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Topological Quantum Field Theories from Compact Lie Groups

This entry is about the article

Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv)

on a central topic in higher category theory and physics: the abstract higher categoretic conception of path integral quantization of classical action functionals to extended quantum field theories.

Preliminaries
The general abstract notion of quantization for discrete theories

Preliminaries

The non-toy example application that gives the paper its title is to Chern-Simons theory.

The notion of quantization discussed builds on the notion of $(∞, n)$ -categories of families of $∞$ -groupoids that appears in some of the later sections of

Jacob Lurie, On the Classification of Topological Field Theories .

Together with a notion of $n$ -vector spaces (the vertical categorification of vector space and 2-vector space) the article sketches a general abstract formalsim making precise the notion of path integral quantization for “finite” theories such as Dijkgraaf-Witten theory.

The development, sketching a rather grand picture, remains somewhat sketchy, though, possibly due to the fact that this is a conference proceedings. Also some of the ideas claimed to now be fully generalized have appeared elsewhere before. Notably the notion of the quantization map ${Fam}_{n} (C) \to C$ (see below) is effectively what John Baez, Jim Dolan call in their program on groupoidification call degroupoidification . The general idea underlying this, that spaces of states are computed as colimits of sections, has been made clear previously by Simon Willerton The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv:math/0503266)

The general abstract notion of quantization for discrete theories

Here is a summary of the general quantization aspect of the article, together with some additional remarks on how to think of all this by an nLab author.

The following is the formalization of the notion of quantization for discrete theories (such as Dijkgraaf-Witten theory) as presented in the article.

Fix some $n \in ℕ$ , the dimension of the quantum field theory to be described.

Jacob Lurie, On the Classification of Topological Field Theories .

are described the following two (∞,n)-categories

the (∞,n)-category of cobordisms ${Bord}_{n}$ : its k-morphisms are roughly $k$ -dimensional manifolds with boundaries and corners;
for $C$ any (∞,n)-category the (∞,n)-category ${Fam}_{n} (C)$ whose
- objects are ∞-groupoids $P$ equipped with functors $F_{P} : P \to C$
- morphisms are spans of ∞-groupoids with natural transformations between the corresponding functors (bi-branes)
  
  $\begin{matrix} P \\ ↙ & ↘ \\ P_{in} & ⇙ & P_{out} \\ _{\exp (S (-))_{in}} ↘ & ↙_{\exp (S (-))_{out}} \\ C \end{matrix}$ \array{ && P \\ & \swarrow && \searrow \\ P_{in} &&\swArrow&& P_{out} \\ & {}_{\mathllap{\exp(S(-))_{in}}}\searrow && \swarrow_{\mathrlap{\exp(S(-))_{out}}} \\ && C }
- “and so on”.

For the application to quantization of sigma-model theories we want to be thinking of the data encoded by these $(∞, n)$ -categories as follows:

An k-morphism $Σ$ in ${Bord}_{n}$ is a piece of $k$ -dimensional “worldvolume” of some extended object, whose quantum dynamics we want to describe; we may roughly think of this as a cospan

$\begin{matrix} Σ \\ ↗ & ↖ \\ Σ_{in} & Σ_{out} \end{matrix}$ \array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} &&&& \Sigma_{out} }

where $Σ_{in}$ and $Σ_{out}$ are pieces of the boundary of $Σ$ . We think of $Σ_{in}$ as the “incoming” piece of the object that we want to describe, which then experiences a self-interaction as described by the topology of $Σ$ and comes out in the shape of $Σ_{out}$ (for instace $Σ$ might be the three-holed sphere, $Σ_{in}$ the disjoint unions of two of its bounding circles and $Σ_{out}$ the remaining one, modelling the interaction where two strings merge to a single one).
An morphism in ${Fam}_{n} (C)$ is to be thought of as
- two configuration spaces of fields $P_{in}, P_{ou}$ of some field theory;
- together with an action functional on it in the form of an higher vector bundle (“gerbe”) $\exp (S_{P} ()) : P \to C$ ; being the component of the natural transformation that assigns to each path $P$ between field configuration a phase ;

In these terms the kinematics of a classical field theory is a choice of $(∞, n)$ -functor

kin : {Bord}_{n} \to {Fam}_{n} (*)

whereas the dynamics of a classical field theory – the specificaton of an action functional** on the given configuration spaces, is a lift of that to ${Fam}_{n} (C)$

\begin{matrix} {Fam}_{n} (C) \\ ^{\exp (S (-))} ↗ & ↓ \\ {Bord}_{n} & \overset{kin}{\to} & {Fam}_{n} (*) \end{matrix}

To illustrate this: specifically, if we consider a sigma-model quantum field theory that is induced from a target space geometry $X$ , such that a field configuration on $Σ$ is a morphism $ϕ : Σ \to X$ , and with a background field $\nabla : X \to C$ , then we think of the corresponding functor

{conf}_{X} : {Bord}_{n} \to {Fam}_{n} (C)

as given by homming a cobordism cospan of the form

\begin{matrix} Σ \times I \\ ↗ & ↖ \\ Σ_{in} & Σ_{out} \end{matrix}

into $X$ to produce a span of path and configuration spaces

\begin{matrix} [Σ \times I, X] \\ ↙ & ↘ \\ [Σ_{in}, X] & [Σ_{out}, X] \end{matrix}

equipped with the transgressed background field as the corresponding action functional

\begin{matrix} [Σ \times I, X] \\ ↙ & ↘ \\ [Σ_{in}, X] & [Σ_{out}, X] \\ ↘ & ↙ \\ C \end{matrix} .

With that in hand, the quantization of the given classical field theory $\exp (S (-)) : {Bord}_{n} \to {Fam}_{n} (C)$ is its “pushforward to the point”, given by postcomopon with a functor

\int : {Fam}_{n} (C) \to C

that over objects $\exp (S (-)) : P \to C$ is given by taking $n$ -categorical colimit

(\exp (S (-)) : P \to C) \mapsto (\lim_{\to} \exp (S (-)))

which in terms of coend-notation is indeed nicely suggestively written as

(\exp (S (-)) : P \to C) \mapsto (\int \exp (S (-))) .

Taking such a colimit may be thought of as forming the space of sections of the action functional $n$ -vector bundle $\exp (S (-)) : P \to C$ . That this is the right general idea was maybe first amplified in

Dan Freed, Higher Algebraic Structures and Quantization (arXiv:hep-th/9212115

a first more categorical formulation of this is in

Simon Willerton The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv:math/0503266)

What exactly the functor $\int : {Fam}_{n} (C) \to C$ does to k-morphisms is apparently left as an exercise for the inclined reader. it requires that in $C$ limits and colimits coincide. This is the case notably for $C = Vect$ .

The authors indicate in section 8 a general recursive procedure for defining higher categories of higher vector spaces, by izterating the bimodule-style definition of 2-vector spaces, as described there. This yields a notion $C = n Vect$ , which should be the right codomain for $n$ -dimensional QFTs. So we end up with a diagram

\begin{matrix} {Fam}_{n} (C) & \overset{\int}{\to} & C \\ ^{\exp (S (-))} ↗ & ↓ \\ {Bord}_{n} & \overset{kin}{\to} & {Fam}_{n} (*) \end{matrix}

whose left bit is the kinematical and dynamical input given by a classical field theory, and whose composition to to the right is supposed to give the corresponding quantum field theory, which by the logic motivating the cobordism hypothesis is a functor $Z : {Bord}_{n} \to n Vact$ :

Z_{S} = \int \exp (S (-)) : {Bord}_{n} \overset{\exp (S (-))}{\to} {Fam}_{n} (n Vect) \overset{\int}{\to} n Vect .

Urs Schreiber: here is a question from me about this.

I am thinking about this kind of stuff from the premise that the right notion of talking about the background field data on the target space $X$ is in terms of differential nonabelian cohomology. Of course this requires me to think of the above setup not in the discrete, but in the smooth setup, but maybe it is still interesting to compare a bit.

In the context of differential nonabelian cohomology, for every space $X$ there is a notion of its path ∞-groupoid $Π (X)$ and differential cohomology of $X$ is effectively the obstruction theory to extended morphisms out of $X$ (which are cocycles for principal ∞-bundles and ∞-vector bundles) to cocycles on $Π (X)$ , which are flat connections or local systems on these.

In general this extension of course does not exists – therefore the interest in its obstruction theory – but lets assume for the moment that we do have a flat cocycle as a background gauge field, a morphism

\nabla : Π (X) \to n Vect .

If we suitably coskeletalize the $∞$ -groupoid $Π (X)$ at degree $n$ , there should be an inclusion

Π_{n} (X) ↪ {Bord}_{n} (X)

into the $(∞, n)$ -category of cobordisms, identifying $Π_{n} (X)$ with the sub-thing of topoloigically trivial cobordisms.

By considering this in the case of $n = 1$ , one sees that extending $\nabla$ through this inclusion amounts to writing down an action functional for the gauge-coupling term of a charged object, so that we may sesibly write this extension as

\begin{matrix} Π_{n} (X) & \overset{\nabla}{\to} & n Vect \\ ↓ & ↗_{\exp (S_{\nabla} (-))} \\ {Bord}_{n} (X) \end{matrix} .

Of course there is also canonically the projection ${Bord}_{n} (X) \to {Bord}_{n} (*) = {Bord}_{n}$ . Staring at this for a second, one can’t help but feel that one should attempt to construct an extension $Z$

\begin{matrix} Π_{n} (X) & \overset{\nabla}{\to} & n Vect \\ ↓ & ↗_{\exp (S_{\nabla} (-))} \\ {Bord}_{n} (X) \\ ↓ & ↗_{Z} \\ {Bord}_{n} \end{matrix} .

I am not sure how exactly that should work. But comparing to the notion of Kan extension one expects that $Z$ should be given on $Σ \in {Bord}_{n}$ by evaluating $\exp (S_{\nabla} (-))$ on all ( $Σ \to X$ ) and “summing up” the result. That certainly begins to look like the path integral prescription.

In fact, it seems to me that the above construction in terms of ${Fam}_{n} (C)$ might be just another way for thinking about this extension.

For notice that $Π_{n} (X)$ is essentially the colimit over all ways of throwing disk-shaped cobordisms into $X$ . Similarly, I imaging that one should be able to conceived ${Bord}_{n} (X)$ as a colimit over a huge diagram of multi-cospans that indicate how all the cobordisms sit inside each other via boundary inclusins (something like the category of cells of ${Bord}_{n} (X)$ )

If we think of ${Bord}_{n} (X)$ as arising as a colimit over the cell-structure of cobordisms this way, then by the universal property of colimits a morphism $\exp (S_{\nabla} (-)) : {Bord}_{n} (X) \to n Vect$ will induce a system of morphisms out spans, as indicated here:

$%20\xymatrix@R=23pt@C=12pt{%20%26E_\Sigma%20\ar[rr]^{E_\mathrm{out}}%20\ar[dd]_{E_\mathrm{in}}%20\ar[ddr]%20%26%26%20E_{\Sigma_{\mathrm{out}}}%20\ar[ddr]%20\\%20\\%20%26E_{\mathrm{in}}%20\ar[ddr]%20%26%20[\Sigma,X]%20\ar[dd]^{\mathrm{in}}%20\ar[rr]^{\mathrm{out}}%20\ar[ddrr]%20%26%26%20[\Sigma_{\mathrm{out}},X]%20\ar[dd]%20\ar@/^1pc/[rrdddd]^{\exp(S_\nabla)|_{\Sigma_\mathrm{out}}}%20\\%20\\%20%26%26%20[\Sigma_{\mathrm{in}},X]%20\ar[rr]%20\ar@/_1pc/[ddrrrr]_{\exp(S_\nabla)|_{\Sigma_\mathrm{in}}}%20%26%26%20\mathrm{Bord}(X)%20\ar[ddrr]|{\exp(S_\nabla)}%20\\%20\\%20%26%26%26%26%20%26%26%20V%20}%20$

Certainly, this system of component morphisms is reminiscent of the bi-brane structure that we see in ${Fam}_{n} (C)$ in the above. So I am wondering:

might the Freed-Hopkins-Lurie-Teleman quantization prescription as indicated above maybe be regarded as a way to construct the “universal” extension $Z$ of $\exp (S_{\nabla} (-)) : {Bord}_{n} (X) \to n Vect$ along the projection ${Bord}_{n} (X) \to {Bord}_{n}$ ?

Domenico Fiorenza: I like this point of view a lot. At first sight, it seemed to me that considering ${Bord}_{n} (X)$ alone would have been a bit restrictive with respect to FHLT construction (at least how it is presented above). Namely, I would have found closer to the FHLT construction to consider a general ‘colimit of spans’ $(∞, n)$ -category $T_{n}$ with functors $\exp (S (-)) : T_{n} \to n Vect$ and $π : T_{n} \to {Bord}_{n}$ , and to wonder how the Kan extension $Z : {Bord}_{n} \to n Vect$ of $\exp (S (-))$ along $π$ is related to FHLT receipt. Then the sigma-model ${Bord}_{n} (X)$ should have been a particularly natural and interesting example of this general construction.

But on second thought, FHLT deals with a sort of presheaf of $∞$ -groupoids over ${Bord}_{n}$ , so it is natural to think of it as a representable functor even when one does knot knows about its representability (or even when it is actually not representable). namely, one could think of a FHLT-presheaves on ${Bord}_{n}$ as ${Bord}_{n} (X)$ for a ‘generalized object’ $X$ .

category: reference

Revised on January 31, 2010 10:45:47 by Domenico Fiorenza (87.21.226.8)

nLab Topological Quantum Field Theories from Compact Lie Groups

Contents

Preliminaries

The general abstract notion of quantization for discrete theories

nLab
Topological Quantum Field Theories from Compact Lie Groups