nLab geometry of physics -- physics in higher geometry

This entry contains one chapter of the material at geometry of physics.

Before we discuss technical details starting in the next chapter here we survey general ideas of theories in fundamental physics and motivate how these are naturally formulated in terms of the higher geometry that we developed in the first part.

Contents

1. Ingredients of fundamental physics

2. Topological local Lagrangian gauge field theory

3. Chern-Simons field theory as a toy example

4. Topological local field theory in higher category theory

5. Intermezzo: Elements of higher cohesive geometry

6. Chern-Simons theory as the archetypical example

7. Topological local Lagrangian field theory in higher geometry

8. 7-dimensional Chern-Simons theory as the next example

9. ∞-Chern-Simons theory as the general example

Ingredients of fundamental physics

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.

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 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 physics and specifically a 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.

standard model of particle physics and cosmology

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

Topological local Lagrangian gauge field theory

While the 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 theory;
2. a gauge theory;
3. a Lagrangian theory;
4. a local theory;
5. a topological theory.

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

• A 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.)

• A 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 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 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 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

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 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 simply connected gauge group, this is mathematically represented by a 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

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 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-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

The formalization of extended topological field theory thus captures the local aspect of field theory. But we saw above 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 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 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.

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

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 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 higher line bundles in higher codimension, namely a 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.

		local/extended Lagrangian field theory		local/extended quantum field theory
closed piece of spacetime of dimension $0 \leq k \leq n$	prequantum field theory	prequantum (n-k)-bundle	geometric quantization	space of $(n-k)$-wave functions
closed manifold of dimension $0 \leq k \leq n$	$\stackrel{local\;prequantum\;field\;theory}{\mapsto}$	circle (n-k)-bundle with connection on moduli stack of field configurations	$\stackrel{polarized\; sections}{\mapsto}$	(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

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

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 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 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 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 $\mathbf{H}$, 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 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

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 has in higher geometry a prequantum (n-k)-bundle associated in codimension $k$.

First consider again $G$ to be a connected and 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 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 as just recalled 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 3-bundle with connection – the “Chern-Simons circle 3-bundle with connection” – on the 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

Here the 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) 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

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) to a differential characteristic class by the Beilinson-Deligne cup product of the 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) 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:

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

The argument in (Witten 98) also says that in view of AdS7/CFT6 duality the 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) the corrected moduli are dicussed and found to involve a coupling of the abelian 7d theory with a nonabelian 7d Chern-Simons theory whose fields are twisted higher spin connections: “twisted differential string structures”.

To see what this is, recall that above 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). These higher obstructions are the (dual) k-invariants of the Whitehead tower of $B O$:

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)

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 differential refinement has been constructed in (FSS diff coc).

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

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 6-bundle-analog of the WZW 2-bundle.

Indeed, the assignment of a higher Wess-Zumino-Witten-type 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 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:

extended prequantum field theory

$0 \leq k \leq n$	(off-shell) prequantum (n-k)-bundle	traditional terminology
$0$	differential universal characteristic map	level
$1$	prequantum (n-1)-bundle	WZW bundle (n-2)-gerbe
$k$	prequantum (n-k)-bundle
$n-1$	prequantum 1-bundle	(off-shell) prequantum bundle
$n$	prequantum 0-bundle	action functional

$\infty$-Chern-Simons theory as the general example

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 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 fields for some $n$-dimensional 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 higher gauge theory of Chern-Simons type:

the curvature $(n+1)$-form of $\mathbf{L}$ (i.e. the pre n-plectic form of the 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 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 ∞-Chern-Simons theory.

This remains true if $\mathbf{Fields}$ is of more general form, so that fields are 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, then there is at most one extended prequantization of a given invariant polynomial.

2. selecting physically natural theories While the traditional formalization of local Lagrangian field theory does capture crucial aspects of those 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 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 ∞-Chern-Simons theories or ∞-Wess-Zumino-Witten theory?

As a hint to the answer of this question, here is a list of names examples of ∞-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-sector

    • topological M-theory

• ∞-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: by the central property of string theory, the action functional for string field theory (or some KK-reduction thereof) contains Einstein-Yang-Mills theory.