# 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 '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 '13 at 15:27

Most of the AQFT toolkit is about algebras of local operators. There aren't any physical local operators in topological QFTs whose interesting observables are global - topological - so AQFT, TQFT have almost nothing to do with each other. TQFT are QFTs that may be made pretty rigorous which is why e.g. Witten could get a Fields medal for such things but AQFT wanted to describe ordinary local QFTs with local physical excitations and TQFT is far from enough for that.

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Thank you Lubos. I am a little puzzled by the terms modular functor and modular group in Tomita-Takesaki Modular Theory. They are not related either? – Hamurabi Mar 13 '13 at 10:58
The modular group is an extremely general and omnipresent construction that appears well outside TQFT, AQFT, too. It's the "functor" that makes things heavily abstract and mathematical. – Luboš Motl Mar 13 '13 at 16:01
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 '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 '13 at 18:22