nLab prequantum field theory

Idea

The notion of quantum field theory exists without reference to any predefined notion of configuration space of quantum fields, action functional, phase space etc.:

A quantum field theory in FQFT-axiomatization is simply a consistent assignment of spaces of quantum states, whereas in AQFT-axiomatization it is a consistent assignment of algebras of quantum observables, and that’s it.

However, most (or maybe all?) quantum field theories of interest in actual physics (as opposed to as devices of pure mathematics) are not random models of these axioms, but do arise under a process called quantization from a (local/extended) Lagrangian, hence from an action functional, defined on a configuration space of quantum fields, or else arise as holographic duals of quantum field theories that arise by quantization. Moreover, the extra information provided by the Lagrangian is commonly used (and is maybe strictly necessary) to interpret the mathematical structure of the axiomatic QFT in actual physics (though notably in AQFT there are results that re-extract at least parts of this data from the axiomatic QFT, for instance the Doplicher-Roberts reconstruction theorem which extract the global gauge group from the local net of quantum observables).

There are in turn two formalizations of the notion of quantization: algebraic deformation quantization and geometric quantization. In the latter one speaks of prequantization when referring to a precursor step to the actual quantization step, in which the symplectic form on phase space is lifted to differential cohomology, hence to a prequantum bundle. But in the context of higher geometry and higher geometric quantization this prequantization step is already part of the data of the Lagrangian itself: an extended Lagrangian already encodes not just the action functional but also the prequantum bundle and all the prequantum (n-k)-bundles in each dimension $k$. The action functional itself is the prequantum 0-bundle in this context.

Therefore, in the refined picture of higher geometry/extended quantum field theory it makes good sense to refer in a unified way to prequantum field theory for all of the data related to Lagrangians that is not yet the final quantum field theory.

In particular, an extended prequantum field theory of dimension $n$ is a rule that assigns

• to every (suitably oriented) closed manifold of dimension $k$ a prequantum (n-k)-bundle

• to every (suitably oriented) compact manifold with boundary $\Sigma_k$ a section of the prequantum $(n-k+1)$-bundle assigned to the boundary $\partial \Sigma_k$ and pulled back to the space of fields over $\Sigma_k$

such that this data is related suitably under transgression.

The actual extended quantum field theory would be obtained from such a data by passing from the assignment of a given prequantum $(n-k)$-bundle to that of the (n-k)-vector space of polarized sections of a suitable associated fiber bundle.

This is summarized in the following table:

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

Details

under construction

We discuss local (“extended”) topological prequantum field theory.

The following originates in the lecture notes (Schreiber Pittsburgh13) and draws on material that is discussed more fully in (Fiorenza-Valentino) and (hCSlpQFT).

After a technical preliminary to set the stage in

• The ambient topos,

the first section gives the the definitions and general properties of

• Local prequantum field theory.

To digest this the reader may first or in parallel want to look at the simplest examples of these general considerations, which we discuss below in the first subsections of

• Higher Dijkgraaf-Witten prequantum field theory.

After that we turn to the general case of examples of

• Higher Chern-Simons prequantum field theory.

Here the pattern of the discussion of examples is the following:

The ambient topos

Prequantum field theory deals with “spaces of physical fields”. These spaces of fields are, in general, richer than just plain sets in two ways

1. Spaces of fields carry geometric structure, notably they may be smooth spaces, meaning that there is a way to determine which collections of fields form a smoothly parameterized collection. This is for instance the structure invoked (often implicitly) when performing variational calculus on spaces of fields in order to find their classical equations of motion.

2. Spaces of fields have gauge transformations between their points and possibly higher gauge transformations between these, meaning that they are in fact groupoids and possibly higher groupoids. In the physics literature this is best known in the infinitesimal approximation to these gauge transformations, in which case the spaces of fields are described by BRST complexes: the dg-algebras of functions on a Lie algebroid or L-∞ algebroid of fields.

Taken together this means that spaces of fields are geometric higher groupoids, such as orbifolds and more generally Lie groupoids, differentiable stacks, Lie 2-groupoids, … smooth ∞-groupoids.

A collection of all such geometric higher groupoids for a chosen flavor of geometry – for instance topology or differential geometry or supergeometry (for the description of fermion fields) or synthetic differential geometry or synthetic differential supergeometry, etc. – is called an ∞-topos.

Not quite every ∞-topos $\mathbf{H}$ serves as a decent context for collections (moduli stacks) of physical fields though. In the following we need at least that $\mathbf{H}$ has a reasonable notion of discrete objects so that we can identify the geometrically discrete spaces in there. We here need this to mean the following

Definition

An ∞-topos $\mathbf{H}$ is called locally ∞-connected and globally ∞-connetced if the locally constant ∞-stack-functor $LConst \colon$ ∞Grpd $\to \mathbf{H}$ is a reflective embedding.

The corresponding reflector we write

$$\Pi \colon \mathbf{H} \to \infty Grpd \hookrightarrow \mathbf{H}$$

and also call the shape modality of $\mathbf{H}$. By the discussion at adjoint triple it follows that $LConst$ is also a coreflective embedding; the corresponding coreflector we write

$$\flat \colon \mathbf{H} \stackrel{\Gamma}{\to} \infty Grpd \hookrightarrow \mathbf{H}$$

and call the flat modality.

Example

Every cohesive (∞,1)-topos is in particular globally and locally $\infty$-connected, by definition. Standard canonical examples to keep in mind are

• $\mathbf{H} =$ ∞Grpd for ∞-Dijkgraaf-Witten theories;

• $\mathbf{H} =$ Smooth∞Grpd for ∞-Chern-Simons theories;

• $\mathbf{H} =$ SuperSmooth∞Grpd for ∞-Chern-Simons theories with fermions and supersymmetry;

• $\mathbf{H} =$ SynthDiff∞Grpd for AKSZ sigma-models.

Local prequantum field theory

After sketching out the general

• Idea

we formulate first

• Bulk field theory,

which concerns the case where the worldvolume/spacetime on which the physical fields propagate has no boundaries with boundary conditions imposed (no “branes” or “domain walls” or “defects”). The point of this section is to see how the “space of fields” – or rather: the moduli stack of fields – on a point induces the corresponding spaces/moduli stacks of fields on an arbitrary closed manifold, and, correspondingly, how the prequantum n-bundle on the space over fields over the point induces the action functional in codimension 0.

However, what makes local prequantum field theory rich is that it naturally incorporates extra structure on boundaries of worldvolume/spacetime. In fact, under suitable conditions there is another local prequantum field theory just over the boundary, which is related to the corresponding bulk field theory possibly by a kind of holographic principle. This general mechanism we discuss in

• Boundary field theory.

But plain boundaries are just the first example of a general phenomenon known as “defects” or “phase dualities” or “singularities” in field theories. Notably the boundary field theory itself may have boundaries, in which case this means that the original theory had corners where different boundary pieces meet. This we discuss in

• Corner field theory.

Generally there are fields theories with general such singularties:

singularity	field theory with singularities
boundary condition/brane	boundary field theory
domain wall/bi-brane	QFT with defects

Idea

A prequantum field theory is, at its heart, an assignment that sends a piece of worldvolume/spacetime $\Sigma$ – technically a cobordism with boundary and corners – to the

1. space of field configurations over incoming and outgoing pieces of worldvolume/spacetime;

2. the space of field configurations over the bulk worldvolume/spacetime – the trajectories of fields;

3. an action functional that assigns to all these field configurations phases in a compatible manner.

These field configurations and spaces of trajectories between them are represented by spans/correspondences of (moduli-)spaces of fields (moduli stacks, really), hence diagrams of the form

$$\mathbf{Fields}_{in} \leftarrow \mathbf{Fields} \rightarrow \mathbf{Fields}_{out} \,.$$

Here $\mathbf{Fields}_{in}$ is to be thought of as the space of incoming fields, $\mathbf{Fields}_{out}$ that of outgoing fields, and $\mathbf{Fields}$ the space of all fields on some cobordism connecting the incoming and the outgoing pieces of worldvolume/spacetime. The left map sends such a trajectory to its starting configuration, and the right one sends it to its end configuration.

Given two such spans/correspondences, that share a common field configuration as in

$$\array{ && \mathbf{Fields}_1 && && \mathbf{Fields}_2 \\ & \swarrow && \searrow && \swarrow && \searrow \\ \mathbf{Fields}_{in_1} && && \mathbf{Fields}_{out_1} = \mathbf{Fields}_{in_2} && && \mathbf{Fields}_{out_2} }$$

can be composed, by forming consecutive trajectories from all pairs of trajectories that match in the middle. The space of these composed trajectories is the fiber product $\mathbf{Fields}_1 \underset{{\mathbf{Fields}_{out_1}} \atop {=\mathbf{Fields}_{in_2}}}{\times} \mathbf{Fields}_2$ which sits in a new span/correspondence

$$\mathbf{Fields}_{in_1} \leftarrow \mathbf{Fields}_1 \underset{{\mathbf{Fields}_{out_1}} \atop {=\mathbf{Fields}_{in_2}}}{\times} \mathbf{Fields}_2 \rightarrow \mathbf{Fields}_{out_2}$$

exhibiting the composite of the previous two. This way, spaces of fields with spans/correspondences between them form a category, which we denote $Span_1(\mathbf{H})$ if $\mathbf{H}$ denotes the ambient context (a topos) in which the spaces of fields live.

If two cobordisms run in parallel, then the field configurations on their union are pairs of the original field configurations, which are elements in the cartesian product of spaces of fields. Hence the operations

$$\left( \mathbf{Fields}_{in} \leftarrow \mathbf{Fields} \rightarrow \mathbf{Fields}_{out} \right) \otimes \left( \tilde \mathbf{Fields}_{in} \leftarrow \tilde \mathbf{Fields} \rightarrow \tilde\mathbf{Fields}_{out} \right) \;\;\coloneqq\;\; \left( \mathbf{Fields}_{in} \times \tilde\mathbf{Fields}_{in} \leftarrow \mathbf{Fields} \times \tilde\mathbf{Fields} \to \mathbf{Fields}_{out} \times \tilde\mathbf{Fields}_{out} \right)$$

make this category of fields and correspondence into a monoidal category.

Then a choice of field configurations for a (not yet localized) field theory in dimension $n \in \mathbb{N}$ is a monoidal functor from a category of cobordisms of dimension $n$ to such a category of spans/correspondences

$$\mathbf{Fields} \colon Bord_n^\otimes \to Span_1(\mathbf{H})^\otimes \,,$$

namely a consistent assignment that to each closed manifold $\Sigma_{n-1}$ of dimension $(n-1)$ assigns a space of field configurations $\mathbf{Fields}(\Sigma_{n-1})$ and that to each cobordism

$$\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$$

assigns a span/correspondence of spaces of field configurations and trajectories

$$\mathbf{Fields}(\Sigma_{in}) \leftarrow \mathbf{Fields}(\Sigma) \rightarrow \mathbf{Fields}_{\Sigma_{out}} \,.$$

Apart from the field configurations themselves, prequantum field theory assigns to each trajectory a “phase” – an element in the circle group $U(1)$ – by a map called the (exponentiated) action functional. In order to nicely relate that to the expression of spaces of trajectories as spans/correspondences as above, it is useful to think of the circle group here as being the automorphisms of something. This is universally accomplished by taking it to be the automorphisms of the unique point in the delooping groupoid $\mathbf{B}U(1) = \{\ast \stackrel{c \in U(1)}{\to} \ast\}$. (A lightning review of groupoid-homotopy theory is below in Groupoids and basic homotopy 1-type theory.) In other words, we think of the group of phases $U(1)$ as the space of homotopies from the point to itself in the Eilenberg-MacLane space $\mathbf{B}U(1)$, expressed by the diagram (a homotopy fiber product diagram)

$$\array{ && U(1) \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}U(1) } \,.$$

Using this, if we assume for simplicity that the in- and outgoing field configurations are sent constantly to the point in $\mathbf{B}U(1)$, then an (exponentiated) action functional on the space of trajectories $\exp(i S) \colon \mathbf{Fields} \to U(1)$ is equivalently a homotopy as shown on the left of the following diagram

$$\array{ && \mathbf{Fields} \\ & \swarrow && \searrow \\ \mathbf{Fields}_{in} && \swArrow && \mathbf{Fields}_{out} \\ & {}_{\mathllap{0}}\searrow && \swarrow_{\mathrlap{0}} \\ && \mathbf{B}U(1) } \;\;\;\; \simeq \;\;\;\; \array{ && \mathbf{Fields} \\ && \downarrow^{\mathrlap{\exp(i S)}} \\ && U(1) \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}U(1) } \,.$$

Hence action functionals are naturally incorporated into spans/correspondences of moduli spaces of fields simply by regarding these to be formed not in the ambient topos $\mathbf{H}$ itself, but in its slice topos $\mathbf{H}_{/\mathbf{B}U(1)}$, where each object is equipped with a map to $\mathbf{B}U(1)$ and each morphism with a homotopy in $\mathbf{B}U(1)$ between the corresponding maps.

We write $\mathrm{Span}_1(\mathbf{H}, \mathbf{B}U(1))$ for the category of spans/correspondences as before, but now equipped with maps to, and transformations over, $\mathbf{B}U(1)$ as in the above diagram.

Then an action functional for a choice of field configurations that itself is given as a monoidal functor $\mathbf{Fields} \colon Bord_n^\otimes \to Span_1(\mathbf{H})$ as above is a monoidal functor

$$S \colon Bord_n^\otimes \to Span_1(\mathbf{H}, \mathbf{B}U(1))$$

such that the spans of spaces of fields are those specified before, hence such that it fits as a lift into the diagram

$$\array{ && Span_1(\mathbf{H}, \mathbf{B}U(1)) \\ & {}^{\mathllap{S}}\nearrow & \downarrow \\ Bord_n &\underset{\mathbf{Fields}}{\to}& Span_1(\mathbf{H}) } \,,$$

where the right vertical functor forgets the phase assignments and just remembers the correspondences of field trajectories.

So far this is a non-local (or: not-necessarily local) prequantum field theory, since it assigns data only to entire $n$-dimensional cobordisms and $(n-1)$-dimensional closed manifolds, but is not guaranteed to be obtained by integrating up local data over little pieces of these manifolds. The latter possibility is however the characteristic property of local quantum field theory, which in turn is the flavor of quantum field theory that seems to matter in nature, and fundamentally.

In order to formalize this localization, we allow the cobordisms to contain higher-codimension pieces that are manifolds with corners. These then form not just a category of cobordisms, but an (∞,n)-category of cobordisms, which we will still denote $Bord_n^\otimes$. If we now have a cobordism with codimension-2 corners, then the field configurations over it now form a span-of-spans

$$\array{ \mathbf{Fields}_{ii} &\leftarrow& \mathbf{Fields}_{ic} &\to& \mathbf{Fields}_{io} \\ \uparrow && \uparrow && \uparrow \\ \mathbf{Fields}_{ci} &\leftarrow& \mathbf{Fields}_{cc} &\to& \mathbf{Fields}_{co} \\ \downarrow && \downarrow && \downarrow \\ \mathbf{Fields}_{oi} &\leftarrow& \mathbf{Fields}_{oc} &\to& \mathbf{Fields}_{oo} } \,.$$

Generally, for $n$-dimensional cobordism that are “localized” all the way to corners in codimension $n$, their field configurations and trajectories-of-trajectories etc. form $n$-dimensional cubes of spans-of-spans this way. We write $Span_n(\mathbf{H})$ for the resulting (∞,n)-category of spans.

In order to still have an action functional on trajectories is codimension-0 associated with this in the above fashion, we need to deloop $U(1)$ $n$-times to the n-groupoid $\mathbf{B}^n U(1)$ (the circle (n+1)-group). Accordingly a local prequantum field theory in dimension $n$ is given by a monoidal (∞,n)-functor

$$S \colon Bord_n^\otimes \to Span_n(\mathbf{H}, \mathbf{B}^n U(1)) \,.$$

The point of local topological (prequantum) field theory is that by the cobordism theorem the above story reverses: the assignment of fields and their action functional in higher dimension is necessarily given by higher traces of the data assigned in lower dimension. Hence the whole assignment $S$ above is fixed by its value on the point, hence by a choice of one single map

$$\left[ \array{ \mathbf{Fields}(\ast) \\ \downarrow^\mathrlap{S} \\ \mathbf{B}U(1) } \right] \,,$$

the fully localized action functional. Or rather, this is the case for pure bulk field theory, with no branes or domain walls. If these are present, then each type of them in dimension $k$ is specified by a k-morphism in $Span_n(\mathbf{H}, \mathbf{B}^n U(1))$.

All this we now describe more formally.

Bulk field theory

We now first consider the formalization of prequantum field theory in the absence of any data such as boundary conditions, domain walls, branes, defects, etc. This describes either field theories in which no such phenomena are taken to be present, or else it describes that part of those field theories where such phenomena are present in principle, but restricted to the “bulk” of worldvolume/spacetime where they are not. Therefore it makes sense to speak of bulk field theory in this case.

Definition

For $n \in \mathbb{N}$, write

$$Bord_n^\otimes \in E_\infty Alg(Cat_{(\infty,n)})$$

for the symmetric monoidal (∞,n)-category of cobordisms with $n$-dimensional framing. For $S \to O(n)$ a homomorphism of ∞-groups (may be modeled by a homomorphism of topological groups) to the general linear group (or homotopy-equivalently its maximal compact subgroup, the orthogonal group), we write

$$(Bord_n^S)^\otimes \in E_\infty Alg(Cat_{(\infty,n)})$$

for the corresponding symmetric monoidal $(\infty,n)$-category of cobordisms equipped with S-structure on their $n$-stabilized tangent bundle.

Remark

In this notation we have an identification

$$Bord_n \simeq Bord_n^{S \coloneqq \ast}$$

because a framing of the $n$-stabilized tangent bundle is a trivialization of that bundle and hence equivalently a G-structure for $G$ the trivial group. In (LurieTFT) this is denoted by “$Bord_n^{fr}$”.

Remark

The cobordism theorem asserts, essentially, that $Bord_n$ is the symmetric monoidal (∞,n)-category with full duals which is free on a single generator, the point. In itself this is a deep statement about the homotopy type of categories of cobordisms. But for the following discussion the reader may just take this as the definition of $Bord_n$. This then makes $Bord_n$ a very simple object, as long as we are just mapping out of it, which we do.

What this means then is that a monoidal (∞,n)-functor

$$Z \colon Bord_n^\otimes \to \mathcal{C}^\otimes$$

sends the point to some fully dualizable object $Z(\ast) \in \mathcal{C}$ and sends

• the circle $S^1$ to the trace of the identity on $Z(\ast)$,

• the sphere $S^2$ to the 2-dimensional higher trace of the identity,

and so on.

Definition

For $\mathbf{H}$ an ∞-topos, and $n \in \mathbb{N}$, write

$$Span_n(\mathbf{H}) \in Cat_{(\infty,n)}$$

for the (∞,n)-category of spans in $\mathbf{H}$. From the cartesian monoidal category structure of $\mathbf{H}$ this inherits the structure of a symmetric monoidal (∞,n)-category which we write

$$Span_n(\mathbf{H})^\otimes \in E_\infty Alg(Cat_{(\infty,n)}) \,.$$

Proposition

Every object in $Span_n(\mathbf{H})$ is a self-fully dualizable object. The evaluation map/coevaluation map $k$-spans in dimension $k$ involve in top degree the spans

$$\ast \leftarrow X \stackrel{}{\to} [\Pi(S^k), X] \,.$$

(…)

Definition

For $B \in Grp(\mathbf{H})$ an abelian ∞-group object in $\mathbf{H}$, spans in the slice (∞,1)-topos $\mathbf{H}_{/B}$ inherits a monoidal structure given on objects by

$$\otimes \; \colon \; \left[ \array{ X \\ \downarrow^{\mathrlap{f}} \\ B } \right] \times \left[ \array{ Y \\ \downarrow^{\mathrlap{g}} \\ B } \right] \mapsto \left[ \array{ X \times Y \\ \downarrow^{\mathrlap{f \circ p_1 + g \circ p_2}} \\ B } \;\;\;\;\;\;\;\;\;\; \right] \,.$$

We write

$$Span_n(\mathbf{H}, B)^\otimes \in E_\infty Alg(Cat_{(\infty,n)})$$

for the resulting symmetric monoidal (∞,n)-category.

Remark

In the case that $\mathbf{H} =$ ∞Grpd this is a special case of (LurieTFT, around prop. 3.2.8), with the abelian ∞-group $B$ regarded as a special case of a symmetric monoidal (∞,1)-category.

Remark

Since the slice (∞,1)-category $\mathbf{H}_{/\flat \mathbf{B}^n U(1)}$ is itself an (∞,1)-topos – the slice (∞,1)-topos – we also have $Span_n(\mathbf{H}_{/\flat \mathbf{B}^n U(1)})$, according to def. . As an (∞,n)-category this is equivalent to $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ from def. , but the monoidal structure is different. The cartesian product in the slice is given by homotopy fiber product in $\mathbf{H}$ over $\flat \mathbf{B}^n U(1)$, not by the addition in the ∞-group structure on $\flat \mathbf{B}^n U(1)$, as in def. .

The central definition in the present context now is the following.

Definition

A local prequantum bulk field in dimension $n \in \mathbb{N}$ (in a given ambient cohesive (∞,1)-topos $\mathbf{H}$) is a monoidal (∞,n)-functor

$$\mathbf{Fields} \;\colon\; Bord^S_n \to Span_n(\mathbf{H})$$

from the (∞,n)-category of cobordisms (with S-structure), def. , to the (∞,n)-category of n-fold correspondences in $\mathbf{H}$.

A local action functional on such a local prequantum bulk field is a monoidal lift $S$ of this in

$$\array{ && Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1)) \\ & {}^{\mathllap{S}}\nearrow & \downarrow^{\mathrlap{Span_n\left(\underset{\flat \mathbf{B}^n U(1)}{\sum}\right)}} \\ Bord_n^S &\underset{\mathbf{Fields}}{\to}& Span_n(\mathbf{H}) } \,.$$

For $\mathbf{H} =$ ∞Grpd this is the perspective in (FHLT, section 3).

Remark

Since a monoidal $(\infty,n)$-functor $\mathbf{Fields} \colon Bord_n \to Span_n(\mathbf{H})$ is determined by its value on the point, we will often notationally identify it with this value and write

$$\mathbf{Fields} = \mathbf{Fields}(\ast) \in \mathbf{H} \hookrightarrow Span_n(\mathbf{H}) \,.$$

As a corollary of prop. we have:

Proposition

Given $\mathbf{Fields} \colon Bord_n^\otimes \to Span_n(\mathbf{H})^\otimes$, it assigns to a k-morphism represented by a closed manifold $\Sigma_k$ the internal hom (mapping stack) from $\Pi(\Sigma_k)$ (the shape modality of $\Sigma_k$, def. ) into the moduli stack of fields

$$\mathbf{Fields} \colon \Sigma_k \mapsto [\Pi(\Sigma_k), \mathbf{Fields}] \,.$$

Remark

By the defining property of the mapping stack construction, this means that if $\mathcal{C}$ is an (∞,1)-site of definition of the (∞,1)-topos $\mathbf{H}$, then $[\Pi(\Sigma_k), \mathbf{Fields}]$ is the ∞-stack which to $U \in \mathcal{C}$ assigns the (∞,1)-categorical hom space

$$[\Pi(\Sigma_k), \mathbf{Fields}](U) \simeq \mathbf{H}(\Pi(\Sigma_k)\times U , \mathbf{Fields}) \,,$$

hence the ∞-groupoid of fields on $\Pi(\Sigma_k) \times U$.

If $\mathbf{Fields}$ is a moduli ∞-stack of gauge fields for some smooth ∞-group $G$, hence of the form $\mathbf{B}G_{conn}$, then this an $\infty$-groupoid of a kind of smoothly (or else geometrically) $U$-parameterized collections of flat ∞-connections on $\Sigma_k$.

(…)

Boundary field theory

(…)

Corner field theory

(…)

Classical mechanics as prequantum field theory

Traditional classical mechanics (Hamiltonian mechanics, Lagrangian mechanics, Hamilton-Jacobi theory) is naturally understood as a special case of – and in fact as deriving from – local prequantum field theory over $\mathbf{B}U(1)_{conn}$. This is discussed in some detail at

• prequantized Lagrangian correspondence.

Higher Dijkgraaf-Witten local prequantum field theory

We discuss here aspects of higher Dijkgraaf-Witten theory-type prequantum field theories, which are those prequantum field theories whose moduli stack $\mathbf{Fields}$ is a discrete ∞-groupoid (and usually also required to be finite, especially if its quantization is considered). This is a special case of the higher Chern-Simons theories discussed below in Higher Chern-Simons local prequantum field theory, and hence strictly speaking need not be discussed separately. We use it here as a means to review some of the relevant homotopy theory by way of pertinent examples.

The original Dijkgraaf-Witten theory is that in dimension 3 (reviewed in 3d Local prequantum field theory below), which was introduced in (Dijkgraaf-Witten 90) as a toy version of standard 3d Chern-Simons theory for simply connected gauge group. A comprehensive account with first indications of its role as a local (extended, multi-tiered) field theory then appeared in (Freed-Quinn 93), and ever since this has served as a testing ground for understanding the general principles of local field theory, e.g. (Freed 94), independently of the subtleties of giving meaning to concepts such as the path integral when the space of fields is not finite. In section 3 of (FHLT 10), the general prequantum formalization as in def. is sketched for Dijkgraaf-Witten type theories, and in section 8 there the quantization of these theories to genuine local quantum field theories is sketched.

1d Dijkgraaf-Witten theory

Dijkgraaf-Witten theory in dimension 1 is what results when one regards a group character of a finite group $G$ as a local

action functional in

the sense of def. . We give now an expository discussion of this simple but instructive example of a local prequantum field theory and in the course of it introduce some of the relevant basics of the homotopy theory of groupoids (homotopy 1-types).

1. Finite gauge groups

2. Essence of gauge theory: Groupoids and basic homotopy 1-type theory

3. Trajectories of fields: Correspondences of groupoids

4. Action functionals on spaces of trajectories: Correspondences of groupoids over the space of phases

The punchline of this section is little theorem at the very end, which states that the 1d local prequantum field theory whose local action functional is the delooping of a group character assigns to the circle the action functional which is again that group character. The proof of this statement is an unwinding of the basic mechanisms of local prequantum field theories.

Finite gauge groups

First some brief remarks, before we dive into the formalism.

A group character on a finite group $G$ is just a group homomorphism $G \to U(1)$ to the circle group (taken here as a discrete group). In order to regard this as an action functional, we are to take $G$ as the gauge group of a physical field theory. The simplest such case is a field theory such that on the point there is just a single possible field configuration, to be denoted $\phi_0$. The reader familiar with basics of traditional gauge theory may think of the fields as being gauge field connections (“vector potentials”), hence represented by differential 1-forms. But on the point there is only the vanishing 1-form, hence just a single field configuration $\phi_0$.

Even though there is just a single such field, that $G$ is the gauge group means that for each element $g \in G$ there is a gauge transformation that takes $\phi_0$ to itself, a state of affairs which we suggestively denote by the symbols

$$\phi_0 \stackrel{g}{\to} \phi_0 \,.$$

Again, the reader familiar with traditional gauge theory may think of gauge transformations as in Yang-Mills theory. Over the point these form, indeed, just the gauge group itself, taking the trivial field configuration to itself.

That the gauge group is indeed a group means that gauge transformations can be applied consecutively, which we express in symbols as

$$\array{ && \phi_0 \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \phi_0 && \underset{g_2 \cdot g_1}{\to} && \phi_0 } \,.$$

Regarded this way, we say the gauge group acting on the single field $\phi_0$ forms a groupoid, whose single object is $\phi_0$ and whose set of morphisms is $G$.

Of course in richer field theories there may be more than one field configuration, clearly, with gauge transformations between them. If $\phi_0$ and $\phi_1$ are two field configurations and $g$ is a gauge transformation taking one to the other, we may usefully denote this by

$$\array{ \phi_0 &\stackrel{g_1}{\to}& \phi_1 } \,.$$

Similarly then for yet another gauge configuration to another field configuration

$$\array{ \phi_1 &\stackrel{g_2}{\to}& \phi_2 }$$

then composing them gives the picture

$$\array{ && \phi_1 \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \phi_0 && \underset{g_2 \cdot g_1}{\to} && \phi_2 } \,.$$

We now discuss this notion of groupoids more formally.

Essence of gauge theory: Groupoids and basic homotopy 1-type theory

The following is a quick review of basics of groupoids and their homotopy theory (homotopy 1-type-theory), geared towards the constructions and fact needed for 1-dimensional Dijkgraaf-Witten theory. For more along these lines see also at geometry of physics – homotopy types

Definition

A (small) groupoid $\mathcal{G}_\bullet$ is

• a pair of sets $\mathcal{G}_0 \in Set$ (the set of objects) and $\mathcal{G}_1 \in Set$ (the set of morphisms)

• equipped with functions

  $$\array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\longrightarrow}& \mathcal{G}_1 & \stackrel{\overset{t}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\longrightarrow}}}& \mathcal{G}_0 }\,,$$

  where the fiber product on the left is that over $\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1$,

such that

• $i$ takes values in endomorphisms;

  $$t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;$$

• $\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{G}_0)$ the identities; in particular

  $s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$;

• every morphism has an inverse under this composition.

Remark

This data is visualized as follows. The set of morphisms is

$$\mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\}$$

and the set of pairs of composable morphisms is

$$\mathcal{G}_2 \coloneqq \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 = \left\{ \array{ && \phi_1 \\ & {}^{\mathllap{k_1}}\nearrow && \searrow^{\mathrlap{k_2}} \\ \phi_0 && \stackrel{k_2 \circ k_1}{\to} && \phi_2 } \right\} \,.$$

The functions $p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1$ are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.

Example

For $X$ a set, it becomes a groupoid by taking $X$ to be the set of objects and adding only precisely the identity morphism from each object to itself

$$\left( X \stackrel {\overset{id}{\longrightarrow}} { \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} } X \right) \,.$$

Example

For $G$ a group, its delooping groupoid $(\mathbf{B}G)_\bullet$ has

• $(\mathbf{B}G)_0 = \ast$;

• $(\mathbf{B}G)_1 = G$.

For $G$ and $K$ two groups, group homomorphisms $f \colon G \to K$ are in natural bijection with groupoid homomorphisms

$$(\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,.$$

In particular a group character $c \colon G \to U(1)$ is equivalently a groupoid homomorphism

$$(\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,.$$

Here, for the time being, all groups are discrete groups. Since the circle group $U(1)$ also has a standard structure of a Lie group, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on

$$\flat U(1) \in Grp$$

to mean explicitly the discrete group underlying the circle group. (Here “$\flat$” denotes the “flat modality”.)

Example

For $X$ a set, $G$ a discrete group and $\rho \colon X \times G \to X$ an action of $G$ on $X$ (a permutation representation), the action groupoid or homotopy quotient of $X$ by $G$ is the groupoid

$$X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right)$$

with composition induced by the product in $G$. Hence this is the groupoid whose objects are the elements of $X$, and where morphisms are of the form

$$x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g)$$

for $x_1, x_2 \in X$, $g \in G$.

As an important special case we have:

Example

For $G$ a discrete group and $\rho$ the trivial action of $G$ on the point $\ast$ (the singleton set), the coresponding action groupoid according to def. is the delooping groupoid of $G$ according to def. :

$$(\ast //G)_\bullet = (\mathbf{B}G)_\bullet \,.$$

Another canonical action is the action of $G$ on itself by right multiplication. The corresponding action groupoid we write

$$(\mathbf{E}G)_\bullet \coloneqq G//G \,.$$

The constant map $G \to \ast$ induces a canonical morphism

$$\array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,.$$

This is known as the $G$-universal principal bundle. See below in for more on this.

Example

The interval $I$ is the groupoid with

• $I_0 = \{a,b\}$;
• $I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}$.

Example

For $\Sigma$ a topological space, its fundamental groupoid $\Pi_1(\Sigma)$ is

• $\Pi_1(\Sigma)_0 =$ points in $X$;
• $\Pi_1(\Sigma)_1 =$ continuous paths in $X$ modulo homotopy that leaves the endpoints fixed.

Example

For $\mathcal{G}_\bullet$ any groupoid, there is the path space groupoid $\mathcal{G}^I_\bullet$ with

• $\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\}$;

• $\mathcal{G}^I_1 =$ commuting squares in $\mathcal{G}_\bullet$ = $\left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,.$

This comes with two canonical homomorphisms

$$\mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet$$

which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.

Definition

For $f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet$ two morphisms between groupoids, a homotopy $f \Rightarrow g$ (a natural transformation) is a homomorphism of the form $\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet$ (with codomain the path space object of $\mathcal{K}_\bullet$ as in example ) such that it fits into the diagram as depicted here on the right:

$$\array{ & \nearrow \searrow^{\mathrlap{f_\bullet}} \\ \mathcal{G} &\Downarrow^{\mathrlap{\eta}}& \mathcal{K} \\ & \searrow \nearrow_{\mathrlap{g_\bullet}} } \;\;\;\; \coloneqq \;\;\;\; \array{ && \mathcal{K}_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{(ev_1)_\bullet}} \\ \mathcal{G}_\bullet &\stackrel{\eta_\bullet}{\to}& \mathcal{K}^I_\bullet \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow^{\mathrlap{(ev_0)_\bullet}} \\ && \mathcal{K} } \,.$$

Definition (Notation)

Here and in the following, the convention is that we write

• $\mathcal{G}_\bullet$ (with the subscript decoration) when we regard groupoids with just homomorphisms (functors) between them,

• $\mathcal{G}$ (without the subscript decoration) when we regard groupoids with homomorphisms (functors) between them and homotopies (natural transformations) between these

  $$\array{ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{g} } \,.$$

The unbulleted version of groupoids are also called homotopy 1-types (or often just their homotopy-equivalence classes are called this way.) Below we generalize this to arbitrary homotopy types (def. ).

Example

For $X,Y$ two groupoids, the mapping groupoid $[X,Y]$ or $Y^X$ is

• $[X,Y]_0 =$ homomorphisms $X \to Y$;
• $[X,Y]_1 =$ homotopies between such.

Definition

A (homotopy-) equivalence of groupoids is a morphism $\mathcal{G} \to \mathcal{K}$ which has a left and right inverse up to homotopy.

Example

The map

$$\mathbf{B}\mathbb{Z} \stackrel{}{\to} \Pi(S^1)$$

which picks any point and sends $n \in \mathbb{Z}$ to the loop based at that point which winds around $n$ times, is an equivalence of groupoids.

Proposition

Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example – a skeleton.

Remark

The statement of prop. becomes false as when we pass to groupoids that are equipped with geometric structure. This is the reason why for discrete geometry all Chern-Simons-type field theories (namely Dijkgraaf-Witten theory-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the AKSZ sigma-models. But even so, Dijkgraaf-Witten theory is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. is not canonical.

Definition

Given two morphisms of groupoids $X \stackrel{f}{\to} B \stackrel{g}{\leftarrow} Y$ their homotopy fiber product

$$\array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B }$$

is the limit cone

$$\array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,,$$

hence the ordinary iterated fiber product over the path space groupoid, as indicated.

Remark

An ordinary fiber product $X_\bullet \underset{B_\bullet}{\times}Y_\bullet$ of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:

$$(X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,.$$

Example

For $X$ a groupoid, $G$ a group and $X \to \mathbf{B}G$ a map into its delooping, the pullback $P \to X$ of the $G$-universal principal bundle of example is equivalently the homotopy fiber product of $X$ with the point over $\mathbf{B}G$:

$$P \simeq X \underset{\mathbf{B}G}{\times} \ast \,.$$

Namely both squares in the following diagram are pullback squares

$$\array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,.$$

(This is the first example of the more general phenomenon of universal principal infinity-bundles.)

Example

For $X$ a groupoid and $\ast \to X$ a point in it, we call

$$\Omega X \coloneqq \ast \underset{X}{\times} \ast$$

the loop space groupoid of $X$.

For $G$ a group and $\mathbf{B}G$ its delooping groupoid from example , we have

$$G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,.$$

Hence $G$ is the loop space object of its own delooping, as it should be.

Proof

We are to compute the ordinary limiting cone $\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast$ in

$$\array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,,$$

In the middle we have the groupoid $(\mathbf{B}G)^I_\bullet$ whose objects are elements of $G$ and whose morphisms starting at some element are labeled by pairs of elements $h_1, h_2 \in G$ and end at $h_1 \cdot g \cdot h_2$. Using remark the limiting cone is seen to precisely pick those morphisms in $(\mathbf{B}G_\bullet)^I_\bullet$ such that these two elements are constant on the neutral element $h_1 = h_2 = e = id_{\ast}$, hence it produces just the elements of $G$ regarded as a groupoid with only identity morphisms, as in example .

Proposition

The free loop space object is

$$[\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X$$

Proof

Notice that $\Pi_1(S^0) \simeq \ast \coprod \ast$. Therefore the path space object $[\Pi(S^0), X_\bullet]^I_\bullet$ has

• objects are pairs of morphisms in $X_\bullet$;

• morphisms are commuting squares of such.

Now the fiber product in def. picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore $X_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X$ is the groupoid whose

• objects are diagrams in $X_\bullet$ of the form

  $$\array{ & \nearrow \searrow \\ x_0 && x_1 \\ & \searrow \nearrow }$$

• morphism are cylinder-diagrams over these.

One finds along the lines of example that this is equivalent to maps from $\Pi_1(S^1)$ into $X_\bullet$ and homotopies between these.

Remark

Even though all these models of the circle $\Pi_1(S^1)$ are equivalent, below the special appearance of the circle in the proof of prop. as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an evaluation map with a coevaluation map.

Example

For $G$ a discrete group, the free loop space object of its delooping $\mathbf{B}G$ is $G//_{ad} G$, the action groupoid, def. , of the adjoint action of $G$ on itself:

$$[\Pi(S^1), \mathbf{B}G] \simeq G//_{ad} G \,.$$

Example

For an abelian group such as $\flat U(1)$ we have

$$[\Pi(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,.$$

Example

Let $c \colon G \to \flat U(1)$ be a group homomorphism, hence a group character. By example this has a delooping to a groupoid homomorphism

$$\mathbf{B}c \;\colon\; \mathbf{B}G \to \mathbf{B}\flat U(1) \,.$$

Under the free loop space object construction this becomes

$$[\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)]$$

hence

$$[\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,.$$

So by postcomposing with the projection on the first factor we recover from the general homotopy theory of groupoids the statement that a group character is a class function on conjugacy classes:

$$[\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to U(1) \,.$$

Trajectories of fields: Correspondences of groupoids

With some basic homotopy theory of groupoids in hand, we can now talk about trajectories in finite gauge theories, namely about spans/correspondences of groupoids and their composition. These correspondences of groupoids encode trajectories/histories of field configurations.

Namely consider a groupoid to be called $\mathbf{Fields} \in$ Grpd, to be thought of as the moduli space of fields in some field theory, or equivalently and specifically as the target space of a sigma-model field theory. This just means that for $\Sigma$ any manifold thought of as spacetime or worldvolume, the space of fields $\mathbf{Fields}(\Sigma)$ of the field theory on $\Sigma$ is the mapping stack (internal hom) from $\Sigma$ into $\mathbf{Fields}$, which means here for DW theory that it is the mapping groupoid, def. , out of the fundamental groupoid, def. , of $\Sigma$:

$$\mathbf{Fields}(\Sigma) = [\Pi_1(\Sigma), \mathbf{Fields}] \,.$$

We think of the objects of the groupoid $[\Pi_1(\Sigma), \mathbf{Fields}]$ as being the fields themselves, and of the morphisms as being the gauge transformations between them.

The example to be of interest in a moment is that where $\mathbf{Fields} = \mathbf{B}G$ is a delooping groupoid as in def. , in which case the fields are equivalently flat principal connections. In fact in the discrete and 1-dimensional case currently considered this is essentially the only example, due to prop. , but for the general idea and for the more general cases considered further below, it is useful to have the notation allude to more general moduli spaces $\mathbf{Fields}$.

The simple but crucial observation that shows why spans/correspondences of groupoids show up in prequantum field theory is the following.

Example

If $\Sigma$ is a cobordism, hence a manifold with boundary with incoming boundary component $\Sigma_{in} \hookrightarrow \Sigma$ and outgoing boundary components $\Sigma_{out} \hookrightarrow \Sigma$, then the resulting cospan of manifolds

$$\array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} && && \Sigma_{out} }$$

is sent under the operation of mapping into the moduli space of fields

$$[\Pi_1(-), \mathbf{Fields}] \;\colon\; Mfds^{op} \to Grpd$$

to a span of groupoids

$$\array{ && [\Pi_1(\Sigma), \mathbf{Fields}] \\ & \swarrow && \searrow \\ [\Pi_1(\Sigma_{in}), \mathbf{Fields}] && && [\Pi_1(\Sigma_{out}), \mathbf{Fields}] } \,.$$

Here the left and right homomorphisms are those which take a field configuration on $\Sigma$ and restrict it to the incoming and to the outgoing field configuration, respectively. (And this being a homomorphism of groupoids means that everything respects the gauge symmetry on the fields.) Hence if $[\Pi_1(\Sigma_{in,out}),\mathbf{Fields}]$ is thought of as the spaces of incoming and outgoing field configurations, respectively, then $[\Pi_1(\Sigma), \mathbf{Fields}]$ is to be interpreted as the space of trajectories (sometimes: histories) of field cofigurations over spacetimes/worldvolumes of shape $\Sigma$.

This should make it plausible that specifying the field content of a 1-dimensional discrete gauge field theory is a functorial assignsment

$$\mathbf{Fields} \;\colon\; Bord_1 \to Span(Grpd)$$

from a category of cobordisms of dimension one into a category of such spans of groupoids. It sends points to spaces of field configurations on the point and 1-dimensional manifolds such as the circle as spaces of trajectories of field configurations on them.

Moreover, for a local field theory it should be true that the field configurations on the circle, says, are determined from gluing the field configurations on any decomposition of the circle, notably a decomposition into two semi-circles. But since we are dealing with a topological field theory, its field configurations on a contractible interval such as the semicircle will be equivalent to the field configurations on the point itself.

The way that the fields on higher spheres in a topological field theory are induced from the fields on the point is by an analog of traces for spaces of fields, and higher traces of such correspondences (the “span trace”). This is because by the cobordism theorem, the field configurations on, notably, the n-sphere are given by the $n$-fold span trace of the field configurations on the point, the trace of the traces of the … of the 1-trace. This is because for instance the 1-sphere, hence the circle is, regarded as a 1-dimensional cobordism itself pretty much manifestly a trace on the point in the string diagram formulation of traces.

$$\array{ && \ast^- \\ & \swarrow & & \nwarrow \\ \downarrow && && \uparrow \\ & \searrow && \nearrow \\ && \ast^+ } \,.$$

Here $\ast^+$ is the point with its potitive orientation, and $\ast^-$ is its dual object in the category of cobordisms, the point with the reverse orientation. Since, by this picture, the construction that produces the circle from the point is one that involves only the coevaluation map and evaluation map on the point regarded as a dualizable object, a topological field theory $Z \colon Bord_n \to Span_n(\mathbf{H})$, since it respects all this structure, takes the circle to precisely the same kind of diagram, but now in $Span_n(\mathbf{H})^\otimes$, where it becomes instead the span trace on the space $\mathbf{Fields}(\ast)$ over the point. This we discuss now.

Before talking about correspondences of groupoids, we need to organize the groupoids themselves a bit more.

Definition

A (2,1)-category $\mathcal{C}$ is

1. a collection $\mathcal{C}_0$ – the “collection of objects”;

2. for each tuple $(X,Y) \in \mathcal{C}_0 \times \mathcal{C}_0$ a groupoid $\mathcal{C}(X,Y)$ – the hom-groupoid from $X$ to $Y$;

3. for each triple $(X,Y,Z) \in \mathcal{C}_0 \times \mathcal{C}_0 \times \mathcal{C}_0$ a groupoid homomorphism (functor)

   $$\circ_{X,Y,Z} \colon \mathcal{C}(X,Y) \times \mathcal{C}(Y,Z) \to \mathcal{C}(X,Z)$$

   called composition or horizontal composition for emphasis;

4. for each quadruple $(W,X,Y,Z,)$ a homotopy – the associator –

   $$\array{ \mathcal{C}(W,X) \times \mathcal{C}(X,Y) \times \mathcal{C}(Y,Z) &\stackrel{}{\to}& \mathcal{C}(W,Y) \times \mathcal{C}(Y,Z) \\ \downarrow &\swArrow_{\alpha_{W,X,Y,Z}}& \downarrow \\ \mathcal{C}(W,X) \times \mathcal{C}(X,Z) &\stackrel{}{\to}& \mathcal{C}(W,Z) }$$

   (…) and similarly a unitality homotopy (…)

such that for each quintuple $(V,W,X,Y,Z)$ the associators satisfy the pentagon identity.

The objects of the hom-groupoid $\mathcal{C}(X,Y)$ we call the 1-morphisms from $X$ to $Y$, indicated by $X \stackrel{f}{\to} Y$, and the morphisms in $\mathcal{C}(X,Y)$ we call the 2-morphisms of $\mathcal{C}$, indicated by

$$\array{ \\ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{\mathrlap{g}} } \,.$$

If all associators $\alpha$ can and are chosen to be the identity then this is called a strict (2,1)-category.

Definition/Example

Write Grpd for the strict (2,1)-category, def. , whose

• objects are groupoids $\mathcal{G}$;

• 1-morphisms are functors $f \colon \mathcal{G} \to \mathcal{K}$;

• 2-morphisms are homotopies between these.

Definition/Example

Write $Span_1(Grpd)$ for the (2,1)-category whose

• objects are groupoids;

• 1-morphisms are spans/correspondences of functors, hence

  $$\array{ A &\leftarrow& X &\rightarrow& B } \,;$$

• 2-morphisms are diagrams in Grpd of the form

  $$\array{ && X_1 \\ & \swarrow &\downarrow& \searrow \\ A &\seArrow& \downarrow^{\mathrlap{\simeq}} &\swArrow& B \\ & \nwarrow &\downarrow& \nearrow \\ && X_2 }$$

• composition is given by forming the homotopy fiber product, def. , of the two adjacent homomorphisms of two spans, hence for two spans

  $$X \stackrel{}{\leftarrow} K \rightarrow Y$$

  and

  $$Y \stackrel{}{\leftarrow} L \rightarrow Z$$

  their composite is the span which is the outer part of the diagram

  $$\array{ && && K \underset{Y}{\times}L \\ && & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ && K && \swArrow && L \\ & \swarrow && \searrow && \swarrow && \searrow \\ X && && Y && && Z } \,.$$

Definition

There is the structure of a symmetric monoidal (2,1)-category on $Span_1(Grpd)$ by degreewise Cartesian product in Grpd.

$$(X \leftarrow K \rightarrow Y) \otimes (\tilde X \leftarrow \tilde K \rightarrow \tilde Y) \;\coloneqq\; X \times \tilde X \leftarrow K \times \tilde K \rightarrow Y \times \tilde Y \,.$$

Definition

An object $X$ of a symmetric monoidal (2,1)-category $\mathcal{C}^\otimes$ is fully dualizable if there exists

1. another object $X^\ast$, to be called the dual object;

2. a 1-morphism $ev_X \colon X^\ast \otimes X \to \mathbb{I}$, to be called the evaluation map;

3. a 1-morphism $coev_X \colon \mathbb{I} \to X \otimes X^\ast$, to be called the coevaluation map;

4. 2-morphisms

   $$\array{ && \rightarrow \\ & \nearrow &\Downarrow^{coev_{tr(X)}}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} }$$

   and

   $$\array{ && \rightarrow \\ & \nearrow &\Uparrow^{ev_{tr(X)}}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} }$$

   and

   $$\array{ && \rightarrow \\ & \nearrow &\Downarrow^{sa(X)}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X }$$

   (the saddle?)

   and

   $$\array{ && \rightarrow \\ & \nearrow &\Uparrow^{cosa(X)}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X }$$

   (the co-saddle)

such that these exhibit an adjunction and are themselves adjoint (…).

Definition

Given a symmetric monoidal (2,1)-category $\mathcal{C}$, and a fully dualizable object $X \in \mathcal{C}$ and a 1-morphism $f \colon X \to X$, the trace of $f$ is the composition

$$tr(f) \;\colon\; \mathbb{I} \stackrel{coev_X}{\to} X \otimes X^\ast \stackrel{f \otimes id_{X^\ast}}{\to} X \otimes X^\ast \stackrel{ev_x}{\to} \mathbb{I} \,.$$

Proposition

Every groupoid $X \in Grpd \hookrightarrow Span_1(Grpd)$ is a dualizable object in $Span_1(Grpd)$, and in fact is self-dual.

The evaluation map $ev_X$, hence the possible image of a symmetric monoidal functor $Bord_1 \to Span_1(Grpd)$ of a cobordism of the form

$$\array{ && \leftarrow & \ast^- \\ & \swarrow \\ \downarrow \\ & \searrow \\ && \rightarrow & \ast^+ }$$

is given by the span

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && && [\Pi_1(S^0),X] &\simeq& X \times X }$$

and the coevaluation map $coev_X$ by the reverse span.

For $X \in Grpd \hookrightarrow Span_1(Grpd)$ any object, the trace (“span trace”) of the identity on it, hence the image of

$$\array{ && \ast^- \\ & \swarrow & & \nwarrow \\ \downarrow && && \uparrow \\ & \searrow && \nearrow \\ && \ast^+ }$$

is its free loop space object, prop. :

$$tr(id_X) \simeq \left( \array{ && [\Pi_1(S^1), X] \\ & \swarrow && \searrow \\ \ast && && \ast } \right) \,.$$

The second order covaluation map on the span trace of the identity is

$$\array{ && && \ast \\ && && \uparrow \\ && && X \\ && && \downarrow \\ && && [\Pi(S^1), X] \\ && & \swarrow & & \searrow \\ \ast &\leftarrow& X &\rightarrow& [\Pi(S^0), X] &\leftarrow& X &\rightarrow& & \ast } \,.$$

Proof

By prop. the trace of the identity is given by the composite span

$$\array{ && && X \underset{[\Pi_1(S^1), X]}{\times} X \\ && & \swarrow && \searrow \\ && X && \swArrow && X \\ & \swarrow && \searrow && \swarrow && \searrow \\ \ast && && [\Pi_1(S^0),X] && && \ast } \,.$$

By prop. we have

$$X \underset{[\Pi_1(S^1), X]}{\times} X \simeq [\Pi_1(S^1), X] \,.$$

Along these lines one checks the required zig-zag identities.

Action functionals on spaces of trajectories: correspondences of groupoids over the space of phases

We have now assembled all the ingredients need in order to formally regard a group character $c \colon G \to U(1)$ on a discrete group as a local action functional of a prequantum field theory, hence as a fully dualizable object

$$S \;\coloneqq\; \left[ \array{ \mathbf{B}G \\ \downarrow^{\mathrlap{c}} \\ \mathbf{B}\flat U(1) } \right] \;\in \; \mathrm{Span}_1(Grpd, \mathbf{B}\flat U(1))$$

in a (2,1)-category of correspondences of groupoids as in def. , but equipped with maps and homotopies between maps to the coefficient over $\mathbf{B}\flat U(1)$. This is described in def. below. Before stating this, we recall for the 1-dimensional case the general story of def. .

Example

Given a discrete groupoid $X$, functions

$$\exp(i S) \colon X \to \flat U(1)$$

are in natural bijection with homotopies of the form

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\phi} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \,,$$

where the function corresponding to this homotopy is that given by the unique factorization through the homotopy fiber product $\flat U(1) \simeq \ast \underset{\mathbf{B}\flat U(1)}{\times} \ast$ (example ) as shown on the right of

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\phi} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \;\;\;\; \simeq \;\;\;\; \array{ && X \\ &\swarrow& \downarrow^{\mathrlap{\exp(i S)}} & \searrow \\ && \flat U(1) \\ \downarrow & \swarrow && \searrow & \downarrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \,,$$

This means that if we have an action functional on a space of trajectories, and if these trajectories are given by spans/correspondences of groupoids as discussed above, then the action functional is naturally expressed as the homotopy filling a completion of the span to a square diagram over $\mathbf{B}\flat U(1)$. Therefore we cosider the following.

Definition

Write $Span_1(Grpd, \flat\mathbf{B}U(1))$ for the (2,1)-category whose

• objects are groupoids $X$ equipped with a morpism

  $$\left[ \array{ X \\ \downarrow^{\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right]$$

• morphisms are spans $X_1 \leftarrow Y \rightarrow X_2$ equipped with a homotopy $\phi$ in

  $$\array{ && Y \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} \\ X_1 && \swArrow_{\phi} && X_2 \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) }$$

• 2-morphisms are morphism of spans compatible with the maps to $\mathbf{B}\flat U(1)$ in the evident way.

The operation of composition is as in $Span_1(Grpd)$, def. on the upper part of these diagrams, naturally extended to the whole diagrams by composition of the homotopies filling the squares that appear.

Proposition

$Span_1(Grpd, \mathbf{B}\flat U(1))$ carries the structure of a symmetric monoidal (2,1)-category where the tensor product is given by

$$\left[ \array{ X_1 \\ \downarrow^{\mathrlap{f_1}} \\ \mathbf{B}\flat U(1) } \right] \otimes \left[ \array{ X_2 \\ \downarrow^{\mathrlap{f_2}} \\ \mathbf{B}\flat U(1) } \right] \;\; \coloneqq \;\; \left[ \array{ X_1 \times X_2 \\ \downarrow^{\mathrlap{f_1 \circ p_1 + f_2 \circ p_2}} \\ \mathbf{B}\flat U(1) } \right] \,.$$

Remark

There is an evident forgetful (2,1)-functor

$$Span_1(Grpd, \mathbf{B}\flat U(1)) \to Span_1(Grpd)$$

which forgets the maps to $\mathbf{B}\flat U(1)$ and the homotopies between them. This is a monoidal (2,1)-functor.

As generalization of prop. we now have the following:

Proposition

Every object

$$\left[ \array{ X \\ \downarrow^{\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right] \in Span_1(Grpd,\mathbf{B}\flat U(1))$$

is a dualizable object, with dual object

$$\left[ \array{ X \\ \downarrow^{-\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right]$$

and with evaluation map given by

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{\simeq}} && X \times X \\ & {}_{\mathllap{0}}\searrow && \swarrow_{\mathrlap{f \circ p_1 - f \circ p_2}} \\ && \mathbf{B}\flat U(1) } \,.$$

In conclusion we may now compute what the 1-dimensional prequantum field theory defined by a group character $c \colon G \to U(1)$ regarded as a local action functional assigns to the circle.

Theorem

The prequantum field theory defined by a group character

$$\left[ \array{ \mathbf{Field} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \flat \mathbf{B}U(1) } \right] \;\; \coloneqq \;\; \left[ \array{ \mathbf{B}G \\ \downarrow^{\mathrlap{\mathbf{B}\mathrlap{c}}} \\ \flat \mathbf{B}U(1) } \right] \in Span_1(Grpd,\mathbf{B}\flat U(1))$$

assigns to the circle the trace of the identity on this object, which under the identifications of example , example , and example is the group character itself:

$$\array{ && && [\Pi_1(S^1), \mathbf{B}G] \\ && & \swarrow && \searrow \\ && \mathbf{B}G && \swArrow && \mathbf{B}G \\ & \swarrow && \searrow && \swarrow && \searrow \\ \ast && && \mathbf{B}G \times \mathbf{B}G && && \ast \\ &{}_0\searrow & && \downarrow^{\mathrlap{\exp(i S \circ p_1 - i S \circ p_2)}} &&& \swarrow_{0} \\ &&&& \mathbf{B}\flat U(1) } \;\;\; \simeq \;\;\; \array{ && G//G \\ && \simeq \\ && [\Pi(S^1), \mathbf{B}G] \\ && \downarrow^{\mathrlap{c}} \\ && \flat U(1) \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \;\;\;$$

Here the action functional on the right sends a field configuration $g \in G = [\Pi(S^1), \mathbf{B}G]_0$ to its value $c(g) \in U(1) = (\flat \mathbf{B}U(1))_1$ under the group character.

Remark

It follows that in a discussion of quantization the path integral for the partition function of 1d DW theory is given by the Schur integral over the group character $c$.

$$\frac{1}{\vert G \vert} \underset{g\in G}{\sum} c(g) = \langle c,1\rangle \,.$$

In conclusion, 1-dimensional Dijkgraaf-Witten theory as a prequantum field theory comes down to be essentially a geometric interpretation of what group characters are and do. One may regard this as a simple example of geometric representation theory. Simple as this example is, it contains in it the seeds of many of the interesting aspects of richer prequantum field theories.

2d Dijkgraaf-Witten theory

The group character $c : G \to U(1)$ which defines 1-dimensional prequantum Dijkgraaf-Witten theory in 1d Dijkgraaf-Witten theory is equivalently a cocycle in degree-1 group cohomology

$$[c] \in H_{\mathrm{Grp}}(G,U(1)) \,.$$

More familiar are maybe cocycles in higher degree. In view of the above it is plausible that one may interpret a cocycle in degree-$n$ group cohomology, for all $n \in \mathbb{N}$ as a higher order action functional $\mathbf{B}G \to \flat\mathbf{B}^n U(1)$ and induce an $n$-dimensional local prequantum Dijkgraaf-Witten-type theory from it.

Here we discuss the case of $n = 2$ where a group 2-cocycle is regarded as the local action functional of a 2-dimensional Digjkgraaf-Witten field theory. We use this as occasion to introduce a bit of the theory of 2-groups and their homotopy theory (homotopy 2-type-theory). Below in 3d DW theory we then turn to the fully general case of ∞-groupoid-theory.

2-Groupoids and basic homotopy 2-type theory

(…)

$n$d Dijkgraaf-Witten theory

In view of the above it is plausible that one may interpret a cocycle in degree-$n$ group cohomology, for all $n \in \mathbb{N}$ as a higher order action functional $\mathbf{B}G \to \flat\mathbf{B}^n U(1)$ and induce an $n$-dimensional local prequantum Dijkgraaf-Witten-type theory from it.

Here we review how to formalize this and then consider the example of DW theory in arbitrary dimension $n$.

Essence of higher gauge theory: $\infty$-Groupoids and basic homotopy theory

We briefly recall here some basic definitions and facts of ∞-groupoids and their homotopy theory, geared towards their use in 3-dimensional Dijkgraaf-Witten theory and generally in ∞-Dijkgraaf-Witten theory.

Higher gauge transformations: Simplicial sets and Kan complexes

An ∞-groupoid is first of all supposed to be a structure that consists of k-morphisms for all $k \in \mathbb{N}$, which for $k \geq 1$ go between $(k-1)$-morphisms.

In the context of Kan complexes, the tool for organizing such collections of k-morphisms is the notion of a simplicial set, which models $k$-morphisms as being of the shape of $k$-simplices – a vertex for $k = 0$, an edge for $k = 1$, a triangle for $k = 2$, a tetrahedron for $k = 3$, and so on.

This means that a simplicial set $K_\bullet$ is a sequence of sets $\{K_n\}_{n \in \mathbb{N}}$ (sets of $k$-simplex shaped $k$-morphisms for all $k$) equipped with functions $d_i \colon K_{k+1} \to K_{k}$ that send a $(k+1)$-simplex to its $i$-th face, and functions $s_i \colon K_k \to K_{k+1}$ that over a $k$-simplex “erects a flat $(k+1)$-simplex” in all possible ways (hence which inserts “identities” called “degeneracies” in this context).

If we write $\Delta$ for the category whose objects are abstract cellular simplices and whose morphisms are all cellular maps between these, then such a simplicial set is equivalently a functor of the form

$$K \colon \Delta^{op} \to Set$$

Hence we think of this as assigning

• a set $[0] \mapsto K_0$ of objects;

• a set $[1] \mapsto K_1$ of morphism;

• a set $[2] \mapsto K_2$ of 2-morphism;

• a set $[3] \mapsto K_3$ of 3-morphism;

and generally

• a set $[k] \mapsto K_k$ of k-morphisms

as well as specifying

• functions $([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n)$ that send $n+1$-morphisms to their boundary $n$-morphisms;

• functions $([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1})$ that send $n$-morphisms to identity $(n+1)$-morphisms on them.

The fact that $K$ is supposed to be a functor enforces that these assignments of sets and functions satisfy conditions that make consistent our interpretation of them as sets of $k$-morphisms and source and target maps between these. These are called the simplicial identities.

But apart from this source-target matching, a generic simplicial set does not yet encode a notion of composition of these morphisms.

For instance for $\Lambda^1[2]$ the simplicial set consisting of two attached 1-cells

$$\Lambda^1[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}$$

and for $(f,g) : \Lambda^1[2] \to K$ an image of this situation in $K$, hence a pair $x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2$ of two composable 1-morphisms in $K$, we want to demand that there exists a third 1-morphisms in $K$ that may be thought of as the composition $x_0 \stackrel{h}{\to} x_2$ of $f$ and $g$. But since we are working in higher category theory (and not to be evil), we want to identify this composite only up to a 2-morphism equivalence

$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\mathrlap{\simeq}}& \searrow^{\mathrlap{g}} \\ x_0 &&\stackrel{h}{\to}&& x_2 } \,.$$

From the picture it is clear that this is equivalent to demanding that for $\Lambda^1[2] \hookrightarrow \Delta[2]$ the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets

$$\array{ \Lambda^1[2] &\stackrel{(f,g)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists h}} \\ \Delta[2] } \,.$$

A simplicial set where for all such $(f,g)$ a corresponding such $h$ exists may be thought of as a collection of higher morphisms that is equipped with a notion of composition of adjacent 1-morphisms.

For the purpose of describing groupoidal composition, we now want that this composition operation has all inverses. For that purpose, notice that for

$$\Lambda^2[2] = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}$$

the simplicial set consisting of two 1-morphisms that touch at their end, hence for

$$(g,h) : \Lambda^2[2] \to K$$

two such 1-morphisms in $K$, then if $g$ had an inverse $g^{-1}$ we could use the above composition operation to compose that with $h$ and thereby find a morphism $f$ connecting the sources of $h$ and $g$. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form

$$\array{ \Lambda^2[2] &\stackrel{(g,h)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists f}} \\ \Delta[2] } \,.$$

Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in $K$.

In order for this to qualify as an $\infty$-groupoid, this composition operation needs to satisfy an associativity law up to coherent 2-morphisms, which means that we can find the relevant tetrahedrons in $K$. These in turn need to be connected by pentagonators and ever so on. It is a nontrivial but true and powerful fact, that all these coherence conditions are captured by generalizing the above conditions to all dimensions as in the definition of Kan complexes.

In order to conceive of the $k$-simplices for higher $k$ as “globular k-morphism” going from a source to a target one needs a bit of combinatorics. This provided by the orientals (due to Ross Street).

The $k$-oriental $O(k)$ is precisely the prescription for how exactly to think of a $k$-simplex as being a k-morphism in an omega-category. The first few look like this:

$$\array{\arrayopts{\rowalign{center}} O(\Delta^0) = & \{ 0\} \\ O(\Delta^1) = & \left\{ 0 \to 1\right\} \\ O(\Delta^2) = & \left\{ \array{\begin{svg} [[!include oriental > Delta2]] \end{svg}} \right\}\\ O(\Delta^3) = & \left\{ \array{\begin{svg} [[!include oriental > Delta3]] \end{svg}}\right\}\\ O(\Delta^4) = & \left\{ \array{\begin{svg} [[!include oriental > Delta4]] \end{svg}} \right\} }$$

In fact, the omega-nerve $N(K)$ of an omega-category $K$ is the simplicial set whose collection of $k$-cells $N(K)_k := Hom(O(k),K)$ is precisely the collection of images of the $k$th oriental $O(k)$ in $K$.

This is fully formally the prescription of how to think of a Kan complex as an $\infty$-groupoid: the Kan complex $C$ is the omega-nerve of an omega-category in which all morphism are invertible:

• the $k$-cells in $C_k$ are precisely the collection of $k$-morphisms ihn the omega-category of shape the $k$th oriental $O(k)$;

• the horn-filler conditions satisfied by these cells is precisely a reflection of the fact that

  1. there exists a notion of composition of adjacent k-morphisms in the omega-category;

  2. under this composition all $k$-morphisms have an inverse.

This is easy to see in low dimensions:

• a 1-cell $\phi \in C_1$ in the simplicial set $C$ has a single source 0-cell $x := d_1 \phi$ and a single target 0-cell $y := d_0 \phi$ and hence may be pictured as a morphism

  $$x \stackrel{\phi}{\to} y \,.$$

• a 2-cell $\phi \in C_2$ in the simplicial set $C$ has two incoming 1-cells $d_2 \phi, d_0 \phi \in C_1$ and one outgoing 1-cell $d_1 \phi \in C_1$, and if we think of the two incoming 1-cells as representing the composite of the corresponding 1-morphisms, we may picture te 2-cell $\phi$ here as a globular 2-morphism

  $$\array{ && x_1 \\ & {}^{\mathllap{d_2 \phi}}\nearrow &\Downarrow^\phi& \searrow^{\mathrlap{d_0 \phi}} \\ x_0 &&\underset{d_1 \phi}{\to}&& x_2 } \,.$$

More in detail, one may think of the incoming two adjacent $1$-cells here as not being the composite of these two morphism, but just as a composable pair, and should think of the existence of the 2-morphism $\phi$ here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism $d_1 \phi$.

So if an $\infty$-groupoid is thought of as a globular ∞-category in which all k-morphisms are invertible, then the corresponding Kan complex is the nerve or rather the ∞-nerve of this ∞-category.

Notably if $C$ is to be regarded as (the nerve of) an ordinary groupoid, every composable pair of morphisms has a unique composite, and hence there should be a unique 2-cell

$$\array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow& \searrow^{g} \\ x_0 &&\underset{h = g \circ f}{\to}&& x_2 }$$

that is the unique identity 2-morphism

$$g \circ f \stackrel{=}{\Rightarrow} h \,.$$

More generally, in a 2-groupoid there may be non-identity 2-morphisms, and hence for any 1-morphism $k _ x_0 \to x_2$ 2-isomorphic to $h$, there may be many 2-morphisms $g \circ f \Rightarrow k$, hence many 2-cells

$$\array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow^{\simeq}& \searrow^{g} \\ x_0 &&\underset{k }{\to}&& x_2 } \,.$$

All we can say for sure is that at least one such 2-cell exists, and that the 2-cells themselves may be composed in some way. This is precisely what the horn-filler conditions in a Kan complex encode.

We have already seen in low dimension how the existence of composites in an $\omega$-category is reflected in the fact that in a Kan-complex certain 2-simplices exist, and how the non-uniqueness of these 2-simplices reflects the existence of nontrivial 2-morphisms.

To see in a similar fashion that the Kan condition ensures the existence of inverses consider an outer horn in $C$, a diagram of 1-cells of the form

$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow \\ x_0 &&\underset{h}{\to}&& x_2 } \,.$$

In general given such a diagram in a category, there is no guarantee that the corresponding triangle as above will exist in its nerve. But if the category is a groupoid, then it is guaranteed that the missing 1-face can be chose to be the inverse of $f$ composed with the morphism $h$, and there is at least one 2-morphism

$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{h \circ f^{-1}}} \\ x_0 &&\underset{h}{\to}&& x_2 } \,.$$

A similar analysis for higher dimensional cells shows that the fact that a Kan complex has all horn fillers encodes precisely the fact that it is the omega-nerve of an omega-category in which all k-morphisms for all $k$ are composable if adjacent and have a weak inverse.

1-Groupoids as Kan complexes

We review how 1-groupoids are incarnated as Kan complexes via their nerve. For more along these lines see at geometry of physics – homotopy types.

Definition

A (small) groupoid $\mathcal{G}_\bullet$ is

• a pair of sets $\mathcal{G}_0 \in Set$ (the set of objects) and $\mathcal{G}_1 \in Set$ (the set of morphisms)

• equipped with functions

  $$\array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\to}& \mathcal{G}_1 & \stackrel{\overset{t}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\to}}}& \mathcal{G}_0 }\,,$$

  where the fiber product on the left is that over $\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1$,

such that

• $i$ takes values in endomorphisms;

  $$t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;$$

• $\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{G}_0)$ the identities; in particular

  $s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$;

• every morphism has an inverse under this composition.

Definition

For $\mathcal{G}_\bullet$ a groupoid, def. , its simplicial nerve $N(\mathcal{G}_\bullet)_\bullet$ is the simplicial set with

$$N(\mathcal{G}_\bullet)_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n}$$

the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$;

with face maps

$$d_k \colon N(\mathcal{G}_\bullet)_{n+1} \to N(\mathcal{G}_\bullet)_{n}$$

being,

• for $n = 0$ the functions that remembers the $k$th object;

• for $n \geq 1$

  • the two outer face maps $d_0$ and $d_n$ are given by forgetting the first and the last morphism in such a sequence, respectively;

  • the $n-1$ inner face maps $d_{0 \lt k \lt n}$ are given by composing the $k$th morphism with the $k+1$st in the sequence.

The degeneracy maps

$$s_k \colon N(\mathcal{G}_\bullet)n \to N(\mathcal{G}_\bullet)_{n+1} \,.$$

are given by inserting an identity morphism on $x_k$.

Remark

Spelling this out in more detail: write

$$\mathcal{G}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\}$$

for the set of sequences of $n$ composable morphisms. Given any element of this set and $0 \lt k \lt n$, write

$$f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}$$

for the comosition of the two morphism that share the $i$th vertex.

With this, face map $d_k$ acts simply by “removing the index $k$”:

$$d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )$$

$$d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n )$$

$$d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,.$$

Similarly, writing

$$f_{k,k} \coloneqq id_{x_k}$$

for the identity morphism on the object $x_k$, then the degenarcy map acts by “repeating the $k$th index”

$$s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,.$$

This makes it manifest that these functions organise into a simplicial set.

Proposition

These collections of maps in def. satisfy the simplicial identities, hence make the nerve $\mathcal{G}_\bullet$ into a simplicial set. Moreover, this simplicial set is a Kan complex, where each horn has a unique filler (extension to a simplex).

(A 2-coskeletal Kan complex.)

Proposition

The nerve operation constitutes a full and faithful functor

$$N \colon Grpd \to KanCplx \hookrightarrow sSet \,.$$

Homotopy theory of $\infty$-groupoids as Kan complexes

Definition

Write

$$KanCplx \hookrightarrow sSet$$

for the category of Kan complexes, which is the full subcategory of that of simplicial sets on the Kan complexes.

Remark

This means that for $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, an element $f_\bullet \colon X_\bullet \to Y_\bullet$ in the hom-set $Hom_{KanCplx}(X_\bullet,Y_\bullet)$ is

• a sequences of functions $f_n \colon X_n \to Y_n$ for all $n \in \mathbb{N}$;

such that

• these respect all the face maps $d_k$ and the degeneracy maps $s_k$.

Definition

For $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, their mapping space

$$Maps(X_\bullet,Y_\bullet)_\bullet \in KanCplx$$

is the simplicial set given by

$$Maps(X_\bullet,Y_\bullet) \colon [k] \mapsto Hom_{sSet}(X_\bullet \times \Delta^n_\bullet, Y_\bullet) \,.$$

Proposition

The construction in def. defines an internal hom of Kan complexes.

Remark

As such it is also common to write $Y^X$ for $Maps(X,Y)$, as well as $[X,Y]$. Notice that the latter notation is sometimes used instead for just the set of connected components of $Maps(X,Y)$.

It follows that the category $KanCplx$ is naturally enriched over itself.

We now have the following immediate generalizations of the corresponding constructions seen above for 1-groupoids.

Example

Write

$$I_\bullet \coloneqq \{0 \stackrel{\simeq}{\to} 1\}$$

for the Kan complex which is 1-groupoid with two objects and one nontrivial morphism and its inverse between them. This comes with two inclusions

$$i_0, i_1 \colon \ast \to I$$

of its endpoints.

Then for $X_\bullet \in KanCplx$ any other Kan complex, the mapping space $[I,X]_\bullet$ from def. is the path space object of $X_\bullet$.

$$X_\bullet \stackrel{[i_0,X_\bullet]}{\leftarrow} [I_\bullet,X_\bullet]_\bullet \stackrel{[i_1,X]}{\to} X_\bullet \,.$$

A 1-cell in the mapping Kan complex $[X_\bullet, Y_\bullet]_\bullet$ is a homotopy between two morphisms of Kan complexes:

Definition

For $f_\bullet, g_\bullet \colon X_\bullet \to Y_\bullet$ two morphisms between two Kan complexes, hence $f_\bullet,g_\bullet \in Hom_{KanCplx}(X,Y)$, a (right-)homotopy $\eta \colon f \Rightarrow g$ is a morphism $\eta_\bullet \colon X_\bullet \to [I_\bullet,X_\bullet]_\bullet$ into the path space object of def. such that we have a commuting diagram

$$\array{ && Y_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{[i_0, X_\bullet]_\bullet}} \\ X_\bullet &\stackrel{\eta_\bullet}{\to}& [I_\bullet, Y_\bullet] \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow_{\mathrlap{[i_1, X_\bullet]_\bullet}} \\ && Y \bullet } \,.$$

Remark

Hence a homotopy between two maps $X_\bullet \to Y_\bullet$ of Kan complexes is precisely a 1-cell in the mapping space $[X_\bullet, Y_\bullet]_\bullet$ of def. .

Definition

We say that a map $X_\bullet \to Y_\bullet$ of Kan complexes is a homotopy equivalence if it has a left and right inverse up to homotopy, hence an ordinary inverse in $\pi_0[X_\bullet, Y_\bullet]$.

Example

For Kan complexes which are 1-groupoids hence which are nerves of groupoids, homotopy equivalence of Kan complexes is equivalently homotopy equivalence of these groupoids according to def. .

Definition

We may write ∞Grpd for $KanCplx$ regarded as a $KanCplx$-enriched category, hence as fibrant sSet-enriched category.

We write $X$ (without the subscript) for a Kan complex $X_\bullet$ regarded as an object of $\infty Grpd$. As such, $X$ (or its equivalence class) is alse called a homotopy type.

The category ∞Grpd itself “is” the canonical homotopy theory. (For more on this see also at homotopy hypothesis.)

The following is the immediate generalization of def. .

Definition

Given two morphisms of Kan complexes $X \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y$ their homotopy fiber product

$$\array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B }$$

is the limit cone

$$\array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,,$$

hence the ordinary iterated fiber product over the path space Kan complex, as indicated.

Higher phases: Homological algebra and abelian $\infty$-groups

An important class of examples of ∞-groupoids are those which are presented under the Dold-Kan correspondence by chain complexes of abelian groups.

Dold-Kan correspondence

Definition

Write $Ch_{\bullet \geq 0}$ for the category of chain complexes (of abelian groups in non-negative degree).

As usual, for $A \in$ Ab an abelian group, we write $A[n]$ for the chain complex with $A$ in degree $n$ and 0 in all other degrees (the suspension of a chain complex).

Definition

Write sAb for the category of simplicial abelian groups, hence simplicial objects in abelian groups. Finally write

$$N \;\colon\; sAb \to Ch_\bullet$$

for the normalized chain complex-functor, which sends a simplicial abelian group $A_\bullet$ to the chain complex whose $n$-chains are the non-degenerate elements of $A_n$ and whose differential is the alternating sum of the face maps of $A_\bullet$:

$$d^{N(A)} = \sum_{i = 0}^n (-1)^i d^A_i \,.$$

Of relevance now are the following two standard facts.

Proposition

1. Dold-Kan: The functor $N \colon Ch_{\bullet \geq 0} \to sAb$ is an equivalence of categories. The inverse functor

   $$\Xi \colon Ch_{\bullet \geq 0} \stackrel{\simeq}{\to} sAb$$

   sends a chain complex to the simplicial abelian group whose $n$-simplices are the images of the normalized chain complex $N(\mathbb{Z}(\Delta^n))$ of chains $\mathbb{Z}(\Delta^n)$ the $n$-simplex.

2. Moore: The forgetful functor $sAb \to sSet$ which sends a simplicial abelian group to its underlying simplicial set factors through Kan complexes

   $$forget \; \colon \; sAb \to KanCplx \hookrightarrow sSet \,.$$

Taken together, this provides us with the following very useful construction.

Definition

We write

$$DK \;\colon\; Ch_{\bullet \geq 0} \underoverset{\simeq}{\Xi}{\to} sAb \stackrel{forget}{\to} KanCplx \hookrightarrow sSet$$

for the composite of the two functors of prop. .

We refer to this as the “Dold-Kan map”, or say “by Dold-Kan”, etc. It provides us with a rich supply of Kan complexes, hence of ∞-groups. In fact, it embeds homological algebra into the homotopy theory of ∞-groupoids in that it is a homotopical functor:

Proposition

The Dold-Kan map of def. sends quasi-isomorphisms of chain complexes to homotopy equivalences of Kan complexes, def. .

Notably we have the following example

Definition

For $n \in \mathbb{N}$ write

$$\mathbf{B}^n \flat U(1) \coloneqq DK(\flat U(1)[n]) \in KanCplx$$

for the $n$-fold suspension of the discrete circle group, regarded by Dold-Kan as the $n$-fold delooping Kan complex of $\flat U(1)$.

Example

Since the first simplex that contains a non-degenerate $n$-cell is $\Delta^n$, it follows that $\mathbf{B}^n\flat U(1)$ is trivial in degrees $\lt n$

$$(\mathbf{B}^n \flat U(1))_{k \lt n} = \ast \,.$$

Then since there is a unique non-degenerate $n$-cell in $\Delta^n$ we next have

$$(\mathbf{B}^n \flat U(1))_{n} = \flat U(1) \,.$$

Next there are $(n+2)$ non-degenrate $n$-simplices in $\Delta^{n+1}$, but known $n+1$ of their images in $\flat U(1)$ determines the last one (as their oriented product), hence next

$$(\mathbf{B}^n \flat U(1))_{n+1} = (\flat U(1))^{\times^{n+1}} \,.$$

Similarly for arbitary $k$ the set $(\mathbf{B}^n \flat U(1))_k$ is some Cartesian power of $\flat U(1)$. But cince here we are mostly interested in $\mathbf{B}^n \flat U(1)$ as an n-groupoid, hence only for mapping $n$-groupoids into it, it is mostly enough to just know its cells up to degree $(n+1)$.

Below in Higher Chern-Simons theory the smooth version of the circle group plays a role. In order to amplify that here we are just dealing with the discrete group underlying $U(1)$, we write

$$\flat U(1) \coloneqq U(1)_{disc}$$

for it.

$$\mathbf{B}^n \flat U(1) \coloneqq DK([U(1)_{disc}[n]]) \,.$$

Proposition

By the Dold-Kan correspondence each $\mathbf{B}^n \flat U(1)$ is naturally an ∞-group and its delooping is indeed $\mathbf{B}^{n+1}\flat U(1)$

$$\Omega \mathbf{B}^{n+1} \flat U(1) \simeq \mathbf{B}^n \flat U(1) \,.$$

Local DW action functionals: Higher group cohomology

Proposition

For $G$ a discrete group and $A$ a discrete abelian group, there is a natural isomorphism

$$H_{Grp}^n(G,A) \simeq \pi_0 Grpd_\infty(\mathbf{B}G, \mathbf{B}^n A)$$

between the degree-$n$ group cohomology of $G$ with coefficients in $A$ and the connected components of maps of ∞-groupoids from the delooping of $G$ to the $n$-fold delooping of $A$.

This means that local action functionals for higher Dijkgraaf-Witten type theories, hence maps of $\infty$-groupoids of the form $\exp(i S_{DW}^n) \colon \mathbf{B} \flat G \to \mathbf{B}^n \flat U(1)$ are equivalently cocycles $[c] \in H^n_{Grp}(\flat G,\flat U(1))$ in degree-$n$ group cohomology:

$$\exp(i S_{DW}^n) = \mathbf{B}c \,.$$

In particular the original 3d Dijkgraaf-Witten theory appears this way as the theory of a group 3-cocycle.

(…)

Higher Chern-Simons local prequantum field theory – Levels

Let $G \in Grp(\mathbf{H})$ be a simply connected compact simple Lie group. and write $\mathbf{B}G \in Smooth\infty Grpd$ for its delooping stack. By the discussion at Lie group cohomology there is a bijection

$$H^n_{Grp}(G,\mathbb{Z}) \simeq \{\mathbf{B}G \to \mathbf{B}^n \mathbb{Z}\}_\sim \,.$$

Write

$$\mathbf{c}_2 \colon \mathbf{B}G\longrightarrow B^4 \mathbb{Z}$$

for a representative of the second Chern class under this bijection. We may regard this map as a local Lagrangian, hence as an object

$$\mathbf{c}_2 \in Corr_3(\mathbf{H}_{/B^4 \mathbb{Z}})^{\otimes_{phased}}$$

in the (∞,n)-category of correspondences in the slice (∞,1)-topos $\mathbf{H}_{/B^4 \mathbb{Z}}$ equipped with the phased tensor product.

Since $Corr_3(\mathbf{H}_{/B^4 \mathbb{Z}})^{\otimes_{phased}}$ is an (∞,n)-category with duals, this defines a framed-topological local prequantum field theory

$$\mathbf{c}_2 \colon (Bord_n^{fr})^\sqcup \longrightarrow Corr_3(\mathbf{H}_{/B^4 \mathbb{Z}})^{\otimes_{phased}}$$

This sends a closed manifold $\Sigma$ of dimension 2 (a surface) to the class of a line bundle

$$\mathbf{Loc}_G(\Sigma) \longrightarrow B^2 \mathbb{Z}$$

on the moduli stack of flat connections (“local systems”) of $\Sigma$, which is the phase space of $G$-Chern-Simons theory.

By the discussion at cobordism hypothesis (this corollary in view of this proposition) an extension of this to an oriented-topological local preqauntum field theory is equivalent to

1. choices of $SO(3)$-∞-actions on $\mathbf{B}G$;

2. choices of equivariant extensions

   $$\array{ \mathbf{B}G &\stackrel{\mathbf{c}_2}{\longrightarrow}& B^4 \mathbb{Z} \\ \downarrow & \nearrow_{\mathrlap{\mathbf{c}_2//SO(3)}} \\ (\mathbf{B}G)//SO(3) } \,.$$

There are in general not too many $SO(3)$-∞-action on $\mathbf{B}G$, so consider the trivial one. Then

$$(\mathbf{B}G)//SO(3) \simeq (\mathbf{B}G)\times (B SO(3))$$

and an equivariant extension is given by a map

$$B SO(3) \longrightarrow B^4 \mathbb{Z}$$

hence an element in $H^4(B SO(3), \mathbb{Z})$. This is essentially given by the first Pontryagin class $p_1$. Hence it follows that the extension of $\mathbf{c}_2$-Chern-Simons local prequantum fields theory (on the level of levels) to oriented cobordisms is given by

$$(c_2 + p_1) \colon (\mathbf{B}G) \times (B SO(3)) \longrightarrow B^4 \mathbb{Z} \,.$$

The moduli space of fields that this assigns to an oriented manifold $\Sigma$ is the space of maps

$$\array{ \Pi(\Sigma) && \longrightarrow && (\mathbf{B}G)\times (B SO(3)) \\ & {}_{\mathllap{T \Sigma \oplus \mathbb{R}^{3-dim(\Sigma)}}}\searrow && \swarrow \\ & B SO(3) } \,.$$

Hence in codimension-0 this is still just $\mathbf{Loc}_G(\Sigma)$.

In order to trivialize the $p_1$-contribution appearing here, consider the homotopy fiber sequence

$$\array{ B^3 \mathbb{Z} &\longrightarrow& B^3 \mathbb{Z}//SO(3) \\ && \downarrow \\ && B SO(3) &\stackrel{p_1}{\longrightarrow}& }$$

exhibiting an $SO(3)$-∞-action on $B^3 \mathbb{Z}$. Then there is an oriented-topological local prequantum field theory with equivariant local Lagrangian

$$(c_2 + p_1) \colon (\mathbf{B}G) \times (B^3 \mathbb{Z}//SO(3)) \longrightarrow B^4 \mathbb{Z}$$

given by the total composite in the diagram

$$\array{ (\mathbf{B}G) \times (B^3 \mathbb{Z}//SO(3)) &\longrightarrow& (B^3 \mathbb{Z}) \\ \downarrow && \downarrow \\ (\mathbf{B}G)\times B SO(3) &\stackrel{(c_2,p_1)}{\longrightarrow}& (B^4 \mathbb{Z}, B^4 \mathbb{Z}) \\ && & \searrow^{\mathrlap{+}} \\ && && B^4 \mathbb{Z} } \,.$$

The moduli spaces of fields of this oriented theory on $\Sigma$ now are

$$\Sigma \mapsto \mathbf{Loc}_G(\Sigma) \times p_1Struct(\Sigma)$$

with 3-stabilized $p_1$-structures as in Atiyah 2-framing.

A further variant of this is given by the model for the supergravity C-field

$$\array{ \mathbf{CField} &\longrightarrow& (B^3 \mathbb{Z}) \\ \downarrow &\searrow& \downarrow \\ (\mathbf{B}G)\times B SO(3) &\stackrel{(c_2,p_1)}{\longrightarrow}& B^4 \mathbb{Z} } \,.$$

(…)

Higher Chern-Simons local prequantum field theory

By def. , a local prequantum bulk field theory in dimension $(n+1)$ is equivalently a morphism of the form

$$\exp(i S) \colon \mathbf{Fields} \to \flat \mathbf{B}^{n+1}U(1) \,,$$

for any object $\mathbf{Fields} \in \mathbf{H}$.

First we now observe in

• Universal topological Yang-Mills theory

that there is a fairly canonical such morphism $S^{n+1}_{tYM}$, namely the “atlas relative to manifolds” of $\flat \mathbf{B}^{n+1}U(1)$ given by the sheaf of closed differential (n+1)-forms. Analyzing what this morphism is like, when regarded as a local prequantum field theory by def. , shows that it is the “universal” higher topological Yang-Mills prequantum field theory. What this means becomes clear when we analyze the possible boundary field theories of this theory.

• Universal boundary condition for $S_{tYM}$: Differential cohomology and Cheeger-Simons field theory

• General boundary conditions: Higher Chern-Weil theory and $\infty$-Chern-Simons theory

where we discuss how the boundary theories for $S^{n+1}_{tYM}$ are precisely the prequantum field theories of higher Chern-Simons theory-type, the ∞-Chern-Simons theories. These include ordinary 3d Chern-Simons theory, higher dimensional Chern-Simons theory on ordinary gauge fields but also higher Chern-Simons theory on higher gauge fields such as the String 2-group 7-dimensional Chern-Simons theory, the AKSZ sigma-models, and also closed string field theory.

This shows how $\infty$-Chern-Simons theories arise canonically as precisely the local boundary field theories of the canonical local field theory which exists in any differential cohesive (∞,1)-topos.

Continuing in this vein we can then work out what all the further higher codimension boundary field theories and defect field theories of this universal higher topological Yang-Mills theory and hence of ∞-Chern-Simons theories are. We find

• Topological Chern-Simons boundaries

which are given by generalized “Bohr-Sommerfeld isotropic subspaces” of the moduli stacks of $\infty$-Chern-Simons fields.

Then…

(…)

$d = n + 1$, Universal topological Yang-Mills theory $S_{tYM}$

There is a special and especially simple map to the coefficient object $\flat \mathbf{B}^{n+1} U(1)$ for flat local action functionals/prequantum n-bundles, namely the map

$$\array{ \Omega^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \mathbf{B}^{n+1} \flat U(1) }$$

which in components is the inclusion of closed differential forms into de Rham hypercohomology cocoycles.

We here introduce and describe this map and then regard it as a local action functional of a local prequantum field theory according to def. . Below in Higher Chern-Simons prequantum field theory we find that this field theory is such that close to its boundaries it looks like (higher) topological Yang-Mills theory for every possible higher gauge group and every possible invariant polynomial on it, as one considers every possible boundary condition. Therefore we here refer to this as the “universal topological Yang-Mills theory”.

Smooth moduli stacks of fields: Smooth $\infty$-groupoids

The notion of a sheaf of chain complexes or equivalently of a chain complex of sheaves over a fixed topological space has a long tradition in homological algebra. Many sheaves however are naturally considered not on one fixed space, but on “all of them”. For instance differential forms in any degree may be “pulled back” along any smooth function between smooth manifolds. Accordingly if we regard the whole category SmoothMfd of smooth manifolds as a replacement for and generalization of the category of open subsets of any given one, then differential forms constitute a sheaf on that site, hence a functor

$$\Omega^{n} \colon SmoothMfd^{op} \to Set \,.$$

In particular for $n = 0$ this is just the sheaf of smooth functions

$$\underline{\mathbb{R}} = C^\infty(-,\mathbb{R}) \colon SmoothMfd^{op} \to Set \,.$$

One way to think of this is that this sheaf is the real line $\mathbb{R}$ first of all as a set – which is the value of $\underline{\mathbb{R}}$ on the point $\ast$ – and secondly equipped with its canonical smooth structure which is encoded by the system of all sets $C^\infty(X,\mathbb{R})$ of smooth functions from any smooth manifold $X$, and the information $C^\infty(\phi, \mathbb{R}) \colon C^\infty(Y, \mathbb{R}) \to C^\infty(X,\mathbb{R})$ of how these functions pull back (precompose) along any smooth function of test manifolds $\phi \colon X \to Y$.

Regarding the smooth manifold $\mathbb{R}$ this way means regarding it as what is sometimes called a diffeological space, and what here more generally call a smooth space.

Therefore we will often just write $\mathbb{R}$ when we really mean the sheaf $\underline{\mathbb{R}}$ represented by it.

The Dold-Kan map of def. directly extends to (pre-)sheaves which regard a (pre-)sheaf of chain complexes as a presheaf of Kan complexes

$$DK \;\colon\; pSh(SmoothMfd, Ch_\bullet) \stackrel{}{\to} pSh(SmoothMfd, KanCplx) \hookrightarrow pSh(SmoothMfd, sSet) \,.$$

By the previous reasoning, a presheaf of Kan complexes on the category of smooth manifolds, we may think of as being a plain Kan complex, hence ∞-groupoid (the value of the presheaf on the point), together with a rule for what the smooth functions into it are, for each smooth testmanifold. Hence we think of it as a smooth ∞-groupoid.

A basic example is a Lie groupoid $\mathcal{G}_\bullet \in Grpd(SmoothMfd)$, which represents a presheaf of Kan complexes on smooth manifolds by

$$U \mapsto N( C^\infty(U,\mathcal{G}_1) \stackrel{\to}{\to} C^\infty(U,\mathcal{G}_0) ) \,.$$

Previously we have considered higher Dijkgraaf-Witten as taking place in the homotopy theory of plain ∞-groupoids (geometrically discrete ∞-groupoids). Now we would like to have “a homotopy theory of smooth ∞-groupoids”.

(A modern term for “a homotopy theory” is “an (∞,1)-category”, but for a heuristic idea of what is going on some readers may find it helpful to think of “a homotopy theory” instead.)

In order to define “a homotopy theory” of smooth ∞-groupoids is, to be denoted Smooth∞Grpd, we need to say what a homotopy and hence what a homotopy equivalence is supposed to be. Since smooth structure should be a local property witnessed on small smooth open balls, we declare that

$$Smooth\infty Grpd \coloneqq L_{lhe} \; pSh(SmoothMfd, KanCplx)$$

is the homotopy theory obtained by “universally turning stalk-wise homotopy equivalences of Kan complexes, def. , into actual homotopy equivalences”. The formal definition of this idea is called the simplicial localization of $pSh(SmoothMfd, KanCplx)$ at the stalkwise homotopy equivalences.

We call ∞Grpd also the differential cohesive ∞-topos of smooth ∞-groupoids. For brevity and since most everything we discuss in the following holds for arbitrary differential cohesive (∞,1)-toposes, we from now on denote it by

$$\mathbf{H} \coloneqq Smooth \infty Grpd \,.$$

We now immediately turn to a simple example that illustrates some basic aspects of this construction.

The canonical local action functional: Differential forms in de Rham hypercohomology

As a first example for how to work in the homotopy theory of smooth ∞-groupoids we have the following.

Proposition/Example

The canonical chain map

$$\array{ \flat U(1) &\to& 0 &\to& 0 &\to& \cdots &\to& 0 \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow \\ C^\infty(-,U(1)) &\stackrel{d log}{\to}& \Omega^1 &\stackrel{d}{\to}& \Omega^2 &\stackrel{d}{\to}& \cdots &\to& \Omega^n_{cl} } \,,$$

where the left map includes the sheaf of (locally) constant $U(1)$-valued functions into that of all $U(1)$-valued smooth functions, is a stalk-wise quasi-isomorphism. Hence under the Dold-Kan correspondence, def. , both present the same smooth ∞-groupoid

$$\flat \mathbf{B}^n U(1) \simeq \mathbf{B}^n \flat U(1) \in Smooth\infty Grpd \,.$$

Proof

By definition and using prop. we need to check that over a small enough smooth open ball, the chain map becomes a quasi-isomorphism. But on an open ball this is the statement of the Poincaré lemma.

Accordingly we have a canonical inclusion:

Definition

For $n \in \mathbb{N}$, write

$$\Omega^{n}_{cl} \in SmoothSpace \hookrightarrow Smooth \infty Grpd$$

for the sheaf of closed smooth differential forms of degree $n$, regarded as a smooth space, regarded as a smooth ∞-groupoid. Write

$$\Omega^{n}_{cl} \to \flat \mathbf{B}^n U(1)$$

for the image in morphisms of Smooth∞Grpd under the Dold-Kan correspondence, def. , of the chain map

$$\array{ 0 &\to& 0 &\to& 0 &\to& \cdots &\to& \Omega^{n}_{cl} \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow^{\mathrlap{id}} \\ C^\infty(-,U(1)) &\stackrel{d log}{\to}& \Omega^1 &\stackrel{d}{\to}& \Omega^2 &\stackrel{d}{\to}& \cdots &\to& \Omega^n_{cl} } \,,$$

Remark

The map $\Omega^n_{cl} \to \flat \mathbf{B}^n U(1)$ in def. may be characterized more abstractly as follows:

for every smooth manifold $\Sigma$, theinduced morphism of mapping spaces

$$[\Sigma, \Omega^n_{cl}] \to [\Sigma, \flat \mathbf{B}^n U(1)]$$

is a 1-epimorphism, hence a stalk-wise epimorphism on connected components.

Definition

By def. we may now regard the map

$$\array{ \Omega^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \mathbf{B}^{n+1} \flat U(1) }$$

of prop. as a local action functional for an $(n+1)$-dimensional local prequantum field theory. We call this the “universal higher topological Yang-Mills theory” for reasons that become clear as we anaylize its boundary field theories now.

$d = n + 0$, Higher Chern-Simons field theories

We consider now the boundary field theories for the “universal topological Yang-Mills theory” of def. .

Universal boundary condition for $S_{tYM}$: Differential cohomology and Cheeger-Simons field theory

Where the plain (∞,n)-category of cobordisms $Bord_n$ is freely generated from the point $\ast$ alone, so the $(\infty,n)$-category of cobordisms with possibly a marked boundary is freely generated from the point and one new morphism

$$\emptyset \to \ast$$

which we think of as being the interval $D^1$ with one end “marked”. Now a local field theory with local action functional according to def. encodes not only the value on the point, which we now take to be

$$\exp(i S) \colon \ast \mapsto \left[ \array{ \Omega^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \mathbf{B}^{n+1} \flat U(1) } \right] \,,$$

but moreover one morphism in $Span_n(\mathbf{H}, \mathbf{B}^{n+1}\flat U(1))$ from the trivial field configuration with trivial action to this data, hence (as amplified in FV) a diagram in $\mathbf{H}$ of the form

$$\array{ && \mathbf{Fields}^{\partial} \\ & \swarrow && \searrow \\ \ast && \swArrow && \Omega^{n+1}_{cl} \\ & \searrow && \swarrow \\ && \mathbf{B}^{n+1}\flat U(1) } \,.$$

Therefore defining such boundary data means defining a moduli stack $\mathbf{Fields}^{\partial}$ of boundary field configurations, together with a homotopy filling the above diagram which encodes the relative action functional on this boundary data.

In order to find all possible such boundary data for $\exp(i S_{tYM})$, we can make use of the homotopy fiber product construction of def. to find the universal such boundary data, the one through which any other one factors.

Proposition

The universal boudnary condition for $\exp(i S_{tYM})$, hence the homotopy fiber product $\ast \underset{\mathbf{B}^{n+1}\flat U(1)}{\times} \Omega^{n+1}_{cl}$, is given by the image under the Dold-Kan map, def. , of the Deligne complex

$$\mathbf{B}^n U(1)_{conn} \coloneqq DK( \underline{U}(1) \stackrel{d log}{\to} \Omega^1 \stackrel{d}{\to} \cdots \to \Omega^{n-1} \stackrel{d}{\to} \Omega^n ) \,,$$

hence the universal boundary data is

$$\array{ && \mathbf{B}^n U(1)_{conn} \\ & \swarrow && \searrow^{\mathrlap{F_{(-)}}} \\ \ast && \swArrow && \Omega^{n+1}_{cl} \\ & \searrow && \swarrow \\ && \mathbf{B}^{n+1}\flat U(1) } \,.$$

The boundary field theory defined this way we may call Cheeger-Simons field theory.

General boundary condition for $S_{tYM}$: Higher Chern-Weil theory and $\infty$-Chern-Simons theory

By the universal property of the homotopy fiber product we then have the following statement.

Proposition

Boundary field theories for $\exp(i S_{tYM})$ are equivalently moduli stacks $\mathbf{Fields} \in \mathbf{H}$ equipped with maps

$$\exp(i S_{CS}) \colon \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}$$

hence equipped with a circle n-bundle with connection (the prequantum n-bundle of the boundary theory).

Moreover, the universal property of the Cheeger-Simons field theory identifies all these boundary theories as being of higher Chern-Simons type, in that they have a curvature associated to them which is an invariant differential form (invariant polynomial) on the moduli stack

$$\array{ \mathbf{Fields} \\ {}^{\mathllap{\exp(i S_{CS})}}\downarrow & \searrow^{\mathrlap{\langle F_{(-)} \wedge \cdots F_{(-)}\rangle}} \\ \mathbf{B}^n U(1)_{conn} &\underset{F_{(-)}}{\to}& \Omega^{n+1}_{cl} } \,.$$

We call these theories of ∞-Chern-Simons theory-type.

Example

For $G$ a simply connected simple Lie group and

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

the moduli stack of $G$-principal connections, the local action functional of ordinary 3d Chern-Simons theory is of the form

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

This prequantum 3-bundle is the Chern-Simons circle 3-bundle.

Many more examples… e.g. 7d Chern-Simons theory, AKSZ sigma-model, etc….

Geometric defects for $S_{tYM}$ from Chern-Simons invariants: Higher holonomy, parallel transport, fiber integration in differential cohomology

While we may think of the (∞,n)-category of cobordisms $Bord_n$ as built from smooth manifolds, the cobordism theorem clearly states that these just serve as a presentation of a structure that is not intrinsically related to smooth or even topological structure. This is made manifest by prop. : the value of a local prequantum field theory on a k-morphism in $Bord_n$ depends only on the homotopy type of the $k$-dimensional manifold that presents this $k$-morphism. Of course this is precisely the property that the term “topological” in topological field theory is referring to.

But boundaries and defects of a topological field theory may add extra structure to the theory which is not “purely topological” in this way. Here we consider a canonical class of defects for universal higher topological Yang-Mills theory, def. , and $\infty$-Chern-Simons theory, def. , which implements the expected higher parallel transport/higher holonomy of the higher Chern-Simons type action functionals over actual smooth manifolds.

Proposition

For $k \leq n$ and for $\Sigma_k$ and oriented closed manifold, there is a morphism of smooth moduli stacks

$$\exp(i \int_{\Sigma_k}(-)) \colon [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k}U(1)_{conn}$$

which is compatible with the standard fiber integration of differential forms and with transgression in ordinary cohomology in that it fits into a commuting diagram

$$\array{ [\Sigma_k, \flat \mathbf{B}^n U(1)] &\stackrel{}{\to}& \flat \mathbf{B}^n U(1) \\ \downarrow && \downarrow \\ [\Sigma_k, \mathbf{B}^n U(1)_{conn}] &\stackrel{\exp(i \int_{\Sigma_k}(-) )}{\to}& \mathbf{B}^{n-k} U(1)_{conn} \\ \downarrow && \downarrow \\ [\Sigma_k, \Omega^{n+1}_{cl}] &\stackrel{\int_{\Sigma_k}}{\to}& \Omega^{n-k+1}_{cl} } \,.$$

More generally, if $\Sigma_k$ is a manifold with boundary then there is a diagram

$$\array{ && [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \\ & {}^{\mathllap{(-)|_{\partial \Sigma}}}\swarrow && \searrow^{\mathrlap{\omega_\Sigma}} \\ [\partial \Sigma_k, \mathbf{B}^n U(1)_{conn}] && \swArrow_{\exp(2 \pi i \int_{\Sigma})} && \Omega^{n-k+1} \\ & {}_{\mathllap{\exp(2 \pi i \int_{\partial \Sigma}(-))}}\searrow && \swarrow \\ && \mathbf{B}^{n-k+1}U(1)_{conn} } \,,$$

where the bottom left map is the fiber integration from before, applied to the closed boundary, and where the homotopy filling the diagram is such that it reproduces this fiber integration in the case that the boundary is empty, in that

$$\array{ && [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \\ & {}^{\mathllap{(-)|_{\partial \Sigma}}}\swarrow && \searrow^{\mathrlap{\omega_\Sigma}} \\ [\emptyset, \mathbf{B}^n U(1)_{conn}] && \swArrow_{\exp(2 \pi i \int_{\Sigma})} && \Omega^{n-k+1} \\ & {}_{\exp(2 \pi i \int_{\emptyset}(-))}\searrow && \swarrow \\ && \mathbf{B}^{n-k+1}U(1)_{conn} } \;\;\; \simeq \;\;\; \array{ && [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \\ && \downarrow \\ && \mathbf{B}^{n-k}U(1)_{conn} \\ & {}^{\mathllap{(-)|_{\partial \Sigma}}}\swarrow && \searrow^{\mathrlal{\omega_\Sigma}} \\ \ast && \swArrow_{\exp(2 \pi i \int_{\Sigma})} && \Omega^{n-k+1} \\ & {}_{}\searrow && \swarrow \\ && \mathbf{B}^{n-k+1}U(1)_{conn} } \,,$$

Proof

This follows by unwinding the traditional formulas for fiber integration in differential cohomology, reformulating them in homotopy theory and observing that they are natural in their arguments, hence extend to morphisms of higher stacks, as discussed here.

Remark

A homotopy/gauge equivalence between a circle n-bundle with connection $(P,\nabla)$ and a trivial circle $n$-bundle with connection given by a globally defined differential form $(0,\omega)$ is equivalently a section/trivialization of the underlying circle n-bundle. Therefore the above says that the fiber integration of an $n$-connection over a manifold with boundary is equivalently a section of the transgression of the underlying bundle to the boundary.

We may combine this with the $\infty$-Chern-Simons action functional:

Definition

Let $\exp(i S_{CS}) \colon \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}$ be an ∞-Chern-Simons theory local action functional as in prop. . Then for $\Sigma_k$ an oriented smooth $k$-dimensional manifold with boundary, the corresponding transgression defect is the pasting-composite

$$\array{ && && [\Sigma, \mathbf{Fields}] \\ && & {}^{\mathllap{(-)|_{\partial \Sigma}}}\swarrow && \searrow^{\mathrlap{[\Sigma_k, \exp(i S_{CS})]}} \\ && [\partial \Sigma, \mathbf{Fields}] && && [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \\ && & \searrow& & {}^{\mathllap{(-)|_{\partial \Sigma}}}\swarrow && \searrow^{\mathrlap{\omega_\Sigma}} \\ && && [\partial \Sigma_k, \mathbf{B}^n U(1)_{conn}] && \swArrow_{\exp(2 \pi i \int_{\Sigma}(-))} && \Omega^{n-k+1} \\ && && & {}_{\exp(2 \pi i \int_{\partial \Sigma}(-))}\searrow && \swarrow \\ && && && \mathbf{B}^{n-k+1}U(1)_{conn} } \,,$$

or rather its further pullback to the shape modality

$$\array{ && [\Pi(\Sigma), \mathbf{Fields}] \\ & \swarrow \\ [\Pi(\partial \Sigma), \mathbf{Fields}] } \,.$$

This is a defect between the boundary transgression, def. , of the $\infty$-Chern-Simons theory and the tautological higher differential higher Chern-Simons theory.

We see below that both the Wess-Zumino-Witten theory as well as Wilson lines in Chern-Simons theory arise from transgression defects this way.

(…)

$d = n-1$, Topological Chern-Simons boundaries

(…)

$d = n-1$, Wess-Zumino-Witten field theories

(…)

defect given by transgression over circle

(…)

$d = n-2$, Wilson loop/Wilson surface field theories

(…)

defect given by transgression over sphere

(…)

Related concepts

• Peierls bracket

• locality of local field theory is obsrvationally related to causal locality

• prequantum geometry

• higher prequantum geometry

• traditional Lagrangian mechanics and Hamiltonian mechanics are naturally embedding into higher prequantum field theory by the notion of prequantized Lagrangian correspondences

• quantization of prequantum field theory is naturally formulated in terms of motivic quantization

References

For intorduction and survey see

• Urs Schreiber, Higher Prequantum Geometry

The idea of formulating local prequantum field theory by spans in a slice over a “space of phases” in higher geometry has been expressed in the unpublished note

• Urs Schreiber, Nonabelian cocycles and their quantum symmetries (2008)

A formulation of the idea for Dijkgraaf-Witten theory-type field theories is indicated in section 3 of

• Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (2010)

based on the considerations in section 3.2 of

• Jacob Lurie, On the Classification of Topological Field Theories (2009).

Based on the general formulation of the more general field theory with defects described in section 4.3 there, in

• Domenico Fiorenza, Alessandro Valentino, Boundary conditions in local TFTs (in preparation)

the structure of such domain walls/defects/branes are analyzed in the prequantum theory, hence with coefficients in an (∞,n)-category of spans.

The study of local prequantum ∞-Chern-Simons theory with its codimension-1 ∞-Wess-Zumino-Witten theory and codimension 2-Wilson line-theory in this fashion, in an ambient cohesive (∞,1)-topos is discussed in

• Urs Schreiber et al., Local prequantum field theory

Much of the content of this entry here serve as, or arose as, lecture notes for

• Urs Schreiber, Higher Chern-Simons theory Introduction, at the workshop Chern-Simons Theory: Geometry, Topology and Physics University of Pittsburgh (May 2013)

Later this was developed further with Igor Khavkine, exposition is here:

• Higher prequantum geometry I: The need for prequantum geometry

• Higher prequantum geometry II: The principle of extremal action – Comonadically

• Higher prequantum geometry III: The global action functional – Cohomologically

• Higher prequantum geometry IV: The covariant phase space – Transgressively

• Higher prequantum geometry V: The local observables – Lie theoretically