Boundary Conditions

String Theory without the Baggage, Episode 23

Hey Friends!

It’s been a tough week getting off the road and trying to reassemble some sense of normalcy around here. Today we’ll look at classical solutions and get our first hint of the applications of complex variable and Riemann surfaces to the study of relativistic strings.

Boundary conditions can seem boring, but as with most things in physics, it’s were a lot of the nuance - and therefore understanding - lives. I’d encourage you to work these arguments out yourself to make sure you know them cold. Hope you’re doing well out there! Oh, and Happy Thanksgiving! 🦃

Best,

Sean

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A quick shot at the river that runs through our town. Sunset hits before 5pm right now. It’s my least favorite time of year.

The Polyakov action for the relativistic string in a Minkowski background reads

\(S = \frac{\kappa}{2} \int d^{2}\sigma \, \sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X_{\mu}.\)

The equations of motion associated with the embedding variables 𝑋^𝜇 are given by the variations:

\(\frac{\delta S}{\delta \partial_{a}X^{\mu}} = \kappa\int d^{2}\sigma \, \sqrt{-\gamma} \gamma^{ab}\partial_{b}X_{\mu},\)

where we have used the fact that

\(\gamma_{ab} = \gamma_{ba}.\)

Up to issues of boundary conditions that we will discuss shortly, this implies that the equations of motion for 𝑋^𝜇 are

\(\partial_{a}\left(\sqrt{-\gamma}\gamma^{ab}\partial_{b}X_{\mu}\right) = 0.\)

Recall that the three continuous, local symmetries of this action allow us to fix the worldsheet metric 𝛾_𝑎𝑏 to be

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

This is called the conformal gauge. In this gauge the equations of motion reduce so

\(\partial^{a}\partial_{a}X^{\mu} = -\ddot{X}^{\mu} + X^{\prime\prime\mu}= 0.\)

This is just the wave equation in one-dimension.

Solving the Wave Equation

The traditional solution to this specific equation is given by changing variables to

\(z = \frac{1}{2}(\tau + \sigma),\quad \bar{z} = \frac{1}{2}(\tau - \sigma),\)

so that

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

We have written these variables suggestively, anticipating the Wick rotation

\(\tau \mapsto -i\tau\)

to the complex plane.

There are lots of gymnastics we can do with this coordinate change. Let f be some function of worldsheet coordinates. We have

\(\frac{\partial f}{\partial z} = \frac{\partial f}{\partial \tau}\frac{\partial \tau}{\partial z} + \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial z} = \left(\partial_{\tau} + \partial_{\sigma}\right)f.\)

Similarly,

\(\frac{\partial f}{\partial z} =\left(\partial_{\tau} - \partial_{\sigma}\right)f.\)

Hence

\(\partial_{z}\partial_{\bar{z}}f = \left(\partial_{\tau}^{2} - \partial_{\sigma}^{2}\right)f.\)

Therefore we see that the equations of motion reduce to

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

where our shorthand now reads

\(\frac{\partial}{\partial z}\rightarrow \partial, \quad \frac{\partial}{\partial \bar{z}}\rightarrow \bar{\partial}.\)

Following our nose, we may write a general solution as

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

suggesting by analogy that the left-moving solution 𝑋_L(z) is essentially a holomorphic function of z. Similarly the right-moving solution 𝑋_R is an antiholomorphic function.

The study of complex variables brings with it the idea of conformal mapping, which is essentially achieved by such holomorphic maps. We will have more to say about this soon, but suffice it to say the Weyl-invariance of the Polyakov action will allow us to port the infrastructure of complex variables and Riemann surfaces to the study of the worldsheet.

Boundary Conditions for the Free String

To get the precise forms of these left-moving and right-moving solutions, we must consider the boundary term from varying the action. Let 𝜎 be bounded between the two endpoints of the string 𝜎_0 and 𝜎_1.

\(\kappa \int d\tau d\sigma\; \partial_{\sigma}\left(X^{\prime \mu}\right) = \kappa\int d\tau \left( X^{\prime}(\sigma_{1}) - X^{\prime}(\sigma_{0}) \right) = 0.\)

We are left with two possibilities. Either 𝑋′ separately vanishes at these endpoints, or it takes the same value at both points.

Open String Boundary Conditions

In the first case, per GSW, this implies that momentum does not flow off the ends of the string, which it would were it attached to something¹. To see this quickly for the case of Dirichlet boundary conditions, consider a simplified model of a standing wave,

\(X\sim\sin\left(2\pi\frac{\sigma-\sigma_{0}}{\sigma_{1}-\sigma_{0}}\right).\)

It's easy to see then that 𝑋′ is maximized at the endpoints.

This case, where 𝑋′ vanishes at the endpoints is called the open string boundary conditions.

Closed String Boundary Conditions

The second case, where the boundary terms cancel each other², can be arranged by demanding that

\(X^{\mu}(\tau,\sigma) = X^{\mu}(\tau,\sigma + \sigma_{1} - \sigma_{0}).\)

That is 𝑋^𝜇 is periodic in 𝜎. These are the closed string boundary conditions. We shall consider both kinds of boundary conditions in this work.

1. Which brings one inevitably to the idea of a Dirichlet brane.

2. Given our toy model X, you might try to arrange for this case by demanding that the string always have an odd number of nodes internally. Once you give the strings something to exchange momentum with, you are necessarily including interactions which we are not presently considering.