arXiv subjects

Luuk Stehouwer

Publications and source records attributed to Luuk Stehouwer.

15 recordsLinked to original sources

arXiv2026

Super $K$-theory and group completion

We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of finitely generated projective graded modules, rather than from spaces of Fredholm operators or Kasparov cycles. We define a connective spectrum $k^{\mathrm{ABS}}_A$ refining the Atiyah--Bott--Shapiro construction as a cofiber, together with its periodification $K^{\mathrm{gr}}_A$, and show that both are lax symmetric monoidal and functorial in bimodules, not merely in homomorphisms. The failure of graded $A$-modules to present all cocycles in $K^{\mathrm{gr}}_0(A)$ is shown to be purely a $\pi_0$-phenomenon on $k^{\mathrm{ABS}}_A$. We obtain a natural equivalence $K^{\mathrm{gr}}_{A\widehat{\otimes} \mathrm{Cl}_{p,q}} \simeq \Sigma^{p-q}K^{\mathrm{gr}}_A$, which links topological Bott periodicity with the Morita equivalence between $\mathrm{Cl}_8$ and $\mathbb{R}$. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor $\operatorname{Pic}(\operatorname{Bim}(\mathrm{sBan}_{\mathbb{R}})^{\mathrm{fd}}) \to \operatorname{Pic}(\mathrm{Mod}(KO))$ which splits off the bottom three Postnikov layers of $\operatorname{Pic}(\mathrm{Mod}(KO))$, giving a direct link between super division algebras and invertible $KO$-modules. We also give spectral refinements of Karoubi's and van Daele's graded $K$-groups, with explicit comparison equivalences, therefore connecting to $KK$-theory. We also provide an extensive general treatment for $K$-theory of ungraded topological rings that might be of independent interest. In particular, we characterize connective topological $K$-theory of ungraded Banach algebras by a universal property.

math.KT↗

arXiv2026

The many faces of higher Hilbert spaces

Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.

math.QA↗

arXiv2026

Unraveling the Bott spiral

We construct and compute a homotopy-theoretic model for the Bott spiral of symmetry-protected topological phases (SPTs) studied by Queiroz--Khalaf--Stern. We model free and interacting fermionic SPTs using K-theory and reflection-positive invertible field theories (IFTs), resp., and define a twisted generalization of the Atiyah--Bott--Shapiro orientation to produce a free-to-interacting map. We also define and compute spiral maps of IFTs to model dimensional reduction in this context, answering a question of Hason--Komargodski--Thorngren. Our analysis highlights two general aspects of homotopical free-to-interacting maps. First, IFTs are more sensitive than K-theory is to the input symmetry data; in particular, the specification of an Altland--Zirnbauer class is insufficient information to define symmetry type for an IFT. Second, the remnant of Bott periodicity on the interacting side relies on an isomorphism of two extraspecial groups of order 32. Our computations use a novel 4-periodic description of a sector of the twisted ko-homology of elementary abelian 2-groups.

math-ph↗

arXiv2025

Free phases of Majorana fermions: Tenfold ways compared

We provide a mathematically rigorous classification of symmetry-protected topological (SPT) phases of neutral free fermions. Our approach utilizes Karoubi triples with negative squares, thought of as polarizations. We prove that neutral free fermion SPT phases protected by a symmetry algebra $A$ are classified by the real $K$-theory group $K_2(A^{op})$, and demonstrate how our classification reproduces known results in the presence of charge. Our formalism also allows for symmetries described by groups, potentially with time-reversal, using the formalism of fermionic groups and their fermionic group $C^*$-algebras. Our classification extends to positive spatial dimensions and includes weak phases using the crystalline equivalence principle. Our approach clarifies and unifies various existing tenfold way classifications by establishing their equivalence through Morita equivalences of symmetry algebras. We expect our classification to be the natural domain for the free-to-interacting map proposed by Freed and Hopkins.

math-ph↗

arXiv2025

Is Crane--Yetter fully extended?

We revisit the question of whether the Crane-Yetter topological quantum field theory (TQFT) associated to a modular tensor category admits a fully extended refinement. More specifically, we use tools from stable homotopy theory to classify extensions of invertible four-dimensional TQFTs to theories valued in symmetric monoidal 4-categories whose Picard spectrum has nontrivial homotopy only in degrees 0 and 4. We show that such extensions are classified by two pieces of data: an equivalence class of an invertible object in the target and a sixth root of unity. Applying this result to the 4-category $\mathbf{BrFus}$ of braided fusion categories, we find that there are infinitely many equivalence classes of fully extended invertible TQFTs reproducing the Crane-Yetter partition function on top-dimensional manifolds, parametrized by a $\mathbb{Z}/6$-extension of the Witt group of nondegenerate braided fusion categories. This analysis clarifies common claims in the literature and raises the question of how to naturally pick out the $SO(4)$-fixed point data on the framed TQFT which assigns the input braided fusion category to the point so that it selects the Crane-Yetter state-sum.

math-ph↗

arXiv2025

SKK groups of manifolds and non-unitary invertible TQFTs

This work considers the computation of controllable cut-and-paste groups $\mathrm{SKK}^ξ_n$ of manifolds with tangential structure $ξ:B_n\to BO_n$. To this end, we apply the work of Galatius-Madsen-Tillman-Weiss, Genauer and Schommer-Pries, who showed that for a wide range of structures $ξ$ these groups fit into a short exact sequence that relates them to bordism groups of $ξ$-manifolds with kernel generated by the disc-bounding $ξ$-sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of $ξ$-manifolds. We are thus led to address two key questions: the existence of $ξ$-manifolds with odd Euler characteristic of a given dimension and conditions for the exact sequence to admit a splitting. We resolve these questions in a wide range of cases. $\mathrm{SKK}$ groups are of interest in physics as they play a role in the classification of non-unitary invertible topological quantum field theories, which classify anomalies and symmetry protected topological (SPT) phases of matter. Applying our topological results, we give a complete classification of non-unitary invertible topological quantum field theories in the tenfold way in dimensions 1-5.

math.AT↗

arXiv2024

Weak topological phases in the presence of interactions

We study weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions. By comparing homotopical free and interacting classifications of these SPTs, we predict their stability under interactions as well as identify potential intrinsically-interacting phases. We mathematically compute the groups of weak phases in dimensions zero through three for all tenfold-way symmetry types using homotopy theory; specifically, we use Atiyah's Real $\mathit{KR}$-theory and the low-energy invertible field theory ansatz of Freed--Hopkins for the free and interacting cases, resp. Our computational techniques involve T-duality, which relates $K$-theory of the spatial torus with $K$-theory of the Brillouin torus, and a binomial formula for computing generalized cohomology of a torus. Our results carry potential implications for theoretical and experimental studies of weak phases.

math-ph↗

arXiv2024

The spin-statistics theorem for topological quantum field theories

We establish the spin-statistics theorem for topological quantum field theories (TQFTs) in the framework of Atiyah. We incorporate spin via spin structures on bordisms, and represent statistics using super vector spaces. Unitarity is implemented using dagger categories, in a manner that is equivalent to the approach of Freed-Hopkins, who employed $\mathbb{Z}/2$-equivariant functors to address reflection-positivity. A key contribution of our work is the introduction of the notion of fermionically dagger compact categories, which extends the well-established concept of dagger compact categories. We show that both the spin bordism category and the category of super Hilbert spaces are examples of fermionically dagger compact categories. The spin-statistics theorem for TQFTs emerges as a specific case of a more general result concerning symmetric monoidal dagger functors between fermionically dagger compact categories.

math-ph↗

arXiv2024

A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories

We prove that every unitary invertible quantum field theory satisfies a generalization of the famous spin-statistics theorem. To formulate this extension, we define a `higher spin' action of the stable orthogonal group $O$ on appropriate spacetime manifolds, which extends both the reflection involution and spin flip. On the algebraic side, we define a `higher statistics' action of $O$ on the universal target for invertible field theories, $I\mathbb{Z}$, which extends both complex conjugation and fermion parity $(-1)^F$. We prove that every unitary invertible quantum field theory intertwines these actions.

math-ph↗

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

Dagger categories via anti-involutions and positivity

Dagger categories are an essential tool for categorical descriptions of quantum physics, for example in categorical quantum mechanics and unitary topological field theory. Their definition however is in tension with the ``principle of equivalence'' that lies at the heart of category theory, thereby inhibiting generalizations to higher categories. In this note we propose an alternative, coherent description of dagger categories based on the well-studied notion of anti-involutions $d\colon \mathcal{C} \to \mathcal{C}^{op}$, which coherently square to the identity functor $\eta\colon d^2 \cong \operatorname{id}_{\mathcal{C}}$. A general anti-involution need not be the identity on objects, but we instead consider certain isomorphisms $dx \cong x$, which we call Hermitian fixed points as they generalize the notion of a Hermitian inner product on a vector space. We define a ``positivity notion" on $(\mathcal{C},d, \eta)$ in terms of such Hermitian fixed points. This terminology is motivated by the dagger category of Hilbert spaces, in which case the positivity notion consists of the positive definite pairings. Our main result is that the $2$-category of anti-involutive categories with a positivity notion is biequivalent to the $2$-category of dagger categories.

math.CT↗

arXiv2023

Reflection Structures and Spin Statistics in Low Dimensions

We give a complete classification of topological field theories with reflection structure and spin-statistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group $G$ which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of $G$. In spacetime dimension two we give a classification in terms strongly $G$-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

math-ph↗

arXiv2022

Interacting SPT phases are not Morita invariant

The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with $K$-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by $K$-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita-equivalent but different symmetry groups. As $K$-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.

hep-th↗

arXiv2021

Some properties of $\operatorname{Pin}^\pm$-structures on compact surfaces

We show that two $\operatorname{Pin}$-structures on a surface differ by a diffeomorphism of the surface if and only if they are cobordant (for comparison, the analogous fact has already been shown for $\operatorname{Spin}$-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of $\operatorname{Pin}$-structures on a surface in a given cobordism class.

math.GT↗

arXiv2018

Classification of crystalline topological insulators through K-theory

Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.

cond-mat.mes-hall↗