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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 involutary $\mathfrak{k}_{1,10} \subset \mathfrak{e}_{11}$) 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 $n = 11$., Comment: 13 pages

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., Comment: 16 pages, some figures -- invited contribution to: Arsiwalla, Elshatlawy & Rickles (eds.) "Quantum Gravity and Computation"

While fractional quantum Hall systems provide the best experimental evidence yet of (abelian) anyons plausibly necessary for future fault-tolerant quantum computation, like all strongly-coupled quantum systems their physics is not deeply understood. However, generally a promising approach is to (holographically) realize such systems on branes in string/M-theory; and specifically an old argument by Hellerman & Susskind gives a sketch of fractional quantum Hall states arising via discrete light cone quantization of M5/M9-brane intersections. Here we present a rigorous derivation of abelian anyon quantum states on M5$\perp$MO9-branes ("open M5-branes") on the discrete light cone, after globally completing the traditional local field content on the M5-worldvolume via a flux-quantization law compatible with the ambient 11d supergravity, specifically taken to be in the unstable co-Homotopy cohomology ("Hypothesis H"). The main step in the proof uses a theorem of Okuyama to identify co-Homotopy moduli spaces with configuration spaces of strings with charged endpoints, and identifies their loop spaces with cobordism of framed links that, under topological light cone quantization, turn out to be identified with the regularized Wilson loops of abelian Chern-Simons theory., Comment: 22 pages

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-embedding of probe M5-branes and M2-branes into their corresponding super-AdS backgrounds. The result seems to be novel and shows that the holographic M-probes, static with respect to the Poincar\'e chart, must sit exactly at the throat radius., Comment: 28 pages

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., Comment: 44 pages; v3: published version, abstract shortened and typos fixed; v2: minor additions

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., Comment: 62 pages; v3: added section 2.1.7 and various minor improvements; v2: fixed lemma 2.48 and added lemma 3.6

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., Comment: 34+6 pages, many figures; v2: published version, commissioned for the Encyclopedia of Mathematical Physics 2nd ed

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"., Comment: 20 pages

While it has become widely appreciated that (higher) gauge theories need, besides their variational phase space data, to be equipped with "flux quantization laws" in generalized differential cohomology, there used to be no general prescription for how to define and construct the resulting flux-quantized phase space stacks. In this short note we observe that all higher Maxwell-type equations have solution spaces given by flux densities on a Cauchy surface subject to a higher Gauss law and no further constraint: The metric duality-constraint is all absorbed into the evolution equation away from the Cauchy surface. Moreover, we observe that the higher Gauss law characterizes the Cauchy data as flat differential forms valued in a characteristic L-infinity-algebra. Using the recent construction of the non-abelian Chern-Dold character map, this implies that compatible flux quantization laws on phase space have classifying spaces whose rational Whitehead L-infinity algebra is this characteristic one. The flux-quantized higher phase space stack of the theory is then simply the corresponding (generally non-abelian) differential cohomology moduli stack on the Cauchy surface. We show how this systematic prescription subsumes existing proposals for flux-quantized phase spaces of vacuum Maxwell theory and of the chiral boson. Moreover, for the case of NS/RR-fields in type II supergravity, the traditional "Hypothesis K" of flux quantization in topological K-theory is naturally implied, without the need, on phase space, for the notorious further duality constraint. Finally, as a genuinely non-abelian example, we consider flux-quantization of the C-field in 11d supergravity/M-theory given by unstable differential 4-Cohomotopy ("Hypothesis H") and emphasize again that, implemented on Cauchy data, this qualifies as the full phase space without the need for a further duality constraint., Comment: 23 pages; v3: published version, with a little more background added to section 3.1; v2: more references added & typos fixed

Formalizations of quantum information theory in category theory and type theory, for the design of verifiable quantum programming languages, need to express its two fundamental characteristics: (1) parameterized linearity and (2) metricity. The first is naturally addressed by dependent-linearly typed languages such as Proto-Quipper or, following our recent observations: Linear Homotopy Type Theory (LHoTT). The second point has received much attention (only) in the form of semantics in "dagger-categories", where operator adjoints are axiomatized but their specification to Hermitian adjoints still needs to be imposed by hand. We describe a natural emergence of Hermiticity which is rooted in principles of equivariant homotopy theory, lends itself to homotopically-typed languages and naturally connects to topological quantum states classified by twisted equivariant KR-theory. Namely, we observe that when the complex numbers are considered as a monoid internal to Z/2-equivariant real linear types, via complex conjugation, then (finite-dimensional) Hilbert spaces do become self-dual objects among internally-complex Real modules. The point is that this construction of Hermitian forms requires of the ambient linear type theory nothing further than a negative unit term of tensor unit type. We observe that just such a term is constructible in LHoTT, where it interprets as an element of the infinity-group of units of the sphere spectrum, tying the foundations of quantum theory to homotopy theory. We close by indicating how this allows for encoding (and verifying) the unitarity of quantum gates and of quantum channels in quantum languages embedded into LHoTT., Comment: 10 pages, some figures

A brief exposition of the point of higher topos theory in (mathematical) physics, commissioned for the Encyclopedia of Mathematical Physics 2nd ed., Comment: 11 pages, some figures

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects., Comment: 120 pages, various figures

Recently Freedman & Hastings asked for a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/bundle-structure via their amalgamation (a hypothetical pushout) along bare quantum (information) theory. As a proposed answer to this question, we first make precise a form of the relevant pushout diagram in monoidal category theory. Then we prove that the pushout produces what is known as the *external* tensor product on vector bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those equipped with monodromy encoding topological Berry phases. The bulk of our result is a further homotopy-theoretic enhancement of the situation to the "derived category" (infinity-category) of flat infinity-vector bundles ("infinity-local systems") equipped with the "derived functor" of the external tensor product. Concretely, we present an integral model category of simplicial functors into simplicial K-chain complexes which conveniently presents the infinity-category of parameterized HK-module spectra over varying base spaces and is equipped with homotopically well-behaved external tensor product structure. In concluding we indicate how this model category serves as categorical semantics for the linear-multiplicative fragment of Linear Homotopy Type Theory (LHoTT), which is thus exhibited as a universal quantum programming language. This is the context in which we recently showed that topological anyonic braid quantum gates are native objects in LHoTT., Comment: 79 pages, various figures; v2: Section 3.4 added, proof of first step in item 3 in thm. 3.23 fixed

Despite the evident necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well as their information-theoretic nature. Building on recent results on defect branes in string/M-theory and on their holographically dual anyonic defects in condensed matter theory, here we explain how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as of real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case, it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware. In a companion article, we will explain how further passage to "dependent linear" homotopy data types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled to verified quantum circuits, complete with quantum measurement gates and classical control., Comment: 88 pages, various figures

Inertia orbifolds homotopy-quotiented by rotation of geometric loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant Tate-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such cyclified stacks has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type. Following our previous formulation of transgression of cohomological charges ("double dimensional reduction"), we explain how cyclification of infinity-stacks is a fundamental and elementary base-change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy type theory). We prove that Ganter/Huan's extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant Tate-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold. As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant Tate-elliptic cohomology; and we close by explaining how this should relate to elliptic M5-brane genera, under our previously formulated Hypothesis H., Comment: 45 pages

While the realization of scalable quantum computation will arguably require topological stabilization and, with it, topological-hardware-aware quantum programming and topological-quantum circuit verification, the proper combination of these strategies into dedicated topological quantum programming languages has not yet received attention. Here we describe a fundamental and natural scheme that we are developing, for typed functional (hence verifiable) topological quantum programming which is topological-hardware aware -- in that it natively reflects the universal fine technical detail of topological q-bits, namely of symmetry-protected (or enhanced) topologically ordered Laughlin-type anyon ground states in topological phases of quantum materials. What makes this work is: (1) our recent result that wavefunctions of realistic and technologically viable anyon species -- namely of su(2)-anyons such as the popular Majorana/Ising anyons but also of computationally universal Fibonacci anyons -- are reflected in the twisted equivariant differential (TED) K-cohomology of configuration spaces of codimension=2 nodal defects in the host material's crystallographic orbifold; (2) combined with our earlier observation that such TED generalized cohomology theories on orbifolds interpret intuitionistically-dependent linear data types in cohesive homotopy type theory (HoTT), supporting a powerful modern form of modal quantum logic. In this short note we give an exposition of the basic ideas, a quick review of the underlying results and a brief indication of the basic language constructs for anyon braiding via TED-K in cohesive HoTT. The language system is under development at the "Center for Quantum and Topological Systems" at the Research Institute of NYU, Abu Dhabi., Comment: 8 pages, 1 figure; extended abstract for contribution to: PlanQC2022 https://icfp22.sigplan.org/home/planqc-2022

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., Comment: 47 pages, various figures; v2: published version with some typos fixed

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities ("inner local systems") that makes the secondary Chern character on a punctured plane inside an A-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman & Varchenko showed realizes sl(2,C)-conformal blocks, here in degree 1 -- in fact it gives the direct sum of these over all admissible fractional levels. The remaining higher-degree conformal blocks appear similarly if we assume our previously discussed "Hypothesis H" about brane charge quantization in M-theory. Since conformal blocks -- and hence these twisted equivariant secondary Chern characters -- solve the Knizhnik-Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of -- and hence of topological quantum computation on -- defect branes in string/M-theory., Comment: 31+8 pages, various figures

In this book we prove unified classification results for equivariant principal bundles when the topological structure group is truncated. The conceptually transparent proof invokes a smooth Oka principle, which becomes available after faithfully embedding traditional equivariant topology into the singular-cohesive homotopy theory of globally equivariant higher smooth stacks. This works for discrete equivariance groups acting properly on smooth manifolds with resolvable singularities, whence we are equivalently describing principal bundles on good orbifolds. In preparation, we re-develop the theory of equivariant principal bundles from scratch by systematic use of Grothendieck's internalization method. In particular we prove that all equivariant local triviality conditions considered in the literature are implied by regarding G-equivariant principal bundles as principal bundles internal to the BG-slice of the ambient cohesive infinity-topos. Generally we find that the characteristic subtle phenomena of equivariant classifying theory all reflect basic "modal" properties of singular-cohesive homotopy theory. Classical literature has mostly been concerned with compact Lie structure groups. Where these are truncated, our classification recovers and generalizes results of Lashof, May, Segal and Rezk. A key non-classical example is the infinite projective unitary structure group, in which case we are classifying degree-3 twists of equivariant KU-theory, recovering results of Atiyah, Segal, Lueck and Uribe. Our theorem enhances this to conjugation-equivariance, where we are classifying the geometric twists of equivariant KR-theory, restricting on "O-planes" to the geometric twists of KO-theory. This is the generality in which equivariant K-theory twists model quantum symmetries of topological phases and the B-field in string theory on orbi-orientifolds., Comment: 242 pages; v2: Lem. 4.1.30 (now .31) fixed and examples expanded; v3: proof of twisted Elmendorf theorem added

Weight systems on chord diagrams play a central role in knot theory and Chern-Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra, and ask which weight systems are positive with respect to this structure; hence we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental gl(n)-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature beta=ln(n) on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature beta; and we prove that the Cayley distance kernel is positive (semi-)definite at beta=ln(n) for all n=1,2,3,... In particular, this proves that all fundamental gl(n)-weight systems are quantum states, and hence so are all their convex combinations. We close with briefly recalling how, under our "Hypothesis H", this result impacts on the identification of bound states of multiple M5-branes., Comment: 21 pages; 1 figure; v2: bound sharpened

In the quest for mathematical foundations of M-theory, the "Hypothesis H" that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory Mf: Stems of homotopy groups correspond to charges of probe p-branes near black b-branes; stabilization within a stem is the boundary-bulk transition; the Adams d-invariant measures G4-flux; trivialization of the d-invariant corresponds to H3-flux; refined Toda brackets measure H3-flux; the refined Adams e-invariant sees the H3-charge lattice; vanishing Adams e-invariant implies consistent global C3-fields; Conner-Floyd's e-invariant is H3-flux seen in the Green-Schwarz mechanism; the Hopf invariant is the M2-brane Page charge (G7-flux); the Pontrjagin-Thom theorem associates the polarized brane worldvolumes sourcing all these charges. Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; the order of the third stable stem implies 24 NS5/D7-branes in M/F-theory on K3. Quaternionic orientations correspond to unit H3-fluxes near M2-branes; complex orientations lift these unit H3-fluxes to heterotic M-theory with heterotic line bundles. In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 10+1., Comment: 99 pages, various figures; v2: p. 7 added with more on brane charges vanishing at infinity (published version)

The celebrated Green-Schwarz mechanism in heterotic string theory has been suggested to secretly underly a higher gauge theoretic phenomenon, embodying a higher Bianchi identity for a higher-degree analog of a curvature form of a higher gauge field. Here we prove that the non-perturbative Horava-Witten Green-Schwarz mechanism for heterotic line bundles in heterotic M-theory with M5-branes parallel to MO9-planes on $A_1$-singularities is accurately encoded in the higher gauge theory for higher gauge group of the equivariant homotopy type of the Z/2-equivariant A-infinity-loop group of twistor space. In this formulation, the flux forms of the heterotic gauge field, the B-field on the M5-brane, and of the C-field in the M-theory bulk are all unified into the character image of a single cocycle in equivariant twistorial Cohomotopy theory; and that cocycle enforces the quantization condition on all fluxes: the integrality of the gauge flux, the half-shifted integrality of the C-field flux and the integrality of the dual C-field flux (i.e., of the Page charge in the bulk and of the Hopf-WZ term on the M5-brane). This result is in line with the Hypothesis H that M-brane charges are quantized in J-twisted Cohomotopy theory. The mathematical essence of our proof is, first, the construction of the equivariant twisted non-abelian character map via an equivariant twisted non-abelian de Rham theorem, which we prove; and, second, the computation of the equivariant relative minimal model of the Z/2-equivariant Sp(1)-parametrized twistor fibration. We lay out the relevant background in equivariant rational homotopy theory and explain how this brings about the subtle flux quantization relations in heterotic M-theory., Comment: 67 pages

We extend the Chern character on K-theory, in its generalization to the Chern-Dold character on generalized cohomology theories, further to (twisted, differential) non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L-infinity algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory to a twisted non-abelian generalization of the de Rham theorem. We show that the non-abelian character reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism as well as its secondary Cheeger-Simons homomorphism on (differential) non-abelian cohomology in degree 1, represented by principal bundles (with connection); and thus generalizes all these to higher (twisted, differential) non-abelian cohomology, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example, we discuss the twisted non-abelian character map on twistorial Cohomotopy theory over 8-manifolds, which can be viewed as a twisted non-abelian enhancement of topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory., Comment: 124 pages; v2: one more example and further references added; v3: several improvements and more details, expanded introduction

We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Horava-Witten's proposal for the extension of the Green-Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in J-twisted Cohomotopy theory., Comment: 30 pages; v2: exposition of real homotopy theory slightly expanded, final version published in RMP

The concept of orbifolds should unify differential geometry with equivariant homotopy theory, so that orbifold cohomology should unify differential cohomology with proper equivariant cohomology theory. Despite the prominent role that orbifolds have come to play in mathematics and mathematical physics, especially in string theory, the formulation of a general theory of orbifolds reflecting this unification has remained an open problem. Here we present a natural theory argued to achieve this. We give both a general abstract axiomatization in higher topos theory, as well as concrete models for ordinary as well as for super-geometric and for higher-geometric orbifolds. Our first main result is a fully faithful embedding of the 2-category of orbifolds into a singular-cohesive infinity-topos whose intrinsic cohomology theory is proper globally equivariant differential generalized cohomology, subsuming traditional orbifold cohomology, Chen-Ruan cohomology, and orbifold K-theory. Our second main result is a general construction of orbifold etale cohomology which we show to naturally unify (i) tangentially twisted cohomology of smooth but curved spaces with (ii) RO-graded proper equivariant cohomology of flat but singular spaces. As a fundamental example we present J-twisted orbifold Cohomotopy theories with coefficients in shapes of generalized Tate spheres. According to "Hypothesis H" this includes the proper orbifold cohomology theory that controls non-perturbative string theory., Comment: 122 pages; v2: some typos fixed

A new super-exceptional embedding construction of the heterotic M5-brane's sigma-model was recently shown to produce, at leading order in the super-exceptional vielbein components, the super-Nambu-Goto (Green-Schwarz-type) Lagrangian for the embedding fields plus the Perry-Schwarz Lagrangian for the free abelian self-dual higher gauge field. Beyond that, further fields and interactions emerge in the model, arising from probe M2- and probe M5-brane wrapping modes. Here we classify the full super-exceptional field content and work out some of its characteristic interactions from the rich super-exceptional Lagrangian of the model. We show that SU(2)xU(1)-valued scalar and vector fields emerge from probe M2- and M5-branes wrapping the vanishing cycle in the A_1-type singularity; together with a pair of spinor fields of U(1)-hypercharge +-1 and each transforming as SU(2) iso-doublets. Then we highlight the appearance of a WZW-type term in the super-exceptional PS-Lagrangian and find that on the electromagnetic field it gives the first-order non-linear DBI-correction, while on the iso-vector scalar field it has the form characteristic of the coupling of vector mesons to pions via the Skyrme baryon current. We discuss how this is suggestive of a form of SU(2)-flavor chiral hadrodynamics emerging on the single (N=1) M5 brane, different from, but akin to, holographic large-$N$ QCD., Comment: 38 pages

We show that charge-quantization of the M-theory C-field in J-twisted Cohomotopy implies emergence of a higher Sp(1)-gauge field on single heterotic M5-branes, which exhibits worldvolume twisted String structure., Comment: 13 pages; v2 has more background and explanations added, to appear in JMP

We highlight what seems to be a remaining subtlety in the argument for the cancellation of the total anomaly associated with the M5-brane in M-theory. Then we prove that this subtlety is resolved under the hypothesis that the C-field flux is charge-quantized in the generalized cohomology theory called J-twisted Cohomotopy., Comment: 20 pages; v1: this is a revised version of Sections 2.5 and 4.5 in v1 of arXiv:1904.10207, which do not appear in the final version there; v2: main section 3 considerably expanded, analogous argument in Sec. 4.1 of arXiv:1910.04166 pointed out, appendix with supplementary proofs added

We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential 4-Cohomotopy theory implies intersecting p/(p+2)-brane moduli given by ordered configurations of points in the transversal 3-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of Dp/D(p+2)-brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, BMN matrix model and fuzzy funnel states, M2-brane 3-algebras, the Hanany-Witten rules, AdS3-gravity observables, supersymmetric indices of Coulomb branches as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is charge-quantized in Cohomotopy theory., Comment: 62 pages; v2: section 4 expanded and introduction re-arranged; v3: discussion of M2/M5-brane states added

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin-Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan-Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges., Comment: 46 pages: v2: orbi-orientifold group extension made more explicit, published version

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D'Auria-Fre's "hidden supergroup" of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the "half" M5-brane locus, super-exceptionally compactified on the Horava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field., Comment: 46 pages, v2: some comments and references added

The full 6d Hopf-Wess-Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess-Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the full 6d Hopf-Wess-Zumino term with a twisted homotopy Whitehead integral formula, which we establish., Comment: 22 pages; v2 fixes a missing theta7-summand in intermediate formulas and has two remarks added, for clarification; v3 fixes typos and makes the slicing over the homotopy fiber space of chi_8 more explicit

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the C-field in Cohomotopy theory implies a list of expected anomaly cancellation conditions, including: shifted C-field flux quantization and C-field tadpole cancellation, but also the DMW anomaly cancellation and the C-field's integral equation of motion., Comment: 55 pages; v2: proof of fluxed tadpole cancellation added; v3: exposition reorganized, discussion of M5 brane backgrounds postponed to a subsequent article arXiv:2002.07737, title and abstract adjusted accordingly; v4: published version

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and $L_\infty$-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context., Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, references updated

We review how core structures of string/M-theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion-subgroups in brane charges and focuses on tangent spaces of super space-time. Already at this level, super homotopy theory discovers all super $p$-brane species, their intersection laws, their M/IIA-, T- and S-duality relations, their black brane avatars at ADE-singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D-branes it recovers twisted K-theory, rationally, but for the M-branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M-theory in super homotopy theory., Comment: 32 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018

The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this "Hypothesis H", here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman's comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox. Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters., Comment: 47 pages, Python code attached as ancillary file; v2: minor polishing & role of table of marks explicit; v3: physics application brought to the forefront, title changed

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan-Paton gauge fields on D-branes have invariant meaning, the problem is really the lift to M-theory of the twisted K-theory classification of D-brane charges. Here we show how this problem has a solution by universal constructions in super homotopy theory, at least rationally. We recall how double dimensional reduction of super M-brane charges is described by the cyclification adjunction applied to the 4-sphere, and how M-theory degrees of freedom hidden at ADE-singularities are induced by the suspended Hopf action on the 4-sphere. Combining these, we demonstrate, at the level of rational homotopy theory, that gauge enhancement in M-theory is exhibited by lifting against the fiberwise stabilization of the unit of this cyclification adjunction on the A-type orbispace of the 4-sphere. This explains how the fundamental D6 and D8 brane cocycles can be lifted from twisted K-theory to a cohomology theory for M-brane charge, at least rationally., Comment: 57 pages, various figures, v2: minor corrections, v3: published version

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf's theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant 4-sphere-valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. We find that on each singular stratum the black brane's instanton contribution, namely its super Nambu-Goto/Green-Schwarz action, appears as the homotopy datum associated to the morphisms in the orbit category., Comment: 87 pages, various figures, v2: minor polishing, v3: published version

By analyzing super-torsion and brane super-cocycles, we derive a new duality in M-theory, which takes the form of a higher version of T-duality in string theory. This involves a new topology change mechanism abelianizing the 3-sphere associated with the C-field topology to the 517-torus associated with exceptional-generalized super-geometry. Finally we explain parity symmetry in M-theory within exceptional-generalized super-spacetime at the same level of spherical T-duality, namely as an isomorphism on 7-twisted cohomology., Comment: 7 pages, concise physics letter companion to the mathematically rigorous treatment arXiv:1803.05634; to appear in Phys. Lett. B

We establish a higher generalization of super L-infinity-algebraic T-duality of super WZW-terms for super p-branes. In particular, we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d super spacetime., Comment: 51 pages, 2 figures, v2 has some typos fixed

We combine Sullivan models from rational homotopy theory with Stasheff's $L_\infty$-algebras to describe a duality in string theory. Namely, what in string theory is known as topological T-duality between $K^0$-cocycles in type IIA string theory and $K^1$-cocycles in type IIB string theory, or as Hori's formula, can be recognized as a Fourier-Mukai transform between twisted cohomologies when looked through the lenses of rational homotopy theory. We show this as an example of topological T-duality in rational homotopy theory, which in turn can be completely formulated in terms of morphisms of $L_\infty$-algebras., Comment: A few inaccuracies fixed and one example added; 24 pages

We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang-Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang-Mills Cauchy problem and show that its well-posedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in [S. Hollander, Israel J. Math. 163, 93-124 (2008)], which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as $\mathrm{B}G_\mathrm{con}$., Comment: 50 pages, final version published in Communications in Mathematical Physics

The "brane scan" classifies consistent Green--Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The "brane bouquet" generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of repeated "maximal invariant central extensions", and show that it discovers the super-Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet., Comment: v2: minor corrections, improved exposition. To appear in Letters in Mathematical Physics

We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields. (abridged), Comment: 68 pages, xypic commutative diagrams

We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\infty$-algebras are naturally related by an $L_\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. In particular this is a derivation of the Buscher rules for RR-fields (Hori's formula) from first principles. Moreover, we show that these $L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced $L_\infty$-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations., Comment: 44 pages; v2: added more discussion of double dimensional reduction via cyclic L-infinity cohomology; v3: added derivation of Buscher rules for RR-fields, expanded on role of curved L-infinity algebras in double dimensional reduction