extended functorial field theory in nLab

Contents

Contents

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(Σ)Z(\Sigma) assigned to dd-dimensional manifolds by cutting these manifolds into a sequence {Σ i}\{\Sigma_i\} of dd-dimensional composable cobordisms with (d−1)(d-1)-dimensional boundaries ∂Σ i\partial \Sigma_i, e.g. Σ=Σ 2∐ ∂Σ 1=∂Σ 2Σ 1\Sigma = \Sigma_2 \coprod_{\partial \Sigma_1 = \partial \Sigma_2} \Sigma_1, then assigning quantities Z(Σ i)Z(\Sigma_i) to each of these and then composing these quantities in some way, e.g. as Z(Σ)=Z(Σ 2)∘Z(Σ 1)Z(\Sigma) = Z(\Sigma_2)\circ Z(\Sigma_1);

we have that

• in extended functorial field theory Z(Σ)Z(\Sigma) may be computed by decomposing Σ\Sigma into dd-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension kk.

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 ZZ regarded as a functor on a higher category of cobordisms.

Definition

The category of extended cobordisms

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

nCob dn Cob_d is an nn-category with smooth compact oriented (d−n)(d-n)-manifolds as objects and cobordisms of cobordisms up to nn-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category nCob nn Cob_n of fully extended nn-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes nn-cobordisms as nn-morphisms and smooth homotopy classes of diffeomorphisms beween them as (n+1)(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 RR, and let CC be the symmetric monoidal nn-category of nn-RR-modules.

An nn-extended CC-valued functorial field theory of dimension dd is a symmetric nn-tensor functor Z:nCob d→CZ: n Cob_d \rightarrow C that maps

• smooth compact oriented dd-manifolds to elements of RR

• smooth compact oriented (d−1)(d-1)-manifolds to RR-modules

• cobordisms of smooth compact oriented (d−1)(d-1)-manifolds to RR-linear maps between RR-modules

• smooth compact oriented (d−2)(d-2)-manifolds to RR-linear additive categories

• cobordisms of smooth compact oriented (d−2)(d-2)-manifolds to functors between RR-linear categories

• etc …

• smooth compact oriented (d−n)(d-n)-manifolds to RR-linear (n−1)(n-1)-categories

• cobordisms of smooth compact oriented (d−n)(d-n)-manifolds to (n−1)(n-1)-functors between RR-linear (n−1)(n-1)-categories

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

Here nn can range between 00 and dd. This generalizes to an arbitrary symmetric monoidal category CC as codomain category.

Examples

Classes of examples by dimension

n=1n=1 gives ordinary TQFT.

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

The most common cases for CC are

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

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

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

3d: Turaev-Viro model

Generic examples

By the cobordism hypothesis-theorem every fully dualizable object in a symmetric monoidal (∞,n)(\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:nCob d→nVect(R)Z: n Cob_d \rightarrow n Vect(R) is an extended TQFT. Since ZZ maps the (d−1)(d-1)-dimensional vacuum to RR as an RR-vector space, by functoriality ZZ is forced to map a dd-dimensional closed manifold to an element of RR. Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of ZZ is enterely determined by its behaviour on (d−n)(d-n)-dimensional manifolds. See details at cobordism hypothesis.

Relation of ETQFT to AQFT

See

• topological chiral homology.

also

• AQFT from n-functorial QFT.

• local quantum field theory

• extended Lagrangian

More on extended QFTs is also at

• FQFT

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]