nLab Topological Quantum Field Theories from Compact Lie Groups



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This entry is about the article


  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 GG-Chern-Simons theory for abelian Lie groups GG.

More on the story of sections 3 and 8 is in



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)(\infty,n)-categories of families of \infty-groupoids that appears in some of the later sections of

Together with a notion of nn-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)CFam_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 nn \in \mathbb{N}, the dimension of the quantum field theory to be described.


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

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

  • An k-morphism Σ\Sigma in Bord nBord_n is a piece of kk-dimensional “worldvolume” of some extended object, whose quantum dynamics we want to describe; we may roughly think of this as a cospan

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

    where Σ in\Sigma_{in} and Σ out\Sigma_{out} are pieces of the boundary of Σ\Sigma. We think of Σ in\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 Σ\Sigma and comes out in the shape of Σ out\Sigma_{out} (for instace Σ\Sigma might be the three-holed sphere, Σ in\Sigma_{in} the disjoint unions of two of its bounding circles and Σ out\Sigma_{out} the remaining one, modelling the interaction where two strings merge to a single one).

  • An morphism in Fam n(C)Fam_n(C) is to be thought of as

    • two configuration spaces of fields P in,P outP_{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()):PC\exp(S_P()) : P \to C; being the component of the natural transformation that assigns to each path PP between field configuration a phase ;

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

kin:Bord nFam n(*) 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)Fam_n(C)

Fam n(C) exp(S()) Bord n kin Fam n(*) \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 XX, such that a field configuration on Σ\Sigma is a morphism ϕ:ΣX\phi : \Sigma \to X, and with a background field :XC\nabla : X \to C, then we think of the corresponding functor

conf X:Bord nFam n(C) conf_X : Bord_n \to Fam_n(C)

as given by homming a cobordism cospan of the form

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

into XX to produce a span of path and configuration spaces

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

equipped with the transgressed background field as the corresponding action functional

[Σ,X] [Σ in,X] [Σ out,X] C. \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 nFam n(C)\exp(S(-)) : Bord_n \to Fam_n(C) is its “pushforward to the point”, given by postcomposition with a functor

:Fam n(C)C \int : Fam_n(C) \to C

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

(exp(S()):PC)(lim exp(S())) (\exp(S(-)) : P \to C) \mapsto (\lim_\to \exp(S(-)))

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

(exp(S()):PC)(exp(S())). (\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 nn-vector bundle exp(S()):PC\exp(S(-)) : P \to C. That this is the right general idea was maybe first amplified in

A first more categorical formulation of this is in

What exactly the functor :Fam n(C)C\int : Fam_n(C) \to C does to k-morphisms is apparently left as an exercise for the inclined reader. it requires that in CC limits and colimits coincide. This is the case notably for C=VectC = 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=nVectC = n Vect, which should be the right codomain for nn-dimensional QFTs. So we end up with a diagram

Fam n(C) C exp(S()) Bord n kin Fam n(*) \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 nnVactZ : Bord_n \to n Vact:

Z S=exp(S()):Bord nexp(S())Fam n(nVect)nVect. 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

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


  • 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:

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 1S^1.

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

Bord 0,1,2,3 Stringsomesymmmon.3category 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

If this can be done, then

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

Theorem Given a modular tensor category AA, there exists a symmetric monoidal 3-category 𝒞 A\mathcal{C}_A containing the fusion categories and a fully dualizable object in X𝒞 AX \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 AA.

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

  • XX AX \otimes X^\vee \simeq A

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


  • i) This is a theory for “string structure” manifolds in the sense that the first Stiefel-Whitney class w 1w_1 and the first Pontryagin class p 1p_1 are trivialized, but not necessarily w 2w_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 XX =AX \otimes X^\vee = A can be solved for any MTC AA.

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

    change of framing: n×exp(2πicn24)n \mapsto \times \exp(\frac{2 \pi i c n }{24}) for cc a “central charge

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

3 24 2 2 1 2 \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) AA generates an invertible 4d extended TQFT for oriented manifolds;

  • ii) AA is a valid boundary condition over itself;

The 3d boundary theory in dim 1-2-3 is equivalent to ReshTur(A)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 TT with braided monoidal functors

    (B)B bopDZ(T)(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 ZZ \Leftrightarrow invertibility of Z(*)Z(*).

so then Z(S 1)1Z(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 01234 StringBord^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 ZZ determined by Z(*)Z(*)

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

For the 3d RT theory this yields for boundary conditions DZ(T)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

category: reference

Last revised on August 7, 2023 at 05:36:51. See the history of this page for a list of all contributions to it.