# Examples of conformal field theories without relevant operators

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In a conformal field theory in $d$ spacetime dimensions, every local operator has a well-defined dimension (which determines its behaviour under the action of the dilatation operator). An operator of dimension strictly greater than $d$ is called irrelevant, an operator of dimension equal to $d$ is called marginal and an operator of dimension stricly less than $d$ and different from the identity operator is called relevant.

The adjectives relevant and irrelevant describe the effect obtained by perturbing the conformal field theory by one of its local operator and then flowing to the infrared under the renormalization group flow. In particular, the relevant operators are the one which, starting from the given conformal field theory, define perturbations generating non-trivial trajectories of the renormalization group flow.

My question is: what are the known conformal field theories without any relevant operators ?

These particular conformal field theories should play a special role in the "space of quantum field theories" as being the "ultimate end points" of the renormalization group flow.

The only examples I know are:

1) for every spacetime dimension $d$, the trivial conformal field theory, without any local operator different from the identity.

2) for $d=2$, the Monster CFT, which has central charge $c=24$, and whis is an orbifold of the toroidal compactification of 24 free bosons on the Leech lattice (the unique even unimodular lattice of rank 24 without roots). The Monster CFT satisfies holomorphic factorization, the partition function of its holomorphic part is $J(q)=j(q)-744 = 1/q + 196 884 q +...$ (the absence of the term in $q$ is related to the absence of relevant operators) and the Monster group, the greatest sporadic finite group, is its group of global symmetries. The Monster CFT has no relevant operators but also no marginal ones: it is an isolated point in the space of conformal field theories. The Monster CFT is conjecturally the CFT dual of pure gravity on $AdS_3$ at minimal radius (see http://arxiv.org/abs/0706.3359 )

Are there other examples?

One reason why it does not seem easy: if a conformal field theory has a Lagrangian description in terms of massless fields, then turning on masses for all these fields define relevant perturbations and the theory flows in the infrared to the trivial conformal field theory.

edited Mar 7, 2015
Dear 40227, would you mind transfering the last sentence asking about the gravity dual of a CFT with no relevant operators into a new question? I am personally interested in this point too and it would then probably get more attention and a nice separate answer.

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There are infinite examples related to your example (2) in $d=2$. Generally, if we just consider CFT obtained from compactification of bosons on a Euclidean lattice, then the criteria that there are no relevant operators translates into that the lattice has no root vectors. The smallest such lattice is the 23-dimensional shorter Leech lattice, in which the shortest vector has length 3. The next one is the Leech lattice, where the shortest vector has length 4. And for each positive integer $n$, there exists a lattice where the shortest vector has length $n$.

One can find other examples in higher dimensions. An easy one is probably the compact $U(1)$ Maxwell electrodynamics in $d=3+1$ (pure gauge theory, no matter). The same CFT in $d=2+1$ has a relevant operator corresponding to proliferation of monopoles, which is the famous result due to Polyakov that $2+1$ compact electrodynamics is confined.

answered Mar 7, 2015 by (550 points)
edited Mar 7, 2015 by Meng

In fact, the number of unimodular lattices without vectors of norm <3 grows very rapidly with the dimension.

@Meng: thanks for your answer. I am confused about your examples in $d=2$. For example, for the shorter Leech lattice, if I have a $p$ with $p^2=3$ then $e^{i p.X}$ will be of dimension 3/2 and so relevant. And for any lattice, I always have the operators $\partial X^i$ which are of dimension 1. It is to eliminate these operators that one needs to do an orbifold in the construction of the Monster CFT.

@40227 I guess you can apply orbifold construction to any of these lattice that I have mentioned, no?  Actually, even before orbifolding, the current operators $\partial X^i$ are quite harmless: they do not directly drive RG flows that change the central charge of the system (i.e. make some of the degrees of freedom massive), all they can possibly do is continuously changing the scaling dimension of operators.

Only operators with equal left and right conformal weights can generate a mass, which means that $e^{i p\cdot X}$  in your example cannot, as it's $(\frac{3}{2},0)$. When augmented with an appropriate right-moving piece, it would have scaling dimension $\frac{3}{2}+\frac{3}{2} = 3$.

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