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Urs Schreiber

Publications and source records attributed to Urs Schreiber.

At least 37 records · Page 2Linked to original sources
arXiv2025

Orientations of Orbi-K-Theory measuring Topological Phases and Brane Charges

Topological Chern phases of quantum materials, as well as brane charges on M-theory orbifolds, have famously been argued to be classified by (orbi) topological K-theory, or possibly by other stable and, notably, complex-oriented cohomology theories, such as elliptic cohomology or Morava K-theory. However, closer inspection reveals that the most fine-grained "fragile" microscopic classification in both cases is in (orbi) Cohomotopy, which is the primordial "unstable" or nonabelian generalized cohomology. Coarsening takes the latter (fragile) to the former (stable) cohomology along an nonabelian cohomology operations. But what then is the role of complex orientation on the stable side? We observe here (i) that over gapped nodal lines in the 2D Brillouin torus and on probe M5-branes in 11D spacetime, the cohomotopical phases/charges lift through the complex/quaternionic Hopf fibration, and (ii) that measuring this fragile situation in stable cohomology means equivalently to ask for universal complex/quaternionic orientation on stable cohomology in four/ten dimensions! Then we give an explicit realization of such unstable four/ten-dimensional complex/quaternionic orientation in U(2)/Sp(2)-equivariant K-theory, using real division-algebraic tools within a new model of twisted orbifold K-theory in cohesive homotopy theory; and we explain this as an extraordinary character map from orbi Cohomotopy-twisted Cohomotopy to relative orbi K-theory. Finally, we discuss an application to the classification of 2-band crystalline topological insulator phases sensitive to the topology in the gapping process of their nodal line semimetal parent phase, and to the measurement of M-string charges inside M5-brane probes.

hep-th↗
arXiv2025

Renormalization of Chern-Simons Wilson Loops via Flux Quantization in Cohomotopy

In the practice of physics model building, the process of renormalization, resummation, and anomaly cancellation is to incrementally repair initially ill-defined Lagrangian quantum field theories. Impressive as this is, one would rather have concisely defined complete theories to begin with, and understand these choices as emergent from fundamental principles. As an instructive example, we recall renormalization choices for Wilson loop observables in abelian Chern-Simons theory. Then we show that these emerge in a novel non-Lagrangian topological completion of 5D Maxwell-Chern-Simons QFT, by means of proper flux quantization in 2-Cohomotopy. This result is a modest cousin, with applications to topologically ordered quantum materials, of the more ambitious flux quantization of 11D supergravity in 4-Cohomotopy ("Hypothesis H").

hep-th↗
arXiv2025

The Character Map in Twisted Equivariant Nonabelian Cohomology

The fundamental notion of non-abelian generalized cohomology gained recognition in algebraic topology as the non-abelian Poincaré-dual to "factorization homology", and in theoretical physics as providing flux-quantization for non-linear Gauss laws. However, already the archetypical example -- unstable Cohomotopy, first studied almost a century ago by Pontrjagin -- has remained underappreciated as a cohomology theory and has only recently received attention as a flux-quantizaton law ("Hypothesis H"). Here we lay out a general construction of the analogue of the Chern character map on twisted equivariant non-abelian cohomology theories (with equivariantly simply-connected classifying spaces) and illustrate the construction by spelling out a twisted equivariant form of Cohomotopy as an archetypical and intriguing running example, essentially by computing its equivariant Sullivan model. We close with an outlook on the application of this result to the rigorous deduction of anyonic quantum states on M5-branes wrapped over Seifert 3-orbifolds.

hep-th↗
arXiv2025

Cohomotopy, Framed Links, and Abelian Anyons

We establish a natural identification of cobordism classes of framed links with the fundamental group of the group-completed configuration space of points in the plane, by appeal to Okuyama's previously underappreciated interval configuration model for the latter. Under Segal's theorem, these classes are integers generated by the Hopf generator in the 2-cohomotopy of the 3-sphere, and we identify these knot-theoretically with the writhe sum of the linking and framing number of links. We observe that this link invariant is the properly regularized Wilson line observable of abelian Chern-Simons theory, and show, in consequence of the main theorem, that this arises equivalently as the expectation value of pure quantum states on the group algebra, under link sum, of cobordism classes of framed links. Observing that these quantum states regard framed links as worldlines of anyons in that they assign a fixed complex phase factor to each crossing (braiding) of strands, we close with an outlook on implications for the identification of anyonic solitons in 2D electron gases as exotic flux quantized in Cohomotopy instead of in ordinary cohomology.

hep-th↗
arXiv2025

Super-$\mathrm{Lie}_\infty$ T-Duality and M-Theory

Super $L_\infty$-algebras unify extended super-symmetry with rational classifying spaces for higher flux densities: The super-invariant super-fluxes which control super $p$-branes and their supergravity target super-spaces are, together with their (non-linear) Bianchi identities, neatly encoded in (non-abelian) super-$L_\infty$ cocycles. These are the rational shadows of flux-quantization laws (in ordinary cohomology, K-theory, Cohomotopy, iterated K-theory, etc). We first review, in streamlined form while filling some previous gaps, double-dimensional reduction/oxidation and 10D superspace T-duality along higher-dimensional super-tori. We do so tangent super-space wise, by viewing it as an instance of adjunctions (dualities) between super-$L_\infty$-extensions and -cyclifications, applied to the avatar super-flux densities of 10D supergravity. In particular, this yields a derivation, at the rational level, of the traditional laws of "topological T-duality" from the super-$L_\infty$ structure of type II superspace. At this level, we also discuss a higher categorical analog of T-duality involving M-branes. Then, by considering super-space T-duality along all 1+9 spacetime dimensions while retaining the 11th dimension as in F-theory, we find the M-algebra appearing as the complete brane-charge extension of the fully T-doubled/correspondence super-spacetime. On this backdrop, we recognize the "decomposed" M-theory 3-form on the "hidden M-algebra" as an M-theoretic lift of the Poincaré super 2-form that controls superspace T-duality as the integral kernel of the super Fourier-Mukai transform. This provides the super-space structure of an M-theory lift of the doubled/correspondence space geometry, which controls T-duality.

hep-th↗
arXiv2025

Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

Recently observed fractional quantum anomalous Hall materials (FQAH) are candidates for topological quantum hardware, but their required anyon states are elusive. We point out dependence on monodromy in the fragile band topology in 2-cohomotopy. An algebro-topological theorem of Larmore & Thomas (1980) then identifies FQAH anyons over momentum space. Admissible braiding phases are 2C-th roots of unity, for C the Chern number. This lays the foundation for understanding symmetry-protected topological order in FQAH systems, reducing the problem to computations in equivariant cohomotopy.

cond-mat.str-el↗
arXiv2025

Flux Quantization on 11-dimensional Superspace

Flux quantization of the C-field in 11d supergravity is arguably necessary for the (UV-)completion of the theory, in that it determines the torsion charges carried by small numbers of M-branes. However, hypotheses about C-field flux-quantization ("models of the C-field") have previously been discussed only in the bosonic sector of 11d supergravity and ignoring the supergravity equations of motion. Here we highlight a duality-symmetric formulation of on-shell 11d supergravity on superspace, observe that this naturally lends itself to completion of the theory by flux quantization, and indeed that 11d super-spacetimes are put on-shell by carrying quantizable duality-symmetric super-C-field flux; the proof of which we present in detail.

hep-th↗
arXiv2025

Flux Quantization on M5-branes

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac's charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5's flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 "super-embeddings". Finally, assuming the flux-quantization law to actually be in co-Homotopy ("Hypothesis H") we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of "open M5-branes" in heterotic M-theory.

hep-th↗
arXiv2025

Holographic M-Brane Super-Embeddings

Over a decade before the modern formulation of AdS/CFT duality, Duff et al. had observed a candidate microscopic explanation by identifying the CFT fields with fluctuations of probe p-branes stretched out in parallel near the horizon of their own black brane incarnation. A profound way to characterize these and more general probe p-brane configurations, especially for M5-branes, is expected to be as "super-embeddings" of their super-worldvolumes into target super-spacetime - but no concrete example of these had appeared in the literature. Here we fill this gap by constructing the explicit holographic super-embeddings of probe M5-branes and M2-branes into their corresponding super-AdS backgrounds.

hep-th↗
arXiv2024

Higher Topos Theory in Physics

A brief exposition of the point of higher topos theory in (mathematical) physics, commissioned for the Encyclopedia of Mathematical Physics 2nd ed.

math-ph↗
arXiv2024

Quantum Observables of Quantized Fluxes

While it has become widely appreciated that defining (higher) gauge theories requires, in addition to ordinary phase space data, also "flux quantization" laws in generalized differential cohomology, there has been little discussion of the general rules, if any, for lifting Poisson-brackets of (flux-)observables and their quantization from traditional phase spaces to the resulting higher moduli stacks of flux-quantized gauge fields. In this short note, we present a systematic analysis of (i) the canonical quantization of flux observables in Yang-Mills theory and (ii) of valid flux quantization laws in abelian Yang-Mills, observing (iii) that the resulting topological quantum observables form the homology Pontrjagin algebra of the loop space of the moduli space of flux-quantized gauge fields. This is remarkable because the homology Ponrjagin algebra on loops of moduli makes immediate sense in broad generality for higher and non-abelian (non-linearly coupled) gauge fields, such as for the C-field in 11d supergravity, where it recovers the quantum effects previously discussed in the context of "Hypothesis H".

hep-th↗
arXiv2024

The M-algebra completes the hierarchy of Super-Exceptional Tangent Spaces

The conjectured symmetries of M-theory famously involve (1.) brane-extended super-symmetry (the M-algebra) and (2.) exceptional duality-symmetry (the $\mathfrak{e}_{11}$-algebra); but little attention has been given to their inevitable combination. In this little note, we highlight (by combining results available in the literature) that the $local$ exceptional duality-symmetry (the hyperbolic involutory $\mathfrak{k}_{1,10} \subset \mathfrak{e}_{11}$) acts on the M-algebra, through the "brane-rotating symmetry" $\mathfrak{sl}_{32}$, in a way which extends the known hierarchy of finite-dimensional $n$-exceptional tangent spaces compatibly beyond the traditional bound of $n \leq 7$ all the way to $n = 11$.

hep-th↗
arXiv2024

Anyons on M5-Probes of Seifert 3-Orbifolds via Flux Quantization

We observe that there is a rigorous derivation of (abelian) anyonic quantum states, hence of "topological order", on the 1+2-dimensional fixed locus of M5-probes wrapped over a trivially Seifert-fibered 3-orbifold singularity. Similar statements have previously been conjectured by appeal to the unknown dynamics of "coincident" M5-branes, but neglecting effects of flux-quantization that, as we highlight, entail anyonic solitons already in the rigorously tractable case of single M5-brane probes. This is possible after globally completing the "self-dual" tensor field on probe M5-branes by flux quantization in the non-abelian cohomology theory called equivariant twistorial Cohomotopy, which is admissible by recent results.

hep-th↗
arXiv2024

The Hidden M-Group

Following arguments that the (hidden) M-algebra serves as the maximal super-exceptional tangent space for 11D supergravity, we make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of the ''decomposed'' M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries. As a simple but consequential application, we highlight how to describe lattice subgroups $\mathbb{Z}^{k \leq 528}$ of the hidden M-group that allow to toroidially compactify also the ''hidden'' dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality. In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the ''decompose'' M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice. Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations is the dimensional reduction of the hidden M-algebra to a ''hidden IIA-algebra'' which in a companion article we explain as the exceptional extension of the T-duality doubled super-spacetime.

hep-th↗
arXiv2024

Topological QBits in Flux-Quantized Super-Gravity

We first give a brief exposition of our recent realization of anyonic quantum states on single M5-brane probes in 11D super-gravity backgrounds, by non-perturbative quantization of the topological sector of the self-dual tensor field on the 6D worldvolume, after its proper flux-quantization. This opens the prospect of holographic models for topological qbits away from the usual but unrealistic limit of large numbers of branes. At the same time, the elementary homotopy-theoretic nature of the construction yields a slick expression of topological quantum gates in homotopically-typed programming languages, opening the prospect of topological-hardware aware quantum programming. In view of these results, we end with some more meta-physical remarks on (cohesive) homotopy (type) theory in view of emergent fundamental physics and, possibly, M-theory.

hep-th↗
arXiv2024

Flux Quantization

Flux- and charge-quantization laws for higher gauge fields of Maxwell type -- e.g. the common electromagnetic field (the "A-field") but also the B-, RR-, and C-fields considered in string/M-theory -- specify non-perturbative completions of these fields by encoding their solitonic behaviour and hence by specifying the discrete charges carried by the individual branes (higher-dimensional monopoles or solitons) that source the field fluxes. This article surveys the general (rational-)homotopy theoretic understanding of flux- and charge-quantization via the Chern-Dold character map generalized to the non-linear (self-sourcing) Bianchi identities that appear in higher-dimensional supergravity theories, notably for B&RR-fields in D=10, for the C-field in D=11 supergravity, and for the B-field on fivebrane worldvolumes.

hep-th↗
arXiv2024

Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory

While the classification of non-interacting crystalline topological insulator phases by equivariant K-theory has become widely accepted, its generalization to anyonic interacting phases -- hence to phases with topologically ordered ground states supporting topological braid quantum gates -- has remained wide open. On the contrary, the success of K-theory with classifying non-interacting phases seems to have tacitly been perceived as precluding a K-theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-theory. Here we provide a detailed argument for the classification of symmetry protected/enhanced su(2)-anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-theory of configuration spaces of points in the complement of nodal points inside the crystal's Brillouin torus orbi-orientifold. We argue, in particular, that: (1) topological 2d semi-metal phases modulo global mass terms are classified by the flat differential twisted equivariant K-theory of the complement of the nodal points; (2) n-electron interacting phases are classified by the K-theory of configuration spaces of n points in the Brillouin torus; (3) the somewhat neglected twisting of equivariant K-theory by "inner local systems" reflects the effective "fictitious" gauge interaction of Chen, Wilczeck, Witten & Halperin (1989), which turns fermions into anyonic quanta; (4) the induced su(2)-anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations.

hep-th↗