Searcharxiv⌕ Search

arXiv subjects

Urs Schreiber

Publications and source records attributed to Urs Schreiber.

At least 55 records · Page 3Linked to original sources
arXiv2024

Flux Quantization on Phase Space

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.

hep-th↗
arXiv2023

M/F-Theory as Mf-Theory

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.

hep-th↗
arXiv2023

Quantum and Reality

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.

quant-ph↗
arXiv2023

The Quantum Monadology

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.

quant-ph↗
arXiv2023

Principal infinity-bundles -- General theory

The theory of principal bundles makes sense in any infinity-topos, such as that of topological, of smooth, or of otherwise geometric infinity-groupoids/infinity-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure infinity-group G these G-principal infinity-bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their further higher and equivariant analogs. The induced associated infinity-bundles subsume the notions of gerbes and higher gerbes in the literature. We discuss here this general theory of principal infinity-bundles, intimately related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize infinity-toposes. We show a natural equivalence between principal infinity-bundles and intrinsic nonabelian cocycles, implying the classification of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber infinity-bundles associated to principal infinity-bundles subsumes a theory of infinity-gerbes and of twisted infinity-bundles, with twists deriving from local coefficient infinity-bundles, which we define, relate to extensions of principal infinity-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice infinity-topos. In a companion article [NSSb] we discuss explicit presentations of this theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by simplicial weakly-principal bundles; and in [NSSc] we discuss various examples and applications of the theory.

math.AT↗
arXiv2023

From Loop Groups to 2-Groups

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras $\mathfrak{g}_k$ each having $\mathrm{Lie}(G)$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having $\mathfrak{g}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. The objects of this 2-group are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group of $G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k = \pm 1$, this topological group can also be obtained by killing the third homotopy group of $G$. Thus, when $G = \mathrm{Spin}(n)$, it is none other than $\mathrm{String}(n)$.

math.QA↗
arXiv2023

Topological Quantum Gates in Homotopy Type Theory

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.

quant-ph↗
arXiv2022

Cyclification of Orbifolds

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.

math.AT↗
arXiv2022

Topological Quantum Programming in TED-K

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.

quant-ph↗
arXiv2022

Equivariant principal infinity-bundles

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.

math.AT↗
arXiv2022

Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation

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.

hep-th↗
arXiv2022

Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory

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.

hep-th↗
arXiv2021

Fundamental weight systems are quantum states

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.

math.GT↗
arXiv2021

The character map in (twisted differential) non-abelian cohomology

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.

math.AT↗
arXiv2021

Twisted Cohomotopy implies M5-brane anomaly cancellation

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.

hep-th↗
arXiv2020

Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane

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.

hep-th↗
arXiv2020

Proper Orbifold Cohomology

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.

math.AT↗