nLab extended functorial field theory

Idea

Extended quantum field theory (or multi-tiered quantum field theory) is the fully local formulation of functorial quantum field theory, stated using higher category theory

Whereas a

• 1-categoricalfunctorial field theory may be regarded as a rule that allows one to compute 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 functorial field theory $Z(\Sigma)$ 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 functorial field theory is also sometimes called local or localized functorial field theory. 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.

Definition

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.)

$n Cob_d$ 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 $n Cob_n$ 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.

Extended functorial field theory

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

An $n$-extended $C$-valued functorial field theory of dimension $d$ is a symmetric $n$-tensor functor $Z: n Cob_d \rightarrow C$ 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

Classes of examples by dimension

$n=1$ gives ordinary TQFT.

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

The most common cases for $C$ are

• $C = n Hilb(R)$, the category of $n$-Hilbert spaces? over a topological field $R$. As far as we know this is only defined up to $n=2$.

• $C = n Vect(R)$, the category of $n$-vector spaces over a field $R$.

• $C = n Mod(R)$, the (conjectured?) category of $n$-modules over a commutative ring $R$.

3d: Turaev-Viro model

Generic examples

By the cobordism hypothesis-theorem every fully dualizable object in a symmetric monoidal $(\infty,n)$-category with duals provides an example.

Specific examples

• Levin-Wen model

• Chern-Simons theory as a fully extended TQFT

• …many more…

See also at TCFT.

Properties

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 $Z: n Cob_d \rightarrow n Vect(R)$ 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

• topological chiral homology.

also

• AQFT from n-functorial QFT.

Related concepts

• local quantum field theory

• extended Lagrangian

More on extended QFTs is also at

• FQFT

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

General

• Ruth J. Lawrence, Triangulations, categories and extended topological field theories. Quantum topology, 191–208, Ser. Knots Everything, 3, World Sci. Publ., River Edge, NJ, 1993. doi. Presented at the AMS Meeting 876, held in Dayton, Ohio, on October 31, 1992.

• Daniel S. Freed, Higher Algebraic Structures and Quantization (arXiv:hep-th/9212115)

• Daniel S. Freed, Quantum Groups from Path Integrals (arXiv:q-alg/9501025)

• John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory (arXiv:q-alg/9503002)

• Daniel S. Freed, Remarks on Chern-Simons theory, arXiv:0808.2507.

• Jacob Lurie, On the Classification of Topological Field Theories. arXiv

• Shawn X. Cui, Higher Categories and Topological Quantum Field Theories, Quantum Topology 10 4 (2019) 593-676 [arXiv:1610.07628, doi:10.4171/QT/128]

With an eye towards the full extension of Chern-Simons theory:

• Daniel S. Freed, Remarks on Fully Extended 3-Dimensional Topological Field Theories (2011) (pdf)

• Daniel S. 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)

Review:

• Luigi Alfonsi, §6 in: Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.07308]

For TQFTs appearing in solid state physics in the context of topological order (see also at K-theory classification of topological phases of matter):

• Daniel S. Freed, Gregory Moore, Twisted equivariant matter (arxiv/1208.5055)

  (uses equivariant K-theory to classify free fermion gapped phases with symmetry)

• Daniel S. Freed, Short-range entanglement and invertible field theories (arXiv:1406.7278)

• Daniel Freed, Michael Hopkins, Reflection positivity and invertible topological phases, Geometry & Topology (arXiv:1604.06527)

• Davide Gaiotto, Theo Johnson-Freyd, Condensations in higher categories (arXiv:1905.09566)

For its relation to the notion of generalized symmetries:

• Daniel S. Freed, Gregory W. Moore, Constantin Teleman, Topological symmetry in quantum field theory [arXiv:2209.07471]

The case of Rozansky-Witten theory:

• Lorenzo Riva, Higher categories of push-pull spans, I: Construction and applications [arXiv:2404.14597]

The $D=6$, $\mathcal{N}=(2,0)$ SCFT as an extended functorial field theory

On the (conjectural) suggestion to view at least some aspects of the D=6 N=(2,0) SCFT (such as its quantum anomaly or its image as a 2d TQFT under the AGT correspondence) as a functorial field theory given by a functor on a suitable cobordism category, or rather as an extended such FQFT, given by an n-functor (at least a 2-functor on a 2-category of cobordisms):

• Edward Witten, Section 1 of: Geometric Langlands From Six Dimensions, in Peter Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings & Lecture Notes Volume: 50, AMS 2010 (arXiv:0905.2720, ISBN:978-0-8218-4777-0)

• Daniel Freed, 4-3-2 8-7-6, talk at ASPECTS of Topology Dec 2012 (pdf, pdf)

• Daniel Freed, p. 32 of: The cobordism hypothesis, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (arXiv:1210.5100, doi:10.1090/S0273-0979-2012-01393-9)

• Daniel Freed, Constantin Teleman: Relative quantum field theory, Commun. Math. Phys. 326 (2014) 459–476 [doi:10.1007/s00220-013-1880-1, arXiv:1212.1692]

• David Ben-Zvi: Theory $\mathcal{X}$ and Geometric Representation Theory, talks at Mathematical Aspects of Six-Dimensional Quantum Field Theories IHES 2014, notes by Qiaochu Yuan (pdf, pdf, pdf)

• David Ben-Zvi, Algebraic geometry of topological field theories, talk at Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014 (web video)

• Lukas Müller, Extended Functorial Field Theories and Anomalies in Quantum Field Theories (arXiv:2003.08217)