I'm looking for a name or references to theories that include a stress energy tensor of the form



Where the fields anti-commute. In particular I'd like to know what kind sensible operators can be formed from normal ordered products of the $\phi$

Edit As commented below, my main interest is $D=4$ and from my calculations it seems that most of the operators than can be formed are only sensible when $B=0$ and there's only one type of operators (of weight 1) that can be formed when $B\neq 0$.

  • $\begingroup$ The second term is very non-generic, if you recall the meaning of $T(z)$. Have you tried finding a Lagrangian description? (Not saying that the latter should exist, of course.) $\endgroup$ – Vibert Mar 25 '13 at 13:15
  • $\begingroup$ @Vibert, I haven't, in fact I omitted the coefficients on the left of both terms. I calculated that there is only three pairs of coefficients that will make the tensor a valid one (weight 2) and one of the solutions is when the second therm is not there. But when the second term is there, there are some normal ordered products of $\phi$ which have extra terms in the $T(z)$ OPE and I'm not sure if I'm making a mistake or if it's a feature. $\endgroup$ – Prastt Mar 25 '13 at 13:46
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    $\begingroup$ Are the $\phi_\alpha$'s weight $\frac 12$ chiral primaries? Naively it seems that the second term does not make sense unless $D=4$. Am I misunderstanding something? By the way, I assume this is 2-dimensional CFT right? $\endgroup$ – Heidar Mar 25 '13 at 18:36
  • $\begingroup$ Heidar: restating what you probably already know, but unless you know that the theory is free, it's impossible to 'predict' the dimension of $:\prod \phi^i:.$ It can have some huge anomalous dimensions, right? (But I don't think OP is aiming for that.) $\endgroup$ – Vibert Mar 25 '13 at 22:10
  • $\begingroup$ @Vibert,@Heidar, in fact I'm interested in the case where $D=4$ but I tried to give it more generality so perhaps others can benefit also. $\endgroup$ – Prastt Mar 26 '13 at 1:25

When you talk about CFTs, it's clear that a theory of this form may only be a CFT, conformal, when $D=4$ because the stress-energy tensor must have dimension 2 which you may only get from 4 fermions that have $h=1/2$ each, and they do as manifested by the first term.

At least an example of a theory with a quartic term in the fermions is known as the Thirring model, and it may be bosonized (is equivalent) to the sine-Gordon model, see e.g.


Even if you meant the IR fixed point of a more general theory with a more general $D$, it's clear that $D$ must be even (superselection sectors, Grassmann segregation) and there are additional constraints.

Let me mention that even for $D=4$, if you write the product of components in the quartic term differently from the Thirring model, you will get a theory that may look conformal at the classical level but it won't be conformal quantum mechanically. The conformality of a theory is an extremely constraining condition. One can't expect that whatever we write down is conformal.

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