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Types of quantum field thories
A set of lecture notes on differential geometry and theoretical fundamental physics, combining an introduction to traditional notions with an exposition of their formulation and refinement by higher geometry and extended prequantum field theory. With an eye towards Hilbert's sixth problem (“Synthetic Quantum Field Theory”).
Divided into two parts:
This page is going to contain an introduction to aspects of differential geometry and their application in fundamental physics: the gauge theory appearing in the standard model of particle physics and the Riemannian geometry appearing in the standard model of cosmology, as well as the symplectic geometry appearing in the quantization of both.
The intended topic scope and readership of the first layer of this page – the Model Layer – is much like that of the book (Frankel), only that here we make use of a more modern and more transparent conceptual toolbox. We also discuss in two other layers, the Semantic Layer and the Syntactic Layer deeper mechanisms at work in the background.
Notably, where traditional expositions of differential geometry proceed by generalizing the geometry of abstract coordinate systems $\mathbb{R}^n$ to smooth manifolds, here we instead begin by generalizing, in Smooth spaces – Model Layer, coordinate systems right away to smooth spaces, which happens to be both more expressive as well as actually much easier. In parallel (and to be read independently or not at all) we discuss in Smooth spaces – Semantic Layer how this means that we are working in the sheaf topos over abstract coordinate systems. Smooth manifolds are then introduced later as an intermediate notion, together with that of diffeological spaces. (Many of the constructions in differential geometry applied in physics do not actually need the notion of a smooth manifold, and, more importantly, for many notions in modern theoretical physics smooth manifolds are not actually sufficiently general.)
In fact we introduce smooth manifolds only after we introduce smooth groupoids (below in Smooth homotopy type - Model Layer - Smooth groupoids), which are differential geometric structures that are still simpler than smooth manifolds, and of course even more expressive than smooth spaces. Moreover, smooth groupoids are at the very heart of the geometry of physics: modern fundamental physics is all based on the “gauge principle” and in Model Layer – Gauge transformations in electromagnetism we explain how, mathematically, this is essentially nothing but the theory of smooth groupoids. As further background information we discuss in Smooth homotopy types - Semantic Layer how this means that we are working in a higher topos over abstract coordinate systems, and in Smooth homotopy type - Syntactic Layer how this means that we are reasoning about physics using the natural deduction rules of homotopy type theory.
From this setup then naturally flow all the many structures and phenomena seen in the geometry of physics:
We discuss each topic below in three stages, in three layers.
The three layers
Model Layer – concrete particular: models
Semantic Layer – concrete general: categorical semantics in higher topos theory
Syntactic Layer – abstract general: syntax in homotopy-type theory
This topos-theoretic perspective on fundamental physics which is discussed here is mostly original in the identifications it makes (Schreiber), but it draws insights and inspiration from (and maybe realizes) a vision already expressed since the 1960s by William Lawvere, one of the central figures in the development of topos theory and categorical logic. Lawvere links the very inception of topos theory to the motivation to axiomatize physics:
My own motivation $[$ for developing topos theory $]$ came from my earlier study of physics. The foundation of the continuum physics of general materials, $[...]$ involves powerful and clear physical ideas which unfortunately have been submerged under a mathematical apparatus including not only Cauchy sequences and countably additive measures, but also ad hoc choices of charts for manifolds and of inverse limits of Sobolev Hilbert spaces, to get at the simple nuclear spaces of intensively and extensively variable quantities. But, as Fichera lamented, all this apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? (Lawvere, 2000)
More historical pointers along these lines and further related material can also be found at higher category theory and physics.
To give a survey of how the exposition below proceeds in the fashion of these three layers, the following section The full story in a few formal words provides what may be read as commented index to the central themes of the following text. Whereas the exposition below is organized to start each topic with the discussion of its concrete models in a Model layer, then pass to a general abstract semantics in a Semantic Layer and then finally to the abstract formal syntax in a Syntactic Layer, these tables indicates how this passage to abstract syntax usefully reflects back onto the concrete theory:
The leftmost columns of the following tables formulate concepts in terms of ordinary language. The second columns translate that ordinary language fairly directly to the formal language of (homotopy) type theory. The third columns then interprets these formal syntactical expressions as universal constructions in a (higher, cohesive) topos by the rules of categorical semantics. Finally, the fourth columns indicate what this universal construction amounts to when concretely realized in the model given by smooth spaces and smooth ∞-groupoids. Finally the rightmost columns point to the chapters in the text below that deal with the given construction.
These tables show that fairly evident and naïve sounding statements in ordinary language turn under this translation into what is generally regarded as fairly sophisticated constructions. In fact some of these constructions have only been found by translating along the categorical semantics dictionary this way. So the following tables also serves to show how the general abstract discussion here is a means to facilitate reasoning about seemingly complicated concepts underlying fundamental physics:
We give an overview in the spirit of Synthetic Quantum Field Theory.
The fundamental physics of the observed world is governed by what is called quantum theory. (This is explicitly so for the standard model of particle physics and induced from this all fundamental physics ever tested in laboratories; but by all that is known also the remaining ingredient of gravity is fundamentally a quantum theory, see at quantum gravity for comments).
Two major axiomatizations of quantum theory are known, namely
FQFT where one axiomatizes the assignment of spaces of states to pieces of worldvolume (the “Schrödinger picture” of quantum theory)
fragments of which involve:
finite quantum mechanics in terms of dagger-compact categories
FRS-construction of 2d CFT from this via holography
AQFT where one axiomatizes the assignment of algebras of observables to pieces of worldvolume (the “Heisenberg picture” of quantum theory)
fragments of which involve:
(For an attempt at a survey of the state of the art as of 2011 see the collection (Sati-Schreiber)).
But all fundamental quantum field theories observed in (or conjectured to underlie) nature arise by a process called quantization from structures in differential geometry (or are induced via a mechanism called the holographic principle from such that do).
This differential geometric data involves
smooth functionals – called action functionals
on smooth "spaces" – called moduli stacks
of differential geometric structures such as fiber bundles and connections – called gauge force fields
as well as sections of associated bundles – called matter fields.
Similar differential geometric structures are involved in the geometric quantization of such an action functional to an actual quantum field theory.
Hence there is a sequence:
differential geometry | $\to$ | geometric quantization | $\to$ | quantum field theory |
---|
We discuss a formalization of central aspects of this entire sequence. Our development proceeds – as befits a theory of physics and hence of nature – via natural deduction from practical foundations.
$\,$
Fundamentally, a language for physics is to be a language about existence; a language in which we can express judgements of the form:
There is a thing $x$ of type $X$.
For instance:
There is a gauge field $\nabla$ in the standard model $[X,\mathbf{B}\left(U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)\right)_{conn}]$ of gauge fields on spacetime $X$.
(Here the square bracket expression for a moduli stack of gauge fields will be incrementally explained in the following.)
To be predictive, a language for physics is moreover to be a language in which we can make natural deductions to deduce further such judgements from given ones. For instance:
Given a gauge field $\nabla$ as above, there is an underlying instanton sector, $UnderlyingBundle(\nabla)$, in the collection $\left[X,\mathbf{B}\left(U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)\right)\right]$ of instanton configurations in the standard model.
Quantum superpositions of such Yang-Mills instantons are the very substrate out of which the vacuum of the observed world is build: the instanton liquid in quantum chromodynamics. (For more see at Yang-Mills theory below.) We consider here a language to reason about such phenomena formally.
The formal language for such natural deduction of judgements about there being terms of some type is called type theory.
Expressions in (dependent) type theory:
(read columns 1+2 first, then 3+4)
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
There is | $\vdash \ldots$ | We speak in the context of a (higher) topos $\mathbf{H}$, a place where things may be. (For the time being a (higher) locally cartesian closed category is sufficient.) | A topos for synthetic differential geometry, such as $\mathbf{H} =$ Sh$($SmthMfd$)$. Eventually a higher such topos: $\mathbf{H} =$ Smooth∞Grpd or SynthDiff∞Grpd or SmoothSuper∞Grpd or | Smooth spaces and Smooth homotopy types |
There is a thing $x$ of type $X$. | $\vdash\; x \colon X$ | An element $\left(* \stackrel{x}{\to} X\right) \in Mor(\mathbf{H})$ of an object $X$ of $\mathbf{H}$. | A point $x$ in a smooth moduli stack $X$. | Judgements about types and terms |
There is a type $X$ of things $x$. | $\vdash\; X \colon Type$ | An element $(* \stackrel{\vdash X}{\to} Obj) \in Mor(\mathbf{H})$ of the small-object classifier $Obj$ of $\mathbf{H}$. | A point in the moduli stack of all small moduli stacks. | Judgements about types and terms |
Given a thing $x$ of type $X$ there is a thing $a(x)$ of type $A(x).$ | $x \colon X\;\vdash\; a(x) \colon A(x)$ | An element of a morphism $(A \to X)$ $\left(\array{ X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id}}\searrow &\swArrow& \swarrow_{} \\ && X }\right)$ in the slice topos $\mathbf{H}_{/X}$. | An $X$-family in a moduli stack bundle $A$ over $X$. | Slice categories and Slice toposes and Slice ∞-Toposes |
There is the collection of all things $a(x)$ for all $x$. | $\vdash\; \left(\sum_{x \colon X} A\left(x\right)\right) \colon Type$ | The dependent sum/left adjoint to the product: $\array{ \mathbf{H}_{/X} &\stackrel{X_!}{\to} & \mathbf{H} \\ (A \to X) &\mapsto& A \in \mathbf{H}}$ | The total space of a bundle. | Natural deduction rules for dependent sum types |
There is a thing $t$ in the collection of all things $a(x)$ for all $x$. | $\vdash\; t \colon \sum_{x \colon X} A(x)$ | An element $*\stackrel{t}{\to} A$ of the total space object. | A point in the moduli stack $A$ over $X$. | |
There is an assignment $f$ of an $a(x)$ to each $x$. | $\vdash \; f \colon \prod_{x \colon X} A(x)$. | An element in the internal object of sections $* \stackrel{f}{\to} [X,A]_X$ | A point in the smooth relative mapping space of smooth sections. | Natural deduction rules for dependent product types |
There is the collection of assignments of an $a(x)$ to each $x$. | $\vdash\; \left( \prod_{x \colon X} A\left(x\right) \right) \colon Type$ | internal space of sections $[X,A]_X \in \mathbf{H}$ | A smooth relative mapping space of smooth sections. | |
In particular, there is the collection of such assignments when $A$ does not depend on $x$, the collection of functions from $X$ to $A$. | $\vdash \; \left(X \to A\right) \coloneqq \left(\prod_{x \colon X} A\right) \colon Type$ | The internal hom object $[X,A] \in \mathbf{H}$. | A smooth mapping space. | Smooth mapping spaces and smooth moduli spaces |
There is a proof $p$ that it is true that there is $x$ of type $X$. | $\vdash \; p \colon [X]$ | An element $* \stackrel{p}{\to}\tau_{-1}(X)$ of the (-1)-truncation of the object $X$. | A point in the smooth space of equivalence classes of points in $X$. | Subobjects |
There is a proof $p$ that it is true that there is an $a(x)$ for some $x$. | $\vdash\; p \colon \left(\exists_{x \colon X} A\left(x\right) \right) \coloneqq \left[ \sum_{x \colon X} A\left(x\right)\right]$ |
In order to describe a structured reality, our language needs to be able to speak about comparison of things.
Fundamental physics rests on the gauge principle: it is meaningless to say that two things – such as two gauge fields $\nabla$ as above – are equal; instead they are gauge equivalent if there is a gauge transformation between them.
So our language needs to express judgements of the form:
There is a gauge equivalence between gauge fields $\nabla_1$ and $\nabla_2$.
And the language needs to be able to make natural deductions from such judgements to arrive at:
Given an equivalence $\lambda \colon \nabla_1 \simeq \nabla_2$ there is an equivalence $UnderlyingBundle(\lambda) \colon UnderlyingBundle(\nabla_1) \simeq UnderlyingBundle(\nabla_2)$ between the underlying instanton sectors.
The formal language based of the dependent type theory which we have so far that contains these statements is type theory with propositional equality. In this language we have judgements such as the following.
Expressions in dependent type theory with propositional equality:
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
Given $x,x'$, there is the collection of equivalences between $x$ and $x'$ equivalent. | $x,x' \colon X \;\vdash \; \left(x \simeq x'\right) \colon Type$. | The mapping cocone object $\array{ P_{x,x'} X &\to& * \\ \downarrow &\swArrow_{e}& \downarrow^{\mathrlap{x}} \\ * &\stackrel{x'}{\to} & X }$ | The moduli stack of gauge transformations between $x$ and $x'$. | Identity types |
There is an equivalence $e$ between $x$ and $x'$. | $\array{\vdash \; e \colon (x \simeq x') \\ or \\ \vdash \; e \colon (x \rightsquigarrow x') }$ | An element of the mapping cocone object. | A gauge transformation between $x$ and $x'$. | |
Given $x,x'$, there is the collection of proofs that it is true that $x$ and $x'$ are equivalent. | $x,x' \colon X \;\vdash \; [x \simeq x'] \colon Type$. | The (-1)-truncation fo the mapping cocone. | The smooth space of equivalence classes of gauge transformations from $x$ to $x'$. |
But the gauge principle reaches deeper: gauge transformations themselves are subject to the gauge principle.
In general it is meaningless to ask if two gauge transformations are equal, but we may ask if there is a higher gauge transformation that transforms one gauge transformation into the other. In the physics literature such gauge-of-gauge transformations are best known in their incarnation as ghost-of-ghost fields in what is called the BRST complex of the given gauge theory.
Careful analysis for instance of the Dirac charge quantization of magnetic charge shows that already quite mundane physical phenomena exhibit such higher gauge transformations. But more famously they are known to arise in various guises in string theory, which is a hypothetical refinement of the standard model of particle physics and gravity.
In either case, our formal language should not allow the deduction that gauge equivalences are themselves either equal or not, but only allow judgements of the following form:
There is a gauge-of-gauge equivalence $\rho \colon (\lambda_1 \simeq \lambda_2)$ between two given gauge equivalences $\lambda_1, \lambda_2 \colon (\nabla_1 \simeq \nabla_2)$ between two given gauge fields $\nabla_1, \nabla_2$.
The flavor of type theory with propositional equality for which this is the case is called intensional type theory.
Since therefore a type $X$ in intensional type theory may contain homotopies between its terms of arbitrary order, we call it a homotopy type.
The homotopy-type nature of the type of gauge connections $[X,\mathbf{B}G_{conn}]$ is most familiar in the physics literature in its infinitesimal approximation, which is the (off-shell) BRST complex of the gauge theory: the $n$-fold ghost-of-ghost fields in the BRST complex correspond to the $n$-fold homotopies in $[X, \mathbf{B}G_{conn}]$.
In particular, in intensional type theory we find the gauge group of a homotopy type, as indicated in the following table.
Expressions in intensional type theory:
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
Given a type $X$, there is (the underlying space) of a group $G$ of ways that $X$ is equivalent to itself. | $X \colon Type \;\vdash \; (X \stackrel{\simeq}{\to} X ) \colon Type$ | A loop space object $\array{ G &\to& * \\ \downarrow &\swArrow& \downarrow^{\mathrlap{X}} \\ * &\stackrel{X}{\to} & Type }$ | A smooth ∞-group. | n-groups |
Given a function between collections of things $X$ and $Y$, and given a thing $y$, there is its preimage-up-to-equivalence. | $\left( f \colon \left(X\to Y\right)\right), \left(y \colon Y\right) \;\vdash\; \sum_{x \colon X} \left(f\left(x\right) \simeq y\right)$ | A homotopy pullback $\array{ X \times_{Y} \{y \} &\to& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ {*} &\underset{y}{\to}& Y }$ | The homotopy fiber of a homomorphism of smooth moduli stacks. |
Suppose then that we have such a map between collections of gauge fields
on two possibly different spacetimes with two possibly different gauge groups.
(For instance we might be looking at Montonen-Olive duality/_S-duality_ or Seiberg duality of super Yang-Mills theory.)
Then we should call $f$ an equivalence - in the physics literature often: a duality – if, while not necessarily being a “bijection”, it is such that the preimage $\phi^{-1}(\nabla) \in [X,\mathbf{B}G_{conn}]$ of a gauge field $\nabla \in [Y, \mathbf{B}H_{conn}]$ consists of gauge fields that are all gauge equivalent to each other, with the gauge equivalences exhibiting this equivalence themselves all being gauge equivalent to each other, etc.
If this is the case one says that all homotopy fibers – all gauge pre-images – of $\phi$ are contractible – are gauge equivalent to a single gauge field – and that $\phi$ is a weak homotopy equivalence.
For consistency we should demand that the notion of equivalence is such that the space of direct equivalences $[X, \mathbf{B}G_{conn}] \simeq [Y, \mathbf{B}H_{conn}]$ is itself equivalent to the space of such weak homotopy equivalences (“dualities”) $[X, \mathbf{B}G_{conn}] \stackrel{\simeq}{\to} [Y, \mathbf{B}H_{conn}]$.
This requirement is called the univalence axiom. The intensional type theory-language considered so far equipped with this axiom is called homotopy type theory.
We indicate now some central judgements that are expressible in homotopy type theory. This involves fundamental judgements in group theory and in representation theory, two of the pillars of modern quantum theory/quantum field theory.
Structures expressible in homotopy type theory:
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
Given a type $X$, there is a group $G$ of ways that $X$ is equivalent to itself. | $X \colon Type \;\vdash \; (X \stackrel{\simeq}{\to} X ) \colon Type$ | A loop space object $\array{ G &\to& * \\ \downarrow &\swArrow& \downarrow^{\mathrlap{X}} \\ * &\stackrel{X}{\to} & Type }$ | A smooth automorphism ∞-group. | n-groups |
Given a type $X$, there is the delooping $\mathbf{B}G$ of $G$, which is the collection of things equipped with equivalences to $X$. | $X \colon Type \; \vdash \; \mathbf{B}G \coloneqq \sum_{Y \colon Type} \left[X \simeq Y\right]$ | The looping and delooping relation $\array{G \simeq &\Omega \mathbf{B}G &\to& * \\ & \downarrow &\swArrow& \downarrow^{\mathrlap{}} \\ & * &\underset{}{\to}& \mathbf{B}G}$ | The smooth moduli stack of smooth $G$-principal ∞-bundles. | Principal n-bundles |
Given a thing in $\mathbf{B}G$, there is a thing $V$. | $\array{* \colon \mathbf{B}G \;\vdash\; V(*) \colon Type \\ or\;with\;more\;emphasis: \\ (*,*',g) \colon \sum_{*,*' \colon \mathbf{B}G} (*\rightsquigarrow *') \;\vdash\; V(* \stackrel{g}{\rightsquigarrow} *') \colon Type }$ | A homotopy fiber sequence $\array{V &\to& V\sslash G \\&& \downarrow^{\overline{\rho}} \\ && \mathbf{B}G }$ with homotopy fiber $V$ over $\mathbf{B}G$. | An ∞-action/∞-representation of $G$ on some $V$, together with its universal $\rho$-associated $V$-fiber ∞-bundle over the moduli stack $\mathbf{B}G$ for $G$-principal ∞-bundles. | Higher actions |
Given a function $g$ classifying a $G$-principal bundle and given a point in the delooping, there is the $G$-principal bundle $P$ itself, being the collection of identifications of the fiber $g(x)$ with $X$ | $\left(g \colon X \to \mathbf{B}G\right), \left(* \colon \mathbf{B}G\right) \;\vdash\; P \coloneqq \sum_{x \colon X} (g(x) \simeq *)$ | $\array{P &\to& * & \simeq \mathbf{E}G \\ \downarrow &\swArrow& \downarrow \\ X &\stackrel{g}{\to} & \mathbf{B}G }$ | The principal ∞-bundle given as the homotopy pullback of the universal principal ∞-bundle. | Principal ∞-bundles |
There is a $G$-equivariant map from the principal bundle to the representation space. | $\vdash\; \sigma \colon \prod_{* \colon \mathbf{B}G} \left(P \to V\right)$ | An element $\array{ X &&\stackrel{\sigma}{\to}&& V \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G}$ of $V$ in the slice topos $\mathbf{H}_{/\mathbf{B}G}$ | A section of the $\rho$-associated $V$-fiber ∞-bundle. |
In gauge theory physics, a representation $\rho$ of the gauge group $G$ encodes the particle-content of the model (in theoretical physics): a section of the $\rho$-associated bundle to the gauge bundle is a matter field in the model.
Therefore all the ingredients so far encode the kinematics of gauge theory, its setup before an actual dynamics is specified.
Dynamics in physics says how things move, hence how they trace out trajectories in a given spacetime or more generally in some phase space.
Our language for reasoning about physics should be able to express this. For $X$ a homotopy type that models spacetime (the collection of all points of spacetime) there should be a homotopy type $\Pi(X)$ whose homotopies and higher homotopies are the smooth trajectories, the smooth paths and higher paths in $X$.
In order to analyse the notion of smoothness here – we will say: the way that points hold together by cohesion – there should also be
an expression $\flat X$ for the discrete collection of points underlying $X$ – detaching all points;
an expression $\sharp X$ which dissolves the cohesion and produces the codiscrete smooth structure on $X$.
There are some natural simple axioms on these constructions. For instance every smooth path in a discrete space $\flat X$ should be constant: $\Pi (\flat X) \simeq \flat X$.
With such natural axioms understood, these three constructions constitute an adjoint triple of modalities $(\Pi \dashv \flat \dashv \sharp)$ in our language. In particular $\Pi$ and $\flat$ are a monad and comonad on the type system, in the sense of computer science and $\sharp$ is even an internal monad.
Equipping the above homotopy type theory with these modalities turns it into what we call cohesive homotopy type theory.
Structures expressible in cohesive homotopy type theory:
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
Given a cohesive homotopy type $X$, there is the dissolved homotopy type $\sharp X$ in which all separate points are collected to one cohesive blob. | $X \colon Type \;\vdash\; \sharp X \colon Type$ | The codiscrete object-monad on a (higher) local topos. | The codiscrete smooth structure on the points of $X$. | Locality of the topos of smooth spaces |
Given a cohesive homotopy type, there is the map that dissolves the cohesion of the points. | $X \colon Type \;\vdash\; DeCoh_X \colon X \to \sharp X$ | The unit of the codiscrete object monad. | The function that sends smooth families in a smooth moduli stack to families of points. | |
Given $X$ there is the collection $\Pi(X)$ of points in $X$ and smooth trajectories between points in $X$. | $\left(X \colon \sharp Type\right) \;\vdash\; \Pi(X) \colon \sharp Type$ | The construction of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos. | The smooth path ∞-groupoid of $X$. | The local ∞-connectedness of the (∞,1)-topos of smooth ∞-groupoids |
Given $X$, there is a canonical map to $\Pi(X)$. | $\left(X \colon \sharp Type\right) \;\vdash\; ConstantPathInclusion_X \colon X \to \Pi(X)$. | The unit of the $\Pi$-monad on a locally ∞-connected (∞,1)-topos. | The inclusion of $X$ into its smooth path ∞-groupoid as the constant paths. | |
Given $X$, there is the result of detaching the points in $X$. | $\left(A \colon \sharp Type\right) \;\vdash\; \flat A \colon \sharp Type$ | The operation of the discrete object comonad on a (higher) local topos. | The moduli stack for flat ∞-connections. | |
Given $A$, there is a map from flat $A$-connections to the underlying $A$-bundles | $\left(A \colon \sharp Type\right) \;\vdash\; UnderlyingBundle_A \colon \flat A \to A$ | The counit of the discrete object-comonad on a (higher) local topos. | The function that sends a flat ∞-connection to its underlying principal ∞-bundle. | Flat connections |
Adding the modalities $(\Pi \dashv \flat \dashv \sharp)$ to the above language of homotopy type theory yields a language that we call cohesive homotopy type theory (following a term introduced by Lawvere).
Fundamental judgements in cohesive homotopy type theory include those indicated in the following table, which capture central concepts of gauge theory and its (higher) geometric quantization.
Structures expressible in cohesive homotopy type theory:
Gauge fields, matter fields, and smooth action functionals on their moduli stacks
| $transport(\nabla) \colon \underset{x,y \colon \Pi X}{\sum} \left( x \rightsquigarrow y \right) \to \underset{*,*' \colon \mathbf{B}G}{\sum} (* \rightsquigarrow *')$ | $\frac{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}$.| The higher parallel transport $trans(\nabla)$ of a flat connection $\nabla$: a (higher) gauge field with vanishing field strength. | Flat connections | | A closed differential form $\omega$ is a flat connection $\nabla$ and a trivialization of the underlying bundle. | $\begin{aligned} & \flat_{dR} \mathbf{B} G \coloneqq \\ & \sum_{\nabla \colon \flat \mathbf{B}G} (UnderlyingBundle(\nabla) \simeq *) \end{aligned}$ | $\begin{matrix} \flat_{dR}\mathbf{B}G & \stackrel{UnderlyingConnection}{\begin{svg} <svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g 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\mathclap{\array{\arrayopts{\align{bottom}}\;\begin{svg}<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.40001pt" height="10.40001pt" viewBox="-0.2 -0.2 10.40001 10.40001 "><g transform="translate(0,10.20001 ) scale(1,-1) translate(0,0.2 )"><g><g stroke="rgb(0.0%,0.0%,0.0%)"><g fill="rgb(0.0%,0.0%,0.0%)"><g stroke-width="0.4pt"><g><path d=" M 0.0 0.0 L 10.00002 0.0 L 10.00002 10.00002 " style="fill:none"/></g></g></g></g></g></g></svg>\end{svg} & \space{10}{0}{30} \\ \space{10}{30}{1} & \swArrow}} & \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}{}^{\mathrlap{Underlying \atop Bundle}} \\ * &\stackrel{}{\begin{svg}<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \mathbf{B}G \end{matrix}$ | The coefficients for de Rham hypercohomology – flat ∞-Lie algebra valued differential forms. | de Rham coefficients | | A general connection $\nabla$ is the equivalence between the curvature $curv(\mathbf{c})$ of a bundle $\mathbf{c}$ and a closed differential form $\omega$. | $\nabla \colon \underset{{\mathbf{c} \colon \mathbf{B}^n \mathbb{G}} \atop { \omega \colon \Omega^{n+1}_{cl} }}\sum \left( curv\left(\mathbf{c}\right) = \omega\right)$ | $\begin{matrix} \mathbf{B}^n \mathbb{G}_{conn} & \stackrel{F_{(-)}}{\begin{svg} <svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \Omega^{n+1}_{cl} \\ \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \mathclap{\array{\arrayopts{\align{bottom}}\;\begin{svg}<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.40001pt" height="10.40001pt" viewBox="-0.2 -0.2 10.40001 10.40001 "><g transform="translate(0,10.20001 ) scale(1,-1) translate(0,0.2 )"><g><g stroke="rgb(0.0%,0.0%,0.0%)"><g fill="rgb(0.0%,0.0%,0.0%)"><g stroke-width="0.4pt"><g><path d=" M 0.0 0.0 L 10.00002 0.0 L 10.00002 10.00002 " style="fill:none"/></g></g></g></g></g></g></svg>\end{svg} & \space{10}{0}{30} \\ \space{10}{30}{1} & \swArrow}} & \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg} \\ \mathbf{B}^n \mathbb{G} &\stackrel{curv}{\begin{svg}<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \flat_{dR} \mathbf{B}^{n+1}\mathbb{G} \end{matrix}$ | The coefficients for smooth differential cohomology: abelian (higher) gauge fields. | Circle principal n-connections | | There is a cohesive function from $G$-gauge fields to higher $\mathbb{G}$-gauge fields. | $\vdash \; \exp(i S) \colon \mathbf{B}G_{conn} \to \mathbf{B}^n \mathbb{G}_{conn}$ | A differential universal characteristic class. | An extended action functional/prequantum n-bundle for extended higher Chern-Simons-type gauge theory. | |
… and their ∞-geometric prequantization (see there for a more comprehensive disctionary):
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
There is a $\mathbb{G}$-equivariant map $\psi$ from the prequantum bundle to the representation space. | $\vdash \; \psi \colon \underset{\nabla \colon \mathbf{B}\mathbb{G}_{conn}}{\prod} \left( P\left(\nabla\right) \to V\left(\nabla\right) \right)$ | $\array{ X &&\stackrel{\psi}{\to}&& V\sslash \mathbb{G}_{conn} \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\overline{\rho}} \\ && \mathbf{B} \mathbb{G}_{conn}}$ | A prequantum state. | Geometric quantization |
There is a differentially $\mathbb{G}$-equivariant equivalence $\exp(\hat O)$ from the prequantum bundle to itself. | $\vdash \; \exp(\hat O) \colon \underset{\nabla \colon \mathbf{B}\mathbb{G}_{conn}}{\prod} \left( P\left(\nabla\right) \stackrel{\simeq}{\to} P\left(\nabla\right) \right)$ | $\array{ X &&\stackrel{\exp(\hat O)}{\to}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\nabla} \\ && \mathbf{B} \mathbb{G}_{conn}}$ | A prequantum operator: an element of the quantomorphism group/Heisenberg group of the quantum system. | Geometric quantization |
Finally, in order to be able to concretely speak about not just about any gauge field, but the concrete particular gauge fields in the observable universe, our language should be able to express the existence of the continuum real line.
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
There is the continuum line. | $\begin{aligned}\vdash\; & \mathbb{R} \colon Type \\ & i \colon \mathbb{Z} \to \mathbb{R} \\ & GeometricallyContract_{\mathbb{R}} \colon (\Pi(\mathbb{R}) \simeq Point) \end{aligned}$ | line object | real line | The continuum real worldline |
This then induces the existence of the circle group $U(1) = \mathbb{R}/\mathbb{Z}$. The electromagnetic field is a gauge field for gauge group $U(1)$. Therefore in the language of cohesive homotopy type theory we can say
Let there be light.
ordinary language | syntax | semantics | model | chapter |
---|---|---|---|---|
general abstract | general concrete | concrete particular | ||
There is the collection of higher $U(1)$-principal connections. | $n\colon \mathbb{N} \; \vdash \; \mathbf{B}^n U(1)_{conn} \colon Type$ | The coefficients for ordinary differential cohomology (with coefficients in an Eilenberg-MacLane object.) | The smooth higher moduli stack of smooth circle n-bundles with connection. | Circle-principal n-connections. |
There is light. | $\vdash \; \nabla_{em} \colon [X,\mathbf{B}U(1)_{conn}]$ | A cocycle in ordinary differential cohomology in degree-2. | A configuration of the electromagnetic field on spacetime $X$. | Circle principal connection |
$\,$
There are of many more constructions in fundamental (quantum) physics that are naturally expressible in cohesive homotopy type theory, but the above should already give an idea and highlight the cornerstones of the following discussion.
$\,$
We now end this introduction and overview and turn to the in-depth account of geometry of physics.
I) Geometry
We begin by laying the foundations of differential geometry. Doing this in th natural abstract way seamlessly leads over to the foundations of higher differential geometry (see also motivation for higher differential geometry). Once this is set up, we discuss the fundamental constructions: groups, actions/representations, fiber bundles, connections, Chern-Weil theory.
This chapter is at geometry of physics -- coordinate systems
This chapter is at geometry of physics -- smooth spaces.
This chapter is at geometry of physics -- differential forms.
This chapter is at geometry of physics -- differentiation.
This chapter is at geometry of physics -- smooth homotopy types.
any Lie group $G$ induces its delooping Lie groupoid
Write
for the functor that sends a chain complex of abelian group objects in smooth spaces first to the simplicial abelian group in smooth spaces given by the Dold-Kan correspondence, then forgets the abelian group structure and finally regards the resulting simplicial smooth space as a smooth ∞-groupoid under simplicial localization.
(…)
This chapter is at geometry of physics -- principal bundles.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos $(\mathbf{\Pi} \dashv \flat \dashv \sharp)$ equipped with differential cohesion $(Red \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf})$.
For $X \in \mathbf{H}$, write
for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on the formally étale maps into $X$, def. \ref{FormallyEtaleMap}.
We call this the petit (∞,1)-topos of $X$.
The petit topos $Sh_{\mathbf{H}}(X)$ of def. 2 is indeed an (∞,1)-topos. Moreover the defining inclusion into the slice (∞,1)-topos is both reflective? as well as coreflective.
This is proven at differential cohesion – structure sheaves.
For $X \in \mathbf{H}$ write
for the (∞,1)-functor which is the composite of the base change to $X$ followed by the co-reflection of prop. 1. We call this the structure sheaf of $X$.
For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a formally étale morphism in $\mathbf{H}_{th}$, we have that
This means that $\mathcal{O}_{X}(A)$ behaves as the sheaf of $A$-valued functions over $X$.
(…)
For $n \in \mathbb{N}$ a manifold of dimension $n$ is an object $X$ that locally looks like a Cartesian space $\mathbb{R}^n$, hence that can be thought of as being glued together from Cartesian spaces by gluing these along diffeomorphisms.
A natural way to make this precisely is to say that a manifold of dimension $X$ is an object such that first of all there is a cover, hence a 1-epimorphism of the form
This encodes that $X$ can surjectively covered by Cartesian spaces, but it does not yet ensure that $X$ is locally equivalent to a Cartesian space in the intended sense. That intended sense is that $p$ is a local diffeomorphism.
Hence a manifold is a smooth space which receives a map out of a coproduct of Cartesian spaces that is a 1-epimorphism and a local diffeomorphism.
By the discussion above at Structure sheaves the general way to say local diffeomorphism is to say formally étale morphism. Hence more generally we can consider the notion of a smooth groupoid which received a map out of a coproduct of Cartesian spaces that is a 1-epimorphism and a formally étale morphism. If here the souce-fibers of the groupoid are in addition compact, then this is what is called an orbifold.
A smooth manifold of dimension $n$
a smooth space with an atlas
of coordinate charts. On each overlap $U_i \cap U_j$ of two charts, the partial derivatives of the corresponding coordinate transformations
form the Jacobian matrix of smooth functions
with values in invertible matrices, hence in the general linear group $GL(n)$. By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations
(here and in the following sums over an index appearing upstairs and downstairs are explicit)
hence the equation
in the group $C^\infty(U_i \cap U_j \cap U_k, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlaps.
This is the cocycle condition for a smooth Cech cocycle in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the sheaf of smooth functions with values in $GL(n)$ ). We write
Formulated as smooth groupoids
$X$ itself is a Lie groupoid $(X \stackrel{\to}{\to} X)$ with trivial morphism structure;
from the atlas $\{U_i \to X\}$ we get the corresponding Cech groupoid
whose objects are the points in the atlas, with morphisms identifying lifts of a point in $X$ to different charts of the atlas;
We discuss how the tangent bundle of a manifold $X$ naturally arises in the above perspecive in terms of the map $\tau_X \;\colon\; X \to \mathbf{B}GL(n)$ that modulates it.
The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form
where
the left morphism is stalk-wise (around small enough neighbourhoods of each point) an equivalence of groupoids (we make this more precise in a moment);
the horizontal functor $\tau_X$ has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.
A transformation of smooth functors $\lambda_1 \Rightarrow \lambda_2 : C(\{U_i\}) \to \mathbf{B} GL(n)$ is precisely a coboundary between two such cocycles.
This defines a morphism of smooth groupoids
The homotopy fiber of this map is a $GL(n)$-principal bundle called the frame bundle of $X$, while the canonically associated bundle via the canonical representation of $GL(n)$ on $\mathbb{R}^n$ is the tangent bundle
Let $\mathbf{H}$ be a cohesive (∞,1)-topos $(\mathbf{\Pi} \dashv \flat \dashv \sharp)$ equipped with differential cohesion $(Red \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf})$. Let
be an line object that exhibits the cohesive structure.
An étale ∞-groupoid of dimension $n$ is an object $X \in \mathbf{H}$ such that there exists a map $p \;\colon\;\left(\coprod_{i} \mathbb{A}^n\right) \to X$ such that
$p$ is a 1-epimorphism;
$p$ is a formally étale morphism, def. \ref{FormallyEtaleMap}
If $X$ here is 0-truncated then we call it it manifold. It $X$ is 1-truncated we call it an orbifold.
(…)
(…)
given a $K$-principal bundle
a reduction of the structure group along $G \to K$ is
reduction of the structure group along
$\mathbf{B}O(n) \to \mathbf{B}GL(n)$
$e$ is vielbein: definition of an orthonormal frame? at each point
example: the other 2 Maxwell equations: $\mathbf{d} \star F = j_{el}$.
(…)
(…)
(…)
(…)
(…)
for the moment see the sub-entry geometry of physics - modules
$X$ connected, $\pi_1(X) \in$Grp its fundamental group for any choice of basepoint, then the holonomy pairing
descends to homotopy classes of (based) loops
to a bijection from equivalence classes of flat? $G$-principal connections to the quotient set of group homomorphisms $\pi_1(X) \to G$ modulo the adjoint action of $G$ on itself.
For $G \in Grp(\mathbf{H})$ and $X \in \mathbf{H}$ a flat $G$-connection $\nabla$ on $X$ is a morphism
We write
and accordingly
for the cohomology of $X \in \mathbf{H}$ with flat coefficients.
By adjunction,
a flat $G$-connection is equivalently a morphism
Since $\Pi(X)$ is the fundamtal infinity-groupoid? of $X$, this manifestly encodes the higher parallel transport of the flat connection.
Write
for the $(Disc \vdash \Gamma)$-counit-
For $\nabla \colon X \to \flat \mathbf{B}G$ the composite
modulates a $G$-principal ∞-bundle on $X$, by def. \ref{spring}. This we call the underlying $G$-principal bundle of $\nabla$.
Let $G$ be a Lie group, and write $\mathfrak{g}$ for its Lie algebra. The set of Lie algebra valued differential 1-forms is the tensor product
flat forms:
(…)
This is a smooth space
For $\mathfrak{g} = Lie(\mathbb{R})$ we have
and we write
Below we see
Below we see that
For $G \in Gpr(\mathbf{H})$, its de Rham coefficient object is the homotopy pullback
in
This pullback diagram expresses that elements of $\flat_{dR}\mathbf{B}G$ are flat $G$-connections $\nabla \colon X \to \flat \mathbf{B}G$, def. 6 equipped with a trivialization of their underlying $G$-principal bundle, def. 8.
Let $\mathbf{H} =$ Smooth∞Grpd. All smooth manifolds and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this $\mathbf{H} = Sh_\infty(CartSp)$.
For $G$ a Lie group, the de Rham coefficient object $\flat_{dR}\mathbf{B}G$, def. 10 of its delooping is given by the sheaf of flat Lie algebra valued differential 1-forms $\Omega^1_{flat}(-,\mathfrak{g})$, def. 9, for $\mathfrak{g}$ the Lie algebra of $G$:
This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group.
Write $U(1)$ for the circle group regared as a Lie group in the standard way.
For $n \in \mathbb{N}$, the de Rham coefficient object $\flat_{dR}\mathbf{B}^n U(1)$, def. 10, of the $n$-fold delooping of $U(1)$ is given by the image under the Dold-Kan correspondence
of the truncated de Rham complex of sheaves of differential forms,
This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups.
Consider
the Maurer-Cartan form on $\mathbb{R}$ is the de Rham differential
Let $\mathbf{H}$ be a cohesive (infinity,1)-topos $(\mathbf{\Pi} \dashv \flat \dashv \sharp)$. We discuss a general formulation of Maurer-Cartan forms on cohesive infinity-groups
Let $G \in Grp(\mathbf{H})$ be a group object.
Use the pasting law together with the fact that $\flat$ is right adjoint and hence preserves limits to get $\theta$ in
This is the Maurer-Cartan form on $G$
For $S \;\colon\; X \to G$ a morphism, write
for its composite with the map of def. 11, hence the pullback of the Maurer-Cartan form along $S$. We also call this the de Rham differential of $S$.
For $G$ a Lie group canonically regarded in $\mathbf{H} =$Smooth∞Grpd the general abstract morphism
is identified, via the identification $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ of prop. 2 and the Yoneda lemma, with the traditional Maurer-Cartan form
The Maurer-Cartan form on the line object
is the de Rham differential,
For $G = \mathbf{B}^n U(1)$
sends a circle $n$-bundle to the curvature of a pseudo-connection on it.
(…)
Dirac charge quantization says that the electromagnetic field is only locally in general a map
globally it is instead a map
where
the smooth groupoid is
quotient of $\Omega^1(-)$ by $U(1)$-gauge transformations
for
a gauge transformation $A \to A'$ is $\lambda : X \to U(1)$ with
There are different equivalent definitions of the classical notion of a connection. One that is useful for our purposes is that a connection $\nabla$ on a $G$-principal bundle $P \to X$ is a rule $tra_\nabla$ for parallel transport along paths: a rule that assigns to each path $\gamma : [0,1] \to X$ a morphism $tra_\nabla(\gamma) : P_x \to P_y$ between the fibers of the bundle above the endpoints of these paths, in a compatible way:
In order to formalize this, we introduce a (diffeological) Lie groupoid to be called the path groupoid of $X$. (Constructions and results in this section are from ([SWI]).
For $X$ a smooth manifold let $[I,X]$ be the set of smooth functions $I = [0,1] \to X$. For $U$ a Cartesian space, we say that a $U$-parameterized smooth family of points in $[I,X]$ is a smooth map $U \times I \to X$. (This makes $[I,X]$ a diffeological space).
Say a path $\gamma \in [I,X]$ has sitting instants if it is constant in a neighbourhood of the boundary $\partial I$. Let $[I,P]_{si} \subset [I,P]$ be the subset of paths with sitting instants.
Let $[I,X]_{si} \to [I,X]_{si}^{th}$ be the projection to the set of equivalence classes where two paths are regarded as equivalent if they are cobounded by a smooth thin homotopy.
Say a $U$-parameterized smooth family of points in $[I,X]_{si}^{th}$ is one that comes from a $U$-family of representatives in $[I,X]_{si}$ under this projection. (This makes also $[I,X]_{si}^{th}$ a diffeological space.)
The passage to the subset and quotient $[I,X]_{si}^{th}$ of the set of all smooth paths in the above definition is essentially the minimal adjustment to enforce that the concatenation of smooth paths at their endpoints defines the composition operation in a groupoid.
The path groupoid $\mathbf{P}_1(X)$ is the groupoid
with source and target maps given by endpoint evaluation and composition given by concatenation of classes $[\gamma]$ of paths along any orientation preserving diffeomorphism $[0,1] \to [0,2] \simeq [0,1] \coprod_{1,0} [0,1]$ of any of their representatives
This becomes an internal groupoid in diffeological spaces with the above $U$-families of smooth paths. We regard it as a groupoid-valued presheaf, an object in $[CartSp^{op}, Grpd]$:
Observe now that for $G$ a Lie group and $\mathbf{B}G$ its delooping Lie groupoid discussed above, a smooth functor $tra : \mathbf{P}_1(X) \to \mathbf{B}G$ sends each (thin-homotopy class of a) path to an element of the group $G$
such that composite paths map to products of group elements
and such that $U$-families of smooth paths induce smooth maps $U \to G$ of elements.
There is a classical construction that yields such an assignment: the parallel transport of a Lie-algebra valued 1-form.
Suppose $A \in \Omega^1(X, \mathfrak{g})$ is a degree-1 differential form on $X$ with values in the Lie algebra $\mathfrak{g}$ of $G$. Then its parallel transport is the smooth functor
given by
where the group element on the right is defined to be the value at 1 of the unique solution $f : [0,1] \to G$ of the differential equation
for the boundary condition $f(0) = e$.
This construction $A \mapsto tra_A$ induces an equivalence of categories
where on the left we have the hom-groupoid of groupoid-valued presheaves and where on the right we have the groupoid of Lie-algebra valued 1-forms whose
objects are 1-forms $A \in \Omega^1(X,\mathfrak{g})$,
morphisms $g : A_1 \to A_2$ are labeled by smooth functions $g \in C^\infty(X,G)$ such that $A_2 = g^{-1} A g + g^{-1}d g$.
This equivalence is natural in $X$, so that we obtain another smooth groupoid.
Define $\mathbf{B}G_{conn} : CartSp^{op} \to Grpd$ to be the (generalized) Lie groupoid
whose $U$-parameterized smooth families of groupoids form the groupoid of Lie-algebra valued 1-forms on $U$.
This equivalence in particular subsumes the classical facts that parallel transport $\gamma \mapsto P \exp(\int_{[0,1]} \gamma^* A)$
is invariant under orientation preserving reparameterizations of paths;
sends reversed paths to inverses of group elements.
There is an evident natural smooth functor $X \to \mathbf{P}_1(X)$ that includes points in $X$ as constant paths. This induces a natural morphism $\mathbf{B}G_{conn} \to \mathbf{B}G$ that forgets the 1-forms.
Let $P \to X$ be a $G$-principal bundle that corresponds to a cocycle $g : C(U) \to \mathbf{B}G$ under the construction discussed above. Then a connection $\nabla$ on $P$ is a lift $\nabla$ of the cocycle through $\mathbf{B}G_{conn} \to \mathbf{B}G$.
This is equivalent to the traditional definitions.
A morphism $\nabla : C(U) \to \mathbf{B}G_{conn}$ is
on each $U_i$ a 1-form $A_i \in \Omega^1(U_i, \mathfrak{g})$;
on each $U_i \cap U_j$ a function $g_{i j} \in C^\infty(U_i \cap U_j , G)$;
such that
on each $U_i \cap U_j$ we have $A_j = g_{i j}^{-1}( A + d_{dR} )g_{i j}$;
on each $U_i \cap U_j \cap U_k$ we have $g_{i j} \cdot g_{j k} = g_{i k}$.
Let $[I,X]_{si}^{th} \to [I,X]^h$ the projection onto the full quotient by smooth homotopy classes of paths.
Write $\mathbf{\Pi}_1(X) = ([I,X]^h \stackrel{\to}{\to} X)$ for the smooth groupoid defined as $\mathbf{P}_1(X)$, but where instead of thin homotopies, all homotopies are divided out.
The above restricts to a natural equivalence
where on the left we have the hom-groupoid of groupoid-valued presheaves, and on the right we have the full sub-groupoid $\mathbf{\flat}\mathbf{B}G \subset \mathbf{B}G_{conn}$ on those $\mathfrak{g}$-valued differential forms whose curvature 2-form $F_A = d_{dR} A + [A \wedge A]$ vanishes.
A connection $\nabla$ is flat precisely if it factors through the inclusion $\flat \mathbf{B}G \to \mathbf{B}G_{conn}$.
For the purposes of Chern-Weil theory we want a good way to extract the curvature 2-form in a general abstract way from a cocycle $\nabla : X \stackrel{\simeq}{\leftarrow }C(U) \to \mathbf{B}G_{conn}$. In order to do that, we first need to discuss connections on 2-bundles.
Write $U(1) \in Smooth0Type$ for the smooth space of the circle group. Write
for the homomorphism of smooth spaces which over an abstract coordinate system $U \in$ CartSp is given by the function
which sends a $U(1)$-valued $f \colon U \to U(1)$ – for which one can always find an $\mathbb{R}$-valued function $\hat f \colon U \to \mathbb{R}$ for whuch $f = \hat f \,mod\, \mathbb{Z}$ – to the differential 1-form $\mathbf{d} \hat f$.
For $n \in \mathbb{N}$, the chain complex of smooth spaces (of sheaves on CartSp)
(regarded as a chain complex of abelian groups and as sitting in degrees $n$ through 0) is called the (smooth) Deligne complex in these degrees.
For $n \in \mathbb{N}$, write
for the smooth ∞-groupoid presented by the Deligne complex under the Dold-Kan correspondence map, def. 1.
For $X \in$ Smooth∞Grpd we say that
We also call $\mathbf{B}^n U(1)_{conn}$ the universal moduli ∞-stack of circle $n$-bundles with connection.
Write
for the canonical forgetful morphism from moduli of $n$-connections to those of the underlying principal bundles.
More generally, we may consider the intermediate stages
Let $G \in Grp(\mathbf{H})$ be a braided ∞-group. Equivalently, let its delooping $\mathbf{B}G \in \mathbf{H}$ be itself equipped with the structure of an ∞-group. Write
for the corresponding double delooping.
Write
for the Maurer-Cartan form on the ∞-group $\mathbf{B}G$, def. 11. We call this the universal curvature characteristic of $G$.
The differential cohomology with coefficients in $\mathbf{B}G$ is cohomology in the slice (∞,1)-topos $\mathbf{H}_{/\flat_{dR} \mathbf{B}^2 G}$ with coefficients in $curv_G$
presented by ordinary differential cohomology
We discuss the moduli stacks of higher principal connections, over a fixed $X \in Smooth\infty Grpd$.
For $n \in \mathbb{N}$ and with $\mathbf{B}^n U(1)_{conn} \in$ Smooth∞Grpd the universal moduli stack for circle n-bundles with connection, def. 16, and for $X \in Smooth\infty Grpd$, one may be tempted to regard the internal hom/mapping space $[X, \mathbf{B}^n U(1)_{conn}]$ as the moduli stack of circle n-bundles with connection on $X$. However, for $U \in$ CartSp an abstract coordinate system, $U$-plots and their k-morphisms in $[X, \mathbf{B}^n U(1)_{\mathrm{conn}}]$ are circle principal $n$-connections and their $k$-fold gauge transformations on $U \times X$, and this is not generally what one would want the $U$-plots of the moduli stack of such connections on $X$ to be. Rather, that moduli stack should have
as $U$-plots smoothly $U$-parameterized collections $\{\nabla_u\}$ of $n$-connections on $X$;
as $k$-morphisms smoothly $U$-parameterized collections $\{\phi_u\}$ of gauge transformations between them.
The first item is equivalent to: a single $n$-connection on $U \times X$ such that its local connection $n$-forms have no legs along $U$. This is essentially the situation of moduli of differential forms which we have discussed above in Smooth moduli space of differential forms).
But the second item is different: a gauge transformation of a single $n$-connection $\nabla$ on $U \times X$ needs to respect the curvature of the connection along $U$, but a family $\{\phi_u\}$ of gauge tranformations between the restrictions $\nabla|_u$ of $\nabla$ to points of the coordinate patch $U$ need not.
In order to capture this correctly, the concretification-process, def. \ref{ConcreteObjectsAndConcretification}, that yielded the moduli spaces of differential forms in above is to be refined to a process that concretifies the higher stack $[X, \mathbf{B}^n U(1)_{conn}]$ degreewise in stages.
We discuss this first for $n = 1$, hence for moduli stacks of circle bundles with connection.
For $X \in Smooth1Typpe \hookrightarrow$ Smooth∞Grpd a smooth groupoids, we write
for the n-image factorization of the canonical morphism $X \to \sharp X$ to the sharp modality, def. \ref{SharpModalityOfLocalTopos}.
If $X \in Smooth0Type$ is just a smooth space then $\sharp_2 X \simeq X$ and $\sharp_1 X = Conc X$ is the concretification of $X$, def. \ref{ConcreteObjectsAndConcretification}.
One might call $\sharp_1 X$ the “1-concretification” and $\sharp_2 X$ the “2-concretification” of $X$. But generally what one might actually want to call a “concretification” of $X$ involves an interplay of both, as we will see now.
For $X \in$ Smooth∞Grpd, write
for the smooth groupoid which is the homotopy pullback in
Here the bottom morphism is the $\sharp_1$-image of the forgetful morphism $U_{\mathbf{B}U(1)_{conn}} \colon \mathbf{B}U(1)_{conn} \to \mathbf{B}U(1)$ from def. 17; and the right morphism is the canonical projection from the 2-image to the 1-image, as discussed there.
The smooth groupoid $U(1)\mathbf{Conn}(X)$ of def. 22 is indeed the smooth moduli object/moduli stack of circle-principal connections on $X$; in that its $U$-plots of are smoothly $U$-parameterized collections of smooth circle-principal connections on $X$ and its morphisms of $U$-plots are smoothly $U$-parameterized collections of smooth gauge transformation between these, on $X$.
By the discussion at n-image and using arguments as for the concretification of moduli of differential forms above, we have:
$\sharp_1 [X, \mathbf{B}U(1)_{conn}]$ has as $U$-plots those connections $\nabla$ on $U \times X$ whose connection 1-forms have no leg along $U$, hence smooth $U$-parameterized families $\{\nabla_u\}$ of connections on $X$, and has as morphisms $\Gamma(U)$-parameterized (hence non-smooth) collections $\{\phi_u\}$ of gauge transformations of connections on $X$;
$\sharp_1 [X, \mathbf{B}U(1)]$ looks similarly, just without the connection information;
$\sharp_1 [X, U_{\mathbf{B}U(1)_{conn}}]$ simply forgets the connection data on the collections of bundles-with-connection; the point to notice is that oveach each chart $U$ it is a fibration(isofibration): given a $\Gamma(U)$-parameterized collection of gauge transformations out of a smoothly $U$-parameterized collection of bundles and then a smooth choice of smooth connections on these bundles, the $\Gamma(U)$ collection of gauge transformations of course also acts on these connections;
$\sharp_2 [X, \mathbf{B} U(1)] \simeq [X, \mathbf{B} U()1]$ (because if two gauge transformations of bundles on $U \times X$ coindide on each point of $U$ as gauge tranformations on $X$, then they were already equal).
From the third item it follows, by the discussion at homotopy pullback, that we may compute equivalently simply the pullback in the 1-category of groupoid-valued presheaves on CartSp. This means that a $U$-plot of $U(1)\mathbf{Conn}(X)$ is a smoothly $U$-parameterized collection $\{\nabla_u\}$ of connections on $X$, and that a morphism between such as a $\Gamma(U)$-parameterized collection of gauge transformations $\{\phi_u\}$ of connections, such that their underlying collection of gauge transformations of bundles is a smoothly $U$-parameterized family. But gauge transformations of 1-connections are entirely determined by the underlying gauge transformation of the underlying bundle, and so this just means that also the morphism of $U$-plots of $U(1)\mathbf{Conn}(X)$ are smoothly $U$-paramezerized collections of gauge transformations.
We now discuss moduli of circle 2-bundles with connection over a given base $X$.
Recall that by def. 18 we write
For $X \in Smooth \infty Grpd$ write
for the homotopy limit in the diagram
Here the horizontal morphisms are those induced under the n-image by those of def. 18 and where the vertical morphisms are the n-image-projections, as discussed there.
We call this the moduli 2-stack of circle-principal 2-connections on $X$.
The smooth 2-groupoid $(\mathbf{B}U(1))\mathbf{Conn}(X)$ of def. 23 is indeed the smooth moduli object of circle-principal 2-connections on $X$; in that its $U$-plots of are smoothly $U$-parameterized collections of smooth circle-principal 2-connections on $X$ and its morphisms of $U$-plots are smoothly $U$-parameterized collections of smooth gauge transformation between these, on $X$, and similarly for its 2-morphisms.
By a variant of the pasting law one sees that the defining homotopy limit may be computed as the pasting of homotopy pullbacks
For each of these smaller homotopy-pullback squares the reasoning is directly analogous to that in the proof of prop. 10:
Starting in the bottom left, $\sharp_1 [X, \mathbf{B}^2 U(1)_{conn}]$ has over $U \in CartSp$ smoothly $U$-parameterized collections of 2-connections on $X$ as objects, but discretely $\Gamma(U)$-parameterized collections of morphisms and 2-morphisms between them. The morphism to the right forgets just the local differential 2-form part of a 2-connection (keeping the 1-form part). This has the effect that where in $\sharp_2 [X,\mathbf{B}^2 U(1)_{conn^1}]$ the 1-morphisms over $U$ are gauge transformations of such truncated 2-connections over $U \times X$, these are now not forced to strictly fix the curvature along $U$, hence these are smoothly $U$-parameterized collections of gauge transformation on $X$ (between truncated 2-connections). So the pullback object $B$ combines these two aspects and hence has as objects and as 1-morphisms now smoothly $U$-parameterized collections of connections and gauge transformations on $X$, respectively, and only the 2-morphisms of $B$ keep beeing discretely $\Gamma(U)$-parameterized collections of 2-gauge transformations.
Similarly then, $A$ is already the correct smooth moduli 2-stack, but of truncated connections, and so finally the fiber product of $A$ with $B$ forces the discrete families of 2-morphisms of untruncated gauge transformations to have underlying smooth families of 2-morphisms of truncated gauge transformations. But since these uniquely fix the untruncated ones, this makes $(\mathbf{B}U(1))\mathbf{Conn}(X)$ have the correct smooth collections of structures in each degree.
The universal curvature characteristic, def. 19, has the syntax
Regarded as a dependent type in the de Rham coefficient context this is
Therefore the syntax for a domain object $F \colon X \to \flat_{dR} \mathbf{B}^2 G$ in this context is
in differential cohomology, def. 20, on $(X,F)$ is hence
a differential characteristic class
is an (extended) Lagrangean for infinity-Chern-Simons theory.
The corresponding action functional is discussed in Semantic Layer - Action functionals from Lagrangeans.
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By the discussion in Differential forms and Principal connections, differential forms and more generally connections may be regarded as infinitesimal measures of change, of displacement. The discussion in Differentiation showed how to extract from a finite but cohesive (e.g. smoothly continuous) displacement all its infinitesimal measures of displacements by differentiation.
Here we discuss the reverse operation: integration is a construction from a differential form of the corresponding finite cohesive displacement. More generally this applies to any connection and is then called the parallel transport of the connection, a term again referring to the idea that a finite displacement proceeds pointwise in parallel to a given infinitesimal displacement.
Under good conditions this construction can proceed literally by “adding up all the infinitesimal contributions” and therefore integration is traditionally thought of as a generalization of forming sums. Therefore one has the notation ”$\int_{\Sigma} \omega$” for the integral of a differential form $\omega$ over a space $\Sigma$, as a variant of the notation ”$\sum_{S} f$” for the sum of values of a function on a set $S$. For the case of integrals of connections the corresponding parallel transport expression is often denoted by an exponentiated integral sign ”$\mathcal{P} \exp(\int_\Sigma \omega)$”, referring to the fact that the passage from infinitesimal to finite quantities involves also the passage from Lie algebra data to Lie group data (“exponentiated Lie algebra data”).
However, both from the point of view of gauge theory physics as well as from the general abstract perspective of cohesive homotopy type theory another characterization of integration is more fundamental: the integral $\int_\Sigma \omega$ of a differential form $\omega$ (or more generally of a connection) is an invariant under those gauge transformations of $\omega$ that are trivial on the boundary of $\Sigma$, and it is the universal such invariant, hence is uniquely characterized by this property.
In traditional accounts this fact is referred to via the Stokes theorem and its generalizations (such as the nonabelian Stokes theorem), which says that the integral/parallel transport is indeed invariant under gauge transformations of differential forms/connections. That this invariance actually characterizes the integral and the parallel transport is rarely highlighted in traditional texts, but it is implicit for instance in the old “path method” of Lie integration (discussed below in Lie integration) as well as in the famous characterization of flat connections, discussed above in Flat 1-connections:
for $X$ a connected manifold and for $G$ a Lie group, the operation of sending a flat $G$-principal connection $\nabla$ to its parallel transport $\gamma \mapsto hol_{\gamma}(\nabla)$ around loops $\gamma\colon S^1 \to X$, hence to the integral of the connection around all possible loops (its holonomy), for any fixed basepoint
exhibits a bijection between gauge equivalence classes of connections and group homomorphisms from the fundamental group $\pi_1(X)$ of $X$ to the gauge group $G$ (modulo adjoint $G$-action from gauge transformations at the base point, hence at the integration boundary). This is traditionally regarded as a property of the definition of the parallel transport $\mathcal{P} \exp(\int_{(-)}(-))$ by integration. But being a bijection, we may read this fact the other way round: it says that forming equivalence classes of flat $G$-connections is a way of computing their integral/parallel transport.
We saw a generalization of this fact to non-closed forms and non-flat connections already in the discussion at Differential 1-forms as smooth incremental path measures, where gauge equivalence classes of differential forms are shown to be equivalently assignments of parallel transport to smooth paths.
This is also implied by the above discussion: for $\nabla \in H^1_{conn}(X,G)$ any non-flat connection and $\gamma \colon S^1 \to X$ a trajectory in $X$, we may form the pullback of $\nabla$ to $S^1$. There it becomes a necessarily flat connection $\gamma^* \nabla \in H^1_{conn, flat}(S^1,G)$, since the curvature differential 2-form necessarily vanishes on the 1-dimensional manifold $S^1$. Accordingly, by the above bijection, forming the gauge equivalence class of $\gamma^* \nabla$ means to find a group homomorphism
modulo conjugation (modulo nothing if $G$ is abelian, such as $G = U(1)$) and since $\mathbb{Z}$ is the free group on a single generator this is the same as finding an element
This total operation of first pulling back the connection and then forming its integration (by taking gauge equivalence classes) is called the transgression of the original 1-form connection on $X$ to a 0-form connection on the loop space $[S^1,X]$.
Below in the Model Layer we discuss the classical examples of integration/parallel transport and their various generalizations in detail. Then in the Semantic Layer we show how indeed all these constructions are obtained forming equivalence classes in the (∞,1)-topos of smooth homotopy types, hence by truncation (followed, to obtain the correct cohesive structure, by concretification, def. \ref{ConcreteObjectsAndConcretification}).
For $n \in \mathbb{N}$ let
be the standard unit cube.
Let
be a differential n-form.
Let $Partitions(C^k)$ be the poset whose elements are partitions of the unit $n$cube $C^n$ into $N^n$ subcubes, for $N \in \mathbb{N}$, ordered by inclusion.
Let
be the function that sends
Then
Let $\Sigma$ be a closed oriented smooth manifold of dimension $k$
For $n \in \mathbb{N}$, $n \geq k$, define the morphism of smooth spaces
by declaring that over a coordinate chart $U \in$ CartSp it is the ordinary integration of differential forms over smooth manifolds
given $A \in \Omega^1(\Delta^1, \mathfrak{g})$
we say $f \in C^\infty(\Delta^1, G)$ is the parallel transport of $A$ if
$f(0) = 1$
$f$ satisfies the differential equation
where on the right we have the differential of the left action of the group on itself.
In this case one writes
and calls it the path ordered integral? of $A$. Here the enire left hand side is primitive notation.
In the case that $G = U(1)$ this reproduces the ordinary integral
There is another way to express this parallel transport, related to Lie integration:
Define an equivalence relation on $\Omega^1(\Delta^1, \mathfrak{g})$ as follows: two 1-forms $A,A'$ are taken to be equivalent if there is a flat 1-form $\hat A \in \Omega^1_{flat}(D^2, \mathfrak{g})$ on the 2-disk such that its restriction to the upper semicircle is $A$ and the restriction to the lower semicircle is $\tilde A$.
If $G$ is simply connected, then the equivalence classes of this relation form
and the quotient map coincides with the parallel transport
Finally yet another perspective is this: consider the equivalence relation on $\Omega^1(\Delta^1, \mathfrak{g})$ where two 1-forms are regarded as equivalent if there is a gauge transformation $\lambda \in C^\infty(\Delta^1, G)$ with $\lambda(0) = e$ and $\lambda(1) = e$, then again
is the parallel transport
if $X$ is connected then forming the holonomy of flat $G$-connections
is an equivalence, $\pi_1(X)$ the fundamental group. If $X$ is not connected then
is an equivalence.
What is called transgression is the combination of
passing a cocycle on some space $X$ with coefficients in some $A$ to a cocycle on a mapping space $[\Sigma,X]$ with coefficients in $[\Sigma,A]$ and
integrating the resulting coefficient over $\Sigma$ to obtain a $B$-valued cocycle on the mapping space, where $B$ is some recipient of an integration map of $A$-cocycles over $\Sigma$.
Let $\Sigma_k$ be a closed smooth manifold of dimension $k$.
For $X \in \mathbf{H}$, the transgression of differential forms on $X$ to the mapping space $[\Sigma,X]$ is the morphism
given on a differential form
as the composition of the mapping space operation with the integration of differential forms, def. 24:
We discuss some examples and applications:
Let $X \in \mathbf{H}$ and consider a circle group-principal connection $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ over $X$. By the discussion in Dirac charge quantization and the electromagnetic field above this encodes an elecrtromagnetic field? on $X$. Assume for simplicity here that the underlying circle principal bundle is trivialized, so that then the connection is equivalently given by a differential 1-form
the electromagnetic potential.
Let then $\Sigma = S^1$ be the circle. The transgression of the electromagnetic potential to the loop space of $X$
is the action functional for an electron or other electrically charged particle in the background gauge field $A$ is $S_{em} = \int_{S^1} [S^1, A]$.
The variation of this contribution in addition to that of the kinetic action of the electron gives the Lorentz force law describing the force exerted by the background gauge field on the electron.
Let $\mathfrak{g}$ be a Lie algebra with binary invariant polynomial $\langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$.
For instance $\mathfrak{g}$ could be a semisimple Lie algebra and $\langle -,-\rangle$ its Killing form. In particular if $\mathfrak{g} = \mathfrak{su}(n)$ is a matrix Lie algebra such as the special unitary Lie algebra, then the Killing form is given by the trace of the product of two matrices.
This pairing $\langle -,-\rangle$ defines a differential 4-form on the smooth space of Lie algebra valued 1-forms
Over a coordinate patch $U \in$ CartSp this sends a differential 1-form $A \in \Omega^1(U)$ to the differential 4-form
The fact that $\langle -, - \rangle$ is indeed an invariant polynomial means that this indeed extends to a 4-form on the smooth groupoid of Lie algebra valued forms
Now let $\Sigma$ be an oriented closed smooth manifold. The transgression of the above 4-form to the mapping space out of $\Sigma$ yields the 2-form
to the moduli stack of Lie algebra valued 1-forms on $\Sigma$.
Over a coordinate chart $U = \mathbb{R}^n \in$ CartSp an element $A \in \mathbf{\Omega}^1(\Sigma,\mathfrak{g})(\mathbb{R}^n)$ is a $\mathfrak{g}$-valued 1-form $A$ on $\Sigma \times U$ with no leg along $U$. Its curvature 2-form therefore decomposes as
where $F_A^{\Sigma}$ is the curvature component with all legs along $\Sigma$ and where
is the variational derivative of $A$.
This means that in the 4-form
only the last term gives a 2-form contribution on $U$. Hence we find that the transgressed 2-form is
When restricted further to flat forms
which is the phase space of $\mathfrak{g}$-Chern-Simons theory, then this is the corresponding symplectic form (by the discussion at Chern-Simons theory – covariant phase spaceheory#CovariantPhaseSpace)).
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integration/higher holonomy is
and higher Chern-Simons action functionals induced from
are
here $\mathbf{L}$ is the Lagrangean.
(…)
The premise in The continuum real world line is now refined to
Premise. The abstract worldline of a fermionic particle is a $\mathbb{Z}_2$-graded formal neighbourhood? $\mathbb{R}^{1|n}$ of the real line, for some $n \in \mathbb{N}$.
For $n = 0$ this is again the real line $\mathbb{R}^{1|0} = \mathbb{R}$.
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II) Physics
Before we discuss technical details starting in the next chapter here we survey general ideas of theories in fundamental physics and motivate how these are naturally formulated in terms of the higher geometry that we developed in the first part.
This chapter is at geometry of physics -- physics in higher geometry.
This chapter is at fields (physics)?.
(…)
Above in Lagrangians and Action functionals we discussed prequantum field theory. Given such there are two directions to go: to the corresponding classical field theory and to a corresponding quantum field theory.
The classical field theory is the study of the critical locus of the action functional, whose points are the solutions to the (Euler-Lagrange-)equations of motion of the system, the conditions which characterize those field configurations that are “physically realized” as asserted by the physical theory that is encoded by the action functional. If the action functional comes from a local Lagrangian then this space carries a canonical presymplectic form and equipped with this form it is called the covariant phase space of the system.
(The term “classical” originates from the time when quantum mechanics was discovered at the beginning of the 20th century. All of the physics that was known until the end of the 19th centure was then called “classical” to distinguish it from the new refinement to quantum theory. Nowadays the term has, strictly speaking, lost its original sense, since nowadays quantum theory is entirely “classical”, but “classical physics” will forever refer to non-quantum physics. )
Here we first discuss the traditional theory of classical equations of motion. Maybe the archetypical example is the geodesic equation which describes the trajectories of particles and of light. Standard examples of equations of motion for spacetime force fields are Maxwell equations and Einstein equations, describing the classical dynamics of the electromagnetic field and gravity, respectively.
Then we reformulate this more abstractly in higher geometry. This yields a notion of derived critical loci of action functionals for which the BV-BRST formalism is a model, a traditional machinery for handling covariant phase spaces while taking care of gauge symmetry and resolving singularities in the critical locus.
Moreover, we discuss how, when interpreted in extended prequantum field theory, the equations of motion are just the codimension-0 piece of a tower of notions which in codimension 1 is the notion of Lagrangian submanifolds of phase space.
Here we discuss the traditional theory of covariant phase spaces and the traditional model of their resolution in higher geometry: BV-BRST formalism.
Here we give a general abstract formulation of higher (“derived”) critical loci in a cohesive (∞,1)-topos.
We had already discussed traditional Variational calculus above. Using this we find:
We now discuss the general abstract formulation of critical loci of action functionals in the context of a cohesive (∞,1)-topos. This generalizes the traditional formulation to critical loci inside higher moduli ∞-stacks of field configuration. In particular, if the ambient (∞,1)-topos is not 1-localic, then this gives a general abstract formulation of derived critical loci.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos $(\mathbf{\Pi} \dashv \flat \dashv \sharp) : \mathbf{H} \to \mathbf{H}$ equipped with differential cohesion $(Red \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf}) \;\colon\; \mathbf{H} \to \mathbf{H}$. We discuss the formalization of critical loci of action functionals and of equations of motion in this context.
Fix
an object that serves as the moduli ∞-stack of physical fields for the theory to be considered, as discussed in Fields above
For $\Sigma \in Mfd_{bdr} \hookrightarrow \mathbf{H}$ a manifold with boundary in $\mathbf{H}$, def. 5, write $[\Sigma,\mathbf{Fields}]_{\partial \Sigma} \in \mathbf{H}$ for the (∞,1)-pullback
where the right vertical morphism is the counit of $\flat$ and where the bottom morphism is the image of the boundary inclusion $\partial \Sigma \to \Sigma$ under the internal hom $[-, \mathbf{Fields}]$.
This implies that for any geometrically contractible $U \in \mathbf{H}$, def. \ref{ShapeTerminology}, then we have
This means that a variation in $[\Sigma, \mathbf{Fields}]_{\partial \Sigma}$ is a variation in $[\Sigma, \mathbf{Fields}]$ which remains constant over the boundary of $\Sigma$.
Fix now
a group object, hence a cohesive ∞-group, to be the object that the action functional is to take values in. In $\mathbf{H} =$ Smooth∞Grpd the standard choice is $\mathbb{G} = U(1)$, the circle group, for “exponentiated action functionals” or $\mathbb{R} = \mathbb{R}$, the additive Lie group of real numbers.
For $S \;\colon\; [\Sigma, \mathbf{Fields}] \to \mathbb{G}$ a map, the variational derivative of $S$ is the restriction of the de Rham differential $S^{-1}\mathbf{d}S$ of def. 12 to variations that keep the boundary data fixed as in def. 26, hence the composite
Since the variational context is clear from the domain of the map, we will often just write $S^{-1} \mathbf{d} S$ for $S^{-1} \mathbf{d}_{var} S$, for convenience.
Under coreflection into structure sheaves, def. 3, this induces a map
in $Sh_{\mathbf{H}}(X)$, which we will denote by the same symbol, as here, when the context is clear. Since $\mathcal{O}_X(\flat_{dR} \mathbf{B}\mathbb{G})$ has the interpretation of the sheaf of flat $Lie(\mathbb{G})$-valued forms on $X$, this may be thought of as realizing $\mathbf{d} S$ as a section of the tangent bundle over $X$.
For $S \;\colon\; [\Sigma, \mathbf{Fields}] \to \mathbb{G}$ a map in $\mathbf{H}$, its critical locus
is the homotopy fiber of the variational derivative $S^{-1} \mathbf{d}S$ over the 0-section, hence the (∞,1)-pullback
in the petit (∞,1)-topos $Sh_{\mathbf{H}}(X)$.
In extended prequantum field theory we may, as discussed in Lagrangians and Action functionals, think of the action functional $S$ as being the prequantum 0-bundle. In this perspective the variational derivative $S^{-1} \mathbf{d}_{var} S$ of def. 27 is the curvature of this 0-bundle. If $\mathbb{G}$ is a 1-group such as $U(1)$ then this is a differential 1-form which is the 0-plectic form. This means that the critical locus in def. 28 the maximal subspace on which the 0-plectic 1-form vanishes.
As the notation above suggests, the critical locus of the function $S\;\colon\; [X, \mathbf{Fields}] \to \mathbb{G}$ is syntactically indeed the dependent sum over the type of fields of the identity type of the variational derivative $S^{-1}\mathbf{d} S \in Sh_{\mathbf{H}}(X)$ and the 0-term in $Sh_{\mathbf{H}}(X)$. This is indeed the standard expression in type theory which formalizes the variations equation of motion:
“The collection of fields for which the variational derivative equals zero.” translates exactly into $\underset{\phi \in [X, \mathbf{Fields}]}{\sum} (S^{-1}\mathbf{d}S \simeq 0)$.
The following is effectively a derivation of, and an introduction to, classical mechanics by studying correspondences in what is called (as we will explain) the slice topos over the moduli stack of prequantum line bundles. One such correspondence in this slice topos is precisely a prequantized Lagrangian correspondence and the reader looking for just these should skip ahead to the section The classical action functional prequantizes Lagrangian correspondences. But for completeness and to introduce the technology used here, we start with introducing also more basic concepts, such as phase space etc.
This chapter is at prequantized Lagrangian correspondence.
This chapter is at geometry of physics -- local prequantum field theory.
In the previous chapters we have set up prequantum field theory and classical field theory in generality. Here we discuss examples of such field theories in more detail.
We introduce a list of important examples of field theories in fairly tradtional terms.
We study the above physical systems with the tools of of cohesive (∞,1)-topos-theory as developed in the previous semantics-layers.
The prequantum field theory which describes the gauge interaction of a single nonabelian charged particle – a Wilson loop – turns out to be equivalent to what in mathematics is called the orbit method. We discuss here the traditional formulation of these matters. Below in Semantics layer – Nonabelian charged particle and Wilson loops we then show how all this is naturally understood from a certain extended Lagrangian which is induced by a regular coadjoint orbit.
A useful review of the following is also in (Beasley, section 4).
Throughout, let $G$ be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.
Write $\mathfrak{g}$ for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write
Since this is non-degenerate, we may equivalently think of this as an isomorphism
that identifies the vector space underlying the Lie algebra with its dual vector space $\mathfrak{g}^*$.
We discuss the coadjoint orbits of $G$ and their relation to the coset space/flag manifolds of $G$.
Write
1 $\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding Cartan subalgebra
In all of the following we consider an element $\langle\lambda,-\rangle \in \mathfrak{g}^*$.
For $\langle\lambda,-\rangle \in \mathfrak{g}^*$ write
for its coadjoint orbit
Write $G_\lambda \hookrightarrow G$ for the stabilizer subgroup of $\langle \lambda,-\rangle$ under the coadjoint action.
There is an equivalence
given by
An element $\langle\lambda,-\rangle \in \mathfrak{g}^*$ is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: $G_\lambda \simeq T$.
For generic values of $\lambda$ it is regular. The element in $\mathfrak{g}^*$ farthest from regularity is $\lambda = 0$ for which $G_\lambda = G$ instead.
We describe a canonical symplectic form on the coadjoint orbit/coset $\mathcal{O}_\lambda \simeq G/G_\lambda$.
Write $\theta \in \Omega^1(G, \mathfrak{g})$ for the Maurer-Cartan form on $G$.
Write
for the 1-form obtained by pairing the value of the Maurer-Cartan form at each point with the gixed element $\lambda \in \mathfrak{g}^*$.
Write
for its de Rham differential.
The 2-form $\nu_\lambda$ from def. 31
satisfies
it descends to a closed $G$-invariant 2-form on the coset space, to be denoted by the same symbol
this is non-degenerate and hence defines a symplectic form on $G/G_\lambda$.
We discuss the geometric prequantization of the symplectic manifold given by the coadjoint orbit $\mathcal{O}_\lambda$ equipped with its symplectic form $\nu_\lambda$ of def. 13.
Assume now that $G$ is simply connected.
The weight lattice $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is isomorphic to the group of group characters
where the identification takes $\langle \alpha , -\rangle \in \mathfrak{t}^*$ to $\rho_\alpha : T \to U(1)$ given on $t = \exp(\xi)$ for $\xi \in \mathfrak{t}$ by
The symplectic form $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. 13 is integral precisely if $\langle \lambda, - \rangle$ is in the weight lattice.
The group $G$ canonically acts on the coset space $G/G_{\lambda}$ (by multiplication from the left). We discuss a lift of this action to a Hamiltonian action with respect to the symplectic manifold structure $(G/T, \nu_\lambda)$ of prop. 13, equivalently a momentum map exhibiting this Hamiltonian action.
Above (…) we discussed how an irreducible unitary representation of $G$ is encoded by the prequantization of a coadjoint orbit $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express Wilson loops/holonomy of $G$-principal connections in this representation as the path integral of a topological particle charged under this background field, whose action functional is that of a 1-dimensional Chern-Simons theory.
Let $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a Lie algebra valued 1-form on the circle, equivalently a $G$-principal connection on the circle.
For
a representation of $G$, write
for the holonomy of $A$ around the circle in this representation, which is the trace of its parallel transport around the circle (for any basepoint). If one thinks of $A$ as a background gauge field then this is alse called a Wilson loop.
Let the action functional
be given by sending $g T : S^1 \to G/T$ represented by $g : S^1 \to G$ to
where
is the gauge transformation of $A$ under $g$.
The Wilson loop of $A$ over $S^1$ in the unitarry irreducible representation $R$ is proportional to the path integral of the 1-dimensional sigma-model with
target space the coadjoint orbit $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the weight corresponding to $R$ under the Borel-Weil-Bott theorem
action functional the functional of def. 32:
See for instance (Beasley, (4.55)).
Notice that since $\mathcal{O}_\lambda$ is a manifold of finite dimension, the path integral for a point particle with this target space can be and has been defined rigorously, see at path integral.
an exposition and survey is in (FSS 13).
For some $n \in \mathbb{N}$ let
be the Lie group homomorphism from the unitary group to the circle group which is given by sending a unitary matrix to its determinant.
Being a Lie group homomorphism, this induces a map of deloopings/moduli stacks
Under geometric realization of cohesive infinity-groupoids this is the universal first Chern class
Moreiver this has the evident differential refinement
given on Lie algebra valued 1-forms by taking the trace
So we get a 1d Chern-Simons theory with $\widehat{\mathbf{B}det}$ as its extended Lagrangian.
We consider now extended Lagrangians defined on fields as above in Nonabelian charged particle trajectories – Wilson loops. This provides a natural reformulation in higher geometry of the constructions in the orbit method as reviewed above in Model layer – Nonabelian charged particle.
We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of smooth moduli stacks of the form
where
$\mathbf{J}$ is the canonical 2-monomorphism;
the left square is a homotopy pullback square, hence $\mathbf{\theta}$ is the homotopy fiber of $\mathbf{J}$;
the bottom map is the extended Lagrangian for $G$-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;
the top map denoted $\langle \lambda,- \rangle$ is an extended Lagrangian for a 1-dimensional Chern-Simons theory;
the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit $\Omega_\lambda \simeq G/T$.
Write $\mathbf{H} =$ Smooth∞Grpd for the cohesive (∞,1)-topos of smooth $\infty$-groupoids.
For the following, let $\langle \lambda, - \rangle \in \mathfrak{g}^*$ be a regular element, def. 30, so that the stabilizer subgroup is identified with a maximal torus: $G_\lambda \simeq T$.
As usual, write
for the moduli stack of $G$-principal connections.
Write
for the canonical map, as indicated.
The map $\mathbf{J}$ is the differential refinement of the delooping $\mathbf{B}T \to \mathbf{B}G$ of the defining inclusion. By the general discussion at coset space we have a homotopy fiber sequence
By the discussion at ∞-action this exhibits the canonical action $\rho$ of $G$ on its coset space: it is the universal rho-associated bundle.
The following proposition says what happens to this statement under differential refinement
The homotopy fiber of $\mathbf{J}$ in def. 33 is
given over a test manifold $U \in$ CartSp by the map
which sends $g \mapsto g^* \theta$, where $\theta$ is the Maurer-Cartan form on $G$.
We compute the homotopy pullback of $\mathbf{J}$ along the point inclusion by the factorization lemma as discussed at homotopy pullback – Constructions.
This says that with $\mathbf{J}$ presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids $hofib(\mathbf{J})$ which is the limit cone in
Unwinding the definitions shows that $hofib(\mathbf{J})$ has
objects over a $U \in$ CartSp are equivalently morphisms $0 \stackrel{g}{\to} g^* \theta$ in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$, hence equivalently elements $g \in C^\infty(U,G)$;
morphisms are over $U$ commuting triangles
in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$ with $t \in C^\infty(U,T)$, hence equivalently morphisms
in $C^\infty(U,G)//C^\infty(U,T)$.
The canonical map $hofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T$ picks the top horizontal part of these commuting triangles hence equivalently sends $g$ to $g^* \theta$.
If $\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^*$ is in the weight lattice, then there is a morphism of moduli stacks
in $\mathbf{H}$ given over a test manifold $U \in$ CartSp by the functor
which is given on objects by
and which maps morphisms labeled by $\exp(\xi) \in T$, $\xi \in C^\infty(-,\mathfrak{t})$ as
That this construction defines a map $*//T \to *//U(1)$ is the statement of prop. 14. It remains to check that the differential 1-forms gauge-transform accordingly.
For this the key point is that since $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the coadjoint action, the gauge transformation law for points $A : U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is
maps for $g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G)$ to the gauge transformation law in $\mathbf{B}U(1)_{conn}$:
The composite of the canonical maps of prop. 17 and prop. 18 modulates a canonical circle bundle with connection on the coset space/coadjoint orbit:
The curvature 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark 12 is the symplectic form of prop. 13. Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a prequantization of the coadjoint orbit $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$.
The curvature 2-form is modulated by the composite
Unwinding the above definitions and propositions, one finds that this is given over a test manifold $U \in$ CartSp by the map
which sends
Let $\Sigma$ be an oriented closed smooth manifold of dimension 3 and let
be a submanifold inclusion of the circle: a knot in $\Sigma$.
Let $R$ be an irreducible unitary representation of $G$ and let $\langle \lambda,-\rangle$ be a weight corresponding to it by the Borel-Weil-Bott theorem.
Regarding the inclusion $C$ as an object in the arrow (∞,1)-topos $\mathbf{H}^{\Delta^1}$, say that a gauge field configuration for $G$-Chern-Simons theory on $\Sigma$ with Wilson loop $C$ and labeled by the representation $R$ is a map
in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$ of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square
in $\mathbf{H}$. In components this is
a $G$-principal connection $A$ on $\Sigma$;
a $G$-valued function $g$ on $S^1$
which fixes the field on the circle defect to be $(A|_{S^1})^g$, as indicated.
Moreover, a gauge transformation between two such fields $\kappa : \phi \Rightarrow \phi'$ is a $G$-gauge transformation of $A$ and a $T$-gauge transformation of $A|_{S^1}$ such that these intertwine the component maps $g$ and $g'$. If we keep the bulk gauge field $A$ fixed, then his means that two fields $\phi$ and $\phi'$ as above are gauge equivalent precisely if there is a function $t \;\colon\; S^1 \to T$ such that $g = g' t$, hence gauge equivalence classes of fields for fixed bulk gauge field $A$ are parameterized by their components $[g] = [g'] \in [S^1, G/T]$ with values in the coset space, hence in the coadjoint orbit.
For every such field configuration we can evaluate two action functionals:
that of 3d Chern-Simons theory, whose extended Lagrangian is $\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$;
that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is $\langle \lambda, -\rangle : \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. 18.
These are obtained by postcomposing the above square on the right by these extended Lagrangians
and then preforming the fiber integration in ordinary differential cohomology over $S^1$ and over $\Sigma$, respectively.
For the bottom map this gives the ordinary action functional of Chern-Simons theory. For the top map inspection of the proof of prop. 18 shows that this gives the 1d Chern-Simons action whose partition function is the Wilson loop observable by prop. 16 above.
In the context of string theory, the background gauge field for the open string sigma-model over a D-brane in bosonic string theory or type II string theory is a unitary principal bundle with connection, or rather, by the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation mechanism, a twisted unitary bundle, whose twist is the restriction of the ambient B-field to the D-brane.
We considered these fields already above. Here we discuss the corresponding action functional for the open string coupled to these fields
The first hint for the existence of such background gauge fields for the open string 2d-sigma-model comes from the fact that the open string’s endpoint can naturally be taken to carry labels $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-gauge field, the $(i,j)$-excitation being the given matrix element of a $U(n)$-valued connection 1-form $A$.
This original argument goes back work by Chan and Paton. Accordingly one speaks of Chan-Paton factors and Chan-Paton bundles .
We discuss the Chan-Paton gauge field and its quantum anomaly cancellation in extended prequantum field theory.
Throughout we write $\mathbf{H} =$ Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids.
For $X$ a type II supergravity spacetime, the B-field is a map
If $X = G$ is a Lie group, this is the prequantum 2-bundle of $G$-Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, form the corresponding associated ∞-bundle and regard the sections of that as the prequantum 2-states? of the theory.
The Chan-Paton gauge field is such a prequantum 2-state.
We discuss the Chan-Paton gauge fields over D-branes in bosonic string theory and over $Spin^c$-D-branes in type II string theory.
We fix throughout a natural number $n \in \mathbb{N}$, the rank of the Chan-Paton gauge field.
The extension of Lie groups
exhibiting the unitary group as a circle group-extension of the projective unitary group sits in a long homotopy fiber sequence of smooth ∞-groupoids of the form
where for $G$ a Lie group $\mathbf{B}G$ is its delooping Lie groupoid, hence the moduli stack of $G$-principal bundles, and where similarly $\mathbf{B}^2 U(1)$ is the moduli 2-stack of circle 2-group principal 2-bundles (bundle gerbes).
Here
is a smooth refinement of the universal Dixmier-Douady class
in that under geometric realization of cohesive ∞-groupoids ${\vert- \vert} \colon$ Smooth∞Grpd $\to$ ∞Grpd we have
By the discussion at ∞-action the homotopy fiber sequence in prop. 20
in $\mathbf{H}$ exhibits a smooth∞-action of the circle 2-group on the moduli stack $\mathbf{B}U(n)$ and it exhibits an equivalence
of the moduli stack of projective unitary bundles with the ∞-quotient of this ∞-action.
For $X \in \mathbf{H}$ a smooth manifold and $\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1)$ modulating a circle 2-group-principal 2-bundle, maps
in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^2 U(1)}$, hence diagrams of the form
in $\mathbf{H}$ are equivalently rank-$n$ unitary twisted bundles on $X$, with the twist being the class $[\mathbf{c}] \in H^3(X, \mathbb{Z})$.
There is a further differential refinement
where $\mathbf{B}^2 U(1)_{conn}$ is the universal moduli 2-stack of circle 2-bundles with connection (bundle gerbes with connection).
Write
for the differential smooth universal Dixmier-Douady class of prop. 23, regarded as an object in the slice (∞,1)-topos over $\mathbf{B}^2 U(1)_{conn}$.
Let
be an inclusion of smooth manifolds or of orbifolds, to be thought of as a D-brane worldvolume $Q$ inside an ambient spacetime $X$.
Then a field configuration of a B-field on $X$ together with a compatible rank-$n$ Chan-Paton gauge field on the D-brane is a map
in the arrow (∞,1)-topos $\mathbf{H}^{(\Delta^1)}$, hence a diagram in $\mathbf{H}$ of the form
This identifies a twisted bundle with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk B-field.
This relation is the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation for the bosonic string or else for the type II string on $Spin^c$ D-branes. (FSS)
If we regard the B-field as a background field for the Chan-Paton gauge field, then remark \ref{PullbackAlongGeneralizedLocalDiffeomorphisms} determines along which maps of the B-field the Chan-Paton gauge field may be transformed.
On the local connection forms this acts as
This is the famous gauge transformation law known from the string theory literature.
The D-brane inclusion $Q \stackrel{\iota_X}{\to} X$ is the target space for an open string with worldsheet $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a field configuration of the open string sigma-model is a map
in $\mathbf{H}^{\Delta^1}$, hence a diagram of the form
For $X$ and $Q$ ordinary manifolds just says that a field configuration is a map $\phi_{bulk} \;\colon\; \Sigma \to X$ subject to the constraint that it takes the boundary of $\Sigma$ to $Q$. This means that this is a trajectory of an open string in $X$ whose endpoints are constrained to sit on the D-brane $Q \hookrightarrow X$.
If however $X$ is more generally an orbifold, then the homotopy filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called “orbifold twisted sectors” of open string configurations.
The moduli stack $[\iota_\Sigma, \iota_X]$ of such field configurations is the homotopy pullback
For $\Sigma$ a smooth manifold with boundary $\partial \Sigma$ of dimension $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a circle n-bundle with connection on some $X \in \mathbf{H}$, then the transgression of $\nabla$ to the mapping space $[\Sigma, X]$ yields a section of the complex line bundle associated to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram
This is the higher parallel transport of the $n$-connection $\nabla$ over maps $\Sigma \to X$.
The operation of forming the holonomy of a twisted unitary connection around a curve fits into a diagram in $\mathbf{H}$ of the form
By the discussion at ∞-action the diagram in prop. 26 says in particular that forming traced holonomy of twisted unitary bundles constitutes a section of the complex line bundle on the moduli stack of twisted unitary connection on the circle which is the associated bundle to the transgression $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ of the universal differential Dixmier-Douady class.
It follows that on the moduli space of the open string sigma-model of prop. 24 above there are two $\mathbb{C}//U(1)$-valued action functionals coming from the bulk field and the boundary field
Neither is a well-defined $\mathbb{C}$-valued function by itself. But by pasting the above diagrams, we see that both these constitute sections of the same complex line bundle on the moduli stack of fields:
Therefore the product action functional is a well-defined function
This is the Kapustin anomaly-free action functional of the open string.
We discuss how an extended Lagrangian for $G$-Chern-Simons theory with Wilson loop defects is naturally obtained from the above higher geometric formulation of the orbit method. In particular we discuss how the relation between Wilson loops and 1-dimensional Chern-Simons theory sigma-models with target space the coadjoint orbit, as discussed above is naturally obtained this way.
More formally, we have an extended Chern-Simons theory as follows.
The moduli stack of fields $\phi : C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the homotopy pullback
in $\mathbf{H}$, where square brackets indicate the internal hom in $\mathbf{H}$.
Postcomposing the two projections with the two transgressions of the extended Lagrangians
and
to yield
and then forming the product yields the action functional
This is the action functional of 3d $G$-Chern-Simons theory on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$.
Similarly, in codimension 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have prequantum bundles given by transgression of the extended Lagrangians to codimension 1
and
and hence a total prequantum bundle
One checks that this is indeed the correct prequantization as considered in (Witten 98, p. 22).
(…)
So far we have discussed extended prequantum field theory: Lagrangians and their induced action functionals and prequantum n-bundles. Now we turn to actual quantum field theory. A prequantum field theory is supposed to induce a quantum field theory under the last step of higher geometric quantization: a choice of polarization (or equivalent) and the passage to the corresponding space of states of polarized sections of the prequantum n-bundles. This step that connects prequantum field theory with quantum field theory we discuss below in Geometric Quantization.
Here we discuss the structure of the outcome of this process
(…)
What is it that higher geometry, higher gauge theory, extended/local field theory and generally higher category theory in physics contribute to open research questions in theoretical physics?
Often when this question is asked the most glaring open question of contemporary theoretical physics is forgotten:
What IS local quantum field theory?
While something going by this name is clearly in use, it is just as clear that the full answer to this question is only being discovered these days, with formalizations such as the cobordism theorem and constructions such as factorization algebras in BV-quantization – both of which are crucially constructions in higher geometry/higher category theory.
Despite the huge success of quantum field theory, it it worthwhile to remember that all the fundamental open questions in present day fundamental physics quite likely require a deeper understanding of what quantum field theory actually is, notably non-perturbatively:
Why is there confinement/chiral symmetry breaking? in non-perturbative QCD/Yang-Mills theory? (The “mass gap problem”.)
What is beyond-the-standard-model physics?
What is quantum gravity?
What is non-perturbative string theory?
For instance the standard model of cosmology says that the bulk of all energy and matter in the observable universe is entirely unknown to us (dark matter, “dark energy”), while at the same time the theoretical prediction what the cosmological constant vacuum energy? should be is entirely off. How glaring an open question about the nature of quantum field theory this actually is is often forgotten due to the success of effective field theory-type of reasoning that allows to neatly wrap up all this unknown energy into a single term in some effective equation. Phenomenologically this may be regarded as a success, but for fundamental theoretical physics it is a glaring open question.
And while there is work going in this direction, it may be worthwhile to recall how relatively primitive the available theoretical tools often still are. For instance it seems clear that “canonical non-covariant quantization” can hardly be an approrpiate tool to approach anything in the direction of quantum gravity. Even so fundamental a notion as that of covariant phase space necessary to make progress here is not widely known in the theoretical physics community. Attempts to refine quantization to a “covariant” and “local” formalism via multisymplectic geometry have mainly got stuck, since local observables just do not form a sensible structure in ordinary Lie theory. This is resolved only in infinity-Lie theory and higher differential geometry, as discussed above (hgp 13, lo 13).
If one assumes that string theory is part of the answer as to what underlies the standard model of particle physics and cosmology, then this situation becomes more drastic even. The fundamental fields of string theory are clearly objects in higher differential geometry, such as the B-field, the RR-field, the supergravity C-field etc. For instance the natural identification of the latter as a homotopy fiber product of moduli stacks in (FSS7dCS, FSSCField) is hardly conceivable when ignoring higher differential geometry. And this is a structure meant to be at the very heart of what makes up string theory. It is unlikely that the landscape of string theory vacua and hence the relation of string theory to phenomenology can really be understood if such basic higher-geometric phenomena of string theory are ignored (see Distler-Freed-Moore 09 on this point).
(…)
A textbook with basic introductions to differential geometry and physics is
A discussion of aspects of quantum field theory with emphasis on the kind of modern tools that we are using here is in
The present discussion corresponds to section “1.2 Geometry of phyics” in
which gives a more comprehensive account.
Another set of lecture notes along the above lines with an emphasis on aspects in gravity and higher gauge theory motivated from string theory is in
An exposition and survey of matters related to Chern-Simons theory and higher geometric quantization is in
The syntactic perspective above is laid out further in
see also at motivic quantization the section General abstract type theoretic summary.
A textbook (really a collection of lecture notes) on quantum field theory and string theory that tries to present material in a conceptually clean way is
A collection trying to summarize the state of the art of the formalization of QFT by FQFT and AQFT as of 2011 is
One of the central figures of topos theory and categorical logic, William Lawvere, has motivated his interest in these subject always with intended application to the formalization of physics (of classical continuum mechanics in his case).
An influential text is
which motivates synthetic differential geometry from differential equations appearing as equations of motion in physics. The early text
already sketches the formulation of cohesive toposes and motivates their axioms with heuristics from geometry and physics.
A review by Lawvere is in
Modern accounts of physics in this spirit includes notably also the book (Paugam) listed above.
An early proposal that the action functional of $n$-dimensional quantum field theory should refine to a structure involving (n-k)-vector spaces in codimension $(n-k)$ is in
The full formalization of this for extended topological field theory is due to
Related comments on the extended quantization of infinity-Dijkgraaf-Witten theory are in
For more pointers see at higher category theory and physics.
The idea of formulating local prequantum field theory by spans in a slice over a “space of phases” in higher geometry has been expressed in the unpublished note
A formulation of the idea for Dijkgraaf-Witten theory-type field theories is indicated in section 3 of
based on the considerations in section 3.2 of
Based on the general formulation of the more general field theory with defects described in section 4.3 there, in
the structure of such domain walls/defects/branes are analyzed in the prequantum theory, hence with coefficients in an (∞,n)-category of spans.
The study of local prequantum ∞-Chern-Simons theory with its codimension-1 ∞-Wess-Zumino-Witten theory and codimension 2-Wilson line-theory in this fashion, in an ambient cohesive (∞,1)-topos is discussed in (lpqft)
Much of the content of this entry here are, or arose as, lecture notes for
For references on the tradtional formulation of physical fields by sections of field bundles as discussed above see there references there.
The formulation of physical fields as cocycles in twisted cohomology in an (∞,1)-topos as in the Definition-section above originates around
Further articles since then are listed at
In particular the general notion of fields as twisted differential c-structures appears in
and the general theory of cohomology and twisted cohomology with local coefficient ∞-bundles as referred to in Relation to twisted cohomology above as well as the theory of associated ∞-bundles as in Sections of associated ∞-bundles is laid out in
Some examples of fields in this sense are called “relative fields” in
The relation between differential 1-forms and smooth incremental path measures as used above is discussed in
For a commented list of related literature see here.
(…)
The discussion of the abelian 7d Chern-Simons theory involved in AdS7/CFT6 duality is due to (Witten 98). A discussion of the non-abelian quantum-corrected and extended refinement is in
Construction of differential cup-product theories is in