geometry of physics -- physics in higher geometry

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> This entry contains one chapter of the material at _[[geometry of physics]]_.

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### Context
#### Physics
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[[!include physicscontents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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Before we discuss technical details starting in the [next chapter](#Fields) here we survey general ideas of [[theory (physics)|theories]] in fundamental [[physics]] and motivate how these are naturally formulated in terms of the [[higher geometry]] that we developed in the [first part](#GEOMETRY).

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**Contents**

1. [Ingredients of fundamental physics](#IngredientsOfFundamentalPhysics)

1. [Topological local Lagrangian gauge field theory](#TopologicalLocalLagrangianGaugeFieldTheory)

1. [Chern-Simons field theory as a toy example](#ChernSimonsTheoryAsAToyExample)

1. [Topological local field theory in higher category theory](#TopologicalLocalFieldTheoryInHigherGeometry)

1. [Intermezzo: Elements of higher cohesive geometry](#IntermezzoElementsOfHigherCohesiveGeometry)

1. [Chern-Simons theory as the archetypical example](#CSAsArchetypicalExample)

1. [Topological local Lagrangian field theory in higher geometry](#TopologcalLocalLagrangianFieldTheoryInHigherGeometry)

1. [7-dimensional Chern-Simons theory as the next example](#7dCSTheoryAsExample)

1. [∞-Chern-Simons theory as the general example](#InfinityCSAsGeneralExample)

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### Ingredients of fundamental physics
 {#IngredientsOfFundamentalPhysics}

Ever since [[Isaac Newton]], [[theories of physics]] are formulated in the language of [[mathematics]]. Modern physics is formulated in terms of modern mathematics in the most intimate way.
We aim to give here an account at least of central parts of modern physics naturally formulated in the mathematics of _[[higher geometry]]_ that we discussed in _[geometry of physics - Geometry](geometry+of+physics#GEOMETRY)_.

The physics that we are concerned with is _fundamental_ physics,
that part of the large subject of physics which is concerned with elementary  processes and phenomena of physics from which all others are thought to _emerge_ as approximations to collective effects of complex configurations of the elementary
entities.

The best [[theory (physics)|theory]] of fundamental physics as presently understood is known as _[[Einstein-Yang-Mills-Dirac-Higgs theory]]_.
While fundamental, this theory has _free parameters_. These index different flavors of the same general mechanism. Notably the species and the masses of the fundamental [[fermion]] [[particles]] are such parameters, as
is their coupling to the [[force fields]].

Specifying these parameters is called constructing a
_[[model (in theoretical physics)|model]]_ in physics  and specifically a [[phenomenology|phenomenological model]] of fundamental physics,
if aimed at making the observations that are predicted by the theory to closely match those that are being made in experiments. The global choice of these parameters that make the predictions  match all available data to the currently best possible degree is called the _standard model_, specifically the  _[[standard model of particle physics]]_ for the sector in which the large-scale description of the
force of [[gravity]] is approximately negligible, and the
_[[standard model of cosmology]]_
which focuses on that large scale structure.

The following table list some key ingredients of the standard theory of fundamental physics, all of which we discuss in detail further below.

[[!include standard model of fundamental physics - table]]

### Topological local Lagrangian gauge field theory
 {#TopologicalLocalLagrangianGaugeFieldTheory}

While the [[standard model of fundamental physics - table|standard model of fundamental physics]] is a specification of the general  framework of [[Einstein-Yang-Mills-Dirac-Higgs theory]], that theory itself is a particular realization of several deep general principles  of theoretical physics: it is

1. a **[[field (physics)|field]]** theory;
1. a **[[gauge theory|gauge]]** theory;
1. a **[[Lagrangian]]** theory;
1. a **[[local quantum field theory|local]]** theory;
1. a **[[topological field theory|topological]]** theory.

We briefly indicate what this means, a formal discussion of these notions will be given in the  main part below.

* A **[[field (physics)|field]]** theory identifies the basic configurations of a physical system with generalized functions on spacetime, such as [[tensor fields]], and more generally with [[sections]] $\phi$ of some [[fiber bundle]] over spacetime

  $$
    \array{
      && E
      \\
      & {}^{\mathllap{\phi}}{\nearrow} & \downarrow^{\mathrlap{field\;bundle}}
      \\
      X &\stackrel{id}{\to}& X
     }
   $$

  (This is the traditional formulation. Below we discuss that the concept of _field [[fiber bundle]]_ needs to be refined to that of field _[[fiber ∞-bundle]]_ in order to accuratly capture all aspects of field theory:   _[The traditional idea of field bundles and its problems](#IdeaOfFieldBundlesAndItsProblems)_.)

* A **[[gauge theory|gauge]]** theory in the broad sense has not just a [[set]] or [[smooth space]] of field configurations but has a [[groupoid]] or [[stack]]/[[smooth groupoid]] of field configurations: there are equivalences between nominally different field configurations

  $$
    \phi \stackrel{\simeq}{\to} \phi'
  $$

  called _[[gauge transformations]]_.

   A gauge theory in the original sense specifically has a groupoid of [[principal connections]] as its configurations. (Notice that gauge transformations are not in general just a redundancy of the description of the theory, but are genuine data: there are in general non-trivial gauge [[automorphisms]] of a field and the stack of gauge field configurations is not in general equivalent to a plain [[sheaf]].)

* A **[[Lagrangian]]** theory is one which is specified by a [[smooth function]] on the space of all its configurations, called the (exponentiated) _[[action functional]]_ $\exp(i S)$. This function is assumed to be the [[integration of differential forms|integral]] of a [[differential form]] $L$ on [[spacetime]] that itself is a function of the field configurations and called the [[Lagrangian]] of the theory.

  $$
    \exp(i S(\phi)) = \exp\left(i \int_X L\left(\phi\right)\left(x\right) \right)
  $$

  The [[critical locus]]

  $$
    \left\{
	 \phi \in \mathbf{Fields}(\Sigma)		|
		d_{\mathrm{var}} S_\phi \simeq 0
	  \right\}
  $$

  of the action functional is called the _[[phase space]]_ of the theory, a point of this is said to be a solution to the _[[equations of motion]]_ of the theory.

* A **[[local quantum field theory|local]]** theory has a Lagrangian form which at each point of spacetime only depends on the values of the fields at that point, and of finitely many derivatives of the fields at that point, hence a [[local action functional]]:

  $$
    \exp(i S(\phi)) = \exp\left(i \int_X L\left(\phi\left(x\right), \nabla \phi\left(x\right)\right)\right)
  $$

  This means that a _[[local Lagrangian]]_ is a horizontal form on the [[jet bundle]] of the field bundle of which the fields are sections.

* A **[[topological field theory|topological]]** theory finally is one where a field bundle $E_X \to X$ is naturally associated to every [[smooth manifold]] $X$, typically of some fixed dimension and possibly equipped with  topological [[G-structure]] such as an [[orientation]], but not equipped with any further geometric structure: all geometric structure such as [[Riemannian metrics]] on spacetime are themselves regarded as parts of the field content in a topological theory.

### Chern-Simons field theory as a toy example
 {#ChernSimonsTheoryAsAToyExample}

Most of this was fairly well understood decades ago, by the 1960s. But even with all this conceptual understanding of the structure of fundamental physics available, there were and are a lot of implications of
_[[topological field theory|topological]] [[local  Lagrangian]] [[gauge field theory]]_ which remained elusive.

In the 1980s attention turned to _other_ topological local Lagrangian gauge field theories than that governing the standard model: it turns out that there are many such  [[theories of physics]] that share crucial properties with the standard model but which  otherwise predict, if considered as phenomenological models, physics  radically different from that
which we actually observe experimentally. For instance there are respectable theories of fundamental
physics which describe physics in dimensions other than the three spatial and one time dimension which we clearly observe, theories that contain no force of gravity, others that contain only the force of gravity, theories that contain mirror partners of every species of matter or contain no matter at all, etc.

In short, it was gradually realized, and realized to be important, that, in the platonic world of mathematics and of theoretical physics, there is a whole space of field theories, in which the one underlying the standard model of observed physics is only one single point, even if regarded with its free parameters.

A crucial step forward happened when around 1989 [[Edward Witten]], in this vein,  introduced the study of what is now called _[[Chern-Simons theory]]_ a topological local Lagrangian gauge field theory which, as such, shares
some crucial principles with Einstein-Maxwell-Yang-Mills theory, while at the  same time being  radically different and most importantly:
much simpler to analyze.

The dimension of Chern-Simons theory is 3 instead of 4, but
Chern-Simons theory has a [[gauge field]] just as [[Yang-Mills theory]] does. For $G$ a [[connected topological space|connected]] [[simply connected topological space|simply connected]] [[gauge group]], this is mathematically represented by a
[[infinity-Lie algebroid-valued differential form|Lie algebra-valued differential 1-form]] $A$.

The [[Lagrangian]] of [[Chern-Simons theory]] is the [[Chern-Simons 3-form]]

$$
  L(A)
  \coloneqq
  \mathrm{CS}_3(A)
  \coloneqq
  \langle A \wedge F_A\rangle
  +
  c
  \langle A \wedge [A \wedge A]\rangle
  \,,
$$

whence the name of the theory, and hence the Chern-Simons [[action functional]] is

$$
  \exp(i S(A))
  =
   \exp\left(
     2 \pi i \int_{\Sigma_X} \mathrm{CS}_3(A)
	\right)
$$

At the same time Chern-Simons theory is still being rich in phenomena,
in fact so rich in phenomena that for instance it helped illuminate deep mathematical problems that had not otherwise been understood as much before (properties called _[[knot invariants]]_).

### Topological local field theory in higher category theory
 {#TopologicalLocalFieldTheoryInHigherGeometry}

Another pleasant effect of the introduction of the Chern-Simons "toy model"  for topological field theory was that,
due to its conceptual simplicity, its basic structure could be appreciated also by mathematicians not trained in theoretical physics, who would otherwise fail to see through all the  physics jargon involved in the available discussion of realistic models. Accordingly, immediately after the realization of Chern-Simons field theory,  the first mathematically precise axiomatizations of an $n$-dimensional _[[topological field theory]]_ was proposed, by [[Michael Atiyah]]: "[[functorial quantum field theory]]"

This axiomatization demands in particular that to each [[closed manifold]] of [[dimension]] $(n-1)$ is assigned a [[vector space]]

$$
  Z
   \;\colon\;
  \Sigma_{n-1} \mapsto Z(\Sigma_{n-1}) \in \mathrm{Vect}
$$

thought of as the _[[space of physical states]]_ of the theory on the
spatial slice $\Sigma_{n-1}$ of spacetime.

However, This original axiomatization did not capture the full _[[local quantum field theory|locality]]_ property: the spaces of states are assigned globally to a space $\Sigma_{n-1}$ and are not required to be obtainable by "integrating up local data" over $\Sigma_{n-1}$.

In order to refine this, one needs an axiomatization of field theory that assigns data also to manifolds $\Sigma_{k}$ of [[dimension]] $0 \leq k \leq n$. This should be such that the data assigned to a torus $\Sigma_{k} \times S^1$ is the [[trace]] in a suitable sense, of the data assigned to $\Sigma_{k}$ itself. It turns out that the way to capture this is to think of $Z(\Sigma_k)$ as being a vector space in [[higher category theory]]/[[directed homotopy type theory]]: an [[n-vector space|(n-k)-vector space]].

$$
  \Sigma_k \mapsto Z(\Sigma_k) \in (n-k)\mathrm{Vect}
$$

That something like this should work was hypothetized by [[Dan Freed]] (in _[[Higher Algebraic Structures and Quantization]]_), the precise formulation was later given by [[Jacob Lurie]] (in _[[On the Classification of Topological Field Theories]]_). In the literature the resulting axiomatization is called _[[extended topological field theory]]_ (with "extended" referring to extending the axioms from [[codimension]] 1 to higher codimension) or _[[multi-tiered field theory]]_ (thinking of the data in higher [[codimension]] as being higher "tiers" of the theory).

But in the [[physics]]-community, at least in the [[algebraic quantum field theory]]-community, there is also the well-established term of _[[local quantum field theory]]_ which refers broadly to what the axioms of "extended TQFT" capture. Hence we should think of this as formalizing _topological local field theory_.

### Topological local *Lagrangian* field theory in higher geometry
 {#TopologcalLocalLagrangianFieldTheoryInHigherGeometry}

The formalization of [[extended topological field theory]] thus captures the _[[local quantum field theory|local]]_ aspect of [[field (physics)|field]] [[theory (physics)|theory]]. But we saw [above](#TopologicalLocalLagrangianGaugeFieldTheory) that the theories of fundamental physics have one more crucial property: they are _[[Lagrangian]]_. Here we indicate how to formulate Lagrangian theories that are also local in the sense of [[extended field theory]].

The step of passing from a [[Lagrangian]] to the corresponding
[[quantum field theory]] is called _[[quantization]]_. There are two broad approaches to formalization of what this means, called _[[deformation quantization|algebraic deformation quantization]]_ and _[[geometric quantization]]_.  The first of these deals explicitly with producing [[algebras of quantum observables]], while the second describes explicitly how to construct the [[spaces of states]] that we considered above. Both are at least roughly dual to each other, in the spirit of [[Isbell duality|the general duality]] between [[algebra]] and [[geometry]].
For the following discussion we follow the ideas of [[geometric quantization]], which naturally fit a discussion of _[[geometry of physics]]_.
The following table surveys some keywords in this context which we will discuss in detail further below.

[[!include Isbell duality - table]]

In its traditional formulation, one considers the [[symplectic form]] $\omega$ which is naturally induced by a [[local Lagrangian]] on its (reduced) [[covariant phase space]] and says that a _[[geometric prequantization]]_ of this form is the choice of a $U(1)$-[[principal connection]] $\nabla$ on the phase space whose [[curvature]] $F_\nabla$ is $\omega$, hence a lift in

$$
  \array{
    &&& \mathbf{B}U(1)_{conn}
    \\
    {{prequantum} \atop {circle\;bundle}} && {}^{\mathllap{\nabla}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} & curvature
    \\
    &PhaseSpace
    &\stackrel{\omega}{\to}&
    \Omega^2_{cl}
    \\
    &&
    sympl.\;form
  }
  \,.
$$

A [[section]] of the [[associated bundle|associated]] [[complex line bundle]]

$$
  \array{
    && P \times_{U(1)}\mathbb{C}
    \\
    & {}^{\mathllap{\Psi}}\nearrow & \downarrow
    \\
    X &\stackrel{id}{\to}& X
  }
$$

is called a _[[wave function]]_ on phase space. One is to choose a [[polarization]] of phase space, equipping it locally with "[[canonical coordinates]]" and "[[canonical momenta]]". Then a _[[quantum state]]_ in geometric quantization is a [[wave function]] $\Psi$ which only depends on the [[canonical coordinates]].

$$
  Z(\Sigma_{n-1})
  \coloneqq
  \{wave\; functions\;of\;canonical\;coordinates\}
  \coloneqq
  \left\{
    polarized\;sections\;of\;prequantum\;line\;bundle
  \right\}
  \,.
$$

From this it is clear that in order to refine this to local (extended, multi-tiered)  [[prequantum field theory]] we are to assign _[[circle n-bundle with connection|higher line bundles]]_ in higher codimension, namely a [[prequantum n-bundle|prequantum (n-k)-bundle]] in [[codimension]] $k$.

As a first indication that this makes good sense, notice that the natural assignment to a field in codimension 0, hence to a field configuration on [[spacetime]], is its [[action functional]]. Being a $U(1)$-valued function, this is naturally thought of as a 0-bundle, the _[[prequantum 0-bundle]]_.

So we are after a picture as indicated in the following table.

| | | [[extended prequantum field theory|local/extended Lagrangian field theory]] |  | [[extended quantum field theory|local/extended quantum field theory]]  |
|--|--|--|--|--|
| closed piece of [[spacetime]] of [[dimension]] $0 \leq k \leq n$ |  [[prequantum field theory]] | [[prequantum n-bundle|prequantum (n-k)-bundle]] | [[geometric quantization]] | [[space of states|space of]] $(n-k)$-[[wave functions]] |
| [[closed manifold]] of [[dimension]] $0 \leq k \leq n$ | $\stackrel{local\;prequantum\;field\;theory}{\mapsto}$ | [[circle n-bundle with connection|circle (n-k)-bundle with connection]] on [[moduli stack]] of [[field (physics)|field configurations]] | $\stackrel{polarized\; sections}{\mapsto}$ | [[n-vector space|(n-k)-vector space of states]] |
| $\Sigma_k$ | $\mapsto$ | $\exp(2 \pi i \int_{\Sigma_k}[\Sigma_k,\mathbf{L}]) : [\Sigma_k, \mathbf{Fields}] \to \mathbf{B}^{n-k} U(1)_{conn}$ | $\mapsto$ | $Z(\Sigma_{n-k})$ |

### Intermezzo: Recalling elements of higher cohesive geometry
 {#IntermezzoElementsOfHigherCohesiveGeometry}

The idea of _[[extended prequantum field theory]]_ indicated [above](#TopologcalLocalLagrangianFieldTheoryInHigherGeometry)  is best illustrated by the example of [[Chern-Simons field theory]]. We talk about this in a moment [below](#CSAsArchetypicalExample), but since in order to do so we need now at least some basic notions from [[cohesion|cohesive]] [[higher geometry]], we very briefly recall these first, just for reference. For more discussion see at _[geometry of physics - smooth homotopy types](geometry+of+physics#SmoothnGroupoids)_  and _[geometry of physics - Maurer-Cartan forms](geometry+of+physics#MaurerCartanForms)_ and _[geometry of physics  - principal connections](geometry+of+physics#PrincipalConnections)_

Fix some [[site]] $\mathcal{C}$ of test spaces which encode some notion of  [[geometry]]. The running example to keep in mind is the site [[SmthMfd]] of [[smooth manifolds]], or equivalently its full [[dense subsite]] of [[CartSp]] on manifolds of the form $\mathbb{R}^n$.  Assume for simplicity that the site has _[[point of a topos|enough points]]_ (which [[SmthMfd]] does).

Then the [[(2,1)-category]] of [[(2,1)-sheaves]] ([[stacks]]) on $\mathcal{C}$ is the [[simplicial localization]]

$$
  Sh_{(2,1)}(\mathcal{C})
  \simeq
  L_{le} Func(\mathcal{C}^{op}, Grpd)
  \,.
$$

of that of [[groupoid]]-valued [[presheaves]] on $\mathbf{C}$ at the _local equivalences_. This means that $Sh_{(2,1)}(\mathcal{C})$ is the result of turning [[stalk]]-wise ("local") [[equivalence of groupoids|equivalence of groupoids]] into actual [[equivalences]].

For the running example of $\mathcal{C} \simeq $ [[SmthMfd]] we may think of an object in $Sh_{(2,1)}(\mathcal{C})$ as a _[[smooth groupoid]]_. Examples include [[smooth manifolds]], [[diffeological space]],  [[orbifolds]], [[Lie groupoids]]/[[differentiable stacks]], sheaves of smooth differential forms, etc.

The higher refinement of groupoids which we need are _[[∞-groupoids]]_. The collection of these is equivalently that of [[simplicial sets]] or (compactly generated weakly Hausdorff) [[topological spaces]] with the [[weak homotopy equivalences]] universally turned into actual [[homotopy equivalences]]:

$$
  \infty Grpd \simeq L_{whe} sSet \simeq L_{whe} Top
$$

The combination of these two aspects, the [[geometry]] and the [[homotopy theory]], is that of [[(∞,1)-sheaves]]/[[∞-stacks]], here we write $\mathbf{H}$ for the collection of [[presheaves]] of [[simplicial sets]] ([[simplicial presheaves]]) with the local ([[stalk]]-wise) [[weak homotopy equivalences]] universally turned into actual homotopy equivalences:

$$
  \array{
    \mathbf{H} \coloneqq
    & Sh_\infty(\mathcal{C})
    &
    \stackrel{
      \overset{\infty-stackif.}{\leftarrow}
    }{
      \hookrightarrow
    }
  &
  PSh_\infty(\mathcal{C})
  \\
  & \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}}
  \\
  & L_{lwhe} Func(\mathcal{C}^{op}, sSet)
  &
  \stackrel{
    \overset{\mathbb{L} id}{\leftarrow}
   }{
     \underset{\mathbb{R} id}{\rightarrow}
  }&
  L_{whe} Func(\mathcal{C}^{op}, sSet)
  }
  \,.
$$

This $\mathbf{H}$ is the _[[(∞,1)-topos]]_ over $\mathcal{C}$. This is the kind of context for the [[higher geometry]] encoded by $\mathcal{C}$ in which all of our constructions take place.

In order to formulate not just [[kinematics]] but also [[dynamics]] in [[physics]], we need to require that $\mathbf{H}$ has an intrinsic notion of what it means to have a _path_ or _[[trajectory]]_ inside one of its objects. This is axiomatized by the notion of  _[[cohesion]]_:

if $\mathcal{C}$ satisfies some properties that make it an an [[infinity-cohesive site]], the the [[global section geometric morphism]] of $\mathbf{H}$ extended to an [[adjoint quadruple]] of [[(∞,1)-functors]]:

$$
  \array{
   \mathbf{H}
    &
   \stackrel{\overset{\Pi}{\to}}{\stackrel{}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}}
   &
  \infty Grpd
  \\
  \uparrow^{\mathrlap{\simeq}}
  &&
  \uparrow^{\mathrlap{\simeq}}
  \\
  L_{lwhe} Func(\mathcal{C}^{op}, sSet)
  &\stackrel{\overset{hocolim_{\mathcal{C}}}{\to}}{\stackrel{\overset{const}{\leftarrow}}{\stackrel{X \mapsto X(*)}{\stackrel{\overset{}{\to}}{\underset{}{\leftarrow}}}}}&
  L_{whe} sSet
  }
$$

with some mild extra condition on $\Pi$ and $coDisc$. This is the case in particular for $\mathcal{C} = $ [[SmthMfd]] in which case we say that $\mathbf{H}$ is the [[cohesive (∞,1)-topos]] of _[[smooth ∞-groupoids]]_.

The defining [[adjoint quadruple]] of [[(∞,1)-functors]] induces an [[adjoint triple]] of [[comonad|co]][[(∞,1)-monads]]:

| [[shape modality]] | $\dashv$ | [[flat modality]] | $\dashv$ |[[sharp modality]] |
|--|--|--|--|--|
| $\mathbf{\Pi} \coloneqq Disc \circ \Pi$ | | $flat \coloneqq Disc \circ \Gamma$ |   | $\sharp \coloneqq coDisc \circ \Gamma$  |

For $G \in Grp(\mathbf{H})$ a [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$, hence a geometric [[∞-group]], we have by these axioms of [[cohesion]] a [[pasting diagram]] of [[(∞,1)-pullbacks]]

$$
  \array{
    G &\to& *
    \\
    \downarrow^{\mathrlap{\theta_G}} && \downarrow
    \\
    \flat_{dR} \mathbf{B}G
    &\to&
    \flat \mathbf{B}G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    \infty-group &\to& *
    \\
    {}^{\mathllap{MC\;form}}\downarrow && \downarrow
    \\
    deRham\, coefficients &\to& flat\;coefficients
    \\
    \downarrow && \downarrow
    \\
    * &\to& principal\,coefficients
  }
  \,.
$$

Here the [[delooping]] $\mathbf{B}G$ is the moduli of geometric $G$-[[principal ∞-bundles]],  $\flat_{dR} \mathbf{B}G$ is that of flat [[infinity-Lie algebroid-valued differential form
|L-∞ algebra valued differential forms]] and $\theta_G$ is the _[[Maurer-Cartan form]]_ on the [[smooth ∞-group]] $G$.

If $\mathbb{G}$ is "slightly abelian" in that it is equipped with the structure of a [[braided ∞-group]], then we write

$$
  curv_{\mathbb{G}}
  \;\colon\;
  \mathbf{B}\mathbb{G}
  \stackrel{\theta_{\mathbf{B}\mathbb{G}}}{\to}
  \flat_{dR}\mathbf{B}^2 \mathbb{G}
$$

for the [[Maurer-Cartan form]] of its [[delooping]] [[∞-group]] and call this the _universal [[curvature]] characteristic_.

Then _$\mathbb{G}$-[[differential cohomology]]_ is
_$curv_{\mathbf{B}\mathbb{G}}$-[[twisted cohomology]]_.
In particular, if we consider globally defined curvature forms

$$
  \Omega_{cl}(-,\mathbb{G}) \to \flat_{dR}\mathbf{B}^2\mathbb{G}
$$

then the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{B}\mathbb{G}_{conn}
    &\to&
    \Omega(-,\mathbb{G})
    \\
    \downarrow && \downarrow^{\mathrlap{F_{(-)}}}
    \\
    \mathbf{B}\mathbb{G}
    &\stackrel{curv_{\mathbf{B}\mathbb{G}}}{\to}&
    \flat_{dR}\mathbf{B}^2 \mathbb{G}
  }
$$

is the moduli for [[principal ∞-connections]].

Now for

$$
  \mathbf{c}
  \;\colon\;
  \mathbf{B}G
  \to
  \mathbf{B}\mathbb{G}
$$

a [[universal characteristic map]] we say that a _differential refinement_ is a choice of $\mathbf{B}G_{conn}$ and $\hat \mathbf{c}$ in

$$
  \array{
    \flat \mathbf{B}G
    &\stackrel{\flat \mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}
    \\
    \downarrow && \downarrow^{}
    \\
    \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}_{conn}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}
  }
  \,.
$$

These are the [[extended Lagrangians]] in our discussion of [[extended prequantum field theory]].

In particular if, in the context of [[smooth ∞-groupoids]], we consider $\mathbb{G} = \mathbf{B}^{n-1}U(1)$ the [[circle n-group]] then

$$
  \mathbf{B}\mathbb{G}_{conn}
  \simeq
  \mathbf{B}^n U(1)_{conn}
  \simeq
  DK(  U(1) \to \Omega^1 \to \cdots \to \Omega^n)
$$

is given as the [[∞-stackification]] of the image under the [[Dold-Kan correspondence]] of the _[[Deligne complex]]_ for [[ordinary differential cohomology]].

In this case there is [[fiber integration in ordinary differential cohomology]] refined to moduli $\infty$-stacks: for $\Sigma_k$ an oriented [[closed manifold]] of [[dimension]] $k$ we have a map of [[moduli ∞-stacks]]

$$
  \exp\left(
   2 \pi i \int_{\Sigma} (-)
  \right)
  \;\colon\;
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \to
  \mathbf{B}^{n-k} U(1)_{conn}
  \,.
$$

Composed with the [[internal hom]] this is _[[transgression]]_ of differential cocycles. The transgressions of an [[extended Lagrangian]] as above are (the moduli for) the higher [[prequantum n-bundles]] that we consider here.

### Chern-Simons theory as the archetypical example
 {#CSAsArchetypicalExample}

For illustration purposes, we survey how the [[Chern-Simons field theory]] considered as a toy model for topological local Lagrangian gauge field theory considered [above](#ChernSimonsTheoryAsAToyExample) has in [[higher geometry]] a [[prequantum n-bundle|prequantum (n-k)-bundle]] associated in [[codimension]] $k$.

First consider again $G$ to be a [[connected topological space|connected]] and [[simply connected topological space|simply connected]] [[simple Lie group]]. Then the [[integral cohomology]] of the [[classifying space]] of $G$ is

$$
  H^4(BG , \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

An element in here is a [[universal characteristic class]] of $G$-[[principal bundles]]. In Chern-Simons theory this is going to be what traditionally is called the [[level (Chern-Simons theory)|level]] of the theory.

We write

$$
  c : B G \to  B^3 U(1) \simeq K(\mathbb{Z},4)
$$

for a representative [[universal characteristic map]] of the corresponding degree-4 [[universal characteristic class]].

Under [[geometric realization of cohesive ∞-groupoids|geometric realization]] as just [recalled](#IntermezzoElementsOfHigherCohesiveGeometry) this has an essentially unique lift to _smooth cohomology_, hence to a morphism of [[moduli ∞-stacks]]

$$
  \mathbf{c}
  \;\colon\;
  \mathbf{B}G
  \to
  \mathbf{B}^3 U(1)
$$

which, in a precise sense, is the [[Lie integration]] of the canonical 3-[[cocycle]]

$$
  \langle -, [-,-]\rangle \;\colon\; \mathfrak{g} \to \mathbf{B}^2 \mathbb{R}
$$

in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ of $G$ in that

$$
  \mathbf{c} \simeq \tau_1 \exp(\mu)
  \,.
$$

Moreover, this smooth universal characteristic map  has a further lift to [[differential cohomology]]

$$
  \hat \mathbf{c}
  \;\colon\;
  \mathbf{B}G_{conn}
  \to
  \mathbf{B}^3 U(1)_{conn}
  \,.
$$

This modulates a [[circle n-bundle with connection|circle 3-bundle with connection]] -- the "[[Chern-Simons circle 3-bundle with connection]]" -- on the [[smooth infinity-groupoid|smooth]] [[moduli stack]] of smooth $G$-[[principal connections]].

$$
  \array{
    \mathbf{B}G_{conn}
    &\stackrel{\hat \mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)_{conn}
    \\
    \downarrow && \downarrow &&&&& \in  \mathbf{H}
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)
    \\
    && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}}
    \\
    B G &\stackrel{c}{\to}& B^3 U(1) &&&&& \in L_{whe} Top
  }
$$

One finds that the [[transgression]] of this to codimension 0 is the Chern-Simons action functional discussed above. The other transgressions are

| [[dimension]] |  | [[prequantum n-bundle|prequantum (3-k)-bundle]] |  |
|---------------|--|-------------------------------------------------|--|
| $k = 0$ | [[twisted differential string structure|differential fractional first Pontryagin class]] | $\hat \mathbf{c} \;\colon\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ |  ([FSS](#FSS))  |
| $k = 1$ | [[WZW model]] [[background gauge field|background]] [[B-field]] | $G \to [S^1, \mathbf{B}G_{conn}] \stackrel{[S^1, \hat \mathbf{c}]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to} \mathbf{B}^2 U(1)_{conn}$ |  |
| $k = 2$ | off-sheff [[prequantum bundle]] | $[\Sigma_2, \mathbf{B}G_{conn}] \stackrel{[\Sigma_2, \hat \mathbf{c}]}{\to} [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_2} (-))}{\to} \mathbf{B} U(1)_{conn}$ | |
| $k = 3$ | [[action functional]] | $[\Sigma_3, \mathbf{B}G_{conn}] \stackrel{[\Sigma_3, \hat \mathbf{c}]}{\to} [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_3} (-))}{\to} U(1)$

Here the [[WZW model|WZW]] [[B-field]] is the topological part of a 2-dimensional local Lagrangian field theory called the _[[Wess-Zumino-Witten model]]_. One says that the WZW model is related by the [[holographic principle]] with [[3d Chern-Simons theory]].

If the [[gauge group]] $G$ is not simple and simply connected, then there are other canonical [[universal characteristic classes]] in $H^4(B G, \mathbb{Z})$. If these have a differential refinement, they give rise to the corresponding Chern-Simons theory.

Notably in the case that $G = U(1)$ is the [[circle group]] (hence not [[simply connected topological space|simply connected]]) there is the [[cup product]] of the universal [[first Chern class]] with itself

$$
  c_1 \cup c_1 \;\colon\; B U(1) \to B^3 U(1)
  \,.
$$

The smooth and differential refinement of the [[first Chern class]] itself $c_1 \;\colon\; B U(1) \to B U(1) \simeq K(\mathbb{Z},2) $ is tautological: this is simply the identity

$$
  \widehat \mathbf{c}_1 \coloneqq id
  \;\colon\;
  \mathbf{B}U(1)_{conn}
  \to
  \mathbf{B}U(1)_{conn}
  \,.
$$

The nontrivial part is that, as one can see, the  [[cup product]] on [[differential cohomology]] refines to [[moduli ∞-stacks]] of [[circle n-bundle with connection]] as the [[∞-stackification]] of what is called the [[Beilinson-Deligne cup product]], and so we have

$$
  \widehat \mathbf{c}_1 \cup \widehat \mathbf{c}_1
  \;\colon\;
  \mathbf{B}U(1)_{conn}
  \to
  \mathbf{B}^3 U(1)_{conn}
  \,.
$$

This is the [[extended Lagrangian]] for abelian Chern-Simons theory. On field configuration which happens to be given by a globally defined 1-form $A$ its value is

$$
  \exp\left(
    2\pi i \int_{\Sigma_3} A \wedge d A
  \right)
  \in
  U(1)
  \,.
$$

But now not every field configuration is necessarily given by a globally defined 1-form anymore, and more generally the action functional has a more sophisticated expression.

### 7-dimensional Chern-Simons theory as the next example
 {#7dCSTheoryAsExample}

So far we have considered general motivation for [[extended prequantum field theory]], and indication how its application to the well-known case of [[3d Chern-Simons theory]] naturally captures the characteristics of the theory. Now we consider further examples whose higher geometry has not necessarily been considered traditionally.

An immediate class of further examples is obtained by abelian [[higher Chern-Simons theory]] induced by the [[cup product]] of the [[first Chern class]] with itself

$$
  [DD_{2k} \cup DD_{2k}] \in H^{4(k+1)}(B^{2k+1} U(1), \mathbb{Z})
  \,.
$$

This has a refinement ([FSS cup](#FSSCup)) to a [[differential characteristic class]] by the
_[[Beilinson-Deligne cup product]]_ of the [[differential characteristic class|differential refinement]] of the higher [[Dixmier-Douady class]] with itself:

$$
  \widehat \mathbf{DD}_{2k}
  \cup
  \widehat \mathbf{DD}_{2k}
  \;\colon\;
  \mathbf{B}^{2k+1} U(1)_{conn}
  \to
  \mathbf{B}^{4k+3} U(1)_{conn}
  \,.
$$

In particular there is the abelian [[7d Chern-Simons theory]]

$$
  \widehat \mathbf{DD}_2
  \cup
  \widehat \mathbf{DD}_2
  \;\colon\;
  \mathbf{B}^3 U(1)_{conn}
  \to
  \mathbf{B}^7 U(1)_{conn}
  \,.
$$

An argument in ([Witten 98](#AdS-CFT#Witten98)) says that this abelian
[[7d Chern-Simons theory]] (or more precisely a [[quadratic refinement]] of it) is related to the _[[self-dual higher gauge theory]]_ sector of the [[6d (2,0)-supersymmetric QFT]] on the [[worldvolume]] of a single [[M5-brane]]_  in analogy to how [[3d Chern-Simons theory]] is related to the [[WZW model]]:

| [[schreiber:∞-Chern-Simons theory]] | $\leftarrow$[[holographic principle]] $\rightarrow$| [[schreiber:∞-Wess-Zumino-Witten theory ]] | [[Kaluza-Klein reduction]] $\to$ |   |
|--|--|--|--|--|
| [[3d Chern-Simons theory]]  | [3dCS-2dCFT](holographic+principle#3dCS-2dCFT) | [[WZW model]] | | |
| [[7d Chern-Simons theory]]  | [AdS7/CFT6 duality](AdS-CFT#AdS7CFT6) | [[6d (2,0)-supersymmetric QFT]] | | [[N=4 D=4 super Yang-Mills theory]] |

The argument in ([Witten 98](#AdS-CFT#Witten98)) also says that in view of [AdS7/CFT6 duality](AdS-CFT#AdS7CFT6) the [[field (physics)|field]] of the 7d theory here is to be identified withe the _[[supergravity C-field]]_ in [[11-dimensional supergravity]]/[[M-theory]].

But the [[supergravity C-field]] is known to have moduli that are in general more complicated than just $\mathbf{B}^3 U(1)_{conn}$ (due to a [[Green-Schwarz anomaly cancellation]] mechanism in [[11-dimensional supergravity]] and a "flux quantization" condition). In ([FSS 7dCD](#FSS7dCS)) the corrected moduli are dicussed and found to involve a coupling of the abelian 7d theory with a nonabelian 7d Chern-Simons theory whose [[field (physics)|fields]] are twisted higher spin connections: "[[twisted differential string structures]]".

To see what this is, recall that [above](#CSAsArchetypicalExample) we interpreted the [[extended Lagrangian]] of [[3d Chern-Simons theory]] for a simply connected compact simple gauge group $G$ as the smooth and moreover differential refinement of the canonical [[universal characteristic class]] in $H^4(B G, \mathbb{Z})$. Specifically for $G$ the [[spin group]] this is the [[first fractional Pontryagin class]], represented by a [[characteristic map]]

$$
  \tfrac{1}{2}p_1 \;\colon\; B Spin \to B^3 U(1)
  \,.
$$

This can be understood as a higher analog of the [[second Stiefel-Whitney class]] $w_2 \;\colon\; B SO \to B^2 \mathbb{Z}$ which is the universal [[obstruction]] to [[spin structure]]. We say that $\tfrac{1}{2}p_1$ is the universal [[obstruction]] to a _[[higher spin structure]]_ called _[[string structure]]_ (this and the motivation for this terminology is discussed in detail in _[geometry of physics - Fields - Examples - Fields of gravity and G-structure](geometry+pf+physics##FieldsOfGravityAndGeneralizedGeometry)_). These higher obstructions are the (dual) [[k-invariants]] of the [[Whitehead tower]] of $B O$:

[[!include higher spin structure - table]]

From the point of view of [[extended prequantum field theory]] we are to regard each horizontal map in this tower as classifying the potential underlying [[instanton sector]] of an [[extended Lagrangian]]/[[prequantum n-bundle]], and we may ask if these lift to actual extended Lagrangian.

The next step up the ladder is the universal [[second fractional Pontryagin class]]. Its [[twisted differential string structure|differential refinement]] has been constructed in ([FSS diff coc](#FiorenzaSchreiberStasheff)).

$$
  \array{
    \mathbf{B}String_{conn}
    &\stackrel{\widehat {\tfrac{1}{6}\mathbf{p}_2}}{\to}&
    \mathbf{B}^7 U(1)_{conn}
    \\
    \downarrow && \downarrow &&&&& \in \mathbf{H}
    \\
    \mathbf{B}G
    &\stackrel{ \tfrac{1}{6}\mathbf{p}_2 }{\to}&
    \mathbf{B}^7 U(1)
    \\
    && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}}
    \\
    B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) &&&&& \in L_{whe} Top
  }
  \,.
$$

This now defines an [[extended Lagrangian]] for a  [[7d Chern-Simons theory]] on [[String 2-group]]-[[2-connections]].

Coming back to the above arguments about the relation of the 7d theory to a 6d theory, these suggest that the 6d theory involves  [[prequantum n-bundle|prequantum 6-bundle]]-analog of the [[WZW model|WZW 2-bundle]].

Indeed, the assignment of a higher Wess-Zumino-Witten-type [[prequantum n-bundle|prequantum (n-1)-bundle]] to an extended prequantum Lagrangian works very generally: the _prequantum $(n-1)$-bundle_ is the WZW-type $(n-1)$-bundle of the theory.
Its underlying $(n-1)$-bundle is simply the [[looping]] of the $n$-bundle underlying the [[extended Lagrangian]]. Specifically, the [[string 2-group]]-[[7d Chern-Simons theory]] has a WZW- prequantum 6-bundle on the [[string 2-group]] which is modulated by the [[looping]] $\Omega(\tfrac{1}{6}\mathbf{p}_2)$ of the differential characteristic map $\tfrac{1}{2}\mathbf{p}_2$

$$
  \array{
    String
    && &\stackrel{\Omega \left( \tfrac{1}{6}\mathbf{p_2}\right)}{\to}& &&
    \mathbf{B}^6 U(1)
    \\
    \downarrow
    && && && \uparrow
    \\
    String //_{Ad} String
    &
    \stackrel{\simeq}{\to}
    &
    [\mathbf{\Pi}(S^1), \mathbf{B}String]
    &
    \stackrel{[\mathbf{\Pi}(S^1), \tfrac{1}{2}\mathbf{p}_2]}{\to}
    &
    [\mathbf{\Pi}(S^1), \mathbf{B}^7 U(1)]
    &\simeq&
    \mathbf{B}^6 U(1) // \mathbf{B}^6 U(1)
  }
  \,.
$$

Here the lower factorization of this map shows canonically that this 6-bundle is [[equivariant]] with respect to the [[adjoint action|adjoint]] [[∞-action]] of the [[string 2-group]] on itself, the analog of the ordinary adjoint equivariance of the ordinary WZW 2-bundle.

Therefore in summary we find the following identifications of [[extended prequantum field theory]] with (semi-)traditional notions:

[[!include extended prequantum field theory - table]]

### $\infty$-Chern-Simons theory as the general example
 {#InfinityCSAsGeneralExample}

In the examples of [[2d Chern-Simons theory]] and [[7d Chern-Simons theory]] that we have discussed above, the fully extended Lagrangian is equivalently the [[prequantum n-bundle]] on the universal [[moduli ∞-stack]] of [[field (physics)|fields]], for $n = 3$ and $n = 7$, respectively.

But in fact this is the general situation for fully extended Lagrangians:

if $\mathbf{Fields} \in \mathbf{H}$ is the object that serves as the [[moduli ∞-stack]] of [[field (physics)|fields]] for some $n$-dimensional [[theory (physics)|theory]] in that for any [[spacetime]]/[[worldvolume]] $\Sigma_n$ of [[dimension]] $n$ the space of fields is the [[mapping stack]] $[\Sigma_n, \mathbf{Fields}]$, then the corresponding [[prequantum n-bundle]] has to be over this object $\mathbf{Fields}$ and hence to be modulated by a map of moduli $\infty$-stacks of the form

$$
  \mathbf{L} \;\colon\;
  \mathbf{Fields}
  \to
  \mathbf{B}^n U(1)_{conn}
$$

hence by a fully extended Lagrangian.

Above we considered the case that $\mathbf{Fields} \simeq \mathbf{B}G_{conn}$ for a [[Lie group]] $G$ or $\mathbf{Fields} \simeq \mathbf{B}String_{conn}$ for the [[string 2-group]]. In general for $G \in Grpd(\mathbf{H})$ a [[smooth ∞-group]] an extended Lagrangian as above
encodes an [[extended prequantum field theory|extended prequantum]] _[[higher gauge theory]] of Chern-Simons type_:

the [[curvature]] $(n+1)$-form of $\mathbf{L}$ (i.e. the [[n-plectic geometry|pre n-plectic form]] of the [[n-plectic geometry|n-plectic]] [[∞-stack]] $\mathbf{Fields}$ )

$$
  F_{\mathbf{L}}
  \;\colon\;
  \mathbf{B}G_{conn}
  \stackrel{\mathbf{L}}{\to}
  \mathbf{B}^n U(1)_{conn}
  \stackrel{F_{(-)}}{\to}
  \Omega^{n+1}_{cl}
$$

is -- by [[∞-Chern-Weil theory]] -- an [[invariant polynomial]] of the [[∞-Lie algebra]] $\mathfrak{g}$ of $G$. Accordingly, the [[action functional]]

$$
  \exp\left(
    2 \pi i \int_{\Sigma_n} [\Sigma_n, \mathbf{L}]
  \right)
  \;\colon\;
  [\Sigma_n, \mathbf{B}G_{conn}]
  \to
  U(1)
$$

locally takes an [[infinity-Lie algebroid-valued differential form
|L-∞ algebra valued differential form]] $A$ (the [[higher gauge fields]]) to a [[Lagrangian]] $CS(A)$ which locally is a differential form with the property that $ d CS(A) = F_A$.
This means that
$\exp\left( 2 \pi i \int_{\Sigma_n} [\Sigma_n, \mathbf{L}] \right)$ is a [[higher gauge theory]] _of Chern-Simons type_. We say that it is an _[[schreiber:∞-Chern-Simons theory]]_.

This remains true if $\mathbf{Fields}$ is of more general form, so that fields are [[infinity-Lie algebroid-valued differential form
|L-∞ algebroid valued differential form]].

This means that fully [[extended prequantum field theory]] is considerably more restrictive than the traditional notion of [[local Lagrangian]] field theory, just as [[extended topological quantum field theory]] is much more restrictive than non-extended [[TFT]].

1. **fewer choices in prequantization** Traditional [[geometric quantization]] involves making considerable choices. Notably the choice of [[prequantum bundle]] lifting the [[symplectic form]]. In [[extended prequantum field theory]] the prequantum bundles in codimension 1 are constrained to all arise by [[transgression]] from a [[prequantum n-bundle]] in codimension $n$. This is a strong coherence condition that drastically reduces the available choices.

   For instance for $G$ a compact Lie group, there is an [[isomorphism]]

   $$
     H^n(B G, \mathbb{Z})
     \simeq
     \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))
     \,.
   $$

   This means that if $H^n(B G, \mathbb{Z})$ has no [[torsion subgroup|torsion]], then there is _at most one_ extended prequantization of a given invariant polynomial.

1. **selecting physically natural theories** While the traditional formalization of [[local Lagrangian]] field theory does capture crucial aspects of those [[theories (physics)|theories]] that one is interested in in [[physics]], it still admits many "unnatural theories". The _generic_ [[local Lagrangian]] (in the sense of a horizontal form on the [[jet bundle]] of the [[field bundle]]) contains many higher order derivatives of fields, non-polynomial interactions etc., properties that the typical [[theory (physics)|theories]] of relevance in theoretical physics do no share.

The remaining question then is how accurately [[extended prequantum field theory]] constrains the space of possible [[local Lagrangians]]. In other words: which theories of interest can not be thought of as [[schreiber:∞-Chern-Simons theories]] or [[schreiber:∞-Wess-Zumino-Witten theory ]]?

As a hint to the answer of this question, here is a list of names examples of [[schreiber:∞-Chern-Simons theories]]. We discuss the items of the list in detail in the following.

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

    * [[Wilson loop]]

  * [[2d Chern-Simons theory]]

    * [[Poisson sigma-model]],

    * [[A-model]], [[B-model]] [[topological string]]

  * [[3d Chern-Simons theory]]

    * [[Chern-Simons theory]]

    * [[Dijkgraaf-Witten theory]]

    * [[Courant sigma-model]]

    * [[3d quantum gravity]]

  * [[4d Chern-Simons theory]]

    * [[Yetter model]]

    * [[BF-theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

    * topological [AdS7/CFT6](http://ncatlab.org/nlab/show/AdS-CFT#AdS7CFT6)-sector

    * [[topological M-theory]]

* [[schreiber:∞-Chern-Simons theory]]

  * [[∞-Dijkgraaf-Witten theory]]

  * [[higher dimensional Chern-Simons gravity]]

  * [[AKSZ sigma-model]]

  * [[string field theory]]

The last example here brings us back full circle to where we started this motivation with [above](#IngredientsOfFundamentalPhysics): by the central property of [[string theory]], the action functional for [[string field theory]] (or some [[KK-reduction]] thereof) contains [[Einstein-Yang-Mills theory]].


geometry of physics -- prequantum gauge theory and gravity

> this entry is going to contain one chapter of _[[geometry of physics]]_

***

#Contents#
* table of contents
{:toc}

## 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 (physics)|field]] [[theory (physics)|theories]] in more detail.

+-- {: bluebox }
###### Contents

1. [Model layer](#GaugeAndGravityModelLayer)

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

1. [Semantics layer](#GaugeAndGravitySemanticsLayer)

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

1. [Syntax layer](#GaugeAndGravitySyntaxLayer)

=--

### Model layer
 {#GaugeAndGravityModelLayer}

+-- {: bluebox }
###### Examples

1. [1d Chern-Simons theory](#1dCSTheory)

1. [Nonabelian charged particle and Wilson loops](#ModLayerNonabelianChargedParticle)

1. [2d Chern-Simons theory](#2dCSTheory)

1. [3d Chern-Simons theory](#3dCSTheory)

1. [(4k+3)d U(1)-Chern-Simons theory](#HigherAbelianCSTheory)

1. [7d Chern-Simons theory](#7dCSTheory)

1. [∞-Chern-Simons theory](#InfinityCSTheory)

   * [String field theory](#StringFieldTheory)

     * [Gauge fields and gravity -- Einstein-Maxwell-Yang-Mills theory](#EinsteinYangMillsTheory)

       * [Kaluza-Klein compactification](#KaluzaKleinCompactification)

       * [Standard model of particle physics](#StandardModelParticlePhyiscs)

       * [standard model of cosmology](#StandardModelCosmology)

=--

#### 1d Chern-Simons theory
 {#1dCSTheory}

* [[1d Chern-Simons theory]]

#### 2d Chern-Simons theory
 {#2dCSTheory}

* [[2d Chern-Simons theory]]

#### Nonabelian charged particle and Wilson loops
 {#ModLayerNonabelianChargedParticle}

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](#GaugeAndGravityWilsonLoops)_ 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](#Beasley)).

##### The group and its Lie algebra

Throughout, let $G$ be a [[semisimple Lie group|semisimple]] [[compact topological group|compact]] [[Lie group]]. For some considerations below we furthermore assume it to be [[simply connected topological space|simply connected]].

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

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

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

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

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

##### The coadjoint orbit and the coset space/ flag manifold

We discuss the [[coadjoint orbits]] of $G$ and their relation to the [[coset space]]/[[flag manifolds]] of $G$.

Write

1. $T \hookrightarrow G$ inclusion of the [[maximal torus]] of $G$.

1 $\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding [[Cartan subalgebra]]

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

+-- {: .num_defn }
###### Definition

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

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

for its [[coadjoint orbit]]

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

Write $G_\lambda \hookrightarrow G$ for the [[stabilizer subgroup]] of $\langle \lambda,-\rangle$ under the coadjoint action.

=--

+-- {: .num_prop}
###### Proposition

There is an equivalence

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

given by

$$
  g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle
  \,.
$$

=--

+-- {: .num_defn #RegularElement}
###### Definition

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

=--

+-- {: .num_example }
###### Example

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

=--

##### The symplectic form

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

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

+-- {: .num_defn #The2FormOnG}
###### Definition

Write

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

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

Write

$$
  \nu_\lambda := d_{dR} \Theta_\lambda
$$

for its [[de Rham differential]].

=--

+-- {: .num_prop #TheSymplecticFormOnTheCoset}
###### Proposition

The 2-form $\nu_\lambda$ from def. \ref{The2FormOnG}

1. satisfies

   $$
     \nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle
     \,.
   $$

1. it descends to a closed $G$-invariant 2-form on the [[coset space]], to be denoted by the same symbol

   $$
     \nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G
     \,.
   $$

1. this is non-degenerate and hence defines a [[symplectic form]] on $G/G_\lambda$.

=--

##### The prequantum bundle

We discuss the [[geometric prequantization]] of the [[symplectic manifold]] given by the [[coadjoint orbit]] $\mathcal{O}_\lambda$ equipped with its [[symplectic form]] $\nu_\lambda$ of def. \ref{TheSymplecticFormOnTheCoset}.

Assume now that $G$ is [[simply connected topological space|simply connected]].

+-- {: .num_prop #WeightsAndCharacters}
###### Proposition

The [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]]

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

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

$$
  \rho_\alpha : \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle)
  \,.
$$

=--

+-- {: .num_prop }

###### Proposition

The [[symplectic form]] $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. \ref{TheSymplecticFormOnTheCoset} is integral precisely if $\langle \lambda, - \rangle$ is in the [[weight lattice]].

=--

##### The Hamiltonian $G$-action / coadjoint moment map

The group $G$ canonically [[action|acts]] on the [[coset space]] $G/G_{\lambda}$ (by multiplication from the left). We discuss a lift of this action to a [[Hamiltonian action]] with respect to the [[symplectic manifold]] structure $(G/T, \nu_\lambda)$ of prop. \ref{TheSymplecticFormOnTheCoset}, equivalently a [[momentum map]] exhibiting this Hamiltonian action.

##### Wilson loops and 1d Chern-Simons $\sigma$-models with target the coadjoint orbit
 {#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}

Above (...) we discussed how an [[irreducible representation|irreducible]] [[unitary representation]] of $G$ is encoded by the [[prequantization]] of a  [[coadjoint orbit]] $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express [[Wilson loops]]/[[holonomy]] of $G$-[[principal connections]] in this representation as the [[path integral]] of a topological particle charged under this background field, whose [[action functional]] is that of a [[1-dimensional Chern-Simons theory]].

Let $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a [[Lie algebra valued 1-form]] on the circle, equivalently a $G$-[[principal connection]] on the circle.

For

$$
  \rho : G \to Aut(V)
$$

a [[representation]] of $G$, write

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

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

+-- {: .num_defn #ActionFunctionalForTopologicalChargedParticle}
###### Definition

Let the [[action functional]]

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

be given by sending $g T : S^1 \to G/T$ represented by $g : S^1 \to G$ to

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

where

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

is the [[gauge transformation]] of $A$ under $g$.

=--

+-- {: .num_prop #WilsonLoopIsPartitionFunctionOf1dCSSigmaModel}
###### Proposition

The [[Wilson loop]] of $A$ over $S^1$ in the unitarry irreducible representation $R$ is proportional to the [[path integral]] of the 1-dimensional [[sigma-model]] with

1. [[target space]] the [[coadjoint orbit]] $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the [[weight (in representation theory)|weight]] corresponding to $R$ under the [[Borel-Weil-Bott theorem]]

1. [[action functional]] the functional of def. \ref{ActionFunctionalForTopologicalChargedParticle}:

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

=--

See for instance ([Beasley, (4.55)](#Beasley)).

+-- {: .num_remark }
###### Remark

Notice that since $\mathcal{O}_\lambda$ is a [[manifold]] of finite [[dimension]], the [[path integral]] for a point particle with this target space can be and has been defined rigorously, see at _[[path integral]]_.

=--

#### 3d Chern-Simons theory
 {#3dCSTheory}

* [[3d Chern-Simons theory]]

#### $(4k+3)$d $U(1)$-Chern-Simons theory
 {#HigherAbelianCSTheory}

* [[higher dimensional Chern-Simons theory]]

#### 7d Chern-Simons theory
 {#7dCSTheory}

* [[7d Chern-Simons theory]]

#### $\infty$-Chern-Simons theory
 {#InfinityCSTheory}

* [[schreiber:infinity-Chern-Simons theory]]

#### String field theory
 {#StringFieldTheory}

* [[string field theory]]

#### Gauge fields and gravity -- Einstein-Maxwell-Yang-Mills theory
 {#EinsteinYangMillsTheory}

* [[gravity]]

* [[Yang-Mills theory]]

  * [[Yang-Mills instanton]] number = [[second Chern class]]

* [[Einstein-Yang-Mills theory]]

#### Kaluza-Klein compactification
 {#KaluzaKleinCompactification}

* [[Kaluza-Klein compactification]]

#### Standard model of particle physics
 {#StandardModelParticlePhyiscs}

* [[standard model of particle physics]]

#### Standard model of cosmology
 {#StandardModelCosmology}

* [[standard model of cosmology]]

### Semantic Layer
 {#GaugeAndGravitySemanticsLayer}

exposition and survey is in ([FSS 13](#FiorenzaSatiSchreiberCSIntroAndSurvey)).

+-- {: bluebox }
###### Examples

1. [1d Chern-Simons theory](#GaugeAndGravity1dCSTheory)

1. [Nonabelian charged particle trajectories -- Wilson loops](#GaugeAndGravityWilsonLoops)

1. [2d CS theory: WZW terms and Chan-Paton gauge fields](#GaugeFieldsSemanticsLayerChanPatonGaugeField)

1. [3d Chern-Simons theory with Wilson loops](#GaugeAndGravity3dCSWithWilson)

1. [Chan-Paton gauge fields on D-branes](#GaugeAndGravityChanPatonGaugeFieldsOnDBranes)

=--

#### 1d Chern-Simons theory
 {#GaugeAndGravity1dCSTheory}

For some $n \in \mathbb{N}$ let

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

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]]

$$
  \mathbf{B}det
   \;\colon\;
  \mathbf{B}U(n) \to \mathbf{B}U(1)
$$

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

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

Moreiver this has the evident differential refinement

$$
  \widehat {\mathbf{B} det}
  \;\colon\;
  \mathbf{B} U(n)_{conn}
  \to
  \mathbf{B} U(1)_{conn}
$$

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

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

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

#### Nonabelian charged particle trajectories -- Wilson loops
 {#GaugeAndGravityWilsonLoops}

We consider now [[extended Lagrangians]] defined on [[field (physics)|fields]] as above in _[Nonabelian charged particle trajectories -- Wilson loops](#NonabelianChargedParticle)_. This provides a natural reformulation in [[higher geometry]] of the constructions in the _[[orbit method]]_ as reviewed above in _[Model layer -- Nonabelian charged particle](#ModLayerNonabelianChargedParticle)_.

+-- {: bluebox }
###### Examples

1. [Survey](#FormulationInHigherGeometrySurvey)

1. [Definitions and constructions](#FormulationInHigherGeometryDefinitions)

1. [Nonabelian charged particle trajectories – Wilson loops](GaugeAndGravityWilsonLoops)

1. [3d Chern-Simons theory with Wilson loops](#ExtendedChern-SimonsTheoryAndWilsonLoops).

=--

##### Survey
 {#FormulationInHigherGeometrySurvey}

We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of the form

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

where

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

1. the left square is a [[homotopy pullback]] square, hence $\mathbf{\theta}$ is the [[homotopy fiber]] of $\mathbf{J}$;

1. the bottom map is the [[extended Lagrangian]] for $G$-[[Chern-Simons theory]], equivalently the universal [[Chern-Simons circle 3-bundle with connection]];

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

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

##### Definitions and constructions
 {#FormulationInHigherGeometryDefinitions}

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

For the following, let $\langle \lambda, - \rangle \in \mathfrak{g}^*$ be a _regular_ element, def. \ref{RegularElement}, so that the [[stabilizer subgroup]] is identified with a [[maximal torus]]: $G_\lambda \simeq T$.

As usual, write

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

for the [[moduli stack]] of $G$-[[principal connections]].

+-- {: .num_defn #InclusionOfModuliStacks}
###### Definition

Write

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

for the canonical map, as indicated.

=--

+-- {: .num_remark }
###### Remark

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

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

By the discussion at _[[∞-action]]_ this exhibits the canonical [[action]] $\rho$ of $G$ on its [[coset space]]: it is the [[universal associated ∞-bundle|universal rho-associated bundle]].

=--

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

+-- {: .num_prop #ThetaAsHomotopyFiberOfJ}
###### Proposition

The [[homotopy fiber]] of $\mathbf{J}$ in def. \ref{InclusionOfModuliStacks} is

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

given over a test manifold $U \in $ [[CartSp]] by the map

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

which sends $g \mapsto g^* \theta$, where $\theta$ is the [[Maurer-Cartan form]] on $G$.

=--

+-- {: .proof}
###### Proof

We compute the [[homotopy pullback]] of $\mathbf{J}$ along the point inclusion by the [[factorization lemma]] as discussed at _[homotopy pullback -- Constructions](homotopy%20pullback#ConstructionsGeneral)_.

This says that with $\mathbf{J}$ presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids $hofib(\mathbf{J})$ which is the [[limit]] [[cone]] in

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

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

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

1. [[morphisms]] are over $U$ [[commuting diagram|commuting triangles]]

   $$
     \array{
       g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta
       \\
       & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}}
       \\
       && 0
     }
   $$

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

   $$
     g_1 \stackrel{t}{\to} g_2
   $$

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

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

=--

+-- {: .num_prop #Extended1dCSLagrangianFromLambda}
###### Proposition

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

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

in $\mathbf{H}$ given over a test manifold $U \in $ [[CartSp]] by the [[functor]]

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

which is given on objects by

$$
  A \mapsto \langle \lambda, A\rangle
$$

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

$$
  \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle )
  \,.
$$

=--

+-- {: .proof}
###### Proof

That this construction defines a map $*//T \to *//U(1)$ is
the statement of prop. \ref{WeightsAndCharacters}. It remains to check that the differential 1-forms gauge-transform accordingly.

For this the key point is that since $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the [[coadjoint action]], the [[gauge transformation]] law for points $A : U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is

$$
  A \mapsto Ad_g A + g^* \theta
  \,,
$$

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

$$
  \begin{aligned}
    \langle \lambda, A \rangle
    & \mapsto
    \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle
    \\
     & = \langle \lambda, A \rangle + d  \langle\lambda, \xi \rangle
  \end{aligned}
$$

=--

+-- {: .num_remark #ThePrequantumBundleFromCanonicalMaps}
###### Remark

The composite of the canonical maps of prop. \ref{ThetaAsHomotopyFiberOfJ} and prop. \ref{Extended1dCSLagrangianFromLambda} modulates a canonical [[circle bundle with connection]] on the [[coset space]]/[[coadjoint orbit]]:

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

=--

+-- {: .num_prop }
###### Proposition

The [[curvature]] 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark \ref{ThePrequantumBundleFromCanonicalMaps} is the [[symplectic form]] of prop. \ref{TheSymplecticFormOnTheCoset}. Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a [[prequantization]] of the [[coadjoint orbit]] $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$.

=--

+-- {: .proof}
###### Proof

The curvature 2-form is modulated by the composite

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

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

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

which sends

$$
  [g] \mapsto d \langle \lambda,  g^* \theta \rangle
  \,.
$$

=--

##### Nonabelian charged particle trajectories – Wilson loops
{#GaugeAndGravityWilsonLoops}

Let $\Sigma$ be an [[orientation|oriented]] [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 3 and let

$$
  C
  \;\colon\;
  S^1 \hookrightarrow \Sigma
$$

be a [[submanifold]] inclusion of the [[circle]]: a [[knot]] in $\Sigma$.

Let $R$ be an [[irreducible representation|irreducible]]
[[unitary representation]] of $G$ and let $\langle \lambda,-\rangle$ be a [[weight (in representation theory)|weight]] corresponding to it by the [[Borel-Weil-Bott theorem]].

Regarding the inclusion $C$ as an object in the [[arrow (∞,1)-topos]] $\mathbf{H}^{\Delta^1}$, say that a [[gauge field]] configuration for $G$-[[Chern-Simons theory]] on $\Sigma$ with
[[Wilson loop]] $C$ and labeled by the [[representation]] $R$ is a map

$$
  \phi
  \;\colon\;
  C
  \to
  \mathbf{J}
$$

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

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

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

* a $G$-[[principal connection]] $A$ on $\Sigma$;

* a $G$-valued function $g$ on $S^1$

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

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

For every such field configuration we can evaluate two [[action functionals]]:

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

1. that of the [[1-dimensional Chern-Simons theory]] discussed [above](#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit) whose [[extended Lagrangian]] is $\langle \lambda, -\rangle : \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. \ref{Extended1dCSLagrangianFromLambda}.

These are obtained by postcomposing the above square on the right by these [[extended Lagrangians]]

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

and then preforming the [[fiber integration in ordinary differential cohomology]] over $S^1$ and over $\Sigma$, respectively.

For the bottom map this gives the ordinary action functional of [[Chern-Simons theory]]. For the top map inspection of the proof of prop. \ref{Extended1dCSLagrangianFromLambda} shows that this gives the 1d Chern-Simons action whose [[partition function]] is the [[Wilson loop]] observable by prop. \ref{WilsonLoopIsPartitionFunctionOf1dCSSigmaModel} above.

#### 2d CS-theory, WZW-term and Chan-Paton gauge fields
 {#GaugeFieldsSemanticsLayerChanPatonGaugeField}

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]] [[connection on a bundle|with connection]], or rather, by the Kapustin-part of the [[Freed-Witten-Kapustin anomaly cancellation]] mechanism, a [[twisted bundle|twisted unitary bundle]], whose twist is the restriction of the ambient [[B-field]] to the [[D-brane]].

We considered these [[field (physics)|fields]] already [above](#ChanPatonGaugeFields). Here we discuss the corresponding [[action functional]] for the [[open string]] coupled to these fields

The first hint for the existence of such [[background gauge fields]] for the [[open string]] 2d-[[sigma-model]] comes from the fact that the open string's endpoint can naturally be taken to carry labels $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-[[gauge field]], the $(i,j)$-excitation being the given [[matrix]] element of a $U(n)$-valued connection 1-form $A$.

This original argument goes back work by Chan and Paton. Accordingly one speaks of _Chan-Paton factors_ and _Chan-Paton bundles_ .

We discuss the Chan-Paton gauge field and its [[quantum anomaly cancellation]] in [[extended prequantum field theory]].

Throughout we write $\mathbf{H} = $ [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]].

+-- {: bluebox }
###### Examples

1. [The B-field as a prequantum 2-bundle](#TheBFieldAsAPrequantum2Bundle)

1. [The Chan-Paton gauge field](#PrequantumGaugeFieldsTheChanPatonGaugeField)

1. [The open string sigma-model](#TheOpenStringSigmaModel)

1. [The anomaly-free open string coupling to the Chan-Paton gauge field](#TheAnomalyFreeOpenStringCouplingToTheChanPatonGaugeField)

=--

##### The $B$-field as a prequantum 2-bundle
 {#TheBFieldAsAPrequantum2Bundle}

For $X$ a [[type II supergravity]] [[spacetime]], the [[B-field]] is a map

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

If $X = G$ is a [[Lie group]], this is the [[prequantum 2-bundle]] of $G$-[[Chern-Simons theory]]. Viewed as such we are to find a canonical [[∞-action]] of the [[circle 2-group]] $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, form the corresponding [[associated ∞-bundle]] and regard the [[sections]] of that as the [[prequantum 2-states]] of the theory.

The Chan-Paton gauge field is such a prequantum 2-state.

##### The Chan-Paton gauge field
 {#PrequantumGaugeFieldsTheChanPatonGaugeField}

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

We fix throughout a natural number $n \in \mathbb{N}$, the _[[rank]]_ of the Chan-Paton gauge field.

+-- {: .num_prop #TheLongSequenceOfTheProjectiveUnitaryExtension}
###### Proposition

The [[extension of groups|extension]] of [[Lie groups]]

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

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

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

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

=--

+-- {: .num_prop}
###### Proposition

Here

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

is a smooth refinement of the universal [[Dixmier-Douady class]]

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

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

$$
  {\vert \mathbf{dd}_n \vert}
  \simeq
  dd_n
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

By the discussion at _[[∞-action]]_ the [[homotopy fiber sequence]] in prop. \ref{TheLongSequenceOfTheProjectiveUnitaryExtension}

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

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

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

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

=--

+-- {: .num_prop }
###### Proposition

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

$$
  \mathbf{c} \to \mathbf{dd}_n
$$

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

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

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

=--

+-- {: .num_prop #DifferentialRefinementOfSMoothDDClass}
###### Proposition

There is a further differential refinement

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

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

=--

+-- {: .num_defn}
###### Definition

Write

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

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

=--

+-- {: .num_defn}
###### Definition

Let

$$
  \iota_X
    \;\colon\;
  Q \hookrightarrow X
$$

be an inclusion of [[smooth manifolds]] or of [[orbifolds]], to be thought of as a [[D-brane]] [[worldvolume]] $Q$ inside an ambient [[spacetime]] $X$.

Then a **field configuration** of a _[[B-field]]_ on $X$ together with a compatible rank-$n$ **Chan-Paton gauge field** on the [[D-brane]] is a map

$$
  \phi
   \;\colon\;
  \iota_X \to \mathbf{Fields}
$$

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

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

=--

This identifies a  [[twisted bundle]] with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk [[B-field]].

This relation is the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] cancellation for the [[bosonic string]] or else for the [[type II string]] on $Spin^c$ D-branes. ([FSS](#FSS))

+-- {: .num_remark }
###### 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.

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

On the local connection forms this acts as

$$
  A \mapsto A + \alpha
  \,.
$$

$$
  B \mapsto B + d \alpha
$$

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

=--

##### The open string sigma-model
 {#TheOpenStringSigmaModel}

+-- {: .num_remark}
###### Remark

The [[D-brane]] inclusion $Q \stackrel{\iota_X}{\to} X$ is the [[target space]] for an [[open string]] with [[worldsheet]] $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a [[field (physics)|field]] configuration of the open string [[sigma-model]] is a map

$$
  \phi \;\colon\; \iota_\Sigma \to \iota_X
$$

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

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

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

If however $X$ is more generally an [[orbifold]], then the [[homotopy]] filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called "orbifold twisted sectors" of open string configurations.

=--

+-- {: .num_prop #TheTypeIIOpenStringSigmaModelModuliStackOfFields}
###### Proposition

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

$$
  \array{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    \\
    \downarrow &\swArrow& \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
  }
  \,.
$$

=--

##### The anomaly-free open string coupling to the Chan-Paton gauge field
 {#TheAnomalyFreeOpenStringCouplingToTheChanPatonGaugeField}

+-- {: .num_prop }
###### Proposition

For $\Sigma$ a [[smooth manifold]] with [[boundary]] $\partial \Sigma$ of [[dimension]] $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a [[circle n-bundle with connection]] on some $X \in \mathbf{H}$, then the [[transgression]] of $\nabla$ to the [[mapping space]] $[\Sigma, X]$ yields a [[section]] of the [[complex line bundle]] [[associated bundle|associated]] to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram

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

=--

+-- {: .num_remark }
###### Remark

This is the _[[higher parallel transport]]_ of the $n$-connection $\nabla$ over maps $\Sigma \to X$.

=--

+-- {: .num_prop #TheTwistedHolonomyMapOnTwistedUnitaryBundles}
###### Proposition

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

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

=--

+-- {: .num_remark}
###### Remark

By the discussion at _[[∞-action]]_ the diagram in prop. \ref{TheTwistedHolonomyMapOnTwistedUnitaryBundles} says in particular that forming traced [[holonomy]] of twisted unitary bundles constitutes a [[section]] of the [[complex line bundle]] on the [[moduli stack]] of twisted unitary connection on the circle which is the [[associated bundle]] to the [[transgression]] $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ of the universal differential [[Dixmier-Douady class]].

=--

It follows that on the moduli space of the open string [[sigma-model]] of prop. \ref{TheTypeIIOpenStringSigmaModelModuliStackOfFields} above there are two $\mathbb{C}//U(1)$-valued [[action functionals]] coming from the bulk field and the boundary field

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

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:

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

Therefore the product action functional is a well-defined function

$$
  [\iota_\Sigma, \iota_X]
  \stackrel{
    \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] )
    \cdot
    hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1}
  }{\to}
  U(1)
  \,.
$$

This is the [[Freed-Witten-Kapustin anomaly|Kapustin anomaly]]-free [[action functional]] of the [[open string]].

#### 3d Chern-Simons theory with Wilson loops
 {#GaugeAndGravity3dCSWithWilson}
 {#ExtendedChern-SimonsTheoryAndWilsonLoops}

We discuss how an [[extended Lagrangian]]
for $G$-[[Chern-Simons theory]] with [[Wilson loop]] [[QFT with defects|defects]] is naturally obtained from the [above](#FormulationInHigherGeometryDefinitions) [[higher geometry|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](#WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit) is naturally obtained this way.

More formally, we have an extended Chern-Simons theory as follows.

The [[moduli stack]] of fields $\phi : C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the [[homotopy pullback]]

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

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

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

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

and

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

to yield

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

and then forming the product yields the action functional

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

This is the action functional of 3d $G$-[[Chern-Simons theory]] on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$.

Similarly, in [[codimension]] 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have [[prequantum bundles]] given by [[transgression]] of the extended Lagrangians to codimension 1

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

and

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

and hence a total prequantum bundle

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

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

#### Chan-Paton gauge fields on D-branes
 {#GaugeAndGravityChanPatonGaugeFieldsOnDBranes}

### Syntactic Layer
 {#GaugeAndGravitySyntaxLayer}

(...)