238 research outputs found
Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology
We study smooth higher symmetry groups and moduli -stacks of generic
higher geometric structures on manifolds. Symmetries are automorphisms which
cover non-trivial diffeomorphisms of the base manifold. We construct the smooth
higher symmetry group of any geometric structure on and show that this
completely classifies, via a universal property, equivariant structures on the
higher geometry. We construct moduli stacks of higher geometric data as
-categorical quotients by the action of the higher symmetries, extract
information about the homotopy types of these moduli -stacks, and prove
a helpful sufficient criterion for when two such higher moduli stacks are
equivalent.
In the second part of the paper we study higher -connections.
First, we observe that higher connections come organised into higher groupoids,
which further carry affine actions by Baez-Crans-type higher vector spaces. We
compute a presentation of the higher gauge actions for -gerbes with
-connection, comment on the relation to higher-form symmetries, and present
a new String group model. We construct smooth moduli -stacks of higher
Maxwell and Einstein-Maxwell solutions, correcting previous such considerations
in the literature, and compute the homotopy groups of several moduli
-stacks of higher - connections. Finally, we show that a
discrepancy between two approaches to the differential geometry of NSNS
supergravity (via generalised and higher geometry, respectively) vanishes at
the level of moduli -stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom
Entanglement of Sections: The pushout of entangled and parameterized quantum information
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: 71 pages, various figure
Smooth generalized symmetries of quantum field theories
Dynamical quantum field theories (QFTs), such as those in which spacetimes
are equipped with a metric and/or a field in the form of a smooth map to a
target manifold, can be formulated axiomatically using the language of
-categories. According to a geometric version of the cobordism
hypothesis, such QFTs collectively assemble themselves into objects in an
-topos of smooth spaces. We show how this allows one to define and
study generalized global symmetries of such QFTs. The symmetries are themselves
smooth, so the `higher-form' symmetry groups can be endowed with, e.g., a Lie
group structure.
Among the more surprising general implications for physics are, firstly, that
QFTs in spacetime dimension , considered collectively, can have -form
symmetries, going beyond the known -form symmetries of individual QFTs
and, secondly, that a global symmetry of a QFT can be anomalous even before we
try to gauge it, due to a failure to respect either smoothness (in that a
symmetry of an individual QFT does not smoothly extend to QFTs collectively) or
locality (in that a symmetry of an unextended QFT does not extend to an
extended one).
Smoothness anomalies are shown to occur even in 2-state systems in quantum
mechanics (here formulated axiomatically by equipping spacetimes with a
metric, an orientation, and perhaps some unitarity structure). Locality
anomalies are shown to occur even for invertible QFTs defined on
spacetimes equipped with an orientation and a smooth map to a target manifold.
These correspond in physics to topological actions for a particle moving on the
target and the relation to an earlier classification of such actions using
invariant differential cohomology is elucidated.Comment: 72 page
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.Comment: 120 pages, various figure
Pregeometry, Formal Language and Constructivist Foundations of Physics
How does one formalize the structure of structures necessary for the foundations of physics? This work is an attempt at conceptualizing the metaphysics of pregeometric structures, upon which new and existing notions of quantum geometry may find a foun- dation. We discuss the philosophy of pregeometric structures due to Wheeler, Leibniz as well as modern manifestations in topos theory. We draw attention to evidence suggesting that the framework of formal language, in particular, homotopy type theory, provides the conceptual building blocks for a theory of pregeometry. This work is largely a synthesis of ideas that serve as a precursor for conceptualizing the notion of space in physical theories. In particular, the approach we espouse is based on a constructivist philosophy, wherein “structureless structures” are syntactic types realizing formal proofs and programs. Spaces and algebras relevant to physical theories are modeled as type-theoretic routines constructed from compositional rules of a formal language. This offers the remarkable possibility of taxonomizing distinct notions of geometry using a common theoretical framework. In particular, this perspective addresses the crucial issue of how spatiality may be realized in models that link formal computation to physics, such as the Wolfram model
Towards non-perturbative BV-theory via derived differential cohesive geometry
We propose a global geometric framework which allows one to encode a natural
non-perturbative generalisation of usual Batalin-Vilkovisky (BV-)theory.
Namely, we construct a concrete model of derived differential cohesive
geometry, whose geometric objects are formal derived smooth stacks, i.e. stacks
on formal derived smooth manifolds, together with a notion of differential
geometry on them. This provides a working language to study generalised
geometric spaces that are smooth, infinite-dimensional, higher and derived at
the same time. Such a formalism is obtained by combining Schreiber's
differential cohesion with the machinery of T\"oen-Vezzosi's homotopical
algebraic geometry applied to the theory of derived manifolds of Spivak and
Carchedi-Steffens. We investigate two classes of examples of non-perturbative
classical BV-theories in the context of derived differential cohesion: scalar
field theory and Yang-Mills theory.Comment: 106 pages, 11 figure
-Bundles
Higher bundles are homotopy coherent generalisations of classical fibre
bundles. They appear in numerous contexts in geometry, topology and physics. In
particular, higher principal bundles provide the geometric framework for
higher-group gauge theories with higher-form gauge potentials and their
higher-dimensional holonomies. An -categorical formulation of higher
bundles further allows one to identify these objects in contexts outside the
worlds of smooth manifolds or topological spaces. This article reviews the
theory of -bundles, focussing on principal -bundles, and
surveys several of their applications. It is an invited contribution to the
Topology section in the second edition of the Encyclopedia of Mathematical
Physics.Comment: 24 pages, several diagram
Ruliology: Linking Computation, Observers and Physical Law
Stephen Wolfram has recently outlined an unorthodox, multicomputational
approach to fundamental theory, encompassing not only physics but also
mathematics in a structure he calls The Ruliad, understood to be the entangled
limit of all possible computations. In this framework, physical laws arise from
the the sampling of the Ruliad by observers (including us). This naturally
leads to several conceptual issues, such as what kind of object is the Ruliad?
What is the nature of the observers carrying out the sampling, and how do they
relate to the Ruliad itself? What is the precise nature of the sampling? This
paper provides a philosophical examination of these questions, and other
related foundational issues, including the identification of a limitation that
must face any attempt to describe or model reality in such a way that the
modeller-observers are include
Thomas Kuhn, Modern Mathematics and the Dynamics of Reason
It is now 30 years since a group of philosophers and historians published a collection of articles (Gillies (ed.) 1992) which took as their central question the usefulness for the philosophy of mathematics of Kuhn's constructions in the philosophy of science. Important work was done to see whether the concepts of `revolution' and `paradigm' made sense there. However, a feature of the collection which should raise doubts about the universality of its findings was the restriction, except in a single case, to work done prior to 1900. In this article I discuss what form a historically-oriented philosophy of modern mathematics should take, and the role it could play in developing Michael Friedman's post-Kuhnian theory as described in his `Dynamics of Reason' (2001) and later publications
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