Notes on string theory: The Virasoro algebra, plus an introduction to the central charge

As a last step toward a climactic study of the string spectrum in the light cone gauge, it is interesting to examine alongside Polchinski the Virasoro algebra. Moreover, some time ago we first encountered the Virasoro constraints in the context of a study of the relativistic string.
It was also previously hinted in our classical definition of the Virasoro ops. that, upon quantisation, they will become operators summed over transverse objects. As we have since invoked the technique of light cone quantisation in our drive for quick access to a study of the physical states of the free bosonic string, let us have a quick look at this.

There are different masses/spins of a centre of mass motion. In the LC gauge, recall P^{-} is the Hamiltonian of the system. Also note, below is the spectrum in the LC gauge for the point particle:

    \[ P^{-} = \mathcal{H} \]

    \[ m^2 = 2P^{+}P^{-} - P^{i}P^{i} \]

    \[ \implies m^2 = 2P^{+}\mathcal{H} - P^{i}P^{i} \]

We think of this result as subtracting from the LC energy the kinetic energy of the string. Further, and more to the point, remember the expression from the Hamiltonian that we’ve already constructed in a previous engagement. We substitute the above,

    \[ m^2 = 2p^{+} [ \frac{P^{i} P^{i}}{2P^{+}} + \frac{1}{2P^{+}\alpha^{\prime}} (\sum_{n=1}^{\infty} a_{-n}^{i} a_{n}^{i} + a)] - P^{i}P^{i} \]

Think, now, of this m^2 as an operator acting on the Fock space. As we have already done the computation, we know that we arrive at the following,

    \[ M^{2} = \frac{1}{\alpha} (\sum_{p=1}^{\infty}) p (\alpha_{p}^{I})^{\dagger}\alpha_{p}^{I} + a \]

Where a \equiv -\frac{(D-2)}{24}.

From all of the work so far, we have observed that when we quantise the bosonic string the \alpha modes become operators. The Virasoro generators L_m, as we documented some time ago, also become operators. This should be fairly obvious at this point, since the generators are constructed from \alpha terms.

But, recall from section 4.2.4 in Polchinski’s book, there is a caveat here about the ordering for L_m. The reason for the ordering ambiguity emerges from the commutation relations. Just recently we worked around the issue without explicitly stating that, as in QFT, we need to use normal ordering for the operators. This concept of normal ordering is, again, one that we’ll also reintroduce and more thoroughly discuss in the next chapter. For now, what we may say is that we can define L_m as

    \[ L_{0}^{\perp} = \frac{\alpha_{0}^{2}}{2} + \sum_{n=1}^{\infty}\alpha_{-n}\alpha_{n} \]

And now, what is interesting, is that we can show the following to be the case,

    \[ L_{0}^{\perp} \equiv \frac{1}{2}\sum_{-\infty}^{\infty} :\alpha_{-n}\alpha_{n}: \]

    \[= \frac{\alpha_{0}^{2}}{2} + :\frac{1}{2}\sum_{n=-\infty}^{-1}\alpha_{-n}\alpha_{n} + : \frac{1}{2}\sum_{n=1}^{\infty}\alpha_{-n}\alpha_{n} : \]

    \[= \frac{\alpha_{0}^{2}}{2} + \frac{1}{2}\sum_{n=-\infty}^{-1}\alpha_{-n}\alpha_{n} + \frac{1}{2}\sum_{n=1}^{\infty}\alpha_{-n}\alpha_{n} \]

We now relabel n \rightarrow -m and then, in the last line, relabel m \rightarrow n.

    \[= \frac{\alpha_{0}^{2}}{2} + \frac{1}{2}\sum_{m=1}^{\infty}\alpha_{-m}\alpha_{m} + \frac{1}{2}\sum_{n=1}^{\infty}\alpha_{-n}\alpha_{n} \]

    \[= \frac{\alpha_{0}^{2}}{2} + \sum_{n=1}^{\infty}\alpha_{-n}\alpha_{n} \]

What we now want to do is investigate the Virasoro operators, as we continue on our path to studying the free string spectrum. Recall, in relation to what is immediately above, the expression:

    \[ L_{0}^{\perp} = \frac{1}{2} \sum_{n \in \mathcal{z}} \alpha_{-n}^{I}\alpha_{n}^{I} + a \]

For a study of the Virasoro ops., we start with (L_{n}^{\perp})^{\dagger} = L_{-n}^{\perp}. And without doing the full computation here, as we read in Polchinski we find,

    \[ [L_{m}^{\perp}, L_{n}^{\perp}] = (m-n)L_{m+n}^{\perp} \]

When m + n = 0 there is a new term (the central charge). And as we’ll see in the chapter on stringy CFTs, the Virasoro algebra can be written below, where it acquires a central extension or conformal anomaly,

    \[ [L_{m}^{\perp}, L_{n}^{\perp}] = (m-n)L_{m+n}^{\perp} + \frac{c}{12}(D-2)(m^3 - m)\delta_{m-n, 0} \]

Where c is the central charge. We will later revisit these notions of the central extension and the conformal anomaly. Their meaning will become much more clear in the context of Conformal Field Theory, as the latter directly implies a quantum mechanical breakdown of classical conformal symmetry. The former also has a nice connection to the Lie algebra, such that the Virasoro algebra is a complex Lie algebra that represents a central extension of the Witt algebra. In the meantime, if one would like to read more about this, a compact summary can be found on p. 62 in Becker, Becker, Schwarz.

Continuing forward, we will eventually grow comfortable with the fact that c is equivalent to the dimension of the spacetime in which our theory lives. In other words, in the quantum case we have found that the Virasoro algebra takes on the above extension. The Virasoro ops. are now the modes of the energy-momentum tensor, and this also means that it is the latter that will generate the sort of conformal transformations that we will study very soon.

It also turns out to be the case that in order to avoid having non-negative norm states it must be that c = 26. (A nice, compact review of this last point can be found in Wray, 2009).

Again, I will reserve deeper discussion about these topics until the next chapter in our discussion (Conformal Field Theories). For now, we just need to note that the first piece on the right-hand side is the Witt algebra. The second piece is the central extension or what is called the central charge.

As a further summary tutorial, the Virasoro ops. are required for the construction of the Lorentz algebra. They also generate reparameterisations of the worldsheet.

    \[ [L_{m}^{\perp}, X^{I}(\tau, \sigma)] = \xi_{m}^{\tau} \dot{X}^{I} + \xi_{m}^{\sigma} X^{\prime I} \]

Where \xi_{m}^{\tau}(\tau, \sigma) = -ie^{im\tau}\cos m\sigma and \xi_{m}^{\sigma} (\tau, \sigma) = e^{im\tau}\sin m\sigma.

As the Virasoro’s generate reparameterisations, it is an infinitesimal action of the Virasoro ops. that change the coordinates on the WS. Note, also, parameters are now complex, when \tau and \sigma should be real. But also recall that in QM only the anti-Hermitian ops. are the ones that generate the correct states/ops. Moreover, anti-Hermitian ops. generate real transformations.

The reason this topic is worth specific mention in an already condensed discussion is because the reparameterisations generated by the Virasoro operators are very special. As we will learn in the future, they are reparameterisations that preserve the constraints of the theory.

One last comment before we focus on the string spectrum: in a study of the Lorentz generators, we end up with the result \mathcal{H} = L_{0}^{\perp} - 1 for the Hamiltonian. I will leave this to the reader to work out.

In the next post, we will study the spectrum of states for the free open bosonic string.

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction”.

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.

Kevin Wray. (2009). “An Introduction to String Theory”.