Why is there an Euler density in SCFT $T_{\mu}^{\mu}$?

Question

The super conformal field theories are above all conformal. Conformal theories are defined on flat space-times. Despite that, if one looks at the stress tensor trace of a SCFT in 4d you get a contribution from the field strength of the gauge sector and the Euler density, i.e. $$T_{\mu}^{\mu} \backsim F_{\mu \nu }^2 - a(R_{\mu \nu \rho \sigma})^2 + \text{other terms}$$ Where does this $R_{\mu \nu \rho \sigma}$ come from in a super-conformal field theory which, from what I know, it is defined in flat space-time? Why we call this $a$ "central charge" despite it looks like some coupling?

This post imported from StackExchange Physics at 2014-12-23 15:08 (UTC), posted by SE-user Marion

Answer 1

There is nothing wrong in putting a conformal field theory in a curved background as long as the conformal anomaly vanishes. A curved background is some Riemannian manifold $\mathcal{M}$ equipped with some metric $g$ from which one calculates both the Euler density and the Weyl tensor. Furthermore one can compute the trace Ward idenity which must vanish and finds that $\langle T_{\mu}^{\mu} \rangle \backsim aE + cW^2 +$ other terms. Check hep-th/9708042.

Comments

I wonder why nobody corrected me yet. Does the trace have to vanish? Hehe.

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It s Christmas ... ;-) Hm I thought that the trace has to vanish for conformal invariance, whereas for just scale invariance to hold it is allowed to corrrspond to some kind of conserved current or something along these lines ...?

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My impression is that the trace does not have to vanish as long as the background is not dynamical. If the trace vanishes then $a$ and $c$ are zero, right? Then does the $a$-theorem in 4d or the $c$-theorem in 2d hold? Of course if there is an anomaly the theory loses its conf. invariance.

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Hm I have read somewhere that the quantities a or c correspond only in specific cases exactly to the anomaly would have to reconsider that ... Maybe @suresh knows?

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$a$ and $c$ correspond to the central charges of the CFT (the corresponding QFT at a fixed point).