nLab 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

  
  

  in P. R. Kotiuga (ed.):

  A celebration of the mathematical legacy of Raoul Bott

  AMS (2010)

  arXiv:0905.0731

  ISBN:978-0-8218-4777-0

on

1. in sections 3 and 8; 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, realized here for finite higher gauge theories Dijkgraaf-Witten theory-type theories (see also at prequantum field theory)

2. the extended TQFT-quantization of $G$-Chern-Simons theory for abelian Lie groups $G$.

More on the story of sections 3 and 8 is in

• Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory, talk at Notre Dame Graduate Summer School on Topology and Field Theories and Harvard lecture 2012 (video part 1, part 2, part 3 part 4, pdf lecture notes by Chris Elliott)

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 $(\infty,n)$-categories of families of $\infty$-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 $n \in \mathbb{N}$, the dimension of the quantum field theory to be described.

In

• 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 of spans over $C$, denoted $Fam_n(C)$,

  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)

    $$\array{ && P \\ & \swarrow && \searrow \\ P_{in} &&\swArrow&& P_{out} \\ & {}_{\mathllap{\exp(S(-))_{in}}}\searrow && \swarrow_{\mathrlap{\exp(S(-))_{out}}} \\ && C }$$

  • “and so on”.

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

• An k-morphism $\Sigma$ 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

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

  where $\Sigma_{in}$ and $\Sigma_{out}$ are pieces of the boundary of $\Sigma$. We think of $\Sigma_{in}$ as the “incoming” piece of the object that we want to describe, which then experiences a self-interaction as described by the topology of $\Sigma$ and comes out in the shape of $\Sigma_{out}$ (for instace $\Sigma$ might be the three-holed sphere, $\Sigma_{in}$ the disjoint unions of two of its bounding circles and $\Sigma_{out}$ 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 $P_{in}, P_{out}$ 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 $(\infty,n)$-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 $Fam_n(C)$

$$\array{ && Fam_n(C) \\ & {}^{\exp(S(-))}\nearrow & \downarrow \\ Bord_n &\stackrel{kin}{\to}& Fam_n(*) }$$

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 $\Sigma$ is a morphism $\phi : \Sigma \to X$, 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

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

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

$$\array{ && [\Sigma,X] \\ & \swarrow && \searrow \\ [\Sigma_{in},X] &&&& [\Sigma_{out},X] }$$

equipped with the transgressed background field as the corresponding action functional

$$\array{ && [\Sigma, X] \\ & \swarrow && \searrow \\ [\Sigma_{in},X] &&&& [\Sigma_{out},X] \\ & \searrow && \swarrow \\ && C } \,.$$

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 postcomposition 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 WillertonThe 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 iterating 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

$$\array{ && Fam_n(C) &\stackrel{\int}{\to}& C \\ & {}^{\exp(S(-))}\nearrow & \downarrow \\ Bord_n &\stackrel{kin}{\to}& Fam_n(*) }$$

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 $Z : Bord_n \to n Vact$:

$$Z_S = \int \exp(S(-)) : Bord_n \stackrel{\exp(S(-))}{\to} Fam_n(n Vect) \stackrel{\int}{\to} n Vect \,.$$

3d Chern-Simons as a fully extended TQFT

A summary of the FHLT-argument about realizing 3d Chern-Simons theory as a fully extended TQFT is given in

• Konrad Waldorf, Chern-Simons theory and the categorified group ring (pdf)

The following are some notes from a talk by Constantin Teleman on joint work with Dan Freed, given at ESI Program on K-Theory and Quantum Fields (2012).

CS as a fully extended TQFT

Goals

• i) describe Chern-Simons theory for compact Lie groups as an extended TQFT, generated by some structure assigned to the point

• ii) relate to chiral WZW model, also down to the point, have a formal framework for this

approach related to:

• Kevin Walker: 3d Chern-Simons theory is best understood as a boundary condition for a 4d theory.

• Chris Douglas, Andre Henriques: conformal nets and their relation to CS theory

construction is special case of Reshetikhin-Turaev construction which to a modular tensor category $\mathcal{B}$ assigns a 1-2-3 extended TQFT that assigns $\mathcal{B}$ to the circle $S^1$.

So the goal here is to extend this down to the point, to a 0-1-2-3 extended TQFT, hence to an $(\infty,3)$-functor

$$Bord^{String}_{0,1,2,3} \to some\;symm\;mon.\;3\; category$$

the standard choice on the right is a 3-category of (multi) fusion categories, (see the reference by Douglas, Schommer-Pries and Snyder there) whose

• objects are fusion categories;

• morphisms are bimodule categories;

• 2-morphisms: bimodule homomorphism functors;

• 3-morphisms: natural transformations of these.

If this can be done, then

• $CS(*) = T$;

• $CS(S^1) =$ Drinfeld center of $T$ (again an MTC)

Witt group? of modular tensor category: many abelian examples of CS give nontrivial classes

Theorem Given a modular tensor category $A$, there exists a symmetric monoidal 3-category $\mathcal{C}_A$ containing the fusion categories and a fully dualizable object in $X \in \mathcal{C}_A$ which generates a 0-1-2-3 extended TQFT whose 1-2-3 part agrees with the Reshetikhin-Turaev construction applied to $A$.

Here $\mathcal{C}_A$ and $X$ are formally constructed form $A$ by means of

• $X \otimes X^\vee \simeq A$

• $\mathcal{C}_A = FusionCat[X, X^\vee]$

Remarks

• i) This is a theory for “string structure” manifolds in the sense that the first Stiefel-Whitney class $w_1$ and the first Pontryagin class $p_1$ are trivialized, but not necessarily $w_2$.

• ii) This is the universal extension: every other one factors through it.

  guess: there is some kind of an “algebraic extension” of fusion categories in which the equation $X \otimes X^\vee = A$ can be solved for any MTC $A$.

• iii) a choice of cube root wil be needed to construct the theory for a framed 3-manifold

  change of framing: $n \mapsto \times \exp(\frac{2 \pi i c n }{24})$ for $c$ a “central charge”

because string bordism group in $dim = 3$ is $\mathbb{Z}_3$, in case of spin structure

$$\array{ 3 & \mathbb{Z}_{24} \\ 2 & \mathbb{Z}_2 \\ 1 & \mathbb{Z}_2 }$$

for spin theories one would need categorical representations of this 3-groupoid, but at the moment not known.

Theorem (Kevin Walker, in Jacob Lurie‘s language)

• i) $A$ generates an invertible 4d extended TQFT for oriented manifolds;

• ii) $A$ is a valid boundary condition over itself;

The 3d boundary theory in dim 1-2-3 is equivalent to $ReshTur(A)$ after a choice of “bulking manifold”

Comments there is a symmetric monoidal 4-category whose

• objects are bimodule tensor categories

• morphisms are bialgebra categories = tensor categories $T$ with braided monoidal functors

  $(B') \otimes B^{bop} \to DZ(T)$

• 2-morphisms are bimodule categories

• 3-morphisms are functors respecting the structure;

• 4-morphisms are natural transformations between these.

Invertibility of a TQFT $Z$ $\Leftrightarrow$ invertibility of $Z(*)$.

so then $Z(S^1) \simeq 1$

Lemma modular tensor categories are invertible in this sense

(4d anomaly theory) + (bulking manifold) = (3d standalone theory)

Result Have a 3d TQFT defined on the full subcategory of $Bord^String_{0-1-2-3-4}$ of manifolds which bound.

So the final step in the construction of the full 0-1-2-3-4 theory is to extend from that to the full category of bordisms.

Boundary conditions: WZW theory

general metaphor:

• TQFT $\leftrightarrow$ algebra

• boundary condition $\leftrightarrow$ bimodule

other metaphor

• TQFT $Z$ determined by $Z(*)$

• boundary condition $\alpha : 1 \to Z(*)$ (left) or $\beta : Z(*) \to 1$ (right)

For the 3d RT theory this yields for boundary conditions $DZ(T)$-algebra categories.

Theorem (Graeme Segal)

Chiral WZW model is a conformal boundary for CS theory

Problem make this work down to the point

Related references

• Dan Freed, 4-3-2 8-7-6

category: reference