Proving conformal invariance of a field theory by property of its stress energy tensor.

Question

I have a question about proving conformal invariance of a field theory by property of its stress energy tensor.

In physics there is argument that when the stress-energy tensor is traceless, symmetry, holomorphic, the QFT is conformal invariance.

But is there a rigorous math proof of this?

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user Qingyun Sun

Comments

What kind of formalization of field theory did you have in mind?

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user j.c.

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In the sense of constructive QFT, namely the probability framework.

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user Qingyun Sun

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This is probably not a necessary criterion, because by taking countably many massless free fields, you get a conformal QFT which is even globally conformal invariant but has no stress energy tensor ("the central charge is infinity"). About how many dimensions are you talking, maybe I can cook you up an argument, when this is sufficient.

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user Marcel Bischoff

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Thanks, I am thinking about the CFT that appears in 2-d stat physics, and mostly c<=1.

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user Qingyun Sun

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For example, for a boundary QFT defined on a simply connected bounded domain, we should be an equivalent condition for the QFT to be conformal invariance.(Say, the "measure" is invariant under the conformal transformation from the domain to disk. Or for the correlation function to be covariant.)

This post imported from StackExchange MathOverflow at 2014-08-29 08:10 (UCT), posted by SE-user Qingyun Sun