For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
physics, mathematical physics, philosophy of physics
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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)
on
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)
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
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
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)
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
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)
“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
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
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 $Fam_n(C)$
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
as given by homming a cobordism cospan of the form
into $X$ to produce a span of path and configuration spaces
equipped with the transgressed background field as the corresponding action functional
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
that over objects $\exp(S(-)) : P \to C$ is given by taking $n$-categorical colimit
which in terms of coend-notation is indeed nicely suggestively written as
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
A first more categorical formulation of this is in
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
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 $Z : Bord_n \to n Vact$:
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).
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
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
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.
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