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And from now on I specialize to two dimensions and I had already started pointing out that in two dimensions many things simplify a lot.

Not only that all the Lorentz indices disappear, but also there is the factorization of space-time into two light-like directions

and also of most of the field theoretical formulas which separate into left-moving and right-moving degrees of freedom.

And maybe I write a few of these simplifying formulas again.

So we have the stress energy tensor. We can take these combinations which I call t plus and this depends only on x plus where x plus is time plus space.

And the same thing, well then we have a minus here and t minus depends only on x minus and x minus is this guy.

And I emphasize that the right-moving stress energy component and the left-moving component of the stress energy tensor must commute with each other.

Whatever the coordinates are and the two field operators moving in the same direction commute whenever the points x plus and y plus are different.

The commutator must be a distribution supported at a point and that means it can only be multiple of delta functions and derivatives.

And from that I had already derived that one can fix the commutator between two stress energy tensors completely up to a single parameter, real parameter, and the commutator must look like this.

So this is t of y times delta prime of x minus y.

And the undetermined term must be a multiple of the third derivative of the delta function and the coefficient is called c.

And it involves also some factors of pi.

So this is derived from the property that the generators of the symmetries are integrals or moments of the stress energy tensor.

So in particular, the p is just the integral over t itself and d is integral over x times t and k is integral x squared times t.

And these here generate the Möbius group or the three one parameter subgroups of the Möbius group in such a way that the commutator with any field or any quasi primary field is just a derivative.

And the commutator of d is xd plus the parameter h, which in four dimensions this was called d.

And here it is called h, the chiral scaling dimension.

And k comma phi is x square, the derivative, plus 2hx times phi of x.

So all these equations now should come with both with a plus sign and with a minus sign because we have in the conformal group two copies of the Möbius group acting on the plus coordinates and acting on the minus coordinates.

But if I have a chiral field that depends only on one of these coordinates, then of course only that chiral set of generators is relevant for that chiral field.

OK, all these formulas are just specializations from the much more complicated formulas involving eta mu nu all the time in four dimensions. They just simplify in this way.

But this formula here is completely new. There is no analog of such a formula in four dimensions because we don't have such a strong statement that the commutator must be localized in a point.

It can be localized anywhere in the future and in the past, a light cone. And then that gets no grip to get such a formula.

OK, this is what I had done last time. And now I continue with new material.

Here I have written the stress energy tensor as a field on the real line. And I already mentioned that one can re-parameterize the real line by a projective transformation called the Cayley transformation to a circle such that the real line is mapped onto the circle except the point minus 1.

And in order to actually make sense of the one parameter groups of conformal transformations which are generated by these generators, one has to add the point at infinity.

In other words, one has to extend the theory from the real line to the circle.

And one can show that these fields which are written here as distributions on the real line do extend to distributions on the circle.

So t of x, which is redefined t hat of z, which is t of x times some factor dz by dx to the minus 2, where z is 1 plus ix over 1 minus ix.

That's this picture here, z and x.

If I redefine the field as a distribution on the circle without the point just by this pullback here, then it turns out that I can extend the field to a field on the entire circle.

So this extends periodically to s1, including the point minus 1.

And as a distribution on the circle, I can just Fourier transform it. And it gives us a discrete Fourier decomposition, t hat of z.

And the convention is that the expansion is given in powers of z to the minus n minus 2.

And then the coefficients are called ln.

There's also a factor of 1 over 2 pi, which is conventional.

So these operators ln are the Fourier components of the operator value distribution t hat.

And then, of course, you may as well write ln as an integral, using Cauchy's formula, an integral of t hat of z times z power n plus 1.

And so you can compute the commutator of ln just by translating this formula into a commutator between the field on the circle and then extracting the Fourier coefficients.

And the result of that is the famous Virasoro algebra.

Which reads ln, lm is n minus m ln plus m.

And then there is a coefficient plus c over 12 n, n square minus 1, delta n plus m, comma 0.

So some people would just start a lecture on conformal field theory with this formula and then discussing its representation theory.

But my intent was to show you that this formula comes from the previous formula.

And the previous formula comes from underlying symmetry and causality principles.

If we don't have this term here, then this algebra is known as the Lie algebra of infinitesimal diffeomorphisms of the circle.

And the extra term here is a multiple of 1.

So that's a central extension of the Lie algebra of the diffeomorphisms.