nLab
geometry of physics

This page is growing incrementally as a series of lecture series proceeds.


Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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:


Contents

About this text

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.

Scope and perspective

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 n\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:

Layers of exposition

We discuss each topic below in three stages, in three layers.

  1. The first layer, called the Model Layer, deals with concrete explicit constructions as familiar from traditional textbooks on differential geometry and physics. This layer is supposed to be readable and useful all in itself and the reader who feels that this is all he or she wants to see can stick to this and ignore the other layers. In particular, while the Model Layer does invoke the basic notion of a category and of a functor – which are as simple as the notions of group or algebra –, it does not use any actual category theory.
  1. The second layer, called the Semantic Layer, makes explicit the (higher) category theory and (higher) topos theory at work in the background. This puts the concrete constructions of the Model Layer into a more general context and helps to see certain organizational patterns that underlie the seemingly different phenomena. It provides some powerful theorems which the Model Layer is secretly benefitting from. For instance this layer gives a systematic rule for generalizing everything at the beginning Model Layers from ordinary differential geometry to what is called supergeometry, which is the context in which fermionic particles are formalized: the matter constituents of the observable universe.
  1. The third layer, called the Syntactic Layer, makes explicit the expression of these phenomena in the formal internal language of the topos of smooth spaces – which is dependent type theory – and of the higher topos of smooth higher groupoids – which is homotopy type theory. This makes more transparent various constructions in (higher) topos theory used in Semantic Layer, and in fact it provides a natural construction principle for objects in a (higher) topos that model some intended meaning – which is precisely what mathematical physics is all about. This is meant for readers who enjoy seeing fundamental physics naturally rooted in genuinely fundamental mathematics, in natural deduction from practical foundations, as it were. Everybody else should ignore this.

The three layers

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:

The full story in a few formal words

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

  1. FQFT where one axiomatizes the assignment of spaces of states to pieces of worldvolume (the “Schrödinger picture” of quantum theory)

    fragments of which involve:

  2. 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

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\togeometric quantization\toquantum 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 xx of type XX.

For instance:

There is a gauge field \nabla in the standard model [X,B(U(1)×SU(2)×SU(3)) conn][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 XX.

(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()UnderlyingBundle(\nabla), in the collection [X,B(U(1)×SU(2)×SU(3))]\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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
There is\vdash \ldotsWe speak in the context of a (higher) topos H\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 H=\mathbf{H} = Sh((SmthMfd)). Eventually a higher such topos: H=\mathbf{H} = Smooth∞Grpd or SynthDiff∞Grpd or SmoothSuper∞Grpd or Smooth spaces and Smooth homotopy types
There is a thing xx of type XX.x:X\vdash\; x \colon XAn element (*xX)Mor(H)\left(* \stackrel{x}{\to} X\right) \in Mor(\mathbf{H}) of an object XX of H\mathbf{H}.A point xx in a smooth moduli stack XX.Judgements about types and terms
There is a type XX of things xx.X:Type\vdash\; X \colon TypeAn element (*XObj)Mor(H)(* \stackrel{\vdash X}{\to} Obj) \in Mor(\mathbf{H}) of the small-object classifier ObjObj of H\mathbf{H}.A point in the moduli stack of all small moduli stacks.Judgements about types and terms
Given a thing xx of type XX there is a thing a(x)a(x) of type A(x).A(x).x:Xa(x):A(x)x \colon X\;\vdash\; a(x) \colon A(x)An element of a morphism (AX)(A \to X) (X a A id X)\left(\array{ X &&\stackrel{a}{\to}&& A \\ & {}_{\mathllap{id}}\searrow &\swArrow& \swarrow_{} \\ && X }\right) in the slice topos H /X\mathbf{H}_{/X}.An XX-family in a moduli stack bundle AA over XX.Slice categories and Slice toposes and Slice ∞-Toposes
There is the collection of all things a(x)a(x) for all xx.( x:XA(x)):Type\vdash\; \left(\sum_{x \colon X} A\left(x\right)\right) \colon TypeThe dependent sum/left adjoint to the product: H /X X ! H (AX) AH\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 tt in the collection of all things a(x)a(x) for all xx.t: x:XA(x)\vdash\; t \colon \sum_{x \colon X} A(x)An element *tA*\stackrel{t}{\to} A of the total space object.A point in the moduli stack AA over XX.
There is an assignment ff of an a(x)a(x) to each xx.f: x:XA(x)\vdash \; f \colon \prod_{x \colon X} A(x).An element in the internal object of sections *f[X,A] X* \stackrel{f}{\to} [X,A]_XA 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)a(x) to each xx.( x:XA(x)):Type\vdash\; \left( \prod_{x \colon X} A\left(x\right) \right) \colon Typeinternal space of sections [X,A] XH[X,A]_X \in \mathbf{H}A smooth relative mapping space of smooth sections.
In particular, there is the collection of such assignments when AA does not depend on xx, the collection of functions from XX to AA.(XA)( x:XA):Type\vdash \; \left(X \to A\right) \coloneqq \left(\prod_{x \colon X} A\right) \colon TypeThe internal hom object [X,A]H[X,A] \in \mathbf{H}.A smooth mapping space.Smooth mapping spaces and smooth moduli spaces
There is a proof pp that it is true that there is xx of type XX.p:[X] \vdash \; p \colon [X] An element *pτ 1(X)* \stackrel{p}{\to}\tau_{-1}(X) of the (-1)-truncation of the object XX.A point in the smooth space of equivalence classes of points in XX.Subobjects
There is a proof pp that it is true that there is an a(x)a(x) for some xx.p:( x:XA(x))[ x:XA(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 1\nabla_1 and 2\nabla_2.

And the language needs to be able to make natural deductions from such judgements to arrive at:

Given an equivalence λ: 1 2\lambda \colon \nabla_1 \simeq \nabla_2 there is an equivalence UnderlyingBundle(λ):UnderlyingBundle( 1)UnderlyingBundle( 2)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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
Given x,xx,x', there is the collection of equivalences between xx and xx' equivalent.x,x:X(xx):Typex,x' \colon X \;\vdash \; \left(x \simeq x'\right) \colon Type.The mapping cocone object P x,xX * e x * x X\array{ P_{x,x'} X &\to& * \\ \downarrow &\swArrow_{e}& \downarrow^{\mathrlap{x}} \\ * &\stackrel{x'}{\to} & X }The moduli stack of gauge transformations between xx and xx'.Identity types
There is an equivalence ee between xx and xx'.e:(xx) or e:(xx)\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 xx and xx'.
Given x,xx,x', there is the collection of proofs that it is true that xx and xx' are equivalent.x,x:X[xx]:Typex,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 xx to xx'.

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 ρ:(λ 1λ 2)\rho \colon (\lambda_1 \simeq \lambda_2) between two given gauge equivalences λ 1,λ 2:( 1 2)\lambda_1, \lambda_2 \colon (\nabla_1 \simeq \nabla_2) between two given gauge fields 1, 2\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 XX 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,BG conn][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 nn-fold ghost-of-ghost fields in the BRST complex correspond to the nn-fold homotopies in [X,BG conn][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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
Given a type XX, there is (the underlying space) of a group GG of ways that XX is equivalent to itself.X:Type(XX):TypeX \colon Type \;\vdash \; (X \stackrel{\simeq}{\to} X ) \colon Type A loop space object G * X * X Type \array{ G &\to& * \\ \downarrow &\swArrow& \downarrow^{\mathrlap{X}} \\ * &\stackrel{X}{\to} & Type } A smooth ∞-group.n-groups
Given a function between collections of things XX and YY, and given a thing yy, there is its preimage-up-to-equivalence.(f:(XY)),(y:Y) x:X(f(x)y)\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 X× Y{y} X f * y Y\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

f:[X,BG conn][Y,BH conn] f \colon [X, \mathbf{B}G_{conn}] \to [Y, \mathbf{B}H_{conn}]

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 ff an equivalence - in the physics literature often: a duality – if, while not necessarily being a “bijection”, it is such that the preimage ϕ 1()[X,BG conn]\phi^{-1}(\nabla) \in [X,\mathbf{B}G_{conn}] of a gauge field [Y,BH conn]\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 fibersall gauge pre-images – of ϕ\phi are contractible – are gauge equivalent to a single gauge field – and that ϕ\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,BG conn][Y,BH conn][X, \mathbf{B}G_{conn}] \simeq [Y, \mathbf{B}H_{conn}] is itself equivalent to the space of such weak homotopy equivalences (“dualities”) [X,BG conn][Y,BH conn][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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
Given a type XX, there is a group GG of ways that XX is equivalent to itself.X:Type(XX):TypeX \colon Type \;\vdash \; (X \stackrel{\simeq}{\to} X ) \colon Type A loop space object G * X * X Type \array{ G &\to& * \\ \downarrow &\swArrow& \downarrow^{\mathrlap{X}} \\ * &\stackrel{X}{\to} & Type } A smooth automorphism ∞-group.n-groups
Given a type XX, there is the delooping BG\mathbf{B}G of GG, which is the collection of things equipped with equivalences to XX.X:TypeBG Y:Type[XY]X \colon Type \; \vdash \; \mathbf{B}G \coloneqq \sum_{Y \colon Type} \left[X \simeq Y\right] The looping and delooping relation G ΩBG * * BG\array{G \simeq &\Omega \mathbf{B}G &\to& * \\ & \downarrow &\swArrow& \downarrow^{\mathrlap{}} \\ & * &\underset{}{\to}& \mathbf{B}G}The smooth moduli stack of smooth GG-principal ∞-bundles.Principal n-bundles
Given a thing in BG\mathbf{B}G, there is a thing VV.*:BGV(*):Type orwithmoreemphasis: (*,*,g): *,*:BG(**)V(*g*):Type\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 V VG ρ¯ BG\array{V &\to& V\sslash G \\&& \downarrow^{\overline{\rho}} \\ && \mathbf{B}G } with homotopy fiber VV over BG\mathbf{B}G.An ∞-action/∞-representation of GG on some VV, together with its universal ρ\rho-associated VV-fiber ∞-bundle over the moduli stack BG\mathbf{B}G for GG-principal ∞-bundles.Higher actions
Given a function gg classifying a GG-principal bundle and given a point in the delooping, there is the GG-principal bundle PP itself, being the collection of identifications of the fiber g(x)g(x) with XX(g:XBG),(*:BG)P x:X(g(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 *) P * EG X g BG\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 GG-equivariant map from the principal bundle to the representation space.σ: *:BG(PV)\vdash\; \sigma \colon \prod_{* \colon \mathbf{B}G} \left(P \to V\right) An element X σ V BG\array{ X &&\stackrel{\sigma}{\to}&& V \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G} of VV in the slice topos H /BG\mathbf{H}_{/\mathbf{B}G}A section of the ρ\rho-associated VV-fiber ∞-bundle.

In gauge theory physics, a representation ρ\rho of the gauge group GG encodes the particle-content of the model (in theoretical physics): a section of the ρ\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 XX a homotopy type that models spacetime (the collection of all points of spacetime) there should be a homotopy type Π(X)\Pi(X) whose homotopies and higher homotopies are the smooth trajectories, the smooth paths and higher paths in XX.

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 X\flat X for the discrete collection of points underlying XXdetaching all points;

  • an expression X\sharp X which dissolves the cohesion and produces the codiscrete smooth structure on XX.

There are some natural simple axioms on these constructions. For instance every smooth path in a discrete space X\flat X should be constant: Π(X)X\Pi (\flat X) \simeq \flat X.

With such natural axioms understood, these three constructions constitute an adjoint triple of modalities (Π)(\Pi \dashv \flat \dashv \sharp) in our language. In particular Π\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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
Given a cohesive homotopy type XX, there is the dissolved homotopy type X\sharp X in which all separate points are collected to one cohesive blob.X:TypeX:TypeX \colon Type \;\vdash\; \sharp X \colon TypeThe codiscrete object-monad on a (higher) local topos.The codiscrete smooth structure on the points of XX.Locality of the topos of smooth spaces
Given a cohesive homotopy type, there is the map that dissolves the cohesion of the points.X:TypeDeCoh X:XXX \colon Type \;\vdash\; DeCoh_X \colon X \to \sharp XThe unit of the codiscrete object monad.The function that sends smooth families in a smooth moduli stack to families of points.
Given XX there is the collection Π(X)\Pi(X) of points in XX and smooth trajectories between points in XX.(X:Type)Π(X):Type\left(X \colon \sharp Type\right) \;\vdash\; \Pi(X) \colon \sharp TypeThe construction of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos.The smooth path ∞-groupoid of XX.The local ∞-connectedness of the (∞,1)-topos of smooth ∞-groupoids
Given XX, there is a canonical map to Π(X)\Pi(X).(X:Type)ConstantPathInclusion X:XΠ(X)\left(X \colon \sharp Type\right) \;\vdash\; ConstantPathInclusion_X \colon X \to \Pi(X).The unit of the Π\Pi-monad on a locally ∞-connected (∞,1)-topos.The inclusion of XX into its smooth path ∞-groupoid as the constant paths.
Given XX, there is the result of detaching the points in XX.(A:Type)A:Type\left(A \colon \sharp Type\right) \;\vdash\; \flat A \colon \sharp TypeThe operation of the discrete object comonad on a (higher) local topos.The moduli stack for flat ∞-connections.
Given AA, there is a map from flat AA-connections to the underlying AA-bundles(A:Type)UnderlyingBundle A:AA\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 (Π)(\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():x,y:ΠX(xy)*,*:BG(**)transport(\nabla) \colon \underset{x,y \colon \Pi X}{\sum} \left( x \rightsquigarrow y \right) \to \underset{*,*' \colon \mathbf{B}G}{\sum} (* \rightsquigarrow *') | Π(X)transport()BGXBG\frac{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}.| The higher parallel transport trans()trans(\nabla) of a flat connection \nabla: a (higher) gauge field with vanishing field strength. | Flat connections | | A closed differential form ω\omega is a flat connection \nabla and a trivialization of the underlying bundle. | dRBG :BG(UnderlyingBundle()*)\begin{aligned} & \flat_{dR} \mathbf{B} G \coloneqq \\ & \sum_{\nabla \colon \flat \mathbf{B}G} (UnderlyingBundle(\nabla) \simeq *) \end{aligned} | dRBG UnderlyingConnection BG UnderlyingBundle * BG\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" 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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}{}^{\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(c)curv(\mathbf{c}) of a bundle c\mathbf{c} and a closed differential form ω\omega. | :c:B n𝔾ω:Ω cl n+1(curv(c)=ω)\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) | B n𝔾 conn F () Ω cl n+1 B n𝔾 curv dRB n+1𝔾 \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 GG-gauge fields to higher 𝔾\mathbb{G}-gauge fields. | exp(iS):BG connB n𝔾 conn\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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
There is a 𝔾\mathbb{G}-equivariant map ψ\psi from the prequantum bundle to the representation space.ψ::B𝔾 conn(P()V())\vdash \; \psi \colon \underset{\nabla \colon \mathbf{B}\mathbb{G}_{conn}}{\prod} \left( P\left(\nabla\right) \to V\left(\nabla\right) \right)X ψ V𝔾 conn ρ¯ B𝔾 conn\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 𝔾\mathbb{G}-equivariant equivalence exp(O^)\exp(\hat O) from the prequantum bundle to itself.exp(O^)::B𝔾 conn(P()P())\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)X exp(O^) X B𝔾 conn\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 languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
There is the continuum line. :Type i: GeometricallyContract :(Π()Point)\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 objectreal lineThe continuum real worldline

This then induces the existence of the circle group U(1)=/U(1) = \mathbb{R}/\mathbb{Z}. The electromagnetic field is a gauge field for gauge group U(1)U(1). Therefore in the language of cohesive homotopy type theory we can say

Let there be light.

ordinary languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
There is the collection of higher U(1)U(1)-principal connections.n:B nU(1) conn:Typen\colon \mathbb{N} \; \vdash \; \mathbf{B}^n U(1)_{conn} \colon TypeThe 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. em:[X,BU(1) conn] \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 XX.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.

Coordinate systems

This chapter is at geometry of physics -- coordinate systems

Smooth spaces

This chapter is at geometry of physics -- smooth spaces.

Differential forms

This chapter is at geometry of physics -- differential forms.

Differentiation

This chapter is at geometry of physics -- differentiation.

Smooth homotopy types

This chapter is at geometry of physics -- smooth homotopy types.

Groups

Model Layer

1-Groups

Discrete
Topological and Lie groups

Simplicial groups

Discrete
Abelian
Topological and Lie simplicial groups

Connected groupoids

any Lie group GG induces its delooping Lie groupoid

BG=*G=(G*). \mathbf{B}G = * \sslash G = \left( G \stackrel{\to}{\to} * \right) \,.

2-Groups

Discrete
Braided and abelian

Abelian \infty-groups

Chain complex
Spectrum
Dold-Kan correspondence
Definition

Write

DK:Ch (Ab(Smooth0Type))ΞsAb(Smooth0Type)Smooth0Type Δ opL wheSmooth0Type Δ opSmoothGrpd DK \colon Ch_\bullet(Ab(Smooth0Type)) \stackrel{\Xi}{\to} sAb(Smooth0Type) \to Smooth0Type^{\Delta^{op}} \to L_{whe} Smooth0Type^{\Delta^{op}} \simeq Smooth \infty Grpd

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.

Cohomology

Nonabelian cohomology
Abelian sheaf cohomology

Semantic Layer

A A_\infty-types

\infty-Groups

Syntactic Layer

Pointed connected types

Identity types of connected types

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Principal bundles

This chapter is at geometry of physics -- principal bundles.

Structure sheaves

Model layer

Semantics layer

Let H\mathbf{H} be a cohesive (∞,1)-topos (Π)(\mathbf{\Pi} \dashv \flat \dashv \sharp) equipped with differential cohesion (RedΠ inf inf)(Red \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf}).

Definition

For XHX \in \mathbf{H}, write

Sh H(X)H /X Sh_{\mathbf{H}}(X) \hookrightarrow \mathbf{H}_{/X}

for the full sub-(∞,1)-category of the slice (∞,1)-topos over XX on the formally étale maps into XX, def. \ref{FormallyEtaleMap}.

We call this the petit (∞,1)-topos of XX.

Proposition

The petit topos Sh H(X)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.

Definition

For XHX \in \mathbf{H} write

𝒪 ():HX *H /XRSh H(X) \mathcal{O}_{(-)} \;\colon\; \mathbf{H} \stackrel{X^*}{\to} \mathbf{H}_{/X} \stackrel{R}{\to} Sh_{\mathbf{H}}(X)

for the (∞,1)-functor which is the composite of the base change to XX followed by the co-reflection of prop. 1. We call this the structure sheaf of XX.

Remark

For X,AH thX, A \in \mathbf{H}_{th} and for UXU \to X a formally étale morphism in H th\mathbf{H}_{th}, we have that

𝒪 X(A)(U)Sh H(X)(U,𝒪 X)H(U,A)=A(U). \mathcal{O}_{X}(A)(U) \coloneqq \Sh_{\mathbf{H}}(X)( U , \mathcal{O}_{X} ) \simeq \mathbf{H}(U,A) = A(U) \,.

This means that 𝒪 X(A)\mathcal{O}_{X}(A) behaves as the sheaf of AA-valued functions over XX.

Syntax layer

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Manifolds and Orbifolds

For nn \in \mathbb{N} a manifold of dimension nn is an object XX that locally looks like a Cartesian space n\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 XX is an object such that first of all there is a cover, hence a 1-epimorphism of the form

p:( ii n)X. p \;\colon\; \left(\coprod_{i \in i} \mathbb{R}^n\right) \to X \,.

This encodes that XX can surjectively covered by Cartesian spaces, but it does not yet ensure that XX is locally equivalent to a Cartesian space in the intended sense. That intended sense is that pp 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.

Model Layer

  1. Smooth manifolds

  2. Tangent bundle and frame bundle

Smooth manifolds

A smooth manifold of dimension nn

a smooth space with an atlas

{ nϕ i 1U iX} \{ \mathbb{R}^n \underoverset{\simeq}{\phi_i^{-1}}{\to} U_i \hookrightarrow X\}

of coordinate charts. On each overlap U iU jU_i \cap U_j of two charts, the partial derivatives of the corresponding coordinate transformations

ϕ jϕ i 1:U iU j n n \phi_j\circ \phi_i^{-1} : U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n

form the Jacobian matrix of smooth functions

((λ ij) μ μ)[ddx νϕ jϕ i 1(x μ)]:U iU jGL n ((\lambda_{i j})^{\mu}{}_{\mu}) \coloneqq \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu) \right] : U_i \cap U_j \to GL_n

with values in invertible matrices, hence in the general linear group GL(n)GL(n). By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations

(λ ij) μ λ(λ jk) λ ν=(λ ik) μ ν, (\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} = (\lambda_{i k})^\mu{}_{\nu} \,,

(here and in the following sums over an index appearing upstairs and downstairs are explicit)

hence the equation

λ ijλ jk=λ ik \lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}

in the group C (U iU jU k,GL(n))C^\infty(U_i \cap U_j \cap U_k, GL(n)) of smooth GL(n)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)GL(n) (precisely: with coefficients in the sheaf of smooth functions with values in GL(n)GL(n) ). We write

[(λ ij)]H smooth 1(X,GL n). [(\lambda_{i j})] \in H^1_{smooth}(X, GL_n) \,.

Formulated as smooth groupoids

  • XX itself is a Lie groupoid (XX)(X \stackrel{\to}{\to} X) with trivial morphism structure;

  • from the atlas {U iX}\{U_i \to X\} we get the corresponding Cech groupoid

    C({U i})=( i,jU iU j iU i)={ (x,j) = (x,i) (x,k)forxU iU jU k}, C(\{U_i\}) = (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i) = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\; for\, x \in U_i \cap U_j \cap U_k \right\} \,,

    whose objects are the points in the atlas, with morphisms identifying lifts of a point in XX to different charts of the atlas;

Tangent bundle

We discuss how the tangent bundle of a manifold XX naturally arises in the above perspecive in terms of the map τ X:XBGL(n)\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

C({U i}) τ X BGL(n). X, \array{ C(\{U_i\}) &\stackrel{\tau_X}{\to}& \mathbf{B} GL(n). \\ {}^{\mathllap{\simeq}}\downarrow \\ X } \,,

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 τ X\tau_X has as components the functions λ ij\lambda_{i j} and its functoriality is the cocycle condition λ ijλ jk=λ ik\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}.

A transformation of smooth functors λ 1λ 2:C({U i})BGL(n)\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

τ X:XBGL(n). \tau_X \;\colon\; X \to \mathbf{B}GL(n) \,.

The homotopy fiber of this map is a GL(n)GL(n)-principal bundle called the frame bundle of XX, while the canonically associated bundle via the canonical representation of GL(n)GL(n) on n\mathbb{R}^n is the tangent bundle

TXX. T X \to X \,.

Semantics Layer

Let H\mathbf{H} be a cohesive (∞,1)-topos (Π)(\mathbf{\Pi} \dashv \flat \dashv \sharp) equipped with differential cohesion (RedΠ inf inf)(Red \dashv \mathbf{\Pi}_{inf} \dashv \flat_{inf}). Let

𝔸 1H \mathbb{A}^1 \in \mathbf{H}

be an line object that exhibits the cohesive structure.

Definition

An étale ∞-groupoid of dimension nn is an object XHX \in \mathbf{H} such that there exists a map p:( i𝔸 n)Xp \;\colon\;\left(\coprod_{i} \mathbb{A}^n\right) \to X such that

  1. pp is a 1-epimorphism;

  2. pp is a formally étale morphism, def. \ref{FormallyEtaleMap}

If XX here is 0-truncated then we call it it manifold. It XX is 1-truncated we call it an orbifold.

Definition

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Syntax Layer

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Reduction of structure groups

Model Layer

GG-Structure

BGBK \mathbf{B}G \to \mathbf{B}K

given a KK-principal bundle

X˜ BK X \array{ \tilde X &\to &\mathbf{B}K \\ \downarrow^{\mathrlap{\simeq}} \\ X }

a reduction of the structure group along GKG \to K is

X˜ BG e BK \array{ \tilde X &&\to&& \mathbf{B}G \\ & \searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}K }

Examples

Vielbein, orthogonal structure, Riemannian geometry

reduction of the structure group along

BO(n)BGL(n)\mathbf{B}O(n) \to \mathbf{B}GL(n)

X˜ BO(n) TΣ e BGL(n) \array{ \tilde X &&\to&& \mathbf{B}O(n) \\ & {}_{\mathllap{\vdash T \Sigma}}\searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}GL(n) }

ee is vielbein: definition of an orthonormal frame? at each point

Electromagnetism in gravitational background

example: the other 2 Maxwell equations: dF=j el\mathbf{d} \star F = j_{el}.

Einstein-Maxwell theory

Almost complex structure
Almost Hermitean structure
  • almost Hermitean structure?
Almost symplectic structure
Metaplectic structure
Metalinear structure
Generalized complex geometry
Type II geometry
Generalized Calabi-Yau structure
Exceptional generalized geometry
Spin structure, String structure, Fivebrane structure

Semantics Layer

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Syntax Layer

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Representations and Associated bundles

Model Layer

Actions

Spin geometry

V VSpin BSpin \array{ V &\to& V \sslash Spin \\ && \downarrow \\ && \mathbf{B}Spin }
X ψ VSpin BSpin \array{ X &&\stackrel{\psi}{\to}&& V \sslash Spin \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}Spin }

Associated bundle

Representations up to coherent homotopy

Semantic Layer

\infty-Actions

∞-action

V VG BG \array{ V &\to& V\sslash G \\ && \downarrow \\ && \mathbf{B}G }

Associated \infty-bundles

E VG pb X˜ BG X \array{ E &\to& V\sslash G \\ \downarrow &pb& \downarrow \\ \tilde X &\to& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }
X σ VG BG \array{ X &&\stackrel{\sigma}{\to}&& V \sslash G \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && \mathbf{B}G }

Syntactic Layer

The context of a pointed connected type: representation theory

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Dependent product over a pointed connected type: invariants

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Dependent sum over a pointed connected type: quotients

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Modules

for the moment see the sub-entry geometry of physics - modules

Flat connections

Model Layer

Flat 1-connections

XX connected, π 1(X)\pi_1(X) \in Grp its fundamental group for any choice of basepoint, then the holonomy pairing

hol:[S 1,X]×H conn 1(X,G)G hol \colon [S^1,X]\times H^1_{conn}(X,G) \to G

descends to homotopy classes of (based) loops

hol:H conn,flat 1(X,G)Hom Grp(π 1(X),G)/G hol \colon H^1_{conn,flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X), G)/G

to a bijection from equivalence classes of flat? GG-principal connections to the quotient set of group homomorphisms π 1(X)G\pi_1(X) \to G modulo the adjoint action of GG on itself.

Semantic Layer

Definition

For GGrp(H)G \in Grp(\mathbf{H}) and XHX \in \mathbf{H} a flat GG-connection \nabla on XX is a morphism

:XBG. \nabla \colon X \to \flat \mathbf{B}G \,.

We write

H flat(X,BG)H(X,BG) \mathbf{H}_{flat}(X, \mathbf{B}G) \coloneqq \mathbf{H}(X, \flat \mathbf{B}G)

and accordingly

H flat 1X,Gπ 0H flat(X,G) H^1_{flat}{X, G} \coloneqq \pi_0 \mathbf{H}_{flat}(X,G)

for the cohomology of XHX \in \mathbf{H} with flat coefficients.

Remark

By adjunction,

XBGΠ(X)transport()BG \frac{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}

a flat GG-connection is equivalently a morphism

transport():Π(X)BG. transport(\nabla) \colon \Pi(X) \to \mathbf{B}G \,.

Since Π(X)\Pi(X) is the fundamtal infinity-groupoid? of XX, this manifestly encodes the higher parallel transport of the flat connection.

Definition

Write

UnderlyingBundle BG:BGBG UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G

for the (DiscΓ)(Disc \vdash \Gamma)-counit-

Definition

For :XBG\nabla \colon X \to \flat \mathbf{B}G the composite

UnderlyingBundle():XBGUnderlyingBundle BGBG UnderlyingBundle(\nabla) \colon X \stackrel{\nabla}{\to} \flat\mathbf{B}G \stackrel{UnderlyingBundle_{\mathbf{B}G}}{\to} \mathbf{B}G

modulates a GG-principal ∞-bundle on XX, by def. \ref{spring}. This we call the underlying GG-principal bundle of \nabla.

ConstantPaths X:XΠ(X) ConstantPaths_{X} \colon X \to \Pi(X)

Syntactic Layer

BG:TypeUnderlyingBundle:BGBG \mathbf{B}G \colon Type \;\vdash \; UnderlyingBundle \colon \flat \mathbf{B}G \to \mathbf{B}G

de Rham Coefficients

Model Layer

Lie-algebra valued differential 1-forms

Definition

Let GG be a Lie group, and write 𝔤\mathfrak{g} for its Lie algebra. The set of Lie algebra valued differential 1-forms is the tensor product

Ω 1(U,𝔤)=Ω 1(U) 𝔤. \Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g} \,.

flat forms:

Ω flat 1(U,𝔤)={ωΩ 1(U,𝔤)F ω=dω+[ω,ω]=0}. \Omega^1_{flat}(U, \mathfrak{g}) = \left\{ \omega \in \Omega^1(U,\mathfrak{g}) | F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0 \right\} \,.

(…)

This is a smooth space

Ω flat 1(,𝔤)Smooth0Type \Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type

For 𝔤=Lie()\mathfrak{g} = Lie(\mathbb{R}) we have

Ω 1(,Lie())=Ω 1 \Omega^1(-,Lie(\mathbb{R})) = \Omega^1

and we write

Ω flat 1(,Lie())=Ω cl 1 \Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}

Below we see

dRBGΩ flat 1(,𝔤) \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})

The de Rham complex

Below we see that

dRB n dRB nU(1)DK[Ω 1()dΩ 2()ddΩ cl n()]. \flat_{dR}\mathbf{B}^n \mathbb{R} \simeq \flat_{dR}\mathbf{B}^n U(1) \simeq DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \,.

Semantic Layer

De Rham coefficient objects

Definition

For GGpr(H)G \in Gpr(\mathbf{H}), its de Rham coefficient object is the homotopy pullback

dRBGBG× BG* \flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *

in

dRBG UnderlyingConnection BG pb UnderlyingBundle * BG. \array{ \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}} \\ * &\to& \mathbf{B}G } \,.
Remark

This pullback diagram expresses that elements of dRBG\flat_{dR}\mathbf{B}G are flat GG-connections :XBG\nabla \colon X \to \flat \mathbf{B}G, def. 6 equipped with a trivialization of their underlying GG-principal bundle, def. 8.

Recovering smooth differential forms from cohesive de Rham coefficients

Let H=\mathbf{H} = Smooth∞Grpd. All smooth manifolds and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this H=Sh (CartSp)\mathbf{H} = Sh_\infty(CartSp).

Proposition

For GG a Lie group, the de Rham coefficient object dRBG\flat_{dR}\mathbf{B}G, def. 10 of its delooping is given by the sheaf of flat Lie algebra valued differential 1-forms Ω flat 1(,𝔤)\Omega^1_{flat}(-,\mathfrak{g}), def. 9, for 𝔤\mathfrak{g} the Lie algebra of GG:

dRBGΩ flat 1(,𝔤). \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group.

Write U(1)U(1) for the circle group regared as a Lie group in the standard way.

Proposition

For nn \in \mathbb{N}, the de Rham coefficient object dRB nU(1)\flat_{dR}\mathbf{B}^n U(1), def. 10, of the nn-fold delooping of U(1)U(1) is given by the image under the Dold-Kan correspondence

DK::Sh(CartSp,Ch )Sh(CartSp,sSet)L lwheSh(CartSp,sSet)H DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet) \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}

of the truncated de Rham complex of sheaves of differential forms,

dRB nU(1) dRB n DK[Ω 1()ddΩ cl n()] DK[Ω cl 1()00]. \begin{aligned} \flat_{dR}\mathbf{B}^n U(1) &\simeq \flat_{dR} \mathbf{B}^n \mathbb{R} \\ & \simeq DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \\ & \simeq DK[\Omega^1_{cl}(-) \to 0 \to \cdots \to 0] \end{aligned} \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups.

Syntactic Layer

dR(BG:Type): Type :BG(UnderlyingBundle()=*) \begin{aligned} \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type \\ \coloneqq & \;\; \sum_{\nabla \colon \flat \mathbf{B}G} ( UnderlyingBundle(\nabla) = * ) \end{aligned}

Maurer-Cartan forms

Model Layer

Maurer-Cartan form on a Lie group

θ G:GΩ flat 1(,𝔤) \theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})

Consider

dRB=Ω cl 1 \flat_{dR} \mathbf{B}\mathbb{R} = \Omega^1_{cl}

the Maurer-Cartan form on \mathbb{R} is the de Rham differential

θ =d:Ω cl 1Ω 1. \theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1 \,.

Semantic Layer

Maurer-Cartan form on a cohesive \infty-group

Let H\mathbf{H} be a cohesive (infinity,1)-topos (Π)(\mathbf{\Pi} \dashv \flat \dashv \sharp). We discuss a general formulation of Maurer-Cartan forms on cohesive infinity-groups

Let GGrp(H)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 θ\theta in

G * θ pb dRBG BG pb * BG \array{ G &\to& * \\ \downarrow^{\mathrlap{\theta}} & pb & \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow \\ * &\to& \mathbf{B}G }
Definition

This is the Maurer-Cartan form on GG

θ:G dRBG. \theta \;\colon\; G \to \flat_{dR} \mathbf{B}G \,.
Definition

For S:XGS \;\colon\; X \to G a morphism, write

S 1dSS *θ G:XSGθ G dRBG S^{-1} \mathbf{d} S \coloneqq S^* \theta_G \;\colon\; X \stackrel{S}{\to} G \stackrel{\theta_G}{\to} \flat_{dR}\mathbf{B}G

for its composite with the map of def. 11, hence the pullback of the Maurer-Cartan form along SS. We also call this the de Rham differential of SS.

Maurer-Cartan forms on smooth \infty-groups

Proposition

For GG a Lie group canonically regarded in H=\mathbf{H} = Smooth∞Grpd the general abstract morphism

θ G:G dRBG \theta_G \colon G \to \flat_{dR}\mathbf{B}G

is identified, via the identification dRBGΩ flat 1(,𝔤)\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) of prop. 2 and the Yoneda lemma, with the traditional Maurer-Cartan form

θ GΩ flat 1(G,𝔤). \theta_G \in \Omega^1_{flat}(G, \mathfrak{g}) \,.

Cohesive differentiation

The Maurer-Cartan form on the line object

θ :Ω cl 1(,) \theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})

is the de Rham differential,

d=θ . \mathbf{d} = \theta_{\mathbb{R}} \,.

Universal curvature characteristic forms

For G=B nU(1)G = \mathbf{B}^n U(1)

curv:B nU(1) dRB n+1U(1) curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)

sends a circle nn-bundle to the curvature of a pseudo-connection on it.

Syntactic Layer

(…)

Principal connections

Model Layer

Circle-principal connections

Dirac charge quantization says that the electromagnetic field is only locally in general a map

Ω 1() A d X ω Ω cl 2 \array{ && \Omega^1(-) \\ & {}^{\mathllap{A}}\nearrow & \downarrow^{\mathrlap{\mathbf{d}}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }

globally it is instead a map

BU(1) conn F () X ω Ω cl 2 \array{ && \mathbf{B}U(1)_{conn} \\ & {}^{\nabla}\nearrow & \downarrow^{F_{(-)}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }

where

BU(1) conn F () Ω cl 2 pb BU(1) diff Ω cl 12 BU(1) \array{ \mathbf{B}U(1)_{conn} &\stackrel{F_{(-)}}{\to}& \Omega^2_{cl} \\ \downarrow &pb& \downarrow \\ \mathbf{B}U(1)_{diff} &\to& \Omega^{1 \leq \bullet \leq 2}_{cl} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}U(1) }

circle bundle with connection

the smooth groupoid is

BU(1) conn=Ω 1()U(1) \mathbf{B}U(1)_{conn} = \Omega^1(-) \sslash U(1)

quotient of Ω 1()\Omega^1(-) by U(1)U(1)-gauge transformations

for

A,A:XΩ 1() A,A' : X \to \Omega^1(-)

a gauge transformation AAA \to A' is λ:XU(1)\lambda : X \to U(1) with

A=A+dlogλ A' = A + \mathbf{d} log \lambda

Dirac charge quantization and the electromagnetic field

Principal 1-connection

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 GG-principal bundle PXP \to X is a rule tra tra_\nabla for parallel transport along paths: a rule that assigns to each path γ:[0,1]X\gamma : [0,1] \to X a morphism tra (γ):P xP ytra_\nabla(\gamma) : P_x \to P_y between the fibers of the bundle above the endpoints of these paths, in a compatible way:

P x tra (γ) P y tra (γ) P z P x γ y γ z X. \array{ P_x &\stackrel{tra_\nabla(\gamma)}{\to}& P_y &\stackrel{tra_\nabla(\gamma')}{\to}& P_z &&& P \\ && && &&& \downarrow \\ x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z &&& X } \,.

In order to formalize this, we introduce a (diffeological) Lie groupoid to be called the path groupoid of XX. (Constructions and results in this section are from ([SWI]).

Definition

For XX a smooth manifold let [I,X][I,X] be the set of smooth functions I=[0,1]XI = [0,1] \to X. For UU a Cartesian space, we say that a UU-parameterized smooth family of points in [I,X][I,X] is a smooth map U×IXU \times I \to X. (This makes [I,X][I,X] a diffeological space).

Say a path γ[I,X]\gamma \in [I,X] has sitting instants if it is constant in a neighbourhood of the boundary I\partial I. Let [I,P] si[I,P][I,P]_{si} \subset [I,P] be the subset of paths with sitting instants.

Let [I,X] si[I,X] si th[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 UU-parameterized smooth family of points in [I,X] si th[I,X]_{si}^{th} is one that comes from a UU-family of representatives in [I,X] si[I,X]_{si} under this projection. (This makes also [I,X] si th[I,X]_{si}^{th} a diffeological space.)

Remark

The passage to the subset and quotient [I,X] si th[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.

Definition

The path groupoid P 1(X)\mathbf{P}_1(X) is the groupoid

P 1(X)=([I,X] si thX) \mathbf{P}_1(X) = ([I,X]_{si}^{th} \stackrel{\to}{\to} X)

with source and target maps given by endpoint evaluation and composition given by concatenation of classes [γ][\gamma] of paths along any orientation preserving diffeomorphism [0,1][0,2][0,1] 1,0[0,1][0,1] \to [0,2] \simeq [0,1] \coprod_{1,0} [0,1] of any of their representatives

[γ 2][γ 1]:[0,1][0,1] 1,0[0,1](γ 2,γ 1)X. [\gamma_2] \circ [\gamma_1] : [0,1] \stackrel{\simeq}{\to} [0,1] \coprod_{1,0} [0,1] \stackrel{(\gamma_2 , \gamma_1)}{\to} X \,.

This becomes an internal groupoid in diffeological spaces with the above UU-families of smooth paths. We regard it as a groupoid-valued presheaf, an object in [CartSp op,Grpd][CartSp^{op}, Grpd]:

P 1(X):U(Diff(U×I,X) si thDiff(U,X)). \mathbf{P}_1(X) : U \mapsto (Diff(U \times I, X)_{si}^{th} \stackrel{\to}{\to} Diff(U,X) ) \,.

Observe now that for GG a Lie group and BG\mathbf{B}G its delooping Lie groupoid discussed above, a smooth functor tra:P 1(X)BGtra : \mathbf{P}_1(X) \to \mathbf{B}G sends each (thin-homotopy class of a) path to an element of the group GG

tra:(x[γ]y)(tra(γ)G) tra : (x \stackrel{[\gamma]}{\to} y) \mapsto ( \bullet \stackrel{tra(\gamma) \in G}{\to} \bullet )

such that composite paths map to products of group elements

tra:( y [γ] = [γ] x [γ][γ] z)( tra(γ) = tra(γ) tra(γ)tra(γ) ) tra : \left( \array{ && y \\ & {}^{\mathllap{[\gamma]}}\nearrow &=& \searrow^{\mathrlap{[\gamma']}} \\ x &&\stackrel{[\gamma']\circ [\gamma]}{\to}&& z } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{tra(\gamma)}}\nearrow &=& \searrow^{\mathrlap{tra(\gamma')}} \\ \bullet &&\stackrel{tra(\gamma)tra(\gamma')}{\to}&& \bullet } \right)

and such that UU-families of smooth paths induce smooth maps UGU \to G of elements.

There is a classical construction that yields such an assignment: the parallel transport of a Lie-algebra valued 1-form.

Definition

Suppose AΩ 1(X,𝔤)A \in \Omega^1(X, \mathfrak{g}) is a degree-1 differential form on XX with values in the Lie algebra 𝔤\mathfrak{g} of GG. Then its parallel transport is the smooth functor

tra A:P 1(X)BG tra_A : \mathbf{P}_1(X) \to \mathbf{B}G

given by

[γ]Pexp( [0,1]γ *A)G, [\gamma] \mapsto P \exp(\int_{[0,1]} \gamma^* A) \; \in G \,,

where the group element on the right is defined to be the value at 1 of the unique solution f:[0,1]Gf : [0,1] \to G of the differential equation

d dRf+γ *Af=0 d_{dR} f + \gamma^*A \wedge f = 0

for the boundary condition f(0)=ef(0) = e.

Theorem

This construction Atra AA \mapsto tra_A induces an equivalence of categories

[CartSp op,Grpd](P 1(X),BG)BG conn(X), [CartSp^{op},Grpd](\mathbf{P}_1(X), \mathbf{B}G) \simeq \mathbf{B}G_{conn}(X) \,,

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 AΩ 1(X,𝔤)A \in \Omega^1(X,\mathfrak{g}),

  • morphisms g:A 1A 2g : A_1 \to A_2 are labeled by smooth functions gC (X,G)g \in C^\infty(X,G) such that A 2=g 1Ag+g 1dgA_2 = g^{-1} A g + g^{-1}d g.

This equivalence is natural in XX, so that we obtain another smooth groupoid.

Definition

Define BG conn:CartSp opGrpd\mathbf{B}G_{conn} : CartSp^{op} \to Grpd to be the (generalized) Lie groupoid

BG conn:U[CartSp op,Grpd](P 1(),BG) \mathbf{B}G_{conn} : U \mapsto [CartSp^{op}, Grpd](\mathbf{P}_1(-), \mathbf{B}G)

whose UU-parameterized smooth families of groupoids form the groupoid of Lie-algebra valued 1-forms on UU.

Remark

This equivalence in particular subsumes the classical facts that parallel transport γPexp( [0,1]γ *A)\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.

Observation

There is an evident natural smooth functor XP 1(X)X \to \mathbf{P}_1(X) that includes points in XX as constant paths. This induces a natural morphism BG connBG\mathbf{B}G_{conn} \to \mathbf{B}G that forgets the 1-forms.

Defintion

Let PXP \to X be a GG-principal bundle that corresponds to a cocycle g:C(U)BGg : C(U) \to \mathbf{B}G under the construction discussed above. Then a connection \nabla on PP is a lift \nabla of the cocycle through BG connBG\mathbf{B}G_{conn} \to \mathbf{B}G.

BG conn C(U) g BG. \array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \,.
Observation

This is equivalent to the traditional definitions.

A morphism :C(U)BG conn\nabla : C(U) \to \mathbf{B}G_{conn} is

  • on each U iU_i a 1-form A iΩ 1(U i,𝔤)A_i \in \Omega^1(U_i, \mathfrak{g});

  • on each U iU jU_i \cap U_j a function g ijC (U iU j,G)g_{i j} \in C^\infty(U_i \cap U_j , G);

such that

  • on each U iU jU_i \cap U_j we have A j=g ij 1(A+d dR)g ijA_j = g_{i j}^{-1}( A + d_{dR} )g_{i j};

  • on each U iU jU kU_i \cap U_j \cap U_k we have g ijg jk=g ikg_{i j} \cdot g_{j k} = g_{i k}.

Definition

Let [I,X] si th[I,X] h[I,X]_{si}^{th} \to [I,X]^h the projection onto the full quotient by smooth homotopy classes of paths.

Write Π 1(X)=([I,X] hX)\mathbf{\Pi}_1(X) = ([I,X]^h \stackrel{\to}{\to} X) for the smooth groupoid defined as P 1(X)\mathbf{P}_1(X), but where instead of thin homotopies, all homotopies are divided out.

Proposition

The above restricts to a natural equivalence

[CartSp op,Grpd](Π 1(X),BG)BG, [CartSp^{op}, Grpd](\mathbf{\Pi}_1(X), \mathbf{B}G) \simeq \mathbf{\flat}\mathbf{B}G \,,

where on the left we have the hom-groupoid of groupoid-valued presheaves, and on the right we have the full sub-groupoid BGBG conn\mathbf{\flat}\mathbf{B}G \subset \mathbf{B}G_{conn} on those 𝔤\mathfrak{g}-valued differential forms whose curvature 2-form F A=d dRA+[AA]F_A = d_{dR} A + [A \wedge A] vanishes.

A connection \nabla is flat precisely if it factors through the inclusion BGBG conn\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 :XC(U)BG conn\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.

Covariant derivatives

(VG) conn BG conn \array{ (V\sslash G)_{conn} \\ \downarrow \\ \mathbf{B}G_{conn} }
X˜ (σ,σ) (VG) conn BG conn \array{ \tilde X &&\stackrel{(\sigma, \nabla \sigma)}{\to}&& (V \sslash G)_{conn} \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G_{conn} }

Deligne cohomology and Cheeger-Simons differential characters

Circle-principal 2-connection

Magnetic charge and the B-field

Circle-principal 3-connection

The 3d Chern-Simons action functional and the C-field

Circle-principal nn-connection

Definition

Write U(1)Smooth0TypeU(1) \in Smooth0Type for the smooth space of the circle group. Write

dlog:U(1)Ω 1 \mathbf{d}log \colon U(1) \to \Omega^1

for the homomorphism of smooth spaces which over an abstract coordinate system UU \in CartSp is given by the function

dlog u:C (U,U(1))Ω 1(U) \mathbf{d} log_u \colon C^\infty(U,U(1)) \to \Omega^1(U)

which sends a U(1)U(1)-valued f:UU(1)f \colon U \to U(1) – for which one can always find an \mathbb{R}-valued function f^:U\hat f \colon U \to \mathbb{R} for whuch f=f^modf = \hat f \,mod\, \mathbb{Z} – to the differential 1-form df^\mathbf{d} \hat f.

Definition

For nn \in \mathbb{N}, the chain complex of smooth spaces (of sheaves on CartSp)

(U(1)dlogΩ 1ddΩ n) (U(1) \stackrel{\mathbf{d} log}{\to} \Omega^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n)

(regarded as a chain complex of abelian groups and as sitting in degrees nn through 0) is called the (smooth) Deligne complex in these degrees.

Definition

For nn \in \mathbb{N}, write

B nU(1) conn=DK(U(1)dlogΩ 1ddΩ n)SmoothGrpd \mathbf{B}^n U(1)_{conn} = DK(U(1) \stackrel{\mathbf{d} log}{\to} \Omega^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n) \in Smooth\infty Grpd

for the smooth ∞-groupoid presented by the Deligne complex under the Dold-Kan correspondence map, def. 1.

For XX \in Smooth∞Grpd we say that

We also call B nU(1) conn\mathbf{B}^n U(1)_{conn} the universal moduli ∞-stack of circle nn-bundles with connection.

Definition

Write

B nU(1) conn DK( U(1) dlog Ω 1 d d Ω n ) U B nU(1) conn DK( id ) B nU(1) DK( U(1) 0 d 0 ) \array{ \mathbf{B}^n U(1)_{conn} & \coloneqq & DK( & U(1) &\stackrel{\mathbf{d} log}{\to}& \Omega^1 &\stackrel{\mathbf{d}}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & \Omega^n & ) \\ \downarrow^{\mathrlap{U_{\mathbf{B}^n U(1)_{conn}}}} & \coloneqq& DK( & \downarrow^{\mathrlap{id}} && \downarrow && && \downarrow ) \\ \mathbf{B}^n U(1) & \coloneqq & DK( & U(1) &\stackrel{}{\to}& 0 &\stackrel{}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & 0 & ) }

for the canonical forgetful morphism from moduli of nn-connections to those of the underlying principal bundles.

Definition

More generally, we may consider the intermediate stages

B nU(1) conn DK( U(1) dlog Ω 1 d d Ω k d Ω k+1 d Ω n) DK( id id id ) B nU(1) conn k DK( U(1) d Ω 1 d d Ω k 0 0) \array{ \mathbf{B}^n U(1)_{conn} & \coloneqq & DK( & U(1) &\stackrel{\mathbf{d} log}{\to}& \Omega^1 &\stackrel{\mathbf{d}}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & \Omega^k & \stackrel{\mathbf{d}}{\to} & \Omega^{k+1} &\to& \cdots &\stackrel{\mathbf{d}}{\to} & \Omega^n) \\ \downarrow^{} & \coloneqq& DK( & \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && && \downarrow^{\mathrlap{id}} & & \downarrow && \cdots & & \downarrow) \\ \mathbf{B}^n U(1)_{conn^k} & \coloneqq & DK( & U(1) &\stackrel{\mathbf{d}}{\to}& \Omega^1 &\stackrel{\mathbf{d}}{\to}& \cdots & \stackrel{\mathbf{d}}{\to} & \Omega^k & \to & 0 &\to& \cdots &\to & 0 ) }

Semantic Layer

Differential cohomology

Let GGrp(H)G \in Grp(\mathbf{H}) be a braided ∞-group. Equivalently, let its delooping BGH\mathbf{B}G \in \mathbf{H} be itself equipped with the structure of an ∞-group. Write

B 2GH \mathbf{B}^2 G \in \mathbf{H}

for the corresponding double delooping.

Definition

Write

curv Gθ B𝔾:BG dRB 2G curv_{G} \coloneqq \theta_{\mathbf{B}\mathbb{G}} \colon \mathbf{B}G \to \flat_{dR} \mathbf{B}^2 G

for the Maurer-Cartan form on the ∞-group BG\mathbf{B}G, def. 11. We call this the universal curvature characteristic of GG.

Definition

The differential cohomology with coefficients in BG\mathbf{B}G is cohomology in the slice (∞,1)-topos H / dRB 2G\mathbf{H}_{/\flat_{dR} \mathbf{B}^2 G} with coefficients in curv Gcurv_G

H / dRB 2G(,curv G). \mathbf{H}_{/\flat_{dR}\mathbf{B}^2 G}(-, curv_G) \,.

Differential-form curvatures

B n𝔾 Ω cl n+1() pb B𝔾 curv dRB 2𝔾 \array{ \mathbf{B}^n \mathbb{G} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow &pb& \downarrow^{\mathrlap{}} \\ \mathbf{B}\mathbb{G} &\stackrel{curv}{\to}& \flat_{dR} \mathbf{B}^2 \mathbb{G} }

presented by ordinary differential cohomology

Differential moduli

We discuss the moduli stacks of higher principal connections, over a fixed XSmoothGrpdX \in Smooth\infty Grpd.

For nn \in \mathbb{N} and with B nU(1) conn\mathbf{B}^n U(1)_{conn} \in Smooth∞Grpd the universal moduli stack for circle n-bundles with connection, def. 16, and for XSmoothGrpdX \in Smooth\infty Grpd, one may be tempted to regard the internal hom/mapping space [X,B nU(1) conn][X, \mathbf{B}^n U(1)_{conn}] as the moduli stack of circle n-bundles with connection on XX. However, for UU \in CartSp an abstract coordinate system, UU-plots and their k-morphisms in [X,B nU(1) conn][X, \mathbf{B}^n U(1)_{\mathrm{conn}}] are circle principal nn-connections and their kk-fold gauge transformations on U×XU \times X, and this is not generally what one would want the UU-plots of the moduli stack of such connections on XX to be. Rather, that moduli stack should have

  1. as UU-plots smoothly UU-parameterized collections { u}\{\nabla_u\} of nn-connections on XX;

  2. as kk-morphisms smoothly UU-parameterized collections {ϕ u}\{\phi_u\} of gauge transformations between them.

The first item is equivalent to: a single nn-connection on U×XU \times X such that its local connection nn-forms have no legs along UU. 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 nn-connection \nabla on U×XU \times X needs to respect the curvature of the connection along UU, but a family {ϕ u}\{\phi_u\} of gauge tranformations between the restrictions u\nabla|_u of \nabla to points of the coordinate patch UU 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,B nU(1) conn][X, \mathbf{B}^n U(1)_{conn}] degreewise in stages.

  1. For circle-principal 1-connections

  2. For circle-principal 2-connections

For circle-principal 1-connections

We discuss this first for n=1n = 1, hence for moduli stacks of circle bundles with connection.

Definition

For XSmooth1TyppeX \in Smooth1Typpe \hookrightarrow Smooth∞Grpd a smooth groupoids, we write

X 2X 1XX X \to \sharp_2 X \to \sharp_1 X \to \sharp X

for the n-image factorization of the canonical morphism XXX \to \sharp X to the sharp modality, def. \ref{SharpModalityOfLocalTopos}.

Remark

If XSmooth0TypeX \in Smooth0Type is just a smooth space then 2XX\sharp_2 X \simeq X and 1X=ConcX\sharp_1 X = Conc X is the concretification of XX, def. \ref{ConcreteObjectsAndConcretification}.

One might call 1X\sharp_1 X the “1-concretification” and 2X\sharp_2 X the “2-concretification” of XX. But generally what one might actually want to call a “concretification” of XX involves an interplay of both, as we will see now.

Definition

For XX \in Smooth∞Grpd, write

U(1)Conn(X) 1[X,BU(1) conn]× 1[X,BU(1)] 2[X,BU(1)] U(1)\mathbf{Conn}(X) \coloneqq \sharp_1[X, \mathbf{B}U(1)_{conn}] \underset{\sharp_1 [X,\mathbf{B}U(1)]}{\times} \sharp_2 [X, \mathbf{B}U(1)]

for the smooth groupoid which is the homotopy pullback in

U(1)Conn(X) 2[X,BU(1)] 1[X,BU(1) conn] 1[X,U BU(1) conn] 1[X,BU(1)]. \array{ U(1)\mathbf{Conn}(X) &\to& \sharp_2 [X, \mathbf{B}U(1)] \\ \downarrow && \downarrow \\ \sharp_1[X, \mathbf{B}U(1)_{conn}] &\stackrel{\sharp_1 [X,U_{\mathbf{B}U(1)_{conn}}] }{\to}& \sharp_1 [X,\mathbf{B}U(1)] } \,.

Here the bottom morphism is the 1\sharp_1-image of the forgetful morphism U BU(1) conn:BU(1) connBU(1)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.

Proposition

The smooth groupoid U(1)Conn(X)U(1)\mathbf{Conn}(X) of def. 22 is indeed the smooth moduli object/moduli stack of circle-principal connections on XX; in that its UU-plots of are smoothly UU-parameterized collections of smooth circle-principal connections on XX and its morphisms of UU-plots are smoothly UU-parameterized collections of smooth gauge transformation between these, on XX.

Proof

By the discussion at n-image and using arguments as for the concretification of moduli of differential forms above, we have:

  • 1[X,BU(1) conn]\sharp_1 [X, \mathbf{B}U(1)_{conn}] has as UU-plots those connections \nabla on U×XU \times X whose connection 1-forms have no leg along UU, hence smooth UU-parameterized families { u}\{\nabla_u\} of connections on XX, and has as morphisms Γ(U)\Gamma(U)-parameterized (hence non-smooth) collections {ϕ u}\{\phi_u\} of gauge transformations of connections on XX;

  • 1[X,BU(1)]\sharp_1 [X, \mathbf{B}U(1)] looks similarly, just without the connection information;

  • 1[X,U BU(1) conn]\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 UU it is a fibration(isofibration): given a Γ(U)\Gamma(U)-parameterized collection of gauge transformations out of a smoothly UU-parameterized collection of bundles and then a smooth choice of smooth connections on these bundles, the Γ(U)\Gamma(U) collection of gauge transformations of course also acts on these connections;

  • 2[X,BU(1)][X,BU()1]\sharp_2 [X, \mathbf{B} U(1)] \simeq [X, \mathbf{B} U()1] (because if two gauge transformations of bundles on U×XU \times X coindide on each point of UU as gauge tranformations on XX, 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 UU-plot of U(1)Conn(X)U(1)\mathbf{Conn}(X) is a smoothly UU-parameterized collection { u}\{\nabla_u\} of connections on XX, and that a morphism between such as a Γ(U)\Gamma(U)-parameterized collection of gauge transformations {ϕ u}\{\phi_u\} of connections, such that their underlying collection of gauge transformations of bundles is a smoothly UU-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 UU-plots of U(1)Conn(X)U(1)\mathbf{Conn}(X) are smoothly UU-paramezerized collections of gauge transformations.

For circle-principal 2-connections

We now discuss moduli of circle 2-bundles with connection over a given base XX.

Recall that by def. 18 we write

B 2U(1) conn 1DK(U(1)dΩ 10). \mathbf{B}^2U(1)_{conn^1} \coloneqq DK(U(1) \stackrel{\mathbf{d}}{\to} \Omega^1 \to 0) \,.
Definition

For XSmoothGrpdX \in Smooth \infty Grpd write

(BU(1))Conn(X) 1[X,B 2U(1) conn]× 1[X,B 2U(1) conn 1] 2[X,B 2U(1) conn 1]× 2[X,B 2U(1)] 3[X,B 2U(1)] (\mathbf{B}U(1))\mathbf{Conn}(X) \coloneqq \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] \underset{\sharp_1 [X, \mathbf{B}^2 U(1)_{conn^1}]}{\times} \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^1}] \underset{\sharp_2 [X, \mathbf{B}^2 U(1)]}{\times} \sharp_3 [X, \mathbf{B}^2 U(1)]

for the homotopy limit in the diagram

(BU(1))Conn(X) 3[X,B 2U(1)] 2[X,B 2U(1) conn 1] 2[X,B 2U(1)] 1[X,B 2U(1) conn] 1[X,B 2U(1) conn 1]. \array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& &\to& \sharp_3 [X, \mathbf{B}^2 U(1)] \\ \downarrow && && \downarrow \\ && \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^1}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\stackrel{}{\to}& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^1}] } \,.

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 XX.

Proposition

The smooth 2-groupoid (BU(1))Conn(X)(\mathbf{B}U(1))\mathbf{Conn}(X) of def. 23 is indeed the smooth moduli object of circle-principal 2-connections on XX; in that its UU-plots of are smoothly UU-parameterized collections of smooth circle-principal 2-connections on XX and its morphisms of UU-plots are smoothly UU-parameterized collections of smooth gauge transformation between these, on XX, and similarly for its 2-morphisms.

Proof (sketch)

By a variant of the pasting law one sees that the defining homotopy limit may be computed as the pasting of homotopy pullbacks

(BU(1))Conn(X) A 3[X,B 2U(1)] B 2[X,B 2U(1) conn 1] 2[X,B 2U(1)] 1[X,B 2U(1) conn] 1[X,B 2U(1) conn 1]. \array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& A &\to& \sharp_3 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow && \downarrow \\ B &\to & \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^1}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\stackrel{}{\to}& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^1}] } \,.

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, 1[X,B 2U(1) conn]\sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] has over UCartSpU \in CartSp smoothly UU-parameterized collections of 2-connections on XX as objects, but discretely Γ(U)\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 2[X,B 2U(1) conn 1]\sharp_2 [X,\mathbf{B}^2 U(1)_{conn^1}] the 1-morphisms over UU are gauge transformations of such truncated 2-connections over U×XU \times X, these are now not forced to strictly fix the curvature along UU, hence these are smoothly UU-parameterized collections of gauge transformation on XX (between truncated 2-connections). So the pullback object BB combines these two aspects and hence has as objects and as 1-morphisms now smoothly UU-parameterized collections of connections and gauge transformations on XX, respectively, and only the 2-morphisms of BB keep beeing discretely Γ(U)\Gamma(U)-parameterized collections of 2-gauge transformations.

Similarly then, AA is already the correct smooth moduli 2-stack, but of truncated connections, and so finally the fiber product of AA with BB 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 (BU(1))Conn(X)(\mathbf{B}U(1))\mathbf{Conn}(X) have the correct smooth collections of structures in each degree.

Higher holonomy

exp(2πi Σ()):[Σ,B nU(1) conn]concτ 0U(1) \exp(2 \pi i \int_{\Sigma}(-)) \colon [\Sigma,\mathbf{B}^n U(1)_{conn}] \stackrel{conc \circ \tau_0}{\to} U(1)

Syntactic Layer

The dependent curvature type

The universal curvature characteristic, def. 19, has the syntax

curv G:BG dRB 2G. \vdash curv_{G} \colon \mathbf{B}G \to \flat_{dR} \mathbf{B}^2 G \,.

Regarded as a dependent type in the de Rham coefficient context this is

ω: dRB 2Gc:BG(curv G(c)ω):Type \omega \colon \flat_{dR}\mathbf{B}^2 G \; \vdash \; \underset{\mathbf{c} \colon \mathbf{B}G}{\sum} \left( curv_G\left(\mathbf{c}\right) \simeq \omega \right) \colon Type

Therefore the syntax for a domain object F:X dRB 2GF \colon X \to \flat_{dR} \mathbf{B}^2 G in this context is

ω: dRB 2Gx:X(F xω):Type \omega \colon \flat_{dR} \mathbf{B}^2 G \;\vdash\; \underset{x \colon X}{\sum} \left( F_x \simeq \omega \right) \colon Type

and the syntax for a cocycle

X P¯ BG F curv G dRB 2G \array{ X &&\stackrel{\bar P}{\to}&& \mathbf{B}G \\ & {}_{\mathllap{F}}\searrow &\swArrow_{\nabla}& \swarrow_{\mathrlap{curv_G}} \\ && \flat_{dR} \mathbf{B}^2 G }

in differential cohomology, def. 20, on (X,F)(X,F) is hence

(P¯,):ω: dRB 2G((x:X(F xω))(c:BG(curv G(c)ω))) \vdash \; (\bar P,\nabla) \colon \underset{\omega \colon \flat_{dR} \mathbf{B}^2 G}{\prod} \left( \left( \underset{x \colon X}{\sum} \left( F_x \simeq \omega \right) \right) \to \left( \underset{\mathbf{c} \colon \mathbf{B}G}{\sum} \left( curv_G(\mathbf{c}) \simeq \omega \right) \right) \right)

Fixed curvature twists

(B n𝔾:Type) conn: Type c:B𝔾 ω:Ω cl n+1(curv(c)=ω) \begin{aligned} (\mathbf{B}^n \mathbb{G} \colon Type)_{conn} \colon & Type \\ \coloneqq & \sum_{\mathbf{c} \colon \mathbf{B}\mathbb{G}} \sum_{\omega \colon \Omega^{n+1}_{cl}} \left( curv(\mathbf{c}) = \omega \right) \end{aligned}

Characteristic classes

Model Layer

Magnetic charge and first Chern class

B() B 2 B B BU(1) \array{ && && \mathbf{B}(\mathbb{R}\sslash \mathbb{Z}) &\to& \mathbf{B}^2 \mathbb{Z} \\ && && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}\mathbb{Z} &\to& \mathbf{B}\mathbb{R} &\to& \mathbf{B}U(1) }
c 1:π 0H(X,BU(1))π 0H(X,B 2)H 2(X,) \mathbf{c}_1 : \pi_0\mathbf{H}(X,\mathbf{B}U(1)) \to \pi_0 \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}) \simeq H^2(X, \mathbb{Z})

Yang-Mills instanton number and second Chern class

Semantic Layer

Gauge theory Lagrangeans

a differential characteristic class

L:BG connB nU(1) conn \mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}

is an (extended) Lagrangean for infinity-Chern-Simons theory.

The corresponding action functional is discussed in Semantic Layer - Action functionals from Lagrangeans.

Syntactic Layer

(…)

Integration

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 ” Σω\int_{\Sigma} \omega” for the integral of a differential form ω\omega over a space Σ\Sigma, as a variant of the notation ” Sf\sum_{S} f” for the sum of values of a function on a set SS. For the case of integrals of connections the corresponding parallel transport expression is often denoted by an exponentiated integral sign ”𝒫exp( Σω)\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 Σω\int_\Sigma \omega of a differential form ω\omega (or more generally of a connection) is an invariant under those gauge transformations of ω\omega that are trivial on the boundary of Σ\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 XX a connected manifold and for GG a Lie group, the operation of sending a flat GG-principal connection \nabla to its parallel transport γhol γ()\gamma \mapsto hol_{\gamma}(\nabla) around loops γ:S 1X\gamma\colon S^1 \to X, hence to the integral of the connection around all possible loops (its holonomy), for any fixed basepoint

hol𝒫exp( ()()):H conn,flat 1(X,G)Hom Grp(π 1(X),G)/G hol \coloneqq \mathcal{P} \exp(\int_{(-)} (-)) \;\colon\; H^1_{conn, flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X) , G)/G

exhibits a bijection between gauge equivalence classes of connections and group homomorphisms from the fundamental group π 1(X)\pi_1(X) of XX to the gauge group GG (modulo adjoint GG-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 𝒫exp( ()())\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 GG-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 H conn 1(X,G)\nabla \in H^1_{conn}(X,G) any non-flat connection and γ:S 1X\gamma \colon S^1 \to X a trajectory in XX, we may form the pullback of \nabla to S 1S^1. There it becomes a necessarily flat connection γ *H conn,flat 1(S 1,G)\gamma^* \nabla \in H^1_{conn, flat}(S^1,G), since the curvature differential 2-form necessarily vanishes on the 1-dimensional manifold S 1S^1. Accordingly, by the above bijection, forming the gauge equivalence class of γ *\gamma^* \nabla means to find a group homomorphism

π 1(S 1)G \mathbb{Z} \simeq \pi_1(S^1) \to G

modulo conjugation (modulo nothing if GG is abelian, such as G=U(1)G = U(1)) and since \mathbb{Z} is the free group on a single generator this is the same as finding an element

hol γ()=𝒫exp( γγ *)G. hol_\gamma(\nabla) = \mathcal{P} \exp(\int_\gamma \gamma^*\nabla) \in G \,.

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 XX to a 0-form connection on the loop space [S 1,X][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}).

Model Layer

Integration

Integration over a coordinate patch

For nn \in \mathbb{N} let

C n{x n i(0x i1)} n C^n \coloneqq \{ \vec x \in \mathbb{R}^n | \forall_i (0 \leq x_i \leq 1) \} \hookrightarrow \mathbb{R}^n

be the standard unit cube.

Let

ωΩ n( n) \omega \in \Omega^n(\mathbb{R}^n)

be a differential n-form.

ω=fdx 1dx 2dx n. \omega = f \mathbf{d} x^1 \wedge \mathbf{d} x^2 \wedge \cdots \wedge \mathbf{d} x^n \,.

Let Partitions(C k)Partitions(C^k) be the poset whose elements are partitions of the unit nncube C nC^n into N nN^n subcubes, for NN \in \mathbb{N}, ordered by inclusion.

Let

()ω:Partitions(C k) \sum_{(-)} \omega \colon Partitions(C^k) \to \mathbb{R}

be the function that sends

1N n x 1=0 N x 2=0 N k n=0 Nf(x 1,,x n). \frac{1}{N^n} \sum_{x^1 = 0}^N \sum_{x^2 = 0}^N \cdots \sum_{k^n = 0}^N f( x^1, \cdots, x^n ) \,.

Then

C kω:lim N Nω. \int_{C^k} \omega \colon \lim_{N} \sum_N \omega \,.
Integration of differential forms over a manifold

Let Σ\Sigma be a closed oriented smooth manifold of dimension kk

Definition

For nn \in \mathbb{N}, nkn \geq k, define the morphism of smooth spaces

Σ:[Σ,Ω n]Ω nk \int_{\Sigma} \colon [\Sigma, \Omega^n] \to \Omega^{n-k}

by declaring that over a coordinate chart UU \in CartSp it is the ordinary integration of differential forms over smooth manifolds

Σ,U:Ω n(Σ×U)Ω nk(U). \int_{\Sigma, U} : \Omega^n(\Sigma\times U) \to \Omega^{n-k}(U) \,.
Integration in ordinary differential cohomology

Holonomy

Parallel transport

given AΩ 1(Δ 1,𝔤)A \in \Omega^1(\Delta^1, \mathfrak{g})

we say fC (Δ 1,G)f \in C^\infty(\Delta^1, G) is the parallel transport of AA if

  1. f(0)=1f(0) = 1

  2. ff satisfies the differential equation

    df=Af \mathbf{d}f = A f

where on the right we have the differential of the left action of the group on itself.

In this case one writes

𝒫exp( Δ 1A)f(1) \mathcal{P} \exp\left(\int_{\Delta^1} A \right) \coloneqq f(1)

and calls it the path ordered integral? of AA. Here the enire left hand side is primitive notation.

In the case that G=U(1)G = U(1) this reproduces the ordinary integral

(G=)Righarrow𝒫exp( Δ 1A)=exp(i Δ 1A)U(1) \left(G = \mathbb{R}\right) \Righarrow \mathcal{P} \exp(\int_{\Delta^1} A) = \exp(i \int_{\Delta^1} A) \in U(1)

There is another way to express this parallel transport, related to Lie integration:

Define an equivalence relation on Ω 1(Δ 1,𝔤)\Omega^1(\Delta^1, \mathfrak{g}) as follows: two 1-forms A,AA,A' are taken to be equivalent if there is a flat 1-form A^Ω flat 1(D 2,𝔤)\hat A \in \Omega^1_{flat}(D^2, \mathfrak{g}) on the 2-disk such that its restriction to the upper semicircle is AA and the restriction to the lower semicircle is A˜\tilde A.

If GG is simply connected, then the equivalence classes of this relation form

Ω 1(Δ 1,𝔤) /G \Omega^1(\Delta^1,\mathfrak{g})_{/\sim} \simeq G

and the quotient map coincides with the parallel transport

𝒫exp( Δ 1()):Ω 1(Δ 1,𝔤)Ω 1(Δ 1,𝔤) /G \mathcal{P} \exp\left(\int_{\Delta^1} \left(-\right)\right) \colon \Omega^1(\Delta^1, \mathfrak{g}) \to \Omega^1(\Delta^1, \mathfrak{g})_{/\sim} \simeq G

Finally yet another perspective is this: consider the equivalence relation on Ω 1(Δ 1,𝔤)\Omega^1(\Delta^1, \mathfrak{g}) where two 1-forms are regarded as equivalent if there is a gauge transformation λC (Δ 1,G)\lambda \in C^\infty(\Delta^1, G) with λ(0)=e\lambda(0) = e and λ(1)=e\lambda(1) = e, then again

𝒫exp( Δ 1()):Ω 1(Δ 1,𝔤)Ω 1(Δ 1,𝔤) /G \mathcal{P} \exp\left(\int_{\Delta^1} \left(-\right)\right) \colon \Omega^1(\Delta^1, \mathfrak{g}) \to \Omega^1(\Delta^1, \mathfrak{g})_{/\sim} \simeq G

is the parallel transport

Holonomy of a flat principal connection

if XX is connected then forming the holonomy of flat GG-connections

hol:GBund ,flat(X)Hom Grp(π 1(X),G) hol \colon G Bund_{\nabla, flat}(X) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X), G)

is an equivalence, π 1(X)\pi_1(X) the fundamental group. If XX is not connected then

hol:GBund ,flat(X)Hom Grpd(Π 1(X),BG) hol \colon G Bund_{\nabla, flat}(X) \stackrel{\simeq}{\to} Hom_{Grpd}(\Pi_1(X), \mathbf{B}G)

is an equivalence.

Transgression

What is called transgression is the combination of

  1. passing a cocycle on some space XX with coefficients in some AA to a cocycle on a mapping space [Σ,X][\Sigma,X] with coefficients in [Σ,A][\Sigma,A] and

  2. integrating the resulting coefficient over Σ\Sigma to obtain a BB-valued cocycle on the mapping space, where BB is some recipient of an integration map of AA-cocycles over Σ\Sigma.

Transgression of differential forms

Let Σ k\Sigma_k be a closed smooth manifold of dimension kk.

Definition

For XHX \in \mathbf{H}, the transgression of differential forms on XX to the mapping space [Σ,X][\Sigma,X] is the morphism

Σ[Σ,]:Ω n(X)Ω nk([Σ,X]) \int_\Sigma [\Sigma,-] : \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])

given on a differential form

(XωΩ n)Ω n(X) (X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X)

as the composition of the mapping space operation with the integration of differential forms, def. 24:

Σ[Σ,ω]:[Σ,X][Σ,ω][Σ,Ω n] ΣΩ nk. \int_{\Sigma} [\Sigma,\omega] \;\colon\; [\Sigma, X] \stackrel{[\Sigma, \omega]}{\to} [\Sigma, \Omega^n] \stackrel{\int_{\Sigma}}{\to} \Omega^{n-k} \,.

We discuss some examples and applications:

Gauge coupling action functional of charged particle

Let XHX \in \mathbf{H} and consider a circle group-principal connection :XBU(1) conn\nabla \colon X \to \mathbf{B}U(1)_{conn} over XX. By the discussion in Dirac charge quantization and the electromagnetic field above this encodes an elecrtromagnetic field? on XX. 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

=A:XΩ 1, \nabla = A \colon X \to \Omega^1 \,,

the electromagnetic potential.

Let then Σ=S 1\Sigma = S^1 be the circle. The transgression of the electromagnetic potential to the loop space of XX

S 1[S 1,A]:[S 1,X][S 1,A][S 1,Ω 1] S 1Ω 0 \int_{S^1} [S^1, A] \;\colon\; [S^1, X] \stackrel{[S^1, A]}{\to} [S^1 , \Omega^1] \stackrel{\int_{S^1}}{\to} \Omega^0 \simeq \mathbb{R}

is the action functional for an electron or other electrically charged particle in the background gauge field AA is S em= S 1[S 1,A]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.

Transgression of Killing form to symplectic form of Chern-Simons theory

Let 𝔤\mathfrak{g} be a Lie algebra with binary invariant polynomial ,:𝔤𝔤\langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}.

For instance 𝔤\mathfrak{g} could be a semisimple Lie algebra and ,\langle -,-\rangle its Killing form. In particular if 𝔤=𝔰𝔲(n)\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

F ()F ():Ω 1(,𝔤)F ()Ω 2(,𝔤)()()Ω 4(,𝔤𝔤),Ω 4 \langle F_{(-)} \wedge F_{(-)} \rangle \colon \Omega^1(-,\mathfrak{g}) \stackrel{F_{(-)}}{\to} \Omega^2(-, \mathfrak{g}) \stackrel{(-)\wedge (-)}{\to} \Omega^4(-, \mathfrak{g}\otimes \mathfrak{g}) \stackrel{\langle-,-\rangle}{\to} \Omega^4

Over a coordinate patch UU \in CartSp this sends a differential 1-form AΩ 1(U)A \in \Omega^1(U) to the differential 4-form

F AF AΩ 4(U). \langle F_A \wedge F_A \rangle \in \Omega^4(U) \,.

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

F ()F ():BG connΩ 4. \langle F_{(-)} \wedge F_{(-)}\rangle \colon \mathbf{B}G_{conn} \to \Omega^4 \,.

Now let Σ\Sigma be an oriented closed smooth manifold. The transgression of the above 4-form to the mapping space out of Σ\Sigma yields the 2-form

ω ΣF ()F ():Ω 1(Σ,𝔤)[Σ,BG conn][Σ,F ()F ()][Σ,Ω 4] ΣΩ 2 \omega \coloneqq \int_{\Sigma} \langle F_{(-)}\wedge F_{(-)}\rangle \colon \mathbf{\Omega}^1(\Sigma,\mathfrak{g}) \hookrightarrow [\Sigma, \mathbf{B}G_{conn}] \stackrel{[\Sigma, \langle F_{(-)}\wedge F_{(-)}\rangle]}{\to} [\Sigma, \Omega^4] \stackrel{\int_{\Sigma}}{\to} \Omega^2

to the moduli stack of Lie algebra valued 1-forms on Σ\Sigma.

Over a coordinate chart U= nU = \mathbb{R}^n \in CartSp an element AΩ 1(Σ,𝔤)( n)A \in \mathbf{\Omega}^1(\Sigma,\mathfrak{g})(\mathbb{R}^n) is a 𝔤\mathfrak{g}-valued 1-form AA on Σ×U\Sigma \times U with no leg along UU. Its curvature 2-form therefore decomposes as

F A=F A Σ+δA, F_A = F_A^{\Sigma} + \delta A \,,

where F A ΣF_A^{\Sigma} is the curvature component with all legs along Σ\Sigma and where

δA i=1 nx iAdx i \delta A \coloneqq - \sum_{i = 1}^n \frac{\partial}{\partial x^i} A \wedge \mathbf{d}x^i

is the variational derivative of AA.

This means that in the 4-form

F AF A=F A ΣF A Σ+2F A ΣδA+δAδAΩ 4(Σ×U) \langle F_A \wedge F_A\rangle = \langle F_A^\Sigma \wedge F_A^\Sigma \rangle + 2 \langle F_A^\Sigma \wedge \delta A\rangle + \langle \delta A \wedge \delta A\rangle \in \Omega^4(\Sigma \times U)

only the last term gives a 2-form contribution on UU. Hence we find that the transgressed 2-form is

ω= ΣδAδA:Ω 1(Σ,𝔤)Ω 2. \omega = \int_\Sigma \langle \delta A \wedge \delta A\rangle \colon \mathbf{\Omega}^1(\Sigma, \mathfrak{g}) \to \Omega^2 \,.

When restricted further to flat forms

Ω 1 flat(Σ,𝔤)Ω 1(Σ,𝔤) \mathbf{\Omega^1}_{flat}(\Sigma,\mathfrak{g}) \hookrightarrow \mathbf{\Omega^1}(\Sigma,\mathfrak{g})

which is the phase space of 𝔤\mathfrak{g}-Chern-Simons theory, then this is the corresponding symplectic form (by the discussion at Chern-Simons theory – covariant phase spaceheory#CovariantPhaseSpace)).

Transgression of circle nn-bundles with connection
Action functionals from transgression

(…)

Lie integration

Semantic Layer

Integration and higher holonomy

integration/higher holonomy is

exp(2πi Σ()):[Σ n,B nU(1) conn]Concτ 0[Σ,B nU(1)]U(1) \exp(2 \pi i \int_{\Sigma}(-)) \colon [\Sigma_n, \mathbf{B}^n U(1)_{conn}] \to Conc \tau_0 [\Sigma, \mathbf{B}^n U(1)] \simeq U(1)

Transgression

transgression is

[Σ k,B nU(1) conn]exp(2πi Σ k)()B nkU(1) conn [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_k}) (-) }{\to} \mathbf{B}^{n-k} U(1)_{conn}

Action functionals from Lagrangeans

and higher Chern-Simons action functionals induced from

L:BG connB nU(1) conn \mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}

are

exp(iS())exp(2π Σ k[Σ k,L]):[Σ k,BG]stackrle[Σ k,L][Σ k,B nU(1)]exp(2πi())B nkU(1) conn \exp\left(i S\left(-\right)\right) \coloneqq \exp(2 \pi \in \int_{\Sigma_k} [\Sigma_k, \mathbf{L}] ) \colon [\Sigma_k, \mathbf{B}G] \stackrle{[\Sigma_k, \mathbf{L}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)] \stackrel{\exp(2 \pi i\left(-\right))}{\to} \mathbf{B}^{n-k} U(1)_{conn}

here L\mathbf{L} is the Lagrangean.

Syntactic Layer

(…)

Super-geometry

Model Layer

The premise in The continuum real world line is now refined to

Premise. The abstract worldline of a fermionic particle is a 2\mathbb{Z}_2-graded formal neighbourhood? 1n\mathbb{R}^{1|n} of the real line, for some nn \in \mathbb{N}.

For n=0n = 0 this is again the real line 10=\mathbb{R}^{1|0} = \mathbb{R}.

Fermion

Semantic Layer

Super cohesion

Syntactic Layer

(…)

Physics in Higher Geometry: Motivation and Survey

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.

Fields

This chapter is at fields (physics)?.

Lagrangians and Action functionals

Model layer

Semantics layer

Syntactic layer

(…)

Equations of motion

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.

Contents:
  1. Model layer

    Here we discuss the traditional theory of covariant phase spaces and the traditional model of their resolution in higher geometry: BV-BRST formalism.

  2. Semantics layer

    Here we give a general abstract formulation of higher (“derived”) critical loci in a cohesive (∞,1)-topos.

  3. Syntax layer

Model layer

Traditional covariant phase space

We had already discussed traditional Variational calculus above. Using this we find:

Higher geometric covariant phase space: BV-BRST formalism

Semantics layer

Contents:
  1. Critical loci

Critical loci

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 H\mathbf{H} be a cohesive (∞,1)-topos (Π):HH(\mathbf{\Pi} \dashv \flat \dashv \sharp) : \mathbf{H} \to \mathbf{H} equipped with differential cohesion (RedΠ inf inf):HH(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

FieldsH \mathbf{Fields} \in \mathbf{H}

an object that serves as the moduli ∞-stack of physical fields for the theory to be considered, as discussed in Fields above

Definition

For ΣMfd bdrH\Sigma \in Mfd_{bdr} \hookrightarrow \mathbf{H} a manifold with boundary in H\mathbf{H}, def. 5, write [Σ,Fields] ΣH[\Sigma,\mathbf{Fields}]_{\partial \Sigma} \in \mathbf{H} for the (∞,1)-pullback

[X,Fields] Σ [Σ,Fields] [Σ,Fields] [Σ,Fields], \array{ [X, \mathbf{Fields}]_{\partial \Sigma} &\to& \flat [\partial \Sigma, \mathbf{Fields}] \\ \downarrow && \downarrow \\ [\Sigma, \mathbf{Fields}] &\to& [\partial \Sigma, \mathbf{Fields}] } \,,

where the right vertical morphism is the counit of \flat and where the bottom morphism is the image of the boundary inclusion ΣΣ\partial \Sigma \to \Sigma under the internal hom [,Fields][-, \mathbf{Fields}].

Rermark

This implies that for any geometrically contractible UHU \in \mathbf{H}, def. \ref{ShapeTerminology}, then we have

H(U,[Σ,Fields Σ])Fields(Σ×U)×Fields((Σ)×U)Field(Σ). \mathbf{H}(U, [\Sigma, \mathbf{Fields}_{\partial \Sigma}]) \simeq \mathbf{Fields}(\Sigma \times U) \underset{\mathbf{Fields}((\partial \Sigma) \times U)}{\times} \mathbf{Field}(\partial \Sigma) \,.

This means that a variation in [Σ,Fields] Σ[\Sigma, \mathbf{Fields}]_{\partial \Sigma} is a variation in [Σ,Fields][\Sigma, \mathbf{Fields}] which remains constant over the boundary of Σ\Sigma.

Fix now

𝔾Grp(H) \mathbb{G} \in Grp(\mathbf{H})

a group object, hence a cohesive ∞-group, to be the object that the action functional is to take values in. In H=\mathbf{H} = Smooth∞Grpd the standard choice is 𝔾=U(1)\mathbb{G} = U(1), the circle group, for “exponentiated action functionals” or =\mathbb{R} = \mathbb{R}, the additive Lie group of real numbers.

Definition

For S:[Σ,Fields]𝔾S \;\colon\; [\Sigma, \mathbf{Fields}] \to \mathbb{G} a map, the variational derivative of SS is the restriction of the de Rham differential S 1dSS^{-1}\mathbf{d}S of def. 12 to variations that keep the boundary data fixed as in def. 26, hence the composite

S 1d varS:[Σ,Fields] Σ[Σ,Fields]S𝔾θ 𝔾 dRB𝔾. S^{-1}\mathbf{d}_{var} S \;\colon\; [\Sigma, \mathbf{Fields}]_{\partial \Sigma} \to [\Sigma, \mathbf{Fields}] \stackrel{S}{\to} \mathbb{G} \stackrel{\theta_{\mathbb{G}}}{\to} \flat_{dR}\mathbf{B}\mathbb{G} \,.

Since the variational context is clear from the domain of the map, we will often just write S 1dSS^{-1} \mathbf{d} S for S 1d varSS^{-1} \mathbf{d}_{var} S, for convenience.

Remark

Under coreflection into structure sheaves, def. 3, this induces a map

S 1d varS:[Σ,Fields] Σ𝒪 Σ( dRBG) S^{-1}\mathbf{d}_{var} S \;\colon\; [\Sigma, \mathbf{Fields}]_{\partial \Sigma} \to \mathcal{O}_{\Sigma}(\flat_{dR} \mathbf{B}G)

in Sh H(X)Sh_{\mathbf{H}}(X), which we will denote by the same symbol, as here, when the context is clear. Since 𝒪 X( dRB𝔾)\mathcal{O}_X(\flat_{dR} \mathbf{B}\mathbb{G}) has the interpretation of the sheaf of flat Lie(𝔾)Lie(\mathbb{G})-valued forms on XX, this may be thought of as realizing dS\mathbf{d} S as a section of the tangent bundle over XX.

Definition

For S:[Σ,Fields]𝔾S \;\colon\; [\Sigma, \mathbf{Fields}] \to \mathbb{G} a map in H\mathbf{H}, its critical locus

ϕ[Σ,Fields] Σ(S 1d varS ϕ0)H \underset{\phi \in [\Sigma, \mathbf{Fields}]_{\partial \Sigma}}{\sum} \left(S^{-1}\mathbf{d}_{var}S_{\phi} \simeq 0\right) \;\;\;\; \in \mathbf{H}

is the homotopy fiber of the variational derivative S 1dSS^{-1} \mathbf{d}S over the 0-section, hence the (∞,1)-pullback

ϕ[Σ,Fields] Σ(S 1d varS ϕ0) 0 [Σ,Fields] Σ S 1d varS 𝒪 X( dRB𝔾) \array{ \underset{\phi \in [\Sigma, \mathbf{Fields}]_{\partial \Sigma}}{\sum} \left(S^{-1}\mathbf{d}_{var}S_{\phi} \simeq 0\right) &\to& 0 \\ \downarrow && \downarrow \\ [\Sigma, \mathbf{Fields}]_{\partial \Sigma} &\stackrel{S^{-1}\mathbf{d}_{var} S}{\to}& \mathcal{O}_X(\flat_{dR}\mathbf{B}\mathbb{G}) }

in the petit (∞,1)-topos Sh H(X)Sh_{\mathbf{H}}(X).

Remark

In extended prequantum field theory we may, as discussed in Lagrangians and Action functionals, think of the action functional SS as being the prequantum 0-bundle. In this perspective the variational derivative S 1d varSS^{-1} \mathbf{d}_{var} S of def. 27 is the curvature of this 0-bundle. If 𝔾\mathbb{G} is a 1-group such as U(1)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.

Syntax layer

As the notation above suggests, the critical locus of the function S:[X,Fields]𝔾S\;\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 1dSSh H(X)S^{-1}\mathbf{d} S \in Sh_{\mathbf{H}}(X) and the 0-term in Sh H(X)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 ϕ[X,Fields](S 1dS0)\underset{\phi \in [X, \mathbf{Fields}]}{\sum} (S^{-1}\mathbf{d}S \simeq 0).

Classical mechanics via prequantized Lagrangian correspondences

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.

Local (topological) prequantum field theory

This chapter is at geometry of physics -- local prequantum field theory.

Prequantum Gauge theory and Gravity

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.

Contents
  1. Model layer

    We introduce a list of important examples of field theories in fairly tradtional terms.

  2. Semantics layer

    We study the above physical systems with the tools of of cohesive (∞,1)-topos-theory as developed in the previous semantics-layers.

  3. Syntax layer

Model layer

1d Chern-Simons theory

2d Chern-Simons theory

Nonabelian charged particle and Wilson loops

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).

The group and its Lie algebra

Throughout, let GG be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.

Write 𝔤\mathfrak{g} for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write

,:𝔤𝔤. \langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,.

Since this is non-degenerate, we may equivalently think of this as an isomorphism

𝔤𝔤 * \mathfrak{g} \simeq \mathfrak{g}^*

that identifies the vector space underlying the Lie algebra with its dual vector space 𝔤 *\mathfrak{g}^*.

The coadjoint orbit and the coset space/ flag manifold

We discuss the coadjoint orbits of GG and their relation to the coset space/flag manifolds of GG.

Write

  1. TGT \hookrightarrow G inclusion of the maximal torus of GG.

1 𝔱𝔤\mathfrak{t} \hookrightarrow \mathfrak{g} the corresponding Cartan subalgebra

In all of the following we consider an element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^*.

Definition

For λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* write

𝒪 λ𝔤 * \mathcal{O}_\lambda \hookrightarrow \mathfrak{g}^*

for its coadjoint orbit

𝒪 λ={Ad g *(λ,)𝔤 *gG}. \mathcal{O}_{\lambda} = \{ Ad_g^*(\langle\lambda,-\rangle) \in \mathfrak{g}^* | g \in G \} \,.

Write G λGG_\lambda \hookrightarrow G for the stabilizer subgroup of λ,\langle \lambda,-\rangle under the coadjoint action.

Proposition

There is an equivalence

G/G λ𝒪 λ G/G_\lambda \stackrel{\simeq}{\to} \mathcal{O}_\lambda

given by

gG λAd g *λ,. g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle \,.
Definition

An element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: G λTG_\lambda \simeq T.

Example

For generic values of λ\lambda it is regular. The element in 𝔤 *\mathfrak{g}^* farthest from regularity is λ=0\lambda = 0 for which G λ=GG_\lambda = G instead.

The symplectic form

We describe a canonical symplectic form on the coadjoint orbit/coset 𝒪 λG/G λ\mathcal{O}_\lambda \simeq G/G_\lambda.

Write θΩ 1(G,𝔤)\theta \in \Omega^1(G, \mathfrak{g}) for the Maurer-Cartan form on GG.

Definition

Write

Θ λ:=λ,θΩ 1(G) \Theta_\lambda := \langle \lambda, \theta \rangle \in \Omega^1(G)

for the 1-form obtained by pairing the value of the Maurer-Cartan form at each point with the gixed element λ𝔤 *\lambda \in \mathfrak{g}^*.

Write

ν λ:=d dRΘ λ \nu_\lambda := d_{dR} \Theta_\lambda

for its de Rham differential.

Proposition

The 2-form ν λ\nu_\lambda from def. 31

  1. satisfies

    ν λ=12λ,[θθ]. \nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle \,.
  2. it descends to a closed GG-invariant 2-form on the coset space, to be denoted by the same symbol

    ν λΩ cl 2(G/G λ) G. \nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G \,.
  3. this is non-degenerate and hence defines a symplectic form on G/G λG/G_\lambda.

The prequantum bundle

We discuss the geometric prequantization of the symplectic manifold given by the coadjoint orbit 𝒪 λ\mathcal{O}_\lambda equipped with its symplectic form ν λ\nu_\lambda of def. 13.

Assume now that GG is simply connected.

Proposition

The weight lattice Γ wt𝔱 *𝔱\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t} of the Lie group GG is isomorphic to the group of group characters

Γ wtHom LieGrp(G,U(1)) \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1))

where the identification takes α,𝔱 *\langle \alpha , -\rangle \in \mathfrak{t}^* to ρ α:TU(1)\rho_\alpha : T \to U(1) given on t=exp(ξ)t = \exp(\xi) for ξ𝔱\xi \in \mathfrak{t} by

ρ α:exp(ξ)exp(iα,ξ). \rho_\alpha : \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle) \,.
Proposition

The symplectic form ν λΩ cl 2(G/T)\nu_\lambda \in \Omega^2_{cl}(G/T) of prop. 13 is integral precisely if λ,\langle \lambda, - \rangle is in the weight lattice.

The Hamiltonian GG-action / coadjoint moment map

The group GG canonically acts on the coset space G/G λG/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,ν λ)(G/T, \nu_\lambda) of prop. 13, equivalently a momentum map exhibiting this Hamiltonian action.

Wilson loops and 1d Chern-Simons σ\sigma-models with target the coadjoint orbit

Above (…) we discussed how an irreducible unitary representation of GG is encoded by the prequantization of a coadjoint orbit (𝒪 λ,ν λ)(\mathcal{O}_\lambda, \nu_\lambda). Here we discuss how to express Wilson loops/holonomy of GG-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 1Ω 1(S 1,𝔤)A|_{S^1} \in \Omega^1(S^1, \mathfrak{g}) be a Lie algebra valued 1-form on the circle, equivalently a GG-principal connection on the circle.

For

ρ:GAut(V) \rho : G \to Aut(V)

a representation of GG, write

W S 1 R(A):=hol S 1 R(A):=Tr R(tra S 1(A)) W_{S^1}^R(A) := hol^R_{S^1}(A) := Tr_R( tra_{S^1}(A) )

for the holonomy of AA around the circle in this representation, which is the trace of its parallel transport around the circle (for any basepoint). If one thinks of AA as a background gauge field then this is alse called a Wilson loop.

Definition

Let the action functional

exp(iCS λ() A):[S 1,G/T]U(1) \exp(i CS_\lambda(-)^A) \;\colon\; [S^1, G/T] \to U(1)

be given by sending gT:S 1G/Tg T : S^1 \to G/T represented by g:S 1Gg : S^1 \to G to

exp(i S 1λ,A g), \exp(i \int_{S^1} \langle \lambda, A^g\rangle ) \,,

where

A g:=Ad g(A)+g *θ A^g := Ad_g(A) + g^* \theta

is the gauge transformation of AA under gg.

Proposition

The Wilson loop of AA over S 1S^1 in the unitarry irreducible representation RR is proportional to the path integral of the 1-dimensional sigma-model with

  1. target space the coadjoint orbit 𝒪 λG/T\mathcal{O}_\lambda \simeq G/T for λ,\langle \lambda, - \rangle the weight corresponding to RR under the Borel-Weil-Bott theorem

  2. action functional the functional of def. 32:

W S 1 R(A) [S 1,𝒪 λ]D(gT)exp(i S 1λ,A g). W_{S^1}^R(A) \propto \int_{[S^1, \mathcal{O}_\lambda]} D(g T) \exp(i \int_{S^1} \langle \lambda, A^g\rangle) \,.

See for instance (Beasley, (4.55)).

Remark

Notice that since 𝒪 λ\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.

3d Chern-Simons theory

(4k+3)(4k+3)d U(1)U(1)-Chern-Simons theory

7d Chern-Simons theory

\infty-Chern-Simons theory

String field theory

Gauge fields and gravity – Einstein-Maxwell-Yang-Mills theory
Kaluza-Klein compactification
Standard model of particle physics
Standard model of cosmology

Semantic Layer

an exposition and survey is in (FSS 13).

1d Chern-Simons theory

For some nn \in \mathbb{N} let

det:U(n)U(1) det \;\colon\; U(n) \to U(1)

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

Bdet:BU(n)BU(1) \mathbf{B}det \;\colon\; \mathbf{B}U(n) \to \mathbf{B}U(1)

Under geometric realization of cohesive infinity-groupoids this is the universal first Chern class

Bdetc 1:BU(n)BU(1)K(,2). {\vert \mathbf{B}det\vert} \simeq c_1 \;\colon\; B U(n) \to B U(1) \simeq K(\mathbb{Z},2) \,.

Moreiver this has the evident differential refinement

Bdet^:BU(n) connBU(1) conn \widehat {\mathbf{B} det} \;\colon\; \mathbf{B} U(n)_{conn} \to \mathbf{B} U(1)_{conn}

given on Lie algebra valued 1-forms by taking the trace

tr:𝔲(n)𝔲(1). tr \;\colon\; \mathfrak{u}(n) \to \mathfrak{u}(1) \,.

So we get a 1d Chern-Simons theory with Bdet^\widehat{\mathbf{B}det} as its extended Lagrangian.

Nonabelian charged particle trajectories – Wilson loops

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.

Survey

We discuss how for λ𝔤\lambda \in \mathfrak{g} a regular element, there is a canonical diagram of smooth moduli stacks of the form

𝒪 λ G/T θ Ω 1(,𝔤)//T λ, BU(1) conn J * BG conn c B 3U(1) conn, \array{ \mathcal{O}_\lambda &\stackrel{\simeq}{\to}& G/T &\stackrel{\mathbf{\theta}}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, - \rangle}{\to}& \mathbf{B} U(1)_{conn} \\ && \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\mathbf{J}}} \\ && * &\stackrel{}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn} } \,,

where

  1. J\mathbf{J} is the canonical 2-monomorphism;

  2. the left square is a homotopy pullback square, hence θ\mathbf{\theta} is the homotopy fiber of J\mathbf{J};

  3. the bottom map is the extended Lagrangian for GG-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;

  4. the top map denoted λ,\langle \lambda,- \rangle is an extended Lagrangian for a 1-dimensional Chern-Simons theory;

  5. the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit Ω λG/T\Omega_\lambda \simeq G/T.

Definitions and constructions

Write H=\mathbf{H} = Smooth∞Grpd for the cohesive (∞,1)-topos of smooth \infty-groupoids.

For the following, let λ,𝔤 *\langle \lambda, - \rangle \in \mathfrak{g}^* be a regular element, def. 30, so that the stabilizer subgroup is identified with a maximal torus: G λTG_\lambda \simeq T.

As usual, write

BG connΩ 1(,𝔤)//GH \mathbf{B}G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G \in \mathbf{H}

for the moduli stack of GG-principal connections.

Definition

Write

J:=(Ω 1(,𝔤)//TΩ 1(,𝔤)//GBG conn)H Δ 1 \mathbf{J} := ( \Omega^1(-,\mathfrak{g})//T \to \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn} ) \in \mathbf{H}^{\Delta^1}

for the canonical map, as indicated.

Remark

The map J\mathbf{J} is the differential refinement of the delooping BTBG\mathbf{B}T \to \mathbf{B}G of the defining inclusion. By the general discussion at coset space we have a homotopy fiber sequence

𝒪 λG/T BT BG. \array{ \mathcal{O}_\lambda \simeq G/T &\to& \mathbf{B}T \\ && \downarrow \\ && \mathbf{B}G } \,.

By the discussion at ∞-action this exhibits the canonical action ρ\rho of GG on its coset space: it is the universal rho-associated bundle.

The following proposition says what happens to this statement under differential refinement

Proposition

The homotopy fiber of J\mathbf{J} in def. 33 is

θ:G/TΩ 1(,𝔤)//T \mathbf{\theta} : G/T \stackrel{}{\to} \Omega^1(-,\mathfrak{g})//T

given over a test manifold UU \in CartSp by the map

θ U:C (U,G/T)Ω 1(U,𝔤) \mathbf{\theta}_U : C^\infty(U,G/T) \to \Omega^1(U,\mathfrak{g})

which sends gg *θg \mapsto g^* \theta, where θ\theta is the Maurer-Cartan form on GG.

Proof

We compute the homotopy pullback of J\mathbf{J} along the point inclusion by the factorization lemma as discussed at homotopy pullback – Constructions.

This says that with J\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(J)hofib(\mathbf{J}) which is the limit cone in

hofib(J) Ω 1(,𝔤) (BG conn) I BG conn * BG conn. \array{ hofib(\mathbf{J}) &\to& &\to& \Omega^1(-, \mathfrak{g}) \\ \downarrow && \downarrow && \downarrow \\ && (\mathbf{B}G_{conn})^I &\to& \mathbf{B}G_{conn} \\ \downarrow && \downarrow \\ * &\stackrel{}{\to}& \mathbf{B}G_{conn} } \,.

Unwinding the definitions shows that hofib(J)hofib(\mathbf{J}) has

  1. objects over a UU \in CartSp are equivalently morphisms 0gg *θ0 \stackrel{g}{\to} g^* \theta in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G), hence equivalently elements gC (U,G)g \in C^\infty(U,G);

  2. morphisms are over UU commuting triangles

    g 1 *θ t g 2 *θ g 1 g 2 0 \array{ g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta \\ & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}} \\ && 0 }

    in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G) with tC (U,T)t \in C^\infty(U,T), hence equivalently morphisms

    g 1tg 2 g_1 \stackrel{t}{\to} g_2

    in C (U,G)//C (U,T)C^\infty(U,G)//C^\infty(U,T).

  3. The canonical map hofib(J)Ω 1(,𝔤)//Thofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T picks the top horizontal part of these commuting triangles hence equivalently sends gg to g *θg^* \theta.

Proposition

If λ,Γ wt𝔤 *\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^* is in the weight lattice, then there is a morphism of moduli stacks

λ,:Ω 1(,𝔤)//TBU(1) conn \langle \lambda, - \rangle \;\colon\; \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}

in H\mathbf{H} given over a test manifold UU \in CartSp by the functor

λ, U:Ω 1(U,𝔤)//C (U,G)Ω 1(U)//C (U,U(1)) \langle \lambda, - \rangle_U \;:\; \Omega^1(U,\mathfrak{g})//C^\infty(U,G) \to \Omega^1(U)//C^\infty(U,U(1))

which is given on objects by

Aλ,A A \mapsto \langle \lambda, A\rangle

and which maps morphisms labeled by exp(ξ)T\exp(\xi) \in T, ξC (,𝔱)\xi \in C^\infty(-,\mathfrak{t}) as

exp(ξ)exp(iλ,ξ). \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle ) \,.
Proof

That this construction defines a map *//T*//U(1)*//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 TG λT \simeq G_\lambda stabilizes λ,\langle \lambda , - \rangle under the coadjoint action, the gauge transformation law for points A:UBG connA : U \to \mathbf{B}G_{conn}, which for gC (U,G)g \in C^\infty(U,G) is

AAd gA+g *θ, A \mapsto Ad_g A + g^* \theta \,,

maps for g=exp(ξ)C (U,T)C (U,G)g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G) to the gauge transformation law in BU(1) conn\mathbf{B}U(1)_{conn}:

λ,A λ,Ad gA+λ,g *θ =λ,A+dλ,ξ \begin{aligned} \langle \lambda, A \rangle & \mapsto \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle \\ & = \langle \lambda, A \rangle + d \langle\lambda, \xi \rangle \end{aligned}
Remark

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:

λ,θ:G/TθΩ 1(,𝔤)//Tλ,BU(1) conn. \langle \lambda, \mathbf{\theta}\rangle : G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \,.
Proposition

The curvature 2-form of the circle bundle λ,θ\langle \lambda, \mathbf{\theta}\rangle from remark 12 is the symplectic form of prop. 13. Therefore λ,θ\langle \lambda, \mathbf{\theta}\rangle is a prequantization of the coadjoint orbit (𝒪 λG/T,ν λ)(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda).

Proof

The curvature 2-form is modulated by the composite

ω:G/TθΩ 1(,𝔤)//Tλ,BU(1) connF ()Ω cl 2. \omega : G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \stackrel{F_{(-)}}{\to} \Omega^2_{cl} \,.

Unwinding the above definitions and propositions, one finds that this is given over a test manifold UU \in CartSp by the map

ω U:C (G/T)Ω cl 2(U) \omega_U : C^\infty(G/T) \to \Omega^2_{cl}(U)

which sends

[g]dλ,g *θ. [g] \mapsto d \langle \lambda, g^* \theta \rangle \,.
Nonabelian charged particle trajectories – Wilson loops

Let Σ\Sigma be an oriented closed smooth manifold of dimension 3 and let

C:S 1Σ C \;\colon\; S^1 \hookrightarrow \Sigma

be a submanifold inclusion of the circle: a knot in Σ\Sigma.

Let RR be an irreducible unitary representation of GG and let λ,\langle \lambda,-\rangle be a weight corresponding to it by the Borel-Weil-Bott theorem.

Regarding the inclusion CC as an object in the arrow (∞,1)-topos H Δ 1\mathbf{H}^{\Delta^1}, say that a gauge field configuration for GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC and labeled by the representation RR is a map

ϕ:CJ \phi \;\colon\; C \to \mathbf{J}

in the arrow (∞,1)-topos H (Δ 1)\mathbf{H}^{(\Delta^1)} of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square

S 1 (A S 1) g Ω 1(,𝔤)//T C g J Σ A BG conn \array{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} }

in H\mathbf{H}. In components this is

which fixes the field on the circle defect to be (A S 1) g(A|_{S^1})^g, as indicated.

Moreover, a gauge transformation between two such fields κ:ϕϕ\kappa : \phi \Rightarrow \phi' is a GG-gauge transformation of AA and a TT-gauge transformation of A S 1A|_{S^1} such that these intertwine the component maps gg and gg'. If we keep the bulk gauge field AA fixed, then his means that two fields ϕ\phi and ϕ\phi' as above are gauge equivalent precisely if there is a function t:S 1Tt \;\colon\; S^1 \to T such that g=gtg = g' t, hence gauge equivalence classes of fields for fixed bulk gauge field AA are parameterized by their components [g]=[g][S 1,G/T][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:

  1. that of 3d Chern-Simons theory, whose extended Lagrangian is c:BG connB 3U(1) conn\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn};

  2. that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is λ,:Ω 1(,𝔤)//TBU(1) conn\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

S 1 (A S 1) g Ω 1(,𝔤)//T λ, BU(1) conn C g J Σ A BG conn c BU(1) conn \array{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, -\rangle}{\to}& \mathbf{B}U(1)_{conn} \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}U(1)_{conn} }

and then preforming the fiber integration in ordinary differential cohomology over S 1S^1 and over Σ\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.

2d CS-theory, WZW-term and Chan-Paton gauge fields

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{1,n}i \in \{1, \cdots n\}. Further analysis then shows that the lowest excitations of these (i,j)(i,j)-strings behave as the quanta of a U(n)U(n)-gauge field, the (i,j)(i,j)-excitation being the given matrix element of a U(n)U(n)-valued connection 1-form AA.

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 H=\mathbf{H} = Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids.

The BB-field as a prequantum 2-bundle

For XX a type II supergravity spacetime, the B-field is a map

B:XB 2U(1). \nabla_B \;\colon\; X \to \mathbf{B}^2 U(1) \,.

If X=GX = G is a Lie group, this is the prequantum 2-bundle of GG-Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group BU(1)\mathbf{B}U(1) on some VHV \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.

The Chan-Paton gauge field

We discuss the Chan-Paton gauge fields over D-branes in bosonic string theory and over Spin cSpin^c-D-branes in type II string theory.

We fix throughout a natural number nn \in \mathbb{N}, the rank of the Chan-Paton gauge field.

Proposition

The extension of Lie groups

U(1)U(n)PU(n) U(1) \to U(n) \to PU(n)

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

U(1)U(n)PU(n)BU(1)BU(n)BPU(n)dd nB 2U(1), U(1) \to U(n) \to PU(n) \to \mathbf{B}U(1) \to \mathbf{B}U(n) \to \mathbf{B}PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1) \,,

where for GG a Lie group BG\mathbf{B}G is its delooping Lie groupoid, hence the moduli stack of GG-principal bundles, and where similarly B 2U(1)\mathbf{B}^2 U(1) is the moduli 2-stack of circle 2-group principal 2-bundles (bundle gerbes).

Proposition

Here

dd n:BPU(n)B 2U(1) \mathbf{dd}_n \;\colon\; \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)

is a smooth refinement of the universal Dixmier-Douady class

dd n:BPU(n)K(,3) dd_n \;\colon\; B PU(n) \to K(\mathbb{Z}, 3)

in that under geometric realization of cohesive ∞-groupoids :{\vert- \vert} \colon Smooth∞Grpd \to ∞Grpd we have

dd ndd n. {\vert \mathbf{dd}_n \vert} \simeq dd_n \,.
Remark

By the discussion at ∞-action the homotopy fiber sequence in prop. 20

BU(n) BPU(n) B 2U(1) \array{ \mathbf{B} U(n) &\to& \mathbf{B} PU(n) \\ && \downarrow \\ && \mathbf{B}^2 U(1) }

in H\mathbf{H} exhibits a smooth∞-action of the circle 2-group on the moduli stack BU(n)\mathbf{B}U(n) and it exhibits an equivalence

BPU(n)(BU(n))//(BU(1)) \mathbf{B} PU(n) \simeq (\mathbf{B}U(n))//(\mathbf{B} U(1))

of the moduli stack of projective unitary bundles with the ∞-quotient of this ∞-action.

Proposition

For XHX \in \mathbf{H} a smooth manifold and c:XB 2U(1)\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1) modulating a circle 2-group-principal 2-bundle, maps

cdd n \mathbf{c} \to \mathbf{dd}_n

in the slice (∞,1)-topos H /B 2U(1)\mathbf{H}_{/\mathbf{B}^2 U(1)}, hence diagrams of the form

X BPU(n) c dd n B 2U(1) \array{ X &&\stackrel{}{\to}&& \mathbf{B} PU(n) \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) }

in H\mathbf{H} are equivalently rank-nn unitary twisted bundles on XX, with the twist being the class [c]H 3(X,)[\mathbf{c}] \in H^3(X, \mathbb{Z}).

Proposition

There is a further differential refinement

(BU(n))//(BU(1)) conn dd^ n B 2U(1) conn (BU(n))//(BU(1)) dd^ n B 2U(1), \array{ (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn} &\stackrel{\widehat \mathbf{dd}_n}{\to}& \mathbf{B}^2 U(1)_{conn} \\ \downarrow && \downarrow \\ (\mathbf{B}U(n))//(\mathbf{B}U(1)) &\stackrel{\widehat \mathbf{dd}_n}{\to}& \mathbf{B}^2 U(1) } \,,

where B 2U(1) conn\mathbf{B}^2 U(1)_{conn} is the universal moduli 2-stack of circle 2-bundles with connection (bundle gerbes with connection).

Definition

Write

((BU(n)//BU(1)) connFieldsB 2U(1) conn)H /B 2U(1) conn \left( \left(\mathbf{B}U\left(n\right)//\mathbf{B}U\left(1\right)\right)_{conn} \stackrel{\mathbf{Fields}}{\to} \mathbf{B}^2 U\left(1\right)_{conn} \right) \;\; \in \mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}

for the differential smooth universal Dixmier-Douady class of prop. 23, regarded as an object in the slice (∞,1)-topos over B 2U(1) conn\mathbf{B}^2 U(1)_{conn}.

Definition

Let

ι X:QX \iota_X \;\colon\; Q \hookrightarrow X

be an inclusion of smooth manifolds or of orbifolds, to be thought of as a D-brane worldvolume QQ inside an ambient spacetime XX.

Then a field configuration of a B-field on XX together with a compatible rank-nn Chan-Paton gauge field on the D-brane is a map

ϕ:ι XFields \phi \;\colon\; \iota_X \to \mathbf{Fields}

in the arrow (∞,1)-topos H (Δ 1)\mathbf{H}^{(\Delta^1)}, hence a diagram in H\mathbf{H} of the form

Q gauge (BU(n)//BU(1)) ι X dd^ n X B B 2U(1) conn \array{ Q &\stackrel{\nabla_{gauge}}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1)) \\ {}^{\iota_X}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{dd}_n}} \\ X &\stackrel{\nabla_B}{\to}& \mathbf{B}^2 U(1)_{conn} }

This identifies a twisted bundle with connection on the D-brane whose twist is the class in H 3(X,)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 cSpin^c D-branes. (FSS)

Remark

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.

Y X (BU(n)//BU(1)) conn B 2U(1) conn. \array{ Y &\stackrel{}{\to}& X &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn} \\ & \searrow & \downarrow & \swarrow \\ &&\mathbf{B}^2 U(1)_{conn} } \,.

On the local connection forms this acts as

AA+α. A \mapsto A + \alpha \,.
BB+dα B \mapsto B + d \alpha

This is the famous gauge transformation law known from the string theory literature.

The open string sigma-model
Remark

The D-brane inclusion Qι XXQ \stackrel{\iota_X}{\to} X is the target space for an open string with worldsheet Σι ΣΣ\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma: a field configuration of the open string sigma-model is a map

ϕ:ι Σι X \phi \;\colon\; \iota_\Sigma \to \iota_X

in H Δ 1\mathbf{H}^{\Delta^1}, hence a diagram of the form

Σ ϕ bdr Q ι Σ ι X Σ ϕ bulk X. \array{ \partial \Sigma &\stackrel{\phi_{bdr}}{\to}& Q \\ \downarrow^{\mathrlap{\iota_\Sigma}} &\swArrow& \downarrow^{\mathrlap{\iota_X}} \\ \Sigma &\stackrel{\phi_{bulk}}{\to}& X } \,.

For XX and QQ ordinary manifolds just says that a field configuration is a map ϕ bulk:ΣX\phi_{bulk} \;\colon\; \Sigma \to X subject to the constraint that it takes the boundary of Σ\Sigma to QQ. This means that this is a trajectory of an open string in XX whose endpoints are constrained to sit on the D-brane QXQ \hookrightarrow X.

If however XX 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.

Proposition

The moduli stack [ι Σ,ι X][\iota_\Sigma, \iota_X] of such field configurations is the homotopy pullback

[ι Σ,ι X] [Σ,X] [S 1,Q] [S 1,X]. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] } \,.
The anomaly-free open string coupling to the Chan-Paton gauge field
Proposition

For Σ\Sigma a smooth manifold with boundary Σ\partial \Sigma of dimension nn and for :XB nU(1) conn\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn} a circle n-bundle with connection on some XHX \in \mathbf{H}, then the transgression of \nabla to the mapping space [Σ,X][\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

[Σ,X] exp(2πi Σ) //U(1) conn [Σ,X] ρ¯ conn [Σ,X] exp(2πi Σ) BU(1) conn. \array{ [\Sigma, X] &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{[\partial \Sigma, X]}} && \downarrow^{\mathrlap{\overline{\rho}}_{conn}} \\ [\partial \Sigma, X] &\stackrel{\exp(2 \pi i \int_{\partial \Sigma})}{\to}& \mathbf{B} U(1)_{conn} } \,.
Remark

This is the higher parallel transport of the nn-connection \nabla over maps ΣX\Sigma \to X.

Proposition

The operation of forming the holonomy of a twisted unitary connection around a curve fits into a diagram in H\mathbf{H} of the form

[S 1,(BU(n))//(BU(1)) conn] hol S 1 //U(1) conn [S 1,dd^ n] ρ¯ conn [S 1,B 2U(1) conn] exp(2πi S 1) BU(1) conn. \array{ [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{[S^1, \widehat\mathbf{dd}_n]}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{\overline{\rho}_{conn}}} \\ [S^1, \mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{S^1})}{\to}& \mathbf{B}U(1)_{conn} } \,.
Remark

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(2πi S 1[S 1,dd^ n])\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 //U(1)\mathbb{C}//U(1)-valued action functionals coming from the bulk field and the boundary field

[ι Σ,ι X] [Σ,X] exp(2πi Σ[Σ, B]) //U(1) conn [S 1,Q] [S 1,X] hol S 1([S 1, gauge]) //U(1) conn. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] &\stackrel{exp(2 \pi i \int_{\Sigma}[\Sigma, \nabla_B] ) }{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] \\ \downarrow^{\mathrlap{hol_{S^1}([S^1, \nabla_{gauge}])}} \\ \mathbb{C}//U(1)_{conn} } \,.

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:

[ι Σ,ι X] [Σ,X] [Σ, B] [S 1,B 2U(1) conn] exp(2πi Σ) //U(1) conn [S 1,Q] [S 1,X] [S 1, gauge] [S 1, B] [S 1,(BU(n))//(BU(1)) conn] [S 1,dd^ n] [S 1,B 2U(1) conn] hol S 1 exp(2πi S 1()) //U(1) conn BU(1) conn. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] &\stackrel{[\Sigma, \nabla_B]}{\to}& [S^1, \mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow &\swArrow& \downarrow && && \downarrow \\ [S^1, Q] &\to& [S^1, X] \\ \downarrow^{\mathrlap{[S^1, \nabla_{gauge}]}} && & \searrow^{\mathrlap{[S^1, \nabla_B]}} & && \downarrow \\ [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] & &\stackrel{[S^1, \widehat \mathbf{dd}_n]}{\to}& & [S^1, \mathbf{B}^2 U(1)_{conn}] \\ \downarrow^{\mathrlap{hol_{S^1}}} && && & \searrow^{\mathrlap{\exp(2 \pi i \int_{S^1}(-))}} \\ \mathbb{C}//U(1)_{conn} &\to& &\to& &\to& \mathbf{B}U(1)_{conn} } \,.

Therefore the product action functional is a well-defined function

[ι Σ,ι X]exp(2πi Σ[Σ, b])hol S 1([S 1,dd^ n]) 1U(1). [\iota_\Sigma, \iota_X] \stackrel{ \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] ) \cdot hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1} }{\to} U(1) \,.

This is the Kapustin anomaly-free action functional of the open string.

3d Chern-Simons theory with Wilson loops

We discuss how an extended Lagrangian for GG-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 ϕ:CJ\phi : C \to \mathbf{J} in H (Δ 1)\mathbf{H}^{(\Delta^1)} as above is the homotopy pullback

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] [Σ,BG conn] [S 1,BG conn] \array{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] }

in H\mathbf{H}, where square brackets indicate the internal hom in H\mathbf{H}.

Postcomposing the two projections with the two transgressions of the extended Lagrangians

exp(2πi Σ[Σ,c]):[Σ,BG conn][Σ,c][Σ,B 3U(1) conn]exp(2πi Σ())U(1) \exp(2 \pi i \int_\Sigma[\Sigma, \mathbf{c}]) \;\colon\; [\Sigma, \mathbf{B}G_{conn}] \stackrel{[\Sigma, \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_\Sigma (-))}{\to} U(1)

and

exp(2πi Σ[S 1,λ,]):[S 1,Ω 1(,𝔤)//T][Σ,λ,][S 1,BU(1) conn]exp(2πi S 1())U(1) \exp(2 \pi i \int_\Sigma[S^1, \langle \lambda, -\rangle]) \;\colon\; [S^1, \Omega^1(-,\mathfrak{g})//T] \stackrel{[\Sigma, \langle \lambda , -\rangle]}{\to} [S^1, \mathbf{B} U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to} U(1)

to yield

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] exp(2πi S 1[S 1,λ,]) U(1) [Σ,BG conn] [S 1,BG conn] exp(2πi Σ 2[Σ 3,c]) U(1) \array{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] &\stackrel{\exp(2 \pi i \int_{S^1} [S^1, \langle \lambda, -\rangle] ) }{\to}& U(1) \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] \\ \downarrow^{\mathrlap{\exp(2\pi i \int_{\Sigma_2} [\Sigma_3, \mathbf{c}])}} \\ U(1) }

and then forming the product yields the action functional

exp(2πi S 1[S 1,])exp(2πi Σ[Σ,c]):Fields(S 1Σ)U(1). \exp(2 \pi i \int_{S^1}[S^1, \langle -\rangle]) \cdot \exp(2 \pi i \int_{\Sigma}[\Sigma, \mathbf{c}]) \;:\; \mathbf{Fields}(S^1 \hookrightarrow \Sigma) \to U(1) \,.

This is the action functional of 3d GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC in the representation determined by λ\lambda.

Similarly, in codimension 1 let Σ 2\Sigma_2 now be a 2-dimensional closed manifold, thought of as a slice of Σ\Sigma above, and let i*Σ 2\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

exp(2πi Σ 2[Σ,c]):[Σ 2,BG conn][Σ 2,c][Σ 2,B 3U(1) conn]exp(2πi Σ 2())BU(1) conn \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma, \mathbf{c}\right]\right) \;\colon\; \left[\Sigma_2, \mathbf{B}G_{conn}\right] \stackrel{\left[\Sigma_2, \mathbf{c}\right]}{\to} \left[\Sigma_2, \mathbf{B}^3 U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\Sigma_2} \left(-\right)\right)}{\to} \mathbf{B}U\left(1\right)_{conn}

and

exp(2πi i*[ i*,λ,]):[ i*,Ω 1(,𝔤)//T][ i*,λ,][ i*,BU(1) conn]exp(2πi i*())BU(1) conn \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \left\langle \lambda, -\right\rangle\right]\right) \;\colon\; \left[\coprod_i {*}, \Omega^1\left(-,\mathfrak{g}\right)//T\right] \stackrel{[\coprod_i {*}, \langle \lambda , -\rangle]}{\to} \left[\coprod_i {*}, \mathbf{B} U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\coprod_i {*}} \left(-\right)\right)}{\to} \mathbf{B}U(1)_{conn}

and hence a total prequantum bundle

exp(2πi i*[ i*,β,])exp(2πi Σ 2[Σ 2,c]):Fields( i*Σ)BU(1) conn. \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \langle \beta, -\rangle\right]\right) \otimes \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma_2, \mathbf{c}\right]\right) \;:\; \mathbf{Fields}\left(\coprod_i {*} \hookrightarrow \Sigma\right) \to \mathbf{B}U\left(1\right)_{conn} \,.

One checks that this is indeed the correct prequantization as considered in (Witten 98, p. 22).

Chan-Paton gauge fields on D-branes

Syntactic Layer

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Quantum mechanics

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

Model Layer

Worldvolumes and cobordisms

Spectral triple

Semantic Layer

Internal categories in an \infty-topos

Simplicial objects in an \infty-topos
Category objects in an \infty-topos

Syntactic Layer

Directed homotopy type theory

Geometric quantization

Model Layer

1-Geometric quantization

Quantum 3d Chern-Simons theory for compact simple gauge group

Higher geometric quantization

Semantics Layer

Syntactic Layer

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Application to open questions in physics

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:

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).

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References

General

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.

Mathematical quantum field theory

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

Topos theory in differential geometry and physics

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

  • William Lawvere, Toposes of laws of notion, Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)

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

  • William Lawvere, Comments on the Development of Topos Theory, Development of Mathematics 1950-2000, 715-734 (2000) Birkhäuser Basel

Modern accounts of physics in this spirit includes notably also the book (Paugam) listed above.

Higher category theory in physics

An early proposal that the action functional of nn-dimensional quantum field theory should refine to a structure involving (n-k)-vector spaces in codimension (nk)(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.

Local prequantum field theory

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

Higher geometric prequantum theory

Further details

Physical fields

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

Differential forms and parallel transport

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.

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3d Chern-Simons theory and Wilson loops

  • Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, , AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

Higher Chern-Simons theories

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

Revised on January 31, 2014 01:02:17 by Urs Schreiber (89.204.138.10)