nLab
extended topological quantum field theory

Idea

Extended quantum field theory (or many-tiered quantum field theory) is the higher categorical version of functorial quantum field theory:

whereas

• 1-categorical TQFT may be regarded as a rule that allows one to compute topological invariants $Z(\Sigma )$ assigned to $d$-dimensional manifolds by cutting these manifolds into a sequence $\{{\Sigma }_i\}$ of $d$-dimensional composable cobordisms with $(d-1)$-dimensional boundaries $\partial {\Sigma }_i$, e.g. $\Sigma ={\Sigma }_2{\coprod }_{\partial {\Sigma }_1=\partial {\Sigma }_2}{\Sigma }_1$, then assigning quantities $Z({\Sigma }_i)$ to each of these and then composing these quantities in some way, e.g. as $Z(\Sigma )=Z({\Sigma }_2)\circ Z({\Sigma }_1)$;

we have that

• in extended TQFT $Z(\Sigma )$ may be computed by decomposing $\Sigma$ into $d$-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension $k$.

For that reason extended QFT is also sometimes called local or localized QFT. In fact, the notion of locality in quantum field theory is precisely this notion of locality. And, as also discussed at FQFT, this higher dimensional version of locality is naturally encoded in terms of n-functoriality of $Z$ regarded as a functor on a higher category of cobordisms.

The category of extended cobordisms

The definition of a $j$-cobordism is recursive. A $(j+1)$-cobordism between $j$-cobordisms is a compact oriented $(j+1)$-dimensional smooth manifold with corners whose the boundary is the disjoint union of the target $j$-cobordism and the orientation reversal of the source $j$-cobordism. (The base case of the recursion is the empty set, thought of as a $(-1)$-dimensional manifold.)

$n{\mathrm{Cob}}_d$ is an $n$-category with smooth compact oriented $(d-n)$-manifolds as objects and cobordisms of cobordisms up to $n$-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category $n{\mathrm{Cob}}_n$ of fully extended $n$-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes $n$-cobordisms as $n$-morphisms and smooth homotopy classes of diffeomorphisms beween them as $(n+1)$-morphisms. Next one iterates this; see details at (∞,n)-category of cobordisms.

See

• extended cobordism.

Definition

Fix a base ring $R$, and let $C$ be the symmetric monoidal $n$-category of $n$-$R$-modules.

An $n$-extended $C$-valued TQFT of dimension $d$ is a symmetric $n$-tensor functor $Z:n{\mathrm{Cob}}_d\to C$ that maps

• smooth compact oriented $d$-manifolds to elements of $R$
• smooth compact oriented $(d-1)$-manifolds to $R$-modules
• cobordisms of smooth compact oriented $(d-1)$-manifolds to $R$-linear maps between $R$-modules
• smooth compact oriented $(d-2)$-manifolds to $R$-linear additive categories
• cobordisms of smooth compact oriented $(d-2)$-manifolds to functors between $R$-linear categories
• etc …
• smooth compact oriented $(d-n)$-manifolds to $R$-linear $(n-1)$-categories
• cobordisms of smooth compact oriented $(d-n)$-manifolds to $(n-1)$-functors between $R$-linear $(n-1)$-categories

with compatibility conditions and gluing formulas that must be satisfied… For instance, since the functor $Z$ is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the $d$-dimensional vacuum? is mapped to the unit element of $R$, the $(d-1)$-dimensional vacuum to the $R$-module $R$, the $(d-2)$-dimensional vacuum to the category of $R$-modules, etc.

Here $n$ can range between $0$ and $d$. This generalizes to an arbitrary symmetric monoidal category $C$ as codomain category.

Examples

$n=1$ gives ordinary TQFT.

The most common case is when $R=ℂ$ (the complex numbers), giving unitary ETQFT.

The most common cases for $C$ are

• $C=n\mathrm{Hilb}(R)$, the category of $n$-Hilbert spaces? over a topological field $R$. As far as we know this is only defined up to $n=2$.
• $C=n\mathrm{Vect}(R)$, the category of $n$-vector spaces? over a field $R$.
• $C=n\mathrm{Mod}(R)$, the (conjectured?) category of $n$-modules? over a commutative ring $R$.

Construction of ETQFT’s

• By generators and relations

• By path integrals (this is Daniel Freed’s approach)

• By modular tensor n-categories?

Classification of ETQFT’s

Assume $Z:n{\mathrm{Cob}}_d\to n\mathrm{Vect}(R)$ is an extended TQFT. Since $Z$ maps the $(d-1)$-dimensional vacuum to $R$ as an $R$-vector space, by functoriality $Z$ is forced to map a $d$-dimensional closed manifold to an element of $R$. Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of $Z$ is enterely determined by its behaviour on $(d-n)$-dimensional manifolds. See details at cobordism hypothesis.

Relation of ETQFT to AQFT

See Urs Schreiber, AQFT from n-functorial QFT.

Related entries

More on extended QFTs is also at

• FQFT

References

• Daniel Freed, Remarks on Chern-Simons theory

• Daniel Freed, Quantum Groups from Path Integrals

• Daniel Freed, Higher Algebraic Structures and Quantization

• John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory. arXiv

• Jacob Lurie, On the Classification of Topological Field Theories

I changed references to bordism here into references to cobordisms, since there's also a notion of bordism (a back-formation) as dual to cobordism, which is not what we want. (Also Wikipedia implies that ‘bordism’ is a mass noun while ‘cobordism’ is a count noun, and these are count nouns, for what that's worth.) —Toby

Rafael: I obviously used a linear category in the definition as C instead of a general C. Do anyone know how to generalize it? Neither do i know the compatibility conditions and gluing formulas, any good explicit references?

Rafael: I am very tired because it is very late. Please check and expand this page, i am not counting very good with my eyes half closed neither am i an expert on ETQFT.

Urs Schreiber: notice that I had a bit on extended QFT over at FQFT. Maybe some of the material needs to be merged. i have now at least added links back and forth.

Rafael: Yes, for the merge. I think of a subsection here construction of ETQFTs and a pointer to the relation between ETQFT to AQFT. I also think you are much better to include the construction of ETQFTs in Nonabelian cocycles and their sigma model QFTs than me.