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Hisham Sati

Publications and source records attributed to Hisham Sati.

At least 37 records · Page 2Linked to original sources
arXiv2026

Synthetic Differential Jet Bundles are Reduced

We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fréchet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.

math.DG↗
arXiv2026

Realization of staircase topological Anderson phase transitions

One-dimensional topological Anderson insulators provide a paradigm for disorder-induced topological phases in which the underlying system turns from a trivial to a topological phase. It is widely recognized that the latter vanishes at large disorder amplitude. Here, and contrary to the general belief, we provide evidence for a successive disorder-driven topological transitions in a single-wall nanotube, culminating in a topological Anderson phase that remains unexpectedly robust at strong disorder. This phenomenon is confirmed by analysis of the corresponding topological invariant, which increases stepwise as disorder increases, giving evidence for the emergence of edge states. We experimentally implement these topological Anderson staircase phase transitions in a one-dimensional topolectrical circuit, where the persistence of edge states is revealed by node-voltage measurements. The robustness of the edge states is corroborated by numerical calculations of their localization properties. Our work opens the road to topological disordertronics, where topological phases can be tuned by disorder.

cond-mat.mes-hall↗
arXiv2026

Higher-Dimensional Anyons via Higher Cohomotopy

We highlight that integer Heisenberg groups at level 2 underlie topological quantum phenomena: their group algebras coincide with the algebras of quantum observables of abelian anyons in fractional quantum Hall (FQH) systems on closed surfaces. Decades ago, these groups were shown to arise as the fundamental groups of the space of maps from the surface to the 2-sphere -- which has recently been understood as reflecting an effective FQH flux quantization in 2-Cohomotopy. Here we streamline and generalize this theorem using the homotopy theory of H-groups, showing that for $k \in \{1,2,4\}$, the non-torsion part of $π_1 \mathrm{Map}\big({(S^{2k-1})^2, S^{2k}}\big)$ is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of $π_{4k-1}(S^{2k})$. This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").

cond-mat.str-el↗
arXiv2025

Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.

cond-mat.str-el↗
arXiv2025

Panhandle polynomials of torus links and geometric applications

We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel of torus knots and links. Due to their panhandle-like intrinsic properties, the HOMFLY-PT polynomial is referred to as a ``panhandle polynomial''. With the help of the $\ell$-invariant, this extends to links the Etnyre-Honda result about the arc index and maximal Thurston-Bennequin invariant of torus knots. It has further geometric consequences, related to the braid index, the existence of minimal string Bennequin surfaces for banded and Whitehead doubled links, the Bennequin sharpness problem, and the equivalence of their quasipositivity and strong quasipositivity. We extend these properties to torus links, which relate to the classification of their component-wise Thurston-Bennequin invariants. Finally, we discuss the definition of the $\ell$-invariant for general links.

math.GT↗
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

Field Theory via Higher Geometry I: Smooth Sets of Fields

Most modern theoretical considerations of the physical world suggest that nature is: (1) field-theoretic, (2) smooth, (3) local, (4) gauged, (5) containing fermions, and (6) non-perturbative. Tautologous as this may sound to experts, it is remarkable that the mathematical notion of geometry which reflects all of these aspects - namely, ``supergeometric homotopy theory'' - has received little attention. Elaborate algebraic machinery is known for perturbative field theories both at the classical and quantum level, but to tackle the deep open questions of the subject, these will need to be lifted to a global geometry of physics. Our aim in this series is to introduce inclined physicists to this theory, to fill mathematical gaps in the existing literature, and to rigorously develop the full power of supergeometric homotopy theory and apply it to the analysis of fermionic (not necessarily super-symmetric) field theories. Secondarily, this will also lead to a streamlined and rigorous perspective we hope would also be desirable to mathematicians. In this first part, we explain how classical bosonic Lagrangian field theory (variational Euler-Lagrange theory) finds a natural home in the ``topos of smooth sets'', thereby neatly setting the scene for the higher supergeometry discussed in later parts of the series. This introductory material will be largely known to a few experts but has never been comprehensively laid out before. A key technical point we make is to regard jet bundle geometry systematically in smooth sets instead of just its subcategories of diffeological spaces or even Fr{é}chet manifolds -- or worse simply as a formal object. Besides being more transparent and powerful, it is only on this backdrop that a reasonable supergeometric jet geometry exists, needed for satisfactory discussion of any field theory with fermions.

math-ph↗
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

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↗