quantization of differential nonabelian cohomology

under construction

This page is concerned with the quest for the mathematical model of what in physics is called the path integral:

the process of quantization of classical σ-model background field data – encoded as differential nonabelian cocycles – to the corresponding extended quantum field theory.


  1. Background and motivation

  2. General procedure

    1. terminology

      1. the notion of space

      2. the notion of path

      3. the notion of trajectory

      4. the notion of background field

      5. the notion of action functional

      6. the notion of section and state

    2. the quantum propagation map

  3. Examples

    1. the electromagnetically charged particle

    2. Dijkgraaf-Witten theory

    3. Yetter model

Background and motivation

The search is on for the abstract formalization of the process of quantization – the process that reads in a “classical field theory” – for instance presented in form of a gauge theory or in form of a σ-model background field data – and spits out the corresponding quantum field theory.

There is an open problem of mathematical (-physics) model building:

One that is to quantization as, say, symplectic geometry is to Hamiltonian mechanics?

The formalism of FQFT clearly suggests that the fundamental description of quantization is some natural operation on higher functors.

While for various aspects and facets of this question there are well-developed formalisms – such as geometric quantization or deformation quantization or BV theory – a full answer is certainly still missing, not the least because the full formalization of the question itself has still to be established.

Considerable progress on this formulation of the question has been achieved with the formalization and proof of the cobordism hypothesis in On the Classification of Topological Field Theories by Jacob Lurie. This at least indicates what the result of any full quantization procedure should be in that it clarifies what exactly a topological quantum field theory is: a morphism from the (∞,n)-category of cobordisms Z:Bord nCZ : Bord_n \to C.

In On the Classification of Topological Field Theories Jacob Lurie indicates some first steps towards finding a similar formalization of “classical field theory” (in terms of his (,n)(\infty,n)-categories of “families”) and a systematic procedure for turning the classical theory into the quantum theory. These thoughts were further developed in the article

(See there also for some comments along the lines here.)

At the same time Stephan Stolz and Peter Teichner in their work on geometric models for elliptic cohomology? match QFTs to cohomology theories and are seeing there the analogy between pushforward in cohomology theory and the quantization process from classical field theories to quantm field theories.

For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. and Stolz-Teichner mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of abstract nonsense answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.

Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here is more in the spirit as indicated to some extent in the work of David Ben-Zvi et. al, which is described in some detail at geometric ∞-function theory. This is essentially based on the idea that a quantum field theory is obtained from a σ-model target space object XX by homming extended cobordism cospans Σ inΣΣ out\Sigma_in \to \Sigma \leftarrow \Sigma_{out} into the target object and then pull-pushing geometric function objects through the resulting spans of configuration space objects [Σ in,X][Σ,X][Σ out,X][\Sigma_{in},X] \leftarrow [\Sigma,X] \to [\Sigma_{out},X].

The resulting pull-push operation is an example or a generalization of what John Baez discusses under the term groupoidification. David Ben-Zvi et al would speak of geometric function theory.

The main result of David Ben-Zvi et. al.’s work on this approach is that they point out that as soon as the geometric function object one uses satisfies the two fundamental theorems of geometric infinity-function theory, a considerable amount of rich structure – that has in parts been known by itself – gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable geometric function theory as induced from the underlying geometry of configuration space objects [Σ,X][\Sigma,X] as well as the loop space objects of XX.

Here in this entry the aim is to discuss in detail aspects and models for the quantization procedure that is suggested by applying geometric function theory to σ-model backgrounds.

But more precisely, the premise of the discussion here is that a fundamental understanding of the situation is obtained after regarding seriously the fact that

  1. background fields are not just encoded by the cohomology with coefficients in the target space object XX, but by the corresponding differential cohomology;

  2. and that the functorial notion of the FQFT that one wants to obtain suggests that these differential cocycles themselves are encoded functorially, as described at differential nonabelian cohomology.

This means that we shall take the classical structure \nabla which the sought-after quantization procedure is supposed to send to an FQFT Z :Bord nCZ_\nabla : Bord_n \to C to be, essentially, a functor

:Π n(X)A \nabla : \Pi_n(X) \to A

from some path n-groupoid of XX to some coefficient object AA.

Here the k-morphisms in Π n(X)\Pi_n(X) are generated from the existing kk-morphisms in XX as well as the kk-dimensional smooth paths in XX. The functor \nabla on this encodes the parallel transport of a connection on a principal ∞-bundle over XX: this assignment of k-morphisms in AA to kk-dimensional paths in XX is the assignment of the classical action to trajectories in XX.

More precisely, this is the classical action assigned to topologically disk-shaped kk-paths. The full classical action of a classical σ-model quantum field theory is an extension of \nabla

Π(X)A V exp(S ) Bord(X) \array{ \Pi(X) \stackrel{\nabla}{\to} A &\to& V \\ \downarrow & \nearrow_{\exp(S_\nabla)} \\ Bord(X) }

from disk-shaped cobordisms to all cobordisms in XX (possibly and usually after an extension of the codomain to some new codomain VV). This extension exp(iS )\exp(i S_\nabla) knows not just the parallel transport, but also the holonomy of the classical background field.

The question to be analyzed here is how this classical action exp(iS ):Bord n(X)F\exp(i S_\nabla) : Bord_n(X) \to F transmutes systematically to an FQFT Z :Bord nCZ_\nabla : Bord_n \to C on abstract bordisms (possibly with extra structure, if we pass from TQFT to richer theories like CFT?). This step should be the path integral:

in some way the assignment of Z Z_\nabla to a cobordism Σ\Sigma is expected to be obtained by “summing” in some way the value of exp(S )\exp(S_\nabla) on elements in the fiber of the forgetful functor Bord n(X)Bord nBord_n(X) \to Bord_n over Σ\Sigma, i.e. over all possible ways to map Σ\Sigma into XX – the sum of exp(S (ϕ))\exp(S_\nabla(\phi)) over all paths or field configurations ϕ:ΣX\phi : \Sigma \to X.

General procedure

This section describes the general formalism that we want to study as a candidate for (aspects of) a fundamental abstract-nonsense? description of quantization?. The next section then picks special (simple) cases and works out the action of the abstract machinery on these.

In rough outline the procedure is this:

  • In a context H\mathbf{H} of sufficiently general spaces;

  • we extend the background field parallel transport :Π(X)A\nabla : \Pi(X) \to A to an action functional exp(S ):Bord(X)V\exp(S_\nabla) : Bord(X) \to V.

  • Then for each cobordism? Σ\Sigma the component of the action functional exp(S ) Σ:[Σ,X]V\exp(S_\nabla)|_\Sigma : [\Sigma,X] \to V on Σ\Sigma defines a total VV-bundle? space E ΣE_\Sigma;

  • the points of E ΣE_\Sigma are the Dirac-δ\delta-distributional sections or quantum states

    – these are picked and added up by the objects in the over category? geometric function object? C(E x):=H/E xC(E_x) := \mathbf{H}/{E_x}.

  • the quantum propagation along a cobordism? cospan? Σ inΣΣ out\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} is the pull-push of geometric function theory?/groupoidification?

    Z (Σ):C(E Σ in)out *in *C(E Σ out) Z_\nabla(\Sigma) : C(E_{\Sigma_{in}}) \stackrel{out_* in^*}{\to} C(E_{\Sigma_{out}})

    through the corresponding span? E Σ inE ΣE Σ outE_{\Sigma_{in}} \leftarrow E_{\Sigma} \to E_{\Sigma_{out}} of total spaces.


The definition of quantum propation defined by a classical background field that we describe below is quite simple once we suitably organize the material by establishing a bit of terminology.

the notion of space

We need to talk about various spaces of sorts:

  • a space XX that is the target space – often, but not always, to be interpreted as physical spacetime – in which the dd-dimensional object propagates – for d=0d=0 for instance a point particle such as an electron – whose dynamics we want to encode.

    This space might be just a manifold? but we want to allow it to be a kind of space a bit more general than that, for instance an orbifold?.

    When we are describing gauge theory? the target space is smooth refinement of a classifying space? BG\mathbf{B}G.

    When we are describing finite gauge theory such as Dijkgraaf-Witten theory?, the target space is just the one-object groupoid? BG\mathbf{B}G that is the delooping? of the gauge group GG.

  • Moreover, for each (d+1)(d+1)-dimensional manifold? Σ\Sigma we need a path space or configuration space [Σ,X][\Sigma,X] of maps from Σ\Sigma into XX.

The general abstract nonsence? for dealing with general spaces of this sort is that of space and quantity?. As described at motivation for sheaves, cohomology and higher stacks, this leads one to describe a space as an ∞-stack? on some site? SS.

In the simplest case, which the reader should keep in mind, the site? in question is the point? and an ∞-stack? on it is just an ∞-groupoid?. In turn in simple cases of this, which the reader should still keep in mind, this ∞-groupoid? will be just a groupoid? or at most a 2-groupoid?.

Whichever choice one makes, the collection of all such generalized spaces modeled on a certain site? SS forms an (∞,1)-topos? H\mathbf{H}.

Despite – possibly – its appearance, the reader should take this statement as suggesting an immense simplification instead of a huge complification: the language of (∞,1)-topos? is a user interface that makes pretty sophisticated and rich notions of generalized spaces have the same look-and-feel as just plain topological space?s. It’s the inner workings of the formalism that will take of things coming out right even when we talk about richer objects.

In particular, it is the machinery of models for ∞-stack (∞,1)-toposes? that provides all the tools for actually performing the operations that we shall consider. The reader unfamiliar with that will suffer little loss from concentrating his or attention on the toy examples where all “spaces” involved are finite groupoid?s and trust that everything goes through analogously also in the other examples.

the notion of path

With an (∞,1)-topos? H\mathbf{H}-fixed we are in a context that allows us to study cohomology?. A classical background field on a target space XX is in parts a cocycle on XX with values in some coefficient object AHA \in \mathbf{H}.

But, as indicated above, it is actually, more: it is a differential cocycle on XX: something that depends not just on XX itself, but also on a notion of paths in XX.

To encode this we may pick a cosimplicial object?

Δ H:ΔH \Delta_H : \Delta \to \mathbf{H}

in our (,1)(\infty,1)-category of spaces, that encodes which object in H\mathbf{H} models the standard kk-simplex? regarded as a space. For instance if H\mathbf{H} is the (∞,1)-topos? of Lie ∞-groupoid?s modeled by ∞-stack?s on Diff?, we would take Δ H n\Delta_H^n to be the standard nn-simplex Δ H n n\Delta^n_H \subset \mathbb{R}^n regarded as a smooth manifold? in the standard way.

Then denote the space of disk-shaped kk-dimensional paths in XX by

[Δ H n,X]. [\Delta_H^n,X] \,.

These spaces glue together to the path ∞-groupoid

Π(X):=lim [Δ H ,X]. \Pi(X) := \lim_\to [\Delta_H^\bullet, X] \,.

This is an ∞-groupoid? whose k-morphism? are generated from the original kk-morphisms of XX and the kk-dimensional path ϕ:Δ H kX\phi : \Delta^k_H \to X.

For instance if X=Y//GX = Y//G is a global orbifold?, the 1-morphisms in Π(X)\Pi(X) are generated from smooth paths γ:y 1y 2\gamma: y_1 \to y_2 in YY as well as orbifold jumps g:y 1g(y 1)g : y_1 \to g(y_1) subject to the relation

y 1 g g(y 1) γ g(γ) y 2 g g(y 2). \array{ y_1 &\stackrel{g}{\to}& g(y_1) \\ \downarrow^{\gamma} && \downarrow^{g(\gamma)} \\ y_2 &\stackrel{g}{\to}& g(y_2) } \,.

Or if X=C(Y)X = C(Y) is realized as the Cech nerve? of some Cech cover? (U= iU i)Y(U = \coprod_i U_i) \to Y then Π(X)\Pi(X) is generated from paths (γ,i):(y 1,i)(y 2,i)(\gamma,i) : (y_1,i) \to (y_2,i) in U iU_i for all ii and transitions (y,i)(y,j)(y,i) \to (y,j) for all yU iU jy \in U_i \cap U_j subject to the condition

(y 1,i) (y 1,j) (γ,i) (γ,j) (y 2,i) (y 2,j). \array{ (y_1,i) &\stackrel{}{\to}& (y_1,j) \\ \downarrow^{(\gamma,i)} && \downarrow^{(\gamma,j)} \\ (y_2,i) &\stackrel{}{\to}& (y_2,j) } \,.

the notion of trajectory

More generally, let DD be some poset? and specify a functor

Σ H:DH. \Sigma_H : D \to \mathbf{H} \,.

We think the image of some kDk \in D under Σ H\Sigma_H as the realization in H\mathbf{H} of some abstract cobordism? whose boundary component inclusions are the images Σ H jΣ H k\Sigma^j_H \to \Sigma_H^k of all morphisms jkj \to k in DD.

So Σ H\Sigma_H encodes a multi-cospan? of cobordisms in HH. In principle Σ H\Sigma_H will in general be supposed to range over all cobordisms in some sense, but for our discussion it will be complete sufficient to concentrate on much less. Most of our discussion concerns a handful of cobordisms, usually just two of them and their composites.

In any case, by slight abuse of notation, we shall write

Bord (X):=Bord(X):=lim [Σ H ,X] Bord_\infty(X) := Bord(X) := \lim_\to [\Sigma_H^\bullet, X]

for the (homotopy?) colimit? over this multi-cospan? of cobordisms and speak of the \infty-groupoid of bordisms in XX. If we assume that Σ H\Sigma_H ranges at least over the disk-shaped cobordisms Δ H k\Delta^k_H we have a canonical inclusion

Π(X)Bord(X). \Pi(X) \hookrightarrow Bord(X) \,.

the notion of background field

A background field that is flat (meaning that it has vanishing field strength?) is a morphism

:Π(X)A \nabla : \Pi(X) \to A

in H\mathbf{H}.

The more general situation is described at …, but need not concern us here for the moment. Its quantization is structurally slightly but not, for our purposes, essentially different from that of the flat case.

This encodes an AA-principal ∞-bundle? with connection?: the value :xP x\nabla : x \mapsto P_x over a point is the fiber? of the principal \infty-bundle, and the value on a path :x 1γ 1x 2P x 1P x 2\nabla : x_1 \stackrel{\gamma_1}{\to} x_2 \mapsto P_{x_1} \to P_{x_2} is the parallel transport along γ\gamma.

In the case that A=BGA = \mathbf{B}G is the delooping? of an \infty-group GG, so that AA has a unique point? pt A:*BGpt_A : {*} \to \mathbf{B}G the total space of this principal ∞-bundle? is the (homotopy?) pullback? PP in

P * X[pt,X] pt BG, \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X \simeq [pt,X] &\stackrel{\nabla|_{pt}}{\to}& \mathbf{B}G } \,,

where pt\nabla|_{pt} is the component of \nabla (as a morphism out of a colimit) on points in XX.

We are interested in forming the associated \infty-bundle induced by a choice of representation?

ρ:AV \rho : A \to V

on some (∞,1)-category? VV. See below for examples.

As a cocycle this is just the composite

ρ *:Π(X)AρV. \rho_* \nabla : \Pi(X) \stackrel{\nabla}{\to} A \stackrel{\rho}{\to} V \,.

This is the background field that encodes the forces acting on the objects propagating in XX whose quantum dynamics we seek.

We will often abuse notation and write just \nabla for ρ *\rho_* \nabla, when the context is clear.

the notion of action functional

A choice of extension of the parallel transport morphism :Π(X)A\nabla : \Pi(X) \to A through the inclusion Π(X)Bord(X)\Pi(X) \to Bord(X) is the corresponding action functional exp(S )\exp(S_\nabla)

Π(X) V exp(S ) Bord(X). \array{ \Pi(X) &\stackrel{\nabla}{\to}& V \\ \downarrow & \nearrow_{\exp(S_\nabla)} \\ Bord(X) } \,.

This extension amounts to choosing (possibly uniquely) traces over parallel transport.

example Let :Π(X)Vect\nabla : \Pi(X) \to Vect encode the parallel transport in a vector bundle? with connection. And consider the extension to Bord 1(X)Bord_1(X) with just 1-dimensional cobordisms in XX. Then the extension amounts to finding a value for exp(S )\exp(S_\nabla) on the circle. To be functorial and symmetric monoidal, it must be true that the value on the circle is the result of cutting the circle open at any point, evaluating \nabla on the resulting 1-disk-shaped cobordism and then applying the trace on the vector space over the endpoint.

For each cobordism Σ\Sigma the component exp(S ) Σ\exp(S_\nabla)|_\Sigma of the action functional will define a total space of an associated bundle E ΣE_\Sigma. These are discussed in the next section.

the notion of section and state

A state of the quantum theory over a cobordism Σ\Sigma is supposed to be a section? of the bundle E Σ[Σ,X]E_\Sigma \to [\Sigma,X] over the configuration space [Σ,X][\Sigma,X] of fields on Σ\Sigma, where we allow generalized sections that may be distributional. This we define now.

In order to define sections and states, we need to fix a point? in the coefficient object VV, i.e. a morphism

pt V:*V pt_V : {*} \to V

from the terminal object? *{*} into VV. The choice of this point determines over which monoid? our quantum theory will be linear:

definition (ground monoid)

Given the pointed object? *pt VV{*} \stackrel{pt_V}{\to} V we say that the respective ground monoid is the endomorphism monoid?

Ω ptV=End V(pt V) \Omega_{pt} V = End_V(pt_V)

i.e. the lax pullback?

Ω ptV * pt V * pt V V. \array{ \Omega_{pt} V &\to& {*} \\ \downarrow && \downarrow^{pt_V} \\ {*} &\stackrel{pt_V}{\to}& V } \,.

example: ordinary linear quantum mechanism

In the special case that V=V = Vect? over some ground field kk the canonical point is that ground field. Then the “ground monoid” is again canonically identified with the ground field: End Vect k(k)kEnd_{Vect_k}(k) \simeq k. Therefore the name “ground monoid” as a generalization of this situation.

More generally, for W:*VW : {*}\to V any other point of VV, we say that the lax pullback?

el pt V(W) * pt V * W V \array{ el_{pt_V}(W) &\to& {*} \\ \downarrow && \downarrow^{pt_V} \\ {*} &\stackrel{W}{\to}& V }

is the object of elements of WW relative to the chosen point.

example: ordinary vectors

Again in the special case that V=V = Vect? with the above standard point, WVW \in V will be some vector space and el k(W)el_{k}(W) is the set – regarded as a discrete category? of its ordinaly elements – of the vectors in WW.

Moreover, given the background field differential cocycle or the action functional

exp(S ):Bord(X)V \exp(S_\nabla) : Bord(X) \to V

we have the lax pullback?

E * pt V Bord(X) exp(S ) V. \array{ E &\to& {*} \\ \downarrow && \downarrow^{pt_V} \\ Bord(X) &\stackrel{\exp(S_\nabla)}{\to}& V } \,.

This EE is really the action groupoid? of the “action?” of Bord(X)Bord(X) on VV: it is much like the Atiyah Lie groupoid? of a principal bundle?, with the difference that its objects are elements in an associated (not a principal bundle) and its morphism connect only elements in the fibers that are related by parallel transport along the corresponding path down in XX.

For each component exp(S ) Σ:[Σ,X]V\exp(S_\nabla)|_\Sigma : [\Sigma,X] \to V we get a corresponding lax pullback

E Σ * pt V [Σ,X] exp(S ) Σ V \array{ E_{\Sigma} &\to& {*} \\ \downarrow && \downarrow^{pt_V} \\ [\Sigma,X] &\stackrel{\exp(S_\nabla)|_\Sigma}{\to}& V }

in particular the fiber over xXx \in X of E ptE_{pt}

el pt v(E x) E pt * * x X exp(S ) pt V \array{ el_{pt_v}(E_x) &\to& E_{pt} &\to& {*} \\ \downarrow &&\downarrow && \downarrow \\ {*} &\stackrel{x}{\to}& X &\stackrel{\exp(S_\nabla)|_{pt}}{\to}& V }

is the fiber of the underlying VV-\infty-bundle over xx-


In the case that V=FinVectV = FinVect and :Π(X)Vect\nabla : \Pi(X) \to Vect is a vector bundle? with connection? and with the canonical point chosen for VectVect as above, this el pt V(E x)el_{pt_V}(E_x) is simply the fiber of that vector bundle over xx.

Picking a point v:*E ptv : {*} \to E_{pt} in the total space is like picking a δ\delta-distributional section of E ptXE_{pt} \to X with support at xx and value vv there. Picking two points v 1v 2:**E ptv_1 \sqcup v_2 : {*} \sqcup {*} \to E_{pt} is naturally interpreted as picking the sum of two such sections.

Picking instead a map v/k:BG*E ptv/k : \mathbf{B}G \to {*} \to E_{pt} for GG a finite group of cardinality kk is like picking a δ\delta-distributional section with value vv and dividing its value by kk.

In light of this, we choose now the geometric function theory? given by over categories? in H\mathbf{H}. This is as described at examples for geometric function objects?

C:=H/():H(,1)Cat C := \mathbf{H}/(-) : \mathbf{H} \to (\infty,1)Cat

Then for Σ\Sigma a cobordism, a a section of the background field or state of the quantum system over Σ\Sigma is an object

ψC(E Σ). \psi \in C(E_{\Sigma}) \,.

special case: degroupoidification for VV-phased groupoids

For some choices of VV this general notion of section may be degroupified – to yield an ordinary notion of (still possibly distributional) section.

In the case that V=FinVectV = FinVect we have that E xE_x is (the discrete category? on the set underlying) the vector space that is the fiber of the background vector bundle over XX at xXx \in X.

For (ψ:ΨE x)C(E x)(\psi : \Psi \to E_x) \in C(E_x) consider the underlying morphism of plain \infty-groupoids and interpret this as the vector

vE xψ 1(v)v \sum_{v \in E_x} |\psi^{-1}(v)|\cdot v

given by the linear combination of all elements vE xv \in E_x weighted by the groupoid cardinality? of the ∞-groupoid? sitting over it.

the propagation map

In total the above discussion yields, starting from a background field :Π(X)V\nabla : \Pi(X) \to V, a multispan? of \infty-bundles E ΣE_\Sigma over configuration spaces [Σ,X][\Sigma,X].

An an object in the over category? C(E Σ):=H/E ΣC(E_{\Sigma}) := \mathbf{H}/E_{\Sigma} encodes a – possibly distributional – section of E Σ[Σ,X]E_\Sigma \to [\Sigma,X]. The quantum propagation we take to be the pull-push of the geometric function object?s C(E Σ)C(E_\Sigma) through this multispan?.

Let Σ inΣΣ out\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} be a cobordism? cospan? regarded as part of the multi-cospan? Σ H\Sigma_H that defines Bord(X)Bord(X) as

Bord(X)=lim [Σ H ,X] Bord(X) = \lim_\to [\Sigma^\bullet_H,X]

this realizes Bord(X)Bord(X) as the universal cocone whose component under Σ\Sigma is

[Σ,X] [Σ in,X] exp(S ) Σ [Σ out,X] exp(S )in exp(S ) out V \array{ && [\Sigma,X] \\ & \swarrow && \searrow \\ [\Sigma_{in},X] &&\downarrow^{\exp(S_\nabla)_\Sigma} && [\Sigma_{out},X] \\ & {}_{\exp(S_\nabla)}_{in}\searrow & & \swarrow_{\exp(S_\nabla)_{out}} \\ && V }

by lax pullback of pt V:*Vpt_V : {*} \to V this induces a span of total spaces

E Σ E in E out E Σ in E Σ out \array{ && E_{\Sigma} \\ & {}^{E_{in}}\swarrow && \searrow^{E_{out}} \\ E_{\Sigma_{in}} && && E_{\Sigma_{out}} }

definition the “quantum propagation along Σ\Sigma” is the functor of geometric function object?s

Z (Σ):C(E Σ in)(E out) *(E in) *C(E Σ out) Z_\nabla(\Sigma) : C(E_{\Sigma_{in}}) \stackrel{(E_out)_* (E_{in})^*}{\to} C(E_{\Sigma_{out}})

remark If instead of the over-category geometric function object? we use its Kan fibrant replacement? Ex (H/())Ex^\infty(\mathbf{H}/(-)) this presciption becomes functorial also for higher morphisms coming from spans of spans?

Concrete examples

Reminder on sections In order to unwrap the above abstract prescription in some special cases recall how we extract from the value ψ xC(E x)\psi_x \in C(E_{x}) of a generalized section ψC(E pt)\psi \in C(E_{pt}) an ordinary vector in the case that EE is a vector bundle?:

given ψ:ΨE Σ\psi : \Psi \to E_\Sigma we extract the corresponding “degroupoidified” vector of the corresponding section over ϕ Σ[Σ,X]\phi|_{\Sigma} \in [\Sigma,X] as the pullback

p *Ψ Ψ ψ ϕ Σ V ϕ Σ p E Σ * ϕ Σ [Σ,X] \array{ p^* \Psi &\to& \Psi \\ \downarrow^{\psi|_{\phi|_{\Sigma}}} && \downarrow \\ V_{\phi|_{\Sigma}} &\stackrel{p}{\to}& E_\Sigma \\ \downarrow && \downarrow \\ {*} &\stackrel{\phi|_{\Sigma}}{\to}& [\Sigma,X] }

electromagnetically charged particle

The background field for the electrically charged particle is the electromagnetic field?. As discussed there, is this modeled by a Deligne cocycle? in degree 2, or equivalently a line bundle with connection.

This is modeled in the present language by choosing as coefficient object A=BU(1)A = \mathbf{B}U(1), the delooping? of the circle Lie group? S 1U(1)S^1 \simeq U(1).

Then an electromagnetic background field over a manifold? XX is encoded in a morphism

:P 1(X)BU(1). \nabla : P_1(X) \to \mathbf{B}U(1) \,.

See connection on a bundle? for more details on this special case.

We choose the standard representation

ρ:BU(1)Vect \rho : \mathbf{B}U(1) \to Vect

of U(1)U(1) on \mathbb{C}.


  • E ptXE_{pt} \to X is the total space of the corresponding line bundle?.

  • for Σ=I=Δ H 1\Sigma = I = \Delta^1_H the interval? the configuration space [Δ H 1,X]=[I,X][\Delta^1_H,X] = [I,X] is the path space of XX;

  • the bundle E I[I,X]E_{I} \to [I,X] can be thought of as having as fiber over a path (γ:xy)[I,X](\gamma : x \to y) \in [I,X] the space of pairs of vectors (v x,v y)E x×E y(v_x, v_y) \in E_x \times E_y that are related by parallel transport along γ\gamma, i.e. those for which v y=ρ(γ)(v x)v_y = \rho\nabla(\gamma)(v_x).

  • The two span maps

    E I E in E out E pt E pt \array{ && E_{I} \\ & {}^{E_{in}}\swarrow && \searrow^{E_{out}} \\ E_{pt} &&&& E_{pt} }

    project on the corresponding component:

    E in:(v x,v y)v x E_{in} : (v_x, v_y) \mapsto v_x
    E out:(v x,v y)v y E_{out} : (v_x, v_y) \mapsto v_y

To see how the corresponding quantum propagation acts on a section, it is sufficient to look at a single δ\delta-distribution state

ψ:*E pt \psi : {*} \to E_{pt}

that hits a vector ψ xE x\psi_x \in E_x. In diagrams this means that

E pt ψ * x X \array{ && E_{pt} \\ & {}^\psi \nearrow & \downarrow \\ {*} &\stackrel{x}{\to}& X }


p ** * ψ x ψ E x p E pt * x X. \array{ p^* {*} &\to& {*} \\ \downarrow^{\psi_x} && \downarrow^{\psi} \\ E_x &\stackrel{p}{\to}& E_{pt} \\ \downarrow && \downarrow \\ {*} &\stackrel{x}{\to}& X } \,.

Then the pull-push propagation functor

Z (I):C(E pt)C(E pt) Z_\nabla(I) : C(E_{pt}) \to C(E_{pt})

sends ψ\psi to the composite right diagonal morphism in

(E in) *Ψ (E in) *ψ *=Ψ E I ψ E in E out E pt E pt. \array{ && (E_{in})^* \Psi \\ & \swarrow && \searrow^{(E_{in})^* \psi} \\ {*}= \Psi&&&& E_{I} \\ &\searrow^{\psi}&& {}^{E_{in}}\swarrow && \searrow^{E_{out}} \\ &&E_{pt} &&&& E_{pt} } \,.

We see that

  • (E in) *Ψ(E_{in})^* \Psi is the space with points the set of pairs (v x,v y)E I(v_x,v_y) \in E_{I} such that v x=ψ xv_x = \psi_x;

  • hence the map

    (E out) *ψ:(E in) *Ψ(E in) *ψE IE outE pt (E_{out})_* \psi : (E_{in})^* \Psi \stackrel{(E_{in})^* \psi}{\to} E_{I} \stackrel{E_{out}}{\to} E_{pt}

    is on the level of sets the map whose fiber over the vector v yE yE ptv_y \in E_y \subset E_{pt} at yy is the set of paths γ:xy\gamma : x \to y such that v y=ρ(γ)v_y = \rho\nabla(\gamma).

This means that, by the above prescription, the degroupoidification of our propagated section is the section Z (I)(ψ)Z_\nabla(I)(\psi) whose value over yXy \in X is formally

Z (I)(ψ) y= γ:xyexp(S (γ))(ψ x). Z_\nabla(I)(\psi)_y = \sum_{\gamma: x \to y} \exp(S_\nabla(\gamma))(\psi_x) \,.

That’s the expression of the Feynman-Kac path integral? for the electron, except, of course, that the Wiener measure? on the space of paths is missing.

Dijkgraaf-Witten theory

Dijkgraaf-Witten theory? is the gauge theory? which, when regarded as a sigma-model? is given by

  • target space XX is the delooping? X:=BGX := \mathbf{B}G of a finite group? GG;

  • the background field is a morphism :Π(BG)B 3U(1)\nabla : \Pi(\mathbf{B}G) \to \mathbf{B}^3 U(1) into the 3-fold delooping? of the Lie group? U(1)U(1) (see Eilenberg-MacLane object?).

This means that the configuration spaces of DW theory over a cobordisms Σ\Sigma are the groupoid?s (orbifold?s) [Σ,BG]=GBund(Σ)[\Sigma,\mathbf{B}G] = G Bund(\Sigma) of GG-principal bundle?s on Σ\Sigma.

Since the cobordisms Δ H k\Delta^k_H are disk-shaped, the groupoids of GG-principal bundles on them are trivial and we have

Π(BG)BG. \Pi(\mathbf{B}G) \simeq \mathbf{B}G \,.

Hence the background field for DW theory is actually just a

:BGB 3U(1). \nabla : \mathbf{B}G \to \mathbf{B}^3 U(1) \,.

This just reflects the fact that GG-principal bundles for discrete group GG have a unique and flat connection?.

By the discussion at group cohomology? this DW background field is nothing but a group cocycle defining a class in degree 3 group cohomology H 3(G,U(1))H^3(G,U(1)). This is the form in which the background data for DW theory was originally and is still usually presented.

The standard representation of the gauge group B 2U(1)\mathbf{B}^2 U(1) (recall that GG is from the sigma-model? perspective not a gauge group but defines the target space!) is on B 21dVectBVectMod\mathbf{B}^2 1dVect \hookrightarrow \mathbf{B} Vect Mod

ρ:B 3U(1)BVectMod. \rho : \mathbf{B}^3 U(1) \to \mathbf{B} Vect Mod \,.

Then for Σ inΣΣ out\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} a cobordism? cospan? the corresponding quantum propagation

Z(Σ):C(E Σ in)C(E Σ out) Z(\Sigma) : C(E_{\Sigma_{in}}) \to C(E_{\Sigma_{out}})

is under the above “degroupoidification” the action of Dijkgraaf-Witten theory, where the 1Aut(P)\frac{1}{|Aut(P)|}-weight comes from the fact that the pulled back state in *ψ{in}^* \psi picks up the automorphisms of PP by the push-forward

in *Ψ in *ψ Ψ E Σ ψ E Σ in E Σ out \array{ && in^* \Psi \\ & \swarrow && \searrow^{in^* \psi} \\ \Psi && && E_{\Sigma} \\ &\searrow^{\psi}& & \swarrow && \searrow \\ && E_{\Sigma_{in}} &&&& E_{\Sigma_{out}} }

Actually, more precisely due to the nature of the homotopy pullback? this picks up another factor of Aut(P in)|Aut(P|_{in})|.

the Yetter model

The Yetter model? is the name for the sigma-model? whose

  • target space XX is the delooping? BG\mathbf{B}G of a strict 2-group? GG coming from a crossed module? (G 2δG 1)(G_2 \stackrel{\delta}{\to} G_1);

  • background field is a morphism :Π(BG)B 4U(1)\nabla : \Pi(\mathbf{B}G) \to \mathbf{B}^4 U(1).

So this is just as Dijkgraaf-Witten theory, discussed above, but in one degree higher.

Accordingly, its discussion is pretty analogous to that of the Dijkgraaf-Witten model discussed above. The important difference is that now the automorphisms of a principal 2-bundle? classified by a map ΣBG\Sigma \to \mathbf{B}G form a 2-group? themselves, so that the Leinster measure? coefficient p 1(v x)|p^{-1}(v_x)| in our degroupoidification formula becomes slightly more subtle.

The right factor was originally found in the literature by requiring the path integral of the Yetter model to be independent of choices made when representing GG-2-bundles as cocycles. The following proposition shows that the factor obtained this way is indeed nothing but the 2-groupoid Leinster measure.

this is an old version of this proof, I’ll upload a better one


The combinatorial weighting for the “categorified DW” theory introduced by Yetter (and studied by Girelli, Pfeiffer, Popescu, Porter, Martins etc.) is indeed the Leinster measure? of the corresponding configuration space.

Recall that for ordinary DW theory for finite group GG config space is the groupoid [Σ,BG][\Sigma,\mathbf{B}G], which, since Σ\Sigma is finite, may be computed by choosing a triangulation of Σ\Sigma, letting P 1(Σ)P_1(\Sigma) denote the free category on the corresponding graph and considering Func(P 1(Σ),BG)Func(P_1(\Sigma),\mathbf{B}G).

Since there are

G ν |G|^\nu

many ways to assign group element to the ν\nu vertices of the triangulation, there are G ν|G|^\nu many natural transformations starting at any functor P 1(Σ)BGP_1(\Sigma) \to \mathbf{B}G and therefore the Leinster measure of the hom-groupoid is

dμ=1G ν. d\mu = \frac{1}{|G|^\nu} \,.

Now, Yetter et al find that when target space is replaced by

BG (2) \mathbf{B}G_{(2)}

for G (2)G_{(2)} the finite strict 2-group? coming from the strict crossed module?

HtGαAut(H), H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H) \,,

then the corresponding combinatorial weight becomes

G νH ν+λ, |G|^{- \nu} |H|^{-\nu + \lambda} \,,

where ν\nu is the number of vertices, as before, and where λ\lambda is the number of edges in XX.

(See for instance theorem 2.13 on p. 127 of Martins-Porter’s On Yetter’s invariants or the discussion below theorem III.2 on p. 6 of Girelli-Pfeiffer-Popescu’s BF to BFCG (while the factor is not manifest in their expression (15))).

The claim here is that this factor is precisely the Leinster measure? on the configuration space

conf=hom 2Cat(P 2(Σ),BG (2))/ conf = \mathrm{hom}_{2Cat}(P_2(\Sigma), \mathbf{B}G_{(2)})/\sim

Here I am taking, to replace awkward pseudofunctors with nicer strict 2-functors, P 2(Σ)P_2(\Sigma) now to be the strict 2-groupoid generated from the triangles in Σ\Sigma.

Then the above hom-groupoid is supposed to be the 1-groupoid obtained by passing to equivalence classes of 1-morphisms in the 2-groupoid whose objects are strict 2-functors P 2(Σ)BG (2)P_2(\Sigma) \to \mathbf{B}G_{(2)}, whose morphisms are pseudonatural transformations of these, and whose 2-morphisms are modifications of those.

To prove this, we need to compute how many 1-morphisms in confconf there start at any given object.

But that’s easy: a 1-morphism

g:AAg : A \to A'

in the 2-groupoid of strict 2-functors P 2(Σ)BG (2)P_2(\Sigma) \to \mathbf{B}G_{(2)} is a transformation whose component is an assignment of squares in BG (2)\mathbf{B}G_{(2)} to edges in XX:

g:(xγy) A(γ) g(x) g(γ) g(y) A(γ) . g : (x \stackrel{\gamma}{\to} y) \;\;\;\; \mapsto \;\;\;\; \array{ \bullet &\stackrel{A(\gamma)}{\to}& \bullet \\ \downarrow^{g(x)} & \Downarrow^{g(\gamma)} & \downarrow^{g(y)} \\ \bullet &\stackrel{A'(\gamma)}{\to}& \bullet } \,.

By the general yoga of strict 2-groups, we know that the square on the right is entirely and precisely fixed by

  • the source 2-functor value A(γ)GA(\gamma) \in G on the top;

  • the two transformation elements g(x),g(y)Gg(x), g(y) \in G on the two sides;

  • and the label g(γ)Hg(\gamma) \in H of the 2-morphism filling the square.

This uniquely determines the value A(γ)A'(\gamma) of the pseudofunctor which is the target of the transformation whose component gg is.

Therefore, clearly, there are precisely

G νH λ |G|^\nu |H|^\lambda

many 1-morphisms in

hom 2Cat(P 1(X),BG (2)) \mathrm{hom}_{2Cat}(P_1(X), \mathbf{B}G_{(2)})

starting at any given object.

But some of them are related by 2-morphisms

To figure out many, we need to compute how many modifications η:gg\eta : g \to g' there are starting at any transformation gg.

But again the same logic applies: such a transformations is given by a component map which sends vertices in XX to 2-cells in BG (2)\mathbf{B}G_{(2)}:

η:x g(x) η(x) g(x) \eta : x \mapsto \array{ & \nearrow \searrow^{g(x)} \\ \bullet & \Downarrow^{\eta(x)}& \bullet \\ & \searrow \nearrow_{g'(x)} }

for η(x)H\eta(x) \in H.

Again one sees, by drawing the relevant naturality tin-can diagram which I won’t bother to do here in MathML, that for each such choice of assignments of elements in HH to vertices in XX there is precisely one modification η:gg\eta : g \to g'.

This means that we have

  • G νH λ|G|^\nu |H|^\lambda 1-morphisms emanating at each object


  • H ν|H|^\nu 2-morphisms emanating at every morphism.

The Leinster measure in this situation is

dμ(A)=G νH νλ, d\mu(A) = |G|^{-\nu} |H|^{\nu - \lambda} \,,

precisely the combinatorial factor which people found makes the Yetter state sum model a topological invariant.

Revised on February 1, 2010 20:05:05 by Urs Schreiber (