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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 17th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: 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 * [[Hessel Posthuma]], _The Heisenberg group and conformal field theory_, [arXiv](https://arxiv.org/abs/0706.4262). ## Kandel * [[Santosh Kandel]], _Functorial Quantum Field Theory in the Riemannian setting_, [arXiv](https://arxiv.org/abs/1502.07219v3). ## Tener * [[James E. Tener]], _Construction of the unitary free fermion Segal CFT_, [arXiv](https://arxiv.org/abs/1608.02095). ## Field theories with interaction ### Pickrell * [[Doug Pickrell]], _P(phi)_2 quantum field theories and Segal's axioms_, [arXiv](https://arxiv.org/abs/math-ph/0702077). $$S(\Phi)=\int_\Sigma 2^{-1}(\|d\Phi\|^2+m^2\Phi^2)+P(\Phi)$$ ### Liouville field theory * Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, _Segal's axioms and bootstrap for Liouville Theory_, [arXiv](https://arxiv.org/abs/2112.14859). <a href="https://ncatlab.org/nlab/revision/list+of+functorial+field+theories/1">v1</a>, <a href="https://ncatlab.org/nlab/show/list+of+functorial+field+theories">current</a>

   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

   • Hessel Posthuma, The Heisenberg group and conformal field theory, arXiv.

   Kandel

   • Santosh Kandel, Functorial Quantum Field Theory in the Riemannian setting, arXiv.

   Tener

   • James E. Tener, Construction of the unitary free fermion Segal CFT, arXiv.

   Field theories with interaction

   Pickrell

   • Doug Pickrell, P(phi)_2 quantum field theories and Segal’s axioms, arXiv.

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

   Liouville field theory

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 17th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexProbably one should count all rational 2d CFTs as examples, via the [[FRS-theorem on rational 2d CFT|FRS construction]].

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

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 21st 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRe #2: Can we pinpoint a specific theorem in their series of papers that actually constructs a (nontopological) functorial field theory?

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 21st 2023
   • (edited Aug 21st 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt's most explicit in the last article in the series: * [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Uniqueness of open/closed rational CFT with given algebra of open states*, Adv. Theor. Math. Phys. **12** (2008) 1283-1375 [[hep-th/0612306](http://arxiv.org/abs/hep-th/0612306), [doi:10.4310/ATMP.2008.v12.n6.a4](https://dx.doi.org/10.4310/ATMP.2008.v12.n6.a4)] 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: * [[Ingo Runkel]], [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Christoph Schweigert]], *Topological and conformal field theory as Frobenius algebras*, in: *Categories in Algebra, Geometry and Mathematical Physics* Contemp. Math. **431** (2007) 225-248 $[$[doi:10.1090/conm/431](http://dx.doi.org/10.1090/conm/431), [arXiv:math/0512076](https://arxiv.org/abs/math/0512076)$]$

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

   • Jens Fjelstad, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Uniqueness of open/closed rational CFT with given algebra of open states, Adv. Theor. Math. Phys. 12 (2008) 1283-1375 [hep-th/0612306, doi:10.4310/ATMP.2008.v12.n6.a4]

   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:

   • Ingo Runkel, Jens Fjelstad, Jürgen Fuchs, Christoph Schweigert, Topological and conformal field theory as Frobenius algebras, in: Categories in Algebra, Geometry and Mathematical Physics Contemp. Math. 431 (2007) 225-248 $[$doi:10.1090/conm/431, arXiv:math/0512076$]$