Notes on string theory: Virasoro operators (defined classically)

Recall the constraints (\dot{X} \pm X^{\prime})^2 = 0. One can also show that in the LC gauge a \cdot a = -2a^{+}a^{-} + \vec{a}_{T} \cdot \vec{a}_{T}. It follows,

    \[ -2(\dot{X}^{+} \pm X^{\prime +})(\dot{X}^{-} \pm X^{\prime -}) + (\dot{X}^{I} \pm X^{\prime I})(\dot{X}^{I} \pm X^{\prime I}) = 0 \]

    \[-2 \cdot 2\alpha^{\prime}p^{+}(\dot{X}^{-} \pm X^{\prime -}) + (\dot{X}^{I} \pm X^{\prime I})(\dot{X}^{I} \pm X^{\prime I}) = 0 \]

    \[ \implies \dot{X}^{-} \pm X^{\prime -} = \frac{1}{2\alpha^{\prime}} \frac{1}{2p^{+}}(\dot{X}^{I} \pm X^{\prime I})^{2} \]

Therefore, we arrive at an expression that gives \dot{X}^{-} and X^{\prime -}. The key realisation here being that if you know \dot{X}^{-} and X^{\prime -}, you can calculate anywhere on the WS.

Notice, also, dX^{-} = \frac{\partial X^{-}}{\partial \tau}d\tau + \frac{\partial X^{-}}{\partial \sigma}d\sigma = \dot{X}^{-}d\tau + X^{\prime -}d\sigma.

For closed strings the situation becomes more complicated.

In the context of the free open string, the aim at this juncture is to arrive classically at the Virasoro operators, or, what will be the Virasoro operators in the quantum theory. The hope is that in accomplishing this task, the discussion in the quantum theory will be more intuitive as we have been building a fairly comprehensive picture.

The essential claim, to start, is that X^{I}(\tau, \sigma), P^{+}, X_{0}^{-} deliver X^{-}(\tau, \sigma) completely. For instance, we know that we can compute:

    \[ \dot{X}^{I} \pm X^{\prime I} = \sqrt{2\alpha^{\prime}} \sum_{n \in \mathbb{Z}} \alpha_{n}^{I} e^{-in(\tau \pm \sigma)} \]

We can also write,

    \[ \dot{X}^{-} \pm X^{\prime -} = \sqrt{2\alpha^{\prime}} \sum_{n \in \mathbb{Z}} \alpha_{n}^{-} e^{-in(\tau \pm \sigma)} \]

And so, it follows,

    \[ X^{-}(\tau, \sigma) = x_{0}^{-} + \sqrt{2\alpha^{\prime}}\alpha_{0}^{-}\tau + i\sqrt{2\alpha^{\prime}} \sum_{n \neq 0} \frac{1}{n} \alpha_{n}^{-} e^{-in\tau}cosn\sigma \]

We have already found (\dot{X}^{-} \pm X^{\prime -}). All we have to do now is make the appropriate substitution and tidy up. Pay attention to the result,

    \[ = \frac{1}{2\alpha^{\prime}} \frac{1}{2P^{\tau}} \sqrt{2\alpha^{\prime}} \sum_{p, q \in \mathbb{Z}} \alpha_{p}^{I} \alpha_{q}^{I} e^{-i(p+q)(\tau \pm \sigma)} \]

Where n = p+q and \{p,q\} = \{n, p\}. It follows,

    \[ \frac{1}{2p^{+}} \sum_{n \in \mathbb{Z}}(\sum_{p \in \mathbb{Z}} \alpha_{p}^{I} \alpha_{n-p}^{I})e^{-in(\tau \pm \sigma)} \]

    \[ \implies \sqrt{2\alpha^{\prime}} \alpha_{n}^{-} = \frac{1}{p^{=}} \frac{1}{2} \sum_{p \in \mathbb{Z}} \alpha_{p}^{I} \alpha_{n-p}^{I} \]

    \[ = \frac{1}{p^{+}} L_{n}^{\perp} \]

Where L_{n}^{\perp} = \frac{1}{2} \sum_{p \in \mathbb{Z}} \alpha_{p}^{I} \alpha_{n-p}^{I}. In the quantum theory, these are the Virasoro operators. For now, in the classical theory, they are just numbers but they will become operators summed over transverse objects (hence the perp. notation).

As an aside, when n=0:

    \[ \sqrt{2\alpha^{\prime}} \alpha_{0}^{-} = \frac{1}{p^{+}}L_{0}^{\perp} \]

As \alpha_{0}^{-} = \sqrt{2\alpha^{\prime}}p^{-},

    \[ 2\alpha^{\prime}p^{-} = \frac{1}{p^{+}}L_{0}^{\perp} \]

    \[ \implies 2p^{+}p^{-} = \frac{1}{\alpha^{\prime}} L_{0}^{\perp} \]

Which is the final complete solution for the free string in terms of all transverse degrees of freedom.

References

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

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Barton Zwiebach. (2009). “A First Course in String Theory”, 2nd ed.