Holomorphicity and Mode Expansions

String Theory without the Baggage, Episode 25

Hey Friends,

I know you know how to solve the wave equation, but we’re going to do it explicitly here anyway. There’s a lot of important nuance. Pay close attention to the details inferred from the boundary conditions. Keep in mind the series expansion format. You might also want to brush up on your studies of complex variables.

Sean

Nothing calls for more coffee like Fourier Series on a Saturday Morning!

A Lightning Review of Lightcone Coordinates

Of late we’ve been using lightcone coordinate to parametrize the worldsheet. They are suggestively written

\(\tau = \left(z + \bar{z}\right),\quad \sigma = \left(z - \bar{z}\right).\)

We saw that the associated metric was

\(\gamma_{ab} \rightsquigarrow -2\left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)\)

Invariance of the action under reparametrizations gave us the constraint equation

\(T^{ab} = 0\)

which in terms of tau and sigma are just

\(T_{zz} = T(z),\quad T_{\bar{z}\bar{z}} = \bar{T}(\bar{z}).\)

The equation of motion are easily stated

\(\partial\bar{\partial}X^{\mu} = 0,\)

which operationally means that

\(X^{\mu}(z,\bar{z}) = X^{\mu}_{L}(z) + X^{\mu}_{R}(\bar{z}).\)

Here X_L and X_R need only be smooth functions of their respective variables¹.

Closed String Solutions

While somewhat peculiar, it is conventional to assign the values to the string endpoints:

\(\sigma_{0} = 0,\quad \sigma_{1} = \pi.\)

The closed string boundary conditions require that X^𝜇 be periodic in the length of the string, so in this case π. Let us attempt to construct a solution based on this fact.

Periodicity can be arranged using the exponential,

\(X_{L} \sim e^{2inz},\)

for any integer n. By linearity then we can add as many of these terms as we like. Additionally, the equations of motion also allow us to add a term proportional to z:

\(X^{\mu}_{L}(z) = A^{\mu}z + \sum_{n\in\mathbb{Z}}B^{\mu}_{n}e^{-2inz}\)

as long as we do the same for X_R

\(X^{\mu}_{R}(\bar{z}) = A^{\mu}\bar{z} + \sum_{n\in\mathbb{Z}}\bar{B}^{\mu}_{n}e^{-2in\bar{z}}\)

so that the 𝜎 dependence cancels between these linear terms, preserving the periodicity.

Lest you worry about the use of exp rather than sin and cos, recall that we are summing over positive and negative values of n, so any imaginary parts will get canceled.

Let’s dress up those constant parameters with a bit of physical intuition.

First we can pull out the n=0 terms into an overall constant, and rescale the mode constants. This gives

\(X^{\mu}_{L}(z) = \frac{1}{2}x^{\mu} + \frac{1}{2}\ell\alpha^{\mu}_{0}z + \frac{i\ell}{2}\sum_{n\neq 0}\frac{1}{n}\alpha^{\mu}_{n}e^{-2inz},\)

and

\(X^{\mu}_{R}(\bar{z}) = \frac{1}{2}x^{\mu} + \frac{1}{2}\ell\alpha^{\mu}_{0}\bar{z} + \frac{i\ell}{2}\sum_{n\neq 0}\frac{1}{n}\bar{\alpha}^{\mu}_{n}e^{-2in\bar{z}},\)

You can probably intuit here that little x represents the center-of-mass position of the string. The dimensionful constant ℓ constraints the relevant scales induced by the action constant κ. We’ll talk more about that later.

We have left the bar off the α_0, as periodicity in 𝜎 demands it be real. Additionally, the reality of X implies that

\(\alpha_{m}^{\mu\dagger} =\alpha_{-m}^{\mu}\quad\mathrm{and}\quad\bar{\alpha}_{m}^{\mu\dagger} =\bar{\alpha}_{-m}^{\mu}.\)

The constraint equation for X_L reads

\(\partial X_{L}^{\mu}\partial X_{L\mu} = 0.\)

This gives us a double sum,

\(T(z) = |\partial X_{L}|^{2} = \ell^{2}\sum_{m,n\in\mathbb{Z}}(\alpha_{m}\cdot\alpha_{n}) e^{-2i(m+n)z} = 0.\)

Dealing with such sums - and their products - will take up a large portion of our discussions moving forward.

Of course, the right-moving sector as a similar constraint.

\(\bar{T}(\bar{z}) = |\partial X_{R}|^{2} = \ell^{2}\sum_{m,n\in\mathbb{Z}}(\bar{\alpha}_{m}\cdot\bar{\alpha}_{n}) e^{-2i(m+n)\bar{z}} = 0.\)

Despite our insistence on using complex notation for our wave solutions, notice that the left and right mode coefficients are independent numbers. In particular,

\(\alpha_{m}^{\mu\dagger} \neq \bar{\alpha}_{m}^{\mu}\)

As you might have expected, X_L is a holomorphic function z, as is T(z). Any other term consistent with the equations of motion can be absorbed into a redefinition of the mode-coefficients. The same is true for X_R.

Open String Solutions

For open strings, we have the additional requirement that

\(\frac{\partial X^{\mu}}{\partial \sigma}\Big|_{\sigma_{0}} = 0 = \frac{\partial X^{\mu}}{\partial \sigma}\Big|_{\sigma_{1}},\)

which imposes a standing wave constraint between them. Effectively this ties the two flavors of mode-coefficients together, resulting in

\(X^{\mu} = X_{L}^{\mu}(z) + X_{R}^{\mu}(\bar{z})= x^{\mu} + \ell \alpha^{\mu}_{0}\tau + \sum_{n\neq 0}\frac{1}{n}\alpha_{n}^{\mu}e^{-in\tau}\left(\frac{e^{in\sigma} + e^{-in\sigma}}{2}\right).\)

Despite this apparently disheveled from, we still have

\(\partial X^{\mu} = \partial X_{L}^{\mu}(z) = \ell \sum_{n\in\mathbb{Z}}\alpha_{n}^{\mu}e^{-inz},\)

and

\(\bar{\partial} X^{\mu} = \bar{\partial} X_{R}^{\mu}(\bar{z}) = \ell \sum_{n\in\mathbb{Z}}\alpha_{n}^{\mu}e^{-in\bar{z}}.\)

Now there is only one set of α’s, and the constraint equations remain

\(T(z) = |\partial X_{L}|^{2} = \ell^{2}\sum_{m,n\in\mathbb{Z}}(\alpha_{m}\cdot\alpha_{n}) e^{-2i(m+n)z} = 0,\)

and

\(\bar{T}(\bar{z}) = |\partial X_{R}|^{2} = \ell^{2}\sum_{m,n\in\mathbb{Z}}(\alpha_{m}\cdot\alpha_{n}) e^{-2i(m+n)\bar{z}} = 0.\)

As you might expect, we’ll tame these unruly sums with integration over 𝜎. But more on that next time.

Holomorphic Coordinate Transformations

In lightcone coordinates, the metric 𝛾_𝑎𝑏 gives the line element

\(ds^2 = -4dzd\bar{z}.\)

Now consider a nonlinear change of coordinates,

\(z\mapsto f(z), \quad\bar{z} \mapsto \bar{f}(\bar{z}).\)

The line element then becomes²

\(ds^2 \rightarrow -4\left|\frac{df}{dz}\right|^{2}dzd\bar{z}.\)

This is just Weyl transformation of the metric 𝛾_𝑎𝑏. We’ve already seen that the action is invariant under such a coordinate transformation.

Now, we’ve already used Weyl invariance to fix our metric via the conformal gauge. What this tells us is that there is a degeneracy amongst such reparameterizations. There are residual symmetries that must be dealt with to truly fix a section of the worldsheet reparametrization gauge bundle.

More precisely, in the conformal gauge, we still have the freedom to make a holomorphic change of coordinates³.

Isolating that freedom, expressing it as a symmetry, further gauge fixing and creating BRST ghosts to account that fixing is going to be a very subtle - but very fascinating - business going forward.

I hope you join us for the rest of it!

1. Or at least differentiable, but we’d prefer our physical solutions to be smooth. But you get the point. They’re pretty arbitrary! The differential equation is linear, so we can always Fourier analyze any smooth function into constitute modes if needed.

2. Here again we are anticipating the Wick rotation of τ to -iτ, making z into a complex variable.

3. This could have been anticipated from the other direction. Given that the composition of two holomorphic functions remains holomorphic, T(f(z)) will remain holomorphic and the constraint equations will remain satisfied. All we are really doing here is just rescaling the mode coefficients.