# What is central charge in a CFT?

Hi. I'm trying to learn CFT on my own, and central charge seems to be a pretty important concept. It seems that I can only find mathematical definitions in terms of the stress-energy tensor, or the Virasoro algebra. I was wondering if someone could give me a physical interpretation of central charge, or at least a definition in plain English.

Thanks.

spaghetti3451

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they're the casimirs of the theory. A casimir of a theory is an operator that commutes with all other symmetry operators of a theory. For angular momentum this is the $$L_i L^i$$ operator. The eigenvalues of the Casimir are used to determine the system, such as the j(j+1) for the angular momentum.

Hope that was of some use

spaghetti3451
Sure. That helps a bunch.

So when we say that a CFT has central charge of, say, c=2, do we really mean that the "eigenvalue" of c is 2?.

spaghetti3451
Yes, the letter c is used to denote both the operator and it's eigenvalue. This confusing state of affairs is the convention

spaghetti3451
So if c is an operator, then I'm guessing that it may carry several eigenvalues. But I've never heard of a CFT with more than one central charge. Maybe I just haven't read far enough yet. Is there such thing as fractional or irrational central charges? Like, say, c=pi?

spaghetti3451
I have to step down at this point since I'm not familiar with CFT I only know about central charges from string theory

spaghetti3451
No problem. Thanks for your help. I guess I just have to keep reading.

spaghetti3451
dextercioby
Homework Helper
they're the casimirs of the theory. A casimir of a theory is an operator that commutes with all other symmetry operators of a theory. For angular momentum this is the $$L_i L^i$$ operator. The eigenvalues of the Casimir are used to determine the system, such as the j(j+1) for the angular momentum.

Hope that was of some use
Does the notion of central charge in the context of the conformal field theory differ so radically from the notion of central charge for any other Lie algebra, like for example the Galilei or Poincare ones ? I'm just asking, because I don't see any reason, even if we complement the CFT algebra of n-dimensional space-time with SUSY generators. In other words the concept of central charge should be uniform (and essentially mathematic) for any physical Lie algebra or superalgebra.

spaghetti3451
fzero
Homework Helper
Gold Member
Does the notion of central charge in the context of the conformal field theory differ so radically from the notion of central charge for any other Lie algebra, like for example the Galilei or Poincare ones ? I'm just asking, because I don't see any reason, even if we complement the CFT algebra of n-dimensional space-time with SUSY generators. In other words the concept of central charge should be uniform (and essentially mathematic) for any physical Lie algebra or superalgebra.
The central charge in a CFT is based on the same principle of central extension as in any other Lie algebra. The Virasoro algebra projects onto the Witt algebra when you divide by the center. Both are subalgebras of the algebra of diffeomorphisms on $$S^1$$.

There was a question about allowed values of the central charge. If we're talking about representations of the Virasoro algebra, the constraint arises from demanding that the representations be unitary (no negative norm states). All values of $$c\geq 1, h\geq 0$$ are allowed ($$h$$ is the conformal weight), while between $$0<c<1$$ there is a discrete set of points

$$c= 1 -\frac{6}{m(m+1)}, ~ m=3,4,\ldots .$$

A blatant example of a fractional value of $$c$$ is that of a free fermion, which has $$c=1/2$$. Ginsparg's lectures http://arxiv.org/abs/hep-th/9108028 are a great reference for many CFT topics.