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 approached via cohesion (“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 \ldots(We 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

ordinary languagesyntaxsemanticsmodelchapter
general abstractgeneral concreteconcrete particular
A flat connection \nabla on XX is a rule for sending paths (xγy)ΠX(x \stackrel{\gamma}{\to} y) \in \Pi X to group elements, respecting composition.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" 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 \mathbf{B}G \\ \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}{}^{\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 dictionary):

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.

Homotopy types

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

Smooth homotopy types

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

Groups

This chapter is at geometry of physics -- groups.

Principal bundles

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

Manifolds and Orbifolds

this chapter is at geometry of physics -- manifolds and orbifolds

GG-Structure and Cartan geometry

this chapter is at geometry of physics -- G-structure and Cartan geometry

Representations and Associated bundles

this chapter is at geometry of physics -- representations and associated bundles

Modules

this chapter is at geometry of physics - modules

Flat connections

this chapter is at geometry of physics -- flat connections

de Rham Coefficients

see geometry of physics -- de Rham coefficients

Principal connections

this chapter is at geometry of physics -- principal connections

Integration

this chapter is at geometry of physics -- integration

Super-geometry

this chapter is at geometry of physics -- supergeometry

Prequantum geometry

this chapter is at geometry of physics -- prequantum geometry

WZW terms

this chapter is at geometry of physics -- WZW terms

BPS charges

this chapter is at geometry of physics -- BPS charges

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. \ref{ManifoldWithBoundaryInSemLayer}, 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. \ref{DifferentiationOfInfinityGroupValuedFunction} to variations that keep the boundary data fixed as in def. 1, 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. \ref{StructureSheafInPetitTopos}, 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. 2 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. 3 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).

Hamilton-Jacobi-Lagrange mechanics via prequantized Lagrangian correspondences

This chapter is at prequantized Lagrangian correspondence.

Hamilton-de Donder-Weyl field theory via Higher correspondences

This chapter is at Local field theory via Higher correspondences.

Local (topological) prequantum field theory

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

Prequantum Gauge theory and Gravity

this chapter is at geometry of physics -- prequantum gauge theory and gravity

Quantum mechanics

This section is at geometry of physics -- quantum mechanics

Geometric quantization

Model Layer

1-Geometric quantization

Geometric quantization with KU-coefficients

this is at geometric quantization with KU-coefficients

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

Higher geometric quantization

Semantics Layer

Syntactic Layer

(…)

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).

(…)

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.

(…)

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 May 21, 2015 05:17:19 by Urs Schreiber (80.92.246.195)