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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
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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 first section gives the the definitions and general properties of
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
After that we turn to the general case of examples of
Here the pattern of the discussion of examples is the following:
Prequantum field theory deals with “spaces of physical fields”. These spaces of fields are, in general, richer than just plain sets in two ways
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.
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
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
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
and call the flat modality.
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.
After sketching out the general
we formulate first
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
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
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 |
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
space of field configurations over incoming and outgoing pieces of worldvolume/spacetime;
the space of field configurations over the bulk worldvolume/spacetime – the trajectories of fields;
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
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
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
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
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
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
assigns a span/correspondence of spaces of field configurations and trajectories
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)
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
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
such that the spans of spaces of fields are those specified before, hence such that it fits as a lift into the diagram
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
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
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
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.
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.
For $n \in \mathbb{N}$, write
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
for the corresponding symmetric monoidal $(\infty,n)$-category of cobordisms equipped with S-structure on their $n$-stabilized tangent bundle.
In this notation we have an identification
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 “$Bord_n^{fr}$”.
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
sends the point to some fully dualizable object $Z(\ast) \in \mathcal{C}$ and sends
the sphere $S^2$ to the 2-dimensional higher trace of the identity,
and so on.
For $\mathbf{H}$ an ∞-topos, and $n \in \mathbb{N}$, write
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
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
(…)
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
We write
for the resulting symmetric monoidal (∞,n)-category.
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.
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.
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
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
For $\mathbf{H} =$ ∞Grpd this is the perspective in (FHLT, section 3).
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
As a corollary of prop. we have:
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
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
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 $\Sigma_k$.
(…)
(…)
(…)
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.
Dijkgraaf-Witten theory in dimension 1 is what results when one regards a group character of a finite group $G$ as a local
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).
Essence of gauge theory: Groupoids and basic homotopy 1-type theory
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.
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
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
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
Similarly then for yet another gauge configuration to another field configuration
then composing them gives the picture
We now discuss this notion of groupoids more formally.
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.
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
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;
$\circ$ 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.
This data is visualized as follows. The set of morphisms is
and the set of pairs of composable morphisms is
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.
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
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
In particular a group character $c \colon G \to U(1)$ is equivalently a groupoid homomorphism
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
to mean explicitly the discrete group underlying the circle group. (Here “$\flat$” denotes the “flat modality”.)
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
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
for $x_1, x_2 \in X$, $g \in G$.
As an important special case we have:
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. :
Another canonical action is the action of $G$ on itself by right multiplication. The corresponding action groupoid we write
The constant map $G \to \ast$ induces a canonical morphism
This is known as the $G$-universal principal bundle. See below in for more on this.
The interval $I$ is the groupoid with
For $\Sigma$ a topological space, its fundamental groupoid $\Pi_1(\Sigma)$ is
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
which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.
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:
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
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. ).
For $X,Y$ two groupoids, the mapping groupoid $[X,Y]$ or $Y^X$ is
A (homotopy-) equivalence of groupoids is a morphism $\mathcal{G} \to \mathcal{K}$ which has a left and right inverse up to homotopy.
The map
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.
Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example – a skeleton.
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.
Given two morphisms of groupoids $X \stackrel{f}{\leftarrow} B \stackrel{g}{\to} Y$ their homotopy fiber product
hence the ordinary iterated fiber product over the path space groupoid, as indicated.
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:
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 $\matrhbf{B}G$:
Namely both squares in the following diagram are pullback squares
(This is the first example of the more general phenomenon of universal principal infinity-bundles.)
For $X$ a groupoid and $\ast \to X$ a point in it, we call
the loop space groupoid of $X$.
For $G$ a group and $\mathbf{B}G$ its delooping groupoid from example , we have
Hence $G$ is the loop space object of its own delooping, as it should be.
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
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 .
The free loop space object is
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
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.
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.
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:
For an abelian group such as $\flat U(1)$ we have
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
Unde the free loop space object construction this becomes
hence
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:
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 $\Sigma$:
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.
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
is sent under the operation of mapping into the moduli space of fields
to a span of groupoids
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
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.
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.
A (2,1)-category $\mathcal{C}$ is
a collection $\mathcal{C}_0$ – the “collection of objects”;
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 $Y$;
for each triple $(X,Y,Z) \in \mathcal{C}_0 \times \mathcal{C}_0 \times \mathcal{C}_0$ a groupoid homomorphism (functor)
called composition or horizontal composition for emphasis;
for each quadruple $(W,X,Y,Z,)$ a homotopy – the associator –
(…) 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
If all associators $\alpha$ can and are chosen to be the identity then this is called a strict (2,1)-category.
Write Grpd for the strict (2,1)-category, def. , whose
1-morphisms are functors $f \colon \mathcal{G} \to \mathcal{K}$;
2-morphisms are homotopies between these.
Write $Span_1(Grpd)$ for the (2,1)-category whose
1-morphisms are spans/correspondences of functors, hence
2-morphisms are diagrams in Grpd of the form
composition is given by forming the homotopy fiber product, def. , of the two adjacent homomorphisms of two spans, hence for two spans
and
their composite is the span which is the outer part of the diagram
There is the structure of a symmetric monoidal (2,1)-category on $Span_1(Grpd)$ by degreewise Cartesian product in Grpd.
An object $X$ of a symmetric monoidal (2,1)-category $\mathcal{C}^\otimes$ is fully dualizable if there exists
another object $X^\ast$, to be called the dual object;
a 1-morphism $ev_X \colon X^\ast \otimes X \to \mathbb{I}$, to be called the evaluation map;
a 1-morphism $coev_X \colon \mathbb{I} \to X \otimes X^\ast$, to be called the coevaluation map;
and
and
(the saddle?)
and
(the co-saddle)
such that these exhibit an adjunction and are themselves adjoint (…).
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
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
is given by the span
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
is its free loop space object, prop. :
The second order covaluation map on the span trace of the identity is
By prop. the trace of the identity is given by the composite span
Along these lines one checks the required zig-zag identities.
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
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. .
Created on January 31, 2014 at 01:01:29. See the history of this page for a list of all contributions to it.