# Algebraic/Axiomatic QFT vs Topological QFT

Can anybody please tell me a good source investigating the relation between Algebraic/Axiomatic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT)? Or is there none?

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There are a few papers in which topological field theories are constructed in terms of nets of algebras. The idea generally is that a net of algebras gives you a model for the higher category associated to a point by an extended TQFT. (Physicists would say that a 2d conformal net describes a 2d CFT which is related to a 3d TQFT.)

The first one that comes to mind is Bartels, Douglas, & Henriques. I'd bet that you'll find others if you dig around in @ursschreiber's nLab.

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Thank you for the reply. It seems there is more relation than initially thought. –  Hamurabi Mar 13 at 14:04
@Hamurabi Yes, what Lubos missed is that global phenomena can arise from the way local nets fit together. In topological field theories, this process is sometimes simple enough that mathematicians can describe it explicitly. –  user1504 Mar 13 at 15:27
The modular group in Tomita-Takesaki theory is a copy of $\mathbb{R}$. It gets its name from the "modular operator", which is related to the modulus of the operator sending an operator to its adjoint. It shares only a name with the modular group in string theory/geometry/number theory, which is a subgroup of $PSL(2,\mathbb{Z})$; they are not related concepts. –  user1504 Mar 13 at 18:05
The modular in modular functor, on the other hand, actually is related to the $PSL(2,\mathbb{Z})$ modular group. Obviously, mathematicians use "modular" to mean too many different things. –  user1504 Mar 13 at 18:22