Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 17th 2023

    Created:

    This article is meant to give an exhaustive list of explicitly constructed nontopological functorial field theories in dimension 2 and higher. All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.

    Free field theories

    Posthuma

    Kandel

    Tener

    Field theories with interaction

    Pickrell

    S(Φ)= Σ2 1(dΦ 2+m 2Φ 2)+P(Φ)S(\Phi)=\int_\Sigma 2^{-1}(\|d\Phi\|^2+m^2\Phi^2)+P(\Phi)

    Liouville field theory

    • Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Segal’s axioms and bootstrap for Liouville Theory, arXiv.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2023

    Probably one should count all rational 2d CFTs as examples, via the FRS construction.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 21st 2023

    Re #2: Can we pinpoint a specific theorem in their series of papers that actually constructs a (nontopological) functorial field theory?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2023
    • (edited Aug 21st 2023)

    It’s most explicit in the last article in the series:

    In section 3.4 there they give a definition of (rational) functorial conformal field theory, then they proceed to show that their construction produces all these.

    They formulate the 2d functorial field theory slightly differently than usual because their primary perspective is the functorial 3d Chern-Simons theory defined by the given modular tensor category, from which the 2d CFT is obtained in a “holographic fashion” as a twisted/boundary field theory of the 3d functorial field theory.

    This way the eponymous “functor” underlying their functorial field theory is the “modular functor” which assigns vector spaces not to 1d boundaries but to cobordisms (these being the spaces of conformal blocks = the spaces of states of the ambient 3d CS theory) and then the 2d CFT proper appears to them as a natural transformation into that functor which picks in each such space an actual correlator — the latter being secretly the linear map that the 2d functorial field theory assigns to the given cobordism.

    In this perspective, the functoriality of the 2d CFT is the “sewing constraints” satisfied by that natural transformation into the modular functor.

    They are more explicit about this perspective on functorial CFT in Section 2 of the review: