nLab geometry of physics -- local prequantum field theory

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

Local (topological) prequantum field theory

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 (lpqft).

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

(…)

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.

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.

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

$$\left( X \stackrel {\overset{id}{\to}} { \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} } X \right) \,.$$

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

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

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

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

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 $X \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y$ 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 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 $\matrhbf{B}G$:

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

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

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

Unde 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. .