Lie groupoids

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski's ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski's conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.

math.KT↗
arXiv2019

Equivariant control data and neighborhood deformation retractions

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \subset X$ is a closed $G$-stratified subspace which is a union of strata of $X$, then the inclusion $i : A \hookrightarrow X$ is a $G$-equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$-invariant analytic subspace of an analytic $G$-manifold $M$ and $A \hookrightarrow X$ is a closed $G$-invariant analytic subspace of $X$.

math.SG↗
arXiv2019

Invariant Whitney Functions

A theorem of Gerald Schwarz [24, Thm. 1] says that for a linear action of a compact Lie group $G$ on a finite dimensional real vector space $V$ any smooth $G$-invariant function on $V$ can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set $Z\subset V$ fulfilling some regularity assumptions. In order to deal with the case when $Z$ is not $G$-stable we use the language of groupoids.

math.SG↗
arXiv2017

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

math.AP↗
arXiv2014

Whitney functions determine the real homotopy type of a semi-analytic set

In this paper, we investigate the Whitney--de Rham complex $Ω^\bullet_\text{W} (X)$ associated to a semi-analytic subset $X$ of an analytic manifold $M$. This complex is a commutative differential graded algebra, that is defined to be the quotient of the de Rham complex of smooth differential forms on $M$ by the differential graded ideal generated by all smooth functions which are flat on $X$. We use Hironaka's desingularization theorem to prove a Poincaré Lemma for $Ω^\bullet_\text{W} (X)$ holds true, which entails that its cohomology is isomorphic to the real cohomology of $X$. Furthermore, we show that this isomorphism is induced by a quasi-isomorphism of differential graded algebras. Thus it preserves the product structure, and is therefore an isomorphism of commutative differential graded algebras. As a consequence we show, when $X$ is simply connected, that the Whitney--de Rham complex determines the real homotopy type of $X$. This allows one further to conclude that the Hochschild homology of the differential graded algebra $Ω^\bullet_\text{W} (X)$ is isomorphic to the cohomology of the free loop space $\mathcal{L} X$.

math.AT↗
arXiv2013

Stratifications of inertia spaces of compact Lie group actions

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.

math.DG↗
arXiv2013

Quantization of Whitney functions and reduction

For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization to the case where the underlying set is a subset of a not necessarily regular Poisson manifold which can be written as the quotient of a regular Poisson manifold on which a compact Lie group acts freely by Poisson maps. Finally, if the quotient Poisson manifold is regular as well, we show a "quantization commutes with reduction" type result. For the proofs, we use methods stemming from both singularity theory and Poisson geometry.

math.DG↗
arXiv2011

Connes-Chern character for manifolds with boundary and eta cochains

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.

math.OA↗